Bacterially Induced Struvite Growth from Synthetic Urine: Experimental

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CRYSTAL GROWTH & DESIGN

Bacterially Induced Struvite Growth from Synthetic Urine: Experimental and Theoretical Characterization of Crystal Morphology

2009 VOL. 9, NO. 8 3538–3543

Jolanta Prywer*,† and Agnieszka Torzewska‡ Institute of Physics, Technical UniVersity of Ło´dz´, ul. Wo´lczan´ska 219, 93-005 Ło´dz´, Poland, and Department of Immunobiology of Bacteria, Institute of Microbiology, Biotechnology and Immunology, UniVersity of Ło´dz´, ul. Banacha 12/16, 90-237 Ło´dz´, Poland ReceiVed March 10, 2009; ReVised Manuscript ReceiVed May 13, 2009

ABSTRACT: We describe the mineralization of struvite crystals from synthetic urine in the presence of Proteus bacteria. The presence of these bacteria causes a successive increase in the pH of the solution of synthetic urine. The effect of pH on the growth morphology of struvite crystals is studied. The results show that for pH in the range approximately from 7.5 to 9.0 struvite crystals take single and hemimorphic morphology. Crystal faces are different at opposite ends of the crystallographic c-axis, and thus polar crystal properties are permitted. At higher pH, many twin and dendritic crystals have been observed. It is suggested that the growth of dendrite is related with the bacterial substances which may serve as the sites for heterogeneous nucleation. The growth of single crystals is analyzed from a theoretical point of view based on relative growth rates.

1. Introduction

O

The formation of urinary stones in human body is a serious clinical problem that affects up to 20% of the population1 with the recurrence after treatment on the level of 50%. Urinary stones form in the course of many physical and chemical processes. Besides having an organic matrix, they consist mainly of crystalline phases of various substances. In the case of the stones related to metabolic disorders, the most predominant crystalline components are calcium oxalate monohydrate (whewellite), carbonate appatite, and calcium oxalate dihydrate (weddellite).2 It happens very often that different components are found in different layers of the same stone. Some of the components are very easy to dissolve, others may grow as crystals with sharp edges that may damage the patient’s tissues. As an example, Figure 1a presents the SEM image of the crystalline structure of kidney stone revealed after its fracture. The crystal habit is very characteristic for weddellite. There are seen very typical sharp edges.

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A separate kind of urinary stones is the so-called infectious stones related to urinary tract infection.4 Several urinary pathogens are isolated from kidney stones very frequently (for example refs 5 and 6). They are mainly the microorganisms from the Proteus species which are isolated in the case of 70% of the bacteria-induced urinary stones.7,8 If the stone formation is the result of an infection by microorganisms producing urease, magnesium ammonium phosphate hexahydrate (NH4MgPO4 · 6H2O), known as struvite, is the most common crystalline component. The urine of a healthy person is undersaturated with regard to struvite formation. However, urease produced by microorganisms plays a main role in the crystallization process and is the important virulence factor of these bacteria.9,10 Urease is an enzyme splitting urea9 into carbon dioxide (CO2) and ammonia (NH3): * Corresponding author. Phone: +48 042 6313653; fax: +48 042 6313639; e-mail: [email protected]. † Technical University of Ło´dz´. ‡ University of Ło´dz´.

urease + H2O

98 CO2 + 2NH3

(1)

H2N-C-NH2 urea Generated ammonia increases urinary pH, which leads to 23elevation of the concentration of the NH+ 4 , CO3 and PO4 ions. These ions together with the ions of calcium Ca2+ and magnesium Mg2+ present in the urine lead to the crystallization of struvite, according to the following reaction:4 pH g 7.2

3Mg2+ + NH+ 4 + PO4 + 6H2O 98

MgNH4PO4 · 6H2O

(2)

Struvite stone may grow very rapidly and may involve the entire renal pelvis and calyces, which may lead to the blockage of the urinary tract. The minor component which crystallizes as a result of the infection is carbonate appatite formed according to the general reaction:4

Figure 1. SEM image of the crystalline structure of a kidney stone revealed after its fracture.3

10.1021/cg900281g CCC: $40.75  2009 American Chemical Society Published on Web 07/06/2009

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pH g 6.8

2+ CO2+ 6PO33 + 10Ca 4 98 Ca10(PO4)6CO3

(3) This component is present as an amorphous precipitate and does not form crystals of defined morphology typical to struvite. The above shows the importance of the analysis of the struvite growth process, notably in the presence of appropriate urinary pathogens. In the present paper, we describe the systematic investigation of the habits of struvite grown from synthetic urine in the presence of Proteus, as this bacterium is mainly isolated from infectious stones. We also present analysis of crystal habits based on relative growth rates for the most typical form of struvite crystals obtained during experiments. The analysis may provide insights into the nature of the struvite growth mechanisms in this specific biological environment.

2. Experimental Procedures The synthetic urine used during crystal growth experiments was made from the following components10,11 (g/L): CaCl2 · 2H2O, 0.651; MgCl2 · 6H2O, 0.651; NaCl, 4.6; Na2SO4, 2.3; KH2PO4, 2.8; KCl, 1.6; NH4Cl, 1.0; urea, 25.0; creatine, 1.1; and tryptic soy broth, 10.0. The content of the mineral components in this synthetic urine corresponds to mean concentration found in 24-h period in normal human urine. In all experiments, a Proteus mirabilis strain isolated from human kidney stone was used. Before the experiment, bacteria were maintained on a slant of tryptic soy agar overnight at 37 °C and then suspended in synthetic urine to the concentration of 5 × 105 CFU per mL (CFUcolony forming unit - is a unit used in microbiology that indicates the number of microorganisms present in a sample). The crystallization occurs after addition of the suspension of bacteria and incubation at 37 °C. The crystallization process occurred at conditions emulating the natural conditions existing in the human body during the infection by Proteus sp. During all the growth experiments, crystals samples were taken at regular intervals and observed by phase-contrast microscopy (Nikon Eclipse TE2000-S). The pH and the concentration of appropriate ions in the synthetic urine solution were screened along the experiments. Crystals were identified as struvite by X-ray diffraction, using an X’pert PRO MPD (PANanalytical) diffractometer. In order to estimate Miller indexes of particular faces, angular measurements were performed, as the obtained crystals are too small to use a single crystal X-ray diffractometer. Additionally, to be sure that the faces are indexed properly, we compared the estimated Miller indexes with those published in literature.12

3. Experimental Results During experiments, we obtained habits typical for crystals growing in living animal13 and human12,14 organisms. The obtained crystals habits are presented in Figure 2. The sizes of most struvite crystals were from 25 to 60 µm (along b-axis) and their habit strongly depends on pH, which increases with time, as explained in the Introduction. The aspect ratio AR defined as the length lb along the b-axis to the length la along a-axis varies from 1 to 2. At low pH (below 7.2; 2 h) we did not observe crystals; Figure 2, panel a1. At this stage, the synthetic urine is undersaturated with regard to struvite formation; however, we can see the concentration of bacteria around places where the nucleation begins; Figure 2, panel a1. For pH values higher than 7.2, we observed individual crystals; Figure 2, panels b1 and b2. With the increase of pH, the amount of crystals was greater. The most basic crystal morphology is typical hemimorphic morphology, that is, the two ends of a crystallographic c-axis are not related by symmetry.15 The crystals are composed of the (001) and (001j) pedions, the {101}, {101j}, and {012} domes, and the {010} pinacoid. A pinacoid is a set of two

symmetrically equivalent and parallel faces, whereas a pedion is a single face that is not symmetrically equivalent to any other faces. A dome is a set of two intersecting faces that are caused by mirroring. All these forms are defined in Figure 3. The parallel (001) and (001j) faces that terminate the c-axis on opposing ends are not symmetrically equivalent. These (001) and (001j) faces are separate pedions, misidentified sometimes as pinacoid faces. In our experiments, usually the (001j) pedion is larger than the (001) pedion. The crystals with the (001) pedion at the end of the c-axis takes a coffin-lid shape; Figure 2, panel c1, arrow 1. The crystals that are terminated by pedions (001) and (001j) at both ends are quite common. However, we have observed also the crystals terminated on one end (-c-axis) by the (001j) pedion and on the other by dome faces. This means that the (001) pedion decreases or disappears totally and the {011} or {012} faces remain. In our experiments the {012} form is expressed more frequently. If the (001) pedion disappears, the remaining {012} faces create an edge (Figure 2, panel e1, arrow 1) or corner (Figure 2, panel c2, arrow). The existence of the {012} form on only one end of the crystals also exemplifies the hemimorphic nature of the struvite crystal. Both the {011} and {012} forms were observed in struvite crystals grown from natural and synthetic urine.12 In some crystals, we have also observed the {010} form; Figure 2, panel e1, arrow 2 - compare with Figure 3b. The shape of the crystal seen from the side of the (001) face is presented in Figure 2, panel c1, arrow 2 compare with Figure 3c. When the pH increases further, the habits of single crystals remain the same, but the crystals very frequently form twins. We may divide them into two groups. First group of twins is composed of two (or three) hemimorphic crystals one rotated 60 or 90 degrees relative to the other; Figure 2, panel d1 and b2. It should be emphasized that twins are observed very frequently at higher pH, approximately equal to 9.5. Therefore, such a twin as shown in Figure 2, panel b2 for pH ) 8.1 is formed rarely. Second kind of twin is a typical penetration twin with the (001) pedions as the penetration plane; Figure 2, panel d2. In some experiments, we have observed X-shaped crystals turn into dendrites; Figure 2, panel e2. This suggests that, at this stage, the growth process is controlled by volume diffusion and the supersaturation is lower on the crystal surface than in the bulk of the solution of synthetic urine. Moreover, the supersaturation is the highest at the corners and edges and is the lowest at the center of a given crystal surface. As a consequence, dendrites and dendritic branches occur. The causes of the formation of dendrites may also be associated with the formation of an increasing number of 3D nuclei and their aggregation. It seems that the bacterial substances may serve as sites for heterogeneous nucleation and subsequently aggregation. This is in agreement with the reports of other researchers.16,17 During the experiments, the pH increases from 5.8 to 9.5. The highest value of pH equal to 9.5 was achieved after 5 h of incubation and is correlated with the bacteria vitality. It was observed that high pH acts bacteriostatic and bactericide after 10 and 18 h of incubation, respectively. The hemimorphic morphology of struvite crystals is correlated with the crystal structure, presented in Figure 4. The structure + 2+ consists of PO34 and NH4 tetrahedra and Mg[H2O]6 octahedra (orange, blue, and green polyhedra, respectively). These molecular units are held together by hydrogen bonds. On the basis of this structure, we can see that the hemimorphism is controlled by the symmetry of the atomic arrangement of the crystal. The

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Figure 2. (a-e) Dependence of growth habits of struvite crystals grown from synthetic urine on pH (time). Scale bar: yellow: 50 µm and red: 20 µm.

+ Figure 4. The structure of struvite: PO34 and NH4 tetrahedra and Mg[H2O]2+ 6 octahedra - orange, blue, and green polyhedra, respectively; O and H atoms - red and white balls, respectively; c-axis is polar; plane of observation: (010).

Figure 3. Schematic representation of struvite crystal showing hemimorphic morphology, plane of observation: (a) (001); (b) (100); and (c) (010). There is no symmetry equivalence between (001) and (001j) pedions and the c-axis is polar.

hemimorphic nature is best seen looking, for example, at the PO43-tetrahedra. Throughout the structure, the apexes of all the tetrahedra are oriented toward the +c-axis, whereas the trigonal bases of the tetrahedra are all oriented toward the direction of -c. Therefore, the (001) and (001j) faces are not the same and not symmetry related. They have different properties and typically grow differently. Such nonequivalence often extends from atomic arrangement to the physical and chemical properties of the ends of the crystal. This may lead, for example, to different activation energies for desorption and desolvation processes for the opposite faces and their different extension. Additionally, different structures of opposite surfaces may influence the binding of additives to either enhance or inhibit the crystallization process. This is related to additives such as citrate,18 heparin sulfate,19 or albumin,20 known to play a role in struvite crystallization. It relates also to bacterial polysaccharides containing negatively charged residues which are able

to bind the Ca2+ and Mg2+ ions.11 Polysaccharide is the most external surface component of Proteus bacterium, which probably plays a role in aggregation precipitated components to form a stone.9,11 These ions accumulate around bacterial cells and the crystallization process appears to be mediated by specific molecular interactions between molecular structures of the crystal surface and molecular arrays around bacterial cells. Research on this subject is our future goal and is in progress.

4. Kinetic Conditions for the Evolution of Struvite Morphology-Theoretical Analysis Struvite belongs to the noncentrosymmetric point group mm2 of the orthorhombic system with space group Pmn21. The unit cell parameters are as follows:21 a ) 6.941 Å, b ) 6.137 Å, c ) 11.199 Å. To check the lattice constants of our crystals, X-ray powder diffraction was used. Measurements were carried out with X’pert PRO MPD diffractometer. The peak indexing was done comparing our XRD pattern with those of the Inorganic Crystal Structure Database (ICSD) card (collection code 014269).

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and is related to the changes in the size of the (001) pedion in relation to the {012} faces. For relative growth rate R(001)/R{012} g 1.35 (region I), the (001) pedion is not present and the {012} faces form the [100] edge (for example, crystals eh1, eh2 in Figure 5); for relative growth rate R(001)/R{012} smaller than 1.35 (regions II and III) the (001) face is present (crystals p1, p2 in Figure 5). Curve 2 is given by

R(001) ) 1.90 R{101}

(5)

and is related to the changes in size of the (001) pedion in relation to the {101} faces. For relative growth rate R(001)/R{101} g 1.90 (region IV), the (001) pedion is absent and the {101} faces create the [010] edge (crystal ev1 in Figure 5); for relative growth rate R(001)/R{101} smaller than 1.90 (region II and III) the (001) face is present (crystals p1, p2, or p3 in Figure 5). Curve 3 is given by the formula: Figure 5. Dependence of the relative growth rate R(001)/R{012} on the relative growth rate R(001)/R{101}. The curves 1, 2, 3, and 4 are given by eqs 4, 5, 6 and 7, respectively, and they delimit four regions, namely, I, II, III, and IV, in which crystals of different shapes grow. All faces, which appear in crystals, are defined in the upper left corner.

Our angular measurements and comparison with published data12 show that the most typical crystal morphology is composed of the (001), (001j), {101}, {101j}, {012}, and {010} faces. The aspect ratio AR defined as the crystal length along the b-axis to the length along the a-axis varies from 1 (see Figure 2, panel c1, arrow 3, aspect ratio for this crystal is 1.10) to 2 (see Figure 2, panel c1, arrow 1, aspect ratio for this crystal is 2). The crystal morphology differs from each other by the (001) pedion. In some cases, the pedion is present and elongated along the b-axis; therefore, the crystals take a coffin-lid shape (Figure 2, panel c1, arrow 1). For crystals with AR close to 1, the (001) pedion takes an almost square shape (Figure 2, panel c1, arrow 3). It happens also that the pedion is not present, but the {012} faces, which remain in crystal morphology form the edge (Figure 2, panel e1, arrow 1) or corner (Figure 2, panel c2, arrow). To explain such changes in crystal morphology, we perform some calculations of relative growth rates of individual faces, as the changes in such rates are responsible for the changes in crystal habit. To perform such calculations, we use a kinetic and geometric approach,22-24 in which a crystal is considered as a set of crystallographic planes intersecting in 3D space. In this formulation, we do not consider the growth process at the molecular level; that is, the incorporation of growth units onto crystal surface and the motion of parallel steps on the surface are not taken into account. Keeping in mind these limitations, we use a kinetic and geometric approach as a first description of growth morphology giving notion about relative growth rates of faces. The details of the derivation are explained in Appendix A. The results of our calculations are illustrated in Figure 5, which presents the dependence of the relative growth rate R(001)/R{012} on the relative growth rate R(001)/R{101}. In Figure 5 we have four curves, first one is given by the expression:

R(001) ) 1.35 R{012}

(4)

R(001) ) R{012}

1 1 0.79 + 0.32 R(001) /R{101}

(6)

For relative growth rate R(001)/R{012} satisfying the above equation, all four edges of the (001) face increase equally fast (crystal p3 in Figure 5). For relative growth rate R(001)/R{012} lying below the curve 3 (region III), the [010] edge of the (001) face increases faster than the [100] edge. As a result, the (001) face is elongated along the b-axis (crystal p4 in Figure 5). If the crystal grows with the relative growth rate R(001)/R{012} within the region II, the edge [100] of the (001) face increases more slowly, and the pedion is elongated along the a-axis (crystals p1 and p2 in Figure 5). Curve 4 is given by the following formula:

R(001) R(001) /R{101} ) R{012} 1.40

(7)

Equation 7 is related to the changes in length of the [100] and [010] edges in the case when the (001) pedion is absent in crystal morphology. Therefore, eq 7 is valid for region I and IV only (for these regions the (001) face is absent in crystal morphology). If the R(001)/R{012} ratio is greater than the right side of eq 7 (region I) the [100] edge exists (for instance crystal eh4 in Figure 5); if eq 7 is satisfied both the [100] and [010] edges disappear forming corner (crystals c1 and c2 in Figure 5); for the R(001)/ R{012} ratio smaller than the right side of eq 7 (region IV) the [010] edge is present (crystal ev1 in Figure 5). As described above, these four curves delimit four regions in which crystals with or without the (001) pedion appear and with an increase of the R(001)/R{101} ratio the habit changes into elongation along the b-axis. It is seen also that with a decreasing value of the R(001)/R{101} ratio, the crystal loses rectangular symmetry and new edges truncating corners appear. This testifies about the interrelation between growth rates of the {012}, {101}, {010}, and {101j} faces, but it does not mean that new faces appear. The crystals with such truncated corners have been observed very rarely in our experiments. Comparing this theoretical analysis and crystals experimentally observed, we may conclude that our crystals grow with relative growth rates R(001)/R{012} and R(001)/R{101} within the III and IV regions. Theoretical analysis in connection with experimental results demonstrates that the crystals elongated along the b-axis with

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Figure 6. Dependence of the aspect ratio AR on relative growth rates R{101}/R{010} and R{101j}/R{010} given by eq 8: (a) 3D graph; (b) 2D graph of the same dependence. The lines 1 and 2 in (b) represent the intersections of the AR surface given by eq 8 with the AR ) 1 and AR ) 2 planes. The AR ) 2 plane is not shown in (a).

the (001) pedion grow with a relative growth rate R(001)/R{012} smaller than 1.2 and at the same time with a relative growth rate R(001)/R{101} greater than 1. The change in relative growth rates implies the disappearance of the (001) pedion during the growth process. For a relative growth rate R(001)/R{101} greater than 1.90 the (001) pedion is absent and the {012} form creates appropriate edge or corner. Hitherto existing analysis does not explain well enough the changes in the aspect ratio of crystals under consideration. In our crystals, we have not observed the {100} form; therefore, the length of the crystal along the a-axis is not so obvious. However, it is easy to derive that in such a case the aspect ratio AR defined as the length lb along the b-axis to the length la along the a-axis is given by

AR )

lb 2cos R ) R{101¯} la R{101} + R{010} R{010}

(8)

where R is the interfacial angle between the (101) and (101j) faces equal to 31.80° (the interfacial angle is defined as an angle between normals to the appropriate faces). Dependence given by eq 8 is presented in Figure 6a. Additionally, the AR ) 1 plane is presented. The line 1 in Figure 6b is the intersection line of the surface given by eq 8 with the AR ) 1 plane. The line 2 in Figure 6b is the intersection of the surface given by eq 8 with the AR ) 2 plane, which is not shown in Figure 6a. These two lines AR ) 1 and AR ) 2 in Figure 6b delimit the range of relative growth rates R{101}/R{010} and R{101j}/R{010} for which the crystals with an aspect ratio 2 > AR > 1 appear. This means that our crystals which achieve AR from a range of 1 to 2, grow with relative growth rates R{101}/R{010} and R{101j}/R{010} which are limited by the lines 1 and 2 (Figure 6b). As an example, we present in Figure 6b the crystals with AR ) 1.1 and AR ) 2, which are very similar to that observed experimentally: compare with crystals shown in Figure 2, panel c1, arrow 3 and 1, respectively.

5. Conclusions Formation and morphology of struvite crystals strongly depend on the pH of synthetic urine, which is correlated with bacteria vitality. For pH in the range from 7.5 to 8.5, the most typical crystal morphology is hemimorphic morphology. The

aspect ratio of such crystals varies from 1 to 2. Crystal faces at opposite ends of the crystallographic c-axis are different. This property may be very important in the case of additive’s (macromolecule’s) interaction with individual surfaces of struvite crystals. At higher pH, many twins and dendritic crystals grow. The growth of dendrite crystals supposedly is associated with the formation of an increasing number of 3D nuclei. We suggest that the bacterial substances may serve as sites for heterogeneous nucleation. Theoretical analysis based on kinetic-geometric approach demonstrates that the crystals which were experimentally observed grow with relative growth rate R(001)/R{012} smaller than 1.2 and at the same time with a relative growth rate R(001)/R{101} greater than 1. These conditions correspond to the growth of crystals with elongated (001) pedion along the b-axis. For a relative growth rate R(001)/R{101} greater than 1.90 the (001) pedion is absent and the {012} form creates an appropriate edge or corner. The results presented in this work are preliminary, but they show that the investigation of bacteria-induced growth process of struvite crystals can increase our understanding of the mechanisms of kidney stone formation in the complex biological environment. Acknowledgment. The authors are grateful to our colleague from Institute of Physics, Technical University of Łódz´, Mr. Włodzimierz Wypych, for the SEM image of the kidney stone (Figure 1).

Appendix A The existence and evolution of individual faces of struvite crystal is analyzed from a macroscopic point of view using a kinetic and geometric approach. Most of the contemporary kinetic and geometric models are based on two quantities:23,24 critical relative crit and tangential growth rate dlhkl/dt. Critical growth rate Rhkl/Rhkl crit growth rate Rhkl is defined as the normal growth rate of the (hkl) face at which a given size lhkl of this face is preserved.25 As it is more convenient to use the relative growth rates rather than absolute rates, we use the critical relative growth rate Rhkl/Rcrit hkl , which is given in the form:25

Rhkl crit Rhkl

)

sin(R + γ) sin γ sin R + Rhkl /Rh1k1l1 Rhkl /Rh2k2l2

(A1)

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Crystal Growth & Design, Vol. 9, No. 8, 2009 3543 edge created by the (001) and (012) faces) and [010] (the edge created by the (001) and (101) faces). We assumed that both these edges increase equally fast, and therefore the tangential growth rates dl[100]/dt and dl[010]/dt are equal to each other. From this condition, we obtain the following equation:

R(001) 1 ) 1 R{012} sin R sin 2γ sin R sin 2R + sin γ R(001) /R{101} 2 sin2 R sin2 γ (A3)

(

Figure A1. Schematic graph illustrating the cross-section of a crystal with the considered (hkl) face represented by lhkl and its neighboring (h1k1l1), (h2k2l2) faces, their normal growth rates Rhkl, Rh1k1l1, Rh2k2l2, respectively, and appropriate interfacial angles R and γ.

where Rhkl, Rh1k1l1, Rh2k2l2 are the normal growth rates of the (hkl), (h1k1l1), and (h2k2l2) faces, respectively, R and γ are the interfacial angles between normals to the pairs of faces (hkl), (h1k1l1) and (hkl), (h2k2l2), respectively, as illustrated in Figure A1. crit The physical meaning of the ratio Rhkl/Rhkl is as follows: (i) for crit crit Rhkl/Rhkl < 1, the size lhkl of the (hkl) face increases; (ii) for Rhkl/Rhkl crit ) 1, the size lhkl of the (hkl) face is preserved; and (iii) for Rhkl/Rhkl > 1, the (hkl) does not appear in crystal habit or if it exists, the size lhkl of the (hkl) face decreases. The tangential growth rate dlhkl/dt describes the instantaneous rate of changes in cross-section size lhkl of a given (hkl) face and is given by26

Rh1k1l1 sin γ + Rh2k2l2 sin R - Rhkl sin(R + γ) dlhkl ) dt sin R sin γ (A2) The positive value of the dlhkl/dt rate corresponds to an increase in the size of the given (hkl) face, negative value means that the (hkl) face decreases, and if dlhkl/dt ) 0, the given (hkl) face does not change. On the basis of the kinetic-geometric formulation, we are able to analyze and predict the evolution of crystal shape, appearance, and disappearance of crystal faces and other phenomena occurring during the crystal growth without taking into account the processes at the molecular level. In order to analyze the relative growth rates, which correspond to the growth of crystals which we have observed experimentally, we apply eq A1 for the (001) pedion. We substitute into this equation appropriate interfacial angles and consider the case crit Rhkl/Rcrit hkl ) 1. From the condition Rhkl/Rhkl ) 1 applied for the (001) pedion, we have obtained eqs 4 and 5. Equations 4 and 5 correspond to the evolution of the (001) pedion in relation to the {012} and {101} forms, respectively. In the case of eqs 4 and 5, we substitute R ) γ ) 42.40° (interfacial angles between normals to the pairs of faces (001), (012) and (001), (01j2) which are equal to each other) and R ) γ ) 58.20° (interfacial angles between normals to the pairs of faces (001), (101) and (001), (1j01) which are equal to each other), respectively. Equation 6 is derived based on eq A2 applied to the edges of the (001) pedion. The (001) pedion possesses four edges, but parallel edges are equivalent. Those we may consider two edges: [100] (the

)

Substituting for R ) 42.40o and γ ) 58.20o, we have obtained the dependence R(001)/R{012} on R(001)/R{101} in the form given by eq 3. Equation 7 describes the interplay between the edges [100] and [010] in the case when the (001) pedion is absent in crystal morphology. It is derived based on the tangential growth rate but in the 3D case. Equation A2 describes the tangential growth rate, that is, the instantaneous rate of changes in cross-section size lhkl of the (hkl) face. This means that eq A2 is related to the 2D case which cannot be applied in our consideration. The precise derivation and analysis of the 3D case are presented in ref 26. All eqs 4-7 were rearranged properly aiming to have the same variables in all equations.

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