Modelling spiral growth at dislocations and determination of critical step lengths from pyramid geometries on calcite {} surfaces

Abstract

To visualise the morphologic changes of a growing spiral pyramid, a mathematical model was developed and implemented in an algorithm that can loop through the sequential time-frames in a growth sequence. The algorithm simulates growth of spiral pyramids with geometries reflecting variation in the step velocity ratio and critical lengths of inequivalent steps. We have developed a new mathematical model that implements a set of equations describing the relationship between time-dependent change of step length and orientation around dislocation sources. These step length relationships can be used for extracting the critical step lengths that restrict step growth at dislocations. This allows determination of the critical thermodynamic parameters from post-growth AFM images that do not rely on real-time AFM fluid-flow experiments. The model was tested on experimental results obtained from calcite {101¯4} surfaces. A shift in pyramid apex geometry resulting from velocity anisotropy for the obtuse and acute steps was observed for pyramids grown from solutions of variable Ca2+ to CO32− activity ratio. Relative step velocity and pyramid apex angle were correlated with activity ratio. AFM images of rhombic spiral pyramids on calcite surfaces were used for measuring the length of sequential steps for the four step orientations. Step length plots show the same trend in the experimental data as was predicted by the theoretical model. Critical step lengths were determined using linear regression analysis of the experimental results. The step edge energy was calculated to be 3.6 · 10−19 J nm−1 for obtuse step edges and 1.3 · 10−19 J nm−1 for acute edges. These values are consistent with previous experimental and computational studies.