4.3 Weakly anchored in-plane switching mode
The one-dimensional approximation of the in-plane switching mode, as described in section 2.5, is limited to strong anchoring. In this section, an extension to weak anchoring is made.
In stead of a rigid surface director, the planar anchoring is now modeled by the surface anchoring energy (4.9). This yields a free en- ergy
F= 1 2
Z d 0
k22
dφ dz
!2
−ε0∆εE2 cos2φ
dz +Wa
2
hsin2
φ(0)−φ0
+sin2
φ(d)−φ0
i. (4.13) The integral consisting of the distortion and electric energy in the bulk is identical to that for strong anchoring. The two additional terms spec- ify the azimuthal anchoring at the top and bottom surfaces. Since the tilt angle of the director is not changed by the applied field along the x-axis,θremains equal to zero and the polar anchoring does not appear in the expression.
An analogous normalization is performed for the electric fieldEand the height z as was done for strong anchoring, using equations (2.7) and (2.8). After division by the common factork22/2d, the normalized expression of the free energy is
Fnorm= Z 1
0
dφ dζ
!2
−π2h2 cos2φ
dζ +π
ρ hsin2
φ(0)−φ0
+sin2
φ(1)−φ0
i, (4.14) with
ρ = πk22
Wad (4.15)
a dimensionless reduced surface-coupling parameter which is a mea- sure for the azimuthal anchoring strength at the surface.handζare the normalized electric field and height.
The principle of Calculus of Variations [40] yields that the minimum energy solution for the twistφ(ζ) in equation (4.14) must fulfill the dif- ferential equation
d2φ
dζ2 −π2h2sinφcosφ=0, (4.16)
with the boundary conditions dφ
dζ ζ=0
+ π 2ρsinh
2
φ(0)−φ0
i = 0 (4.17a)
dφ dζ
ζ=1 − π 2ρsinh
2
φ(1)−φ0
i = 0. (4.17b)
The differential equation (4.16) is identical to (2.12) that was obtained for strong anchoring, but the boundary conditions have changed.
The differential system is again solved numerically, using the Matlab solver. Also for weak anchoring, the number of independent vari- ables is strongly reduced. Besides the normalized electric field h and the azimuthal alignment direction φ0, the reduced surface-coupling parameterρmust be specified additionally.
As a first example, the one-dimensional simulation of the in-plane switching mode of liquid crystals in Figure 2.12 is repeated with weak anchoring instead of strong anchoring. The twist angle φis given in Figure 4.1 as a function of the height ζ for φ0 = 85◦, ρ = 0.25 and different equally spaced values of the applied fieldh.
Figure 4.1:The twist angleφin the weakly anchored one-dimensional approximation of the in-plane switching mode as a function of the rel- ative height ζ, for different equally spaced values of the electric field h =0,0.2, . . . ,10,ρ=0.25 andφ0 =85◦. Forh=0, the twistφis con- stant along the alignmentφ0 = 85◦and for increasinghthe midplane twist decreases toward 0. The arrow indicates the direction in which the field increases.
4.3 Weakly anchored in-plane switching mode 57
The plots of the twist as a function of the heightζ show a similar behavior as in Figure 2.12 for ρ = 0. Forh = 0 the twistφ(ζ) is a con- stant, equal to the alignment φ0 of 85◦. For increasing values of the normalized fieldh, the midplane director rotates toward the field along x-axis and the twist angleφdecreases. Because of the weak surface an- choring, not only the bulk director is reoriented, but also the surface director has rotated toward the electric field.
The influence of the anchoring strength ρ becomes clearer in Fig- ure 4.2. The midplane and surface twist are plotted as a function of the applied fieldhforφ0=85◦and different values of the reduced surface- coupling parameterρ. The largest influence of the anchoring strengthρ
Figure 4.2: The midplane twistφ(1/2) and the surface twistφ(0) for weak and strong anchoring as a function of the applied fieldhforρ = 0, 0.125, . . . , 5 andφ0 =85◦.
is visible on the surface director. Above the threshold, the surface twist varies almost linearly toward zero. The midplane twist shows an anal- ogous variation as in the strongly anchored case (ρ = 0), with a slight reduction of the threshold field.
The reduced surface-coupling parameterρis related to the extrap- olation length [11, 94, 104]
ξa = k22
Wa
= ρd
π , (4.18)
which is the ratio of surface rotation of the director to the director gra- dient at the interface for small changes of the surface twist [94, 105].
The extrapolation lengthξa is an equivalent way, often used in lit- erature, to specify the anchoring strength. It is proportional to the di-
mensionless surface-coupling parameterρand inversely proportional to the weight parameter Wa. If both polar and azimuthal anchoring are important, separate extrapolation lengthsξa andξpmust be speci- fied [11, 92].
Figure 4.3: Twistφas a function of the heightzwith indication of the extrapolation length ξa, forh = 1.0 andh = 1.2 in caseφ0 = 85◦ and ρ=0.25.
Physically,ξacorresponds to an increment of the cell thicknessd, as illustrated in Figure 4.3, in such a way that there is no change of the twist at the effective boundaries.
The effective cell thickness d +2ξa must be used for the calcula- tion of the threshold voltage in the weakly anchored in-plane switch- ing mode [3, 104]. Therefore, the threshold voltage Vth for alignment perpendicular to the electric fieldEis reduced to
Vth = πg d+2ξa
r k22
|∆ε|ε0. (4.19) The electro-optic characteristic of the weakly anchored in-plane switching mode is given in Figure 4.4 for different values of the surface- coupling parameterρ as a function of the applied field andφ0 = 85◦. The calculations are carried out with the Jones Matrix Method using a layer thicknessdof 2.1àm, a wavelengthλof 600 nm and the parame- ters of the liquid crystal E7.
The influence of the anchoring strength is clearly visible. The threshold field is reduced and the rotation of the surface director in- creases the optical response at weaker fields compared to the strongly anchored case plotted in Figure 3.5. The steeper increase of the electro-