A phase transition is the transformation of
a thermodynamic system from one phase or state
of matter to another one by heat transfer.
The term is most commonly used to describe
transitions between solid, liquid and gaseous
states of matter, and, in rare cases, plasma.
A phase of a thermodynamic system and the
states of matter have uniform physical properties.
During a phase transition of a given medium
certain properties of the medium change, often
discontinuously, as a result of the change
of some external condition, such as temperature,
pressure, or others. For example, a liquid
may become gas upon heating to the boiling
point, resulting in an abrupt change in volume.
The measurement of the external conditions
at which the transformation occurs is termed
the phase transition. Phase transitions are
common in nature and used today in many technologies.
Types of phase transition
Examples of phase transitions include:
The transitions between the solid, liquid,
and gaseous phases of a single component,
due to the effects of temperature and/or pressure:
(see also vapor pressure and phase diagram)
A eutectic transformation, in which a two
component single phase liquid is cooled and
transforms into two solid phases. The same
process, but beginning with a solid instead
of a liquid is called a eutectoid transformation.
A peritectic transformation, in which a two
component single phase solid is heated and
transforms into a solid phase and a liquid
phase.
A spinodal decomposition, in which a single
phase is cooled and separates into two different
compositions of that same phase.
Transition to a mesophase between solid and
liquid, such as one of the "liquid crystal"
phases.
The transition between the ferromagnetic and
paramagnetic phases of magnetic materials
at the Curie point.
The transition between differently ordered,
commensurate or incommensurate, magnetic structures,
such as in cerium antimonide.
The martensitic transformation which occurs
as one of the many phase transformations in
carbon steel and stands as a model for displacive
phase transformations.
Changes in the crystallographic structure
such as between ferrite and austenite of iron.
Order-disorder transitions such as in alpha-titanium
aluminides.
The dependence of the adsorption geometry
on coverage and temperature, such as for hydrogen
on iron.
The emergence of superconductivity in certain
metals and ceramics when cooled below a critical
temperature.
The transition between different molecular
structures, especially of solids, such as
between an amorphous structure and a crystal
structure, between two different crystal structures,
or between two amorphous structures.
Quantum condensation of bosonic fluids. The
superfluid transition in liquid helium is
an example of this.
The breaking of symmetries in the laws of
physics during the early history of the universe
as its temperature cooled.
Isotope fractionation occurs during a phase
transition, the ratio of light to heavy isotopes
in the involved molecules changes. When water
vapor condenses, the heavier water isotopes
become enriched in the liquid phase while
the lighter isotopes tend toward the vapor
phase.
Phase transitions occur when the thermodynamic
free energy of a system is non-analytic for
some choice of thermodynamic variables. This
condition generally stems from the interactions
of a large number of particles in a system,
and does not appear in systems that are too
small.
At the phase transition point the two phases
of a substance, liquid and vapor, have identical
free energies and therefore are equally likely
to exist. Below the boiling point, the liquid
is the more stable state of the two, whereas
above the gaseous form is preferred.
It is sometimes possible to change the state
of a system diabatically in such a way that
it can be brought past a phase transition
point without undergoing a phase transition.
The resulting state is metastable, i.e., less
stable than the phase to which the transition
would have occurred, but not unstable either.
This occurs in superheating, supercooling,
and supersaturation, for example.
Classifications
Ehrenfest classification
Paul Ehrenfest classified phase transitions
based on the behavior of the thermodynamic
free energy as a function of other thermodynamic
variables. Under this scheme, phase transitions
were labeled by the lowest derivative of the
free energy that is discontinuous at the transition.
First-order phase transitions exhibit a discontinuity
in the first derivative of the free energy
with respect to some thermodynamic variable.
The various solidgas transitions are classified
as first-order transitions because they involve
a discontinuous change in density, which is
the first derivative of the free energy with
respect to chemical potential. Second-order
phase transitions are continuous in the first
derivative but exhibit discontinuity in a
second derivative of the free energy. These
include the ferromagnetic phase transition
in materials such as iron, where the magnetization,
which is the first derivative of the free
energy with respect to the applied magnetic
field strength, increases continuously from
zero as the temperature is lowered below the
Curie temperature. The magnetic susceptibility,
the second derivative of the free energy with
the field, changes discontinuously. Under
the Ehrenfest classification scheme, there
could in principle be third, fourth, and higher-order
phase transitions.
Though useful, Ehrenfest's classification
has been found to be an inaccurate method
of classifying phase transitions, for it does
not take into account the case where a derivative
of free energy diverges. For instance, in
the ferromagnetic transition, the heat capacity
diverges to infinity.
Modern classifications
In the modern classification scheme, phase
transitions are divided into two broad categories,
named similarly to the Ehrenfest classes:
First-order phase transitions are those that
involve a latent heat. During such a transition,
a system either absorbs or releases a fixed
amount of energy. During this process, the
temperature of the system will stay constant
as heat is added: the system is in a "mixed-phase
regime" in which some parts of the system
have completed the transition and others have
not. Familiar examples are the melting of
ice or the boiling of water. Imry and Wortis
showed that quenched disorder can broaden
a first-order transition in that the transformation
is completed over a finite range of temperatures,
but phenomena like supercooling and superheating
survive and hysteresis is observed on thermal
cycling.
Second-order phase transitions are also called
continuous phase transitions. They are characterized
by a divergent susceptibility, an infinite
correlation length, and a power-law decay
of correlations near criticality. Examples
of second-order phase transitions are the
ferromagnetic transition, superconducting
transition and the superfluid transition.
In contrast to viscosity, thermal expansion
and heat capacity of amorphous materials show
a relatively sudden change at the glass transition
temperature which enable quite exactly to
detect it using differential scanning calorimetry
measurements. Lev Landau gave a phenomenological
theory of second order phase transitions.
Apart from isolated, simple phase transitions,
there exist transition lines as well as multicritical
points, when varying external parameters like
the magnetic field, composition,...
Several transitions are known as the infinite-order
phase transitions. They are continuous but
break no symmetries. The most famous example
is the Kosterlitz–Thouless transition in
the two-dimensional XY model. Many quantum
phase transitions, e.g., in two-dimensional
electron gases, belong to this class.
The liquid-glass transition is observed in
many polymers and other liquids that can be
supercooled far below the melting point of
the crystalline phase. This is atypical in
several respects. It is not a transition between
thermodynamic ground states: it is widely
believed that the true ground state is always
crystalline. Glass is a quenched disorder
state, and its entropy, density, and so on,
depend on the thermal history. Therefore,
the glass transition is primarily a dynamic
phenomenon: on cooling a liquid, internal
degrees of freedom successively fall out of
equilibrium. Some theoretical methods predict
an underlying phase transition in the hypothetical
limit of infinitely long relaxation times.
No direct experimental evidence supports the
existence of these transitions.
Characteristic properties
Phase coexistence
A disorder-broadened first order transition
occurs over a finite range of temperatures
with the fraction of the low-temperature equilibrium
phase grows from zero to one as the temperature
is lowered. This continuous variation of the
coexisting fractions with temperature raised
interesting possibilities. On cooling, some
liquids vitrify into a glass rather than transform
to the equilibrium crystal phase. This happens
if the cooling rate is faster than a critical
cooling rate, and is attributed to the molecular
motions becoming so slow that the molecules
cannot rearrange into the crystal positions.
This slowing down happens below a glass-formation
temperature Tg, which may depend on the applied
pressure., If the first-order freezing transition
occurs over a range of temperatures, and Tg
falls within this range, then there is an
interesting possibility that the transition
is arrested when it is partial and incomplete.
Extending these ideas to first order magnetic
transitions being arrested at low temperatures,
resulted in the observation of incomplete
magnetic transitions, with two magnetic phases
coexisting, down to the lowest temperature.
First reported in the case of a ferromagnetic
to anti-ferromagnetic transition, such persistent
phase coexistence has now been reported across
a variety of first order magnetic transitions.
These include colossal-magnetoresistance manganite
materials, magnetocaloric materials, magnetic
shape memory materials, and other materials.
The interesting feature of these observations
of Tg falling within the temperature range
over which the transition occurs is that the
first order magnetic transition is influenced
by magnetic field, just like the structural
transition is influenced by pressure. The
relative ease with which magnetic field can
be controlled, in contrast to pressure, raises
the possibility that one can study the interplay
between Tg and Tc in an exhaustive way. Phase
coexistence across first order magnetic transitions
will then enable the resolution of outstanding
issues in understanding glasses.
Critical points
In any system containing liquid and gaseous
phases, there exists a special combination
of pressure and temperature, known as the
critical point, at which the transition between
liquid and gas becomes a second-order transition.
Near the critical point, the fluid is sufficiently
hot and compressed that the distinction between
the liquid and gaseous phases is almost non-existent.
This is associated with the phenomenon of
critical opalescence, a milky appearance of
the liquid due to density fluctuations at
all possible wavelengths.
Symmetry
Order parameters
An order parameter is a measure of the degree
of order across the boundaries in a phase
transition system; it normally ranges between
zero in one phase and nonzero in the other.
At the critical point, the order parameter
susceptibility will usually diverge.
An example of an order parameter is the net
magnetization in a ferromagnetic system undergoing
a phase transition. For liquid/gas transitions,
the order parameter is the difference of the
densities.
From a theoretical perspective, order parameters
arise from symmetry breaking. When this happens,
one needs to introduce one or more extra variables
to describe the state of the system. For example,
in the ferromagnetic phase, one must provide
the net magnetization, whose direction was
spontaneously chosen when the system cooled
below the Curie point. However, note that
order parameters can also be defined for non-symmetry-breaking
transitions. Some phase transitions, such
as superconducting and ferromagnetic, can
have order parameters for more than one degree
of freedom. In such phases, the order parameter
may take the form of a complex number, a vector,
or even a tensor, the magnitude of which goes
to zero at the phase transition.
There also exist dual descriptions of phase
transitions in terms of disorder parameters.
These indicate the presence of line-like excitations
such as vortex- or defect lines.
Relevance in cosmology
Symmetry-breaking phase transitions play an
important role in cosmology. It has been speculated
that, in the hot early universe, the vacuum
possessed a large number of symmetries. As
the universe expanded and cooled, the vacuum
underwent a series of symmetry-breaking phase
transitions. For example, the electroweak
transition broke the SU(2)×U(1) symmetry
of the electroweak field into the U(1) symmetry
of the present-day electromagnetic field.
This transition is important to understanding
the asymmetry between the amount of matter
and antimatter in the present-day universe
Progressive phase transitions in an expanding
universe are implicated in the development
of order in the universe, as is illustrated
by the work of Eric Chaisson and David Layzer.
See also Relational order theories.
Critical exponents and universality classes
Continuous phase transitions are easier to
study than first-order transitions due to
the absence of latent heat, and they have
been discovered to have many interesting properties.
The phenomena associated with continuous phase
transitions are called critical phenomena,
due to their association with critical points.
It turns out that continuous phase transitions
can be characterized by parameters known as
critical exponents. The most important one
is perhaps the exponent describing the divergence
of the thermal correlation length by approaching
the transition. For instance, let us examine
the behavior of the heat capacity near such
a transition. We vary the temperature T of
the system while keeping all the other thermodynamic
variables fixed, and find that the transition
occurs at some critical temperature Tc. When
T is near Tc, the heat capacity C typically
has a power law behavior:
Such a behaviour has the heat capacity of
amorphous materials near the glass transition
temperature where the universal critical exponent
α = 0.59 A similar behavior, but with the
exponent instead of , applies for the correlation
length.
The exponent is positive. This is different
with . Its actual value depends on the type
of phase transition we are considering.
For -1 < α < 0, the heat capacity has a "kink"
at the transition temperature. This is the
behavior of liquid helium at the lambda transition
from a normal state to the superfluid state,
for which experiments have found α = -0.013±0.003.
At least one experiment was performed in the
zero-gravity conditions of an orbiting satellite
to minimize pressure differences in the sample.
This experimental value of α agrees with
theoretical predictions based on variational
perturbation theory.
For 0 < α < 1, the heat capacity diverges
at the transition temperature. An example
of such behavior is the 3-dimensional ferromagnetic
phase transition. In the three-dimensional
Ising model for uniaxial magnets, detailed
theoretical studies have yielded the exponent
α ∼ +0.110.
Some model systems do not obey a power-law
behavior. For example, mean field theory predicts
a finite discontinuity of the heat capacity
at the transition temperature, and the two-dimensional
Ising model has a logarithmic divergence.
However, these systems are limiting cases
and an exception to the rule. Real phase transitions
exhibit power-law behavior.
Several other critical exponents - β, γ,
δ, ν, and η - are defined, examining the
power law behavior of a measurable physical
quantity near the phase transition. Exponents
are related by scaling relations such as , . It
can be shown that there are only two independent
exponents, e.g. and .
It is a remarkable fact that phase transitions
arising in different systems often possess
the same set of critical exponents. This phenomenon
is known as universality. For example, the
critical exponents at the liquid-gas critical
point have been found to be independent of
the chemical composition of the fluid. More
amazingly, but understandable from above,
they are an exact match for the critical exponents
of the ferromagnetic phase transition in uniaxial
magnets. Such systems are said to be in the
same universality class. Universality is a
prediction of the renormalization group theory
of phase transitions, which states that the
thermodynamic properties of a system near
a phase transition depend only on a small
number of features, such as dimensionality
and symmetry, and are insensitive to the underlying
microscopic properties of the system. Again,
the divergency of the correlation length is
the essential point.
Critical slowing down and other phenomena
There are also other critical phenomena; e.g.,
besides static functions there is also critical
dynamics. As a consequence, at a phase transition
one may observe critical slowing down or speeding
up. The large static universality classes
of a continuous phase transition split into
smaller dynamic universality classes. In addition
to the critical exponents, there are also
universal relations for certain static or
dynamic functions of the magnetic fields and
temperature differences from the critical
value.
Percolation theory
Another phenomenon which shows phase transitions
and critical exponents is percolation. The
simplest example is perhaps percolation in
a two dimensional square lattice. Sites are
randomly occupied with probability p. For
small values of p the occupied sites form
only small clusters. At a certain threshold
pc a giant cluster is formed and we have a
second order phase transition. The behavior
of P∞ near pc is, P∞~(p-pc)β, where β
is a critical exponent.
Phase transitions in biological systems
Phase transitions play many important roles
in biological systems. Examples include the
lipid bilayer formation, the coil-globule
transition in the process of protein folding
and DNA melting, liquid crystal-like transitions
in the process of DNA condensation, and cooperative
ligand binding to DNA and proteins with the
character of phase transition.
In biolgical membranes, gel to liquid crystalline
phase transitions play a very critical role
in physiological functioning of biomembranes.
In gel phase, due to low fluidity of membrane
lipid fatty-acyl chains, membrane proteins
have restricted movement and thus are restrained
in exercise of their physiological role. Plants
depend critically on photosynthesis by chloroplast
thylakoid membranes which are exposed cold
environmental temperatures. Thylakoid membranes
retain innate fluidity even at relatively
low temperatures because of high degree of
fatty-acyl disorder allowed by their high
content of linolenic acid, 18-carbon chain
with 3-double bonds. Gel-to-liquid crystalline
phase transition temperature of biological
membranes can be determined by many techniques
including calorimetry, flouorescence, spin
label electron paramagnetic resonance and
NMR by recording measurements of the concerned
parameter by at series of sample temperatures.
A simple method for its determination from
13-C NMR line intensities has also been proposed.
The relevance of phase transitions in neural
networks has been pointed out, because of
the complex and emergent nature of neural
interactions. A point of view can be found
in the very recent paper by Tkačik et al.
See also
Allotropy
Autocatalytic reactions and order creation
Crystal growth
Abnormal grain growth
Differential scanning calorimetry
Diffusionless transformations
Ehrenfest equations
Jamming
Kelvin probe force microscope
Landau theory of second order phase transitions
Laser-heated pedestal growth
List of states of matter
Micro-Pulling-Down
Percolation theory
Continuum percolation theory
Phase separation
Superfluid film
Superradiant phase transition
References
Further reading
Anderson, P.W., Basic Notions of Condensed
Matter Physics, Perseus Publishing.
Fisher, M.E., "The renormalization group in
the theory of critical behavior", Rev. Mod.
Phys. 46, 597–616.
Goldenfeld, N., Lectures on Phase Transitions
and the Renormalization Group, Perseus Publishing.
Ivancevic, Vladimir G; Ivancevic, Tijana T,
Chaos, Phase Transitions, Topology Change
and Path Integrals, Berlin: Springer, ISBN 978-3-540-79356-4,
retrieved 14 March 2013  e-ISBN 978-3-540-79357-1 
Kogut,J. and Wilson,K, "The Renormalization
Group and the epsilon-Expansion", Phys. Rep.
12, 75.
Krieger, Martin H., Constitutions of matter :
mathematically modelling the most everyday
of physical phenomena, University of Chicago
Press, 1996. Contains a detailed pedagogical
discussion of Onsager's solution of the 2-D
Ising Model.
Landau, L.D. and Lifshitz, E.M., Statistical
Physics Part 1, vol. 5 of Course of Theoretical
Physics, Pergamon Press, 3rd Ed..
Kleinert, H., Gauge Fields in Condensed Matter,
Vol. I, "Superfluid and Vortex lines; Disorder
Fields, Phase Transitions,", pp. 1–742,
World Scientific; Paperback ISBN 9971-5-0210-0
Kleinert, H. and 
Verena Schulte-Frohlinde, Critical Properties
of φ4-Theories, World Scientific; Paperback
ISBN 981-02-4659-5.
Mussardo G., "Statistical Field Theory. An
Introduction to Exactly Solved Models of Statistical
Physics", Oxford University Press, 2010.
Schroeder, Manfred R., Fractals, chaos, power
laws : minutes from an infinite paradise,
New York: W. H. Freeman, 1991. Very well-written
book in "semi-popular" style—not a textbook—aimed
at an audience with some training in mathematics
and the physical sciences. Explains what scaling
in phase transitions is all about, among other
things.
Yeomans J. M., Statistical Mechanics of Phase
Transitions, Oxford University Press, 1992.
H. E. Stanley, Introduction to Phase Transitions
and Critical Phenomena.
External links
Interactive Phase Transitions on lattices
with Java applets
