Free energy difference of pitch variation and calculation of the order-disorder transition in block copolymer systems using thermodynamic integration

Thermodynamic integration and the calculations involved in using thermodynamic integration in a coarse-grained molecular dynamics model are explained and are used to measure the energy penalty of altering the pitch of polymers at various xN values. The minimum free energy occurred at a pitch that corresponded to the calculated natural pitch from scattering calculations. The penalty for pitches lower than the natural pitch was found to scale with χ and the proportional decrease in pitch, indicating a purely enthalpic driving force and resulting in a universal energy penalty curve. The penalty for pitches larger than the natural pitch is likely entropically driven. The order-disorder transition (ODT) was also measured in a manner similar to work done previously by calculating the free energy in the mixed state and of the demixed state. The intersection of these curves is the ODT at the pitch used. The measured ODT depends on the pitch used in the demixed calculation. If the pitch is too large, then the χN at the ODT will be overestimated. The ODT measurement was taken as a function of pitch and found to approach the expected ODT of 10.5 as the correct pitch was approached, thus improving the work done previously.