Quantitative Analysis of the Regenerating Single-Crystal Ball Evolution

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Quantitative analysis of the regenerating single-crystal ball evolution Valentin N. Kovalev, Victor G. Thomas, and Dmitry A. Fursenko Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.9b00065 • Publication Date (Web): 17 Apr 2019 Downloaded from http://pubs.acs.org on May 9, 2019

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Crystal Growth & Design

Quantitative analysis of the regenerating single-crystal ball evolution Valentin N. Kovalev a, Victor G. Thomas,b,c*, Dmitry A. Fursenkob a

Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russia Sobolev Institute of Geology and Mineralogy, Siberian Branch, the Russian Academy of Sciences, 3, Academician Koptyug Pr., Novosibirsk, 630090, Russia. c Novosibirsk State University, 2, Pirogova st., Novosibirsk, 630090, Russia. b

*

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

Abstract A quantitative analysis of the evolution of a spherical surface artificially prepared from a potassium alum single-crystal during its regeneration is carried out in real growth experiments. The results are compared with the results of numerical simulation based on the proposed regeneration process kinematical model. It is demonstrated that for many boundary conditions there is a close quantitative correspondence between the model and real experiments on the regeneration of single-crystal balls. It is shown that the most important parameter that has a quantitative effect on the growing regeneration surface evolution is the geometry of the roughness (protrusions and depressions), initially present on this surface. The rare quantitative discrepancies between the results of real growth and numerical experiments are due to the discrepancies between the geometry of the roughness set in the model and those actually present on the regenerating single-crystal ball. The proximity of the model to real regeneration processes allows us to accept, as there are proven, the basic postulates of the model. Namely: (i) the genetic predecessors of subindividuals are the protrusions that are initially presented on the regeneration surface; (ii) the rate of growth of all faces within the same crystallographic form is equal, regardless of whether the latest faces present on the polyhedral crystal, or they are localized on the subindividual's surfaces; (iii) the effects observed during the growth of the regeneration surface are the result of geometric selection either between the faces of each subindividual or between adjacent subindividuals in the course of which some subindividuals absorb their neighbors. Random differences in the initial protrusions geometry is the driving force for the latter. Introduction The crystal regeneration mechanism and the phenomena observed during this process has attracted the attention of crystallographers and specialists in the field of crystal growth for more than a century. Despite the extensive empirical base on this issue 1,2, until recently there was no single theory that quantitatively describes the processes of regeneration surfaces*growth. For this reason, we have recently proposed the kinematical model for the growth of such surfaces 3-5, which qualitatively describes most of the phenomena observed during this process. In particular, the model explains the decrease in the number of subindividuals** on the growing regeneration surface, changing its growth velocity during regeneration processes and some other phenomena. The model is based on the assumption that the genetic predecessor of subindividuals is the roughness (protrusions and depressions), which initially presents on the regeneration surface (arising, for example, by sawing and grinding of seed plates, selective etching of such a surface 6, etc.). In addition, the model assumes constancy of the growth rates of all faces belonging to the _____________ * A regeneration surface is a surface that is not parallel to any of the possible crystal faces under the given conditions. This kind of surface can occur, for example, as a result of partial dissolution of the crystal. ** The growing regeneration surface is characterized by a macroscopically rough growth front, where steps, pyramids and other elements similar to each other can be distinguished. The subindividual is a single convex piece of the growth front. ACS Paragon Plus Environment

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same crystallographic form, regardless of whether the face is localized on the polyhedral crystal, or on the surface of subindividual. According to the model, the effects observed during the growth of the regeneration surface are the result of two types of geometric competition: (a) the competition between the faces of the same subindividual proceeding in accordance with the Borgstrom criterion 7; (b) the competition between adjacent subindividuals in the course of which some subindividuals “consume” their neighbors. The driving force for the last type of competition is random differences in the geometry of the initial protrusions on the regeneration surface and, consequently, differences in the sets of faces, these protrusions may be covered as tangent planes. The numerical simulation of the regeneration process for flat surface of the potassium alum, KAl(SO4)212H2O (hereinafter referred to as “alum”), with the Miller indices (30.30.19), i.e., weakly inclined from face (332), undertaken on the basis of the proposed kinematical model showed good qualitative correspondence of the simulation results and the results of growth experiments 5. At the same time, it seems relevant to check the qualitative and quantitative compliance of the model with the real regeneration processes at other sites. One of such promising objects is regenerating balls made of single crystal 8. Recall, during regeneration in supersaturated solution, most of the surface of the monocrystalline ball becomes macroscopically rough; at the same time, in some places (in the position of the alum polyhedral crystal’s faces), flat spots are formed (Fig. 1). The flat spots corresponding to the faces {111} of alum continue to grow tangentially; in contrast, those corresponding to the faces {001} grow up to a certain size, after which they remain unchanged until they intersect with the faces {111}. Most of the flat spots corresponding to {112}, {221} and {210} grow tangentially to a certain size, and then begin to be consumed by the surrounding rough surface until complete disappearance. In general, the numerical simulation of the regeneration process of the alum singlecrystal ball on the basis of the kinematical model 9 at the qualitative level completely coincides with the described phenomenology.

Figure 1. The evolution of the shape of a single-crystal ball of potassium alum during its regeneration. During the process, there is a disappearance of flat spots corresponding to {112} and {221} (they present in Fig. 1a and are absent in Fig. 1b) and the beginning of absorption of the face (110) by the surrounding macroscopically rough surface. Exposure in aqueous solution (supersaturation ~0.5%, temperature 42°C): a – 1.5 hours, b – 7 hours. Photo 9.

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At the same time, the kinematical model predicts that the rate of tangential growth of flat spots, as well as their appearance, is very sensitive to the initial surface roughness of the regenerating ball, which is determined by the ratio of the roughness parameters Sm/Rz*. This fact is very convenient for checking the quantitative correspondence of the proposed kinematical model 5,9 to the real processes of regeneration of a single-crystal ball. This work assumes in real experiments on the regeneration of single-crystal balls of alum, characterized by different initial surface roughness, to trace for the tangential growth of flat spots corresponding to the faces of various crystallographic forms belonging to the diagonal zone of alum crystal. The obtained data will be compared with the data of numerical simulation of the ball regeneration process, on the basis of which the conclusion about the degree of compliance of the kinematical model with the real processes of balls regeneration will be made. Experiments and methods For the convenience of subsequent measurements, the direct object of research was not balls of alum (space symmetry Pa3), but segments of the spherical surface of radius 21 mm. In order to obtain such segments, the flat plates with a thickness of ~ 5 mm, oriented parallel to one of the faces of crystallographic shapes {111}, {001}, {110}, {112} and {221}, were cut from the pregrown bulk single crystals of alum. The sawing was done using the saw with a two-axes goniometrical crystal holder; alignment of the crystal was carried out by the reflection from the faces {111}. The error of the plate’s orientation for both polar angles did not exceed ±0.5°. Rectangular fragments of the obtained plane-parallel plates with dimensions of ~ 15x10 mm on one side were abrasive grinded in a jasper mortar (the radius of the mortar deepening is 21 mm) with silicon carbide powders of different grain size in the medium of 96% ethyl alcohol at a standardized pressure on the plate. Some of the plates were then lightly polished with a felt cloth dampened with distilled water. The duration of processing of all plates for the last case was the same. Table 1 shows the surface roughness parameters Rz and Sm, measured with Taylor-Hobson profilometer, for all four kinds of treatments. The growth experiments on the regeneration of single-crystal segments of alum were carried out by their growth from the aqueous solution supersaturated by lowering the temperature, which was prepared as follows. The aqueous solution, slightly supersaturated at 48°C, was overheated to 60°C, filtered into a glass thermostated crystallizer (the volume of the solution is 1.5 l) and cooled again Table 1. Surface roughness parameters of the spherical segments prepared from single crystal of alum treated in different ways.

No.

Type of treatment

3

Abrasive grinding by powder with particles size of 80 – 100 μm, high load Abrasive grinding by powder with particles size of 80 – 100 μm, low load Abrasive grinding by powder with particles size of 5 – 7 μm, low load

4

Polishing with a wet felt cloth

1 2

Rz, μm

Sm, μm

Sm/Rz

31.7

112.6

3.6

6.4

113.6

17.8

1.0

23.6

23.6

0.4

45.0

112.5

___________ * Roughness parameters: Rz is maximal vertical amplitude of protrusions and depressions on a rough surface, Sm is average horizontal step between protrusions. More information about the roughness parameters can be found in 10.



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to 48°C, after which a large alum crystal was placed in the crystallizer and exhibited in the solution for 1 to 3 days. Thus, due to the growth of the crystal, the solution approached the saturated one. On the replaceable teflon cover of the crystallizer was mounted 8 alum segments with the same orientation and uniform kind of the surface treatment. In addition, a flat rectangular seed parallel to {111} was also mounted onto the cover. The butt-ends of this seed were covered with waterresistant varnish in order to avoid tangential growth. Before the beginning of the growth experiment, the large crystal was extracted from the crystallizer, the solution temperature was reduced to 47°C, thereby creating a supersaturation of  = (C48 – C47) / C47  0.04, where C48 and C47 are the concentrations of the saturated solution at 48 and 47°C 11, respectively. After that, the crystallizer was covered by the lid with seeds mounted on it, and the growth experiment began. After a given period of time, the seed plates were removed (the exposure of the first one was 100 seconds, the exposure of each subsequent one was twice as large as the previous one). At the end of the experiment (through 12800 sec) both the last segment and the seed plate parallel to {111} have been removed; the absolute growth rate of the octahedron faces was determined by weighting of the last: V{111}=Δm/(2·s·ρ)/t, where Δm is weight gain per 12800 sec of experiment (g), s is the area of the seed's surface not covered with varnish (cm2), ρ = 1.757 g/cm3 is the specific gravity of alum crystals 11, t = 12800 sec is the exposure seed {111} in the supersaturated solution. According to our estimates, the reduction of supersaturation during all experiment does not exceed 5 rel. % and therefore we believe it is constant. The need to determine the absolute values of V{111} is due to a possible error in the setting of supersaturation of the alum solution. To compare the results of different series of experiments with each other, we used an adjusted time scale, the magnitude of which was set based on the ratio: tc = tr V{111} /Vc,, where tc is an adjusted segment exposure in the supersaturated solution, tr is its real exposure, V{111} – the real growth rate of {111}, specific for each series of experiments, Vc = 11.6 nm/sec = 1 mm/day – the growth rate of {111} accepted for numerical simulation (close to the real growth rates of this face). The correctness of this approach is quite obvious, and therefore will not be considered in this paper. The regenerated alum segments were photographed under an optical microscope Olympus BX51 with a camera Olympus ColorView III (IGM SB RAS, Novosibirsk), using a lens with 2.5x magnification in reflected light. Figure 2 presents the example of the constructed evolutionary ruler for a segment (treatment #3, Table1) with a face (111) at the zenith orientation*. The experimental data processing was reduced to the measurement of the average size of the flat spot by calculating the arithmetic mean of the maximum and minimum lengths (see Fig. 2g). In the cases when the flat spot were very large and the capabilities of the lens did not allow to capture them entirely, several photos of one flat spot were shot, and then this set was processed in STOIK Panorama Maker. The result was a complete panoramic image of the flat spot, the dimensions of which are easily determined as it is described above. Numerical simulation of the growth of regeneration surfaces, the results of which were compared with the results of real growth, was carried out using the original program previously used in 9 for 2D-examination of the evolution of the diagonal zone on the regenerating alum sphere. The effect of diffusion on growth processes was not taken into account. In this program, the initial roughness on the surface of the sphere was approximated by arcs of circles of radii Ri, whose values were randomly and uniformly distributed on the interval [Rmin, Rmax]. The location of the arcs that emulate the protrusions on the spherical surface was produced in such a way that the end of the i-th arc coincided with the beginning of the (i+1)-th and the size of the recesses between any two adjacent protrusions was the same. The procedure of the numerical simulation results processing is __________ * Zenith orientation of a face on the regenerating segment is the position when the normal to the face is parallel to the normal to the base of the segment.


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Figure 2. The example of the surfaces evolution of the alum balls segments (initial roughness of the surface # 3, Table 1), flat spots of faces {111} are oriented to the zenith. The exposure of segment (a) is 100 seconds, the subsequent exposure (b-h) is twice as large as the previous one. In the images a-b the emergence of the flat spot is observed; further, it continuously expands tangentially (c - h). Fig. 2g presents the rule of determining of the average length (L) of flat spot from the maximum (XX') and minimum (YY') values of the lengths of the spot: L=(XX'+YY')/2. The scale of photos (a - e) is 500 µm.



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also considered in detail in 9. The input parameters for the calculations were Rmin and Rmax calculated on the basis of Rz and Sm (Rz ≡ Rmin, Sm = 2 𝑅min ∙ 𝑅max, see 5), radius of a curvature of segments (21 mm), hkl of zenith face, and the growth rates of the faces of the various crystallographic shapes (under consideration were the faces with hkl: 0 ≤ |h|=|k|,|l| ≤ 10). The growth velocities V{001} and V{110} were estimated based on their Borgstrom criterion for the joint existence of two faces A and B on a polyhedral crystal [7]: 1/cos()  VA/VB  cos() (α is the angle between the normals to the faces A and B, VA and VB – their growth velocity) and the fact that the {001} face exists simultaneously with {111} on alum crystals and even grows in its size slightly, and {110}, on the contrary, almost never appears. Hence V{111} : V{001} : V{110} = 1 : 1.7 : 2.0. For the growth rates of the other crystallographic forms faces were used the data of 5: V{111} : V{112} : V{221} : V{h,k,l >2} = 1 : 11 : 9.5 : 15. Results and Discussion The results of growth experiments on the regeneration of single-crystal segments of potassium alum are qualitatively consistent with the previously described results9. The flat spots corresponding to the habitus crystallographic forms {111}, {001} and {110} continue to grow tangentially within the studied time interval (Fig. 2). The faces of the other crystallographic forms, for example, {112} (Fig. 3), grow tangentially to a certain size and then the surrounding rough surface begins to absorb them until their complete disappearance. The results of the quantitative measurements of the length of the flat spots corresponding to the different crystallographic forms of the diagonal zone of alum for different exposures of segments and different initial roughness of their surface (different ratios of Sm/Rz) are given in Table 2. It is seen that due to the initial surface roughness with a ratio Sm/Rz, the flat spots of the nonhabitus crystallographic forms appear on the surface of the segment what was predicted by the model. For example, the flat spots corresponding to {112} and {221}, which do not appear on the roughly treated segment surface (Sm/Rz = 3.6) at any exposure in supersaturated solution, are observed in the cases of use of slightly rough (Sm/Rz = 23.6) and polished (Sm/Rz = 112.5) surfaces. Let us consider separately the results obtained for various crystallographic forms. Octahedron {111} is the main crystallographic form on the potassium alum crystals. A comparison of the changes in the size of the flat spots corresponding to {111} on the surface of the regenerated segment in the real growth experiment with the results of numerical simulation for the cases of different initial surface roughness is shown in Fig. 4 in a double (both the abscissa and ordinate axis) logarithmic coordinates. First of all, it is necessary to note that for all three considered cases, the results of growth experiments in double logarithmic coordinates form straight lines (the coefficient of determination is R20.98), i.e. the dependence of the face length (L) on time (tc) has a power form: L = ktca. The values of the parameters k and a in this equation, found by regression analysis, together with the errors of their determination are given in Table 3. The absence of data for small exposures (Fig. 4 a, b) is the result of the fact that at this period separate adjacent subindividuals do not have time to join together flat spot, i.e. its size is not defined. Thus, there is an induction period for the process of formation of such spot, the duration of which is determined, from the standpoint of the model, by the average size of the subindividuals which is equal to Sm. According to the Table 1, it is about 0.1 mm for the cases of a, b and about0.025 mm for the case of c. Therefore, if we continue the approximating line to the value of L = 0.1 mm (cases a, b) and to L = 0.025 mm (case c), we obtain that the duration of the induction period is ~500 sec (>100 sec, loss of points at small exposures), ~ 30 sec and ~ 1 sec ( Vmin is close to the parabolic one with the parabola axis coinciding with the abscissa axis, i.e. to the function L(t) = ktc0

Quantitative Analysis of the Regenerating Single-Crystal Ball Evolution

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