Phenomenological description of grain growth stagnation for nanocrystalline films and powders
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Eric Stach National Center for Electron Microscopy, Lawrence Berkeley Laboratory, Berkeley, California 94720
Joanna R. Groza Chemical Engineering/Materials Science Department, UC Davis, Davis, CA 95616 (Received 21 August 2000; accepted 26 January 2001)
We provide a phenomenological grain growth stagnation force incorporating a near-linear temperature dependence of stagnated grain sizes and irreversible growth. The resulting law captures the observation of the restart of grain growth in the size versus time plateau on temperature increases. This description also reduces to standard laws commonly used for data fitting. The law may be useful for workers who wish to characterize a nanocrystalline film or powder annealing process predictively from a limited number of measurements or may be useful in a designed experiment. Other laws are discussed and compared. Fits to size versus time data from the literature are successfully made. I. INTRODUCTION
For clarity, we must begin by providing reasonable definitions for two types of grain sizes. Stagnated grain size is grain diameter or radius where there has been a rapid and obvious reduction in the size versus time characteristics at a fixed temperature. Examples of this may be seen in Figs. 5 and 6. Endpoint grain size is the grain size reported at a specified time interval in either an experiment or a theoretical prediction. For nanocrystalline systems at fixed hold temperatures, grain growth usually stagnates indefinitely after sufficiently long hold times, Figs. 5 and 6.1–8 The change in the growth behavior can be very abrupt, resulting in a sudden plateau in plots of grain size versus time at fixed temperature. In thin films, workers have reported that grain growth may be restarted with sufficient temperature increases from an apparently stagnated state, Figure 5.1,2 Growth then rapidly restagnates at the new hold temperature. Standard grain growth laws do not formally predict the rapid onset of stagnation, merely a gradual slow down of grain growth to imperceptible rates. Therefore, the grain size in the plateau regions for long hold times is typically not in agreement with that predicted with kinetic variables derived from the size versus time curves for short hold times, where there is pronounced curvature. With standard laws, the restart of the growth process from slow growth on temperature increases leads to endpoint grain sizes with strong dependencies on the hold times. However, the experimental observation is a nearly linear temperature dependence that in many cases is independent of the hold times after a sufficient duration, 1090
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J. Mater. Res., Vol. 16, No. 4, Apr 2001 Downloaded: 14 Mar 2015
Fig. 1.1–8 For growth laws including size-dependent opposing forces, endpoint grain sizes are predicted to be either independent of temperature or exponentially temperature dependent with thermodynamic reversibility, the latter an impossibility. II. STANDARD GROWTH LAWS
The simplest growth laws are based on the rigorously derived fac
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