DOI: 10.1021/cg100774q
Effect of the Surface Relaxation on the Theoretical Equilibrium Shape of Calcite. 1. The [001] Zone
2010, Vol. 10 4096–4100
Massaro Francesco Roberto,† Bruno Marco,‡ and Aquilano Dino*,‡ †
Dipartimento di Scienza dei Materiali, Universit a degli Studi di Milano Bicocca, Via Roberto Cozzi 53, a degli I-20125 Milano, Italy, and ‡Dipartimento di Scienze Mineralogiche e Petrologiche, Universit Studi di Torino, Via Valperga Caluso 35, I-10125 Torino, Italy Received June 9, 2010; Revised Manuscript Received July 12, 2010
ABSTRACT: A comparison is made between the {1120} and {1010} hexagonal prisms of calcite crystal. The first one exhibits a strong flat (F) character, while the second one is a stepped (S) form, in the sense of Hartman-Perdok theory. Athermal surface energies were calculated by means of the Rohl interatomic potential for calcite, which reproduces very successfully the equilibrium geometries and the surface energy values of the {0112} and {1014} forms previously obtained from ab initio calculations at the DFT level, along with the observations of the {1014} surface relaxation. From calculation, it ensues that the surface relaxation effect allows the stepped {1010} prism to belong to the crystal equilibrium shape. This result is at variance with the theoretical expectations and represents a prerequisite needed to understand the morphological importance of the {1010} prism, as it follows from the observation of the growth morphology on natural calcites. Finally, the athermal equilibrium shape of calcite is redrawn through the recollected values of the surface energy of the {1014}, {0112}, {1010}, and {0001} relaxed surfaces, showing the sharp differences with the equilibrium shape evaluated without considering the relaxation effect.
1. Introduction In a recent work on the surface reconstruction and relaxation effects in a center-symmetrical crystal, it was demonstrated that the {0112} and {0001} forms of calcite can enter the athermal equilibrium shape (E.S.; equilibrium shape of the crystal at 0 K),1 at variance with all the previous works,2-12 where the {1014} form was believed to be the only one belonging to the equilibrium morphology. The reason why the athermal E.S. of calcite results to be morphologically enriched is due to the lowering of the surface energies γ0112 and γ0001 that are obtained when (i) the surface reconstruction of the {0112} and {0001} forms is carried out respecting1,13,14 the symmetry elements of the underlying bulk structure of the crystal and (ii) the surface relaxation is allowed to occur on such reconstructed surfaces. The above-mentioned result was a little bit surprising because the ideal surfaces of both the {0112} and {0001} forms are electrically polar, being built by a stacking of alternating planes of opposite charge density. Further, while {0112} is a flat (F) form in the sense of Hartman-Perdok theory15 (H.P. hereinafter), {0001} is a kinked (K) form and, consequently, hardly can belong either to the equilibrium or to the growth shape of the crystal. More recently,16 dealing with the theoretical equilibrium and growth shapes of gypsum (CaSO4 3 2H2O), we found that two stepped (S) forms, {100} and {122}, can belong to the athermal E.S. of the crystal, especially when their surface relaxation is taken into account. Then also, this finding violates the well consolidated thinking that only the F forms can belong to the E.S. of crystals. Besides these considerations, there is a striking observation coming from the experimental occurrence frequency of the main crystallographic forms of natural calcite, as it comes out *Corresponding author. E-mail:
[email protected]. Telephone: þ390116705125. Fax: þ390116705128. pubs.acs.org/crystal
Published on Web 08/09/2010
from a descriptive statistic we compiled on the monumental work of Goldschmidt17 and a sound Sunagawa’s study.18 As a matter of fact and contrarily to the common opinion, the form with the highest occurrence frequency (O.F.) is not the wellknown {1014} cleavage rhombohedron (O.F. = 36%) but the principal {1010} prism (O.F. = 46%). This is in plain contradiction with the {1010} stepped character, as it results when one rigorously applies the method of the PBC analysis.14 Moreover, the secondary prism {1120}, which has a sharp F character, exhibits a much lower occurrence frequency (O.F. = 15%), in natural crystals. In this paper we will confine our attention to the equilibrium morphology of the calcite crystal in the [001] zone to highlight the apparent discrepancy between the observation and the theoretical expectations and to outline how this difficulty could be overcome if surface relaxation is taken into account. In a forthcoming paper we will deal with the development of the Æ441æ zone axis: this is the main structural direction in the calcite crystal and belongs both to two important flat forms, {1014} and {1120}, and to the stepped {0118} rhombohedron and {2134} scalenohedron. 2. Character of the Forms in the [001] Zone As shown in Figure 1, two elementary slices can be drawn from the [001] projection of the calcite crystal (hexagonal frame; space group, R3c; a0 = b0 = 4.9897 A˚, c0 = 17.0610 A˚; R = β = 90; γ = 120). Their thicknesses, d1120 = 2.495 A˚ and d3030 = 1.440 A˚, fulfill the systematic extinction rules. Within the slice d1120, three kinds of PBCs run along the repeat periods: 1 /3Æ441æ = 12.85 A˚, 1/3Æ421æ = 8.10, and [001] = 17.06 A˚. Their strength is quite different, as it follows from a revision of a preceding paper,19 because the values of their “end chain energy” are as follows: -0.195 10-10, -0.181 10-10, and -0.136 10-10 erg ion-1, respectively (see Appendix, Table A1). Hence, {1120} is a F form. Instead, {1010} is a S form, since only the [001] PBC develops within the slice of thickness d3030 r 2010 American Chemical Society
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and no bond can be found between two adjacent [001] PBCs (Figure 2). Then, one might conclude that the {1120} prism should be much more stable than the {1010} one, at variance with the observed occurrence frequency on natural crystals. 3. Materials and Methods 3.1. Computational Details. Calculations (optimizations of slab geometries and surface energy estimates) were performed by using the interatomic potential for calcite developed by Rohl et al.20 (Rohl potential hereinafter) and the General Utility Lattice Program (GULP) simulation code,21 which, being based on force field methods, allows the calculation of structures and properties of minerals from a given set of empirical potentials. The parameters of the Rohl potential were obtained by fitting structural data for both calcite and aragonite, as well as physical properties (elastic and dielectric constants) and phonon frequencies. This potential reproduced very successfully the equilibrium geometries and the surface
Figure 1. Schematic drawing of the calcite structure projected along the [001] direction. Triangles and small spheres represent the carbonate groups and the calcium ions, respectively. The [001] PBCs are bound within the slice of thickness d1120, while no bond can be drawn between the same PBCs within the d3030 slice, allowed by the systematic extinction rules of the space group of the crystal (see also Figure 2).
Figure 2. Comparison between the slices of thickness d3030 (left side) and d1120 (right side) viewed along their perpendicular directions [210] and [110], respectively. On the left side one can see that no bond can be found between two adjacent [001] PBCs (built only by δ (Ca-O) bonds). On the right side, the [001] PBCs are bound through the strong [441] PBC (built by R, β, and γ (Ca-O) bonds) and the [221] PBC (built by partial segments of both [441] and [001] PBCs). The lengths of the vectors represent the repeat period of each PBC. The [221] PBC, lying in the face (110) belongs to the set of the three equivalent Æ421æ PBCs.
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energy values of the {0112} and {1014} faces obtained from ab initio calculations at the DFT (density functional theory; B3LYP Hamiltonian22) level,1,14 as well as the experimental observations of the (1014) surface relaxation.20 Geometry optimization is considered converged when the gradient tolerance and the function tolerance (gtol and ftol adimensional parameters in GULP) are smaller than 0.0001 and 0.00001, respectively. 3.2. Slab Geometry Optimization. The (1120) and (1010) surfaces were studied by using the 2D-slab model.23 (1120) and (1010) slabs of varying thickness were generated by separating the bulk structure along the plane of interest. The calculations were performed by considering the original (1 1) cells. The geometry optimization was performed by means of the Newton-Raphson method and by considering the slab subdivided into two regions: region 1, which contains both the surface and the underlying atomic layers that are allowed to relax, and region 2, which has the same number of layers as region 1 and contains the rest of the slab material, where no relaxation with respect to the bulk crystal structure is assumed to occur. The calculations were done by considering slabs with thickness up to ten layers (in both regions 1 and 2), which are sufficient to reproduce bulklike properties at the center of the slab and to obtain a careful description of the surface. 3.3. Calculation of the Surface Energy. According to the standard two-regions strategy employed by GULP, the specific surface energy (γ, erg/cm2) was evaluated from the energy of the surface block (Us, region 1) and the energy of a portion of bulk crystal (Ub) containing the same number of atoms as the surface block. Both energies have been referred to as A, the common surface area of the primitive unit cell: Us - Ub γ ¼ A A ten layers slab (in both regions 1 and 2) was sufficient to reach convergence on the γ1010 and γ1120 values.
4. Surface Energies of the {1120} and {1010} Prisms Contrary to the literature data quoted in Table 1, the athermal relaxed surface energies (γr; the relaxed surface energies at 0 K) of the {1120} and {1010} forms we calculated in this work reverse the conclusions drawn from our application of the H.P. analysis (unrelaxed γ values, γu). As a matter of fact, the ranking of the unrelaxed γ values, which is coherent with the H.P. analysis of the face character, is reversed when surface relaxation is allowed to occur. This is not surprising when comparing the profiles of both the {1120} and {1010} forms, before and after relaxation (Figure 3a and b) and the dispersion of the bond distances occurring when going from the bulk structure to the outmost layers of both prisms (Figure 3c). Surface relaxation affects the {1120} form a little bit more than the most compact {1014} rhombohedron, which corresponds to the first cleavage system of the calcite crystal. However, it should be worth outlining that the differences between unrelaxed and relaxed γ1120 and γ1014 values are markedly higher (-32.0 and -24.2%, respectively) than those usually occurring in ionic crystals made by rigid spheres, with this behavior being attributable to the strong structural
Table 1. Surface Energies (γ, erg cm-2; the Values in Parentheses Are Expressed in J mol-1 A˚-2) Calculated at 0 K for Unrelaxed and Relaxed Surface Structures of the {1120} and {1010} Prisms of the Calcite Crystala form
character
PJr
PP(RIM)r
PP(SM)r
PW*r
γu(this work)
γr(this work)
Δγ (%)
{1010} {1120} {1014}
S F F
520 (3131) 500 (3011) 230 (1385)
1050 (6323) 940 (5661) 590 (3553)
920 (5540) 840 (5058) 530 (3192)
1590 (9575) 1430 (8611) 600 (3613)
2906 (17500) 1812 (10912) 707 (4258)
722 (4348) 1232 (7419) 536 (3228)
-75.1 -32.0 -24.2
The γ values of the most compact form, i.e. the {1014} cleavage rhombohedron, are quoted as a reference. The meanings of the labels PJ, PP(RIM), PP(SM), and PW* along with the origin of the potential functions are extensively explained in ref 24. r and u stay for relaxed and unrelaxed values, respectively. a
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Roberto et al.
unrelaxed relaxed Table 2. Comparison between Unrelaxed (Eslice ) and Relaxed (Eslice ) Slice Energies of the Infinite Layers Fulfilling the Extinction Rules and Parallel to the {1010}, {1120}, and {1014} Formsa
slice thickness d3030 d1120 d1014
E unrelaxed (erg ion-1) slice -10
-0.117 10 -0.187 10-10 -0.223 10-10
E relaxed (erg ion-1) slice -10
-0.163 10 -0.197 10-10 -0.224 10-10
E unrelaxed (%) slice
E relaxed (%) slice
ΔE (%)
49.4 78.9 94.1
68.8 83.1 94.5
þ39.3 þ5.3 þ0.4
a unrelaxed E slice (%) and E relaxed (%) are the percentage values of the unrelaxed and relaxed slice energies, respectively, referred to the lattice energy of the slice bulk (-0.237 10-10 erg ion-1).
Figure 4. Theoretical athermal equilibrium shape of calcite crystal built by the surface energy values calculated for relaxed surfaces. The forms entering the equilibrium morphology are the following: two rhombohedra {1014} and {0112} having F character, the principal prism {1010} with S character and {0001} with K character. The surfaces of both {0112} and {0001} forms have been reconstructed before relaxation, due to their surface polarity. Figure 3. Surface profiles of the {1120} and {1010} forms, viewed along the [001] direction, (a) before and (b) after relaxation. (c) The dispersion of the shortest Ca-O bond distance and of the C-O distance within the carbonate groups is represented, from the outmost layer to the inner (bulk) ones, for both {1120} and {1010} forms.
anisotropy introduced by the triangular shaped carbonate ion. On the contrary, the two outmost layers of the {1010} prism are strongly rearranged by relaxation, as follows from the visible compactness of the {1010} surface structure and by the deep dispersion of the Ca-O bond distances (Figure 3b and c). The consequences on the surface properties are dramatic: (i) the Δγ1010 due to relaxation reaches -75.1%, which is the highest lowering ever calculated for a surface energy; (ii) the variation in the slice energy, Eslice (see Appendix), when going from the {1120} to the {1010} prism and from unrelaxed to relaxed surface structures (Table2). From Table 2 it follows that the athermal slice energy is scarcely modified by relaxation as concerns the {1014} and {1120} forms that show very strong and strong F character, respectively. This is an expected result since the compactness of the unrelaxed slices is hardly improvable. On the contrary, the surface structure of the {1010} is such that the complex relaxation of the three outmost layers induces an increase of nearly 40% of the slice energy of the outmost d3030 slice, which implies a sensible improvement of the slice compactness. On this ground we cannot hypothesize, at the time being, the possibility of formation of 2D nuclei (of thickness d3030) on the {1010} relaxed surfaces, since searching for the stability of these nuclei is beyond the scope of the present work. Nevertheless, one should take into account that the primary slip system in the {1014} planes can be made by the successive glide of two partial dislocations with Burgers vectors [2/3τ1, 1 /3τ2, 1/6τ3], i.e. half of 7.7 A˚ in length, which is the basic translation vector in the {1014} planes.25 This means that a dislocation line crossing a (1010) face can have a screw
component (perpendicular to the surface) higher than twice the d3030 thickness. This is sufficient to generate an exposed ledge of 2d3030 thickness which is stable due to its high slice energy and is able to spread as a step on the (1010) face, so winding in a spiral growth. When summarizing, we can reasonably affirm that the {1010} form of calcite can grow layer by layer, in spite of its S character, as determined through the structural H.P. analysis. Thanks to this growth mechanism, the {1010} prism can compete with the {1120} one, within the [001] zone. 5. Conclusions If relaxation is allowed to occur on the {1010} and {1120} surface profiles determined from the H.P. analysis, the {1010} prism enters the athermal E.S. of the calcite crystal, at variance with the usual thinking that the {1010} form can be stabilized only through the surface adsorption of Mg2þ ions.26 This result can be easily obtained when applying the Wulff’s construction27 to the relaxed values of the athermal surface energies (erg cm-2) of the different forms of calcite we calculated1 up to now: γ1014 = 536; γ0001 = 711; γ1010 = 722; γ0112 = 750; γ1120 = 1232 (Figure 4). On the contrary, the E.S. of calcite is built only by the {1014} cleavage rhombohedron if the surface relaxation is not considered. First, this proves the discriminating role played by the relaxation when a structure of a mainly ionic crystal is built by strongly asymmetric ionic groups. Second, two considerations follow from Figure 4: (i) The relative areas of the forms belonging to the E.S. (Figure 4) allow estimation of the weighted value X X Ahkil γrhkil Þ= Ahkil ¼ 578 erg cm- 2 γrweighted ¼ ð of the athermal relaxed surface energy of calcite; Ahkil is the surface area of the {hkil} form. This value is lowered by -18.2% with respect to that one might have obtained (γuweighted = 707 erg cm-2) from the unrelaxed athermal surface energy. Then, it is worth considering this sensible difference when predicting
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the 3D nucleation frequency of calcite, even if in this case one should look at a more realistic E.S., in which configurational and vibrational entropies are to be taken into account along with the solvent adsorption. There are only two experimental ways to obtain information on the E.S. of calcite in solution: i.e. to allow the evolution toward the equilibrium either of a microscopic calcite crystal in a microscopic isolated solution droplet, according to the method proposed by Lemmlein28 and successfully carried out by Bienfait and Kern,29 or of a liquid inclusion trapped in a calcite crystal. Both these ways are very elegant but scarcely feasible due to the very low solubility of calcite and hence to the impractical observation time. Hence, if one aims at predicting the E.S. in solution, the calculation of the adhesion energy of the water solution adsorbed on the different {hkil} forms of calcite is to be done. To do this, the minimum energy profiles are needed for every crystallographic form, in order to simulate the water adsorption on them. The work previously carried out and improved in the present paper represents the first step of this complex and unavoidable path.
Acknowledgment. The authors acknowledge two anonymous reviewers for helpful comments. Appendix. Building Units and the PBCs in the [001] Zone of Calcite. The periodic bond chains (PBCs) that can be found in the calcite structure are built up by two types of building units (BU); here, the term BU means a way by which the Ca2þ and the CO3- ions are bound in the bulk structure. BU1 is the stronger one, being made by three kinds of Ca-O bonds: R = 2.359 A˚, β = 3.459 A˚, and γ = 4.285 A˚. Having considered the repulsion due to the Ca-C distance of 3.212 A˚, the calculated formation energy of this BU amounts to E(BU1) = -0.284 10-10 erg. The Æ441æ PBC is made by a periodic and nonpolar stacking of the type ABABAB..., owing to the antiparallelism of two successive carbonate groups. The weaker BU2 is symmetric with respect to the A3 axis parallel to the [001] direction and is made by three equivalent Ca-O bonds: δ = 4.453 A˚. The Ca-C distance increases in this case to 4.265 A˚, and the formation energy of this BU lowers to E(BU2) = -0.201 10-10 erg. The [001] PBC is built by a CDCDCD... sequence of BU2s, fulfilling the same rules as for the Æ441æ PBC. The Æ421æ PBC is made by segments of composition (BU1 þ BU2) originating an ACACAC... (or BDBDBD...) sequence that respects the nonpolarity of the alternating carbonate groups. “End Chain Energy” (ECE). This is the energy released when an ion enters, in a crystallographic position, at one end of a semi-infinite chain. As an example, for a given PBC made by the sequence ...ABABAB..., 1 1 -1 ðE PBC þ E AB Þ þ EBU ECE ðerg ion Þ ¼ 2 2 where EPBC represents the interaction energy between the AB pair and the remaining part of the semi-infinite chain and where EAB is the interaction energy between two consecutive building units (A and B) and is the energy of formation of the considered building unit. Hence, the ECE values indicate the hierarchy of the strength of the PBCs.
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Table A1. End Chain Energies (ECEs) of the PBCs Running within the Forms of the [001] Zonea PBC direction
EPBC
EAB
EBU
ECE
Æ441æ Æ421æ Æ001æ
-0.122 -0.171 -0.083
-0.091 -0.067 -0.060
-0.284 -0.242 -0.201
-0.195 -0.181 -0.136
a EPBC, EAB, and EBU are expressed in 10-10 erg, while ECE is expressed in 10-10 erg ion-1.
Calculating the ECE Values in the [001] Zone. To obtain a rough estimation of the strength of the PBCs developing in the forms of the [001] zone (Table A1), the Rohl et al.20 potential for calcite was used. Slice Energy. The slice energy is a way of measuring the compactness of a slice of thickness dhkl. Let us consider a given {hkl} form and the corresponding slice of thickness dhkl fulfilling the extinction rules due to the space group of the crystal structure. Within this slice, a 2D cell can be drawn, containing one (or more) building units, according to the multiplicity of the cell. There is a well-known relationship30 between the crystallization, the slice, and the attachment energies: Ecrystallization ¼ E slice þ E attachment where: (i) Ecrystallization is a constant, for a given crystal species, and coincides with the energy of a kink, which does not vary from face to face of the crystal; (ii) Eattachment represents the energy released (per growth unit of the 2D cell) when the 2D cell is “adsorbed” on the underlying face. Consequently, Eslice is the complementary quantity of Eattachment. This means that the lower the attachment energy, the higher the corresponding slice energy. Remembering what is a kink, one can say that Eslice is nothing else than one half of the interaction energy of the entire 2D cell with its surrounding infinite layer of thickness dhkl. Accordingly, Eslice measures the compactness of the dhkl slice.
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