content_image/339565.jpg
parameter profiles ( φ$_{1}$ and φ$_{2}$ ) are obtained each is fit to the following function: 

<img src='content_image/339565.jpg'>

where a , b , x$_{0}$ and d are fitting parameters, and the boundary width is derived from the difference in values of x$_{0}$ determined from the φ$_{1}$ and φ$_{2}$ profiles. 

The analysis used to compute grain-boundary widths is illustrated graphically in Fig. 2 for a snapshot of the Σ9 twist boundary at an undercooling 2 K below the bulk melting temperature. In Fig. 2 the top panel shows the atoms, color coded according to the magni- tude of the difference in the order parameter values, lighter colors (red in the on-line figure) denoting atoms in an environment corresponding to either of the two grains, and darker colors (blue in the on-line figure) indicating an environment that does not correspond to either grain. The corresponding order parameter profiles with the fits to Eq. 5 (solid lines) are shown in the bottom of Fig. 2. The width is defined as the difference in the center positions of the fits to φ$_{1}$ and φ$_{2}$ . In this example, a width of 13.5 ˚ A is obtained by the analysis. 

It should be emphasized that the choice of an order parameter used to compute the grain boundary width is not unique (this point was discussed also in a recent related study by Williams and Mishin [33]). Other choices for the structural order parameter used to define an interface width would include the so-called centrosymmetry parameter [44], the order-parameter used by Morris in studies of solid-liquid interfaces [45], excess mass used by Mellenthin[19], a parameter introduced by Williams and Mishin [33] based on the local structure factor, and an order parameter introduced by von Alfthan et al. [34] based on structural units. Different choices for the order parameter are expected to lead to slightly different values for the interface width, and the quantitative values of the disjoining potential derived from them. For example, in the sharp-interface formalism we describe the limit of Ψ( w ) for small w as the interfacial free energy of a hypothetical “dry” grain boundary - we would expect that this limiting value will depend on the way in which the width is defined, which is not unreasonable since with any choice of this definition a real boundary at zero temperature will have some small finite width. The ambiguities associated with the different choices for the definition of width is inherent in the use of a sharp-interface theoretical formalism to describe the properties of systems such as these with diffuse interfaces. In 

