Cahn-Hilliard Equations and Phase Transition Dynamics for Binary Systems

The process of phase separation of binary systems is described by the Cahn-Hilliard equation. The main objective of this article is to give a classification on the dynamic phase transitions for binary systems using either the classical Cahn-Hilliard equation or the Cahn-Hilliard equation coupled with entropy, leading to some interesting physical predictions. The analysis is based on dynamic transition theory for nonlinear systems and new classification scheme for dynamic transitions, developed recently by the authors.


Introduction
Cahn-Hilliard equation describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. The main objective of this article is to provide a theoretical approach to dynamic phase transitions for binary systems.
The work was supported in part by the Office of Naval Research and by the National Science Foundation.
Classically, phase transitions are classified by the Ehrenfest classification scheme, based on the lowest derivative of the free energy that is discontinuous at the transition. In general, it is a difficult task to classify phase transitions of higher order, which appears in many equilibrium phase transition systems, such as the PVT system, the ferromagnetic system, superfluids as well as the binary systems studied in this article.
For this purpose, a new dynamic transition theory is developed recently by the authors. This new theory provides an efficient tool to analyze phase transitions of higher order. With this theory in our disposal, a new dynamic classification scheme is obtained, and classifies phase transitions into three categories: Type-I, Type-II and Type-III, corresponding mathematically to continuous, jump, and mixed transitions, respectively; see the Appendix as well as two recent books by the authors [3,4] for details.
There have been extensive studies in the past on the dynamics of the Cahn-Hilliard equations. However, very little is known about the higher order transitions encountered for this problem, and this article gives a complete classification of the dynamics transitions for binary systems. The results obtained lead in particular to various physical predictions. First, the order of phase transitions is precisely determined by the sign of a nondimensional parameter K such that if K > 0, the transition is first-order with latent heat and if K < 0, the transition is secondorder. Second, a theoretical transition diagram is derived, leading in particular to a prediction that there is only second-order transition for molar fraction near 1/2. This is different from the prediction made by the classical transition diagram. Third, a critical length scale is derived such that no phase separation occurs at any temperature if the length of the container is smaller than the critical length scale. These physical predictions will be addressed in another article.
This article is organized as follows. In Section 2, both the classical Cahn-Hilliard equation and the Cahn-Hilliard equation coupled with entropy are introduced in a unified fashion using a general principle for equilibrium phase transitions outlined in Appendix B. Sections 3-6 analyze dynamic transitions for the Cahn-Hilliard equation in general domain, rectangular domain, with periodic boundary conditions, and for the Cahn-Hilliard equation coupled with entropy. Physical conclusions are given in Section 7, and the dynamic transition theory is recalled in Appendix A.

Dynamic Phase Transition Models for Binary Systems
Materials compounded by two components A and B, such as binary alloys, binary solutions and polymers, are called binary systems. Sufficient cooling of a binary system may lead to phase separations, i.e., at the critical temperature, the concentrations of both components A and B with homogeneous distribution undergo changes, leading to heterogeneous distributions in space. Phase separation of binary systems observed will be in one of two main ways. The first is by nucleation in which sufficiently large nuclei of the second phase appear randomly and grow, and this corresponds to Type-II phase transitions. The second is by spinodal decomposition in which the systems appear to nuclear at once, and periodic or semi-periodic structure is seen, and this corresponds to Type-I phase transitions.
Since binary systems are conserved, the equations describing the Helmholtz process and the Gibbs process are the same. Hence, without distinction we use the term "free energy" to discuss this problem.
Let u A and u B be the concentrations of components A and B respectively, then u B = 1 − u A . In a homogeneous state, u B =ū B is a constant, and the entropy density S 0 =S 0 is also a constant. We take u, S the concentration and entropy density deviations: u = u B −ū B , S = S 0 −S 0 By (B.1) and (B.2), the free energy is given by (2.1) Since entropy is increasing as u → 0, and by δ δS F (u, S) = −µ 2 ∆S + β 1 S + β 0 u + β 2 u 2 = 0, we have which implies that S is a decreasing function of |u|. According (B.11) and (B.5), we derive from (2.1) and (2.2) the following equations governing a binary system: with Ω ⊂ R n (1 ≤ n ≤ 3) being a bounded domain, or the periodic boundary condition: with Ω = [0, L] n , K = (k 1 , · · · , k n ), 1 ≤ n ≤ 3.
For simplicity, in this section we always assume that p = 2. Thus, function (2.4) is rewritten as Based on Theorem A.1, we have to assume that there exists a temperature T 1 > 0 such that b 1 = b 1 (T ) satisfies If we ignore the coupled action of entropy in (2.1), then the free energy F is in the following form

Phase Transition in General Domains
In this section, we shall discuss the Cahn-Hilliard equation from the mathematical point of view. We start with the nondimensional form of equation. Let where l is a given length, u 0 =ū B is the constant concentration of B, and γ 3 > 0. Then the equation (2.9) can be rewritten as follows (omitting the primes) , For the Neumann boundary condition (2.5) we define and for the periodic boundary condition (2.6) we define Then we define the operators L λ = −A + B λ and G : Thus, the Cahn-Hilliard equation (3.1) is equivalent to the following operator equation It is known that the operators defined by (3.2) satisfy the conditions (A.2) and (A.3).
We first consider the case where Ω ⊂ R n (1 ≤ n ≤ 3) is a general bounded and smooth domain. Let ρ k and e k be the eigenvalues and eigenfunctions of the following eigenvalue problem: The eigenvalues of (3.4) satisfy 0 < ρ 1 ≤ ρ 2 ≤ · · · ≤ ρ k ≤ · · · , and lim k→∞ ρ k = ∞. The eigenfunctions {e k } of (3.4) constitute an orthonormal basis of H. Furthermore, the eigenfunctions of (3.4) satisfy Hence, {e k } is also an orthogonal basis of H 1 under the following equivalent norm We are now in position to give a phase transition theorem for the problem (3.1) with the following Neumann boundary condition: Theorem 3.1. Assume that γ 2 = 0 and γ 3 > 0 in (3.1), then the following assertions hold true: (1) If the first eigenvalue ρ 1 of (3.4) has multiplicity m ≥ 1, then the problem (3.1) with (3.5) bifurcates from (u, λ) = (0, ρ 1 ) on λ > ρ 1 to an attractor Σ λ , homeomorphic to an (m−1)-dimensional sphere S m−1 , and Σ λ attracts H \ Γ, where Γ is the stable manifold of u = 0 with codimension m. (2) Σ λ contains at least 2m singular points. If m = 1, Σ λ has exactly two steady states ±u λ , and if m = 2, Σ λ = S 1 has at most eight singular points. (3) Each singular point u λ in Σ λ can be expressed as where w is an eigenfunction corresponding to the first eigenvalue of (3.4).
Proof. We proceed in several steps as follows.
Step 1. It is clear that the eigenfunction {e k } of (3.4) are also eigenvectors of the linear operator L λ = −A + B λ defined by (3.2) and the eigenvalues of L λ are given by It is easy to verify the conditions (A.4) and (A.5) in our case at λ 0 = ρ 1 . We shall prove this theorem using the attractor bifurcation theory introduced in [3].
We need to verify that u = 0 is a global asymptotically stable singular point of (3.3) at λ = ρ 1 . By γ 2 = 0, from the energy integration of (3.1) we can obtain where C > 0 is a constant, u = v + w, and Ω vwdx = 0, and w is a first engenfunction. It follows from (3.7) that u = 0 is global asymptotically stable. Hence, for Assertion (1), we only have to prove that Σ λ is homeomorphic to S m−1 , as the rest of this assertion follows directly from the attractor bifurcation theory introduced in [3].
Let g(u) = −∆u − λu + γ 3 u 3 . Then the stationary equation of (3.1) is given by which is equivalent, by the maximum principle, to By the Lagrange multiplier theorem, (3.8) is the Euler equation of the following functional with zero average constraint: Since F is an even functional, by the classical Krasnoselskii bifurcation theorem for even functionals, (3.9) bifurcates from λ > ρ 1 at least to 2m mini-maximum points, i.e., equation (3.8) has at least 2m bifurcated solutions on λ > ρ 1 . Hence, the attractor Σ λ contains at least 2m singular points.
Step 3. To complete the proof, we reduce the equation (3.3) to the center manifold near λ = ρ 1 . By the approximation formula given in [3], the reduced equation of (3.3) is given by: x i e 1i , and {e 11 , · · · , e 1m } are the first eigenfunctions of (3.4). Equations (3.10) can be rewritten as x j e 1j .
Then for any x ∈ R m , for some constant C > 0. Thus by the attractor bifurcation theorem [3], it follows from (3.11) and (3.12) that the attractor Σ λ is homeomorphic to S m−1 . Hence, Assertion (1) is proved. The other conclusions in Assertions (2) and (3) can be derived from (3.11) and (3.12). The proof is complete.
For this purpose, let the multiplicity of the first eigenvalue ρ 1 of (3.4) be m ≥ 1, and {e 1 , · · · , e m } be the first eigenfunctions. We introduce the following quadratic equations Theorem 3.2. Let γ 2 = 0, γ 3 > 0, and x = 0 be an isolated singular point of (3.13). Then the phase transition of (3.1) and (3.5) is either Type-II or Type-III. Furthermore, the problem (3.1) with (3.5) bifurcates to at least one singular point on each side of λ = ρ 1 , and has a saddle-node bifurcation on λ < ρ 1 . In particular, if m = 1, then the following assertions hold true: (1) The phase transition is Type-III, and a neighborhood U ⊂ H of u = 0 can be decomposed into two sectorial regionsŪ =D 1 (π) +D 2 (π) such that the phase transition in D 1 (π) is the first order, and in D 2 (π) is the n-th order with n ≥ 3. (2) The bifurcated singular point u λ on λ > ρ 1 attracts D 2 (π), which can be expressed as where, by assumption, a = Ω e 3 1 dx = 0. (3) When |γ 2 a| = ε is small, the assertions in the transition perturbation theorems (Theorems A.8 and A.9) hold true.
Remark 3.1. We shall see later that when Ω is a rectangular domain, i.e., then a = Ω e 3 1 dx = 0. However, for almost all non-rectangular domains Ω, the first eigenvalues are simple and a = 0. Hence, the Type-III phase transitions for general domains are generic.
Proof of Theorem 3.2. Assertions (1)-(3) can be directly proved using Theorems A.5, A.8, and A.9. By assumption, u = 0 is a second order non-degenerate singular point of (3.3) at λ = ρ 1 , which implies that u = 0 is not locally asymptotically stable. Hence, it follows from Theorems A.3 and the steady state bifurcation theorem for even-order nondegenerate singular points [3] that the phase transition of (3.1) with (3.5) is either Type-II or Type-III, and there is at least one singular point bifurcated on each side of λ = ρ 1 .
Finally, we shall apply Theorem A.6 to prove that there exists a saddle-node bifurcation on λ < ρ 1 .
It is known that Moreover, since L λ + G defined by (3.2) is a gradient-type operator, we can derive that ind(L ρ1 + G, 0) ≤ 0.

Phase Transition in Rectangular Domains
The dynamical properties of phase separation of a binary system in a rectangular container is very different from that in a general container. We see in the previous section that the phase transitions in general domains are Type-III, and we shall show in the following that the phase transitions in rectangular domains are either Type-I or Type-II, which are distinguished by a critical size of the domains.
(2) If γ 3 > 2L 2 9π 2 γ 2 2 , then the transition is Type-I. In particular, the problem bifurcates on λ > π 2 /L 2 to exactly two attractors u T 1 and u T 2 which can be expressed as Proof. With the spatial domain as given, the first eigenvalue and eigenfunction of (3.4) are given by ρ 1 = π 2 /L 2 , e 1 = cos πx 1 L .
The eigenvalues and eigenfunctions of L λ = −A+B λ defined by (3.2) are as follows: By the approximation of the center manifold obtained in [3], the reduced equation of (3.3) to the center manifold is given by where y ∈ R 1 , Φ(y) is the center manifold function, and  It follows that H Ω e K e 2 1 dx.

Notice that
Then we have Inserting Φ(y) into (4.9) we find Finally, by (4.7), (4.8) and (4.10), we derive from (4.5) the following reduced equation of (3.3): Near the critical point λ 0 = π 2 /L 2 , the coefficient Thus, by Theorem A.2 we derive from (4.11) the assertions of the theorem except the claim for the saddle-node bifurcation in Assertion (1), which can be proved in the same fashion as used in Theorem 3.2. The proof is complete.
(2) If , then the transition is Type-II. In particular, the problem has a saddle-node bifurcation on λ < λ 0 = π 2 /L 2 , and bifurcates on both side of λ = λ 0 to exactly 3 m − 1 singular points which are non-degenerate.
Proof. We proceed in several steps as follows.
Step 1. Consider the center manifold reduction. It is known that the eigenvalues and eigenfunctions of L λ = −A + B λ are given by (4.3) and (4.4) with L 1 = · · · = L m . As before, the reduced equations of (3.3) are given by where y = (y 1 , · · · , y m ) ∈ R m , β 1 (λ) is as in (4.6), and Here e i = cos πx i /L for 1 ≤ i ≤ m, L is given by (4.12 and Φ is the center manifold function. Direct computation shows that We need to compute the center manifold function Φ(y). As in [3], we have

By (4.3) and (4.4) we have
Direct computation gives that Putting (4.14) and (4.17) in (4.13), we get the reduced equations in the following form Step 2. It is known that the transition type of (3.3) at the critical point λ 0 = π 2 /L 2 is completely determined by (4.18), i.e., by the following equations (4.20) It is easy to see that (4.21) Step 3. We consider the case where m = 2. Thus, the transition type of (4.20) is equivalent to that of the following equations We can see that on the straight lines (4.23)

equations (4.22) satisfy that
Hence the straight lines (4.23) are orbits of (4.22) if σ 0 1 + σ 0 2 = 0. Obviously, the straight lines (4.24) y 1 = 0 and y 2 = 0 are also orbits of (4.22). There are four straight lines determined by (4.23) and (4.24), and each of them contains two orbits. Hence, the system (4.22) has at least eight straight line orbits. Hence it is not hard to see that the number of straight line orbits of (4.22), if finite, is eight.
Thus by (4.21), for m = 2 we prove that the transition is Type- Step 4. Consider the case where m = 3. Thus, (4.20) are written as (4.25) It is clear that the straight lines consist of orbits of (4.25). There are total 13 straight lines in (4.26) and (4.27), each of which consists of two orbits. Thus, (4.25) has at least 26 straight line orbits. We shall show that (4.25) has just the straight line orbits given by (4.26) and (4.27).
In fact, we assume that the line is a straight line orbit of (4.25). Then z 1 , z 2 satisfy (4.28) , .
In the same fashion, we can prove that the straight line orbits of (4.25) given by y 1 = α 1 y 3 , y 2 = α 2 y 3 , and y 1 = β 1 y 2 , y 3 = β 2 y 2 have to satisfy that Thus, we prove that when σ 0 1 = σ 0 2 , the number of straight line orbits of (4.25) is exactly 26.
In this case, it is clear that y = 0 is an asymptotically stable singular point of (4.25). Hence, the transition of (4.18) at λ 0 = π 2 /L 2 is I-type.
Obviously, there are only finite number of λ > π 2 /L 2 satisfying Hence, for any λ − π 2 /L 2 > 0 sufficiently small the Jacobian matrices (4.32) at the singular points (4.31) are non-degenerate. Thus, the bifurcated solutions of (4.30) are regular. Since all bifurcated singular points of (3.1) with (3.5) are non-degenerate, and when Σ λ is restricted on x i x j -plane (1 ≤ i, j ≤ m) the singular points are connected by their stable and unstable manifolds. Hence all singular points in Σ λ are connected by their stable and unstable manifolds. Therefore, Σ λ must be homeomorphic to a sphere S m−1 .
Step 6. Proof of Assertions (2) and (3). When m = 2, by Step 5, Σ λ = S 1 contains 8 non-degenerate singular points. By a theorem on minimal attractors in [3], 4 singular points must be attractors and the others are repellors, as shown in When m = 3, we take the six singular points are attractors, which implies that Σ λ contains only six minimal attractors as shown in The claim for the saddle-node bifurcation in Assertion (3) can be proved by using the same method as in the proof of Theorem 3.2, and the claim for the singular point bifurcation can be proved by the same fashion as used in Step 5.
The proof of this theorem is complete. provided π 2 /L 2 < λ 1 , where λ 1 is the first eigenvalue of the equation Remark 4.2. In Theorem 4.2, the minimal attractors in the bifurcated attractor Σ λ can be expressed as where e is a first eigenfunction of (3.4). The expression (4.34) can be derived from the reduced equations (4.18). We address here that the exponent β = 1/2 in (4.34), called the critical exponent in physics, is an important index in the phase transition theory in statistical physics, which arises only in the Type-I or the continuous phase transitions. It is interesting to point out that the critical exponent β = 1 in (3.14) is different from these β = 1/2 appearing in (4.2) and (4.34). The first one occurs when the container Ω ⊂ R 3 is a non rectangular region, and the second one occurs when Ω is a rectangle or a cube. We shall continue to discuss this problem later from the physical viewpoint.

Phase Transitions Under Periodic Boundary Conditions
When the sample or container Ω is a loop, or a torus, or bulk in size, then the periodic boundary conditions are necessary. In this section, we shall discuss the problems in a loop domain and in the whole space Ω = R n .
Proof. Let v = y cos θ + z sin θ, u = v + Φ(y, z), and Φ is the center manifold function. Then the reduced equations of (3.1) with (5.1) are given by Direct computation shows that (5.4) can be rewritten as and the center manifold function Φ = Φ(y, z) is ).
It is clear that the first eigenvalue β 1 (λ) = λ − 1 of L λ has multiplicity 2n, and the first eigenfunctions are Theorem 5.2.
(2) If then the transition is Type-II.
Proof. We only have to prove Assertion (2), as the remaining part of the theorem is essentially the same as the proof for Theorem 4.2.
Since the space of all even functions is an invariant space of L λ + G defined by (3.2), the problem (3.1) with (5.7) has solutions given in (4.31) with m = n in the space of even functions.
Obviously, for a fixed (j 1 , · · · , j k ), the 2 n−k steady state solutions of (3.1) and (5.7) associated with (4.31) are in the same singular torus T n−k . Furthermore, for two different index k-tuples (j 1 , · · · , j k ) and (i 1 , · · · , i k ), the two associated singularity tori are different. Hence, for each 0 ≤ k ≤ n − 1, there are exactly C k n (n − k)-dimensional singularity tori in Σ λ . Thus the proof is complete.

Cahn-Hilliard Equations Coupled with Entropy
When a phase separation takes place in a binary system, the entropy varies, and if the phase transition is Type-II, it will yield latent heat. Hence, it is necessary to discuss the equations (2.3), which are called the Cahn-Hilliard equations coupled with entropy.
To make the equations (2.3) non-dimensional, let Omitting the primes, equations (2.3) are in the following form

By assumptions (2.2) and (2.4), the coefficients satisfy
(1) For the case where m = 1, let (a) If σ < 0, then the phase transition of (6.1) at λ = π 2 /L 2 is Type-II and Assertion (1) (a) If σ > 0, the phase transition of (6.1) at λ = π 2 /L 2 is Type-I and Assertions (1) and (2)  Proof. It suffices to compute the reduced equations of (6.1) on the center manifold. Similar to (4.18), the second order approximation of the reduced equation can be expressed as where β 1 , σ 1 and σ 2 are as in (4.18), and the center manifold function Φ 1 (y) derived from the first equation in (6.1) can be expressed as where λ K and ϕ K are the eigenvalues and eigenfunctions of the following equation which are given by Let K i = (δ i1 /L 1 , · · · , δ in /L n ).
Then we find for some 1 ≤ r, l ≤ m. Thus we derive that Putting Φ 0 and Φ K in (6.3) we obtain Then, inserting Φ 1 into (6.2), we derive the following reduced equations: Then the remaining part of the proof can be achieved in the same fashion as the proofs for Theorems 4.1 and 4.2. The proof is complete.

Physical remarks
We now address the physical significance for the phase transition theorems obtained in the previous sections. 7.1. Equation of critical parameters. For a binary system, the equation describing the control parameters T, p, Ω at the critical states is simple.
We first consider the critical temperature T c . There are two different critical temperatures T 1 and T 0 in the Cahn-Hilliard equation. T 1 is the one given by (2.8), at which the coefficient b 1 (T, p) or λ = −l 2 b 1 (T, p)/k will change its sign, and T 0 satisfies that λ 0 < λ 1 , and for fixed p, where ρ 1 is the first eigenvalue of (3.4), which depends on the geometrical properties of the material such as the size of the container of the sample Ω. When Ω = (0, L) m × D ⊂ R n is a rectangular domain with L > diameter of D, ρ 1 = π 2 /L 2 .
Hence, in general at the critical temperature T 1 a binary system does not undergo any phase transitions, but the phase transition does occur at T = T 0 . At T 1 and T 0 we know that For a rectangular domain, ρ 1 = π 2 /L 2 , therefore from (7.2) we see that T 1 is a limit of the critical temperature T 0 of phase transition as the size of Ω tends to infinite. In fact, for a general domain, it is easy to see that the first eigenvalue ρ 1 of the Laplace operator is inversely proportional to the square of the maximum diameter of Ω: where L represents the diameter scaling of Ω.
Thus the equation of critical parameters in the Cahn-Hilliard equation, by (7.2) and (7.3), is given by (7.4) λ(T, p) = C L 2 , where C > 0 is a constant depending on the geometry of Ω. According to the Hildebrand theory (see Reichl [7]), the function λ(T, p) can be expressed in a explicit formula. If regardless of the term |∇u| 2 , the molar Gibbs free energy takes the following form where µ A , µ B are the chemical potential of A and B respectively, R the molar gas constant, a > 0 the measure of repel action between A and B. Therefore, the coefficient b 1 in (2.7) with constant p is where u 0 =ū B is the constant concentration of B. Hence Thus, equation (7.4) is expressed as Equation (7.6) gives the critical parameter curve of a binary system with constant pressure for temperature T and diameter scaling L of container Ω. Because T ≥ 0, from (7.5) we can deduce the following physical conclusion.
Physical Conclusion 7.1. Under constant pressure, for any binary system with given geometrical shape of the container Ω, there is a value L 0 > 0 such that as the diameter scaling L < L 0 , no phase separation takes place at all temperature T ≥ 0, and as L > L 0 phase separation will occur at some critical temperature T 0 > 0 satisfying (7.6).
We shall see later that it is a universal property that the dynamical properties of phase transitions depend on the geometrical shape and size of the container or sample Ω. 7.2. Physical explanations of phase transition theorems. We first briefly recall the classical thermodynamic theory for a binary system. Physically, phase separation processes taking place in an unstable state are called spinodal decompositions; see Cahn and Hilliard [1] and Onuki [6]. When consider the concentration u as homogeneous in Ω, then by (7.5) the dynamic equation of a binary system is an ordinary differential equation: Let u 0 (0 < u 0 < 1) be the steady state solution of (7.7). Then, by the Taylor expansion at u = u 0 , omitting the nth order terms with n ≥ 4, (7.7) can be rewritten as

RT.
It is easy to see that It is clear that the critical parameter curve λ = 0 in the T − u 0 plane is given by which is schematically illustrated in the classical phase diagram; see the dotted line in Figure 7.1. We obtain from (7.8) the following transition steady states: By Theorem 3.2, we see that there is T * = T * (u 0 ) satisfying that b 2 2 − 4b 3 λ = 0; namely, is illustrated by the solid line in Figure 7.1. This shows that the region T 0 (u 0 ) < T < T * (u 0 ) is metastable, which is marked by the shadowed region in Figure 7.1. See, among others, Reichl [7], Novick-Cohen and Segal [5] , and Langer [2] for the phase transition diagram from the classical thermodynamic theory. In the following we shall discuss the spinodal decomposition in a unified fashion by applying the phase transition theorems presented in the previous sections.
As mentioned in the Introduction, phase separation processes of binary systems occur in two ways, one of which proceeds continuously depending on T , and the other one does not. Obviously, the classical theory does not explain these phenomena. In fact, the first one can be described by the Type-I phase transition, and the second one can be explained by the Type-II and Type-III phase transitions.
We first consider the case where the container Ω = Π n i=1 (0, L i ) with L = L 1 = · · · = L m > L j (j > m) is a rectangular domain. Thus, by Theorems 4.1 and 4.2 (or Theorem 6.1) there are only two phase transition types: Type-I and Type-II, with the type of transition depending on L. We see that if then the transition is Type-I, i.e., the phase pattern formation gradually varies as the temperature decreases. In this case, no meta-stable states and no latent heat appear. The phase diagram is given by Figure 7  If L satisfies that then the phase transition is Type-II. Namely, there is a leaping change in phase pattern formation at the critical temperature T c . The phase diagram for Type-II transition is given by Figure 7.3.
In Figure 7.3, T 0 is the critical temperature as in (7.1), T * is defined by (7.9) and is the saddle-node bifurcation point of (7.8). The constant concentration u = u 0 is stable in T c < T , is meta-stable in T 0 < T < T c , and is unstable in T < T 0 . The two bifurcated states U T 1 and U T 2 from T * are meta-stable in T 0 < T < T * , and are stable in T < T 0 . Here for i = 1, 2, U T i = u T i + u 0 , and u i are the separated solutions of (3.1) with (3.5) from T * .
There is a remarkable difference between Type-I and Type-II transitions. The Type-I phase transition occurs at T = T 0 and Type-II does in T 0 < T < T * . Furthermore, latent heat is accompanied the Type-II phase transition. Actually, when a binary system undergoes a transition from u 0 to U T i (i = 1, 2), there is a gap |U T i − u 0 | 2 = |u T i | 2 > ε > 0 for any T 0 < T < T * . By the first equation in (6.1) it yields a jump of entropy between u 0 and U T i : where S = S i −S 0 represents the entropy density deviation. Hence the latent heat is given by which implies that the process from u 0 to U T i is exothermic, and the process from U T i to u 0 is endothermic. Now, we consider the case where the container Ω is non rectangular. Thus, by Theorem 3.2 the transition is Type-III, and its phase diagram is given by Figure  7.4.
In Figure 7.4, T 0 and T * are the same as those in Figure 7.3. The state u 0 is stable in T * < T , is metastable in T 0 < T < T * , and unstable in T < T 0 . The equilibrium state U T 1 separated from T * is metastable in T 0 < T < T * , and is stable in T < T 0 . However, the equilibrium state U T 2 separated from T * is unstable in T 0 < T < T * , and is metastable in T < T 0 . Similar to the Type-II, the Type-III phase transition has also latent heat, which occurs in T 0 < T < T * . But the difference between Type-II and Type-III is that Type-II has 2m (m ≥ 1) stable equilibrium states separated from T = T * , but Type-III has just one. The 2m stable states of a Type-II transition are of some symmetry caused by Ω, and we shall investigate it later. A particular aspect of Type-III is that there is a state U T 2 bifurcated from (u, T ) = (u 0 , T 0 ), which is rarely observed in experiments. 7.3. Symmetry and periodic structure. Physical experiments have shown that in pattern formation via phase separation, periodic or semi-periodic structure appears. From Theorems 5.1 and 5.2 we see that for the loop domains and bulk domains which can be considered as R n or R m × D (D ⊂ R n−m ) the steady state solutions of the Cahn-Hilliard equation are periodic, and for rectangular domains they are semi-periodic, and the periodicity is associated with the mirror image symmetry.
Only these elements in A 1 or in A 3 are stable, and they are determined by the following criterion elements in A 1 is stable ⇔ 22 9 7.4. Critical exponents. From (4.2) and (4.34) we see that for Type-I phase transition of a binary system the critical exponent β = 1 2 . In this case, it is a second order phase transition with the Ehrenfest classification scheme, and there is a gap in heat capacity at critical temperature T 0 . To see this, by (4.2) and (4.34) we have and the free energy for (3.1) at u T is Thus, the heat capacity C at T = T 0 satisfies It is known that dλ/dT = 0; hence the heat capacity at T = T 0 has a finite jump. From (3.14) we know that for the Type-III case, the critical exponent β = 1. Thus, it is not hard to deduce that the continuous phase transition in Type-III is of the 3rd order.

Appendix A. Dynamic Transition Theory for Nonlinear Systems
In this appendix we recall some basic elements of the dynamic transition theory developed by the authors [3,4], which are used to carry out the dynamic transition analysis for the binary systems in this article.
A.1. New classification scheme. Let X and X 1 be two Banach spaces, and X 1 ⊂ X a compact and dense inclusion. In this chapter, we always consider the following nonlinear evolution equations where u : [0, ∞) → X is unknown function, and λ ∈ R 1 is the system parameter. Assume that L λ : X 1 → X is a parameterized linear completely continuous field depending continuously on λ ∈ R 1 , which satisfies (A.2) In this case, we can define the fractional order spaces X σ for σ ∈ R 1 . Then we also assume that G(·, λ) : X α → X is C r (r ≥ 1) bounded mapping for some 0 ≤ α < 1, depending continuously on λ ∈ R 1 , and Hereafter we always assume the conditions (A.2) and (A.3), which represent that the system (A.1) has a dissipative structure.
Obviously, the attractor bifurcation of (A.1) is a type of transition. However, bifurcation and transition are two different, but related concepts. Definition A.1 defines the transition of (A.1) from a stable equilibrium point to other states (not necessary equilibrium state). In general, we can define transitions from one attractor to another as follows.
Let the eigenvalues (counting multiplicity) of L λ be given by The following theorem is a basic principle of transitions from equilibrium states, which provides sufficient conditions and a basic classification for transitions of nonlinear dissipative systems. This theorem is a direct consequence of the center manifold theorems and the stable manifold theorems; we omit the proof.
Theorem A.1. Let the conditions (A.4) and (A.5) hold true. Then, the system (A.1) must have a transition from (u, λ) = (0, λ 0 ), and there is a neighborhood U ⊂ X of u = 0 such that the transition is one of the following three types: (1) Continuous Transition: there exists an open and dense set U λ ⊂ U such that for any ϕ ∈ U λ , the solution u λ (t, ϕ) of (A.1) satisfies In particular, the attractor bifurcation of (A.1) at (0, λ 0 ) is a continuous transition.
(2) Jump Transition: for any λ 0 < λ < λ 0 + ε with some ε > 0, there is an open and dense set U λ ⊂ U such that for any ϕ ∈ U λ , where δ > 0 is independent of λ. This type of transition is also called the discontinuous transition. (3) Mixed Transition: for any λ 0 < λ < λ 0 + ε with some ε > 0, U can be decomposed into two open sets U λ 1 and U λ 2 (U λ i not necessarily connected): With this theorem in our disposal, we are in position to give a new dynamic classification scheme for dynamic phase transitions.
Definition A.1 (Dynamic Classification of Phase Transition). The phase transitions for (A.1) at λ = λ 0 is classified using their dynamic properties: continuous, jump, and mixed as given in Theorem A.1, which are called Type-I, Type-II and Type-III respectively.
An important aspect of the transition theory is to determine which of the three types of transitions given by Theorem A.1 occurs in a specific problem. By reduction to the center manifold of (A.1), we know that the type of transitions for (A.1) at (0, λ 0 ) is completely dictated by its reduction equation near λ = λ 0 , which can be expressed as: where J λ is the m × m order Jordan matrix corresponding to the eigenvalues given by (A.4), Φ(x, λ) is the center manifold function of (A.1) near λ 0 , P : X → E λ is the canonical projection, and is the eigenspace of L λ . By the spectral theorem, (A.6) can be expressed into the following explicit form where (A.8) g(x, λ) = (g 1 (x, λ), · · · , g m (x, λ)), Here e j and e * j (1 ≤ j ≤ m) are the eigenvectors of L λ and L * λ respectively corresponding to the eigenvalues β j (λ) as in (A.4).
When x = 0 is an isolated singular point of g k (x, λ), in general the transition of (A.1) is determined by the first-order approximate bifurcation equation of (A.10) as follows: The following theorem is useful to distinguish the transition types of (A.1) at (u, λ) = (0, λ 0 ).
where W s is the stable set, W u is the unstable set, and D is the hyperbolic set of (A.7). Then we have the following theorem. (1) The transition of (A.1) at (u, λ) = (0, λ 0 ) is continuous if and only if u = 0 is locally asymptotically stable at λ = λ 0 , i.e., the center manifold is stable: M c = W s . Moreover, (A.1) bifurcates from (0, λ 0 ) to minimal attractors consisting of singular points of (A.1). (2) If the stable set W s of (A.1) has no interior points in M c , i.e., M c = W u +D, then the transition is jump.
Let Φ(x, λ) be the center manifold function of (A.1) near λ = λ 0 . We assume that where k ≥ 2 an integer and α = 0 a real number.  (1) (A.1) has a mixed transition from (0, λ 0 ). More precisely, there exists a neighborhood U ⊂ X of u = 0 such that U is separated into two disjoint open sets U λ 1 and U λ 2 by the stable manifold Γ λ of u = 0 satisfying the following properties: (a) U = U λ 1 + U λ 2 + Γ λ , (b) the transition in U λ 1 is jump, and (c) the transition in U λ 2 is continuous. The local transition structure is as shown in Figure A.3.
(2) (A.1) bifurcates in U λ 2 to a unique singular point v λ on λ > λ 0 , which is an attractor such that for any ϕ ∈ U λ 2 , lim where u(t, ϕ) is the solution of (A.1). A.3. Singular Separation. In this section, we study an important problem associated with the discontinuous transition of (A.1), which we call the singular separation.
(1) An invariant set Σ of (A.1) is called a singular element if Σ is either a singular point or a periodic orbit.
To derive a general time-dependent model, first we recall that the classical le Châtelier principle amounts to saying that for a stable equilibrium state of a system Σ, when the system deviates from Σ by a small perturbation or fluctuation, there will be a resuming force to restore this system to return to the stable state Σ. Second, we know that a stable equilibrium state of a thermal system must be the minimal value point of the thermodynamic potential.
By the mathematical characterization of gradient systems and the le Châtelier principle, for a system with thermodynamic potential H(u, λ), the governing equations are essentially determined by the functional H(u, λ). When the order parameters (u 1 , · · · , u m ) are nonconserved variables, i.e., the integers Ω u i (x, t)dx = a i (t) = constant. and φ j (u, λ) is a function depending on the other components u i (i = j). When