Let us now examine the stability of the antiferroelectric state when subject to an applied electric field. That is, we will calculate the conditions under which an applied electric field can cause a homogeneously nucleated transition of the system from the antiferroelectric alignment shown in Fig. 2a to the ferroelectric alignment shown in Fig. 2b. For this analysis, which expands on the brief treatment already presented in Ref. , we take to be a function of z alone in the even-numbered smectic layers, and write . For the odd layers, which contain molecules with dipole moments initially aligned along the applied electric field, we put . These assumptions are illustrated in Fig. 2c.
When an electric field is applied of sufficient strength to make the
state unstable, will begin to deviate from zero. We accordingly
energy of the system when . To second order in
, the energy per unit length of the smectic layer can be
obtained as an expansion of Eq. (1):
In order to find the requirements for the Fréedericksz transition to
must find the conditions under which there exists a nonzero
that minimizes Eq. (2) in the bulk of the sample
and also makes the total energy negative.
Minimization of Eq. (2) in the interior of the sample, where W(z)
vanishes, gives a differential Euler-Lagrange equation for , whose
solution is with
A nonzero will occur only when the total energy is negative. Substituting into Eq. (2) and applying the condition , we obtain
This condition enables us to extract the electric field, , at which the antiferroelectric configuration becomes unstable. This critical field is always larger than the field at which the ferroelectrically aligned state becomes energetically preferred, and may even become much larger than if the thickness of the AFLC cell becomes small. The phase diagram of the AFLC in this model is shown in Fig. 3 as the line labeled ``uncoupled case".
Equation (4) bears some similarity to a closely related result of Handschy and Clark  (their Eq. (13)), who studied the threshold fields in ferroelectric liquid crystals. The difference between the two cases lies in the fact that our theory refers to discrete layers rather than a continuum and includes the effects of dielectric anisotropy, but does not include the sensitivity of the anchoring potential to the direction of the electric polarization.
In the limit of strong anchoring () Eq. (4)
reduces to the condition
For the case of positive dielectric anisotropy, , the condition (5) can always be achieved at sufficiently strong electric fields. The critical field for infinitely strong anchoring is
If, on the other hand, the dielectric anisotropy, , is
there is a maximum value that q can attain. As a result, there exists a
critical cell thickness,
such that for thicknesses , there is no possible electric field that can destabilize the antiferroelectric configuration. For the cell with , the critical field for the Fréedericksz transition at infinitely strong anchoring is: