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
Minimization of this free energy in the bulk of the
layer now gives us a pair of coupled Euler-Lagrange equations, whose
Here self-consistency dictates that
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
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. 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 1m 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.