. ▁As ▁a ▁result , ▁there ▁is ▁a ▁therm od ynamic ▁driving ▁force ▁for ▁the ▁total ▁area ▁of ▁boundary ▁to ▁be ▁reduced . ▁If ▁the ▁gra in ▁size ▁increases , ▁accompanied ▁by ▁a ▁reduction ▁in ▁the ▁actual ▁number ▁of ▁gra ins ▁per ▁volume , ▁then ▁the ▁total ▁area ▁of ▁gra in ▁boundary ▁will ▁be ▁reduced . ▁ ▁The ▁local ▁velocity ▁of ▁a ▁gra in ▁boundary ▁at ▁any ▁point ▁is ▁proportional ▁to ▁the ▁local ▁curv ature ▁of ▁the ▁gra in ▁boundary , ▁i . e .: ▁ ▁, ▁ ▁where ▁ ▁is ▁the ▁velocity ▁of ▁gra in ▁boundary , ▁ ▁is ▁gra in ▁boundary ▁mob ility ▁( gener ally ▁depends ▁on ▁orientation ▁of ▁two ▁gra ins ), ▁ ▁is ▁the ▁gra in ▁boundary ▁energy ▁and ▁ ▁is ▁the ▁sum ▁of ▁the ▁two ▁principal ▁surface ▁curv atures . ▁For ▁example , ▁shr ink age ▁velocity ▁of ▁a ▁spher ical ▁gra in ▁embedded ▁inside ▁another ▁gra in ▁is ▁ ▁, ▁ ▁where ▁ ▁is ▁radius ▁of ▁the ▁sphere . ▁This ▁driving ▁pressure ▁is ▁very ▁similar ▁in ▁nature ▁to ▁the ▁Lap lace ▁pressure ▁that ▁occurs ▁in ▁fo ams . ▁ ▁In ▁comparison ▁to ▁phase ▁transformations ▁the ▁energy ▁available ▁to ▁drive ▁gra in ▁growth ▁is ▁very ▁low ▁and ▁so ▁it ▁tends ▁to ▁occur ▁at ▁much ▁slower ▁rates ▁and ▁is ▁easily ▁slow ed ▁by ▁the ▁presence ▁of ▁second ▁phase ▁particles ▁or ▁sol ute ▁atoms ▁in ▁the ▁structure . ▁ ▁Ide al ▁gra in ▁growth ▁ ▁Ide al ▁gra in ▁growth ▁is ▁a ▁special ▁case ▁of ▁normal ▁gra in ▁growth