(4)

Here the average correlation factor is calculated for all molecules together.

Thus is 0 is the isotropic liquid phase and has a value >0 in the R_{I} and R_{II} phases.

3. Free Energy

Thus we take and as two order parameters involved in the R_{I}-R_{II}, IL-R_{II} and IL-R_{I} phase transitions. For simplicity we neglect the weak interlayer interaction between the stacking layers in different rotator phases so that the problem becomes two dimensional. The distortion is a two component order parameter; its components are expressed the distortion amplitude and the azimuthal angle. The multiplier 2 comes from the fact that the distortion is a symmetric traceless tensor. Since the the free energy is a scalar quantity, negative and positive and result in inequivalent structures, causing the Landau free energy expansion contain and terms, thus resulting in a first order transition. Expanding the total free energy in terms of the above mentioned order parameters yields

(5)

where is free energy of the isotropic liquid phase. The coefficients a and are assumed to vary strongly with an external parameter. For, free energy (5) describe a first order R_{I}-R_{II} transition for b > 0 and c > 0. In this case the minimum free energy occurs at, for b > 0 and at for b < 0. According to the experimental observations, in the R_{I} phase,. H is the coupling constant., , b, c and H are chosen positive.

The material parameters a and can be assumed as and. and are virtual transition temperatures. and are constants.

From the experimental phase diagrams [3] one observes, and can be portrayed as

andwhere is some specific chain length of the molecules. u and v are positive constants.

Minimization of Equation (5) with respect to and yields the following phases:

1) Isotropic liquid (IL) phase:,.

This phase exists for and a > 0.

2) R_{II} phase:,.

The R_{II} phase exists when and a > 0.

3) R_{I} phase:,.

The R_{I} phase exists for and a < 0.

Thus it is clear from the solutions that three types of transition are possible: 1) IL-R_{II}; 2) IL-R_{I}; 3) R_{II}-R_{I}.

The sufficient condition for the R_{I} phase to be stable are

(6)

(7)

(8)

These three conditions determine the stability of the R_{I} phase explicitly. The sufficient condition for the stability of the R_{II} phase reads

(9)

For the IL phase the stability conditions are and a > 0.

By lowering the temperature from the isotropic liquid phase, the R_{II} and R_{I} phases can appear sequentially or in partial sequence. The R_{II} and R_{I} phases can arise either directly from the IL phase along the curves IL-R_{II} and IL-R_{I} or along the curve R_{II}-R_{I}. In the spirit of Landau theory all the phase transitions IL-R_{II}, IL-R_{I} and R_{II}-R_{I} are first order because of the cubic invariant in the free energy expansion. If all the phase transitions involved are first order ones, then one can observe the IL-R_{II}-R_{I} triple point as observed in experiment [3].

The conditions for the first order R_{II}-R_{I} transition can be obtained as

(10)

The conditions for the first order IL-R_{I} transition are given by

(11)

The conditions for the first order IL-R_{II} transition read

(12)

Solving (10)-(12) simultaneously will determine the various phase transition lines. Figure 1 shows a typical phase diagram for the IL-R_{I}, IL-R_{II} and R_{II}-R_{I} phase transitions. As can be seen from the Figure 1, the R_{II} and R_{I} phases arise from the isotropic phase along the curves IL-R_{II} and IL-R_{I} or along the curve R_{II}-R_{I} respectively. The IL-R_{II} and IL-R_{I} transitions are first order because of the cubic invariant in the free energy expansion. The line of the R_{II}-R_{I} transition starts at the IL-R_{II}-R_{I} triple point as shown in Figure 1. When the temperature of the IL-R_{II} and of R_{II}-R_{I} transitions coincide, a triple point appears.

Figure 1. Possible chain length (n)-temperature (T) phase diagram in the vicinity of the Liquid-R_{II}-R_{I} triple point..

The region of the R_{I} phase shrinks and finally disappears when the IL-R_{II} transition takes place. In experimental studies [3], alkanes C20-C27 show such a IL-R_{II} transition via triple point. Thus the above analysis of the IL-R_{II}-R_{I} triple point agrees well with the experimental observations [3]. Thus there is always a direct IL-R_{II} transition is possible within the framework of our model free energy (6) satisfying the above stability conditions. Of course the IL-R_{I} transition could proceed before the IL-R_{II} the transition temperature is reached. To prevent this, has to be larger than the IL-R_{I} transition temperature Thus we have always . The cubic coefficient in the free energy (5) yields a first order IL-R_{II} transition at

with an order parameter jump and a latent heat of. The so-called temperature hysteresis is related to the existence of metastable states within a certain temperature range. The above analysis qualitatively agrees with experimental observations[3,4,7,8].

4. Conclusion

A simple model free energy has been constructed to describe the IL-R_{II}-R_{I} phase sequence and transitions between them. The order parameters are identified for different phase transitions. The model predicts the first order character of the IL-R_{II}, IL-R_{I} and R_{II}-R_{I} transitions and IL-R_{II}-R_{I} triple point in the phase diagram. The proposed interpretation of the IL-R_{II} transition allow us to explain the various types of phase behavior observed experimentally. These results are in qualitative agreement with all experiments reported so far.

5. Acknowledgements

PKM thanks the Alexander von Humboldt Foundation for equipment and book grant.

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NOTES

^{*}Corresponding author.