The Cholesteric Energy

It is known from the previous Section that k = ku = 0 for both nematics and cholesterics, and so, by (2.37) i, Sq = 0 also. The invariance condition (2.6) must hold for cholesterics also, showing that the constants ki must again be of the form given in equation (2.22) but these values for ki must necessarily be inequivalent to those given in (2.47) because of the presence of enantiomorphy in cholesterics. Noting that kz must be zero, we are therefore forced to conclude that k2 0 for cholesterics. Hence, by (2.37)2, to 0 and, employing the notation of the previous Section, the energy for cholesterics based upon (2.44) and (2.48) can now be written as 

It is instructive at this point to insert the natural cholesteric state ric from equation (2.69) into Wchoi to find that 

Clearly, Wchoii c) is minimised, with Wchoii c) = 0 when 

The quantities derived above in equations (2.51) to (2.54) can also be used again to gain information about the elastic constants for cholesterics. Inserting these quantities into equation (2.70) gives 

Each of the three lines on the right-hand side of (2.73) is a quadratic in the variables ai and so applying the results from (2.56) and (2.57) to the first and third lines shows that if each line is to represent a non-negative quantity then, as in the nematic 

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