Crystals characterized by oscillatory zoning show a quasi-cyclic alternation in the chemical composition in the growth direction, from a few tens of nanometers to several tens of micrometers in thickness. Such mineral zoning is a common phenomenon in magmatic rocks, hydrothermally altered rocks, mineralized rocks, and carbonate sequences [1]. Oscillatory zoning may originate from an intrinsic mechanism —complex diffusion-attachment processes at the solution-crystal interface— even under stable growth conditions. An important factor that affects the crystal growth kinetics is impurity, which is known to inhibit crystal growth. Pinning is a well-known effect of impurities on the kinetics of crystal growth in solution [2-4]. The pinning mechanism has been discussed successfully for various crystallizing systems with different impurities. However, the role of impurities on oscillatory growth has not been modeled on a physical basis to date.

We consider the adsorption and desorption of the impurities on the growing crystal surface along with the pinning mechanism. If the steps barely pass beyond a certain area on the crystal surface, the adsorption and desorption of impurities reach an equilibrium state (Langmuir isotherm). In contrast, repeated passage of steps shortens the exposure time of adsorption sites for impurities, and this tends to decrease the density of the adsorbed impurities [5]. Assuming for simplicity that once the step has passed beyond this area, the impurities adsorb in the crystal and do not seriously obstruct the advance of subsequent steps [2]. Frequent step passages will result in less impurities at the surface, as if the crystal surface is swept by the advancing steps. Because of impurity sweeping coupled with pinning, there is a potential feedback on the change in the step velocity through the change in the impurity density. The feedback loop causes periodic oscillation of the step velocity.

We formulated these two mechanisms, pinning mechanism and impurity sweeping, as relations between the adsorbed impurity density θ and the step velocity V. We found steady solutions that satisfy both of two relations simultaneously when the supersaturation at interface σ is larger than a critical value (see Fig. 1). The steady solution is unique when σ is large enough. There are, however, two different steady solutions for intermediate σ. One of the solutions suggests a steady growth with large step velocity under less effect of impurity. Another is an unstable solution that would not appear in actual systems. The multiple-valued feature may result in oscillatory behavior as suggested previously [6-8].

Refs. [1] Shore & Fowler 1996, Can. Miner. 34, 1111-1126. [2] Cabrera & Vermilyea 1958, in Growth and Perfection of Crystals, Proceedings, pp. 393-410. [3] Burton et al. 1951, Phil. Trans. Roy. Soc. London, 243, pp. 299-358. [4] Kubota & Mullin 1995, JCG 152, 203-208. [5] van Driessche et al. 2009, Crys. Growth Des. 9, 3062-3071. [6] Haase et al. 1980, Science 209, 272-274. [7] Ortoleva 1990, Earth Sci. Rev. 29, 3-8. [8] Kalischewski et al. 2007, Phys. Rev. E 75, 021601.

Fig. 1: Steady solution that satisfies both of two relationships: pinning mechanism and impurity sweeping by step advancement. The horizontal axis is the supersaturation at crystal-liquid interface and the vertical one is the normalized step velocity.

• Legal notice:
  

  Copyright (c) Pielaszek Research, all rights reserved.
  The above materials, including auxiliary resources, are subject to Publisher's copyright and the Author(s) intellectual rights. Without limiting Author(s) rights under respective Copyright Transfer Agreement, no part of the above documents may be reproduced without the express written permission of Pielaszek Research, the Publisher. Express permission from the Author(s) is required to use the above materials for academic purposes, such as lectures or scientific presentations.
  In every case, proper references including Author(s) name(s) and URL of this webpage: https://science24.com/paper/29467 must be provided.

1. Growth Mechanism of Lysozyme Crystals in The International Space Station Based on The Analysis  of  In-Situ Interferometric Observation
2. Nucleation experiment in vapor phase under microgravity using the sounding rocket S-520-28
3. Submerged atomic resolution imaging of hydration structure on calcite
4. Two Pathways Determining Chirality in NaClO₃ Crystals Grown from Solution via Achiral Precursors