▁the ▁variation ▁in ▁free ▁energy ▁is ▁zero ▁at ▁equilibrium : ▁▁▁▁ ▁The ▁variation ▁in ▁the ▁pressure ▁at ▁the ▁free ▁liquid - v apor ▁boundary ▁is ▁due ▁to ▁Lap lace ▁pressure , ▁which ▁is ▁proportional ▁to ▁the ▁mean ▁Gaussian ▁curv ature . ▁Sol ving ▁the ▁above ▁equation ▁for ▁both ▁convex ▁and ▁con ca ve ▁surfaces ▁yields : ▁▁▁▁ ▁where ▁, ▁ ▁and ▁. ▁ ▁This ▁equation ▁rel ates ▁the ▁contact ▁angle , ▁a ▁geometric ▁property ▁of ▁a ▁sess ile ▁dro ple t ▁to ▁the ▁bulk ▁therm od ynam ics , ▁the ▁energy ▁at ▁the ▁three ▁phase ▁contact ▁boundary , ▁and ▁the ▁mean ▁curv ature ▁of ▁the ▁dro ple t . ▁ ▁For ▁the ▁special ▁case ▁of ▁a ▁sess ile ▁dro ple t ▁on ▁a ▁flat ▁surface ▁: ▁▁▁▁ ▁In ▁the ▁above ▁equation , ▁the ▁first ▁two ▁terms ▁are ▁the ▁modified ▁Young ’ s ▁equation , ▁while ▁the ▁third ▁term ▁is ▁due ▁to ▁the ▁Lap lace ▁pressure . ▁This ▁non linear ▁equation ▁correctly ▁predict s ▁the ▁sign ▁and ▁magnitude ▁of ▁ κ , ▁the ▁fl atten ing ▁of ▁the ▁contact ▁angle ▁at ▁very ▁small ▁scales , ▁and ▁contact ▁angle ▁h yst er esis . ▁ ▁Contact ▁angle ▁h yst er esis ▁A ▁given ▁substr ate - liqu id - v apor ▁combination ▁yields ▁a ▁continuous ▁range ▁of ▁contact ▁angle ▁values ▁in ▁practice . ▁The ▁maximum ▁contact ▁angle ▁is ▁referred ▁to ▁as ▁the ▁adv ancing ▁contact ▁angle ▁and ▁the ▁minimum ▁contact ▁angle ▁is ▁referred ▁to ▁as ▁the ▁re ced ing ▁contact ▁angle . ▁The ▁adv ancing