The Wulff construction is a method to determine the
equilibrium shape of a
droplet
A drop or droplet is a small column of liquid, bounded completely or almost completely by free surfaces. A drop may form when liquid accumulates at the lower end of a tube or other surface boundary, producing a hanging drop called a pendant ...
or
crystal of fixed volume inside a separate phase (usually its saturated solution or vapor).
Energy minimization
In the field of computational chemistry, energy minimization (also called energy optimization, geometry minimization, or geometry optimization) is the process of finding an arrangement in space of a collection of atoms where, according to some com ...
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 (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
proposed that a droplet or crystal will arrange itself such that its surface
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and p ...
is minimized by assuming a shape of low
surface energy
In surface science, surface free energy (also interfacial free energy or surface energy) quantifies the disruption of intermolecular bonds that occurs when a surface is created. In solid-state physics, surfaces must be intrinsically less energe ...
. He defined the quantity
:
Here
represents the surface (Gibbs free) energy per unit area of the
th crystal face and
is the area of said face.
represents the difference in energy between a real crystal composed of
molecules with a surface and a similar configuration of
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 Russian scientist
George Wulff
George (Georgy/Yuri Viktorovich) Wulff (russian: Георгий (Юрий) Викторович Вульф) (22 June 1863, Nizhyn (Russian Empire, nowadays Ukraine) – 25 December 1925, Moscow) was a Russian crystallographer.
The Wulff con ...
stated (without proof) that the length of a vector drawn normal to a crystal face
will be proportional to its surface energy
:
. The vector
is the "height" of the
th face, drawn from the center of the crystal to the face; for a spherical crystal this is simply the radius. This is known as the Gibbs-Wulff theorem.
In 1953
Herring
Herring are forage fish, mostly belonging to the family of Clupeidae.
Herring often move in large schools around fishing banks and near the coast, found particularly in shallow, temperate waters of the North Pacific and North Atlantic Oceans, ...
gave a proof of the theorem and a method for determining the equilibrium shape of a crystal, consisting 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,
Laue
Max Theodor Felix von Laue (; 9 October 1879 – 24 April 1960) was a German physicist who received the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals.
In addition to his scientific endeavors with cont ...
, Herring, and a rather extensive treatment by Cerf. The following is after the method of R. F. Strickland-Constable.
[R. F. Strickland-Constable: ''Kinetics and Mechanism of Crystallization,'' page 77, Academic Press, 1968.]
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. This is minimized for a given volume when
:
Surface free energy, being an
intensive property, does not vary with volume. We then consider a small change in shape for a constant volume. If a crystal were nucleated to a thermodynamically unstable state, then the change it would undergo afterward to approach an equilibrium shape would be under the condition of constant volume. By definition of holding a variable constant, the change must be zero,
. Then by expanding
in terms of the surface areas
and heights
of the crystal faces, one obtains
:
,
which can be written, by applying the
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
, as
:
.
The second term must be zero, that is,
This is because, if the volume is to remain constant, the changes in the heights of the various faces must be such that when multiplied by their surface areas the sum is zero. If there were only two surfaces with appreciable area, as in a pancake-like crystal, then
. In the pancake instance,
on premise. Then by the condition,
. This is in agreement with a simple geometric argument considering the pancake to be a cylinder with very small
aspect ratio. The general result is taken here without proof. This result imposes that the remaining sum also equal 0,
:
Again, the surface energy minimization condition is that
:
These may be combined, employing a constant of proportionality
for generality, to yield
:
The change in shape
must be allowed to be arbitrary, which then requires that
, which then proves the Gibbs-Wulff Theorem.
References
{{reflist
Thermodynamics
Crystallography