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This assumes that each of the individual mechanisms, represented by the subscript i, provides an instantaneous contribution to the total neck growth. Accordingly, computer simulations are employed to continuously calculate the contributions from each transport mechanism at a given time. Once the rates of mass flow are determined, the sum is used to enlarge the neck by determining its new volume and shape for the next instant of time. With reposition of the transported mass, the geometry is recalculated. Time is advanced by a small step size, and the corresponding microstructure gradients are used to recalculate the fluxes and total mass flow. This process goes on with millions of small steps to eventually determine the geometry after a reasonable sintering time. Only in the past few years, has the speed of such calculations for just two particles exceeded the speed of actual sintering. (In the 1970s it would have taken 10 times longer to simulate the process than the actual experimental time; computer times of 600 min would be required for simulation of 60 min of sintering for two spheres.)

In calculating the rate of sintering, the evolution has been from simple geometric approximations to sophisticated multiple mechanism computer simulations. The usefulness of such simulations is evident in several of the proprietary codes, where the agreement with the experiment is excellent. Surface diffusion dominates sintering, while evaporation condensation is a minor contributor.

A consequence of these computer-based sintering calculations is the realization that the process is very involved and not easily treated using simple models. The logical extension has been the synthesis of sintering diagrams that provide guidelines on the interplay of the processing parameters with rigorous geometric models.

A sintering diagram proves useful in condensing and representing sintering behavior. These are based on computer simulations, wherein multiple mechanisms contribute to mass flow, and accurate geometric representations are used for the microstructure. Contributions are included from each of the mass transport mechanisms, and the geometric approximations are minimized to accurately reflect microstructure evolution during sintering. The more recent simulations include a change in grain size with time and recognize densification results in new particle contacts. A sintering diagram can have any of several forms, but typically shows the density or neck size versus temperature for isothermal sintering at various times. The underlying simulations rely on material data to combine mechanisms and determine the effects of the main process variables (time, temperature, green density, particle size, and grain size). Data needed for these calculations are collected in various books on sintering theory. The diagrams are sensitive to input material characteristics. The output might be density, neck size, dimensional change, or final compact shape (includes distortion from green density gradients) as a function of compaction conditions, sintering conditions, composition, or powder characteristics. The graphical output is most impressive, since the plots show the interactions between the key processing variables.

The effect of simultaneous transport mechanisms is to increase the overall rate of neck growth, but possibly with a decreased rate of shrinkage. One merit of computer simulated diagrams is that such combined effects are included in the calculations. When the sintering diagram is based on density, the surface transport contributions are less evident.

Today, the complexity of sintering is recognized, and realistic sintering calculations rely on sophisticated computer calculations. Early models failed to appreciate the multiple mechanisms and complex geometric changes associated with solid-state sintering. In modern industrial sintering situations, the systems are complex, and many proprietary codes exist that include nonlinear heating, wide particle size distributions, liquid formations, atmosphere interactions, and the behavior of mixed powders. Sintering models have only recently embraced some of the practical realities.

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