Free energy bistable

Figure 9 (Top) schematic of bistability in 1,3,2-dithiazolyl radicals arising from a solid-solid phase transition between regular and Peierls distorted n-stacks (bottom) free energy diagram of the two structural phases present...

Fig. 1.1.6. Dotplots for two bistable RNA sequences generated by the RNAfold program (Vienna RNA package) [29]. The lower left triangle shows the base pairs of the minimum free energy structure. The upper right triangle represents the frequencies of each base pair...

We summarize as follows [54]. The free energy i)mV(M), Eq. (81), has a bistable structure with minima at ni and nj separated by a potential barrier containing a saddle point [54] at Dq. If (aj denote the direction... 

Figure 1.16. Two illustrative two dimensional free energies v> q, q ) depending on two variables q and q, and their corresponding reduced free energy functions w q) = —k Thi dq exp[ —j3w( , )] In both cases, w(q) has the same bistable form, but in (a) the coordinate q is a reasonable reaction coordinate, as the transition state surface coincides with q = q. In (6), on the other hand, q is not at a reasonable reaction coordinate. The orthogonal variable q" is crucial for the mechanism of T -> S transitions, and the maximum in w(q) at q = q does not coincide with the transition state surface. The trajectories initiated at configurations with q(r) = q and all ending in B illustrate this.

Fig. 12.16 Bistable twist cell. Right- and left-handed twist-structures of a nematic liquid crystal with the same elastic energy (a) and the angular dependence of total free energy (b)...

The smectic A is an untilted phase in which the mass density wave is parallel to the director. The cost in free energy of buckling the layers into saddle-shaped deformations is low, with the result that it is relatively easy to construct devices that show bistability between a scattering focal conic director configuration in which the layers are buckled and a clear homeotropic configuration in which the director is perpendicular to the cell walls and the layers parallel to the walls. Transitions between these two textures have been exploited in laser-written projection displays and in both thermo-optic and electrooptic matrix displays. The various mechanisms employed are summarized in Fig. 12. 

The parameters controlling the bistability are the angle (j) between the aUgning directions of the alignment layers and intrinsic pitch P of the liquid crystal. So far bistability has been observed for -nil liquid crystal should be chosen heuris-tically in such a way that the (j) + n) twist state has the minimum free energy, that is ( + ji) is the intrinsic twist. Hence InQilP) = + n[16. ... 

FIGURE 6.16. Bistable switching in long-pitch cholesterics with a tilt of the director o- (a) Tilted states with n/2 turns in zero field, = 55 . (b) Free energies g as functions of thickness to pitch ratio d/Po at zero field, dlPoY = 0.89 is the operating point, (c) g(d/PoY in an electric field versus reduced volteige U/Ufi Uf is the Frederiks transition threshold, Ae > 0. 

Consider a system with a bistable free energy F q) as shown in Fig. 3. Such a free energy might be characteristic for a molecule in solntion undergoing isomerizations between two stable conformations A and B, i.e., for a unimolecular chemical reaction... 

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