Antiskeletal morphology of crystals as a possible result of their

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Antiskeletal morphology of crystals as a possible result of their regeneration. Victor G. Thomas, and Dmitry A. Fursenko Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b01761 • Publication Date (Web): 20 Mar 2018 Downloaded from http://pubs.acs.org on April 10, 2018

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Crystal Growth & Design

Antiskeletal morphology of crystals as a possible result of their regeneration. Victor G. Thomas a,b*, Dmitry A. Fursenko a a

Sobolev Institute of Geology and Mineralogy, Siberian Branch, the Russian Academy of Sciences, 3, Academician Koptyug Pr., Novosibirsk, 630090, Russia. b Novosibirsk State University, 2, Pirogova St., Novosibirsk, 630090, Russia. *

Corresponding author, e-mail: [email protected], [email protected].

Abstract The paper presents a possible mechanism of forming crystals with the antiskeletal morphology due to their regeneration after the partial dissolution. The consideration is carried out by numerical 2Dsimulating the coordinate zone evolution of a single crystal ball using the kinematic model of regeneration crystal surface growth. According to this model, the genetic predecessors of subindividuals on the regenerated crystal are protrusions formed on its surface during the stage of its partial dissolution. It has been shown that the main parameter responsible for the antiskeletal morphology of regenerated crystals is the ratio of depression depths (l) between adjacent protrusions and protrusion radii (r), 0 < l/r < 1. When l/r ≤ 0.1, the stationary shape of the regenerating ball is a polyhedron. If l/r > 0.6, there is a ball with a rough surface covered by flat areas on the most slowly growing faces. The crystal with the antiskeletal morphology grows at intermediate values of l/r. Introduction Alongside free-grown crystals having the shape of a convex polyhedron, other morphological types with the substantially violated convex shape are quite common (Figure 1). The so-called skeletal crystals 1 (Fig.1b) are one of these types, they are produced by the preferential growth of crystal edges and vertices, with negative relief elements being formed in the position of convex polyhedron faces. Crystals where negative elements develop in the position of convex polyhedron edges and vertices refer to the opposite morphological type (Fig. 1c). Following O. M. Ansheles 2 crystals with such morphology are called antiskeletal. The genesis of skeletal crystals and criteria for such a morphology type occurrence are investigated in sufficient details (see, e.g., 3–6), but similar issues related to antiskeletal crystals are developed significantly worse. There is only one model explaining the growth of antiskeletal crystals according to BCF-mechanism under small supersaturations. Such antiskeletal crystal morphology formation is directly caused by inhibiting the tangential propagation of step bunches from the dislocation in the center to the face edge due to some adsorbed impurities 7,8. The examples of implementing such a mechanism are diamonds crystals grown from the solution of carbon in metal melt in presence of water 9-11 or magnesium oxide 12. It is the admixtures of H2O or MgO that act as inhibitors of the tangential expansion of step butts on such crystals. Without denying the possibility of realizing such a mechanism, we doubt its uniqueness. Indeed, big antiskeletal fluorite (CaF2) crystals up to 5 cm can be quite symmetric 13. Considering this fact together with the mechanism mentioned above, it is easy to conclude that outputs of screw dislocations must be located exactly in the face centers. At the same time, there are many facts when the outputs of screw dislocations are significantly shifted from the centers of the faces, which suggest implementing another formation mechanism of antiskeletal crystals in these cases.

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Figure 1. A cube as example of a convex polyhedral crystal (a) and the violated forms, skeletal (b) and antiskeletal (c), developed on its base.

In this regard, we reckon the results of numerical modeling of the single-crystal ball regeneration† in accordance with the kinematic model developed in 15–17 to be promising, as they can explain the genesis of antiskeletal crystals in a different way. The results presented in the article show that crystals with the antiskeletal morphology can be formed due to regenerating rounded crystals under some specific boundary parameters. They are of greater interest as the quantitative consistency of the proposed model and real growth experiment results has been demonstrated in case of a flat surface regeneration 16, and, at least, the semi-quantitative one in case of the curved surface regeneration 17. The kinematic model of the regeneration process and its application for the antiskeletal crystal formation The kinematic model of the regeneration process. Regeneration surfaces‡ are known to be growing in a macroscopically rough front 14. The physical model of the regeneration process is based on the assumption that bumps initially present on the surface are the genetic predecessors of irregularities (subindividuals§) on the growing regenerative surface front. The possible genesis of initial bumps is considered in 16. In the framework of the model these asperities are approximated by arcs of circles having radius ri; they are so constructed that the end of the i-th one coincides with the beginning of the i+1-th one, all depressions between adjacent arcs being the same. The values ri are uniformly and randomly distributed on the interval [rmin, rmax]. By the beginning of the regeneration all arcs-bumps are instantly facetted by the faces from the number of possible crystal faces as tangents to these arcs. The further regeneration process is to † The crystal regeneration is the process of recovering the polyhedral crystal shape during its subsequent growth if this shape was previously violated 14. ‡ The regeneration surface means any crystal surface, the orientation of which does not match any of the possible crystal faces (under given conditions) at the initial moment of regeneration. § Subindividual is a unit convex polyhedral area of a growing regeneration surface. Generally speaking, vicinal sculptures on real crystal faces may be also covered by this definition, but it is not important for the kinematic model considered.


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transfer faces covering subindividuals parallelly to themselves at distances corresponding to growth rates of these faces. The faces of the same crystallographic shapes are growing with the same growth rates regardless of the fact whether they are implemented on a subindividual or a polyhedral crystal. Two kinds of geometric competitions are realized during the parallel transfer of subindividual faces: (I) selection among faces within each subindividual and (II) competition among adjacent subindividuals, as observed in real processes of the crystal regeneration 15. The reason for competition II is the fact that the set of faces covering these arcs may be different due to different radii of the arcs approximating adjacent bumps. Because of this, the adjacent subindividuals will move at different speeds, and they will be able to absorb ("consume") each other. The implementation of the model presented is considered in more detail as well as with examples in16, 17. The application of the model in case of the antiskeletal crystal growth. The mentioned above differences in the radii of adjacent arcs are the necessary competition condition of subindividuals while modelling the growth of flat seeds sawn parallel to regeneration surfaces. When simulating the sphere regeneration in order to reproduce the sequence of their taking a polyhedral shape 17, the dimension differences of irregularities were also set by different radii of the arcs approximating them. At the same time, bumps on curved surfaces can differ by the set of faces initially facetting them even if they are characterized by the equal geometric dimensions. Figure 2 presents 2D-examination of the coordinate zone [100] evolution of a hemisphere made from a cubic crystal where, for simplicity, there are only two crystallographic shapes: cube and

Figure 2. 2D-illustration of a single-crystal hemisphere (light gray color) evolution during its regeneration (cubic symmetry, coordinate zone). The hemisphere surface is initially covered with protrusions approximated by arcs of circles (dark grey color). For clarity, protrusion dimensions (all bumps are numbered) are exaggerated. The ratio of the hemisphere radius (R) and those of the arcs (r) approximating the bumps is R : r = 6 : 1. The depth of all depressions between any pair of adjacent arcs (l) is constant: l : r = 5 : 7. The set of possible crystallographic shapes facetting the arcs includes, for simplicity, only cube h{100} and rhombic dodecahedron d{110}, with the growth rate being Vh : Vd = 1 : 5. The dotted broken lines are growth isochrones, an external solid broken line is the stationary shape of the antiskeletal crystal. ACS Paragon Plus Environment

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rhombic dodecahedron. Despite the identity of all protrusion dimensions, a set of faces covering bumps can be different at the initial moment of regeneration. Indeed, arc #6 is covered only by faces h1, d1 and d2, since it is impossible to construct tangents that are parallel, for example, to h2 or h3. Adjacent arc # 5 contains an additional face h3, and arc #7 has an additional face h2. It is possible to see similar differences for other pairs of neighboring protrusions. Inasmuch each subindividual can be covered by faces growing with different rates, the geometrical competition-I within each subindividual become possible. In accordance with Bergström criterion18, two faces, a and b, coexist on a polyhedron without absorbing one another, if their growth rates satisfy the inequality cos( β ) ≤ Va Vb ≤ (cos( β )) −1 , where -90˚ Vh (cos54˚)-1 = 1.74. Since o(111) symmetrically dulls the trihedral angle formed by faces of h(001), h(010) and h(100), the absorption of octahedron faces by all cube faces will simultaneously occur, to form a vertex. In other words, the application of the model for the diagonal zone allows reproducing antiskeletal crystal shape as well, but it is necessary to use 3D-modeling to illustrate it. Numerical 2D-simulation of forming antiskeletal crystals due to the regeneration of a single crystal sphere. It is clear from the model considered above that there are some possible parameters influencing the shape of the regenerating sphere: ACS Paragon Plus Environment

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1. the full set of possible crystallographic forms involved into building of subindividuals; 2. the ratios of the growth rates of faces belonging to different crystallographic forms; 3. the depth of recesses between adjacent protrusions; 4. the radius value of arcs approximating original bumps on the sphere surface. Let us investigate the influence of these parameters by the numerical 2D-simulation of the zone [100] evolution on the regenerating single crystal ball (cubic symmetry), as it was done in 17. 1. The influence of the full set of possible crystallographic forms. In addition to the already used cube h{100} and rhombic dodecahedron d{110} shapes in Fig.2, some other crystallographic forms with Miller indices {hk0} (depending on the symmetry class, e.g. different pyritohedrons or tetrahexahedrons) are possible in the zone under consideration. Let's consider one of them with the minimum sum of Miller indices – e{210}, and suppose that face growth rates correspond to Ve > Vd > Vh. The angle between the normals to the e-face and the nearest h-face is 26.6˚, and to the nearest d-face is 18.4˚. Therefore, the face e coexists with h and d in a very narrow range of growth rates of 1 < Ve/Vh ≤ 1.12, and 1 < Ve/Vd ≤ 1.05. The numerical simulation shows that in case both inequalities hold true together with 1 < Vd/Vh ≤ 1.41, the faces of all three crystallographic shapes will coexist on the regenerating sphere up to its the conversion into a convex polyhedron, with no antiskeletal crystal morphology appearing. In case of those inequality failures faces e{210} are "consumed" either by cube faces or rhombic dodecahedron ones, or both almost instantly. This is consistent with increasing rate of "consuming" the neighboring face at decreasing the angle between the normals to them, when Bergström criterion holds false 22. Thus, the influence of the set completeness of crystallographic shapes involved in building subindividuals can be neglected in the case under consideration. 2. The effect of the ratio of growth rates for faces belonging to different crystallographic shapes. Fig. 4 presents the results of numerical 2D-simulation of the regenerating sphere evolution in case all the protrusions on its original surface (the radius of arcs, r = 500µm) are identical; the recesses between adjacent protrusions have the same depth, l = 150µm). For all cases the relative growth rate of the cube face Vh = 1. If the growth rate of the rhombic dodecahedron face Vd = 1, i.e. when Bergström criterion holds true 0.71 ≤ Vd/Vh ≤ 1.41, the antiskeletal crystals are not formed (Fig.4a). Flat areas are formed on the points where normals exit to faces h(001), h(010) and d(011), they are tangentially expanding, which will eventually lead to the formation of a convex polyhedral crystal. If the growth rate Vd = 1.5 (i.e. 6% higher than the critical Vd = 1.41) flat areas that are tangentially expanding are still formed on cube faces, whereas macroscopically rough surface (Fig.4b) covers rhombic dodecahedron faces; the latter is typical to a growing regeneration surface. Some further increase of Vd up to 2 leads to the formation of an antiskeletal (Fig. 4c). The fundamental increase of Vd up to 10 (Fig.4d) in comparison with the case of Vd = 2 (Fig.4c) weakly affects the morphology of the antiskeletal crystal stationary shape. Thus, the ratio of the growth rates of the faces belonging to different crystallographic shapes significantly affects the antiskeletal crystal morphology only in the region of little differences from the boundary values of rate ratios according to Bergström criterion. 3. The influence of depths of recesses between adjacent protrusions. First, let’s consider that as the influence of ratio l/r. The simulation results indicate (Fig. 5) that when varying the ratio l/r in the interval (0,1), the stationary shape of the regenerated crystal undergoes substantial changes from an ideal polyhedron when l/r ~ 0 (Fig.5a) to a classic curved macroscopically rough surface cut off by flat areas in h{100} position (Fig.5d). Large negative relief patterns are formed at intermediate values in the position of d{110}, i.e. they are antiskeletals by definition 2 (Figs.5b,c). ACS Paragon Plus Environment

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Figure 4. Numerical 2D modeling of the effect of the ratio of growth rates of cube (Vh = 1) and rhombic dodecahedron (Vd = 1 (a), 1.5 (b), 2 (c), 10 (d)) faces to the formation of antiskeletal crystals as a result of regenerating a fragment (the arc length is 120˚) single crystal sphere. Figure 4a shows the directions of normals to faces. The radius of the arcs approximating the original bumps on the ball surface, r = 500µm, constant depth recesses between adjacent bumps, l = 150µm.

If we fix l/r = const, it is possible to trace the influence of protrusion magnitudes on the surface morphology of the regenerating ball by the time it reaches its stationary form (Fig. 5c, e, f). The decrease in r values can be seen not to cause any significant alterations in the amplitude of macrochanges when positive relief forms of an antiskeletal crystal transfer into the negative ones. Only subindividual dimensions on the curved surface near the position of face d(110) change (decrease). The possibility of implementing the presented mechanism in real crystal growth processes Although the possibility of forming antiskeletal crystals during the regeneration of a curved surface has been demonstrated, it is still necessary to consider the genesis of such a surface. Indeed, how could a rounded crystal with the surface evenly covered by bumps of similar geometry be formed? In this paper, we consider only one of a possible number of ways to form such a surface. From this point of view, the reason for the formation of such crystals is alternating acts of their growth and dissolution. In the course of dissolution crystal vertices and edges are the areas which are primarily eliminated from the crystal surface 14,23. Also, the preferential dissolution occurs from surfaces of small faces which cut off those vertices and edges and have a higher free energy. As a result, a crystal gets a ACS Paragon Plus Environment

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Figure 5. Numerical 2D modeling of the influence of l/r value (Figs. a-d) and r value (Figs. c, e, f) for the original sphere surface protrusions on its evolution during the regeneration. Conditions: a set of crystallographic shapes is h{100}, d{110}, e{210}; growth rates are Vh = 1, Vd = 10, Ve = 20; the radii of arcs approximating the original bumps on the sphere surface, r (µm) – 500 (a-d), 250 (e), 125 (f); recesse depths l (µm) – 50 (a,f), 100 (b,e), 150 (c), 300 (d).

quasi-spherical shape. In parallel, the surface etching is taking place along its defects, and the surface is covered with depressions separated by protrusions facetted by curved faces with high values of Miller indices 23. If defects are assumed to be absolutely identical in their energy and uniformly distributed in the crystal, the crystal dissolution will result in the desired curved surface evenly covered with bumps of the same geometry.


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Of course, it is more likely that the distribution of defects both by their energy and concentration of exits on the crystal surface is described by the normal law. However, it is necessary to note that the uniform distribution of protrusions according to their geometric dimensions considered in 17 and in the present work as well as the distribution in the form of ∆-function are extreme cases of the normal distribution (at the standard deviation σ→∞ normal distribution is degenerated into the first one, and at σ→0 in to the second one). Thus, depending on the value of σ(ri) the whole set can be obtained from the classic regeneration of single crystal balls 17 to the formation of antiskeletal crystals. Conclusions The possible mechanism of forming antiskeletal crystals as a consequence of their regeneration after the partial dissolution has been considered in accordance with the kinematic model of growth processes of regenerated crystals surfaces in the work presented. The consideration has been carried out by the numerical 2D-simulation of the coordinate zone evolution (cubic symmetry) of a single-crystal sphere. The proposed model has been shown to allow reproducing an antiskeletal crystal morphology, since it assumes similar protrusions (with identical sizes) initially existing on the ball surface instantly facetted at the beginning of the regeneration to be the genetic predecessor of subindividuals on the growth front of the regenerated surface. In accordance with the model the following input parameters control the regeneration process: the full set of possible crystallographic shapes involved in facetting subindividuals; the ratio of growth rates of faces belonging to different crystallographic forms; the radius value of arcs (r) approximating initial bumps on the sphere surface; the depths of recesses between adjacent protrusions (l). The value of l/r (0 < l/r < 1) has been shone to be of the greatest influence. When l/r ≤ 0.1 the stationary shape of a regenerating ball is a polyhedron. When l/r > 0.6, the ball has a rough surface covered by flat areas in the positions of the most slowly growing faces. The crystal with the antiskeletal morphology grows at intermediate values of l/r. The ratio of the growth rates of faces belonging to different crystallographic forms and faceting subindividuals on the growing regeneration surface produces a significant influence only in case the ratio of adjacent face growth rates slightly differs from their Bergström criterion critical values.

Conflicts of interest There are no conflicts to declare. Acknowledgements This work was supported by the state assignment project No. 0330-2016-0016 with the financial support of TAIRUS Company (Novosibirsk, Russia). We thank Prof. Yu. N. Palyanov, Prof. A.F. Khokhryakov, Prof. S.Z. Smirnov and Dr. P.N. Gavriushkin (Sobolev Institute of Geology and Mineralogy SB RAS, Novosibirsk, Russia) for useful scientific discussions. We kindly thank both Reviewers for their helpful comments and suggestions, allowing us to improve the manuscript and make it clearer. We are very grateful to Dr. E. V. Tomas (Higher School of Economics, Moscow, Russia) for the great work on the design of the manuscript. Special thanks the authors would like to express to our teacher, Prof. I. T. Bakumenko (Lviv, Ukraine) for help in the work on the paper. References (1) Shafranovskii I. I. Lectures on Crystal Morphology. Balkema, Rotterdam, 1973, 174 p.) (translated from Russian version - 1968). (2) Ansheles O.M. The Foundations of Crystallography. Leningrad, LSU, 1952, 283 p. (In Russian). ACS Paragon Plus Environment

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(3) Berg W.F. Crystal growth from solutions. Proc. R. Soc. Lond., 1938, A, 164, pp. 79-95. (4) Jackson К.А., Uhlmann D.H., Hunt J.D. On the nature of crystal from melt. J. Crystal Growth, 1967, v. 1, 1, p.1. (5) Cabrera N., Vermilyea D.A. The growth of crystals from solution. Growth and perfection of crystals. N. Y.: John Wiley & Sons, Inc.; London: Chapman& Hall, Ltd., 1958, pp. 393-408. (6) Sunagawa I. Growth of Crystals in Nature. Materials Science of the Earth's Interior. Tokyo: «Terrapub», 1984, pp.63-105. (7) Punin Yu.O., Petrov T.G., Treivus E.B. Low-temperature modeling of mineral formation processes. Zap. Vses. Mineral. O-va, 1980, 109 (5), pp. 517-529. (In Russian). (8) Treivus E.B. Kinetics of growth and dissolution of crystals. Leningrad, LSU, 1979, 248 p. (In Russian). (9) Palyanov Yu.N., Chepurov A.I., Khokhryakov A.F., Growth and morfology of antiskeletal crystals of synthetic diamond. Miner. Zh., 1985, 7 (5), p.50-61. (In Russian)) (10) Palyanov Yu.N., Borzdov Yu.M., Kupriyanov I.N., Khokhryakov A.F. Eﬀect of H2O on Diamond Crystal Growth in Metal−Carbon Systems. Crys. Growth Des., 2012, 12, pp. 5571-5578. (11) Palyanov Yu.N., Khokhryakov A.F., Borzdov Yu.M., Kupriyanov I.N. Diamond Growth and Morphology under the Inﬂuence of Impurity Adsorption. Crys. Growth Des., 2013, 13, pp. 5411-5419. (12) Khokhryakov A.F., Sokol A.G., Borzdov Yu.M., Palyanov Yu.N. Morphology of diamond crystals grown in magnesium-based systems at high temperatures and high pressures. J. Crys. Growth, 2015, 426, pp. 276 – 282. (13) Chuhrov F.V., Bonshtedt-Kupletskaya E. M. (Eds.) Minerals – Reference book, Moscow, Nauka, 1965, v.II (1), 296 p. (In Russian). (14) Askhabov A. M. Regeneration of Crystals, Moscow, Nauka, 1979, 174 p. (in Russian). (15) Gavryushkin P.N., Thomas V.G. Growth Kinematics of the Regeneration Surfaces of Crystals. Crystallography Reports, 2009, Vol. 54, No. 2, pp. 334–341. (16) Thomas V. G. Gavryushkin P. N., Fursenko D. A. 2D Modeling of the Regeneration Surface Growth on Crystals. Crystallography Reports, 2012, Vol. 57, No. 6, pp. 962–974. (17) Thomas V. G., Gavryushkin P. N., Fursenko D. A. 2D Modeling of Regeneration Surface Growth on a Single-Crystal Sphere. Crystallography Reports, 2014, Vol. 60, No. 4, pp. 583–593. (18) Prywer J. Theoretical analysis of changes in habit of growing crystals in response to variable growth rates of individual faces. J. Crystal Growth, 1999, 197, pp. 271-285. (19) Van Enckevort W.J.P. Contact nucleation of steps: theory and Monte Carlo simulation. J. Crystal Growth, 2003, 259, pp.190-207. (20) A.F. Gorodetsky, D.D. Saratovkin. The dendritic shape of crystals formed by the way of antiskeletal growth. In: Crystal Growth, v.1, (Eds: A.V. Shubnikov, N.N. Sheftal’), 1957, pp. 190 - 198. (in Russian). (21) The website "All about Geology". URL. http://geo.web.ru/mindraw/skelet1b-2.jpg (date of access: 20.09.2017). (22) Gavryushkin P. N. The kinematic model of the growth of regeneration surfaces of crystals. PhD thes., IGM SB RAS, 2009, 165 p. (in Russian).. (23) Heimann R.B. Auflösung von Kristallen. Spr.-Ver., Wien – New York, 1975, 272 p.


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Antiskeletal morphology of crystals as a possible result of their regeneration. Thomas V.G. and Fursenko D.A.

By the numerical 2D-simulation based on the proposed kinematic model of the regeneration process of crystals we demonstrated that the antiskeletal morphology of crystals can be explained by their regeneration after partial dissolution.


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Antiskeletal morphology of crystals as a possible result of their

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