Spinodal Decomposition

Spinodal decomposition

Spinodal decomposition is one thermodynamic phase decomposing into two phases, when there is no nucleation barrier to this decomposition. Thus at least some fluctuations in the system spontaneously grow as they reduce the free energy, and so there is no waiting, as there typically is when there is a nucleation barrier. Spinodal decomposition can occur, for example, when mixtures of polymers are unstable as a mixture and separate into two coexisting phases, each one rich in one polymer, and poor in the other. It can also occur in metal alloys. When the two phases forming above approximately the same volume (or area) then characteristic intertwined structures form and coarsen - see animation on this page. The dynamics of spinodal decomposition are commonly modelled using the Cahn-Hilliard equation. Spinodal decomposition can be compared with another mechanism where one thermodynamic phase can decompose into two phases. This different mechanism is called nucleation and growth, and there, in contrast to spinodal decomposition, there is a nucleation barrier which typically takes a time to overcome before the new phase appears. As there is no barrier (by definition) to spinodal decomposition, at least some fluctuations start growing instantly. These fluctuations start growing throughout the volume, whereas nucleation typically involves the formation of a small number of nuclei of a new phase, at random points in the volume. Spinodal decomposition occurs for phases that are thermodynamically unstable. An unstable phase lies at a maximum in free energy. In contrast, nucleation and growth occurs in a metastable phase, which is a phase that lies at a local but not global minimum in free energy, and 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..

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Relation to polar decomposition

Consider g l n ( R ) displaystyle mathfrak gl_n(mathbb R ) with the Cartan involution ( X ) = X T displaystyle theta (X)=-X^T . Then k = s o n ( R ) displaystyle mathfrak k=mathfrak so_n(mathbb R ) is the real Lie algebra of skew-symmetric matrices, so that K = S O ( n ) displaystyle K=mathrm SO (n) , while p displaystyle mathfrak p is the subspace of symmetric matrices. Thus the exponential map is a diffeomorphism from p displaystyle mathfrak p onto the space of positive definite matrices. Up to this exponential map, the global Cartan decomposition is the polar decomposition of a matrix. The polar decomposition of an invertible matrix is unique.

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Decomposition reactions

This white polycrystalline solid was found to be stable under standard conditions but is extremely shock sensitive causing it to violently decompose when ground with a mortar. The thermodynamic properties of cyanuric triazide were studied using bomb calorimetry with a combustion enthalpy (H) of 2234 kJmol1 under oxidizing conditions and 740 kJmol1 otherwise. The former value is comparable to the military explosive RDX, (C3N3)(NO3)3H6, but is not put into use due to its less than favorable stability. Melting point examination showed a sharp melting range to clear liquid at 94-95 C, gas evolution at 155 C, orange to brown solution discoloration at 170 C, orange-brown solidification at 200 C and rapid decomposition at 240 C. The rapid decomposition at 240 C results from the formation of elemental carbon as graphite and the formation of nitrogen gas.

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About second uniqueness primary decomposition theorem

The two statements you cite are related - the claim in Atiyah-Macdonald implies the second claim. Both address the uniqueness of certain primary components, rather than the primes themselves (which depend only on $alpha$ in the sense that given $alpha$, they can be fixed). Let me address them in order: $DeclareMathOperatorAssAss$ $DeclareMathOperatorMinMin$1) Recall that in Atiyah-Macdonald, an isolated set of primes of $mathfrak a$ is a downward-closed subset of $Ass(mathfrak a)$, i.e. $S subseteq Ass(mathfrak a)$ is isolated if for $p, q in Ass(mathfrak a)$, $p subseteq q$, then $q in S implies p in S$. The claim in Atiyah-Macdonald is that if $S$ is isolated, then the intersection of the primary components, corresponding to primes in $S$, depends only on $mathfrak a$. 2) The claim in your other source is the following: each primary component, corresponding to a prime in $Min(mathfrak a)$, depends only on $mathfrak a$. This follows from the claim in Atiyah-Macdonald, by taking $S = p$ for $p in Min(mathfrak a)$, which is an isolated set.The point is that generally speaking, the primary components are not unique: for a given ideal $alpha$ and $p in Ass(mathfrak a)$, there may be (infinitely) many $p$-primary ideals $q$ that can appear in (valid) minimal primary decompositions of $mathfrak a$. In other words, there is no way in general to uniquely fix a $p$-primary component of $mathfrak a$, even after giving $mathfrak a$ and $p$. However, the second claim says that this behavior can only happen for embedded (i.e. non-minimal) primes.