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Soliton-like molecular deformations in a nematic liquid crystal film
Physics Letters A 372 (2008) 2623–2633 www.elsevier.com/locate/pla
Soliton-like molecular deformations in a nematic liquid crystal film M. Daniel a,∗ , K. Gnanasekaran b a Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirapalli 620 024, Tamil Nadu, India b Department of Physics, Nehru Memorial College, Puthanampatti, Tiruchirapalli 621 007, Tamil Nadu, India
Received 9 August 2006; received in revised form 21 November 2007; accepted 27 November 2007 Available online 23 December 2007 Communicated by C.R. Doering
Abstract We study the nature of molecular deformations in a nematic liquid crystal film with elastic energy under homeotropic boundary conditions. The deformation in terms of splay, twist and bend fields of the director axis is found to be governed by the completely integrable Davey–Stewartson-I (DS-I) equation in (2 + 1) dimensions. Using the line soliton and breather solutions of the DS-I equation, the director axis is constructed, the components of which exhibit damped spatial oscillations. However, the splay and bend fields of the director axis exhibit localized structures of deformation. © 2008 Elsevier B.V. All rights reserved. PACS: 61.30-v; 61.30.Gd; 02.30.Ik
1. Introduction Liquid crystals are interesting anisotropic materials that have been investigated because of their importance in displays and also they serve as an interesting nonlinear model exhibiting different kinds of molecular deformations [1–5]. In particular, the theoretical study of nematic liquid crystals play a very important role because of their simple geometric structure. In nematic liquid crystal, the molecules are considered as elongated rods, which are positionally disordered but ordered in orientation. The average direction of orientation of the liquid crystal molecules along a particular direction is represented by a vector field n(r) known as director axis [6,7]. However, director reorientation or molecular excitation in nematic liquid crystals takes place due to elastic deformations such as splay, twist and bend [8]. In addition to the above hydrodynamic flow, externally applied electric, magnetic or optic field can also induce director axis deformations [8]. In general, director axis deformation in nematic liquid crystals is nonlinear in nature and it was studied by Legar [9] by taking into account the twist energy deformation alone and under the influence of a static magnetic field. The nonlinear behaviour leads to soliton in nematic liquid crystal, which was studied by Lei et al. [10] under uniform shear. Further, soliton and pattern formation in nematic liquid crystals were obtained by Migler and Meyer [11] in the presence of a rotating magnetic field. The above studies were carried out in one dimension and by assuming that the elastic coefficients due to splay, twist and bend are equal (one constant approximation). The effect of nonlinearity on molecular deformations in nematic liquid crystal even in the absence of any externally applied field is also equally important. Recently, the present authors studied the director dynamics in a quasi-one-dimensional nematic liquid crystal under elastic deformations in the absence of any externally applied field without imposing the one constant approximation [12,13]. The molecular deformations in terms of a rotational director axis field (related to bend mode) in this case, is found to exhibit localized behaviour in the form of pulse, hole and shock as well as soliton and finally leading to damped oscillations of the director axis. The above studies were restricted to one spatial dimension of the material. * Corresponding author.
E-mail address: [email protected] (M. Daniel). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.11.068
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Further, in the case of nematic liquid crystal films (two spatial dimensions) director dynamics is represented in terms of planar solitons in the presence of a constant magnetic field [14,15]. Also, two-dimensional behaviour of director axis in a nematic liquid crystal has been studied under spherical approximation in the presence of a rotating magnetic field and stationary solitons were obtained [16]. Also, molecular reorientation dynamics induced by circularly polarized, and spatially confined light beams leads to the formation of various chaotic regimes such as periodic, quasi-periodic, and intermittent-types [17,18]. In the presence of a Gaussian-type electric field, two-dimensional spatial soliton has been found within the inner region of the nematic film [19,20]. In addition, it has been proved experimentally that when a linearly polarized argon laser beam in the Gaussian mode is applied in a direction normal to the nematic sample of N − 4 -methoxybenzylinden and 4-cyano–4 − N -hexylbiphenyl, strong self focusing action occurs leading to the formation of soliton in planar geometry [21]. Further, it has been demonstrated experimentally that nematic phase exhibits two-dimensional spatial solitary waves when acted upon by a few milliwatt power beam [3,22]. In the present Letter guided by an experience in the case of one-dimensional nematics with pure elastic energies, we try to generate soliton-like molecular reorientations in the absence of any externally applied field in a nematic film. The plan of the Letter is as follows. After deriving the dynamical equation by considering the elastic free energy in Section 2, we rewrite the same in terms of a well-known completely integrable soliton equation in (2 + 1) dimensions, using a multi-scale expansion technique in Section 3. After presenting the soliton solutions of the nonlinear evolution equation in Section 4, we construct the director axis using Green’s function in Section 5. The results are concluded in Section 6. 2. Model and dynamical equation We consider a nematic liquid crystal film of finite size (0l × 0m) with homeotropic boundary conditions (see Fig. 1). Thus on the boundary walls, all the nematic liquid crystal molecules are arranged to point normal to the boundaries and are expected to remain invariant during the dynamical motion due to strong surface anchoring. However, the molecules in the inner region of the film are free to undergo spatial variation. The collective behaviour of the molecules may be described in terms of the dynamics of the director axis field n(r), with the constraint n · n = 1. The fluidity of the molecules gives rise to elastic deformations such as splay, twist and bend. The Frank free energy density associated with the above elastic deformations in the nematic film is given by [8] 2 1 K1 (∇ · n)2 + K2 (n · ∇ ∧ n)2 + K3 n ∧ (∇ ∧ n) , (1) 2 where K1 , K2 and K3 are the elastic constants associated with the splay, twist and bend deformations respectively and ∇ = ∂ ∂ + eˆ2 ∂y . The molecular field h corresponding to the above free energy can be obtained by minimizing the free energy density (1) eˆ1 ∂x f=
∂f ∂f + ∂j ∂g , i, j = x, y, z and gj i = ∂j ni . Thus, we obtain the expression for the molecular filed as using hi = − ∂n i ji h = K1 ∇(∇ . n) − K2 S(∇ ∧ n) + (∇ ∧ Sn) + K3 B ∧ (∇ ∧ n) + ∇ ∧ (n ∧ B) ,
(2)
where S = n . ∇ ∧ n and B = n ∧ (∇ ∧ n). The thermodynamic force h˜ acting on the director is given by h˜ = h − (h · n)n. The term (h · n)n is the Lagrange multiplier term (projection of the molecular field on the director) which has been introduced in the field equation to take care of the condition n · n = 1 that explicitly states that in equilibrium the director must be at each point parallel to the molecular field. Any component of the field parallel to the director only serves the purpose of determining the Lagrange multiplier but has no significance in the dynamics of the director which is the main objective of the study in this Letter [23]. Further, the torque associated with this contribution of the field vanishes and hence we drop this term in our study because it is not of interest to us. In addition to the molecular field, there exists a viscous field due to pure rotational effect and another field arising ˆ from the coupling of the director axis to the fluid motion given respectively by γ1 ( ∂n ∂t − ω ∧ n) and γ2 An where γ1 and γ2 are the ˆ viscosity coefficients. Here ω is the angular velocity of the background fluid and A is the velocity gradient tensor. However, the fluid
Fig. 1. A nematic liquid crystal film of finite size (0l × 0m) with homeotropic boundaries.
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velocity and its gradients are small enough such that the term proportional to γ2 can be neglected and hence the director motion is largely unaffected by the hydrodynamic flow. Also, the effect of background flow is negligible. Under these circumstances the director dynamics is described by the equation ∂n (3) = K1 ∇(∇ . n) − K2 S(∇ ∧ n) + (∇ ∧ Sn) + K3 B ∧ (∇ ∧ n) + ∇ ∧ (n ∧ B) . ∂t The director dynamics is thus governed by a highly nontrivial vector nonlinear partial differential equation and solving the same in its natural vector form is a challenging task. Hence, we rewrite Eq. (3) by taking curl and divergence on both sides separately and by introducing two new fields namely A and Φ such that A ≡ (Ax , Ay , Az ) = (∇ ∧ n) and Φ = ∇ · n which are related to the bend and splay modes respectively γ1
∂A (r, t) = K3 ∇ 2 A − A2 A , (4a) ∂t ∂Φ γ1 (4b) (r, t) = K1 ∇ 2 Φ − K3 A2 Φ. ∂t While deriving Eqs. (4), it is assumed that the rotation of the director axis (A) is always normal to n so that n · A = 0 which makes the term proportional to K2 to vanish. In order to associate Eqs. (4) with more standard evolution equations, we introduce a new complex scalar field in terms of the components of A. For this, we make A to be a two component vector in the xy-plane, i.e., x ∂ny A = (Ax , Ay , 0) and define a new complex scalar field as U = Ax + iAy . When Az ≡ (∇ ∧ n)z = 0, we have ∂n ∂y = ∂x and the γ1
x
constraint n · A = 0 leads to Ay = − nny Ax . Also in this process, the number of unknowns that appear in the resultant equations (Eqs. (5a), (5b)) are reduced by one. Using the above, Eqs. (4) after suitable rescaling of time can be written as ∂U 2 = ∇ U − |U |2 U , ∂t ∂Φ K1 2 ∇ Φ − |U |2 Φ. = ∂t K3
(5a) (5b)
The above set of coupled equations (5) describes the director dynamics in terms of two new fields namely U and Φ which are respectively related to the bend and splay modes. The advantage of rewriting the dynamical equation in terms of these new fields lies in associating these set of new equations with a family of known integrable nonlinear evolution equations. 3. Molecular deformation dynamics Having derived the dynamical equation in a convenient form, the problem now boils down to solving Eqs. (5) to understand the director dynamics. As we have considered a nematic liquid crystal film for our study, we fix the film to lie in the xy-plane ∞ α ∂2 ∂2 − 2 pψ(x,y)t and Φ = p so that ∇ 2 ≡ ∇⊥ 2 = ∂x 2 + ∂y 2 . We assume the solutions of Eqs. (5) in the form U = p=0 qp (x, y, t)e ∞ α − 2 pψ(x,y)t , where is an arbitrary small parameter and α ’s are real integers. We substitute the above expansions p p p=0 vp (t) e √ √ for U and Φ in Eqs. (5), rescale the time and spatial variables as t → − 2 t , x → 2x and y → 2y, and equate the coefficients 2 2 of every e pψ(x,y)t separately for different p values to zero. Thus, the coefficients of the terms proportional to e nψ(x,y)t where n is an integer give
1 1 ∂qn ∂ψ ∂ψ 2 ∂ψ 2 ∂qn ∂ψ ∂qn + + nψqn + ∇⊥ 2 qn − 2 n ∇⊥ 2 ψ qn + + t + 2 2 nt qn 2+αn ∂t 2 2 ∂x ∂x ∂y ∂y ∂x ∂y
− (6a) αs αn−r αr−s qn−r qr−s qs∗ = 0, s
and
r
K1 2 ∂ψ 2 ∂ψ 2 ∂vn nvn t ∇⊥ 2 ψ + 2 2 nt + + nψvn + ∂t 2K3 ∂x ∂y ∞ ∞
αs αn−r αr−s qn−r q ∗ r−s vs = 0. −
2+αn
s=0 r=0 2
∂ Operating Eq. (6b) by the operator ( αn ∂y 2 −
2(1+αn )
∂2 ) ∂x 2
we obtain
2 ∂v ∂2 K1 2 ∂ψ 2 ∂ψ 2 n 2 2+αn ∂ 2 − nvn t ∇⊥ ψ + 2 nt + + nψvn + 2K3 ∂x ∂y ∂y 2 ∂x 2 ∂t
(6b)
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−
∞ ∞
αs αn−r αr−s
s=0 r=0
2 ∂2 αs αn−r αr−s ∂ − qn−r q ∗ r−s vs = 0. ∂x 2 ∂y 2
(6c)
Eqs. (6) are in the form of an infinite series in powers of and we assume that for every value of n, → 0. However, the terms corresponding to the minimal power of the parameter vanishes and consequently we have [24] ∂q1 1 2 + ∇⊥ q1 + |q1 |2 q1 + ψq1 = 0, ∂t 2 ∂ 2 |q1 |2 ∂ 2ψ ∂ 2ψ − − 2 = 0. ∂y 2 ∂x 2 ∂x 2
(7a)
i
(7b)
While writing the above equations we have transformed time variable as t → it. Also we have chosen qr−s = −qs−r , qs = q−s , vs = v−s and further we have assumed αn = α−n = n for n 1 and α0 = 2. The set of coupled equations (7) are together known as Davey–Stewartson-I (DS-I) equation, which was first derived in fluid dynamics describing the evolution of waves of slowly varying amplitude on a two-dimensional water surface under gravity [25]. The above equation later appeared in several other contexts as well including stability of gravity waves [26], nonlinear optics [27], plasma [28], spin systems [29], etc. 4. Soliton in a nematic film The DS-I equation given in Eqs. (7) is a two-dimensional integrable generalization of the completely integrable cubic nonlinear Schrödinger (NLS) equation and admits solutions in the form of line soliton, dromion and breather which were originally obtained by solving the equation using dressing method in nonlocal Riemann–Hilbert problems [30]. It is also possible to construct the N -soliton solutions for the equation using Hirota’s bilinearization procedure [29,31,32]. However, as our problem involves only time-independent boundary conditions, the dromion solution (which involves time-dependent boundary conditions) is not relevant to us here. Therefore in the following, we write down the explicit form of line one soliton and (1, 1) breather solutions obtained through Hirota’s bilinear method. 4.1. Line one soliton The line one soliton solutions of the DS-I equation (7) is written as q1 = 4 λR μR sech χR + ln(4 λR μR ) exp(iχI ),
(8a)
where (0)
χR = (λR + μR )x + (λR − μR )y − 2[λR λI + μR μI ]t + χR , (0) χI = (λI + μI )x + (λI − μI )y + λR 2 + μR 2 − λI 2 + μI 2 t + χI .
(8b) (8c) (0)
Here χR and χI are the real and imaginary parts of χ = (λ + μ)x + (λ − μ)y + i[λ2 + μ2 ]t + χ (0) and λR (λI ), μR (μI ) and χR (0) (χI ) are the real (imaginary) part of the complex constants λ, μ and χ respectively. Using the line one soliton solution given in Eq. (8a) in Eq. (7b) the field ψ(x, y) can be obtained by solving the resultant equation. The result reads as 2 2 tanh (λR + μR )x + (λR − μR )y + δ − sech (λR + μR )x + (λR − μR )y + δ , ψl = −16λR μR (9) λR + μR √ where δ = ln(4 λR μR ). Knowing q1 and ψ , the bend field U and hence A can be constructed using the relation U ≡ Ax + iAy = q1 e−ψt . Similarly, the splay field Φ can be obtained using the relation Φ = v1 e−ψt . However, since we have switched to an imaginary time variable (t → it) while going from Eq. (6) to Eq. (7), this will arise problem in the case of the solution with real time. This can be avoided by choosing |λR | = |λI | and |μR | = |μI |. Therefore, Eq. (8c) must be replaced by χI = (λI + μI )x + (0) (λI − μI )y + χI . Thus, using Eqs. (8) and (9) we obtain the bend and splay fields as U ≡ Ax + iAy 2 = 4 λR μR sech χR + ln(4 λR μR ) exp 16λR μR tanh (λR + μR )x + (λR − μR )y + δ λR + μR
− 16 sech2 (λR + μR )x + (λR − μR )y + δ t exp iχI ,
(10a)
M. Daniel, K. Gnanasekaran / Physics Letters A 372 (2008) 2623–2633
and
Φ = v1 exp 16λR μR
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2 2 tanh (λR + μR )x + (λR − μR )y + δ − sech (λR + μR )x + (λR − μR )y + δ t . λ R + μR
(10b) Eq. (10a) represents the bend field (i.e., rotation of the director axis) and (10b) the splay field (i.e., divergence of the director axis) of the nematic film corresponding to the line soliton solution of DS-I equation. In Fig. 2(a), we have given a snapshot (at t = 1) of the line one soliton of the DS-I equation as found in Eq. (8) by choosing λR = 0.012, μR = 1.0, λI = 0.012 and μI = 1.0. In Fig. 2(b), we have plotted |A|2 (Eq. (10a)) which is related to the stationary bend mode corresponding to the line one soliton for the same parameter values used to plot the line one soliton in Fig. 2(a). In Fig. 2(c), Φ, which is related to the stationary splay mode corresponding to the line soliton (Eq. (10b)) is plotted by choosing λR = 0.012, μR = 1, λI = 0.012, μI = 1 and v1 = 0.001. From Figs. 2(b), (c) we observe that the bend field |A|2 which represents the rotation of the director axis and the splay field that is related to the divergence of the director axis corresponding to the line one soliton exhibit localized behaviour in the form of pulse and kink respectively. 4.2. (1, 1) breather solution The (1, 1) breather solution of the DS-I equation (7a) and (7b) which is localized spatially and oscillatory in time is given by
q1 =
ρe(λR +μR )x+(λR −μR )y 1 + j e2λR (x+y)
+ ke2λR (x−y)
+ le(2λR (x+y)+2λR (x−y))
2 +μ 2 )t R
ei(λR
(11)
,
where ρ is a complex constant and j , k, l and λR , μR , are real positive constants such that λR 2 + μR 2 1. Knowing q1 , the field ψ(x, y) can be constructed by substituting the solution q1 in Eq. (7b) and the resultant equation solved. Thus, the field ψb (x, y) corresponding to the (1, 1) breather solution given in Eq. (11) is written as 2 2λR x N λR x λR x −2 cosh λR y + e −4 1+e , ψb = −ρ e (12a) D
where N = tanh(λR y) + 2{(eλR x − 2) + (1 + 9λR 2
1 2
−e
5λR 2
λR
5λR 2
x
sinh
λR
λR 2 x
− 4 cosh λR x) sech2 λR y} arctan[
tanh λR y
3 λ x 2(1+e 2 R
cosh
λR 2
] and D = x)
x sinh 2 x − 2e x cosh 2 x − 5e2λR x ) sech2 λR y − (e4λR x + 3eλR x − 4)}. Having found q1 and ψ , the 2{( 19 2 − 4cosh λR x − 4e field U and hence A can be constructed using the relation U ≡ Ax + iAy = q1 e−ψt . Similarly, the field Φ can be obtained using the relation Φ = v1 e−ψ(x,y)t . Thus, in the case of (1, 1) breather solution (Eqs. (11) and (12a)), we obtain
U=
and
ρe(λR +λR )x+(λR −λR )y
1 + j e2λR (x+y) + ke2μR (x−y) + le2(λR +λR )x+2(λR −λR )y 2 2 2λR x N λR x λR x −2 2 cosh λR y + e × exp ρ e −4 1+e + i λR + λR t , D
2 2λR x N λR x λR x −2 Φ = v1 exp ρ e cosh λR y + e −4 1+e t . D
(13a)
(13b)
In Fig. 2(d), we have plotted a snapshot (at t = 1) of the (1, 1) breather solution of the DS-I equation as found in Eq. (11) by choosing λR = 0.012, μR = 0.01 and ρ = 0.001. In Fig. 2(e), |A|2 (Eq. (13a)) which is related to the stationary bend mode corresponding to (1, 1) breather solution of the DS-I equation has been plotted for the same parameter values. In Fig. 2(f), Φ, which is related to the stationary splay mode (Eq. (13b)) corresponding to (1, 1) breather solution of DS-I equation is plotted by choosing λR = 0.012, μR = 1.0 and v1 = 0.001. The (1, 1) breather solution of the DS-I equation is known to exhibit a highly localized pattern (see Fig. 2(d)). From Fig. 2(e) and 2(f) we observe that the bend field |A|2 (rotation of the director axis) is still exhibiting a localized behaviour similar to breathers with an added effect of a localized spike at a specified location (which is evident from Fig. 2(f)), that represents the splay field Φ (divergence of the director axis) corresponding to (1, 1) breather solution of the DS-I equation. 5. Construction of director axis Having found the fields A = ∇ ∧ n and Φ = ∇ · n which correspond to the bend and splay modes respectively in terms of the line one soliton and (1, 1) breather solutions of the DS-I equation we now construct the director axis n using these results. The
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(a)
(b)
(c)
(d)
(e)
(f)
Fig. 2. A snapshot of (a) line one soliton |q1 |2 (Eq. (8)) of DS-I equation for λR = 0.012, μR = 1, λI = 0.012 and μI = 1, (b) of the bend field (|A|2 ) (Eq. (10a)) corresponds to the line one soliton for the same parameter values as given above and (c) splay mode (Φ) (Eq. (10b)) corresponding to the line one soliton of DS-I equation for λR = 0.012, μR = 1, λI = 0.012, μI = 1 and v1 = 0.001, (d) (1, 1) breather solution of DS-I equation |q1 |2 (Eq. (11)), for λR = 0.012, μR = 0.01 and ρ = 0.001. (e) the bend mode (|A|2 ) (Eq. (13a)) corresponding to (1, 1) breather solutions of DS-I equation for the same parameter values as in (d) and (f) splay mode (Φ) (Eq. (13b)) corresponding to (1, 1) breather solution of DS-I equation for λR = 0.012, μR = 0.01 and v1 = 0.001.
equations defining A and Φ can be combined and written as the following two-dimensional Poisson equation: ∇⊥ 2 n = ∇Φ − ∇ ∧ A, n = nx , ny , nz .
(14)
The homeotropic boundary conditions for our problem is written as nx = 1,
ny = 1,
nz = 0|x=(0,l);y=(0,m) .
(15)
The above inhomogeneous boundary conditions (associated with the x- and y-components of the director axis) can be converted into homogeneous ones by defining a new vector field p ≡ (p x , p y , p z ) such that p x = nx − 1, p y = ny − 1, and p z = nz . In terms of the new vector field, the two-dimensional Poisson equation is written in the component form as ∂Φ ∂Φ ∂Ax ∂Ay (16) , ∇⊥ 2 p y = , ∇⊥ 2 p z = − . ∂x ∂y ∂y ∂x The above Poisson equations with the set of new homogeneous boundary conditions can be solved using the Green’s function method of eigenfunction expansion. ∇⊥ 2 p x =
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The equation governing the Green’s function G is written as ∇⊥ 2 G = −δ(x − ζ )δ(y − β),
(17)
with G satisfying the boundary conditions G(x, ζ ; y, β) = 0|(x=0,l);(y=0,m) . Here ζ and β are arbitrary points along x and y directions respectively. The Green’s function solution of Eq. (17) can be expressed using the method of eigenfunction expansion as [33]
∞ ∞ 4lm sin sπl ζ sin smπ β sin sπl x sin smπ y . G(x, ζ ; y, β) = 2 π (ms )2 + (ls)2
(18)
s=1 s =1
Using the value of the above Green’s function G, the components of the vector field p namely p x , p y and p z can be constructed using the right-hand sides of Eq. (16) as follows: x y
dx dy G(x , y ; ζ, β)Ri ,
p (x, y) = i
i = x, y, z,
(19)
−∞ −∞
where Ri ’s are the right-hand sides of the p x , p y and p z equations in Eqs. (16). On substituting the Green’s function G from Eq. (18) and the values of Φ, Ax and Ay from Eqs. (10) corresponding to the line one soliton of the DS-I equation in Eq. (19), we obtain the values of p x , p y and p z as follows: p x = γss hx (x, y, t) − Hx (x, y, t) , (20a) y p = γss hy (x, y, t) − Hy (x, y, t) , (20b) z p = γss hz (x, y, t) + hz (x, y, t) − Hz (x, y, t) − Hz (x, y, t) . (20c) ∞ ∞ π 4lm sπ s In the above equations γss = π 2 s=1 s =1 sin l ζ sin m β. Defining sπ 2 sπ tanh (λR + μR )x + (λR − μR )y + δ x sin y exp 16λR μR f (x, y, t) = sin l m λR + μR
− sech2 (λR + μR )x + (λR − μR )y + δ t the functions hi (x, y, t) and Hi (x, y, t) (i = x, y, z) are written as y hx (x, y, t) = v1
f (x, y , t) dy ,
−∞ x
−∞ −∞ x y
Hy (x, y, t) = v1
−∞ y
hz (x, y, t) =
−∞ x
f (x , y, t) dx ,
(21a)
f (x , y , t) sin
n π y dx dy , m
(21b)
f (x , y , t) sin
nπ x dx dy , l
(21c)
−∞ −∞
hz (x, y, t) =
hy (x, y, t) = v1 −∞
y
Hx (x, y, t) = v1
x
x
f (x , y, t) sech2 (λR + μR )x + (λR − μR )y + δ cos (λR + μR )x + (λR − μR )y dx ,
(21d)
f (x, y , t) sech2 (λR + μR )x + (λR − μR )y + δ sin (λR + μR )x + (λR − μR )y dy ,
(21e)
nπ nπ f (x , y , t) sin x cos y cos (λR + μR )x + (λR − μR )y Hz (x, y, t) = l m −∞ −∞ × sech2 (λR + μR )x + (λR − μR )y + δ dx dy , x y nπ n π Hz (x, y, t) = f (x , y , t) cos x sin y sin (λR + μR )x + (λR − μR )y l m −∞ −∞ × sech2 (λR + μR )x + (λR − μR )y + δ dx dy . y
(21f)
(21g)
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As it is difficult to evaluate the above integrals in the present form, we expand the exponential function that appear in the function f (x, y, t). After using the expansion and on evaluating the integrals at the lowest order (dropping higher order terms) and using the results in Eqs. (20), we obtain the components of the director axis in the form nx ≡ p x + 1
sπ sπ γss v1 m l μ ) sin (1 − 16λ x cos y =1+ R R sπ sπ l m
2 tanh (λR + μR )x + δ − sech2 (λR + μR )x + δ t , + exp 16λR μR λR + μR
γss v1 m m sπ sπ y y n ≡p +1=1+ (1 − 16λR μR ) 1 − cos x sin y sπ sπ l m
2 2 tanh (λR + μR )x + δ − sech (λR + μR )x + δ t , + exp 16λR μR λR + μR
and nz ≡ p z =
γss l m sπ sπ (16λR μR − 1) 1 − cos x sin y sπ sπ l m m sπ sπ + sin (λR + μR )x sin x cos y sech (λR + μR )x + δ s π l m
2 tanh (λR + μR )x + δ − sech2 (λR + μR )x + δ t . × exp 16λR μR λR + μR
(22a)
(22b)
(22c)
The above solutions must satisfy the constraint n · A ≡ n · ∇ ∧ n = 0, which is equivalent to nx Ax + ny Ay = 0. Therefore, to verify this, we evaluate the quantities nx Ax and ny Ay separately using Eqs. (22a), (22b) and (10a) and write the final form of the expression nx Ax + ny Ay as nx Ax + ny Ay ∞ ∞
l 16lm πζ s πβs λ = μ (1 − 16λ μ ) sin sin R R R R 2 πs l m π s=1 s =1
πs π s 2 2 × exp 16λR μR tanh (λR + μR )x + δ − sech (λR + μR )x + δ t + sin x cos y cos χ1 λ R + μR l m
πs m 2 + tanh (λR + μR )x + δ − sech2 (λR + μR )x + δ t 1− exp 16λR μR l π s λR + μR πs πs + cos x sin y sin χ1 l m 2 × sech χR + ln(4 λR μR ) exp 16λR μR tanh (λR + μR )x + (λR − μR )y + δ λ R + μR
− sech2 (λR + μR )x + (λR − μR )y + δ t , (23) ∞ l πβs πζ s which should vanish. The right-hand side of Eq. (23) vanishes when ∞ s=1 s =1 π s sin l sin m = 0 which is equivalent to ∞ ∞ l πβs πβs πζ s πζ s s=1 s =1 πs [cos( l + m ) − cos( l − m )] = 0. The left-hand side of the above condition averages out to be zero quickly when l and m are large, that is in the case of an infinite film. Thus, the solutions satisfy the constraint n · A = 0 in the limit of an infinite film. We repeat the entire calculations for the case of (1, 1) breather solution of the DS-I equation. However, in this case, the various resultant expressions are very complicated and due to the unwieldy nature of the final results, we are not presenting the details here. In Figs. 3(a)–(c), we have given a snapshot of the x-, y- and z-components of the director axis namely nx , ny and nz for s = s = 1.0, λR = 0.012, μR = 1.0, λI = 0.012, μI = 1.0, ζ = β = 0.1, and l = m = 1.0 at time t = 1. From the figures it is observed that the x- and y-components of the director axis namely nx and ny slowly gain spatial oscillations in the xy-plane starting from the equilibrium configuration. On the other hand, the z-component of the director axis (nz ) exhibits uniform periodic oscillations along the y-direction. However, along the x-direction the amplitude grows and gets saturated. Further, when we analyze the long time behaviour of the director axis oscillations along different directions, we find that the oscillations are getting damped as shown in Figs. 3(d)–(i).
M. Daniel, K. Gnanasekaran / Physics Letters A 372 (2008) 2623–2633
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
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(i) Fig. 3. Snapshots of the x-, y- and z-components (nx , ny , nz ) of the director axis (Eqs. (22a)–(22c) respectively), exhibiting spatial oscillations for s = s = 1.0, λR = 0.012, μR = 1.0, λI = 0.012, μI = 1.0, v1 = 0.001, ζ = β = 0.1 and l = m = 1.0 at t = 1. (d)–(i): Long-time behaviour of the x-, y- and z-components (nx , ny and nz ) of the director axis along x and y axes showing damped oscillations for the same values of the parameters.
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M. Daniel, K. Gnanasekaran / Physics Letters A 372 (2008) 2623–2633
6. Conclusions In this Letter, we have investigated the nature of director dynamics in a nematic liquid crystal film, with homeotropic boundary conditions. Starting from the Frank free energy density for splay, twist and bend-type elastic deformations, we constructed the associated field equation by minimizing the free energy density. In order to solve the resultant field equation for understanding the director dynamics, we introduced two new fields in terms of splay and bend modes (i.e., divergence and rotation of the director axis respectively). A multi-scale expansion was found useful in finally writing the resultant equation in terms of the well-known completely integrable Davey–Stewartson equation in (2 + 1) dimensions which admits line soliton and breather solutions for our problem. Using these soliton solutions of DS-I equation, the components of the director axis were constructed by solving the Poisson equation associated with the definitions of splay and bend fields of our model by using Green’s function technique. From the results, we observe that the splay and bend fields of the director axis exhibit localized structures of reorientation due to elastic deformations in close analogy with the localized soliton and breather solutions of the DS-I equation. Spatial soliton in nematic liquid crystal was experimentally observed when the nematic film is allowed to pass through high intense coherent laser beam [3,21,34]. However, there is no experimental evidence available in the literature for the generation of solitons induced through elastic deformations. This may be due to the fact that solitons generated through elastic deformations are expected to be less stable compared to solitons generated through externally applied optic field. Hence, it may be very difficult to observe soliton induced through elastic deformation. It is planned to carry out stability analysis of the elastic solitons and the results will be published elsewhere. From Fig. 3, we notice that at a given time the x- and y-components of the director axis gain uniform oscillations form their equilibrium configuration. However, the z-component of the director axis shows a slightly different behaviour with the amplitude growing steadily and gets saturated along x-direction but exhibiting uniform oscillations along y-direction. However, the long time behaviour of oscillation of the director axis components in different directions is shown to exhibit damping feature of the oscillations, which is due to the fact that the two-dimensional Poisson equation is solved using the eigen function expansion method for Green’s function. Acknowledgements The work of M.D. forms part of a major DST project. K.G. would like to thank UGC for a Teacher Fellowship under the FIP programme. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
D.W. McLaughlin, D.J. Murkai, M.J. Shelly, Physica D 97 (1996) 471. M.A. Karpierz, M. Sierakowski, T.R. Wolinski, Mol. Cryst. Liq. Cryst. 375 (2002) 313. M. Peccianti, C. Conti, G. Assanto, A. De Luca, C. Umeton, Nature 432 (2004) 733. S. Kralj, R. Rosso, E.G. Virga, Eur. Phys. J. E 17 (2005) 37. N. Mukohzaka, Appl. Opt. 33 (1994) 2785. H. Zocher, Trans. Faraday Soc. 29 (1933) 945. P.G. de Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1974. F.C. Frank, Discuss. Faraday Soc. 25 (1958) 19. L. Legar, Solid State Commun. 10 (1972) 697. L. Lei, S. Changqing, S. Juelian, P.M. Lam, H. Yun, Phys. Rev. Lett. 49 (1982) 1335. K.B. Migler, R.B. Meyer, Phys. Rev. Lett. 66 (1991) 1487. M. Daniel, K. Gnanasekaran, Physica A 346 (2005) 250. M. Daniel, K. Gnanasekaran, Phys. Lett. A 332 (2004) 17. H. Arodz, J. Stelzer, Phys. Rev. E 57 (1998) 3007. J. Stelzer, H. Arodz, Mol. Cryst. Liq. Cryst. 330 (1999) 221. A. Polimeno, A.F. Martins, Liq. Cryst. 25 (1998) 545. E. Brasselet, B. Doyon, T.V. Galstian, L.J. Dube, Phys. Rev. E 67 (2003) 031706; E. Brasselet, B. Doyon, T.V. Galstian, L.J. Dube, Phys. Rev. E 69 (2004) 021701. E. Brasselet, B. Doyon, T.V. Galstian, L.J. Dube, Mol. Cryst. Liq. Cryst. 421 (2004) 69. M.A. Karpierz, M. Sierakowski, T.R. Wolinski, Mol. Cryst. Liq. Cryst. 375 (2002) 313. M.A. Karpierz, Q.V. Nguyen, Opto-Electron. Rev. 10 (2002) 59. E. Braun, L.P. Faucheux, A. Libchaber, Phys. Rev. E 48 (1993) 611. M. Peccianti, G. Assanto, Phys. Rev. E 65 (2002) 035603(R). I. Dahl, A. De Meyere, Liq. Cryst. 18 (1995) 683. V.E. Zakharov, E.A. Kuznetsov, Physica D 18 (1986) 455. A. Davey, K. Stewartson, Proc. R. Soc. London A 338 (1974) 101. V.D. Djordjevic, L.G. Redekopp, J. Fluid Mech. 79 (1977) 703. V.E. Zakharov, E.A. Kuznetsov, Math. Phys. Rev. 4 (1984) 167. K. Nishinari, K. Abe, J. Satsuma, J. Phys. Soc. Jpn. 62 (1993) 2021. R. Radha, S. Vijayalakshmi, M. Lakshmanan, J. Nonlinear Math. Phys. 6 (1999) 120.
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[30] [31] [32] [33] [34]
A.S. Fokas, V.E. Zakharov, J. Nonlinear Science 2 (1992) 109. R. Radha, M. Lakshmanan, Inverse Problems 10 (1994) L29. J. Hietarinta, Phys. Lett. A 149 (1990) 113. E. Zauderer, Partial Differential Equations of Applied Mathematics, Wiley, New York, 1983. J.F. Henninot, J.F. Blach, M. Warenghem, J. Opt. A: Pure Appl. Opt. 9 (2006) 20.
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Soliton-like molecular deformations in a nematic liquid crystal film M. Daniel a,∗ , K. Gnanasekaran b a Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirapalli 620 024, Tamil Nadu, India b Department of Physics, Nehru Memorial College, Puthanampatti, Tiruchirapalli 621 007, Tamil Nadu, India
Received 9 August 2006; received in revised form 21 November 2007; accepted 27 November 2007 Available online 23 December 2007 Communicated by C.R. Doering
Abstract We study the nature of molecular deformations in a nematic liquid crystal film with elastic energy under homeotropic boundary conditions. The deformation in terms of splay, twist and bend fields of the director axis is found to be governed by the completely integrable Davey–Stewartson-I (DS-I) equation in (2 + 1) dimensions. Using the line soliton and breather solutions of the DS-I equation, the director axis is constructed, the components of which exhibit damped spatial oscillations. However, the splay and bend fields of the director axis exhibit localized structures of deformation. © 2008 Elsevier B.V. All rights reserved. PACS: 61.30-v; 61.30.Gd; 02.30.Ik
1. Introduction Liquid crystals are interesting anisotropic materials that have been investigated because of their importance in displays and also they serve as an interesting nonlinear model exhibiting different kinds of molecular deformations [1–5]. In particular, the theoretical study of nematic liquid crystals play a very important role because of their simple geometric structure. In nematic liquid crystal, the molecules are considered as elongated rods, which are positionally disordered but ordered in orientation. The average direction of orientation of the liquid crystal molecules along a particular direction is represented by a vector field n(r) known as director axis [6,7]. However, director reorientation or molecular excitation in nematic liquid crystals takes place due to elastic deformations such as splay, twist and bend [8]. In addition to the above hydrodynamic flow, externally applied electric, magnetic or optic field can also induce director axis deformations [8]. In general, director axis deformation in nematic liquid crystals is nonlinear in nature and it was studied by Legar [9] by taking into account the twist energy deformation alone and under the influence of a static magnetic field. The nonlinear behaviour leads to soliton in nematic liquid crystal, which was studied by Lei et al. [10] under uniform shear. Further, soliton and pattern formation in nematic liquid crystals were obtained by Migler and Meyer [11] in the presence of a rotating magnetic field. The above studies were carried out in one dimension and by assuming that the elastic coefficients due to splay, twist and bend are equal (one constant approximation). The effect of nonlinearity on molecular deformations in nematic liquid crystal even in the absence of any externally applied field is also equally important. Recently, the present authors studied the director dynamics in a quasi-one-dimensional nematic liquid crystal under elastic deformations in the absence of any externally applied field without imposing the one constant approximation [12,13]. The molecular deformations in terms of a rotational director axis field (related to bend mode) in this case, is found to exhibit localized behaviour in the form of pulse, hole and shock as well as soliton and finally leading to damped oscillations of the director axis. The above studies were restricted to one spatial dimension of the material. * Corresponding author.
E-mail address: [email protected] (M. Daniel). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.11.068
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Further, in the case of nematic liquid crystal films (two spatial dimensions) director dynamics is represented in terms of planar solitons in the presence of a constant magnetic field [14,15]. Also, two-dimensional behaviour of director axis in a nematic liquid crystal has been studied under spherical approximation in the presence of a rotating magnetic field and stationary solitons were obtained [16]. Also, molecular reorientation dynamics induced by circularly polarized, and spatially confined light beams leads to the formation of various chaotic regimes such as periodic, quasi-periodic, and intermittent-types [17,18]. In the presence of a Gaussian-type electric field, two-dimensional spatial soliton has been found within the inner region of the nematic film [19,20]. In addition, it has been proved experimentally that when a linearly polarized argon laser beam in the Gaussian mode is applied in a direction normal to the nematic sample of N − 4 -methoxybenzylinden and 4-cyano–4 − N -hexylbiphenyl, strong self focusing action occurs leading to the formation of soliton in planar geometry [21]. Further, it has been demonstrated experimentally that nematic phase exhibits two-dimensional spatial solitary waves when acted upon by a few milliwatt power beam [3,22]. In the present Letter guided by an experience in the case of one-dimensional nematics with pure elastic energies, we try to generate soliton-like molecular reorientations in the absence of any externally applied field in a nematic film. The plan of the Letter is as follows. After deriving the dynamical equation by considering the elastic free energy in Section 2, we rewrite the same in terms of a well-known completely integrable soliton equation in (2 + 1) dimensions, using a multi-scale expansion technique in Section 3. After presenting the soliton solutions of the nonlinear evolution equation in Section 4, we construct the director axis using Green’s function in Section 5. The results are concluded in Section 6. 2. Model and dynamical equation We consider a nematic liquid crystal film of finite size (0l × 0m) with homeotropic boundary conditions (see Fig. 1). Thus on the boundary walls, all the nematic liquid crystal molecules are arranged to point normal to the boundaries and are expected to remain invariant during the dynamical motion due to strong surface anchoring. However, the molecules in the inner region of the film are free to undergo spatial variation. The collective behaviour of the molecules may be described in terms of the dynamics of the director axis field n(r), with the constraint n · n = 1. The fluidity of the molecules gives rise to elastic deformations such as splay, twist and bend. The Frank free energy density associated with the above elastic deformations in the nematic film is given by [8] 2 1 K1 (∇ · n)2 + K2 (n · ∇ ∧ n)2 + K3 n ∧ (∇ ∧ n) , (1) 2 where K1 , K2 and K3 are the elastic constants associated with the splay, twist and bend deformations respectively and ∇ = ∂ ∂ + eˆ2 ∂y . The molecular field h corresponding to the above free energy can be obtained by minimizing the free energy density (1) eˆ1 ∂x f=
∂f ∂f + ∂j ∂g , i, j = x, y, z and gj i = ∂j ni . Thus, we obtain the expression for the molecular filed as using hi = − ∂n i ji h = K1 ∇(∇ . n) − K2 S(∇ ∧ n) + (∇ ∧ Sn) + K3 B ∧ (∇ ∧ n) + ∇ ∧ (n ∧ B) ,
(2)
where S = n . ∇ ∧ n and B = n ∧ (∇ ∧ n). The thermodynamic force h˜ acting on the director is given by h˜ = h − (h · n)n. The term (h · n)n is the Lagrange multiplier term (projection of the molecular field on the director) which has been introduced in the field equation to take care of the condition n · n = 1 that explicitly states that in equilibrium the director must be at each point parallel to the molecular field. Any component of the field parallel to the director only serves the purpose of determining the Lagrange multiplier but has no significance in the dynamics of the director which is the main objective of the study in this Letter [23]. Further, the torque associated with this contribution of the field vanishes and hence we drop this term in our study because it is not of interest to us. In addition to the molecular field, there exists a viscous field due to pure rotational effect and another field arising ˆ from the coupling of the director axis to the fluid motion given respectively by γ1 ( ∂n ∂t − ω ∧ n) and γ2 An where γ1 and γ2 are the ˆ viscosity coefficients. Here ω is the angular velocity of the background fluid and A is the velocity gradient tensor. However, the fluid
Fig. 1. A nematic liquid crystal film of finite size (0l × 0m) with homeotropic boundaries.
M. Daniel, K. Gnanasekaran / Physics Letters A 372 (2008) 2623–2633
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velocity and its gradients are small enough such that the term proportional to γ2 can be neglected and hence the director motion is largely unaffected by the hydrodynamic flow. Also, the effect of background flow is negligible. Under these circumstances the director dynamics is described by the equation ∂n (3) = K1 ∇(∇ . n) − K2 S(∇ ∧ n) + (∇ ∧ Sn) + K3 B ∧ (∇ ∧ n) + ∇ ∧ (n ∧ B) . ∂t The director dynamics is thus governed by a highly nontrivial vector nonlinear partial differential equation and solving the same in its natural vector form is a challenging task. Hence, we rewrite Eq. (3) by taking curl and divergence on both sides separately and by introducing two new fields namely A and Φ such that A ≡ (Ax , Ay , Az ) = (∇ ∧ n) and Φ = ∇ · n which are related to the bend and splay modes respectively γ1
∂A (r, t) = K3 ∇ 2 A − A2 A , (4a) ∂t ∂Φ γ1 (4b) (r, t) = K1 ∇ 2 Φ − K3 A2 Φ. ∂t While deriving Eqs. (4), it is assumed that the rotation of the director axis (A) is always normal to n so that n · A = 0 which makes the term proportional to K2 to vanish. In order to associate Eqs. (4) with more standard evolution equations, we introduce a new complex scalar field in terms of the components of A. For this, we make A to be a two component vector in the xy-plane, i.e., x ∂ny A = (Ax , Ay , 0) and define a new complex scalar field as U = Ax + iAy . When Az ≡ (∇ ∧ n)z = 0, we have ∂n ∂y = ∂x and the γ1
x
constraint n · A = 0 leads to Ay = − nny Ax . Also in this process, the number of unknowns that appear in the resultant equations (Eqs. (5a), (5b)) are reduced by one. Using the above, Eqs. (4) after suitable rescaling of time can be written as ∂U 2 = ∇ U − |U |2 U , ∂t ∂Φ K1 2 ∇ Φ − |U |2 Φ. = ∂t K3
(5a) (5b)
The above set of coupled equations (5) describes the director dynamics in terms of two new fields namely U and Φ which are respectively related to the bend and splay modes. The advantage of rewriting the dynamical equation in terms of these new fields lies in associating these set of new equations with a family of known integrable nonlinear evolution equations. 3. Molecular deformation dynamics Having derived the dynamical equation in a convenient form, the problem now boils down to solving Eqs. (5) to understand the director dynamics. As we have considered a nematic liquid crystal film for our study, we fix the film to lie in the xy-plane ∞ α ∂2 ∂2 − 2 pψ(x,y)t and Φ = p so that ∇ 2 ≡ ∇⊥ 2 = ∂x 2 + ∂y 2 . We assume the solutions of Eqs. (5) in the form U = p=0 qp (x, y, t)e ∞ α − 2 pψ(x,y)t , where is an arbitrary small parameter and α ’s are real integers. We substitute the above expansions p p p=0 vp (t) e √ √ for U and Φ in Eqs. (5), rescale the time and spatial variables as t → − 2 t , x → 2x and y → 2y, and equate the coefficients 2 2 of every e pψ(x,y)t separately for different p values to zero. Thus, the coefficients of the terms proportional to e nψ(x,y)t where n is an integer give
1 1 ∂qn ∂ψ ∂ψ 2 ∂ψ 2 ∂qn ∂ψ ∂qn + + nψqn + ∇⊥ 2 qn − 2 n ∇⊥ 2 ψ qn + + t + 2 2 nt qn 2+αn ∂t 2 2 ∂x ∂x ∂y ∂y ∂x ∂y
− (6a) αs αn−r αr−s qn−r qr−s qs∗ = 0, s
and
r
K1 2 ∂ψ 2 ∂ψ 2 ∂vn nvn t ∇⊥ 2 ψ + 2 2 nt + + nψvn + ∂t 2K3 ∂x ∂y ∞ ∞
αs αn−r αr−s qn−r q ∗ r−s vs = 0. −
2+αn
s=0 r=0 2
∂ Operating Eq. (6b) by the operator ( αn ∂y 2 −
2(1+αn )
∂2 ) ∂x 2
we obtain
2 ∂v ∂2 K1 2 ∂ψ 2 ∂ψ 2 n 2 2+αn ∂ 2 − nvn t ∇⊥ ψ + 2 nt + + nψvn + 2K3 ∂x ∂y ∂y 2 ∂x 2 ∂t
(6b)
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−
∞ ∞
αs αn−r αr−s
s=0 r=0
2 ∂2 αs αn−r αr−s ∂ − qn−r q ∗ r−s vs = 0. ∂x 2 ∂y 2
(6c)
Eqs. (6) are in the form of an infinite series in powers of and we assume that for every value of n, → 0. However, the terms corresponding to the minimal power of the parameter vanishes and consequently we have [24] ∂q1 1 2 + ∇⊥ q1 + |q1 |2 q1 + ψq1 = 0, ∂t 2 ∂ 2 |q1 |2 ∂ 2ψ ∂ 2ψ − − 2 = 0. ∂y 2 ∂x 2 ∂x 2
(7a)
i
(7b)
While writing the above equations we have transformed time variable as t → it. Also we have chosen qr−s = −qs−r , qs = q−s , vs = v−s and further we have assumed αn = α−n = n for n 1 and α0 = 2. The set of coupled equations (7) are together known as Davey–Stewartson-I (DS-I) equation, which was first derived in fluid dynamics describing the evolution of waves of slowly varying amplitude on a two-dimensional water surface under gravity [25]. The above equation later appeared in several other contexts as well including stability of gravity waves [26], nonlinear optics [27], plasma [28], spin systems [29], etc. 4. Soliton in a nematic film The DS-I equation given in Eqs. (7) is a two-dimensional integrable generalization of the completely integrable cubic nonlinear Schrödinger (NLS) equation and admits solutions in the form of line soliton, dromion and breather which were originally obtained by solving the equation using dressing method in nonlocal Riemann–Hilbert problems [30]. It is also possible to construct the N -soliton solutions for the equation using Hirota’s bilinearization procedure [29,31,32]. However, as our problem involves only time-independent boundary conditions, the dromion solution (which involves time-dependent boundary conditions) is not relevant to us here. Therefore in the following, we write down the explicit form of line one soliton and (1, 1) breather solutions obtained through Hirota’s bilinear method. 4.1. Line one soliton The line one soliton solutions of the DS-I equation (7) is written as q1 = 4 λR μR sech χR + ln(4 λR μR ) exp(iχI ),
(8a)
where (0)
χR = (λR + μR )x + (λR − μR )y − 2[λR λI + μR μI ]t + χR , (0) χI = (λI + μI )x + (λI − μI )y + λR 2 + μR 2 − λI 2 + μI 2 t + χI .
(8b) (8c) (0)
Here χR and χI are the real and imaginary parts of χ = (λ + μ)x + (λ − μ)y + i[λ2 + μ2 ]t + χ (0) and λR (λI ), μR (μI ) and χR (0) (χI ) are the real (imaginary) part of the complex constants λ, μ and χ respectively. Using the line one soliton solution given in Eq. (8a) in Eq. (7b) the field ψ(x, y) can be obtained by solving the resultant equation. The result reads as 2 2 tanh (λR + μR )x + (λR − μR )y + δ − sech (λR + μR )x + (λR − μR )y + δ , ψl = −16λR μR (9) λR + μR √ where δ = ln(4 λR μR ). Knowing q1 and ψ , the bend field U and hence A can be constructed using the relation U ≡ Ax + iAy = q1 e−ψt . Similarly, the splay field Φ can be obtained using the relation Φ = v1 e−ψt . However, since we have switched to an imaginary time variable (t → it) while going from Eq. (6) to Eq. (7), this will arise problem in the case of the solution with real time. This can be avoided by choosing |λR | = |λI | and |μR | = |μI |. Therefore, Eq. (8c) must be replaced by χI = (λI + μI )x + (0) (λI − μI )y + χI . Thus, using Eqs. (8) and (9) we obtain the bend and splay fields as U ≡ Ax + iAy 2 = 4 λR μR sech χR + ln(4 λR μR ) exp 16λR μR tanh (λR + μR )x + (λR − μR )y + δ λR + μR
− 16 sech2 (λR + μR )x + (λR − μR )y + δ t exp iχI ,
(10a)
M. Daniel, K. Gnanasekaran / Physics Letters A 372 (2008) 2623–2633
and
Φ = v1 exp 16λR μR
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2 2 tanh (λR + μR )x + (λR − μR )y + δ − sech (λR + μR )x + (λR − μR )y + δ t . λ R + μR
(10b) Eq. (10a) represents the bend field (i.e., rotation of the director axis) and (10b) the splay field (i.e., divergence of the director axis) of the nematic film corresponding to the line soliton solution of DS-I equation. In Fig. 2(a), we have given a snapshot (at t = 1) of the line one soliton of the DS-I equation as found in Eq. (8) by choosing λR = 0.012, μR = 1.0, λI = 0.012 and μI = 1.0. In Fig. 2(b), we have plotted |A|2 (Eq. (10a)) which is related to the stationary bend mode corresponding to the line one soliton for the same parameter values used to plot the line one soliton in Fig. 2(a). In Fig. 2(c), Φ, which is related to the stationary splay mode corresponding to the line soliton (Eq. (10b)) is plotted by choosing λR = 0.012, μR = 1, λI = 0.012, μI = 1 and v1 = 0.001. From Figs. 2(b), (c) we observe that the bend field |A|2 which represents the rotation of the director axis and the splay field that is related to the divergence of the director axis corresponding to the line one soliton exhibit localized behaviour in the form of pulse and kink respectively. 4.2. (1, 1) breather solution The (1, 1) breather solution of the DS-I equation (7a) and (7b) which is localized spatially and oscillatory in time is given by
q1 =
ρe(λR +μR )x+(λR −μR )y 1 + j e2λR (x+y)
+ ke2λR (x−y)
+ le(2λR (x+y)+2λR (x−y))
2 +μ 2 )t R
ei(λR
(11)
,
where ρ is a complex constant and j , k, l and λR , μR , are real positive constants such that λR 2 + μR 2 1. Knowing q1 , the field ψ(x, y) can be constructed by substituting the solution q1 in Eq. (7b) and the resultant equation solved. Thus, the field ψb (x, y) corresponding to the (1, 1) breather solution given in Eq. (11) is written as 2 2λR x N λR x λR x −2 cosh λR y + e −4 1+e , ψb = −ρ e (12a) D
where N = tanh(λR y) + 2{(eλR x − 2) + (1 + 9λR 2
1 2
−e
5λR 2
λR
5λR 2
x
sinh
λR
λR 2 x
− 4 cosh λR x) sech2 λR y} arctan[
tanh λR y
3 λ x 2(1+e 2 R
cosh
λR 2
] and D = x)
x sinh 2 x − 2e x cosh 2 x − 5e2λR x ) sech2 λR y − (e4λR x + 3eλR x − 4)}. Having found q1 and ψ , the 2{( 19 2 − 4cosh λR x − 4e field U and hence A can be constructed using the relation U ≡ Ax + iAy = q1 e−ψt . Similarly, the field Φ can be obtained using the relation Φ = v1 e−ψ(x,y)t . Thus, in the case of (1, 1) breather solution (Eqs. (11) and (12a)), we obtain
U=
and
ρe(λR +λR )x+(λR −λR )y
1 + j e2λR (x+y) + ke2μR (x−y) + le2(λR +λR )x+2(λR −λR )y 2 2 2λR x N λR x λR x −2 2 cosh λR y + e × exp ρ e −4 1+e + i λR + λR t , D
2 2λR x N λR x λR x −2 Φ = v1 exp ρ e cosh λR y + e −4 1+e t . D
(13a)
(13b)
In Fig. 2(d), we have plotted a snapshot (at t = 1) of the (1, 1) breather solution of the DS-I equation as found in Eq. (11) by choosing λR = 0.012, μR = 0.01 and ρ = 0.001. In Fig. 2(e), |A|2 (Eq. (13a)) which is related to the stationary bend mode corresponding to (1, 1) breather solution of the DS-I equation has been plotted for the same parameter values. In Fig. 2(f), Φ, which is related to the stationary splay mode (Eq. (13b)) corresponding to (1, 1) breather solution of DS-I equation is plotted by choosing λR = 0.012, μR = 1.0 and v1 = 0.001. The (1, 1) breather solution of the DS-I equation is known to exhibit a highly localized pattern (see Fig. 2(d)). From Fig. 2(e) and 2(f) we observe that the bend field |A|2 (rotation of the director axis) is still exhibiting a localized behaviour similar to breathers with an added effect of a localized spike at a specified location (which is evident from Fig. 2(f)), that represents the splay field Φ (divergence of the director axis) corresponding to (1, 1) breather solution of the DS-I equation. 5. Construction of director axis Having found the fields A = ∇ ∧ n and Φ = ∇ · n which correspond to the bend and splay modes respectively in terms of the line one soliton and (1, 1) breather solutions of the DS-I equation we now construct the director axis n using these results. The
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(a)
(b)
(c)
(d)
(e)
(f)
Fig. 2. A snapshot of (a) line one soliton |q1 |2 (Eq. (8)) of DS-I equation for λR = 0.012, μR = 1, λI = 0.012 and μI = 1, (b) of the bend field (|A|2 ) (Eq. (10a)) corresponds to the line one soliton for the same parameter values as given above and (c) splay mode (Φ) (Eq. (10b)) corresponding to the line one soliton of DS-I equation for λR = 0.012, μR = 1, λI = 0.012, μI = 1 and v1 = 0.001, (d) (1, 1) breather solution of DS-I equation |q1 |2 (Eq. (11)), for λR = 0.012, μR = 0.01 and ρ = 0.001. (e) the bend mode (|A|2 ) (Eq. (13a)) corresponding to (1, 1) breather solutions of DS-I equation for the same parameter values as in (d) and (f) splay mode (Φ) (Eq. (13b)) corresponding to (1, 1) breather solution of DS-I equation for λR = 0.012, μR = 0.01 and v1 = 0.001.
equations defining A and Φ can be combined and written as the following two-dimensional Poisson equation: ∇⊥ 2 n = ∇Φ − ∇ ∧ A, n = nx , ny , nz .
(14)
The homeotropic boundary conditions for our problem is written as nx = 1,
ny = 1,
nz = 0|x=(0,l);y=(0,m) .
(15)
The above inhomogeneous boundary conditions (associated with the x- and y-components of the director axis) can be converted into homogeneous ones by defining a new vector field p ≡ (p x , p y , p z ) such that p x = nx − 1, p y = ny − 1, and p z = nz . In terms of the new vector field, the two-dimensional Poisson equation is written in the component form as ∂Φ ∂Φ ∂Ax ∂Ay (16) , ∇⊥ 2 p y = , ∇⊥ 2 p z = − . ∂x ∂y ∂y ∂x The above Poisson equations with the set of new homogeneous boundary conditions can be solved using the Green’s function method of eigenfunction expansion. ∇⊥ 2 p x =
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The equation governing the Green’s function G is written as ∇⊥ 2 G = −δ(x − ζ )δ(y − β),
(17)
with G satisfying the boundary conditions G(x, ζ ; y, β) = 0|(x=0,l);(y=0,m) . Here ζ and β are arbitrary points along x and y directions respectively. The Green’s function solution of Eq. (17) can be expressed using the method of eigenfunction expansion as [33]
∞ ∞ 4lm sin sπl ζ sin smπ β sin sπl x sin smπ y . G(x, ζ ; y, β) = 2 π (ms )2 + (ls)2
(18)
s=1 s =1
Using the value of the above Green’s function G, the components of the vector field p namely p x , p y and p z can be constructed using the right-hand sides of Eq. (16) as follows: x y
dx dy G(x , y ; ζ, β)Ri ,
p (x, y) = i
i = x, y, z,
(19)
−∞ −∞
where Ri ’s are the right-hand sides of the p x , p y and p z equations in Eqs. (16). On substituting the Green’s function G from Eq. (18) and the values of Φ, Ax and Ay from Eqs. (10) corresponding to the line one soliton of the DS-I equation in Eq. (19), we obtain the values of p x , p y and p z as follows: p x = γss hx (x, y, t) − Hx (x, y, t) , (20a) y p = γss hy (x, y, t) − Hy (x, y, t) , (20b) z p = γss hz (x, y, t) + hz (x, y, t) − Hz (x, y, t) − Hz (x, y, t) . (20c) ∞ ∞ π 4lm sπ s In the above equations γss = π 2 s=1 s =1 sin l ζ sin m β. Defining sπ 2 sπ tanh (λR + μR )x + (λR − μR )y + δ x sin y exp 16λR μR f (x, y, t) = sin l m λR + μR
− sech2 (λR + μR )x + (λR − μR )y + δ t the functions hi (x, y, t) and Hi (x, y, t) (i = x, y, z) are written as y hx (x, y, t) = v1
f (x, y , t) dy ,
−∞ x
−∞ −∞ x y
Hy (x, y, t) = v1
−∞ y
hz (x, y, t) =
−∞ x
f (x , y, t) dx ,
(21a)
f (x , y , t) sin
n π y dx dy , m
(21b)
f (x , y , t) sin
nπ x dx dy , l
(21c)
−∞ −∞
hz (x, y, t) =
hy (x, y, t) = v1 −∞
y
Hx (x, y, t) = v1
x
x
f (x , y, t) sech2 (λR + μR )x + (λR − μR )y + δ cos (λR + μR )x + (λR − μR )y dx ,
(21d)
f (x, y , t) sech2 (λR + μR )x + (λR − μR )y + δ sin (λR + μR )x + (λR − μR )y dy ,
(21e)
nπ nπ f (x , y , t) sin x cos y cos (λR + μR )x + (λR − μR )y Hz (x, y, t) = l m −∞ −∞ × sech2 (λR + μR )x + (λR − μR )y + δ dx dy , x y nπ n π Hz (x, y, t) = f (x , y , t) cos x sin y sin (λR + μR )x + (λR − μR )y l m −∞ −∞ × sech2 (λR + μR )x + (λR − μR )y + δ dx dy . y
(21f)
(21g)
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As it is difficult to evaluate the above integrals in the present form, we expand the exponential function that appear in the function f (x, y, t). After using the expansion and on evaluating the integrals at the lowest order (dropping higher order terms) and using the results in Eqs. (20), we obtain the components of the director axis in the form nx ≡ p x + 1
sπ sπ γss v1 m l μ ) sin (1 − 16λ x cos y =1+ R R sπ sπ l m
2 tanh (λR + μR )x + δ − sech2 (λR + μR )x + δ t , + exp 16λR μR λR + μR
γss v1 m m sπ sπ y y n ≡p +1=1+ (1 − 16λR μR ) 1 − cos x sin y sπ sπ l m
2 2 tanh (λR + μR )x + δ − sech (λR + μR )x + δ t , + exp 16λR μR λR + μR
and nz ≡ p z =
γss l m sπ sπ (16λR μR − 1) 1 − cos x sin y sπ sπ l m m sπ sπ + sin (λR + μR )x sin x cos y sech (λR + μR )x + δ s π l m
2 tanh (λR + μR )x + δ − sech2 (λR + μR )x + δ t . × exp 16λR μR λR + μR
(22a)
(22b)
(22c)
The above solutions must satisfy the constraint n · A ≡ n · ∇ ∧ n = 0, which is equivalent to nx Ax + ny Ay = 0. Therefore, to verify this, we evaluate the quantities nx Ax and ny Ay separately using Eqs. (22a), (22b) and (10a) and write the final form of the expression nx Ax + ny Ay as nx Ax + ny Ay ∞ ∞
l 16lm πζ s πβs λ = μ (1 − 16λ μ ) sin sin R R R R 2 πs l m π s=1 s =1
πs π s 2 2 × exp 16λR μR tanh (λR + μR )x + δ − sech (λR + μR )x + δ t + sin x cos y cos χ1 λ R + μR l m
πs m 2 + tanh (λR + μR )x + δ − sech2 (λR + μR )x + δ t 1− exp 16λR μR l π s λR + μR πs πs + cos x sin y sin χ1 l m 2 × sech χR + ln(4 λR μR ) exp 16λR μR tanh (λR + μR )x + (λR − μR )y + δ λ R + μR
− sech2 (λR + μR )x + (λR − μR )y + δ t , (23) ∞ l πβs πζ s which should vanish. The right-hand side of Eq. (23) vanishes when ∞ s=1 s =1 π s sin l sin m = 0 which is equivalent to ∞ ∞ l πβs πβs πζ s πζ s s=1 s =1 πs [cos( l + m ) − cos( l − m )] = 0. The left-hand side of the above condition averages out to be zero quickly when l and m are large, that is in the case of an infinite film. Thus, the solutions satisfy the constraint n · A = 0 in the limit of an infinite film. We repeat the entire calculations for the case of (1, 1) breather solution of the DS-I equation. However, in this case, the various resultant expressions are very complicated and due to the unwieldy nature of the final results, we are not presenting the details here. In Figs. 3(a)–(c), we have given a snapshot of the x-, y- and z-components of the director axis namely nx , ny and nz for s = s = 1.0, λR = 0.012, μR = 1.0, λI = 0.012, μI = 1.0, ζ = β = 0.1, and l = m = 1.0 at time t = 1. From the figures it is observed that the x- and y-components of the director axis namely nx and ny slowly gain spatial oscillations in the xy-plane starting from the equilibrium configuration. On the other hand, the z-component of the director axis (nz ) exhibits uniform periodic oscillations along the y-direction. However, along the x-direction the amplitude grows and gets saturated. Further, when we analyze the long time behaviour of the director axis oscillations along different directions, we find that the oscillations are getting damped as shown in Figs. 3(d)–(i).
M. Daniel, K. Gnanasekaran / Physics Letters A 372 (2008) 2623–2633
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
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(i) Fig. 3. Snapshots of the x-, y- and z-components (nx , ny , nz ) of the director axis (Eqs. (22a)–(22c) respectively), exhibiting spatial oscillations for s = s = 1.0, λR = 0.012, μR = 1.0, λI = 0.012, μI = 1.0, v1 = 0.001, ζ = β = 0.1 and l = m = 1.0 at t = 1. (d)–(i): Long-time behaviour of the x-, y- and z-components (nx , ny and nz ) of the director axis along x and y axes showing damped oscillations for the same values of the parameters.
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6. Conclusions In this Letter, we have investigated the nature of director dynamics in a nematic liquid crystal film, with homeotropic boundary conditions. Starting from the Frank free energy density for splay, twist and bend-type elastic deformations, we constructed the associated field equation by minimizing the free energy density. In order to solve the resultant field equation for understanding the director dynamics, we introduced two new fields in terms of splay and bend modes (i.e., divergence and rotation of the director axis respectively). A multi-scale expansion was found useful in finally writing the resultant equation in terms of the well-known completely integrable Davey–Stewartson equation in (2 + 1) dimensions which admits line soliton and breather solutions for our problem. Using these soliton solutions of DS-I equation, the components of the director axis were constructed by solving the Poisson equation associated with the definitions of splay and bend fields of our model by using Green’s function technique. From the results, we observe that the splay and bend fields of the director axis exhibit localized structures of reorientation due to elastic deformations in close analogy with the localized soliton and breather solutions of the DS-I equation. Spatial soliton in nematic liquid crystal was experimentally observed when the nematic film is allowed to pass through high intense coherent laser beam [3,21,34]. However, there is no experimental evidence available in the literature for the generation of solitons induced through elastic deformations. This may be due to the fact that solitons generated through elastic deformations are expected to be less stable compared to solitons generated through externally applied optic field. Hence, it may be very difficult to observe soliton induced through elastic deformation. It is planned to carry out stability analysis of the elastic solitons and the results will be published elsewhere. From Fig. 3, we notice that at a given time the x- and y-components of the director axis gain uniform oscillations form their equilibrium configuration. However, the z-component of the director axis shows a slightly different behaviour with the amplitude growing steadily and gets saturated along x-direction but exhibiting uniform oscillations along y-direction. However, the long time behaviour of oscillation of the director axis components in different directions is shown to exhibit damping feature of the oscillations, which is due to the fact that the two-dimensional Poisson equation is solved using the eigen function expansion method for Green’s function. Acknowledgements The work of M.D. forms part of a major DST project. K.G. would like to thank UGC for a Teacher Fellowship under the FIP programme. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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