Role of a Free Energy Action Principle in Predicting Microstructure Formation
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6fGdt=0"
G= f g (i, V71) dV V
ti
where the Lagrangian density is the free energy density g, and qIis an appropriate order parameter. To illustrate the universality of the formulation we survey in some detail its degree of equivalence to twenty-one currently adopted or promoted procedures for resolving pattern degeneracies, and critically reappraise theoretical descriptions of grain growth, ordinary diffusion, spinodal decomposition and Ostwald Ripening and their predictive success. The importance of a hierarchy of kinetic scaling laws is discussed. The physics of fluctuations is essential to understanding. INTRODUCTION The maturity of a branch of science can be associated with the validation of broadly encompassing action principles and variational methods since a maximum of predictive power is subsumed by their capability of dealing with singularities and by the efficiency of the mathematical symbolism concomitantly with a concise and transparent pedagogy. Examples are Hamilton's Principle of Least Action in Mechanics' (1834), Gibbs' Thermo2 (1878), Onsager's Principle of dynamic Principal of Minimum Free Energy (G) on the Isotherm 3 Maximum Path Probability in Irreversible Thermodynamics (1931), Pellew and Southwall's variational principle in hydrodynamics4' 5 (1940) and Feynman's Path Integral Method in Quantum Mechanics and Relativistic Quantum Field Theory6 (1965), all of which are relevant to the establishment ofthe following isothermal principle for dissipative heterogeneous systems in a classical or quantum manifold t2
68fGdt = 0;
G =f g(Tl(),V11,...)dV
II
(1)
V
where t 2-t 1 >> the collision time but otherwise small, g is the free energy density and special case of is not original' and is inclusive the generic Lagrangian in the variational calculus7 . This proposition 2 of the static case described by the Gibbs stability principle (2) 8G=0 The power of the method lies in the fact that from a general comprehension of system symmetries, statistical or otherwise (e.g., isotropy), one can write down g as a Taylor expansion, and this in turn according to its symmetries associates conserving currents and global invariants with the solutions
of the Euler-Lagrange equation(s) (Noether's Theorem9).
Sharing the general aim within Materials Science theory of generating predictive software
supporting new microstructural design and process development one seeks an adaptive phenomeno3 Mat. Res. Soc. Symp. Proc. Vol. 580 © 2000 Materials Research Society
logy which is as accurate and efficient as possible. In consideration of irreversible heterogeneous mechanical systems the theoretician immediately encounters the problems of order parameter specification and incomplete initial and boundary conditions, which forces her to confront what are often 0 designated as free boundaries" . In formal terms, the intuitively favoured order parameters and associated difference or differential equations define degeneracies,which is to say they allow a finite or infinite set of solutions which at worst can encompass a set
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