Crystal Growth Shape of Whewellite Polymorphs: Influence of

Crystal Growth Shape of Whewellite Polymorphs: Influence of Structure Distortions on Crystal Shape Angel Millan* Solid State Chemistry, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands and Instituto de Ciencia de Materiales de Arago´ n, CSIC-Universidad de Zaragoza, 50009 Zaragoza, Spain

CRYSTAL GROWTH & DESIGN 2001 VOL. 1, NO. 3 245-254

Received December 28, 2000

ABSTRACT: The theoretical and experimental crystal growth shape has been determined for the two whewellite polymorphs. Theoretical studies consisted of PBC analysis, Donnay-Harker, Ising temperature, and Attachment Energy calculations. These calculations were based on accurate energy values, which were specially derived for the whewellite crystal structure by ab initio quantum mechanics. The morphology of crystals grown below and above the polymorph transition temperature is similar. However, theoretical morphologies are different, due to the doubling of the unit cell dimensions in the low-temperature structure. These results suggest a refinement of the thickness rule in Hartman-Perdock (H-P) theory. The Attachment Energy method is the most accurate of the three theoretical morphology methods used in this report. The high performance of the Ising model for this ionic crystal is probably due to the directionality of Ca-oxalate bonds. It has been found that crystal shape changes substantially along growth, massive penetration twinning occurs below the transition temperature, and the action of growth inhibitors is mainly directed to the (100) face. These phenomena are related to the crystal structure. The shape of whewellite natural crystals has been revised with the aim to show that the study of crystal shape can give clues about the growth conditions of natural crystals, with emphasis on renal calculi. Introduction The high interest of calcium oxalates as major components of renal calculi is a well-known subject.1 Consequently, a huge amount of publications have been devoted to the study of the kinetics of nucleation, growth, and agglomeration of calcium oxalates. To understand the mechanisms of formation of renal calculi and how to avoid them, it is also necessary to improve our knowledge of the morphology and habit of these crystals. Calcium oxalate presents three degrees of hydration: monhydrate (COM, whewellite), dihydrate (COD, weddellite), and trihydrate (COT). The growth shapes of the last two salts have been extensively studied.8,9 With respect to whewellite, the information on real crystal shape is spread over reports focusing on other items, and the only theoretical study on the subject consists of a PBC analysis.10 However, whewellite is by far the most abundant and undesirable component in renal calculi.1 Besides, this is the most frequent calcium oxalate phase in minerals11 and plants.12,13 The analysis of periodic bond chains (PBCs) on crystal structure, based on Hartman-Perdock (H-P),2 has become a standard initial step in the prediction of crystal morphology. It yields a set of F-faces that should be likely to appear in crystals grown from their vapor. Whewellite is an interesting case for H-P theory. This compound presents an enantiotropic transformation between 38 and 45 °C. This transformation is only distortional, and it hardly affects the bonds network. Hence, it should not make any significant difference on * To whom correspondence should be addressed. Angel Millan, Instituto de Ciencia de Materiales de Arago´n, CSIC-Universidad de Zaragoza, 50009 Zaragoza, Spain. Phone: 34 976 762301. Fax: 34 976 76 1229. E-mail: [email protected].

crystal surface energy, and we should expect a similar shape on crystals grown above and below the transition temperature. On the other hand, the key requisite of H-P theory for a face to be flat is to contain two nonparallel interconnected PBCs within a slice of dhkl thickness. Whewellite structural transition implies a doubling of the b-axis. Presumably, this change of lattice dimensions will make a difference on the F-character of some faces and consequently on crystal shape prediction. Thus, whewellite offers a good opportunity to check the applicability of the thickness criterion of PBC theory. A further step in crystal shape studies is the calculation of the relative morphological importance of F-faces. The most simple procedure for establishing a hierarchy of F-faces is that of Donnay-Harker (D-H),3 which relates the growth rate of a face to its interplanar distance. Other models involve energy calculations on crystal slices parallel to the face. One of them is the Ising model that is based on the order-disorder transition temperature of two-dimensional nets of bonds contained in a crystal slice.4-6 Because it only accounts for first neighbors interactions, it works better for covalent crystals. Another one is the attachment energy model that comprises Coulomb and van der Waals energy summations on infinite crystal slices.7 Therefore, it is especially suitable for ionic crystals. As it is shown in this paper, calcium-oxalate interactions are ionic, but they are also directional. Thus, it is interesting to compare the performance of the two methods on whewellite crystal shape prediction. This paper is concerned with the real and theoretical shape of whewellite crystals. A PBC analysis has been carried out on both whewellite’s polymorphic structures. The relative morphological importance of F-faces has been estimated by means of D-H, Ising temperature,

10.1021/cg0055530 CCC: $20.00 © 2001 American Chemical Society Published on Web 03/30/2001

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Crystal Growth & Design, Vol. 1, No. 3, 2001

and Attachment Energy methods. Crystal binding has been studied by ab initio quantum mechanics. A pointcharge model for the oxalate ion has been derived from the quantum potential at the very atomic sites in whewellite lattice. Our own computer program based on the Ewald summation method has calculated the attachment energy of F-faces. This paper also presents observations of the crystal growth shape on a variety of experimental conditions. We followed the evolution of crystal shape along growth, and we studied the influence of temperature, supersaturation, and additives. Theoretical crystal shapes are compared to natural and synthetic crystal shapes. The thickness rule for F-slices is discussed in the light of these studies. The shape of natural whewellite crystals is critically revised, with the aim to see how the study of crystal shape can teach about their growth history. Some conclusions concerning renal calculi formation mechanisms have been presented. Finally, the use of no less than seven different unit cell choices for whewellite crystal structure has occasionally caused some confusion among urolithiasis researchers on the indexing of crystal faces and twin planes. This report tries to tackle this problem to some extent. Experimental Methods Whewellite crystal samples were grown in semibatch experiments by addition of stock solutions of CaCl2 and Na2C2O4 or H2C2O4, under mechanical stirring and controlled temperature. Two different crystallizers were used in the experiments with 1 and 7 L capacity, respectively. The mother liquor was either pure water or an aqueous solution containing 10-2 mol/L of CaCl2 and/or NaCl in several amounts ranging from 10-4 to 10-1 mol/L. Run times ranged from 2 to 24 h. Experiments were carried out at 25, 35, 50, and 70 °C. Several reactant addition rates were used in a range between 250 and 1000 mL/h. The concentration of the stock solutions was varied between 3 × 10-3 and 2 × 10-2 mol/L. The pH and the concentration of Ca2+ ions in the growing solution were screened all along the run by potentiometry. In another set of experiments, whewellite crystals were grown at conditions simulating the functioning of human kidneys in the absence and in the presence of citric acid and mucin. This procedure consists of sliding down a calcium oxalate supersaturated solution on a porous substrate emulating the dropping of urine from the renal papillae. Precipitated whewellite crystals grow stuck to the substrate. More details about the conditions used in these experiments can be found elsewhere.14 During all the growth runs, crystal samples were taken at regular intervals and observed by optical microscope and scanning electron microscope (SEM). Theoretical Calculations. (a) Whewellite Crystal Structure. Calcium oxalate monohydrate presents three polymorphs. One of them15 has an orthorhombic structure, it is only obtained by dehydration of the dihydrate salt, and therefore it is not relevant to this paper. The other two phases (whewellite) transform reversibly in the temperature range 38-45 °C. Both of them have a monoclinic structure. The hightemperature structure (basic)16 belongs to space group I2/m with unit cell: a ) 9.978, b ) 7.295, c ) 6.292, β ) 107.07°. According also to Deganello,17 the low-temperature structure belongs to space group P21/n, and it has a double unit-cell with dimensions: a ) 9.9763, b ) 14.5884, c ) 6.2913, β ) 107.05°. However, we have followed Tazzolli’s unit cell choice18 for convenience (P21/c, a ) 6.290, b ) 14.583, c ) 10.116, β ) 109.46°). To avoid confusion with other works, we offer a listing of relevant faces indexes according to the three most widely used notations,11,17,18 in Table 1.

Millan Table 1. Whewellite Crystal Face Indexes in Several Notation Systems

a

Tazzollia

Goldschmidtb

Deganelloc

{100} {010} {001} {101 h} {011} {111 h} {102 h} {021} {121} {121 h} {112} {013} {031} {131} {131 h} {123 h} {140} {141 h} {151 h} {161} {161 h}

e {1 h 01} b {010} c {001} a {100} y {012} r {210} µ {101} x {011} φ {112 h} m {110} p {216 h} w {016} i {032} {234 h} n {230} f {112} δ {121 h} u {120} d {250} s {132 h} l {130}

{1 h 01} {010} {100} {001} {110} {011} {101} {120} {221 h} {021} {311 h} {310} {130} {231 h} {031} {221} {141 h} {041} {261 h} {261 h} {061}

Ref 18. b Ref 11. c Ref 17.

Drawings about the main features concerning basic crystal structure are given in Figure 1. Calcium ions occupy two different positions in the first sphere of coordination of oxalate ions (Figure 1). Ca2+ ions at position (I) are in close distance to two oxygen atoms forming a chelato bond with oxalate. The second position (II) corresponds to a single bond. A third kind of interaction in COM crystals is the hydrogen bond. It is formed between two different water molecules, and between water molecules and oxalate ions. The first coordination sphere of calcium ions consists of eight oxygen atoms making two chelato bonds, three single bonds, and a water hydration bond. There are two types of oxalate ions present in equal amounts (Figure 1). Ox1 ions are disposed in planes parallel to the (100) face together with all of the Ca2+ ions. Each Ox1 ion binds six coplanar Ca2+ ions, two of them by chelato bond, and the other four by single bond. Ox2 ions are situated between (100) layers parallel to (010) planes. Each Ox2 ion binds four Ca2+ ions, two of them by chelato bonds and the other two by single bonds. Hydration water molecules are contained within Ox2 layers, and they form hydrogen bonds with Ox2 ions and among themselves. The derivative structure is almost identical to the basic structure with respect to ion coordination. The main difference concerns Ox1 ions. In the derivative structure, they suffer a slight distortion of bond angles (a few tenths of a degree) and bond distances (differences lower than 0.01 Å). At the same time, they get slightly off the (100) crystal plane, and they are no longer coplanar. In this way, they lose the 2/m local symmetry and the dimensions of the b-axis are doubled. PBC Analysis of the Basic Structure. Note that, in the following, we name faces according to Tazzolli’s convention, even if they are not appropriate for the basic structure. We realized a revision of PBCs on 3 × 3 × 3 tridimensional crystal graphs, in which oxalate ions and water molecules were reduced to a point located at their center of mass. We found PBCs along directions: [100], [010], [001], [101], [012], [101 h ], [012], [412], [4h 12h ] and those symmetrically equivalent. The directions and thickness of the first five PBCs are represented in Figure 1. PBC [101] is just built by chelato bonds. At this stage, it must be pointed out that chelato bonds also constitute the backbone in COD9 and COT8 crystal structures. PBC [012] (and equivalent [012h ]) is built by alternating links of two single bonds and a chelato bond. We have revised the crystal structure of all the metal oxalates included in the Cambridge Crystal Structural Database, and we found PBCs of both kinds in all the metal oxalates apart from silver oxalate. A PBC running along the [100] direction is composed of double links of chelato bond and single bond in alternating sequence. Other relevant PBCs go along [001] and [010] directions. These

Crystal Growth Shape of Whewellite Polymorphs

Crystal Growth & Design, Vol. 1, No. 3, 2001 247

Figure 1. Whewellite basic crystal structure and binding. Molecular graphs correspond to slices with a dhkl thickness cut along real crystal edges (surrounding polygons). Directions and thickness of PBCs are indicated by arrow ending lines and slashed lines, respectively. At the top, the two kinds of calcium-oxalate bonds and the coordination of each type of oxalate ion are represented.

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Crystal Growth & Design, Vol. 1, No. 3, 2001 Table 2. Binding Energies in Whewellite Basic Structure (kcal/mol)

bond type

QM (6-31G*)

charge model

vdW

Ox1-Ca1 Ox2-Ca1 Ox-Ca1a Ox2-Ca2 Ox1-Ca2 Ox-Ca2a Ox-H2O H2O-H2O

-460.5 -460.1 -460.3 -387.9 -388.5 -388.2

-431.1 -432.2 -430.1 -362.5 -362.1 -362.3

2.0 3.7 2.9 1.8 1.8 1.8

a

Millan Table 3. Theoretical Morphological Importance of F-Slices in Whewellite Crystalsa

H-bond

-3.0 -4.9

Average value.

strong PBCs determine the F-slices: {100}, {040}, {002}, {021}, and {121 h } (Figure 1). Slices with a dhkl thickness, parallel to the first five faces, are represented in Figure 1. The rest of PBCs are weakest as they contain links of one single bond or just one H-bond. After an inspection of all the possible crossings PBCs that conform a connected net, we found only another two F-slices: {102 h } and {102}. We also performed a computer search of connected nets for (hkl) slices with h, k, and l smaller than 5, using the program FIND•NETS developed by Geertman.19 This search confirmed our eye inspection. PBC Analysis of the Derivative Structure. According to Hartman-Perdock,2 the main requisite for a F-face is to contain at least two nonparallel interconnected PBCs in a slice of thickness dhkl. The doubling of the unit-cell dimensions in the derivative structure with respect to the basic structure allows for new PBCs and F-faces. In a previous work, Franchini et al.10 found the following PBCs for the derivative structure: [100], [010], [001], [101], [101h ], [011] and [012]. According to the same report, these PBCs determine the F-faces: {100}, {010}, {001}, {101 h }, {011}, {021} and {121 h }. We perform a revision of this PBC analysis and we found other PBCs along directions [211], [2 h 11 h ], [212], [2 h 12 h ], [412] and [4 h 12 h ]. These PBCs originate the F-faces: {102}, {102 h }, and {122 h }. Therefore, as expected, there are three new F-faces in the derivative structure that were not allowed in the basic structure: {101 h }, {011}, and {122 h }. Bond Energies. Considering the very small differences in bonding distances and coordination with respect to the derivative structure, we have studied the chemical interactions in whewellite just from the basic crystal structure. The energy of interaction between calcium and oxalate ions has been calculated by ab initio quantum mechanics methods using the program GAMESS20. In a first stage, a geometry optimization was carried out on oxalate ions surrounded by punctual double charges located at the positions of Ca2+ ions in the crystal structure. Several orbital basis sets were consecutively employed in this geometry optimization, starting with an STO basis and ending with a 6-31G* basis. The obtained geometry was used in self-consistent field (SCF) energy calculations on C2O42+ ions and C2O42+-Ca2+ ion pairs. SCF calculations were also carried out on isolated Ca2+. Then, the energy of C2O42+Ca2+ interactions was estimated from the difference between the energy of ion pairs and that of isolated ions. The values obtained are presented in Table 3. The energies of van der Waals interactions between oxalate and calcium ions were calculated using parameters from Teleman et al.21 The values obtained are also presented in Table 2. It can be seen that the contribution of van der Waals forces to crystal binding is negligible. About the nature of C2O42+-Ca2+ interactions, Ca-O interatomic distances, ultraviolet and infrared absorption spectroscopy,22 and the contribution of calcium atomic orbitals to molecular orbitals on calcium-oxalate pairs are all indicating that they are purely ionic. On the other hand, calculations of quantum potential around the oxalate ion performed by us showed that Ca2+ ions in the vicinity of oxalate ions are located at points of oxalate potential minimum. Moreover, the sp2 hybridation of O atoms in oxalate implies an accumulation of electronic density in the directions of the two free electron pairs of the O, which are very close to the directions of O-Ca bonds.

Eatt F-slices

1/dhkl

1/θc

basic

derivative

{100} {040} {002} {202 h }b {011}b {102} {102 h} {021} {121} {121 h} {122 h }b

0.17 0.27 0.21 0.34 0.13 0.28 0.22 0.17 0.26 0.22 0.26

0.478 0.71 0.332-0.766 0.731 0.413-0.820 0.931-0.954 0.872-0.896 0.599 0.767 0.711 0.711-1.114

-1.3443 -1.5466 -1.9208

-1.0985 -1.8359 -1.3336 -1.6633 -1.5035 -2.1320 -2.2012 -1.7753 -1.8487 -1.6409 -2.6598

-2.5288 -2.7534 -1.9372 -2.3212 -1.1909

a θc, order-disorder transition temperature (relative); E , att energy of attachment (e2/Å). b Only allowed in the derivative structure.

Table 4. Point Charges Used in the Calculations of Attachment Energies Ca charge

2

C 0.992

O (oxalate)

O (water)

-0.996

-0.733

H 0.3665

Furthermore, the revision of metal-oxalate mentioned above reveals that this is a very common scheme. Therefore, although ionic in character, C2O42+-Ca2+ bonds resemble covalent bonds in that they are directional. The energy of hydrogen bonds (Table 2) was estimated from the expression:

EHB ) E12(5(R12/r12)12 - 6(R12/r12)10)cos4R (kcal/mol)

(1)

extracted from Brooks et al.,23 where E12 ) 9.5, R12 ) 2.75, r12 is the distance between the two oxygen atoms involved in the hydrogen bond, and R is the angle (OsH‚‚‚O). Derivation of a Point-Charge Model for the Oxalate Ion. The point-charge model used in lattice energy calculations has been specially derived for whewellite crystal structure. Ab initio SCF calculations using a 6-31G* orbital basis were performed on an oxalate ion surrounded by a grid of charges. The geometry of the oxalate ion was that of Ox1 in whewellite basic crystal structure. The grid was a reproduction of the crystal structure around and Ox1 ion, within a radius of three unit cells. The values of the charges in the grid were 2.0, -0.898, 0.797, -0.733, and 0.3665 at the positions of Ca, O, C, O(water), and H, respectively. These values were taken from previous reports.24,25 The resulting orbitals were used for the calculation of electrostatic potential at the crystal atomic sites. Then, a least squares optimization routine was used to fit a charge model to the ab initio potential. This charge model consisted of four identical charges situated at the oxygen centers and another two identical charges at the carbon centers. The obtained charge values are presented in Table 4. The mean quadratic deviation of the model from ab initio values is 0.19%. The largest deviations are found at the nearest points, and they were always below 3%. Table 2 shows a comparison of the energy values for Ca-oxalate bond from the point-charge model and from the ab initio calculations. The differences are quite acceptable considering the simplicity of the model. Calculation of the Relative Morphological Importance of F-Faces. The relative morphological importance of F-faces was estimated by three different methods. The D-H approach3 relates the linear growth rate of crystal faces to the inverse of their lattice interplanar distance, dhkl. These values are presented in Table 3. The Ising morphology model is based on order-disorder transition temperatures (θc) of two-dimensional nets comprised within dhkl slices. Crystal shape results from making the linear growth rate (Rhkl) of a face proportional to its 1/θc value. The method for the calculation of θc was originally developed for rectangular nets.26 The application of this model to crystal



Figure 2. Theoretical growth morphology of whewellite crystals according to Donnay-Harker, Ising Temperature, and Attachment Energy models. slices requires that nonrectangular nets are previously transformed into rectangular. Some tools have been developed to perform transformations of this kind.4-6 However, for some nets with crossing bonds, an exact transformation is not possible. Then, maximum and minimum values for net strength are obtained, respectively, by considering the energy of the stronger bond as infinite, and by deleting the weakest bond. This was the case of slices: (002), (011), (021), (102 h ), and (102). Then, we used mean values in the derivation of the crystal shape. The results are presented in Table 2. The third method relates linear growth rate of a face (hkl) to the energy of attachment (Eatt) of a crystal slice parallel to this face, with a dhkl thickness. The Eatt values are obtained from the difference between the lattice energy and the slice energy. Coulomb energy calculations were carried out by means of a computer program developed by us according to the method proposed by Heyes et al.27 and Strom et al.,28 which is based on the Ewald lattice summation method.29 The pointcharge models used in these calculations are shown in Table 4. The origin of the oxalate point-charge model has been explained above, and the water model was taken from Woods et al.25 The charge distribution function σ employed in the Coulomb calculations was one of the four proposed by Heyes: 27

σ ) u exp(-u/η)/(24πη4)

(2)

where u is the radial distance, and η is an adjustable parameter. The summation was made cell by cell for 30 concentric spherical layers with a cell thickness. The variation of the total potential over a unit cell in the last 10 spheres was always less than 0.15%, and often less than 0.01%. After subtracting the internal energy of oxalate ions and water molecules, the lattice energy values for the basic and the derivative whewellite crystal structures are 2126.1 and 2210.9 kcal/mol, respectively. Replacing the values for nearest neighbors interactions of the point-charge model for the more accurate ab initio values yields 2290.1 and 2374.9 kcal/mol. As expected, the energy of the low-temperature polymorph (derivative) is higher than that of the high-temperature polymorph. However, this gain of lattice energy is achieved at the cost of oxalate ion vibration energy. Thus, it is not surprising that, for a moderate increment of the thermal

Figure 3. SEM pictures of whewellite crystals grown at 70 °C: (A) from pure solutions and (B) from solutions containing a high ionic strength and an excess of Ca2+ ions. energy, the derivative structure is already transformed into the basic structure, in which oxalate ions have a higher symmetry. Calculated values of Eatt for F-slices are given in Table 3. Theoretical Growth Shapes of “Basic Crystals” and “Derivative Crystals”. The theoretical growth shape of whewellite crystals with a derivative structure and a basic structure is depicted in Figure 2 , panels A and B, respectively. It can be appreciated that according to the D-H model, derivative and basic crystals should have a very different morphology. Differences arise from the thickness rule of H-P theory for selecting F-faces. According to this rule, the thickness of slices containing connected nets parallel to flat faces should be dhkl or shorter. Following this rule, {101 h } and {011} faces have an S-character for the basic structure. However, due to the doubling of the lattice parameter b, they have an F-character in the derivative structure. According to the attachment energy model, these faces would not be very relevant, and from the Ising model their area should be small. However, the D-H model, which is exclusively dependent on lattice parameters, attributes them a predominant role. The D-H model differs from the rest also in that {021}, {121 h } faces should not appear in “derivative crystals” and the {010} face should not appear in “basic crystals”. The attachment energy and the Ising model are quite coincident especially for the basic structure. For both models, the most relevant face should be the {100} followed by {121 h }, {021}, {001}, and {010}.

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Millan

Figure 4. SEM pictures of whewellite crystals grown at 37 °C from pure solutions: (A) Flat penetration twin crystals, shortly after nucleation (bar is 2 µm). (B) Detail of the end of a prismatic penetration twin crystal showing striations along the [101] crystallographic direction. (C) Penetration twin crystals after a 2 h growing period; faces (021) and (121 h ) have been developed at the crystal ends (bar is 10 µm).

Results To observe the shape of whewellite crystals at near equilibrium conditions, some samples were grown during 6 h, in 7 L vessels, from pure solutions, at 70 °C and very low reactant addition rates. At these conditions, the final precipitate consists of crystals with smooth faces and perfect edges. Most of the crystals are single, and a few of them are (100) contact twins. The crystals are bounded by {100}, {010}, {121 h }, and {021} faces with surface areas decreasing in this order (Figure 3A). When the precipitation temperature is lowered to 50 °C, crystal perfection diminishes and aggregation augments, but the crystal shape hardly changes. However, when the precipitation is carried out at 37 or 25 °C, most of the crystals are penetration twins (Figure 4). In this case, growing crystals show a prismatic habit (Figure 4A). Side faces are {100} and {010}, and the faces at the ends are rough with striations going along the [101] direction (Figure 4B). End faces meet at reentrant angles, and they form acute dihedral angles with {100} faces, with which they share [021] and [021 h] edges. Macrosteps are observed on {100} faces advancing from the ends. As the crystals grow, they develop {021} faces over the sharp edges and indentations with a {121 h } orientation on the central part of the crystal (Figure 4B). These events coincide with a decrease of the growth rate along the direction of elongation, thus the shape of individual crystals becomes nearly isometric. The final crystal morphology is identical to that of crystals grown at high temperatures. Experiments were performed at 25, 37, and 70 °C using several rates of addition of reactants. In general, an increase of the reactant addition rate produces an enhancement of nucleation and crystal aggregation. At temperatures of 25 or 37 °C, and reactant addition rates of 500 mL/h or higher, crystal intergrowth and agglomeration are massive, and surface roughening takes

Figure 5. SEM pictures of whewellite crystals grown by a method emulating kidney stone functioning: (A) in the absence of additives, (B) in the presence of citric acid, (C) in the presence of mucin.

place especially at the crystal edges. However, it has little influence on the final crystal morphology. Semibatch experiments were carried out in a medium with a high ionic strength and an excess of calcium ions. Resulting crystals are flattened along the [100] direction, and they do not show {021} faces (Figure 3B). The influence of discrete polyanions, such as citrate, and that of macromolecules with charged groups, such as mucin, were studied in a system emulating the process of formation of stones in the kidneys. Using a pure aqueous mixture of the reactants, crystals have the typical morphology and habit found on semibatch precipitation (Figure 5A). However, crystals precipitated from solutions with 0.2 M ionic strength and 2.5 × 10-4 mol/L of citrate are elongated hexagonal platelets (Figure 5B). The tabular faces are {100} and the lateral faces are {010} and {121 h }, and occasionally {021} also. When the crystals are grown in the presence of mucin (a mucoprotein), the lateral faces get rough and the crystals show a lentil-like shape, often with serrated sides (Figure 5C). Apart from that, nucleation is very much enhanced, and intergrowth is extremely frequent in the form of typical rosette aggregates. It was observed in every experiment that the crystal shape changes significantly as the crystals grow. This



Figure 6. Evolution of the crystal shape with the growing time for whewellite single crystals (1), contact twins (2), and penetration twins (3). Table 5. Morphology of Whewellite Mineral Crystals from Goldschmidta frequency (%) form

total

dominant

large

small

{100} {010} {001} {121 h} {021} {131 h} {123 h} {141 h} {161} {140} {011} {161 h} [Ω]b {151 h} {013} {121} {112} {031}

78 75 69 67 53 50 46 46 39 24 24 19 18 11 11 11 10 10

9 11 2 10 2 3 1 1 2 4 0 0 1 0 0 0 1 0

18 9 17 17 17 12 9 6 6 7 1 4 6 1 1 0 4 0

8 14 12 4 11 6 9 10 7 1 2 4 1 3 1 2 1 0

a Ref 11. b Nomenclature by Kolbeck et al. Beitr. Kryst. I 1918, Taf. 21.

process is represented in Figure 6 for single crystals, contact twins, and four part penetration twins. Early, crystals are rodlike, elongated along the [001] direction (Figure 4A). Gradually, they become tabular on {100} faces (Figure 4B). Then, crystals are converted into prisms by enlargement of {010} faces (Figure 4C). Finally, they adopt an isometric habit after the development of {121 h } and {021} faces (Figure 4D). Discussion The Real Crystal Growth Shape. Our crystal growth experiments at near-equilibrium conditions and high temperature, and experiments performed by other authors using slow growth methods,10,30 indicate that the main forms in synthetic crystals are {100}, {010}, {021}, and {121 h }. These are also some of the most frequent and dominant faces in mineral crystals (Table 5). However, the morphology of mineral crystal presents several features that are rarely found in synthetic

Figure 7. (A) Possible equilibrium shape of whewellite crystals, as deduced from synthetic and natural samples. (B) Crystal drawing showing the most usual habit modification caused by additives. (C) Shape of crystals grown in the presence of mucoproteins.

crystals. The most remarkable is the high frequency of the {001} form, which is rarely the dominant face, but it is usually one of the largest. Another important feature is the occurrence of faces belonging to the [101] zone. The absence of the {001} form in synthetic crystals may be due to growth kinetics. In fact, we have observed that the development of faces in synthetic crystals is gradual: first {100}, then {010}, and later {121 h } and {021} faces. Thus, it is possible that development of {001} faces in synthetic crystals occurs much later, after a growing time out of the reach of our experiments. That would explain their occurrence in mineral crystals. Besides, the d002 slice cuts a larger number of chelato bonds than d021 and d12-1 slices, and chelato bonds may play an important role in the kinetics of growth of whewellite crystals. In any case, {001} should be considered as a relevant face in whewellite real crystals. Thus, putting together observations on synthetic and natural crystals we can infer that the real “equilibrium” morphology of whewellite crystals is composed of {100}, {010}, {121 h }, {001}, and {021} faces. On average, the first three show a similar morphological importance, and the other two are a little less relevant. On the other hand, the relative surface area of these faces varies even among crystals from the same precipitate. Thus, the crystal shape on Figure 7A can be taken as a reference for the equilibrium morphology of whewellite crystals. Faces in the [101] zone are far less frequent than the five mentioned above, and their surface area is usually small. These S-faces contain the PBC [101], which is built by chelato bonds, and it is the strongest of all. They could form by adsorption of foreign molecules on [101] steps. Actually, we have observed [101] steps on the end faces of synthetic crystals grown at low temperatures (Figure 4B). Crystal Twinning. It has been observed that (100) penetration twinning is much more frequent among

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“derivative crystals”. It is significant that the structure transition affects mainly to oxalate ions situated in this plane (Ox1). They are perfectly aligned within (100) planes in the high-temperature polymorph and get off the (100) plane in the low-temperature polymorph. Penetration twinning could be related to this off-plane distortion of oxalate ions. A molecular dynamics simulation of this phenomenon could be an interesting exercise to understand the causes of twinning. Comparison of Real and Theoretical Crystal Shape. The PBC analysis on the basic structure yields seven F-forms. Observations on real crystals have shown that five of these faces compose the equilibrium morphology of whewellite crystals. At this stage, we can already state that H-P theory has been successful in the prediction of whewellite crystal morphology. On the other hand, the theoretical importance of crystal forms depends on the method used in the calculations. Comparing Figure 2B and Figure 7A, one can see that the Attachment Energy model is very accurate. The Ising model predicts the apparition of the {102 h } face, which is usually absent on real crystals, it gives an excessive importance to {021} and {001} faces, and too little importance to the {010} face. However, taking into account that we used mean values for the Ising temperature of some faces, the precision of the Ising method is remarkable. This success arises from the fact that metal-oxalate bonds are quite directional, and it outlines the importance of short-range interactions in calcium oxalate crystal growth. The D-H model yields also a realistic crystal shape. It fails in predicting the presence of {102 h } face and the absence of {010} face. The Effect of Structure Distortions on Crystal Shape. The slight structural distortions occurring on the derivative structure carry out a doubling of the b lattice parameter. That makes a considerable change on the PBC analysis of crystal structure. Three new F-slices are now allowed, {101h }, {110}, and {122h }, which have a forbidden thickness in the basic structure. The consequences on crystal shape are different for each method. Within the D-H framework, based on lattice geometry, these new faces have a large relative importance. This leads to a crystal shape that is completely different from the one derived from the basic structure (Figure 2A.1 and Figure 2B.1). However, for the Ising and the Attachment Energy models, which are rooted on binding energy, the importance of the new faces is small, especially in the first case (compare Figure 2A.2 and Figure 2B.2, and Figure 2A.3 and Figure 2B.3). It has been observed experimentally that the real growth morphology of crystals is not affected by the structural phase transition. It is very clear for the three theoretical models that predictions based on the basic structure are much more precise than those based on the derivative structure. Thus, concentrating exclusively on the derivative structure, it can be stated that the dimensions of the b-axis should be reduced to a half to perform a realistic PBC analysis. This is an important conclusion for crystal shape prediction. Geometrical criteria could be insufficient for shape prediction of crystals that have their symmetry reduced by small molecular distortions. In these cases, energy criteria should be considered in PBC analysis prior to the application of the thickness rule.

Millan

Evolution of Crystal Shape during Growth. The shape of whewellite crystals changes along growth. The development of a needlelike shape at the initial stages of precipitation may be related to the strong tendency of oxalate ions to form chelato chains with metal ions. Chains of this type are found in most of metal oxalate crystals. The structure of oxalate double salts with divalent and alkaline metal ions usually consist of linear polymeric chelato complexes of the divalent metal and oxalate ions surrounded by the alkaline metal ions. Considering that supersaturation is very high at the initial stages of precipitation,31 complexes of this type may be present in the solution at this stage. These complexes would form linear nuclei rather than spherical ones. A random condensation of these linear nuclei would explain the spherolite formations of needle crystals that are so typical in precipitation by fast mixing of concentrated reactant solutions. The formation of stepped faces at the ends of crystals grown at low temperature can be related to kinetics. At moderate to high supersaturation, whewellite crystal growth rate is determined by a surface reaction mechanism in which dehydration of the calcium solvates is the slowest step.32 Thus, the formation of stepped faces during rapid growth should not be surprising. Once crystal growth rate decreases, the stable {121 h } and {021} faces can develop at the crystal ends covering the stepped faces. Obviously, the linear advance of these faces is slower than that of stepped faces and the crystal finally reaches an isometric habit. Habit Modifications. About the modifications of crystal habit on crystals grown from solution, the first factor to be considered is the effect of the solvent. The needlelike habit of urea crystals in water solutions is a well-explained example of this effect.33 In the case of whewellite, both Ca2+ and C2O42- ions interact extensively with water molecules. Ca2+ ions form stable hydrates and C2O42- ions form strong H-bonds with water molecules. Thus, one should expect an important wetting effect on whewellite crystals grown in water. To establish the relative importance of solvent wetting on F-faces would involve extensive computer calculations.33 On the other hand, a rough inspection of surface structure of the main whewellite crystal faces shows that structural hydration water molecules are pointing outward for all of them. This restricts the possibilities of strong differences on wetting energies between faces. Besides, the good agreement between theoretical crystal shape from the vapor phase and that of crystals grown from pure water solutions indicates that the wetting is apparently similar for the main crystal faces. The main modification of whewellite crystal habit induced by additives is the flattening of the crystals along the [100] direction. There are many reports in the literature about the effect of additives on whewellite crystal growth shape. We can only cite a few representative examples.34-39 They all find a similar flattening effect on {100} faces. This modification does not seem very selective because it can be induced by a high ionic strength, as we showed here. The origin of crystal flattening could be the polar character of the {100} face. The stacking of ions along the [100] direction consist of alternating layers of Ox2 ions and Ca-Ox1 ions in a 2:1 ratio. Thus, a full (100) surface will have either a


negative or a positive net charge. It is expected that in an ion-rich environment, such as a high ionic strength solution, a double charge layer would be formed at the (100) crystal interface surface. This double charge layer would restrain the growth of the face by hindering the diffusion of add-atoms from the bulk. Therefore, any ionic species in enough amounts can cause this effect. From another point of view, it can also be said that most of ionic species would be preferably adsorbed on this face. Nevertheless, some substances with an appropriate charge and geometry may adsorb selectively onto the growth sites and cause the same growth restrain effect in lower amounts. This seems to be the case for citrate, glutamic acid, cholate, poly(acrylic acid),34 aspartic-rich proteins,35 sodium diisooctyl sulfosuccinate,36 and others.37 All these substances have several charged groups and a relatively flexible structure so they can link to several Ca2+ sites on the (100) surface. The case of mucoproteins is special because they are large tridimensional polymers with abundant side chains ending in carboxylate and sulfate groups. These macromolecules can attach firmly to the (100) whewellite crystal surface and also to the crystal side faces. The result is a flat and oval crystal shape. Khan et al.38 reported that mucoproteins from membrane vesicles initially produce a crystal flattening, but they can reach a polyhedral shape when they are kept in contact with the mother phase for a long period (72 h). Tabular crystal shapes have also been observed in the presence of phospholipid micelles.39 Concerning the treatment of urolithiasis decease, many substances may be capable of restraining whewellite crystal growth. However, their effect is mostly directed to the (100) face. The design of a whole inhibitor should (mainly?) regard growth blockage along the [001] direction also. The Shape of Natural Crystals and Their Growth History. The knowledge of crystal morphology and habit can be useful to infer the conditions of formation of natural crystals. We give a few examples. For instance, we can guess certain features about the process of formation of whewellite renal calculi from the results of this work. There are two types of whewellite renal stones that show a different texture.40 In stones of the first type, crystals have a polyhedral shape, and they are randomly arranged. Because they nearly reach the equilibrium morphology, they must grow slowly and permanently immersed in the mother phase. Actually, this type of stone is formed in the renal calyx where the urine can be retained. In stones of the second type, crystals have a tabular habit. We can expect that crystals in stones of type II grew in the presence of additives such as polyanions, and that they spent a short time in contact with the mother phase. In fact, these types of stones grow hanging from the renal papillae under a constant flow of urine that drops from the renal tubules. The cause of their tabular shape is multiple, because urine contains a high ionic strength, polyanions, and macromolecules detached from damaged epithelium. Other biomaterials with a structure resembling that of whewellite kidney stones such as eggshells and eel’s otholiths41 are probably developed in a similar environment. Most of whewellite mineral crystals are single or


contact twins with a shape that can be isometric, prismatic, or spear-point-like.11 Thus, they were probably nucleated at moderate supersaturation (otherwise crystal intergrowth would be observed), at high temperature (or penetration twins would be frequent) and with long times in contact with their mother phase (or they would not reach an isometric habit). Some mineral crystals have a simple morphology composed of {100}, {021}, {010}, and {121 h } forms. Therefore, they were grown at moderate rates from pure solutions with a low ionic strength. Many of them show a rich morphology, which is consistent with the presence of additives and very low growth rates. Whewellite is also present in plants, either in the leaves,12 or near the roots associated with fungus hiphae.13 Crystals in the leaves usually show a needlelike shape and occasionally a tabular habit. It can be inferred that they were formed at high supersaturations, and they grew for a brief period. Their tabular shape implies a high content of polyanions in the growing medium (citrate and malate are abundant in the leaves). On the other hand, crystals found in the plant roots are penetration twins, obviously because they are formed at a moderate temperature. Their habit is prismatic, and they form compact aggregates. Thus, they probably grew at moderate to high supersaturation, and for relatively long periods. Actually, fungus associated to the roots exudes oxalic acid in a regular way. Conclusions The main forms in whewellite crystals are {100} and {010}, which appear in all kind of whewellite crystals. Following in importance are {121 h } and {021}, which appear in synthetic crystals after a growing period and are present in most of the natural whewellite crystals. Next, it is the {001} form that is absent in synthetic crystals but very abundant in mineral crystals. These five forms constitute the equilibrium morphology of whewellite crystals. The structure distortion of the low temperature polymorph does not have any apparent effect on crystal morphology. However, it makes an important difference in the outcome of PBC analysis. This difference arises from a strict application of the thickness rule in the selection of F-slices. The slight molecular distortions hardly affect to the intermolecular interactions. However, the doubling of the unit cell dimensions is introducing an artifact in the selection of F-faces. It is suggested that binding energy criteria should be considered when applying the thickness rule. From the three crystal shape prediction models assayed, the Attachment Energy model is the most accurate. The Ising model is more successful than expected for an ionic crystal, probably owing to the high directionality of oxalate-calcium interactions. Synthetic whewellite crystals adopt several habits during growth: elongated, prismatic, and finally a coffin-like shape. The main habit modification induced by additives is the flattening along the [100] direction. It is due to the polar stacking of ions in along this direction and apparently can be caused by any electrolyte. However, polyanions with a flexible chain are very effective. Clues about the growth conditions of natural whewellite crystals from mines, plants, and renal calculi can be derived from the study of their morphology and habit.

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Acknowledgment. The author is indebted to Prof. P. Bennema, Dr. C. Strom, Dr. R. Geertman, and the staff of the CAOS-CAMM centre of the University of Nijmegen. References (1) Schneider, H. J. In Urolithiasis Ethiology, Diagnosis; Springer-Verlag: Berlin, 1985. (2) Hartman, P. In Crystal Growth An Introduction; NorthHolland: Amsterdam, 1973. (3) Donnay, J. D. H.; Harker, G. Am. Mineral. 1937, 22, 446. (4) Bennema, P.; Gies, E. A.; Weidenborner, J. E. J. Cryst. Growth 1983, 62, 41. (5) van der Eerden, J. P.; Bennema, P. J. Cryst. Growth 1983, 61, 45. (6) Rijpkema, J. J. M.; Knops, H. J. F.; Bennema, P.; van der Eerden, J. P. J. Cryst. Growth 1983, 61, 295. (7) Hartman, P.; Bennema, P. J. Cryst. Growth 1980, 49, 145. (8) Heijnen, W. M. M. J. Cryst. Growth 1982, 57, 216. (9) Heijnen, W. M. M.; van Duijneveldt, F. B. J. Cryst. Growth 1984, 67, 324. (10) Franchini-Angela, M.; Aquilano, D. Phys. Chem. Miner. 1984, 10, 114. (11) Goldschmidt, V. M. In Atlas der Krystallformen; Winters Universita¨tsbuchhandlung: Heidelberg, 1923; band IX. (12) Ishii, Y.; Takiyama, K. J. Electron Microsc. 1989, 38, 423. (13) Graustein, W. C.; Cromack, K., Jr.; Sollins, P. Science 1977, 198, 1252. (14) Grases, F.; Costa-Bauza, A.; March, J. G. Br. J. Urol. 1994, 74, 298. (15) Schubert, G.; Ziemer, B. Cryst. Res. Technol. 1981, 16, 1025. (16) Deganello, S. Acta Crystallogr. B 1981, 37, 826. (17) Deganello, S.; Piro, O. E. N. Jb. Miner. Mh. H 1981, 2, 81. (18) Tazzoli, V.; Domeneghetti, C. Am. Mineral. 1980, 65, 327. (19) Geertman, R. FIND•NETS Program, University of Nijmegen, Holland, 1992.

Millan (20) Program GAMESS, CAOS/CAMM Center, University of Nijmegen. (21) Teleman, O.; Ahlstroem, P. J. Am. Chem. Soc. 1986, 108, 4333. (22) Krishnamurty, K. V.; Harris, G. M. Chem. Rev. 1961, 61, 3. (23) Brooks, B. R.; Bruccoleri, R. E.; Olafson, B. D.; States, D. J.; Swaminthan, S.; Karplus, M. J. Comput. Chem. 1983, 4, 187. (24) Strom, C. S.; Grimbergen, R. F. P.; Hiralal, I. D. K.; Koenders, B. G.; Bennema, P. J. Cryst. Growth 1995, 149, 96. (25) Woods, R. J.; Khalil, M.; Pell, W.; Moffat, S. H.; Smith, V. H., Jr. J. Comput. Chem. 1990, 11, 297. (26) Onsager, L. Phys. Rev. 1944, 65, 117. (27) Heyes, D. M.; van Swol, F. L. J. Chem. Phys. 1981, 75, 5051. (28) Strom, C. S.; Hartman, P. Acta Crystallogr. A 1989, 45, 371. (29) Ewald, P. P. Ann. Phys. (Leipzig) 1921, 64, 253. (30) Cody, A. M., Jr.; Cody, R. D. J. Cryst. Growth 1987, 83, 485. (31) Millan, A.; Grases, F.; Sohnel, O.; Krivankova, I. Cryst. Res. Technol. 1992, 27, 31. (32) Nielsen, A. R.; Toft, J. M. J. Cryst. Growth 1984, 67, 289. (33) Liu, X. Y.; Boek E. S.; Briels, W. J.; Bennema P. J. Chem. Phys. 1995, 103, 3747. (34) Sarig, S.; Kahana, F.; Fuchs, J. J. Cryst. Growth 1988, 87, 431. (35) Addadi, L.; Weiner, S. Proc. Natl. Acad. Sci. U.S.A. 1985, 82, 4110. (36) Cody, A. M.; Cody, R. D. J. Cryst. Growth 1994, 135, 235. (37) Tunik, L.; Addadi, L.; Garti, N.; Furedimilhofer, H. J. Cryst. Growth 1996, 167, 748. (38) Khan, S. R.; Whalen, P. O.; Glenton, P. A. J. Cryst. Growth 1993, 134, 211. (39) Brown, C. M.; Novin, F.; Purich, D. L. J. Cryst. Growth 1994, 135, 523. (40) Millan, A. J. Mater. Sci. Mater. Med. 1997, 8, 247. (41) Lecomte-Finiger, R. Nature 1994, 370, 424.

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Crystal Growth Shape of Whewellite Polymorphs: Influence of

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