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Modeling Precipitation as a Sharp-Interface Phase Transformation
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Modeling Precipitation as a Sharp-Interface Phase Transformation - - E r n s t Kozeschnik
During phase transformations, a new phase grows at the expense of an existing phase. The new phase and the existing phase, or parent phase, are distinguished by either a different state of matter (e.g., liquid water droplets in a saturated H 2 0 vapor, solid crystals in a liquid melt), different crystal structure (e.g., solid bcc iron (ferrite) in solid fcc iron (austenite) in steels) and/or chemical composition (e.g., coherent L12-ordered fcc Ni3A1 precipitates in an fcc Ni-A1 alloy). In this chapter, it is assumed that the growing and the shrinking phases are clearly and unambiguously separated by a phase boundary, the interface. The thickness of this interface is considered to be infinitely small (sharp-interface limit). This assumption is in contrast to what is assumed, for instance, in the phase-field model, which is discussed in Chapter 7, and where the transition from one phase into the other is assumed to be continuous. The term sharp implies that the interfacial region is sufficiently small to allow for a theoretical treatment of phase transformations with a stepwise change of material properties from one phase into the other. During phase transformation, the interface between the two phases migrates from the growing phase into the parent phase. Simultaneously, physical processes such as transport of energy (heat conduction) or transport of matter (atomic diffusion) occur in the bulk of the two phases as well as across the interface. This fact makes sharp-interface phase transformations a classical moving boundary problem, a class of problems which is generally known as Stefan problem and which is named after the Austrian-Slovene physicist and mathematician Jo~ef Stefan (1835-1893) in the analysis of simultaneous liquid-solid interface movement and heat transfer during solidification of water (ice formation). Precipitation is a special case of phase transformation, where the spatial extension of the new phase is usually small (few nanometers to few micrometers) compared to the parent phase (micrometers to millimeters). Usually, the parent phase remains widely unaltered during the precipitation process. In this context, the parent phase is also often denoted as the matrix phase. Typical precipitates can be intermetallic phases or oxides, carbides, and nitrides. The term "phase transformation" is often, yet not exclusively, used in the context of phase changes that occur on the scale of the polycrystalline microstructures, that is, typically micrometers to millimeters. Otherwise, in the context of this book, phase transformation and precipitation denote the same class of problem.
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Modeling Precipitation as a Sharp-Interface Phase Transformation - - E r n s t Kozeschnik
During phase transformations, a new phase grows at the expense of an existing phase. The new phase and the existing phase, or parent phase, are distinguished by either a different state of matter (e.g., liquid water droplets in a saturated H 2 0 vapor, solid crystals in a liquid melt), different crystal structure (e.g., solid bcc iron (ferrite) in solid fcc iron (austenite) in steels) and/or chemical composition (e.g., coherent L12-ordered fcc Ni3A1 precipitates in an fcc Ni-A1 alloy). In this chapter, it is assumed that the growing and the shrinking phases are clearly and unambiguously separated by a phase boundary, the interface. The thickness of this interface is considered to be infinitely small (sharp-interface limit). This assumption is in contrast to what is assumed, for instance, in the phase-field model, which is discussed in Chapter 7, and where the transition from one phase into the other is assumed to be continuous. The term sharp implies that the interfacial region is sufficiently small to allow for a theoretical treatment of phase transformations with a stepwise change of material properties from one phase into the other. During phase transformation, the interface between the two phases migrates from the growing phase into the parent phase. Simultaneously, physical processes such as transport of energy (heat conduction) or transport of matter (atomic diffusion) occur in the bulk of the two phases as well as across the interface. This fact makes sharp-interface phase transformations a classical moving boundary problem, a class of problems which is generally known as Stefan problem and which is named after the Austrian-Slovene physicist and mathematician Jo~ef Stefan (1835-1893) in the analysis of simultaneous liquid-solid interface movement and heat transfer during solidification of water (ice formation). Precipitation is a special case of phase transformation, where the spatial extension of the new phase is usually small (few nanometers to few micrometers) compared to the parent phase (micrometers to millimeters). Usually, the parent phase remains widely unaltered during the precipitation process. In this context, the parent phase is also often denoted as the matrix phase. Typical precipitates can be intermetallic phases or oxides, carbides, and nitrides. The term "phase transformation" is often, yet not exclusively, used in the context of phase changes that occur on the scale of the polycrystalline microstructures, that is, typically micrometers to millimeters. Otherwise, in the context of this book, phase transformation and precipitation denote the same class of problem.
179