The anchoring phenomenon between liquid crystal (LC) and solid is usually described by the anchoring energy, ^[1] which can be seen as an anisotropic part of the interface energy. The expression of anchoring energy and its physical effects in the LC cell have been an important content of the LC surface physics.^[2]

Physical reasons for the anchoring phenomenon and its microscopic mechanism are important subjects.^[3] Recently, Yang et al.^[4] have proposed that interface energy should be seen as the total sum of potentials between molecules of LC and the substrate surface. They gave out an expression of the anchoring energy, which turns into an RP formula in the one-dimensional (1D) case. However, the anchoring energy may originate from other physical reasons, such as adsorptive ions, surface polarization P_s and flexoelectric polarization P_f, which induce the effective ahchoring energy on the substrate surface. The problem of effective anchoring energy induced by surface polarization or flexoelectric polarization has aroused the interest of many investigators.^{[5– 14]} Although adsorption ion exists inevitably in the real LC cell, the density of adsorptive ions is strongly dependent on the clean technology and cleaning time for the substrate. For a long cleaning time the density of adsorptive ions can be ignored. Hence, it is necessary to study these effects on the nematic liquid crystal (NLC) cell when the density of adsorptive ions is ignored.

In this paper, we investigate the physical effects of surface and flexoelectric polarization on the weak anchoring NLC cell for the case of ignoring the density of adsorptive ions, and put emphasis on the contributions of these two polarizations to the effective anchoring energy. Flexoelectric polarization P_f can be expressed as^[15]

where e₁₁ and e₃₃ are splay and bend flexoelectric coefficients, respectively, and n is the director of LC. Surface polarization P_s can be induced by many reasons, ^[16] in which the order-electric polarization can be expressed as^[17]

where r₁ and r₂ are the order-electric coefficients, and S is the surface order parameter.

Based on the Frank elastic theory and Eqs. (1) and (2), the expression of the Gibbs free energy G for this NLC cell can be deduced. It consists of elastic, dielectric, polarization free energy, and anchoring energy. By calculating the variation of the Gibbs free energy G under a given voltage U using the method indicated in Ref. [18], we obtain the differential equation and the boundary conditions of the director. Then, the mathematical method to solve this equation with boundary conditions is given. Furthermore, we obtain the expressions of effective anchoring energy g_s1eff and g_s2eff for the lower and upper substrate respectively. We find that the anchoring strength and easy direction of effective anchoring energy vary with the applied voltage U no matter what the original anchoring is. For an original weak anchoring hybrid aligned nematic (HAN) cell, it may be equivalent to a planar cell for small values of U and has a threshold voltage.

We consider a weak anchoring NLC cell with two substrates perpendicular to the z-axis. An external voltage U is applied to the cell. Assuming that the director of LC is limited in the xoz-plane, the tilt angle θ of the director is only a function of z. The anchoring energy densities of the lower substrate (placed at z = 0) and upper substrate (placed at z = l) are g_s1 = A₁ sin²(θ ₀ − Θ ₁)/2 and g_s2 = A₂ sin²(θ _l − Θ ₂)/2, where A₁ and A₂ are the anchoring strengths, θ ₀ and θ _l are the tilt angles of the director, and Θ ₁ and Θ ₂ are the tilt angles of the easy direction e₁ and e₂ at the lower and upper substrate, respectively.

Suppose that the surface polarizations are P_s1 and P_s2 for the lower and the upper substrate, and they arise in very thin surface layers with thickness values δ ₁ and δ ₂ (δ ₁, δ ₂ ≪ l) respectively, i.e., they satisfy the following conditions: P_s1 = 0 for z ≥ δ ₁ and P_s2 = 0 for z ≤ l − δ ₂. The total polarization is P = P_f + P_s1 + P_s2. The dielectric displacement D should be D = ε _⊥ E + Δ ε (n· E)n + P, where Δ ε = ε _// − ε _⊥ is the anisotropy of the dielectric constant, ε _// and ε _⊥ are the dielectric constants parallel and perpendicular to the director respectively. Because ∇ · D = 0, the z component of the dielectric displacement, D_z = (ε _⊥ + Δ ε sin²θ )E_z + P_z, is a constant for 0 ≤ z ≤ l. The applied voltage can be expressed as

where g(θ ) = ε _⊥ + Δ ε sin²θ .

The total Gibbs free energy per unit area of substrate can be expressed as

where f_elas, f_diel, and f_polar denote the free energy density of elastic, dielectric, and polarization respectively, and can be expressed as f_elas = f(θ )θ ^{′ 2}/2, , f_polar = − P_z E_z, θ ′ = dθ /dz, f(θ ) = k₁₁ cos²θ + k₃₃ sin²θ , k₁₁ and k₃₃ are the splay and bend elastic constants, respectively. Substituting the expressions of g_s1 and g_s2 into Eq. (4), the expression of Gibbs free energy becomes

where

The differential equation and the boundary conditions of θ (z) can be derived by calculating the variation of G for given voltage U using the method explained in Ref. [18]. Suppose that is the solution which makes G minimal and then θ (z) can be expressed as

where ξ (z) is an arbitrary function of z, and η is a common variable. Substituting Eq. (6) into Eq. (5), the resulting G becomes a common function of η . According to ∂ G/∂ η | _{η = 0} = 0, we obtain the equation and the boundary conditions of θ (z) as follows:

where , , and we neglect all terms proportional to δ ₁ or δ ₂ by considering that the values of P_f and P_s have the same order of magnitude and δ ₁ ≪ l, δ ₂ ≪ l. The mathematical progress is presented in Appendix A. Equations (7)– (9) are the fundamental equations of NLC cell.

Now we discuss the solution of Eq. (7) with boundary conditions Eqs. (8) and (9). From Eq. (7) if θ ′ ≠ 0 we obtain(¹⁾If Θ ₁, Θ ₂ = 0, π /2, then both θ ≡ 0 and θ ≡ π /2 may be the solutions also. For these cases, θ ′ = 0.)

then

where C is a constant, which can be expressed as or . If θ ′ = 0 at z = d (0 < d < l) and θ _d = θ _max, then C can be expressed as . If we put

then

where “ ± ” can be determined according to the actual case. Then θ can be obtained by integrating Eq. (13).

The concrete expression of M(θ ) is determined by the expression of constant C. There are four different cases as follows:

i) θ ′ > 0 for 0 ≤ z ≤ l, the value of θ _l is largest and function θ can be expressed as

ii) θ ′ < 0 for 0 ≤ z ≤ l, the value of θ ₀ is largest and function θ can be expressed as

iii) θ ′ > 0 for 0 ≤ z < d, θ ′ < 0 for d < z ≤ l, and θ ′ = 0 at z = d, the value of θ at z = d is the maximum, denoted by θ _max. M(θ ) can be expressed as

and function θ can be expressed as

for 0 ≤ z < d, and

for d < z ≤ l;

iv) θ ′ < 0 for 0 ≤ z < d, θ ′ > 0 for d < z ≤ l, and θ ′ = 0 at z = d, the value of θ at z = d is the minimum, denoted by θ _min. The expression of M(θ ) is the same as Eq. (14) and function θ can be expressed as

for 0 ≤ z < d, and

for d < z ≤ l.

Equations (8) and (9) can be seen as the balance equations of “ forces” acting on the director at the interfaces. From the expressions of these “ forces” , one can deduce the expressions of corresponding “ potential” . In this way, we may obtain the expressions of effective anchoring energy.

Equations (8) and (9) may be re-expressed as

By comparing Eqs. (15) and (16) with the boundary conditions of the same NLC cell without these two polarizations P_s and P_f, the effective anchoring energy g_s2eff and g_s1eff satisfy the following equations respectively

Therefore, g_s2eff and g_s1eff can be obtained by means of the indefinite integral, expressed as

where r = r₁/r₂,

and

Note that D_z is a function of θ ₀ and θ _l, and its expression is complex (see Appendix B), so that the corresponding terms of expressions (19) and (20) are expressed as an indefinite integral form for the time being.

Now we discuss the effective easy direction e_1eff and e_2eff as well as the effective anchoring strength A_1eff and A_2eff. We use ω ₁ and ω ₂ to represent the tilt angles of effective easy direction e_1eff and e_2eff respectively. According to Ref. [3], they can be determined by the following equations:

According to Ref. [19], for a small deviation of the director n from easy direction e, the anchoring energy may be expressed as g_s = A sin²(θ − ω )/2, where θ and ω are the tilt angles of n and e respectively, A is the anchoring energy strength at the substrate surface. Generally speaking, the anchoring energy may be expressed as g_s = c₀ + c₁ sin²(θ − ω ) + c₂ sin⁴(θ − ω ) + … , where c₀, c₁, c₂, … are constants. If the value of θ − ω is small, the anchoring strength A is equal to 2c₁.

Consider g_s1eff first. θ and ω in the expression of g_s = A sin²(θ − ω )/2 are written as θ ₀ and ω ₁ at present. In this case, by making a variable transformation from θ ₀ to t, t = sin(θ ₀ − ω ₁), g_s1eff is a function of t. It can be expanded in Taylor’ s series at t = 0 (i.e. θ ₀ = ω ₁) as

Because

and

the effective anchoring strength of the lower substrate can be expressed as

meanwhile the second equation in Eq. (21) is equivalent to A_1eff > 0.

Similarly, for g_s2eff, ω ₂ and A_2eff may be solved from the following two equations:

and the second equation in Eq. (22) is equivalent to A_2eff > 0.

By using the method stated above, we may obtain the effective anchoring strength and easy direction. For g_s1eff, there are two equations to determine ω ₁ and A_1eff, i.e.,

Their solutions are relevant to Θ ₁. As an example, if Θ ₁ = 0, equation (27) has three solutions: ω ₁ = 0, π /2, and sin^{− 1}R₁, where

if . For each value of ω ₁, the corresponding effective anchoring strength A_1eff may be calculated from Eq. (24). The results are shown in Table 1 and the mathematical process is explained in Appendix B. For the case Θ ₁ = π /2, A_1eff and ω ₁ can be obtained also (see Table 1).

Table 1.

Table 1.

Table 1. Effective anchoring strengths and easy directions for gs1eff.

Table 1. Effective anchoring strengths and easy directions for gs1eff.

Adopting the same method, we obtain the tilt angle ω ₂ and the effective anchoring strength A_2eff for g_2eff, see Table 2, where

Table 2.

Table 2.

Table 2. Effective anchoring strengths and easy directions for gs2eff.

Table 2. Effective anchoring strengths and easy directions for gs2eff.

We have explained the mathematical method to solve Eq. (7) under the boundary conditions Eqs. (8) and (9) in Section 2. Now, we consider an original weak anchoring HAN cell with Θ ₁ = 0 and Θ ₂ = π /2. There are four cases discussed in Section 2. Case iii) and case iv) are more particular, and these two cases are similar. Thus, we consider case iii). In this case, and , from Eqs. (8) and (9) we have

If equations (31) and (32) hold for 0 ≤ z ≤ l, these two equations mean that the two effective anchoring strengths are

and the tilt angles of the easy direction e_1eff and e_2eff of the effective anchoring are ω ₁ = 0 and ω ₂ = 0, as stated in Section 3 (see Tables 1 and 2). This is to say, the original HAN cell is equivalent to a planar cell in this situation.

Now we return to discuss the conditions Eqs. (33) and (34). Several quantities , , and D_z, are represented as

Equations (33) and (34) may be rewritten as

According to Ref. [6], r₂ = − (e₁₁ + e₃₃)/6S, and r = r₁/r₂ = − 3. From Eq. (36) we see that the case [dS/dz]_{z = l} < 0 is favourable, which means that it should be for e₁₁ + e₃₃ > 0, and for e₁₁ + e₃₃ < 0. If these equations hold for a small value of U (up to U = 0), then there is a threshold voltage for this cell.

Substituting Eq. (14) into Eqs. (8) and (9), the boundary conditions can be re-expressed as

In order to calculate the threshold voltage, we take a transformation sin α = sin θ /sin θ _max. The threshold voltage U_th may be obtained from Eqs. (14), (37), and (38) by using the condition θ _max → 0. Introducing some reduced parameters: reduced voltage , reduced threshold voltage , reduced anchoring strength a₁ = A₁l/2k₁₁, a₂ = A₂l/2k₁₁, reduced flexoelectric coefficient and reduced strength of surface polarization and , the reduced threshold voltage can be expressed as

According to Eq. (39), we calculate the dependences of the reduced threshold voltage u_th on the surface polarization strength p₁ under different parameters. The results are shown in Fig. 1. The material parameters of LC are k₁₁ = 13.2 pN, k₃₃ = 18.3 pN, and Δ ε = 4.8. The thickness of the NLC cell is l = 10 μ m.

Fig. 1.

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	Fig. 1. Variations of the reduced threshold voltage uth with the reduced strength of surface polarization p1 under different parameters: (a) e11 + e33 = − 4.5 × 10− 11 C/m, , A1 = 1.0 × 10− 5 J/m2, A2 = 1.0 × 10− 6 J/m2; (b) e11 + e33 = − 4.5 × 10− 11 C/m, , A1 = 4.0 × 10− 5 J/m2, A2 = 1.0 × 10− 5 J/m2; (c) e1 + e3 = 4.5 × 10− 11 C/m, , A1 = 4.0 × 10− 6 J/m2, A2 = 1.0 × 10− 6 J/m2.

Now, we continue the discussion of the original weak anchoring HAN cell. We calculate the values of A_1eff and A_2eff as well as ω ₁ and ω ₂ with different values of U by using Eqs. (7)– (9). About the measurement of the flexoelectric coefficient, different research groups have been done using different methods.^{[20– 24]} Even for the same kind of LC materials, the value of flexoelectric coefficient is not the same. However, Castles et al.^{[25, 26]} have proved that the flexoelectric coefficient in NLC has a clear fundamental limit to the conventional order of magnitude of 10^{− 11} C/m from conservation of energy considerations. On the selection of surface polarization, we adopt the measurement value given in Ref. [27]. In order to put emphasis on the effects of two polarizations, especially the surface polarization, we adopt three sets of parameters in calculation: (i) e₁₁ + e₃₃ = − 45 pC/m, A₁ = 1.0 × 10^{− 5} J/m², A₂ = 1.0 × 10^{− 6} J/m², and , ; (ii) e₁₁ + e₃₃ = − 45 pC/m, A₁ = 4.0 × 10^{− 5} J/m², A₂ = 1.0 × 10^{− 5} J/m², and , ; (iii) e₁₁ + e₃₃ = 45 pC/m, A₁ = 4.0 × 10^{− 6} J/m², A₂ = 3.5 × 10^{− 6} J/m², and , . The results are shown in Figs. 2 and 3.

Fig. 2.

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	Fig. 2. Variations of the effective anchoring strength with applied voltage. The material parameters of LC are k11 = 13.2 pN, k33 = 18.3 pN, are Δ ε = 4.8. The thickness of NLC cell is l = 10 μ m. Panels (a), (b), and (c) correspond to the three sets of parameters respectively, as given in the text.

Fig. 3.

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	Fig. 3. Variations of tilt angles of the effective anchoring with applied voltage. The material parameters of LC are k11 = 13.2 pN, k33 = 18.3 pN, and Δ ε = 4.8. The thickness of NLC cell is l = 10 μ m. Panels (a), (b), and (c) correspond to the three sets of parameters respectively, as given in the text.

From these figures, we see that

1) For a small value of U, if the magnitude of surface polarization is suitable, the case both ω ₁ = 0, ω ₂ = 0 or π /2 may occur. Then the original HAN cell turns into a planar cell or homeotropic cell effectively, which is similar to the conclusion presented in Ref. [13]. However, if ω ₁ = 0, ω ₂ = π /2 or ω ₁ = π /2, ω ₂ = 0, the cell remains an HAN cell.

2) For a certain value of U (in a certain interval), ω ₁ (or ω ₂) equals a certain definite value (not including 0 and π /2), then the effective anchoring of this substrate has a pretilt angle and its maximal value is close to 42° in the cases concerned by these figures above.

3) For a large value of U, if e₁₁ + e₃₃ > 0, we have ω ₁ = 0 and ω ₂ = π /2, then it remains an HAN cell, in which the effective anchoring strengths A_1eff and A_2eff grow up linearly with U. However, if e₁₁ + e₃₃ < 0, we have ω ₂ = 0 and ω ₁ = π /2. Although the cell remains an HAN cell, it seems that the upper and the lower substrate are placed upside down. The effective anchoring strengths A_1eff and A_2eff also grow up linearly with U.

These results can be directly obtained from Tables 1 and 2 in Section 4. In spite of planar or homeotropic anchoring of a substrate, the effective anchoring may be either planar or homeotropic, or has a pretilt angle especially the magnitude of surface polarization and the value of U are suitable. It means that surface and flexoelectric polarization change the anchoring property of the substrate to produce the effective anchoring energy. Essentially, the influences of surface and flexoelectric polarization on the anchoring property of the substrate in a weak anchoring HAN cell are due to the director deformation of the surface of the substrate. So that, these two polarizations should be considered when investigating the anchoring property of the substrate in detail.

In this paper, we derived the concrete expressions of the effective anchoring energy. Both the effective anchoring strength and the corresponding easy direction are related to the magnitudes of two polarizations as well as the applied voltage. This result is obtained in the case of ignoring the adsorptive ions of the substrate surface. That is to say, the surface polarization may make a contribution to the effective anchoring energy, even in the absence of adsorptive ions, by reason that there is flexoelectric polarization.

In a factual problem, there exist not only surface and flexoelectric polarization but also the adsorptive ions. In this case, the energy term contributed by adsorptive ions as mentioned in Refs. [5], [6], and [8], should be added to the Gibbs free energy. Therefore, our work should be generalized.

Substituting Eq. (6) into Eq. (5), one can obtain

Some terms in Eq. (A1) can be denoted as the following form

Substituting Eqs. (A2)– (A7) into Eq. (A1), equation (7) and boundary conditions (8) and (9) may be obtained. Here, some terms proportional to δ ₁ or δ ₂ are ignored because they are very small compared with the other term. The reasons are as follows.

This term containing δ ₁ can be written as

Comparing it with the term (e₁₁ + e₃₃)/2· (P_s1z sin 2θ ₀)/g(θ ₀), we can obtain

in which the term may be neglected with respect to the term [(e₁₁ + e₃₃)/2] · [(P_s1z sin 2θ ₀)/g(θ ₀)], and the term may also be neglected compared with the term [(e₁₁ + e₃₃)/2] · [(P_s2z sin 2θ _l)/g(θ _l)].

It is very notable that the rigorous expression of U_eff is given by

However, the latter two terms in Eq. (A10) are neglected when deducing the fundamental equation. If these two terms are considered, the boundary conditions should increase two terms D_z (∂ /∂ θ )[P_s1z/g (θ )]_{θ = θ 0}δ ₁ and D_z(∂ /∂ θ )[P_s2z/g(θ )]_{θ = θ l}δ ₂ additionally. Taking the term D_z (∂ /∂ θ )[P_s1z/g(θ )]_{θ = θ 0}δ ₁ for example, comparing it with the term [(e₁₁ + e₃₃)/2] · D_z · [sin 2θ ₀/g(θ ₀)], we can obtain

So the term D_z(∂ /∂ θ )[P_s1z/g(θ )]_{θ = θ 0}δ ₁ may be neglected. Similarly, the term D_z(∂ /∂ θ )[P_s2z/g(θ )]_{θ = θ l}δ ₂ may also be neglected.