The Wulff construction is a method for determining the equilibrium

Equilibrium

Equilibrium is the condition of a system in which competing influences are balanced. The word may refer to:-Biology:* Equilibrioception, the sense of a balance present in human beings and other animals...

 shape of a droplet or crystal

Crystal

A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...

 of fixed volume inside a separate phase (usually its saturated solution or vapor). Energy minimization

Energy minimization

In computational chemistry, energy minimization methods are used to compute the equilibrium configuration of molecules and solids....

 arguments are used to show that certain crystal planes are preferred over others, giving the crystal its shape.

Theory

In 1878 Josiah Willard Gibbs

Josiah Willard Gibbs

Josiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...

 proposed that a droplet or crystal will arrange itself such that its Gibbs free energy

Gibbs free energy

In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...

 is minimized by assuming a configuration of low Surface energy

Surface energy

Surface energy quantifies the disruption of intermolecular bonds that occur when a surface is created. In the physics of solids, surfaces must be intrinsically less energetically favorable than the bulk of a material, otherwise there would be a driving force for surfaces to be created, removing...

. He defined the quantity


where represents the surface energy per unit area of the jth crystal face, and is the area of said face. represents the difference in energy between a real crystal composed of i molecules with a surface, and a similar configuration of i molecules located inside an infinitely large crystal. This quantity is therefore the energy associated with the surface. The equilibrium shape of the crystal will then be that which minimizes the value of

In 1901, Georg Wulff stated -without proving- that the length of a vector drawn normal to a crystal face will be proportional to its surface energy : . This is known as the Gibbs-Wulff theorem.

In 1953 Conyers Herring

Conyers Herring

Conyers Herring was an American physicist. He was Professor of Applied Physics at Stanford University and the Wolf Prize in Physics recipient in 1984/5.-Academic career:...

gave a proof of the theorem and a method for determining the equilibrium shape of a crystal, which consists of two main exercises. To begin, a polar plot of surface energy as a function of orientation is made. This is known as the gamma plot and is usually denoted as where denotes the surface normal, e.g. a particular crystal face. The second part is the Wulff construction itself in which the gamma plot is used to determine graphically which crystal faces will be present. It can be determined graphically by drawing lines from the origin to every point on the gamma plot. A plane perpendicular to the normal is drawn at each point where it intersects the gamma plot. The inner envelope of these planes forms the equilibrium shape of the crystal.

Proof

Various proofs of the theorem have been given by Hilton, Liebman, von Laue, Herring and a rather extensive treatment by Cerf. The following is after the method of R. F. Strickland-Constable.
We begin with the surface energy for a crystal
which is the product of the surface energy per unit area times the area of each face, summed over all faces, which is minimized for a given volume when
We then consider a small change in shape for a constant volume
which can be written as
the second term of which must be zero, as it represents the change in volume, and we wish only to find the lowest surface energy at a constant volume (i.e. without adding or removing material.) We are then given from above
and
which can be combined by a constant of proportionality as
The change in shape must be allowed to be arbitrary, which then requires that which then proves Gibbs-Wulff Theorem.