8 Fréedericksz-Like Positional Transition Triggered by An External Electric Field
349
Fig. 8.18 Equilibrium position x 0 for different a radii (2.2 µm, 2.35 µm, 2.5 µm and 3.0 µm); b
densities (0.99, 1.0 and 1.02 g · cm −3 ) of microparticle with K = 8 pN and L = 10 µm, showing
the same critical value E c of electric field triggering positional transition. The dependence of E c
and
√
K /L for different c radii (2.2 µm, 2.35 µm, 2.5 µm and 3.0 µm); d densities (0.99, 1.0
and 1.02 g · cm −3 ) of microparticle, obeying strictly a master curve given by theoretical prediction
Eq. (8.37)
value is correlated with the size and density of the microparticle. The dependence
of the equilibrium position on the applied electric field for different microparticle
sizes and densities is shown in Fig. 8.18a and b, where the strict overlapping of
equilibrium position in the figures implies that the critical electric value is, as shown
in the previous section, independent of microparticle size and density. For a better
understanding of the dynamic behaviors of the microparticle, we further plot the
threshold value against
√
K /L in Fig. 8.18c and d, with a Fréedericksz transition
curve (black) shown as well. It is found that the critical electric field triggering a positional transition for a microparticle suspended in a NLC cell follows a Fréedericksz
master curve irrelevant to microparticle size and density.
More precisely, by comparing the numerical calculation results with the Fréedericksz effect curve (π
√
4π/|
√
K /L) in Fig. 8.18c and d, it is found that the slope
difference between them is by a prefactor of ∼ 3/π , leading to a proposed theoretical
prediction for the critical electric field. Such a prediction, as shown by straight line
(red) in Fig. 11c and d, agrees very well for different radii (2.2, 2.35, 2.5, and 3.0
µm) and densities (0.99, 1.0 and 1.02 g · cm
−3 ) of microparticle.
349
Fig. 8.18 Equilibrium position x 0 for different a radii (2.2 µm, 2.35 µm, 2.5 µm and 3.0 µm); b
densities (0.99, 1.0 and 1.02 g · cm −3 ) of microparticle with K = 8 pN and L = 10 µm, showing
the same critical value E c of electric field triggering positional transition. The dependence of E c
and
√
K /L for different c radii (2.2 µm, 2.35 µm, 2.5 µm and 3.0 µm); d densities (0.99, 1.0
and 1.02 g · cm −3 ) of microparticle, obeying strictly a master curve given by theoretical prediction
Eq. (8.37)
value is correlated with the size and density of the microparticle. The dependence
of the equilibrium position on the applied electric field for different microparticle
sizes and densities is shown in Fig. 8.18a and b, where the strict overlapping of
equilibrium position in the figures implies that the critical electric value is, as shown
in the previous section, independent of microparticle size and density. For a better
understanding of the dynamic behaviors of the microparticle, we further plot the
threshold value against
√
K /L in Fig. 8.18c and d, with a Fréedericksz transition
curve (black) shown as well. It is found that the critical electric field triggering a positional transition for a microparticle suspended in a NLC cell follows a Fréedericksz
master curve irrelevant to microparticle size and density.
More precisely, by comparing the numerical calculation results with the Fréedericksz effect curve (π
√
4π/|
√
K /L) in Fig. 8.18c and d, it is found that the slope
difference between them is by a prefactor of ∼ 3/π , leading to a proposed theoretical
prediction for the critical electric field. Such a prediction, as shown by straight line
(red) in Fig. 11c and d, agrees very well for different radii (2.2, 2.35, 2.5, and 3.0
µm) and densities (0.99, 1.0 and 1.02 g · cm
−3 ) of microparticle.