A growing 2D spherulite and calculus of variations Part I: A 2D spherulite in a linear field of growth rate

We propose to take the calculus of variations in order to compute the shape of a growing 2D spherulite in an uniaxial field of growth rate. We are concerned with the growth line (a path that is traveled in the shortest possible time from nucleus to a point (x₁, y₁), where a molecule just crystallizes) and the growth front (the times between spherulite and supercooled material). The Euler differential equation—a result of the calculus of variations—is derived for all uniaxial growth ratesv (x). Here we especially investigatev(x)=px+q.