Spinodal decomposition is a mechanism by which a single thermodynamic
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
spontaneously separates into two phases (without
nucleation).
Decomposition
Decomposition or rot is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and is e ...
occurs when there is no
thermodynamic
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
barrier to phase separation. As a result, phase separation via decomposition does not require the nucleation events resulting from thermodynamic fluctuations, which normally trigger phase separation.
Spinodal decomposition is observed when mixtures of metals or
polymer
A polymer (; Greek '' poly-'', "many" + ''-mer'', "part")
is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
s separate into two co-existing phases, each rich in one species and poor in the other. When the two phases emerge in approximately equal proportion (each occupying about the same volume or area), characteristic intertwined structures are formed that gradually coarsen (see animation). The dynamics of spinodal decomposition is commonly modeled using the
Cahn–Hilliard equation The Cahn–Hilliard equation (after John W. Cahn and John E. Hilliard) is an equation of mathematical physics which describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pu ...
.
Spinodal decomposition is fundamentally different from nucleation and growth. When there is a nucleation barrier to the formation of a second phase, time is taken by the system to overcome that barrier. As there is no barrier (by definition) to spinodal decomposition, some fluctuations (in the
order parameter
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
that characterizes the phase) start growing instantly. Furthermore, in spinodal decomposition, the two distinct phases start growing in any location uniformly throughout the volume, whereas a nucleated phase change begins at a discrete number of points.
Spinodal decomposition occurs when a homogenous phase becomes thermodynamically unstable. An unstable phase lies at a maximum in
free energy. In contrast, nucleation and growth occur when a homogenous phase becomes
metastable
In chemistry and physics, metastability denotes an intermediate energetic state within a dynamical system other than the system's state of least energy.
A ball resting in a hollow on a slope is a simple example of metastability. If the ball i ...
. That is, another biphasic system becomes lower in free energy, but the homogenous phase remains at a local minimum in
free energy, and so is resistant to small fluctuations.
J. 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 ...
described two criteria for a metastable phase: that it must remain stable against a small change over a large area.
History
In the early 1940s, Bradley reported the observation of sidebands around the Bragg peaks in the X-ray diffraction pattern of a Cu-Ni-Fe alloy that had been quenched and then annealed inside the
miscibility gap A miscibility gap is a region in a phase diagram for a mixture of components where the mixture exists as two or more phases – any region of composition of mixtures where the constituents are not completely miscible.
The IUPAC Gold Book defines ...
. Further observations on the same alloy were made by Daniel and Lipson, who demonstrated that the sidebands could be explained by a periodic modulation of composition in the <100> directions. From the spacing of the sidebands, they were able to determine the wavelength of the modulation, which was of the order of 100 angstroms.
The growth of a composition modulation in an initially homogeneous alloy implies uphill diffusion or a negative diffusion coefficient. Becker and Dehlinger had already predicted a negative diffusivity inside the spinodal region of a binary system. But their treatments could not account for the growth of a modulation of a particular wavelength, such as was observed in the Cu-Ni-Fe alloy. In fact, any model based on
Fick's law
Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion equ ...
yields a physically unacceptable solution when the diffusion coefficient is negative.
The first explanation of the periodicity was given by
Mats Hillert
Mats Hillert (28 November 1924 – 2 November 2022) was a Swedish metallurgist who was an emeritus professor in metallography (physical metallurgy) at the Royal Institute of Technology (KTH).Thermodynamics and Phase Transformations - The Selecte ...
in his 1955 Doctoral Dissertation at
MIT
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...
. Starting with a regular solution model, he derived a flux equation for one-dimensional diffusion on a discrete lattice. This equation differed from the usual one by the inclusion of a term, which allowed for the effect of the interfacial energy on the driving force of adjacent interatomic planes that differed in composition. Hillert solved the flux equation numerically and found that inside the spinodal it yielded a periodic variation of composition with distance. Furthermore, the wavelength of the modulation was of the same order as that observed in the Cu-Ni-Fe alloys.
[Hillert, M., ''A Theory of Nucleation for Solid Metallic Solutions'', Sc. D. Thesis (MIT, 1955)]
Building on Hillert's work, a more flexible continuum model was subsequently developed by
John W. Cahn
John Werner Cahn (January 9, 1928 – March 14, 2016) was an American scientist and recipient of the 1998 National Medal of Science. Born in Cologne, Weimar Germany, he was a professor in the department of metallurgy at the Massachusetts Institu ...
and John Hilliard, who included the effects of coherency strains as well as the gradient energy term. The strains are significant in that they dictate the ultimate morphology of the decomposition in anisotropic materials.
Cahn–Hilliard model for spinodal decomposition
Free energies in the presence of small amplitude fluctuations, e.g. in concentration, can be evaluated using an approximation introduced by
Ginzburg and
Landau
Landau ( pfl, Landach), officially Landau in der Pfalz, is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town (since 1990) ...
to describe magnetic field gradients in superconductors. This approach allows one to approximate the free energy as an expansion in terms of the concentration gradient
, a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
. Since free energy is a scalar and we are probing near its minima, the term proportional to
is negligible. The lowest order term is the quadratic expression
, a scalar. Here
is a parameter that controls the free energy cost of variations in concentration
.
The Cahn–Hilliard free energy is then
:
where
is the bulk free energy per unit volume of the homogeneous solution, and the integral is over the volume of the system.
We now want to study the stability of the system with respect to small fluctuations in the concentration
, for example a sine wave of amplitude
and wavevector
, for
the wavelength of the concentration wave. To be thermodynamically stable, the free energy change
due to any small amplitude concentration fluctuation
, must be positive.
We may expand
about the average composition c
o as follows:
:
:and for the perturbation
the free energy change is
:
:when this is integrated over the volume
, the
gives zero, while
and
integrate to give
. So, then
: