Abstract

Point defects created by the collisions of high-energy particles and their subsequent evolutions form the foundation for all observed irradiation effects. Qualitative analysis is performed for the local and global behaviors of the planar system of nonlinear ordinary differential equations for the point defects balance model. The results indicated that the evolution behavior for the in-pile process is qualitatively very similar to the more simple annealing process, but very different from the degenerated systems that possess analytical solutions. However, quantitatively, irradiations in the in-pile process will shift the stable node away from the defect free state and change the local behaviors. A too strong irradiation may result in a nonphysical stable node and produce amorphous states, thus making the model inadequate.

1. Introduction

In irradiate environments, solid atoms can be displaced from their lattice sites and become interstitial atoms by the collisions of high-energy particles [1, 2]. The generated point defects (vacancies and interstitials, known as the Frenkel pairs) and their subsequent evolutions form the foundation for all observed irradiation effects on the physical and mechanical properties of materials. Notably irradiation-induced mechanical degradations include the volumetric swelling, creep instabilities, increase in hardness, loss of ductility, and also cracking [1, 3]. Thus, a proper understanding of their evolution behavior is at the central of the radiation damage.

Experimental and theoretical studies have shown that the point defects evolution is mainly governed by three different mechanisms, the production rate, the recombination rate, and the annihilation rate [4–12]. The rate of point defects production depends on the irradiation conditions and the material types. The irradiation generated pairs of vacancies and interstitials can get lost due to their recombination and the recombination rate is mainly determined by their diffusions. They can also be lost to sinks such as voids, dislocation loops, and grain boundaries. The annihilation rates depend on their diffusions and the sink concentrations and strengths [1, 13].

By using the defect reaction rate theory, a planar system of nonlinear ordinary differential equations (ODEs), named the point defect balance equations (PDBEs), has been proposed by Lomer [9] for the evolution of the spatially averaged point defects concentrations (, ) for materials under irradiations. The system is quadratic in and with a constant production rate and four additional material constants representing the recombination and the annihilation mechanisms, respectively. The system has been used widely as a standard model for the evolution behavior of the irradiation-induced point defects. Analytical solutions were obtained for several degenerated systems in which some mechanisms are neglected and the corresponding material constants are assumed null. Although such analytical solutions are very helpful, they cannot be directly extrapolated to the original nondegenerated (UD) system. Many numerical solutions have been obtained and can provide useful information [5, 14, 15]. However, a proper understanding of the solution behaviors of PDBEs and the effect of the three different mechanisms are still lacking.

In this paper, the qualitative (geometrical) method of ODEs [16, 17] is utilized to study the solution behavior of PDBEs. After introducing the planar system, the singular points are obtained and their local behaviors are discussed in the next section. In Section 3, the global behavior is analyzed in the phase plane. Considering the behavior at infinity, the global phase portraits are also obtained on the Poincaré disk. Possible physical admissible conditions are discussed in Section 4. A numerical example is also presented before the conclusion to illustrate the difference between degenerated and nondegenerated systems.

2. Balance Equations and Local Behaviors Near Singular Points

2.1. Point Defects Balance Equations (PDBEs)

The Frenkel defects are generated by collisions cascade and can be lost through either recombination or reaction with defect sinks. So the defect concentration of vacancy and interstitial is the balance between (1) local production rate, (2) reaction with other species, and (3) diffusion into or out of the local volume. The balance equations of the irradiation-induced point defects are [1, 5]where(1) is the atom fraction of vacancy/interstitial, or vacancy/interstitial concentration;(2) is the defect production rate by radiation;(3) is the recombination rate coefficient of vacancy and interstitial;(4) is the vacancy/interstitial-sink reaction rate coefficients;(5) is the sink density.

The parameters , , , and in (1) are nonnegative constants. And we have for sinks absorb interstitials at faster rates than vacancies.

Let and ; if , system (1) becomeswhere

For simplicity, the superscript of , , and in the following contents will be neglected.

We will discuss the following two different processes separately.

(1) Annealing: , i.e., , no point defect source. Thus, the defects could only annihilate by recombination or in sinks.

(2) In-pile irradiation: , i.e., . Thus, the point defects are generated as the material is exposed in irradiate environments.

2.2. Singular Points and Local Behaviors

2.2.1. Annealing:

First, we will discuss the properties of the singular points and the local behaviors near them during annealing process. In system (2), by letting , we can get the singular points.

Theorem 1. If and , system (2) has two singular points

The local behaviors near the two singular points are easily obtained as follows.

Theorem 2. If and , the singular point (4) is a stable node. And the local behaviors of system (2) near the stable node areand the trajectories satisfy

Proof. At , the Jacobi matrix of system (2) isBoth eigenvalues of are negative real number and is a stable node. The local behaviors could be easily obtained.

The phase portraits are shown in Figure 1(a); the decrease of is much faster than , because sinks absorb interstitials at faster rates than vacancies, i.e., . Besides, as shown in Figure 1(b), versus under the different value of , it is easy to find that the smaller leads to the steeper curve for the faster decrease of .

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Figure 1 

(a) Phase portraits near stable node and (b) versus under different if .

Theorem 3. If and , the singular point (4) is a saddle point. And the local behaviors of system (2) near the saddle point arewhereand the trajectories satisfy

Proof. At , the Jacobi matrix of system (2) isThe eigenvalues of are : one is a positive and the other is a negative real number, so is a saddle point. And the local behaviors could be easily obtained.

The phase portraits are shown in Figure 2(a). For the same constant, the trajectories of different in the first quadrant are shown in Figure 2(b). It could be found that the larger leads to the larger slope of the asymptotic line and the larger for the same .

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Figure 2 

(a) Phase portraits near saddle point and (b) local trajectories under different if .

2.2.2. In-Pile Irradiation:

After introducing the source term, the singular points become as follows.

Theorem 4. If and , system (2) has two singular points

Compared with the annealing process, after introducing the source term, we know that the amount of the singular point has not been changed. But the position of the stable node is shifted to the first quadrant. And is still in the third quadrant. The local behaviors near the singular points are as follows.

Theorem 5. If and , the singular point (12) of system (2) is a stable node in the first quadrant, and the local behaviors near the stable node arewhere and the eigenvalues and eigenvectors of the Jacobi matrix arewhere

Proof. At (12), the Jacobi matrix of system (2) isThe trace and the determinant of areSo we have , . And it is easy to find that for and . The eigenvalues of Jacobi matrix areBy definition, we have and . So the singular point is a stable node.

Theorem 6. If and , the singular point (12) of system (2) is a saddle point in the third quadrant. And the local behaviors near the saddle point arewhere and the eigenvalues and eigenvectors of the Jacobi matrix arewhere

Proof. Similarly, at (12), the eigenvalues of the Jacobi matrix arewhereIt is easy to find that and . Besides, , , and . So the singular point is a saddle point.

As shown in Figure 3, in the in-pile process, or will no longer approach or deviate the singular points along the coordinate axis. Because the eigenvectors are different from the annealing process, also, decreases faster than for sinks absorb interstitials at faster rates than vacancies. The trajectories tend to the eigenvector near the stable node . The trajectories tend to near the saddle point .

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Figure 3 

Phase portraits near (a) stable node and (b) saddle point if .

Next, we analyze the effects of defects production rate on the eigenvalues and eigenvectors. As shown in Figure 4, with the increase of the defects production rate , will always decrease. But will decrease first and then remain almost constant for large enough. Define as the angle between and axis. We found that, with the increase of , will decrease first and then remain almost constant for large enough. The effects of on and are similar, as shown in Figure 5.

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Figure 4 

(a) Eigenvalue and (b) angle between eigenvector and axis versus .

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Figure 5 

(a) Eigenvalue and (b) angle between eigenvector and axis versus .

2.3. Some Degenerated Systems

Under some extreme conditions, one or more effects in system (2) are small enough and can be neglected. We get some degenerated systems.

(1) Sink-Free Limit (SFL): , i.e., , negligible defects-sink reactions. Thus, the point defects can only annihilate due to recombination and system (2) becomes symmetric.

(2) Low Temperature Limit (LTL): , i.e., , negligible vacancy-sink reaction. Thus, the vacancies can only annihilate due to recombination.

(3) High Temperature Limit (HTL): . The defect annihilation rate at the sinks keeps the concentration of interstitial low, such that recombination does not contribute much.

Compared to UD, the singular points degenerate to singular lines in SFL.

Theorem 7. (i) If , the singular lines of system (2) are and ; ii) if and , the singular lines of system (2) are .

Theorem 8. If , the singular lines of system (2) are asymptotically stable only if and unstable if . is a saddle-node.

Corollary 9. If , the trajectories of system (2) satisfy , where . And the solutions are as follows:
(i) if and ,(ii) if and ,where ,
(iii) if , the solutions have the same form as (30), but .

For SFL system, the balance equations (2) are symmetric. Lines and are the same, which is . The stable node degrades to asymptotic stable part of the singular line , and the saddle-node degrades to unstable part of . The local behaviors near the singular lines are shown in Figure 6.

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Figure 6 

Phase portraits near singular lines of system (2) if (a) and (b) .

If only vacancy-sink reaction is negligible, the singular line is quite different compared to the previous system.

Theorem 10. If , , the singular line of system (2) is if . System (2) does not have any singular point if .

Corollary 11. If , , and , the singular line is asymptotically stable only if and unstable if . And the trajectories satisfy

For LTL system, for vacancy-sink reactions are too small and negligible, we have and . During annealing process, the stable node degrades to asymptotic stable part of the singular line , and the saddle-node degrades to unstable part . In in-pile process, the stable node and the saddle-node could not be found. The local behaviors near the singular line are shown in Figure 7.

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Figure 7 

Phase portraits near singular lines or points of system (2) if (a) and (b) .

At high temperatures, the recombination does not contribute much and could be neglected. System (1) becomes a linear system, and we can easily get the following analytical solutions.

Theorem 12. If , and
(i) , the only stable node is and the solutions arethe trajectories satisfy(ii) , there is no singular point; the solutions and the trajectories are(iii) , , and , the singular line is if and there is no singular point if . The solutions are

3. Global Behaviors and Phase Plane Portraits

3.1. Nonexistence of Closed Trajectories

Before we discuss the global behaviors of system (2), we should prove the nonexistence of closed trajectory first.

Lemma 13. There is no closed trajectory of system (2) that can cross the or axis.

Proof. At axis, , we have in system (2). And always increases if and unchanged if .
Assuming there exists a closed trajectory crossing axis, must decrease at one point and increase at another point, and is no longer satisfied, such that there is no closed trajectory crossing axis. Similarly, there is no closed trajectory crossing axis.

Corollary 14. The closed trajectory of system (2) must be completely inside one of the quadrants of plane; i.e.,

Theorem 15. If and , system (2) has no closed trajectory.

Proof. If , the stable node is and the saddle point is . By Lemma 13, the closed trajectory could only exist inside the quadrant which contains singular point. Besides, the singular point in is a saddle point, so system (2) has no closed trajectory.

Theorem 16. If and , system (2) has no closed trajectory.

Proof. If , the stable node and the saddle point . There is no closed trajectory in , , and . Inside , we haveSo there is no closed trajectory in .

3.2. Phase Plane Portraits

If , the phase plane portraits of system (2) are shown in Figure 8. For annealing () and in-pile () process, the portraits are qualitative similar. Only if and are small enough (lower than the asymptotic line ), the trajectories will approach to negative infinity. Others will approach to the stable node (12).

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Figure 8 

Phase plane portraits of system (2) with and (a) and (b) .

The phase plane portraits of SFL system are shown in Figure 9. During annealing process, as shown in Figure 9(a), in the first quadrant, if , the trajectories evolve to axis and only vacancies are left at last. Otherwise if , only interstitials are left at last. Only if , the system becomes defect free in the end. During the in-pile process, for the introducing of source term, both defects always exist at last.

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Figure 9 

Phase plane portraits of system (2) with and (a) and (b) .

The phase plane portraits of LTL system are shown in Figure 10. During annealing process, in the first quadrant, if , the trajectories evolve to axis and only vacancies are left at last. And if , the trajectories will evolve to origin, as shown in Figure 10(a). But during in-pile process, for there is no singular point, we could not find any stable phenomenon in the finite condition, and more information will be presented in the next section.

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Figure 10 

Phase plane portraits of system (2) with and (a) and (b) .

Phase plane portraits of HTL system are shown in Figure 11. If , the stable node is , so any trajectory will approach when .

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Figure 11 

Phase plane portraits of system (1) if and (a) , (b) , (c) , and (d) .

The global phase portraits for annealing and in-pile processes cannot be easily constructed from the phase plane behaviors of degenerated systems.

3.3. Global Phase Trajectory on Poincaré Disk

Next, we analyze the behaviors at infinity of system (2). First, transform and in Poincaré and timeSystem (2) becomesLet and the singular points of system (39) are

Theorem 17. If , the singular point of system (39) is a saddle-node. And is an unstable node.

Proof. At , the Jacobi matrix of (39) isThe trace of (41) is and the determinant , so is nonhyperbolic singular point. Similarly, at , the trace is and the determinant , so is an unstable node.
For nonhyperbolic singular point , system (39) can be written asLetting , we haveInserting (43) into (42b), we have the expansionSo, if
(1) , the minimal order of in is and is a saddle-node;
(2) and , it has and is an unstable node.