▁where ▁boundary ▁motion ▁is ▁driven ▁only ▁by ▁local ▁curv ature ▁of ▁the ▁gra in ▁boundary . ▁It ▁results ▁in ▁the ▁reduction ▁of ▁the ▁total ▁amount ▁of ▁gra in ▁boundary ▁surface ▁area ▁i . e . ▁total ▁energy ▁of ▁the ▁system . ▁Add itional ▁contributions ▁to ▁the ▁driving ▁force ▁by ▁e . g . ▁el astic ▁stra ins ▁or ▁temperature ▁grad ients ▁are ▁neglect ed . ▁If ▁it ▁holds ▁that ▁the ▁rate ▁of ▁growth ▁is ▁proportional ▁to ▁the ▁driving ▁force ▁and ▁that ▁the ▁driving ▁force ▁is ▁proportional ▁to ▁the ▁total ▁amount ▁of ▁gra in ▁boundary ▁energy , ▁then ▁it ▁can ▁be ▁shown ▁that ▁the ▁time ▁t ▁required ▁to ▁reach ▁a ▁given ▁gra in ▁size ▁is ▁approxim ated ▁by ▁the ▁equation ▁ ▁where ▁d 0 ▁is ▁the ▁initial ▁gra in ▁size , ▁d ▁is ▁the ▁final ▁gra in ▁size ▁and ▁k ▁is ▁a ▁temperature ▁dependent ▁constant ▁given ▁by ▁an ▁exponential ▁law : ▁ ▁where ▁k 0 ▁is ▁a ▁constant , ▁T ▁is ▁the ▁absolute ▁temperature ▁and ▁Q ▁is ▁the ▁activation ▁energy ▁for ▁boundary ▁mob ility . ▁The oret ically , ▁the ▁activation ▁energy ▁for ▁boundary ▁mob ility ▁should ▁equal ▁that ▁for ▁self - diff usion ▁but ▁this ▁is ▁often ▁found ▁not ▁to ▁be ▁the ▁case . ▁ ▁In ▁general ▁these ▁equations ▁are ▁found ▁to ▁hold ▁for ▁ult ra - high ▁pur ity ▁materials ▁but ▁rapidly ▁fail ▁when ▁even ▁tiny ▁concentr ations ▁of ▁sol ute ▁are ▁introduced . ▁ ▁Self - similar ity ▁▁ ▁An ▁old - standing ▁topic ▁in ▁gra in ▁growth ▁is ▁the ▁evolution