Crystal morphology of octacalcium phosphate: Theory and observation

416

Journal of Crystal Growth 82 (1987) 416—426 North-Holland, Amsterdam

CRYSTAL MORPHOLOGY OF OCTACALCIUM PHOSPHATE: THEORY AND OBSERVATION R.A. TERPSTRA

*

Institute of Denial Materials Science, University of Nijmegen, P.O. Box 9101, 6500 HB N(/megen, The Netherlands

and P. BENNEMA RIM Laboratory of Solid State Chemistry, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands Received 8 November 1986; manuscript received in final form 20 October 1986

The crystal growth of octacalcium phosphate (OCP) (Ca

8H2(P04)6 . 5H20) is thought to be of great importance for the growth of calcified tissues like enamel, dentin and bone. The growth form of OCP is predicted on the basis of the Hartman—Perdok theory. Thirty-four f slices are identified and the order of morphological importance, based on the attachment energy, is given and compared to the few literature data on the morphology of synthetic OCP crystals. The predicted and observed morphology agree well. The suggestion in the literature [Brown et al., Progr. Crystal Growth Characterization 4 (1981) 59] for the choice of boundary of the important (100} form, which plays a key role in the proposed mechanism for the growth of OCP and hydroxyapatite, can now be supported by physical arguments based on the attachment energy.

1. Introduction The crystal growth of octacalcium phosphate (OCP), Ca8H2(P04)6 5H~O,is thought to be of importance for the growth of renal stones, dental calculus and the mineral particles in calcified tissues like enamel, dentin and bone [1—6]and is also gaining interest in industries in connection with the precipitation (scaling) of calcium phosphates in general due to the increased phosphate concentrations in rivers and lakes [7]. The prototype for the mineral of calcified tissues like enamel, dentin and bone is hydroxyapatite (OHA), CA10(PO4)6(OH)2, with hexagonal space group P63/m. (The space group of OCP is triclinic P1). The crystal morphology of bone, dentin and young actively calcifying enamel crystallites however is plate or ribbon like and the thin plates are elongated along the crystallographic c axis [8—141,which is not consistent with the apatite .

*

Present address: Netherlands Energy Research Foundation, ECN, P.O. Box 1, 1755 ZG Petten, The Netherlands.

symmetry. This plate like habit is thought to be due to the involvement of OCP in the first stages of crystal growth of bone and tooth tissues. Although OCP is thermodynamically metastable with respect to OHA and calcium deficient apatites, OCP forms first due to kinetical reasons and is then (partly) hydrolized to apatite [4]. Alternatively it has been suggested that the low symmetry of biological apatitic crystals is due to a lowering of the apatite symmetry due to the ordering of OH and OH-columns to P21/b [15], but this is unlikely because the Ca—OH bonds are not very important for the morphology of apatite crystals [16] and there still exists a strong hexagonal pseudosymmetry in monoclinic hydroxyapatite. In this paper we will determine the growth form of OCP on the basis of the Hartman-Perdok theory [17—20].Essential in this theory are the concepts of periodic bond chain (PBC) and F face. A PBC is an uninterrupted path of strong bonds in the crystal structure having a period [uvw] of the lattice. An F face is a crystallographic face (hkl) in which two or more sets of PBCs are

0022-0248/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

R.A. Terpstra, P. Bennemu

/ Crystal morphology of octacalcium phosphate

interconnected. The morphology of a crystal structure is determined by the set of F faces of the crystal structure. We will determine the most important PBCs and F faces of OCP and the order of morphological importance (MI) of each F face. The MI is defined as a statistical measure for the relative frequency of occurrence and/or the relative size of the faces of the form { hkl } (the set of faces which are equivalent through the symmetry operations of the space group of the crystal structure under consideration). There are different criteria to the used to determine the order of the F faces of a crystaistructure in which they contribute to the morphology of the crystal. We will use in this paper the criterion of the attachment energy Eatt [17,21]. The smaller E,~, the higher the MI of the F face (hk/). The attachment energy is defined as the energy which is released when from the vacuum a stoichiometric unit attaches to a crystal face. The slice energy, ~ is the energy of a stoichiometric unit and is complementary to Eatt. The sum of both is the crystallization energy, Ecr, which is a bulk property and, unlike Eatt and Esl~ independent of the F face under consideration. To calculate the attachment energy we use a broken bond model which will only give reliable results for the larger dhki values in case of highly ionic substances like OCP. But it will be shown that the crystallographic faces that are important for the morphology are those with relatively large dhkl values so that the values calculated for the attachment energy will be quite reliable. Other criteria which can be used to determine the order of morphological importance of F faces are the classical interplanar distance dhki [21—24], the sum and the product of the edge energies c~ and ~ (see ref. [25]) and the Ising critical temperature 9~[26] of an F face (see for example ref. [27] where all these criteria are used and compared to each other). In order to use the criterion of the Ising critical temperature the growth units are reduced to centres of gravity and the crystal structure is in fact reduced to a crystal graph. A crystal graph is defined as a set of an infinite number of interconnected points, which fulfils the symmetry of one 0C

417

of the 230 space groups [28,29]. In such a crystal graph connected nets, also called F slices, which run parallel to F faces can be determined. Connected nets can be transformed into planar rectangular Ising nets by allowing for bonds of strength zero and infinite. For each net an Ising temperature can be c~iculated below which the F face parallel to the net grows flat and above which the surface of the F face roughens up [29]. the higher O~of an F face (connected net) the higher the MI. In this paper we will also use the concept of crystal graph but no attempt is made to calculate values. Generally, the connected nets of OCP are much more complex than those of the apatite structure which we analysed before [16]. For the most complex (double) nets of the apatite structure we had to make an over- and an underestimation of 9C in order to be able to make the net planar. This was done by leaving out crossingbonds (under estimation) or by taking some bonds infinitely strong (over estimation). This gave a sometimes great range for the actual O~value. For OCP this range would even be greater. But on the other hand it appeared from the apatite analysis that the prediction based on the criterion of Eatt runs quite parallel to that based on G~,so that we may expect t~hiscriterion to also give good results for the OCP structure which resembles the apatite structure quite a lot. In the end we will compare the prediction of the morphology of OCP crystals with the few data from the literature on the morphology of synthetic OCP crystals. 9C

2. The crystal structure of octacalcium phosphate 2.]. Chemical composition, structure and space group

Octacalcium phosphate (OCP) can contain various amounts of water of crystallization which can easily enter or leave the structure through channels within the hydrate layer (see below). With the maximum amount of water of crystalization its chemical formula is Ca8H2(PO)4)6 5H2O. The crystal structure of OCP resembles the structure of calciumhydroxyapatite with formula Ca10(P04)6(OH)2. In ref. [1] it was first shown .

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R.A. Terpstra, P. Bennema

418

Crystal morphology of octacalc,’um phosphate

that OCP consists of apatitic and hydrated layers.

OCP this means that only Ca—P04 bonds will be considered since P04 as a whole is a growth unit. The OCP unit cell contains 84 Ca—P04 bonds of which 42 are essentially different (see figs. lb and lc). For apatite we have shown [16] that on the basis of the coulombic interaction the Ca—P04 bonds are very close to each other. This will also apply to the apatitic layer of OCP. The Ca—PO4 bonds related to the hydrated layer of OCP are of the same order of magnitude as those in the apatitic layer except for the bonds between Ca(3) and P04 with P(14) (see ref. [1] and fig. la). We will “replace” the two weaker bonds between Ca(3)

This is also shown in figs. la and 2 by projections of the crystal structure of OCP and apatite in the [001] directions, The cell parameters of OCP are a 19.87 A, b 9.63 A, c 6.87 A, x 89°17’, $ 92°13’ and y 108 °57’[1]. =

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Fig. 1. (a) Projection in the [001] direction of the OCP crystal structure. One unit cell is shown and the so-called apatitic (compare fig. 2) and hydrated layer have been indicated. The position of the centres of symmetry is shown. P(14), Ca(3), Ca(4) and the term “strong bond” refer to ref. [1] and/or section 2.2 (b) The pseudo z-coordinates of Ca and P04 ions in the [001] projection of the OCP structure. In case of two coinciding ions, two z coordinates are shown. Note that the origin is different form (a). (c) The [001] projection of OCP showing Ca—P04 bonds. Each line represents one Ca—P04 bond. Using the symmetry —1 it can be counted that there are 84 Ca—P04 bonds per unit cell. Note that the origin is different from (a).

and two equivalent P04 ions with P(14) for one strong bond between Ca(3) and P(14) as indicated in fig. la. So in fact Ca(3) is shifted to a position exactly on top of Ca(4) which also has a strong bond with the nearest P04 having P(14). When the growth units of the crystal structure are reduced to centres of gravity and when the cell

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constants a and $ are both taken 900 (as we have also done in our projections) instead of 89°17’ and 92°13’ respectively a pseudomirror plane is perpendicular to the crystallographic c-axis and runs through the plane of the centres of gravity representing the phosphate groups (approximately). The centres of symmetry of the OCP crystal structure are exactly in between the pseudomirror planes so that the symmetry of the crystalgraph is now P21/m. This will simplify the otherwise even more complex PBC analysis of the OCP structure since fewer PBC directions need to be investigated. The approximation of one strong bond instead of two weaker bonds will not create additional F slices. In figs. ic and 6 the different bonds are shown (see also section 5). In figs. 4, 5 and 6 the length and direction of the crystallographic axes are shown starting from the same origin (different from ref. [1]) which will help to visualize the crystal structure of OCP.

420

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R.A. Terpstra, P. Bennema

Crystal morphology of octacalcium phosphate

3. Application of the Hartinan—Perdok theory of OCP

It is expected that the shortest translation distances of the lattice are potential PBC directions and these are investigated systematically. The PBCs that have been used in the analysis have been indicated in fig. 3.

3.]. Geometrical criterion to search for the most important PBC directions

According to the Hartman—Perdok theory [17—20],the morphology of a crystal is governed by uninterrupted chains of strong bonds running through the structure and having a period [uvw] of the lattice. Such a periodic bond chain (PBC) is not necessarily stoichiometric (primitive PBC) but can be made stoichiometric (complete PBC) by adding growth units to the PBC. Only the strong bonds that are formed during the crystallization process are relevant.

3.2. Geometrical criterion for the search of F slices

Crystallographic faces which consist of two or more sets of interconnected PBCs are called F faces and these F faces determine the crystal morphology [17—20].F slices run parallel to F faces. We will investigate the crystal forms { hkl) of OCP with decreasing interplanar spacing dhk, which are parallel to two or more PBC directions (see ref. [30], table 1 and also section 3.3).

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R..4. Terpstra, P. Bennema

/

Crystal morphology of octacalcium phosphate

Table 1 The interplanar distance dhk/ (ref. [30]) of the faces which were investigated is given as well as their character (F for a face with two or more sets of interconnected PBCs, S for a face with only one set of parallel PBC5); for the F slice the attachment energy is given and also the morphological importance based on the attachment energy; the attachment energy is expressed in number of Ca—P0 4 bonds per cell content hkl

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421

and the points of intersection are faces that are parallel to two or more PBC directions and so they are potential F faces (see fig. 3). These potential F faces are examined in one or more of the projections in three different PBC directions (seetion 4) in order to find out whether the PBCs are interconnected. We will use the criteria which were developed in ref. [29] for so-called connected nets which are also F slices. We will recall these criteria here: (1) the whole crystal graph, defined as an infinite set of points and connections between these points which fulfils the symmetry of one of the 230 space groups must be partitioned into independent connected nets and (2) the set of basic connected nets must be invariant under those symmetry operations which leave the direction of the reciprocal vector Hflhflkfll, perpendicular to the F slice (nhnknl) invariant. This second criterion implies that the symmetry elements meant above can only be at the border of

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using the position of the relevant symmetry elements. It is then investigated whether real connected nets or F slices exist between the border lines. For each alternative real F slice more alternatives can be found by attaching or detaching ions along the borderline, but a s soon as assumptions have been made about the bond strength in the crystal it is clear how the borderline of a slice has to be drawn and what the position of a growth unit (along the border) in the slice has to be to make the slice as strong as possible. In the case of two alternative connected nets, in general one will be the stronger.

—

3.4. Labeling of PBCs and F slices In principle PBCs can be labelled [uvw] ~ to the period of crystal structure, where

3.3. Survey of potential F slices using a stereographic projection.

according

The most important PBC directions from section 3.1 are drawn in a stereographic projection

x distinguishes between different PBCs with the same period [16,25].

/ Crystal morphology of oc’tacalciurn phosphate

R.A. Terpstra, P. Bennema

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/ Crystal morphology of octacalcium phosphate

R.A. Terpstra, P. Bennema

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Since we are only interested in the F slices or connected nets it is not necessary to label the PBCs. In the projections of figs. 4, 5 and 6 there are many primitive PBCs which have been taken together in an arbitrary way. From the choice of boundary of the PBC is clear which growths units (ions) belong to the slice or net under consideration. The PBCs drawn in the figs. 4, 5 and 6 are seen and on, and all the (primitive) PBCs that have been indicated between the borders (dotted lines) of a slice are interconnected. It is to be noted that for the determination of connected nets only primitive PBCs are needed. For this reason it can be seen, e.g. in fig. 4, that some PBCs do not fill the slice. The F slices are labeled (hkl)~where h, k and 1 are Miller indices, and z 1 or 2 but only used in case of two alternative F slices, =

4. Projections in three different PBC directions 4.]. Projection of the [001] direction It can be seen from fig. 4 that 12 F slices are determined in this projection. These slices are: (120), (100)12, (010)12, (310), (110), (110)12, (210)12, (320). The slices (100)12, (010)12, (110)32 and (210)1,2 are shifted ‘~d ‘-‘ 1d2 210’ respectively, in reference 100 to 2”QlO’ each ‘d110 2other. and ,

4.2. Projection in the [010] direction

In fig. 5 it is shown that 10 F slices are determined in this projection. These slices are: (201)1,2, (301)1,2, (001)1,2, (101)1,2, (100)12 which are shifted ~d 1d 1d 201, ~d 301’ 2 301’ 2 001, ~d101 and Sd 2 ~~00’ respectively, in reference to each other. The slices (100)12 had been determined already.

424

R.A. Terpstra, P. Bennema

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Crystal morphology of octacalcium phosphate

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R.A. Terpstra, P. Bennema

/

425

Crystal morphology of octacalcium phosphate

4.3. Projection in the [011] direction In fig. 6 the projection in the [011] direction is shown and it can be seen that 16 F slices are

of bonds between two neighbouring slices (100)2 and equals 12, as can also be seen from fig. 6. The crystallization energy, ~r, is the sum of ~ and Eatt and equals 84 (Ca—P04 bonds per

determined. These slices are (100)12, (111)12,

cell content).

—

(211)1,2, (311)1,2, (411)1,2, (011)1,2, (111)1.2, (211)1,2. These slices are shifted ~d100,~d111,~d511,

~d511, ~d311,~d~11,~d011,~d111and ~d211,respectively, in reference to each other. The slices (100)12 had been determined already.

5. Order of morphological importance of F slices on the basis of the attachment energy In table 1 the attachment energy is given for each F slice and on this basis its order of MI. In section 1 it was mentioned that E alt

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— —

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The crystallization energy (ECr) of OCP equals 84 (mean Ca—P04) bonds per unit cell contents, i.e. 2 x Ca5H2(PO4)6 5H~O; see also section 2.2). The crystallization energy is constant and for the sake of control we have also calculated the slice energy for each slice, 1~ give an In example how E~ andBelow Eatt we are will calculated. fig. 6 of showing the projection in the [011] direction, the 42 essentially different Ca—P0 4 bonds have been shown (see also figs. lb and lc). In case of “clustering” of ions in the projection, only the number of bonds with the cluster has been given. This type of figure obtained with the use of a model of the OCP structure was used to count the number of bonds in order to obtain Est1 and Eatt. Without a structural model this is difficult and complex but it is not impossible. From thisslice fig. (100)2 6 it can howPart Eshhi~ tIfor the canbebeseen found. of and the E~ [011] of the (100)2 slice has been drawn and PBC it is to be noted that this PBC is centrosymmetric and that the identical PBCs of (100)2 are related through a centre of symmetry. E~’~is calculated from the number of bonds in the PBC [011] of (100)2 (i.e. 2 X 25) and the number of bonds between two identical PBCs [011] of (100)2 (i.e. 2 x 11). So E~1~ = 72. Eatt is equal to the number .

6. Conclusion and discussion There are very few data in the literature on the morphology of synthetic OCP crystals [1,31,32]. These data can be summarized as follows: the synthetic OCP crystals are (100) blades which are elongated along the c axis and bordered by the forms {010}, (001), and (0l1}. Our analysis also predicts (100) blades and the bordering forms mentioned in the literature are all F faces according to our analysis. For (0l0}, (001) and (011) the predicted order of morphological importance according to our analysis is respectively 3, 4/5/6 and 10/11/12. The (110) form with a MI of 2 is missing in the literature data while the {0ll} form scores relatively low in our analysis compared to the literature data. Generally the comparison between the predicted and observed morphology is quite good, but the litera-

01

100

110

Fig. 7. Growth form of OCP on the basis of the attachment energy.

426

R.A. Terpstra, P. Bennema

/

Crystal morphology of oclacalcium phosphate

ture data are too few to make a more than global comparison. If a computer drawing is made in which the linear growth rate R of each F face is taken equal to the attachment energy Eatt of that F face, a crystal with a very poor morphology is obtained. The crystal is elongated along the c axis and the (100) blades are bordered by (110) and (001) (see fig. 7) It has to be noted here that, according to ref. [4], the choice of the boundary of the important {l00} from of OCP has major implications for the (proposed) growth of OCP and OHA. The choice of boundary of ref. [4] agrees with what we have called (lOO)~in our analysis and it was based on the argument that this slice contains the complete apatitic layer. We can know provide physical arguments, based on the attachment energy for the choice of (l00)~ because since E~O) < will determine the boundary of the F face (100).

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[21] Acknowledgements

[22] [23]

R.A. Terpstra was supported by HGO—TNO Project No. 13-52-19. We thank Drs. C.F. Woensdregt for making the computer drawing of the OCP growth form.

[24] [25] [26] [27]

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