▁angle . ▁Gib bs ▁post ulated ▁the ▁existence ▁of ▁a ▁line ▁t ension , ▁which ▁acts ▁at ▁the ▁three - phase ▁boundary ▁and ▁accounts ▁for ▁the ▁excess ▁energy ▁at ▁the ▁confl u ence ▁of ▁the ▁solid - liqu id - gas ▁phase ▁interface , ▁and ▁is ▁given ▁as : ▁▁▁▁ ▁where ▁ κ [ N ] ▁is ▁the ▁line ▁t ension ▁and ▁a [ m ] ▁is ▁the ▁dro ple t ▁radius . ▁ ▁Although ▁experimental ▁data ▁valid ates ▁an ▁aff ine ▁relationship ▁between ▁the ▁cos ine ▁of ▁the ▁contact ▁angle ▁and ▁the ▁inverse ▁line ▁radius , ▁it ▁does ▁not ▁account ▁for ▁the ▁correct ▁sign ▁of ▁ κ ▁and ▁over est im ates ▁its ▁value ▁by ▁several ▁orders ▁of ▁magnitude . ▁ ▁Contact ▁angle ▁prediction ▁while ▁account ing ▁for ▁line ▁t ension ▁and ▁Lap lace ▁pressure ▁▁ ▁With ▁improvements ▁in ▁meas uring ▁techniques ▁such ▁as ▁atomic ▁force ▁micro sc opy , ▁conf ocal ▁micro sc opy , ▁and ▁sc anning ▁electron ▁mic ros cope , ▁research ers ▁were ▁able ▁to ▁produce ▁and ▁image ▁dro ple ts ▁at ▁ever ▁smaller ▁scales . ▁With ▁the ▁reduction ▁in ▁dro ple t ▁size ▁came ▁new ▁experimental ▁observations ▁of ▁wet ting . ▁These ▁observations ▁confirmed ▁that ▁the ▁modified ▁Young ’ s ▁equation ▁does ▁not ▁hold ▁at ▁the ▁micro - n ano ▁scales . ▁Jas per ▁proposed ▁that ▁including ▁a ▁V   d P ▁term ▁in ▁the ▁variation ▁of ▁the ▁free ▁energy ▁may ▁be ▁the ▁key ▁to ▁solving ▁the ▁contact ▁angle ▁problem ▁at ▁such ▁small ▁scales . ▁Given ▁that