328
K. Xiao and C.-X. Wu
where 10
−3
− 10
−4 mJ/m
2 is considered as weak anchoring and 1 − 10
−1 mJ/m
2 is
regard as strong anchoring. In most cases the inclusion of particles into an NLC cell
tends to create LC alignment singularities around the suspended substances, which
in general are determined by surface anchoring conditions, particle size, boundary
conditions, and external fields etc. [17, 49–52]. It has been widely accepted and
confirmed that when a spherical particle is immersed in NLC, there are three possible types of defect configurations [53–55]. Dipole and quadrupolar configurations
are usually seen around a spherical particle with strong vertical surface anchoring,
whereas boojum defect is formed by a micro-sphere with tangential surface anchoring. In addition, recently B. Senyuk et al. assumed that conically degenerate boundary condition gives rise to the so-called elastic hexadecapole [56], and then Y. Zhou
reported that the dipole-hexadecapole transformation can be achieved via tuning the
preferred tilt angle of LC molecules anchoring on colloidal particle surface [57].
Through experimental observations it has been found that, when an external field is
applied, there exists a transition between elastic dipole and quadrupolar configuration, which depends on particle size and surface anchoring strength [58–60].
As an application of the above theory, let us first of all consider a system that
a particle is embedded in a uniform NLC without confinement. There are now two
contributions to the free energy. The first contribution is the elastic deformation of
the LC and can be accounted by the well known Frank-Oseen free energy. With one
constant approximation K 11 = K 22 = K 33 = K , the bulk deformation energy can be
written as
F b =
K
2
dV [(∇ · n)
2
+ (∇ × n)
2
].
(8.13)
The second contribution is the surface anchoring free energy which is in the RapiniPopula form Eq. (8.12), and the integration is over the particle surface. To determine
the distribution of the director field n(r) for a particle embedded in a NLC, the goal is
solve the Euler-Lagrange equations arising from the variation of the total free energy
F = F b + F anchoring . Unfortunately, the Euler-Lagrange equations with subjected
boundary conditions at the surface of the particle and parallel boundary conditions
at infinity are highly nonlinear, and analytical solutions are quite difficult to found.
However, utilising the multipole expansion method similar to electrostatic [61], we
can obtain analytic solutions for the director field far from the particle. We assume
that the director field at infinite approach the undeformed director field n 0 = (0,0,1)
when there is no other confinement. The deviation of n(r) from n 0 induced by
the embedded particle is small at the large distance but not infinite, and n(r) ≈
(n x , n y , 1). Therefore, at large r , the nonlinear bulk free energy of deformation can
be replaced by the harmonic free energy [53, 55]
F har =
K
2
dV (∇n μ )
2
(8.14)
K. Xiao and C.-X. Wu
where 10
−3
− 10
−4 mJ/m
2 is considered as weak anchoring and 1 − 10
−1 mJ/m
2 is
regard as strong anchoring. In most cases the inclusion of particles into an NLC cell
tends to create LC alignment singularities around the suspended substances, which
in general are determined by surface anchoring conditions, particle size, boundary
conditions, and external fields etc. [17, 49–52]. It has been widely accepted and
confirmed that when a spherical particle is immersed in NLC, there are three possible types of defect configurations [53–55]. Dipole and quadrupolar configurations
are usually seen around a spherical particle with strong vertical surface anchoring,
whereas boojum defect is formed by a micro-sphere with tangential surface anchoring. In addition, recently B. Senyuk et al. assumed that conically degenerate boundary condition gives rise to the so-called elastic hexadecapole [56], and then Y. Zhou
reported that the dipole-hexadecapole transformation can be achieved via tuning the
preferred tilt angle of LC molecules anchoring on colloidal particle surface [57].
Through experimental observations it has been found that, when an external field is
applied, there exists a transition between elastic dipole and quadrupolar configuration, which depends on particle size and surface anchoring strength [58–60].
As an application of the above theory, let us first of all consider a system that
a particle is embedded in a uniform NLC without confinement. There are now two
contributions to the free energy. The first contribution is the elastic deformation of
the LC and can be accounted by the well known Frank-Oseen free energy. With one
constant approximation K 11 = K 22 = K 33 = K , the bulk deformation energy can be
written as
F b =
K
2
dV [(∇ · n)
2
+ (∇ × n)
2
].
(8.13)
The second contribution is the surface anchoring free energy which is in the RapiniPopula form Eq. (8.12), and the integration is over the particle surface. To determine
the distribution of the director field n(r) for a particle embedded in a NLC, the goal is
solve the Euler-Lagrange equations arising from the variation of the total free energy
F = F b + F anchoring . Unfortunately, the Euler-Lagrange equations with subjected
boundary conditions at the surface of the particle and parallel boundary conditions
at infinity are highly nonlinear, and analytical solutions are quite difficult to found.
However, utilising the multipole expansion method similar to electrostatic [61], we
can obtain analytic solutions for the director field far from the particle. We assume
that the director field at infinite approach the undeformed director field n 0 = (0,0,1)
when there is no other confinement. The deviation of n(r) from n 0 induced by
the embedded particle is small at the large distance but not infinite, and n(r) ≈
(n x , n y , 1). Therefore, at large r , the nonlinear bulk free energy of deformation can
be replaced by the harmonic free energy [53, 55]
F har =
K
2
dV (∇n μ )
2
(8.14)