Renormalization Theory for Heterogeneous Materials

Professor John Wilkins

Ohio Sate University

Department of Physics

174 West 18^th Avenue

Columbus, OH 43210

(614) 292-5193

(614) 688-3871 FAX

wilkins@pacific.mps.ohio-state.edu

The renormalization group was first developed for understanding second-order phase transitions. The method was driven by the observation that sufficiently close to the phase transitions properties could be described by a universal free-energy with only a few degrees of freedom. In contrast, the simplest model Hamiltonian had infinite degree of freedom in the thermodynamic limit. The challenge was to reduce such models to a few degrees of freedom that could quantitatively describe the phase transition.

In the renormalization group method, the degrees of freedom are reduced by iteratively integrating out the highest energy states. At each stage new terms in the model could arise. If the procedure were stable, the form of the model stopped evolving after a few iterations, although the coefficients of the terms in the model were continuously renormalized. Mostly, evolved forms of the model could be anticipated by limiting cases of the original, unrenormalized model. Occasionally and more interestingly, the renormalized model developed an unexpected term.

A special role for the renormalization group is deriving models effective on larger, more relevant length scales. In phase transitions, that model is the Ginzburg-Landau free energy -- a set of terms in some continuous variable characterizing the possible phases. The coefficients multiplying these terms can be derived from the renormalization group.

The Ginzburg-Landau approach is widely used by materials scientists to model the spatial distribution of phases on the micron or microstructural scale. For example, GL amazingly mimics the discontinuous rafting structure observed in nickel-based superalloys.

Typically the GL free energy will depend on both the concentration and the order-parameter, itself dependent on concentration. The GL form is chosen on phenomenological grounds with parameters adjusted to fit the material. Concentration dynamics are described by the diffusion equation with the diffusion constant, either fit or extracted from experiments on homogeneous materials. GL dynamics are phenomenological.

The natural question a physicist asks is: would it be possible to derive these equations from a microscopic starting point? Just thinking about that question inspires the suggestion of a new field: renormalized theory for heterogeneous materials. Consider these topics:

Defected single-phase. Even a single-phase material contains defects: point (interstitial, vacancy), line (dislocations) and surface(differently oriented crystals). In any renormalization process how are these structures handled? Most have high energies and so their degrees of freedom should be integrated out. The resulting single-phase must have a description that somehow reflects the effects of these defects.

Multi-phase material. More interestingly, multi-phase materials also have defects that sometimes stay in one phase and sometimes pass through both phases. Such questions are currently ignored in the phenomenological GL descriptions. But this is a mistake. For example, for Ni-based superalloys, mechanical properties depend on the motion of dislocations through the gamma and gamma-prime "coherent" material.

Dissipation. Current renormalization theory neglects how dissipation properties develop. In real materials dissipation in the mechanical and thermal processes affect how the phase develops and their resulting properties. In integrating out high-energy degrees of freedom, can renormalization theory develop terms corresponding to dissipation on the micron scale?

I am not arguing that renormalization theory is necessary to develop effective models at the micron scale but it does provide a physics-oriented way of stating the problems.