Spinodal Decomposition

Transcription

1 Spinodal Decomposition Lauren Ayers December 17, Introduction Spinodal decomposition is one process by which an alloy decomposes into equilibrium phases. It occurs when there is no thermodynamic barrier to the decomposition; thus, it occurs solely by diffusion. The process results in a very finely dispersed microstructure, exhibiting a characteristic wavelength. The evolution of the microstructure with spinodal decomposition results in changes of the materials properties of the alloy, such as strength, hardness, magnetism, fracture toughness, and ductility; while this can result in problems in industrial applications, such as in boiling water reactor (BWR) piping, the process of spinodal decomposition can be applied through carefully engineered aging processes to improve the properties of an alloy as well. 2 The Mechanism of Spinodal Decomposition The spinodal is defined as the point where the curvature of the Gibb s free energy with respect to concentration is zero, or: d 2 G dc 2 = 0 (1) [1]The condition for stability is that the curvature of the Gibb s free energy must be equal to or greater than zero; the unstable region is defined by the locus of the spinodal [1]. Above the spinodal region, or when the curvature of the Gibb s free energy is positive, the two phase region decomposes by nucleation and growth, and below this region, the two phase region decomposes by spinodal decomposition. A diagram of 1

2 Figure 1: Variation of the chemichal and coherent spinodal with composition. The region within which two phases are stable is called the miscibility gap. Reproduced from Reference [3] the spinodal can be seen in Figure 1. A concentration gradient causes uphill diffusion, or diffusion in the direction against the gradient, if the composition is inside the spinodal, and downhill diffusion otherwise; the change in free energy acts as the driving force for diffusion [2]. Spinodal decomposition is one mechanism by which an alloy decomposes into equilibrium phases. It was first proposed by Hillert s thesis in 1956 [2] as a method to explain the growth of a composition modulation in an alloy that begins with a homogenous compisition. Such a mechanism implies a negative diffusion coefficient. Hillert derived a new diffusion equation for cases where the third derivative of concentration is large; this equation accounts for the fact that a concentration gradient does not always cause downward diffusion. Hillert also developed a model from the zeroth approximation of nearest-neighbor interactions. The model predicts the existance of periodically modulated structures in ordering, as well as in precipitation systems [4] by deriving an expression for the free energy of the system. A diffusion equation was also derived, which took into account the discontinuity in composition between planes in crystalline solids [4]. This model predicts the formation of many wavelengths during the first stage of transformation [4]. 2

3 Figure 2: The progression of the sinusoidal composition profile through time. Modified from [8]. 2.1 The Evolution of Concentration Profiles During spinodal decomposition, the two phases separate from each other into a sinusoidal composition profile [5]. This takes place over two stages. First, a primary wavelength forms, and the difference in concentration between the two phases increases exponentially until the limit as set by the miscibility gap, or the two phase region where both phases are stable [6], is reached. If aging continues, the coarsness of the composition increases, but the concentration of each phase will be constant. The process is illustrated in Figure 2. This results in a sinusoidal composition profile, with a wavelength of approximately five nm or less [7]. Cahn determined that true spinodal decomposition possesses two properties. It occurs everywhere within the sample, with the exception of near structural imperfections where the rate or mechanism may be different, and the amplitude of composition fluctuations should grow continuously until a metastable equilibrium is reached, with a preferential amplification of certain wavelength components [1]. 3

4 Figure 3: A typical evolution sequence for the Cahn-Hilliard equation, where u represents the concentration. As time progresses the concentration approaches a sinusoidal composition profile, as that of spinodal decomposition. Reproduced from Figure 2 of Reference [10]. 2.2 The Cahn-Hilliard Equation Cahn and Hilliard determined a governing equation to describe the process of phase separation: [9] dc dt = D 2 (c 3 c γ 2 c) (2) where D is the diffusion coefficient, c is the concentration, and γ is the surface energy. If a linear analysis of the Cahn-Hilliard equation is performed, it suggests that spinodal decomposition occurs [10]. A typical evolution sequence can be seen in Figure 3. As time progresses, the concentration approaches a sinusoidal composition profile, as that of spinodal decomposition. An example of a numerical solution to the Cahn-Hilliard equation can be seen in Figure 4. The solution begins at an unstable concentration (a), and decomposes into two distinct phases with a characteristic length scale (through c). As time continues to progress, the length scale is coarsened while maintaining fixed phase concentration fractions [11]. 4

5 Figure 4: A numerical solution to the Cahn-Hilliard equation, demonstrating spinodal decomposition. The system begins at an unstable concentration (a), and decomposes into two distinct phases with a characteristic length scale (b, c); as time progresses, the length scale coarsens while maintaining fixed phase concentration fractions. Reproduced from Reference [KINETICS OF MATERIALS] The initial stage composition can be expressed by the equation : X X s = Acos(βX) (3) [12], where A is the amplitude of the fluctuating wave, and β = 2π λ is the wavenumber, characterized by a wavelength λ. The critical wavelength is given by: λ c = ( 8π2 κ d 2 G/dX 2 )1/2 [12], where κ is a materials constant known as the gradient energy coefficient. All wavelengths longer than this critical wavelength are unstable, and will amplify to trigger the spinodal decomposition process. For a one-dimensional system, the solution to the Cahn-Hilliard equation takes the form: c < c >= e R(β)t cos(βx) (4) where R(β) is the amplification factor [12]. The amplification factor can be plotted as a function of wave number or wavelength, to predict the wavelength that grows most rapidly with time. 5

6 Figure 5: The sequences of formation of a two-phase mixture by nucleation and growth (above) and spinodal decomposition (below). Reproduced from Figure 2 of [13] 2.3 Spinodial Decomposition vs. Nucleation and Growth Nucleation and growth is another method of decomposition into equilibrium phases; it occurs in the two phase region when the curvature of the Gibb s free energy is positive, or above the spinodal. Nucleation thus occurs in the metastable region of the phase diagram [11]. The system initiates a discontinuous phase transformation, as seen in Figure 5. Nucleation requires localized fluctuations in composition that are large enough that the free energy decreases from the chemichal driving force, enough to offset all terms that resist the transformation, such as the elastic strain energy. The transformation begins at discrete nucleation sites, and evolves by outward growth of the nuclei of the new phase, as seen in Figure 5. Nucleation and growth can thus be catagorized as a discontinuous phase transformation, where spinodal decomposition is a continuous phase transformation. This distinction occurs as spinodal decomposition arises from systems that are thermodynamically unstable, as is indicated by the curvature of the Gibb s free energy, where nucleation and growth occurs in metastable systems [11]. The evolution of the phase separation differs between the two processes. The second-phase structure that evolves by nucleation and growth results in more spherical nuclei, where spinodal decomposition has a worm-like or tweed-like interconnectivity [12]. 6

7 2.4 Spinodal Decomposition in Cubic Crystals Cahn furthered his research on spinodal decomposition by examining cubic crystals. He found that no anisotropy is introduced by the incipient surface introduced by cubic symmetry, and spinodal decomposition gives rise to plane waves primarily either on the {100} planes, or the {111} planes, depending on the relative concentrations [14]. The kinetics are such that after some time, the fastest growing waves predominate; this gives rise to almost pure sinusoidal fluctuations which lie in the <100> or <111> directions [14]. 3 Properties of Spinodal Alloys 3.1 Magnetic Properties Influence of Spinodal Decomposition on Magnetic Properties of Fe28Cr16Co Spinodal decomposition has been shown to result in the aquisition of magnetic properties; for example, in iron-chromium-cobalt alloys, the property of permanent magnetism is aquired through spinodal decomposition of ferromagnetic phase α1, the solid solution of alloying elements in α-iron in two spinodal phases: particles of phase α-1 enriched in cobalt, and the matrix phase α-2 enriched in chromium. Both phases conserve the initial BCC lattice observed in the original alloy. However, the difference in the chemichal compisitions in the two phases causes a difference in the lattice parameter, and the two phases accomodate through elastic internal stress which stabilise the orientation of elemetary magnetic domains imposed by an external magnetic field; this produces a permanent magnet. The anisotropy of the magnetic properties depend on the form of the elementary domains. It can be modified by either annealing in a magnetic field, or cold deformation. 7

8 o C Embrittlement - Application to Boiling Water Reactors Stainless steels in particular are particularly susceptible to embrittlement when aged at temperatures around 475 o C. This embrittlement occurs by spinodal decomposition. The ferrite phase of the steel decomposes into a chromium rich phase, known as the α phase, and an iron-rich phase, known as the α phase [15]. The process is known as the 475 o C embrittlement because it occurs most rapidly at 475 o C; however, the process also occurs at lower temperatures, particularly at the 288 C operating temperature of a boiling water reactor (BWR), where reductions in ductility have been observed after several tens of thousands of hours [16]. Stainless steels are used in the piping of boiling water reactors (BWR) because of their ability to retain their mechanical properties at high temperatures. When the piping in BWRs is welded, the chemistry is controlled to require a specified amount of δ-ferrite in the weld to prevent hot cracking; however, this δ-ferrite undergoes spinodal decomposition over time in a BWR environment. In addition to this, the α phase precipitates several orders of magnitude larger than the sinusoidal composition profile after extended aging [17]. This phase separation, and the resulting precipitation of the alpha phase, causes embrittlement of the weld [17]. Spinodal decomposition occuring in the stainless steel piping of BWRs is a concern not only for the embrittlement it causes; there are many other consequences to materials that undergo spinodal decomposition over time. It has been suggested that spinodal decomposition is largely responsible for the reduction in fracture toughness observed in BWR piping [16]. As a result, thermal aging of type 316L stainless steel welds shows an increase in hardness, yield strength and tensile strength, as well as a decrease in ductility [18]. The embrittlement caused by the precipitation of the α phase after spinodal decomposition can result in accelerated stress corrosion cracking (SCC) [18]. Spinodal decomposition of the ferrite phase can also result in a loss of corrosion resistance [15]. Considering that the typical lifetime of a BWR is at least years, the effect of spinodal decomposition on piping cannot be ignored. 3.3 Applications Spinodal decomposition can, however, be engineered to improve the properties of a material. Kim et al have designed a procedure to produce a bulk metallic glass matrix composite [19]. The process, which involved 8

9 two melting steps, resulted in a Zr-Ta binary alloy, which formed a two-phase mixture of Zr- and Ta-rich solid solution by spinodal decomposition. The composite showed an improvement of strength and plasticity, indicating the interfaces between the Ta-rich particles and the matrix formed by phase separation are strong enough to provide a high interfacial cohesion [19]. In metallic alloy systems, spinodal decomposition in Cu-Ni-Sn results in alloys that exhibit high performance characteristics, including high strength, resistance to elevated temperatures, and good tribological properties [12]. These properties are attained, yet the alloys retain the qualities of copper alloys, such as high thermal conductivity and resistance to corrosion. There are also single phase alloys, such as Nicomet [12], that gain their strength due to spinodal structure; in Nicomet, nickel and tin atoms are arranged in a wave-like configuration. References [1] J.W. Cahn. On spinodal decomposition. Acta Metallurgica, 9: , [2] M. Hillert. A Theory of Nucleation for Solid Metallic Solutions. Praca doktorska, Massachusets Institute of Technology, [3] R.J. Bishop R.E. Smallman. Modern Physical Metallurgy & Materials Engineering. Butterworth- Heinemann, [4] M. Hillert. A solid-solution model for inhomogenous systems. Acta Metallurgica, 9: , [5] W.C. Carter R.W. Balluffi, S. Allen. Kinetics of Materials. John Wiley & Sons, [6] R.E. Reed-Hill R. Abbaschian, L. Abbaschian. Physical Metallurgy Principles. Cengage Learning, [7] M.K. Miller et al. Spinodal decomposition in fe-cr alloys: Experimental study at the atomic level and comparison with computer models - i. introduction and methodology. Acta M, 1995: , 43. [8] W.C. Carter. Thermodynamics of materials, lecture 32 notes. MIT Open Courseware, [9] J.E. Hilliard J.W. Cahn. Free energy of a nonuniform system. i. interfacial free energy. Journal of Chemical Physics, 28: , [10] C.P. Grant. Spinodal decomposition for the cahn-hilliard equation. Communications in Partial Differential Equations, 18: , [11] S. Puri. Kinetics of Phase Transitions. CRC Press, [12] S.H. Risbud. Materials Processing Handbook, Chapter 4. CRC Press,

10 [13] F. Findik. Improvements in Spinodal Alloys from Past to Present. Materials & Design, 42: , [14] J.W. Cahn. On spinodal decomposition in cubic crystals. Acta Metallurgica, 10: , [15] M.R. da Silva J.M. Neto S. Pairis S.S.M. Tavares, R.F. de Noronha. 475C Embrittlement in a Duplex Stainless Steel UNS S Materials Research, 4, [16] T.R. Lucas. The Effect of Thermal Aging and Boiling Water Reactor Environment on Type 316L Stainless Steel Welds. Praca doktorska, Massachusetts Institute of Technology, [17] H.M. Chung, O.K. Chopra. Kinetics and Mechanism of Thermal Aging Embrittlement of Duplex Stainless Steels. June [18] R.G. Ballinger J.H. Kim. Stress Corrosion Cracking Crack Growth Behavior of Type 316L Stainless Steel Weld Metals in Boiling Water Reactor Environments. NACE, [19] M.U. Kim et. al. Application of spinodal decomposition to produce metallic glass matrix composite with simultaneous improvement of strength and plasticity. Met. Mater. Int., 15: ,