Nikiya Anton Bettey 2021: 09-21-2021-1403

Tuesday, September 21, 2021

09-21-2021-1403 - Spinodal decomposition

Spinodal decomposition occurs when one thermodynamic phase spontaneously (i.e., without nucleation) separates into two phases.^[1] Decomposition occurs in the absence of nucleation because certain fluctuations in the system reduce the free energy. As a result, the phase change occurs immediately. There is no waiting, as there typically is when there is a nucleation barrier.

Spinodal decomposition is observed, for example, when mixtures of metals or polymers separate into two coexisting phases, each rich in one species and poor in the other.^[2]

When the two phases emerge in approximately equal proportion (each occupying about the same volume or area), they form characteristic intertwined structures that gradually coarsen - see the animation on this page. The dynamics of spinodal decomposition are commonly modeled using the Cahn–Hilliard equation.

Spinodal decomposition is fundamentally different from nucleation and growth. When there is a nucleation barrier to the formation of a second phase, the system takes time to overcome that barrier. As there is no barrier (by definition) to spinodal decomposition, some fluctuations (in the order parameter that characterizes the phase) start growing instantly. Furthermore, in spinodal decomposition fluctuations start growing everywhere, uniformly throughout the volume, whereas a nucleated phase form 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 occurs when a homogenous phase becomes metastable. That is, another new phase 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 described two criteria for a metastable phase: that it must remain stable against a small change over a large area, and that it must remain stable against a large change over a small area.^[3]

Microstructural evolution under the Cahn–Hilliard equation, demonstrating distinctive coarsening and phase separation.

Order parameters[edit]

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 (usually above the critical point) and nonzero in the other.^[22] 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.^[23]

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[edit]

Symmetry-breaking phase transitions play an important role in cosmology. 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 explain the asymmetry between the amount of matter and antimatter in the present-day universe, according to electroweak baryogenesis theory.

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^[24] and David Layzer.^[25]

Critical exponents and universality classes[edit]

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 T_c. When T is near T_c, the heat capacity C typically has a power law behavior:

C\propto |T_{\text{c}}-T|^{-\alpha }.

The heat capacity of amorphous materials has such a behaviour near the glass transition temperature where the universal critical exponent α = 0.59^[26] A similar behavior, but with the exponent ν instead of α, applies for the correlation length.

The exponent ν is positive. This is different with α. Its actual value depends on the type of phase transition we are considering.

It is widely believed that the critical exponents are the same above and below the critical temperature. It has now been shown that this is not necessarily true: When a continuous symmetry is explicitly broken down to a discrete symmetry by irrelevant (in the renormalization group sense) anisotropies, then some exponents (such as $\gamma$ , the exponent of the susceptibility) are not identical.^[27]

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.^[28] This experimental value of α agrees with theoretical predictions based on variational perturbation theory.^[29]

For 0 < α < 1, the heat capacity diverges at the transition temperature (though, since α < 1, the enthalpy stays finite). An example of such behavior is the 3D 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

\beta =\gamma /(\delta -1),\quad \nu =\gamma /(2-\eta ).

It can be shown that there are only two independent exponents, e.g. ν and η.

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 impressively, but understandably 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 divergence of the correlation length is the essential point.

Critical slowing down and other phenomena[edit]

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[edit]

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 p_c a giant cluster is formed, and we have a second-order phase transition.^[30] The behavior of P_∞ near p_c is P_∞ ~ (p − p_c)^β, where β is a critical exponent. Using percolation theory one can define all critical exponents that appear in phase transitions.^[31]^[30] External fields can be also defined for second order percolation systems^[32] as well as for first order percolation^[33] systems. Percolation has been found useful to study urban traffic and for identifying repetitive bottlenecks.^[34]^[35]

Phase transitions in biological systems[edit]

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.^[36]

In biological membranes, gel to liquid crystalline phase transitions play a 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.^[37] Gel-to-liquid crystalline phase transition temperature of biological membranes can be determined by many techniques including calorimetry, fluorescence, 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.^[38]

It has been proposed that some biological systems might lie near critical points. Examples include neural networks in the salamander retina,^[39] bird flocks^[40]gene expression networks in Drosophila,^[41] and protein folding.^[42] However, it is not clear whether or not alternative reasons could explain some of the phenomena supporting arguments for criticality.^[43] It has also been suggested that biological organisms share two key properties of phase transitions: the change of macroscopic behavior and the coherence of a system at a critical point.^[44]

The characteristic feature of second order phase transitions is the appearance of fractals in some scale-free properties. It has long been known that protein globules are shaped by interactions with water. There are 20 amino acids that form side groups on protein peptide chains range from hydrophilic to hydrophobic, causing the former to lie near the globular surface, while the latter lie closer to the globular center. Twenty fractals were discovered in solvent associated surface areas of > 5000 protein segments.^[45] The existence of these fractals proves that proteins function near critical points of second-order phase transitions.

In groups of organisms in stress (when approaching critical transitions), correlations tend to increase, while at the same time, fluctuations also increase. This effect is supported by many experiments and observations of groups of people, mice, trees, and grassy plants.^[46]

Experimental[edit]

A variety of methods are applied for studying the various effects. Selected examples are:

Thermogravimetry (very common)
X-ray diffraction
Neutron diffraction
Raman Spectroscopy
SQUID (measurement of magnetic transitions)
Hall effect (measurement of magnetic transitions)
Mössbauer spectroscopy (simultaneous measurement of magnetic and non-magnetic transitions. Limited up to about 800–1000 °C)
Perturbed angular correlation (simultaneous measurement of magnetic and non-magnetic transitions. No temperature limits. Over 2000 °C already performed, theoretical possible up to the highest crystal material, such as tantalum hafnium carbide 4215 °C.)

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