Up to this point we have considered an idealized special case, in
which a transition is initiated by a rotation of each layer in the sense shown
in Fig. 2c. All odd layers remain unchanged while the orientation
of the even layers alternates in direction. This arrangement was chosen
as a reasonable approximation for a thin cell, in which anchoring forces
dominate, and
because of its mathematical convenience. In a thick cell, however, the
scenario of the
Fréedericksz transition may not be that simple, so we now consider a more
probable
path for this transition. Such a path is shown in Fig. 2d. Now the
odd layers
are allowed to rotate as well as the even ones. The path of lowest energy
is then for
the even layers to all rotate in the same direction, while the odd layers also
undergo a small rotation in this direction. This results in an effective
softening of the system, which permits it to have a lower energy than in the
previous, uncoupled, arrangement.
We now calculate the critical electric field 
at
which the system leaves the ideally aligned state shown in Fig. 2a to
begin the transformation into the state shown in Fig. 2d. We
will show
that taking into account the cooperative motion of even and odd smectic
layers
may result in a significant reduction of the critical field when the cell
thickness is not very small. Furthermore, this field does not necessarily have
to be larger than 
. As we will show, for an infinitely thick cell and
in the
absence of any tendency to form a helical structure, the AFLC leaves the
ideally
aligned antiferroelectric configuration even when a very weak electric field is
applied.
For our analysis we take 
for the even layers, and
for the odd layers, and assume both 
and 
to be much smaller than unity. Expansion of Eq. (1) to
second order in and 
gives, with omission of all
angle-independent terms:
Minimization of this free energy in the bulk of the
layer now gives us a pair of coupled Euler-Lagrange equations, whose
solutions yield:
Here self-consistency dictates that
and
As in the uncoupled case, nonzero and occur only when the
total energy, Eq. (9), is negative. After substitution of
Eqs. (10)-(11) into Eq. (9) we obtain the
condition
for this to be true:
In the limit of strong anchoring (
) Eq. (13)
reduces to the condition
In the limit of small cell thickness, and hence large electric fields,
for which 
, both Eq. (13) and Eq. (14)
reduce to the corresponding equations describing the uncoupled case,
Eqs. (4) and (5). For moderate
cell thicknesses, however, Eq. (13) predicts a Fréedericksz
transition to occur at much smaller electric fields than before. For such
fields we can neglect the dielectric anisotropy, and find the following
expression for the critical field in the case of infinitely strong anchoring:
We note that the critical field , at which the system leaves an ideally
aligned antiferroelectric state, may be much smaller than the field
, at
which the purely ferroelectric state shown in Fig. 2b becomes
energetically preferred to the purely antiferroelectric one shown in
Fig. 2a. This is shown schematically
in Fig. 3, which indicates the critical field for homogeneous
nucleation in each model as a function of cell thickness. The thickness
at
which a field of strength sufficient to cause ferroelectric
alignment can
also cause homogeneous nucleation is given by 
.
To place these results in context, we can substitute typical values for the
parameters involved[7]. With 
N, 
C m
,
J m
, 
, and
infinitely strong anchoring we find the critical cell thickness 
to be
as small as
25nm. At the more typical experimental cell thickness of 1
m the electric
field at which homogeneous nucleation is predicted to occur is 
Vm
in the coupled model, and more than an order of magnitude
higher, 
Vm in the uncoupled model. We should note, however, that the
magnitude of
the interlayer interaction U is known only from indirect estimates, and
so these
figures may be subject to considerable revision.