Derivation and analysis of a phase field model for alloy solidification [Elektronische Ressource] / vorgelegt von Björn Stinner

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• Physik

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• Publié par: universitat_regensburg
• Publié le: 01 janvier 2006
• Nombre de lectures: 10
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Derivation and Analysis of a
Phase Field Model
for Alloy Solidification
Dissertation zur Erlangung des
Doktorgrades der Naturwissenschaften
(Dr. rer. nat.)
der Naturwissenschaftlichen Fakult¨at I - Mathematik
der Universit¨at Regensburg
vorgelegt von
Bj¨orn Stinner
Regensburg, Oktober 2005Promotionsgesuch eingereicht am 10. Oktober 2005
Die Arbeit wurde angeleitet von Prof. Dr. H. Garcke
Pru¨fungsausschuss: Vorsitzender: Prof. Dr. Jannsen
1. Gutachter: Prof. Dr. Garcke
2. Gutachter: Priv.-Doz. Dr. Eck
weiterer Pru¨fer: Prof. Dr. Finster ZirkerContents
1 Alloy Solidification 11
1.1 Irreversible thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1 Thermodynamics for a single phase . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.2 Multi-phase systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.3 Derivation of the Gibbs-Thomson condition . . . . . . . . . . . . . . . . . . . 18
1.2 The general sharp interface model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3 Non-negativity of entropy production . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4.1 Phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4.2 Mass diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Phase Field Modelling 33
2.1 The general phase field model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 Non-negativity of entropy production . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.1 Possible choices of the surface terms . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.2 Relation to the Penrose-Fife model . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.3 A linearised model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.4 Relation to the Caginalp model . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.5 Relation to the Warren-McFadden-Boettinger model . . . . . . . . . . . . . . 43
2.4 The reduced grand canonical potential . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.1 Motivation and introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.3 Reformulation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Asymptotic Analysis 49
3.1 Expansions and matching conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 First order asymptotics of the general model . . . . . . . . . . . . . . . . . . . . . . 55
3.2.1 Outer solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.2 Inner expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.3 Jump and continuity conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.4 Gibbs-Thomson relation and force balance . . . . . . . . . . . . . . . . . . . . 60
3.3 Second order asymptotics in the two-phase case . . . . . . . . . . . . . . . . . . . . . 64
3.3.1 The modified two-phase model . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3.2 Outer solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.3 Inner solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3.4 Summary of the leading order problem and the correction problem . . . . . . 69
3.4 Numerical simulations of test problems . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.1 Scalar case in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.2 Scalar case in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.3 Binary isothermal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
33.4.4 Binary non-isothermal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 Existence of Weak Solutions 79
4.1 Quadratic reduced grand canonical potentials . . . . . . . . . . . . . . . . . . . . . . 81
4.1.1 Assumptions and existence result . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1.2 Galerkin approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.1.3 Uniform estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.1.4 First convergence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.1.5 Strong convergence of the gradients of the phase fields . . . . . . . . . . . . . 90
4.1.6 Initial values for the phase fields . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.1.7 Additional a priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Linear growth in the chemical potentials . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2.1 Assumptions and existence result . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2.2 Solution to the perturbed problem . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2.3 Properties of the Legendre transform . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.4 Compactness of the conserved quantities . . . . . . . . . . . . . . . . . . . . . 104
4.2.5 Convergence statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.3 Logarithmic temperature term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3.1 Assumptions and existence result . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.2 Solution to the perturbed problem . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3.3 Estimate of the conserved quantities . . . . . . . . . . . . . . . . . . . . . . . 114
4.3.4 Strong convergence of temperature and chemical potentials . . . . . . . . . . 118
4.3.5 Convergence statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A Notation 123
B Equilibrium thermodynamics 125
C Facts on evolving surfaces and transport identities 129
D Several functional analytical results 131
4Introduction
The subject of the present work is the derivation and the analysis of a phase field model to de-
scribesolidificationphenomenaonamicroscopiclengthscaleoccurringinalloysofiron,aluminium,
copper, zinc, nickel, and other materials which are of importance in industrial applications. Me-
chanicalproperties of castings and the quality of workpiecescan be traced back to the structure on
an intermediate length scale of some μm between the atomic scale of the crystal lattice (typically
of some nm) and the typical size of the workpiece. This so-called microstructure consists of grains
which may only differ in the orientation of the crystal lattice, but it is also possible that there are
differences in the crystalline structure or the composition of the alloy components. In the first case
the system is named homogeneous, in the latter case heterogeneous. The homogeneous parts in
heterogeneous systems are named phases. These phases itself are in thermodynamic equilibrium
but the boundaries separating the grains of the present phases are not in equilibrium and comprise
excess free energy. Following [Haa94], Chapter 3, the microstructure is defined to be the totality of
all crystal defects which are not in thermodynamic equilibrium.
The fact that the thermodynamic equilibrium is not attained results from the process of solid-
ification. When a melt is cooled down solid germs appear and grow into the liquid phase. The
type of the solid phase and the evolution of the solid-liquid phase boundaries depends on the local
concentrations of the components and on the local temperature. But also the surface energy of
the solid-liquid interface plays an important role. Not only the typical size of the microstructure is
determined by the surface energy. Its anisotropy, together with certain (possibly also anisotropic)
mobilitycoefficients, andthefactthatthesolid-liquidinterfaceisunstableleadstotheformationof
dendrites as in Fig. 1. The properties as the number of tips, the tip velocity, and the tip curvature
are of special interest in materials science.
During the growth, the primary solid phases can meet forming grain boundaries which involve
surface energies of their own. In eutectic alloys, lamellar eutectic growth as in Fig. 2 on the left
can be observed, i.e., layers of solid phases enriched with two different components grow into a
melt of an intermediate composition. The strength and robustness of workpieces thanks to that
fine microstructure make such alloys of particular interest in industry. The typical width of the
grains and its dependence on composition and cooling rate is of interest as well as the appearance
of patterns like, for example, eutectic colonies (cf. Fig. 2 on the right). At an even later stage
of solidification, when essentially the whole melt is solidified, coarsening and ripening processes
involving a motion of the grain boundaries on a larger timescale are observed.
In the following, the distinction between phase and grain will be dropped, and the notation
”phase”will be used for an atomic arrangementin thermodynamic equilibrium as well as a domain
occupied by a certain phase, i.e., a grain of the phase. As a consequence, the notation ”phase
boundary” will be used for int