4) The Wulff construction is a method to determine the equilibrium shape of a droplet or crystal of fixed volume inside a separate phase (usually its saturated solution or vapor). Free energy minimization arguments can be used to show that certain crystal planes are preferred over others, giving the crystal its shape. Specifically, in 1901 Russian scientist George Wulff stated (without proof) that the length of a vector drawn normal to a crystal face h; will be proportional to its surface energy y;. This is often referred to as the \"Wulff Rule\" or the \"Wulff construction\" in the literature. Here, h; is the \"height\" of the j h face, drawn from the center of the crystal to the face (for a spherical shape this would just be the radius), and y; is the surface Gibbs free energy per unit area of the crystal face. a) Provide a derivation of the Wulff Rule. (25 points) Hint: Write down the total surface energy of the crystal and minimize this energy with the constraint of the crystal volume held fixed (you can use the method of Lagrange multipliers if you wish). b) Often, nanowire growth does not obey Wulff's Rule, since these structures can have very high aspect ratios and their shapes may not be determined by the Wulff construction described above. Provide an explanation for why Wulff's rule fails for such systems. (5 points)