This diagram shows the nomenclature for the different phase transitions.
In
thermodynamics, 'phase transition' or 'phase change' is the transformation of a thermodynamic system from one
phase to another. The distinguishing characteristic of a phase transition is an abrupt change in one or more physical properties, in particular the
heat capacity, with a small change in a thermodynamic variable such as the
temperature.
In the English vernacular, the term is most commonly used to describe transitions between
solid,
liquid and
gaseous
states of matter, in rare cases including
plasma.
Types of phase transition
A typical phase diagram. The dotted line gives the anomalous behaviour of water
A small piece of rapidly melting argon ice simultaneously shows the transitions from solid to liquid to gas.
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.
★ 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 emergence of
superconductivity in certain
metals when cooled below a critical temperature.
★ The transition between different molecular structures (
polymorphs or
allotropes), especially of solids, such as between an
amorphous structure and a
crystal structure or between two different crystal structures.
★ Quantum condensation of
bosonic fluids, such as
Bose-Einstein condensation and the
superfluid transition in liquid
helium.
★ The breaking of
symmetries in the laws of physics during the early history of the universe as its temperature cooled.
★ Phase transitions in intractable
computational complexity problems such as
NP-complete or
PSPACE problems. For example it has been noticed in
k-SAT problems that the transition from solvable to unsolvable instances exhibits
threshold behavior depending on the ratio of number of clauses to number of variables. Moreover, the amount of computational time required to solve the problem or determine it to be unsolvable increases drastically around the threshold. This line of research comes mostly from investigating similarities between computational complexity and statistical physics.
Phase transitions happen when the
free energy of a system is
non-analytic for some choice of thermodynamic variables - see
phases. This non-analyticity generally stems from the interactions of an extremely large number of particles in a system, and does not appear in systems that are too small.
It's sometimes possible to change the state of a system non-
adiabatically in such a way that it can be brought past a phase transition without undergoing a phase transition. The resulting state is
metastable i.e. not theoretically stable, but quasistable. See
superheating,
supercooling and
supersaturation.
Classification of phase transitions
Ehrenfest classification
The first attempt at classifying phase transitions was the '
Ehrenfest classification scheme', which grouped phase transitions based on the degree of non-analyticity involved. Though useful, Ehrenfest's classification is flawed, as will be discussed in the next section.
Under this scheme, phase transitions were labelled 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 a thermodynamic variable. The various solid/liquid/gas 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' have a 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 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 classication scheme, there could in principle be 'third', 'fourth', and higher-order phase transitions.
Modern classification of phase transitions
The Ehrenfest scheme is an inaccurate method of classifying phase transitions, for it does not take into account the case where a
derivative of
free energy diverges (which is only possible in the thermodynamic limit). For instance, in the
ferromagnetic transition, the
heat capacity diverges to
infinity.
In the modern classification scheme, phase transitions are divided into two broad categories, named similarly to the Ehrenfest classes:
The 'first-order phase transitions' are those that involve a
latent heat. During such a transition, a system either absorbs or releases a fixed (and typically large) amount of energy. Because energy cannot be instantaneously transferred between the system and its environment, first-order transitions are associated with "mixed-phase regimes" in which some parts of the system have completed the transition and others have not. This phenomenon is familiar to anyone who has boiled a pot of
water: the water does not instantly turn into gas, but forms a
turbulent mixture of water and
water vapor bubbles. Mixed-phase systems are difficult to study, because their dynamics are violent and hard to control. However, many important phase transitions fall in this category, including the solid/liquid/gas transitions and
Bose-Einstein condensation.
The second class of phase transitions are the ''continuous phase transitions'', also called 'second-order phase transitions'. These have no associated latent heat. Examples of second-order phase transitions are the ferromagnetic transition and the
superfluid transition.
Several transitions are known as the 'infinite-order phase transitions'.
They are continuous but break no symmetries (see 'Symmetry' below). The most famous example is the
Kosterlitz-Thouless transition in the two-dimensional
XY model. Many
quantum phase transitions in two-dimensional
electron gases belong to this class.
Properties of phase transitions
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 (including those of visible light).
During a phase change, the temperature of the sample does not change until the phase change is complete.
Symmetry
Phase transitions often (but not always) take place between phases with different
symmetry. Consider, for example, the transition between a fluid (i.e. liquid or gas) and a
crystalline solid. A fluid, which is composed of atoms arranged in a disordered but homogeneous manner, possesses continuous translational symmetry: each point inside the fluid has the same properties as any other point. A crystalline solid, on the other hand, is made up of atoms arranged in a regular
lattice. Each point in the solid is ''not'' similar to other points, unless those points are displaced by an amount equal to some lattice spacing.
Generally, we may speak of one phase in a phase transition as being more symmetrical than the other. The transition from the more symmetrical phase to the less symmetrical one is a 'symmetry-breaking' process. In the fluid-solid transition, for example, we say that continuous translation symmetry is broken.
The ferromagnetic transition is another example of a symmetry-breaking transition, in this case the symmetry under reversal of the direction of electric currents and magnetic field lines. This symmetry is referred to as "up-down symmetry" or "time-reversal symmetry". It is broken in the ferromagnetic phase due to the formation of magnetic domains containing aligned magnetic moments. Inside each domain, there is a magnetic field pointing in a fixed direction chosen spontaneously during the phase transition. The name "time-reversal symmetry" comes from the fact that electric currents reverse direction when the time coordinate is reversed.
The presence of symmetry-breaking (or nonbreaking) is important to the behavior of phase transitions. It was pointed out by
Landau that, given any state of a system, one may unequivocally say whether or not it possesses a given symmetry. Therefore, it cannot be possible to analytically deform a state in one phase into a phase possessing a different symmetry. This means, for example, that it is impossible for the solid-liquid phase boundary to end in a critical point like the liquid-gas boundary. However, symmetry-breaking transitions can still be either first- or second-order.
Typically, the more symmetrical phase is on the high-temperature side of a phase transition, and the less symmetrical phase on the low-temperature side. This is certainly the case for the solid-fluid and ferromagnetic transitions. This happens because the
Hamiltonian of a system usually exhibits all the possible symmetries of the system, whereas the low-energy states lack some of these symmetries (this phenomenon is known as
spontaneous symmetry breaking.) At low temperatures, the system tends to be confined to the low-energy states. At higher temperatures, thermal fluctuations allow the system to access states in a broader range of energy, and thus more of the symmetries of the Hamiltonian.
Order parameters
When symmetry is broken, 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. Such variables are examples
of 'order parameters'. An order parameter is a measure for the degree of order in a system with extremes 0 for total disorder and 1 for complete order
[1]. For example an order parameter can indicate the degree of order in a
liquid crystal. However, note that order parameters can also be defined for symmetry-nonbreaking transitions.
There exist also
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 for cosmology
Symmetry-breaking phase transitions play an important role in
cosmology. It has been speculated that, in the
hot early universe, the vacuum (i.e. the various
quantum fields that fill space) 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 (see
electroweak baryogenesis.)
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. 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 ''T
c''. When ''T'' is near ''T
c'', the heat capacity ''C'' typically has a
power law behaviour:
:
The constant α is the critical exponent associated with the heat capacity. It is not difficult to see that it must be less than 0 in order for the transition to have no latent heat. 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 (
see here). This experimental value of α agrees with theoretical predictions
based on
variational perturbation theory (
see here).
For 0 < α < 1, the heat capacity diverges at the transition temperature (though, since α < 1, the divergence is not strong enough to produce a latent heat.) 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 this 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 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 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, 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 is insensitive to the underlying microscopic properties of the system.
Phase-change data storage
Several data-storage technologies use
chalcogenide glass, which can be "switched" between two states, crystalline or amorphous, with the application of heat.
Phase change and optical disc technology
Phase change technology is also used to write to
optical discs, such as
CD-RW or
DVD-RW discs. This is accomplished by including both a
read laser and a more powerful
write laser inside the drive. The discs are made of a
crystalline material that, when hit by a pulse of
laser light from the write laser, changes to an
amorphous state, thus changing its
reflectivity. The read laser is not powerful enough to induce a phase change, but can be used by the drive to tell whether a
bit is "on" or "off" based on an area of the disc's reflectivity.
History of phase change optical disc technology
★ 1990: LF 7010 by
Panasonic, store 472
MB per side.
★ 1995:
PD (Phasewriter Dual) by
Panasonic, store 650
MB.
★ 1996:
CD-RW (Compact Disc ReWritable) by
Philips,
Sony,
Hewlett-Packard,
Mitsubishi Chemical Corp. and
Ricoh, store initially 650 MB and later 700 MB.
★ 1998:
DVD-RAM (DVD-Random Access Memory) by
Panasonic, store initially 2.6
GB and later 4.7 GB.
★ 199x:
DVD±RW (DVD-ReWritable) by supplier consortium, store 4.7 GB.
★ 2004:
PDD (Professional Disc for Data) by
Sony, store 20.5 GB.
★ 2004:
UDO (Ultra Density Optical) by
Plasmon, store 28 GB.
★ 2006:
BD-RE (Blu-ray Disc Rerecordable) by
Sony, store 50 GB.
Phase-change memory
Main articles: phase-change memory
Phase-change memory (PRAM) is a kind of
non-volatile computer memory.
Prototype PRAM devices have demonstrated higher density and faster write times than flash memory.
PRAM uses chalcogenide glass, the same material utilized in re-writable optical media (such as CD-RW and DVD-RW).
The amorphous, high resistance state is used to represent a binary 1, and the crystalline, low resistance state represents a 0.
Samsung, Intel, and STMicroelectronics demonstrated prototype PRAM devices in 2006, and announced plans for commercial productions.
See also
★
Differential scanning calorimetry
★
Lambda transition universality class
★
superfluid film
References
1. Gold Book definition Link
General references
★
Anderson, P.W., ''Basic Notions of Condensed Matter Physics'', Perseus Publishing (1997).
★ Goldenfeld, N., ''Lectures on Phase Transitions and the Renormalization Group'', Perseus Publishing (1992).
★ 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, 3rd Ed. (1994).
★
Kleinert, H., ''Critical Properties of φ
4-Theories'',
World Scientific (Singapore, 2001); Paperback ISBN 9810246595'' (readable online
here).''
★
Kleinert, H. and Verena Schulte-Frohlinde, ''Gauge Fields in Condensed Matter'', Vol. I, "
SUPERFLOW AND
VORTEX LINES;
Disorder Fields,
Phase Transitions,", pp. 1--742,
World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 '' (readable online
here)
★ 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.
External links
★
Interactive Phase Transitions on lattices with Java applets
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