Grain boundary diffusion model

Figure 9. Plot of apparent temperature recorded between a mineral with slow oxygen diffusion (pyroxene) and a mineral with fast diffusion (plagioclase) vs. the modal percentage of plagioclase in a bi-mineral rock. Cooling is at 5°C/m.y. after peak equilibration at 750°C. In feldspar dominated rocks, Ta may preserve peak temperature, but for a rock dominated by pyroxene, Ta can be hundreds of degrees below peak T approaching 250°C, Tc(plag). The mode effect is modeled by the Fast Grain Boundary diffusion model and predicts that refractoiy accessory mineral (RAM) thermometers are most rehable (from Eiler et al. 1993).

Figure 10. Contour plots of apparent temperatures and fractionations (%o in italics) as a function of modal percentages in an idealized three-mineral rock composed of quartz, feldspar, and hornblende, predicted by the Fast Grain Boundary diffusion model to be recorded by (a) quartz and hornblende, (b) feldspar and hornblende, and (c) quartz and feldspar. These figures show the relation of discordance and the mode of each mineral in a rock (from Eiler et al. 1993).

Question The goal of this problem is to use the simple grain bulk/grain boundary diffusion model discussed above to estimate the temperature above which bulk diffusion becomes more important than grain boundary diffusion. 

Fig. 1. Shrinkage isotherms for Alcoa A-14 plotted according to the grain boundary diffusion model.

Figure C2.11.6. The classic two-particle sintering model illustrating material transport and neck growtli at tire particle contacts resulting in coarsening (left) and densification (right) during sintering. Surface diffusion (a), evaporation-condensation (b), and volume diffusion (c) contribute to coarsening, while volume diffusion (d), grain boundary diffusion (e), solution-precipitation (f), and dislocation motion (g) contribute to densification.

Several models for diffusive transport in and among minerals have been discussed in the literature one is the fast grain boundary (FGB) model of Eiler et al. (1992, 1993). The FGB model considers the effects of diffusion between non-adjacent grains and shows that, when mass balance terms are included, closure temperatures become a strong function of both the modal abundances of constituent minerals and the differences in diffusion coefficients among all coexisting minerals. 

The constitutive equation of the A-V model, when lattice and grain boundary diffusion are taken into account, is written ... 

Gribb, T. T. Cooper, R. F. 1998. Low-frequency shear attenuation in polycrystalline oUvine grain boundary diffusion and the physical significance of the Andrade model for viscoelastic rheology. Journal of Geophysical Research, 103, 27 267-27 279. 

Figure 4.10 A schematic model for tracer diffusion in a compacted nanocrystalline sample. Dg isthe bulk lattice diffusion coefficient, Dg/, is the grain boundary diffusion coefficient, Da is the inter-agglomerate diffusion coefficient, tf is the grain size, 5 is the width ofthe grain boundary, and 5a is the separation between agglomerates.

If the mechanism of material transport in sintering is grain boundary diffusion, the viscosity can be expressed as a function of the temperature and the grain size. From a model for boundary diffusion controlled creep proposed by Coble [7] and the temperature dependence of diffusion coefficients, we obtain... 

Figure 10.15 Geometric constructions used to model porosity elimination, assuming (a) bulk diffusion, where the vacancies are assumed to be eliminated at the grain surface, and (6) grain boundary diffusion, where vacancies are restricted to diffusing along a grain boundary of width gb-...

Watson EB, Cherniak DJ (1997) Oxygen diffusion in zircon. Earth Planet Sci Lett 148 527-544 Wendlandt RW (1991) Oxygen diffusion in basalt and andesite melts Experimental results and discussion of chemical versus tracer diffusion. Contrib Mineral Petrol 108 463-471 West AR (1984) Solid State Chemistry and Its Applications. John Wiley and Sons, New York Whipple RTP (1954) Concentration contours in grain boundary diffusion. Phil Mag 45 1225-1236 White AF, Peterson MI (1990) Role of reactive-surface area characterization in geochemical kinetic models. In Melchior DC, Bassett RL (eds) Chemical Modeling of Aqueous Systems. II. Am Chem Soc Symp 416 461-475... 

Herzig C, Mishin Y (2005) Grain boundary diffusion in metals. In Heitjans P, Kclrger J (eds) Diffusion in condensed matter methods, materials, models. Springer, Berlin, p 337... 

Fig. 3.11. Model for elimination of porosity. Vacancy flux by a) grain boundary diffusion, (b) volume diffusion.

Because it has been assumed in the model that the pore geometry is uniform, the nondensifying mechanisms are not present. This is attributed to the fact that the chemical potential is the same everywhere on the surface of the pores. Therefore, only two densifying mechanisms should be considered lattice difiusion and grain boundary diffusion. 

The geometrical model for lattice diffusion can be modified to derive the flux equations for grain boundary diffusion, which is... 

Figure 5.20 shows a model consisting of a row of cylinders with same radius, whose neck contours are shown in Fig. 5.21 [40]. Two situations are simulated for matter transport by (i) surface diffusion only and (ii) both surface diffusion and grain boundary diffusion, as shown in Fig. 5.21a, b, respectively. It is found that numerical simulation predicts undercutting and a continuous change in curvature of the neck surface, which is different from that given by the analytical models with the circle approximation. The region of the neck surface influenced by the matter transport also extends far beyond that predicted by the circle approximation. However, if both the surface diffusion and grain boundary diffusion are considered, the extension is less pronounced.

A simple geometrical model consisting of spherical particles with a diameter a, arranged as a simple cubic pattern, as shown in Fig. 5.26a, is used to derive the sintering equations for the mechanism of grain boundary diffusion [70]. It is... 

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