Shear Flow of a Nematic Liquid Crystal near a Charged Surface

Physics of the Solid State, Vol. 45, No. 6, 00, pp. 9 96. Translated from Fizika Tverdogo Tela, Vol. 45, No. 6, 00, pp. 5 40. Original Russian Text Copyright 00 by Zakharov, Vakulenko. POLYMERS AND LIQUID CRYSTALS Shear Flow of a Nematic Liquid Crystal near a Charged Surface A. V. Zakharov and A. A. Vakulenko Institute for Problems of Mechanical Engineering, Russian Academy of Sciences, Bol shoœ pr. 6, Vasil evskiœ ostrov, St. Petersburg, 9978 Russia Received May 0, 00; in final form, October, 00 Abstract The maximum alignment angle θ and the ective rotational viscosity coicient of polar liquid crystals, such as 4-n-octyloxy-4'-cyanobiphenyl (8OCB), are investigated in the vicinity of charged bounding surfaces. The quantities θ and are calculated in the framework of the Ericksen Leslie theory. The results of calculations demonstrate that, for a homeotropic alignment of molecules on charged indium tin γ oxide surfaces, the ective rotational viscosity coicient can increase by 7.8% as compared to the bulk rotational viscosity coicient γ. 00 MAIK Nauka/Interperiodica. γ i γ i. INTRODUCTION In a shear laminar flow of a nematic liquid crystal between two bounding surfaces, the flow field v(r) is one of the main factors affecting the director field n(r). In the bulk of a nematic liquid crystal, the influence of bounding surfaces can be ignored and the hydrodynamic properties can be described in terms of the classical Ericksen Leslie theory allowing for the interaction of the fields n(r) and v(r). For a Couette flow between two plane-parallel surfaces when one (lower) surface is stationary and the other (upper) surface moves with a constant velocity v, the flow field takes the form v(r) = v (y)i. Here, the x axis is aligned with the unit vector i oriented parallel to the bounding surfaces and the y axis is directed along the unit vector j perpendicular to the bounding surfaces. At high flow rates, the equilibrium angle θ bulk between the vectors n and v can be determined from the condition that the hydrodynamic torques acting on an elementary volume of the nematic liquid crystal T vis = (/)(γ + γ cosθ bulk ) γ are equal to zero [, ]. As a result, we have θ bulk = -- cos ( /λ), () where λ = γ /γ, γ, and γ are the rotational viscosity coicients of the nematic liquid crystal and γ = v (y)/ y is the strain rate. However, in the vicinity of charged bounding surfaces, the hydrodynamic description of anisotropic systems such as liquid crystals should take into account the ect of elastic and surface forces acting on the material over a depth ξ. Since these forces make an additional contribution to the balance of the torques acting on the elementary volume of the nematic liquid crystal, they also affect the angle θ bulk and the rotational viscosity coicients γ i (i =, ). In this case, the surface forces are responsible for the coupling of nematic liquid-crystal molecules with the surface and the surface coupling energy can be written in the form [] f 0 = --ω 0 ( n s n 0 ) = --ω 0 cos ( θ s θ 0 ), () where ω 0 is the coupling constant having the dimension of the surface energy density, θ s is the angle between the vectors n s and n 0, and θ 0 is the angle between the vectors n s and v. Here, n s stands for the director n on the bounding surface and n 0 is the unit vector aligned along the easy orientation axis [] and characterizing the surface anisotropy. The surface energy f 0 is localized in a narrow surface layer of depth λ s ~ 0 00 nm [4]. In this range, the order parameter of the nematic liquid crystal varies from the surface to bulk value. When the solid bounding surface is in contact with the nematic liquid crystal, the surface selectively interacts with ions involved in the nematic phase. For example, a negatively charged surface with the charge density σ attracts positive ions and repels negative ions. If the number N + of positive ions is equal to the number N of negative ions, the depth of penetration of the electric field E(y) induced by the surface charge density σ is equal to the Debye length λ D [5] (the case of a weak electrolyte). The spatial dependence of this electric field for the bulk screening can be represented in the following form: E( y) = E 0 exp( y/λ D )j, () 06-784/0/4506-9$4.00 00 MAIK Nauka/Interperiodica

9 ZAKHAROV, VAKULENKO where E 0 = σ/ 0 is the electric field strength of the charged surface, 0 is the permittivity of free space, = ( + )/ is the mean permittivity, and and are the permittivities parallel and perpendicular to the director n, respectively. In the present work, we applied the Ericksen Leslie theory [, ] to analyze how the surface charge density σ affects the angle θ (y) and the rotational viscosity coicient γ at distances of the order of ξ. Since experimenters have encountered considerable difficulties in measuring these parameters in the vicinity of bounding surfaces [6], theoretical data on θ (y) and γ are undeniably of particular importance. This paper is organized as follows. The basic equations of the hydrodynamic theory for nematic liquid crystals in the vicinity of charged bounding surfaces are presented in Section. The results of numerical calculations of the angles θ (y) and the rotational viscosity coicients γ are summarized in Section.. BASIC EQUATIONS OF THE HYDRODYNAMIC THEORY FOR NEMATIC LIQUID CRYSTALS NEAR A CHARGED SURFACE Within the classical Ericksen Leslie approach to considering the viscosity of nematic liquid crystals, a preferential orientation of molecules in the vicinity of points r is described by the director vector field n(r, t). This orientation can vary from point to point. In the incompressible-fluid approximation ( v = 0), the balances of momentum and torques acting on an elementary volume are represented by the relationships [, ] ρ dv ----- = σ, (4) dt T vis + T el + T elast = 0. (5) Here, ρ = N/V is the particle number density and σ is the stress tensor with the components defined by the expressions [] σ ij = α n l n m M lm n i n j + α n i N j + α N i n j (6) + α 4 M ij + α 5 n i n l M lj + α 6 M im n m n j, where N m dn m = -------- + -- ( v dt mk, v k, m )n k, M ij = -- ( v i, j + v j, i ). (7) In relationships (7), M i, j are the components of the symmetric part of the Euler tensor. The formula describing the director kinetics includes the antisymmetric part W of this tensor (the components of the vector N) in the form of the convolution W n, which is the vector with components (/)(v m, k v k, m )n k (m =,, ). The material derivatives of the components of the vector n dn can be written as m n -------- = -------- m + v l n m, l. dt t The Leslie coicients α i (i =,,, 6) satisfy the Onsager Parodi relation α + α = α 6 α 5, and, hence, only five out of six coicients α i are independent. The rotational viscosity coicients γ i (i =, ) and the Leslie coicients α i are related by the expressions γ = α α and γ = α 6 α 5. The torque with respect to the director due to the hydrodynamic forces has the form T vis = n ( γ N + γ M n), (8) the torque associated with the electric forces is given by the relationship T el = ( a / 0 )n E( E n), (9) and the torque caused by the elastic forces is defined as T elast = n h. (0) Here, the components of the vector N are determined by relationships (7) and M n is the convolution of the symmetric part M of the Euler tensor with the vector n. This convolution represents the vector with components (v i, j + v j, i )n j / (i =,, ). The molecular field associated with the gradients of the director n [] in the strained nematic liquid crystal is specified by the vector h = h s + h t + h b, where h s = K ( n), h t = K [(n l)l + (n l)n], h b = K [n l l + (n (n l))], and l = n. Here, the Frank elastic constants K i (i =,, ) describe three types of strains in the nematic liquid crystal, namely, the lateral bending, torsional, and buckling strains, respectively, and a = is the dielectric anisotropy of the nematic liquid crystal. In the case of a Couette flow with a planar geometry, the vectors v(r) and n(r) are represented by the expressions v = (v (y), 0, 0) and n = (cosθ, sinθ, 0). Making allowance for the sole nonzero component of the flow field gradient γ = v (y)/ y, expression (5) can be transformed into the form [6] θ ---------- + -- [ + ( γ τ /γ ) cosθ ] -- sinθ B () h( θ ) θ ------------ --h'( θ y ) θ ---------- = 0, y where B = B /γ γ = a σ /( 0 γ γ )exp( y ), y = y/λ D is the dimensionless coordinate, τ = tγ is the dimensionless time, h(θ ) = (K cos θ + K sin θ )/( γ γ λ D ), and h'(θ ) is the derivative of the function h(θ ) with respect to θ. At high strain rates, the director orientation is determined by the balance of PHYSICS OF THE SOLID STATE Vol. 45 No. 6 00

SHEAR FLOW OF A NEMATIC LIQUID CRYSTAL 9 only the hydrodynamic forces and the electric forces affect the material over a depth 0 ξ µm [6]. In particular, for 4-n-octyloxy-4'-cyanobiphenyl (8OCB) at γ = 800 s and σ = 0 C/m, we obtain T elast ~ 0.5 N/m, T vis ~ 4.0 N/m, T el (y = 0. µm) ~ N/m, and T el (y = µm) ~ 0. N/m. Taking into consideration that = T elast / T vis ~ 0.5, the contribution of the elastic forces to the momentum balance can be disregarded at strain rates γ 800 s. Therefore, Eq. () can be rewritten in the following form: θ ---------- + -- [ + ( γ () τ /γ ) cosθ ] -- sinθ B = 0. For a stationary Couette shear flow and arbitrary strain rates γ, Eq. () can be represented in the form θ ( y) -------------------- Aexp( y)θ y ( y) + D = 0, where A = a σ λ D / K and D = (γ + γ ) γ λ D /K. () Equation () was written with due regard for the fact that the last term in Eq. () can be ignored in comparison with the other terms, because we have θ 4 ( θ < 0.8) for all liquid crystals [, 6] and θ '' ( y ) 0.0. The boundary conditions for Eq. () are as follows: θ ( y ) = 0 and θ ' ( y ) = 0 at y = y a = λ s /λ D and θ ( y ) = θ bulk and θ ' ( y ) = 0 at y = y b. Below, we will demonstrate that σ = 0 C/m and λ D = 0.545 µm for the homeotropic orientation of the 8OCB molecules of the nematic liquid crystal on the indium tin oxide bounding surface at a temperature of 40 K and σ = 0 C/m. The theoretical calculations performed in terms of the statistical mechanical theory [4] indicate that it is reasonable to choose λ s ~ 0. µm for 8OCB molecules in the vicinity of the indium tin oxide surfaces. On this basis, the interval boundaries are determined to be y a = 0.8 and y b = 5.5. At A, Eq. () without the last term D possesses the asymptotic solution [7] θ y ( y) q /4 iδ q / = exp ± dy [ + O ( δ )], 0 (4) where q = exp( y ) and δ = i A. For the 8OCB nematic liquid crystal, we find A 7.7 at σ = 0 C/m. Since D 0.004, the last term in Eq. () can be ignored. Then, the asymptotic solution of Eq. () with allowance made for the boundary conditions takes the form θ ( y) = C exp[ y/ + A( exp( y) ) ] (5) + C exp[ y/ A( exp( y) ) ], where C i = θ bulk β i /β (i =, ), β = β β β β 4, β = exp{ y a / A [exp( y a ) ]}, β = exp{ y a / + A [exp( y a ) ]}, β = exp{ y b / A [exp( y b ) ]}, and β 4 = exp{ y b / + A [exp( y b ) ]}. At high strain rates ( γ 800 s ), Eq. () in the vicinity of the charged bounding surface has the form γ [ γ + γ cosθ ] = B sinθ (6) and the equation for θ can be written as follows: θ = -- cos ( /λ ) = -- cos C, λ (7) where C = = α 0 κ[ + ( + ( )κ ) / ], α 0 = λ = γ /γ, and κ = + α 0 B 4. According to the theory proposed by Kuzuu and Doi [8], the rotational viscosity coicient can be represented in the form P γ = ( f /λ)p. (8) Here, is the order parameter of the nematic liquid crystal (the second-degree Legendre polynomial averaged over molecular orientations [8]), f = ρk B Tp/D, D is the coicient of rotational diffusion with respect to the short axes of the molecules of the nematic liquid crystal, k B is the Boltzmann constant, p = (a )/(a + ), and a = σ /σ is a geometric parameter equal to the length-to-width ratio of molecules forming the nematic liquid crystal. It should be noted that the bounding surface affects the order parameter P only in a very thin layer (~0 00 nm [4]). Moreover, the broadband dielectric spectroscopic data obtained by Rozanski et al. [9] indicate that the relaxation time τ 00 of molecular rotation about the short axes for cyanobiphenyls (including 8OCB) in pores with a diameter of up to 0. µm is virtually identical to that in the bulk of samples. Since the relaxation time τ 00 and the rotational diffusion coicient D are related through the expression P τ 00 = D -----------------, P there are strong grounds to believe that the quantities P and D depend only on the temperature. Therefore, the ratio of the rotational viscosity coicients in the B 4 PHYSICS OF THE SOLID STATE Vol. 45 No. 6 00

94 ZAKHAROV, VAKULENKO Magnitudes of the rotational viscosity coicients γ and γ in the bulk of the nematic phase of 8OCB according to the data taken from [0] T, K 40 45 50 γ, kgm/s 0.046 0.0 0.0 γ, kgm/s 0.049 0.05 0.0 vicinity of the charged bounding surface bulk of the sample has the form γ γ /γ = λ/λ = ( γ /γ ) cosθ ( y). and in the (9) Knowing the rotational viscosity coicients in the bulk of the nematic liquid crystal, for example, 8OCB [0] (see table), we can calculate the ective rotational viscosity coicients γ and the angle θ ( y ). In the case of high strain rates ( γ 800 s ), the above ratio of the rotational viscosity coicients can be written as follows []: γ /γ = λ/λ = κ[ + ( + ( B 4 )κ ) / ]. θ (y) 8 4 0 4 0 Fig.. Spatial dependences of the angle θ in a shear flow of the 8OCB nematic liquid crystal at a temperature of 40 K for a homeotropic alignment of molecules (θ s = 0) on a charged bounding surface (σ = 0 C/m ). Strain rates: (, ) 80 and (, 4) 00 s. Calculations are performed according to relationships (, 4) (5) and (, ) (7). (0) channels at strain rates γ 800 s and lower should be calculated with allowance made for the elastic forces. The spatial dependences of the ratio γ /γ for the homeotropic orientation of 8OCB molecules on the indium tin oxide bounding surface (θ s = 0) at two sur-. RESULTS OF NUMERICAL CALCULATIONS AND DISCUSSION The derived relationships allow us to calculate the profiles θ ( y ) and the dependence of the ratio γ /γ of the rotational viscosity coicients on the distance to the charged bounding surface. Hereafter, we will analyze the shear flow of the anisotropic system formed by 8OCB molecules near the indium tin oxide surface in the temperature range from 40 to 50 K corresponding to the nematic phase [6]. Taking into account that the bulk concentration of ions in the liquid-crystal phase is given by n bulk = N + /V = N /V 5 0 0 m [6, ], the Debye screening length can be calculated from the formula [5, ] λ 0 k B T D = ----------------- /, e n bulk where e =.60 0 9 C is the proton charge and = 0 ( cos θ s + sin θ s ). The Debye screening length was determined to be λ d = 0.55 µm for the homeotropic orientation of the director on the bounding surface (θ s = 0) and λ D = 0.8 µm for the planar alignment of the director (θ s = π/). In our calculations, the surface charge density was chosen to be σ = 0 C/m. This surface charge density σ = en surf corresponds to the surface concentration of charge carriers n surf = 0 6 m and agrees well with the experimental data n ~ 0 5 0 6 m [5, 4]. The strain rate in our calculations varied from 0 to 800 s. Figure depicts the dependences θ (y) calculated from formulas (5) and (7) for the homeotropic orientation of the director on the bounding surface (θ s = 0) at the surface charge density σ = 0 C/m and a temperature of 40 K. At this temperature, the equilibrium angle θ (y) monotonically increases to the bulk value θ (y ~ µm) = θ bulk = 0.5 with an increase in the distance to the bounding surface. It was found that, for flows at high strain rates γ 800 s, the angles θ (y) calculated from relationship (5) (Fig., curve ), which takes into account the contributions of the hydrodynamic, elastic, and electric forces to the momentum balance, are almost identical to those calculated with expression (7) (Fig., curve ) allowing only for the contribution of the hydrodynamic forces. As the strain rate γ decreases, the contribution of the elastic forces increases; as a result, the dependence θ (y) calculated from relationship (7) (Fig., curve ) is shifted with respect to the dependence θ (y) calculated using relationship (5) (Fig., curve 4). Therefore, shear flows of nematic liquid crystals in charged PHYSICS OF THE SOLID STATE Vol. 45 No. 6 00

SHEAR FLOW OF A NEMATIC LIQUID CRYSTAL 95 γ /γ.08 γ /γ.06.04.0.00 (a) 0.98 (a).08.04.08.04.00 (b) 0 Fig.. Dependences of the ratio γ /γ of the rotational viscosity coicients on the distance y to the bounding surface for the 8OCB nematic liquid crystal at the homeotropic alignment of the director (θ s = 0) for the strain rate γ = 00 s (a) at two surface charge densities () 0 and () 0 4 C/m and the temperature T = 40 K and (b) at three temperatures T = () 40, () 45, and () 50 K. Calculations are performed according to relationship (9). face charge densities σ = 0 C/m (curves ) and σ = 0 4 C/m (curves ), a temperature of 40 K, and strain rates γ = 00 and 800 s are displayed in Figs. a and a, respectively. The behavior of the dependences of the ratio γ /γ suggests that the electric forces are involved in the bulk of the shear flow over a depth of ~ µm. As the surface is approached, the ective rotational viscosity coicient γ increases by 7.8% as compared to the bulk coicient γ. The surface charge density affects the depth of penetration of the electric forces. For shear flows at the strain rate γ = 00 s, an increase in σ by an order of magnitude from 0 4 to 0 C/m results in an increase in the depth of penetration of the electric field by ~.5 µm. The calculated ratios γ /γ for the homeotropic orientation of the director on the bounding surface at three temperatures T = 40 K (curves ), 45 K (curves ), and 50 K (curves ) and the strain rates γ = 00 and 800 s are presented in Figs. b and b, respectively. It can be seen that, in both cases, an increase in the temperature is.00 (b) 0 Fig.. The same as in Fig. at the strain rate γ = 800 s. γ /γ.08.04.00 (a).08.04.00 (b) 0 Fig. 4. Dependences of the ratio γ /γ of the rotational viscosity coicients on the distance y to the bounding surface for the 8OCB nematic liquid crystal at the planar alignment of the director (θ s = π/) for the surface charge density σ = 0 C/m (a) at three temperatures T = () 40, () 45, and () 50 K and the strain rate γ = 00 s and (b) at three strain rates γ = () 00, () 00, () 00 s and the temperature T = 40 K. PHYSICS OF THE SOLID STATE Vol. 45 No. 6 00

96 ZAKHAROV, VAKULENKO accompanied by a decrease in the ective rotational viscosity coicient γ. An analysis of the results presented in Figs. a and a demonstrates that the depth of penetration of the electric field into the shear flow increases with a decrease in the strain rate. In turn, this leads to an increase in the ective rotational viscosity coicient γ. The spatial dependences of the ratio /γ calculated for the planar alignment of the director on the bounding surface (θ s = π/) are plotted in Fig. 4. The dependences at three temperatures (T = 40, 45, and 50 K), the surface charge density σ = 0 C/m, and the strain rate γ = 00 s are depicted in Fig. 4a. Figure 4b shows the dependences at the temperature T = 40 K, the surface charge density σ = 0 C/m, and three strain rates ( γ = 00, 00, 00 s ). In all cases, the depth of penetration of the electric field into the bulk of the shear flow is approximately equal to.5 µm, which is half as large as the depth for the homeotropic orientation. Knowing the magnitudes of the bulk coicients γ i (i =, ) calculated recently in [0] for the nematic phase of 8OCB (see table), it is easy to obtain the magnitudes of the coicients γ. It should be noted that, according to relationship (0), an infinite increase in the surface potential density σ results in lim γ /γ = σ γ /γ and an infinite increase in the depth of penetration of the electric field into the bulk of the sample. Therefore, the maximum increment in the coicient γ with respect to the coicient γ is approximately equal to 6 8% [0, 5]. In the other limiting case σ 0, we have lim γ /γ =. σ γ 4. CONCLUSIONS Thus, the influence of the charged bounding surface on the rotational viscosity in the shear flow of the nematic liquid crystal was investigated in the framework of the Ericksen Leslie theory. The contribution of the long-range electric forces to the rotational viscosity coicient and the maximum alignment angle between the direction of the Couette shear flow and the director was analyzed. For this purpose, the terms associated with the hydrodynamic, elastic, and electric forces were included in the equation for the balance of the torques acting on the elementary volume. It was established that the ective rotational viscosity coicient increases as the surface is approached. In the case of the 8OCB nematic liquid crystal, this coicient can increase by 7.8% as compared to the bulk rotational viscosity coicient γ. It was found that the contribution of the elastic forces to the torque balance can be ignored for shear flows at strain rates γ 800 s. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (project no. 0-0-084) and the Natural Science Foundation (project no. E00-5.0-54). REFERENCES. J. L. Ericksen, Arch. Ration. Mech. Anal. 4, (960).. F. M. Leslie, Arch. Ration. Mech. Anal. 8, 65 (968).. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, nd ed. (Oxford Univ. Press, Oxford, 995; Mir, Moscow, 98), p. 60. 4. A. V. Zakharov, Phys. Rev. E 5, 5880 (995). 5. J. N. Israelachvili, Intermolecular and Surface Forces, nd ed. (Academic, London, 99). 6. A. V. Zakharov and R. Y. Dong, J. Chem. Phys. 6, 648 (00). 7. V. F. Zaœtsev and A. D. Polyanin, A Handbook on Ordinary Differential Equations (Fiz. Mat. Lit., Moscow, 00), p. 576. 8. N. Kuzuu and M. Doi, J. Phys. Soc. Jpn. 5, 486 (98). 9. S. A. Rozanski, R. Stunnarius, H. Groothues, and F. Kremer, Liq. Cryst. 0, 59 (996). 0. A. V. Zakharov and R. Dong, Phys. Rev. E 6, 0704 (00).. A. V. Zakharov and R. Dong, Eur. Phys. J. E 7, 67 (00).. S. Ponti, P. Ziherl, C. Ferrero, and S. Zumer, Liq. Cryst. 6, 7 (999).. R. N. Thurston, J. Cheng, R. B. Meyer, and G. D. Boyd, J. Appl. Phys. 56, 6 (984). 4. S. Ponti, P. Ziherl, C. Ferrero, and S. Zumer, Liq. Cryst. 6, 7 (999). 5. A. G. Chmielewski, Mol. Cryst. Liq. Cryst., 9 (986). Translated by O. Borovik-Romanova PHYSICS OF THE SOLID STATE Vol. 45 No. 6 00