Phase diagrams of grafted molecules under external fields: A mean field model

A mean field theory is extended to investigate the phase behavior of grafted molecules under external fields. In the model, each grafted molecule contains two states, similar to the Ising model, and the applied field interacts with only one of the two states. The free energy of this model is formulated as a function of composition, field strength, and a parameter to account for the intermolecular interactions of neighboring molecules. Our calculations show that uniform fields exhibit no significant effect on phase diagrams. In contrast, linear gradient fields affect critical temperature but show no direct effect on critical composition. Under external fields with quadratic spatial gradient, both critical constants become sensitive to field direction and strength. Moreover, an analysis of field dependent critical constants suggests that quadratic fields affect miscibility and critical temperature. A brief comparison of the current model and the Ising model is also given.