Energy minimization method, packing

In the case of crystals, both intramolecular (conformational) and packing energies should be taken into account simultaneously. Such a total energy minimization method, with suitable crystallographic constraints, has been applied in different steps of the analysis of crystalline structures of three different synthetic polymers. Structures of these molecules, namely, isotactic trans-1,4-poly-penta-1,3-diene (ITPP), poly-pivalolactone (PPVL), and isotactic cis-1,4-poly(2-methyl-penta-1,3-diene)(PMPD), do not have troublesome features such as charged groups, counterions, and solvent molecules. 

Figure 6. Application of packing energy minimization method to poly(ethylene oxide) (16). (a) Starting model of uniform helix (b) stable crystal structure model obtained by energy minimization calculations and (c) the structure determined by x-ray analysis.

Experimental tilt angles have usually an accuracy of at best 3°, leading to an error of about 0.1 A in cell axes. The calculated third cell axis will show a higher deviation. If possible an internal standard should be used for calibration purposes but a higher accuracy will be obtained with a Pawley fit (e.g. fit forP CuPc in Fig. 6) from x-ray powder diffraction data [11]. Especially for packing energy minimization used for simulation methods it is essential to determine the cell parameters as precise as possible. In the case of polymorphism, it is essential to use x-ray powder diffraction to ensure that bulk and investigated nano crystals represent the same modifications. 

Figure 2. Crystal packing in polyethylene (a, h projection), one coordination shell about a central chain is shown. Unit cell boundaries also are shown. In defect calculations, center chain (1) is replaced by one containing a conformational defect (see Figure 1). In the hybrid Newton-Raphson relaxation method, one chain at a time has its energy minimized with the other chains fixed. In the free chain, all internal degrees of freedom participate (torsional angles, etc.) as well as the position and orientation of the chain. After a cycle through all of the chains, the total energy of the assembly is computed. The cycles are repeated until the energy is stable.

McDonald and Floudas (1994) and Michelsen (1994) also present methods to find the phase conditions for a given mixture. Both methods discover the number of phases and the compositions of these phases. McDonald and Floudas globally minimize the Gibbs free energy of the mixture using a computer pack-... 

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