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The (100) Contact Twin of Gypsum M. Rubbo,* M. Bruno, and D. Aquilano Dipartimento di Scienze Mineralogiche e Petrologiche dell’ Universita di Torino, via Valperga Caluso 35, I-10125 Torino. Italy ABSTRACT:
We investigated the structure of the interface between two gypsum crystals forming a (100) contact twin. The athermal twinning and adhesion free energies were calculated using empirical potential functions. By minimizing the twin energy, the optimal interface configuration was obtained and carefully described. The layer expansion and in plane deformation occurring at the twin interface is analyzed in terms of the atomic relaxation displacements. A comparison with previous studies on this twin has been made. A discussion on the formation of the (100) contact twin follows where we show that this twin can be formed by two-dimensional nucleation.
1. INTRODUCTION Regular associations13 of bi(poly)-crystals can be studied from different and complementary perspectives. There are topological, geometrical, and structural aspects. All of these are needed in order to understand the cause of twinning and to calculate the energy of its formation. A study based on an energy balance was proposed in a paper on twinning by Fleming et al.4 In that work, the differences between the attachment energies of twinned and untwinned growth slices were calculated; a small difference of the attachment energies indicates that contact twins could form. We, instead, calculated the difference of energy between twinned and untwinned slabs consisting of the same number of layers. The thickness of the slabs was progressively increased to approach the asymptotic value of the twin interface energy; successively the adhesion energy of the two crystals forming the twin was calculated. Then the structure of the interface between the two crystals making the twin could be analyzed, as will be described in the following. We are interested in a particular class of contact twins, those growing in a crystal/solution heterogeneous system. There are relatively few studies on these twins, while many works focus on the formation of twinned regions due to martensitic transformations and to the growth mechanism of deformation twins (see, for instance, some recent works59 and the classical books by Christian,10 Khachaturian,11 Sutton and Balluffi12). This study is limited to the (100) contact twin of gypsum and to the thermodynamic conditions of its formation. For sake of r 2011 American Chemical Society
convention, the De Jong and Bouman reference frame is used for gypsum twin nomenclature in the results;13 however, the calculations, in this as in our previous works,14,15 are based on the structure by Boyens and Icharam16 (S.G. C2/c) which is adopted in the reference structure of Adam’s17 work on the optimization of the force field for sulfates (see Figure 1). A lucid exposition18 “Sur la formation des macles de croissance” guides us. In his work, Kern reviews the knowledge on growth twins and puts forward ideas later quantitatively exploited by Simon;1921 however, it was limited to the computational power available in the 60s of the last century. Today, the power of small workstations allows us to have a look at structural details hardly detectable with sophisticated instruments. However, Frey and Monier22 and Lacmann have carried out some remarkable computations.23 Obviously, experimental progress is needed and welcome. In our opinion, one key idea proposed to describe the topology and genesis of twins is that a lattice plane exists on which a stacking fault of growth units may occur at some stage of the growth.24,25 It was named “original composition plane”: OCP. This statement is an induction inspired by the works by Stranski, Krastanov, Dankov, and van der Merwe on nucleation and epitaxy. The affinity of formation of a nucleus in faulty Received: January 19, 2011 Revised: March 21, 2011 Published: April 07, 2011 2351
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Figure 1. The (100) contact twin viewed along the [010] direction. The main forms and directions of the black (B) and white (W) crystals are indicated. The reference frame is by De Jong and Bouman.13
position should then be calculated to evaluate the probability of it occurring. As the probability of nucleation of twinned and untwinned edifices is independent, the production of twins and stacking faults implies a higher rate of entropy production. However, a two-dimensional nucleation of an edifice in faulty position can occur on a face only if the face itself can form during growth and if the affinity of crystallization is higher than a critical value.18 This requires a careful analysis of the stability of equilibrium and growth form which should take into account surface relaxation, which is most important when nonsingular faces (S and K in the Hartman’s classification24) are considered, as in the case of gypsum, where some of the S forms can enter the equilibrium morphology.14,15 We will come back to other points proposed by Kern in the 1961 work18 in a following section. For the time being, we are going to present our calculation.
2. BUILDING THE CONTACT TWINS Making a slab composed of the two crystals to produce a twin requires several steps. At first, there is evidence of a movement which is not a symmetry of the crystal. This requires specifying a minimum number of macroscopic geometric variables or degrees of freedom.12 Two degrees of freedom are needed to specify the normal to the interface, while three degrees are needed to specify the movement producing the twin, for example, the orientation of the rotation axis and the rotation angle. These movements determine the boundary conditions far from the interface and the relationships between the lattices of the two crystals. Moreover, there can be a translation of one crystal relative to the other. Because the periodicity of the interface, three degrees of freedom are sufficient to define this rigid body displacement represented by a vector T, in the following. However, to determine the local structure of the interface requires assessing the atomic movements which can be analyzed as the composition of the collective translation T and of atomic deviations from it. A small, bold t will indicate atomic movements at the interface. It ensues, from the discussion, that the measurements of the angular relations between the faces of the two crystals making the twin are not sufficient to describe the bicrystal, and, in some cases, it may be misleading.1
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3. THE (100) CONTACT TWIN In the case of the (100) contact twin, the normal to the plane is τ*100, and the symmetry operation we used is a reflection with respect to a plane parallel to a (100) acting as OCP or, what is equivalent, a rotation by π about τ*100 followed by inversion with respect to a center at the intersection of the 2-fold axis with the plane. This twin is often also described by a π rotation about [001]; the geometric interface can be identified with a reflection plane relating the two individuals before the translations T and t. The reflection generates a slab, 2D periodic, limited in the direction τ*100, made of two crystals. In the following the lower half of the slab is named black (B) and the upper one white (W). The relationships between the B and W lattices are described by a 2D coincidence site lattice (CSL) whose vectors are [001] τ3 and [010] τ2. Parallel to the plane 010 through the B crystal, periodic bond chains (PBCs24) develop in the direction [301] and continue in the W one. The misalignment between the B and W PBCs is 2.28° so that Simon21 could work out a supercell of multiplicity 3 on the common 010 plane and hence a 3D CSL could be identified. However, the lattice coincidence does not have a structural counterpart. The strict alignment of the PBCs would certainly give rise to short-range repulsions of atoms with their images at the B/W interface and to a long-range elastic field whose energy would diverge with increasing crystal size. Obviously, this does not occur as the PBCs [301] in the B and W crystals are translated one with respect to the other by atomic movements occurring on both sides of the geometrical interface. In order to determine these movements, a translation is initially made by inspecting the structure and facing the two crystals in a sensible way, in order to avoid evident repulsion (in this work this is done using the program GIDS26). Successively, the energy minimization (here performed using the GULP program;27 for computational details see Massaro et al.14,15) refines the value of the three components of the rigid body translation and determines the atomic movements in the interface t. The atomic displacements (t) occur in a small volume extending over several d200 layers, defining the interface phase. There are two kinds (A and B) of layers d200, as shown in the [001] projection (Figure 2b). This can be easily appreciated when looking at the couples of water molecules symmetry related, through the 010 glide plane, in the A and B layers, respectively. Hence, in the d100 layers there are four water molecules, repeated by translations and partitioned in two couples: W11, W12 in the layer W1 and W21, W22 in the layer W2, related by the diad axis in the bulk. Because of the energy degeneration of the two A and B configurations, two variants of the layers at the interface between B and W crystals are possible. Only the one shown in Figure 2 will be described. In A or B, the coplanar Ca atoms and SO4 groups are separated by the vector 1/2 [τ2 þ τ3]; the particles in A are related by the inversion to those in B layers. Before energy minimization the sequence of layers in the B crystal up to the mirror plane (bold lines in Figure 2) is (....A-BA-B)B and the image (B-A-B-A-.....)W, in the W crystal. In order to determine the optimal structure (that shows the minimum energy), twinned slabs of increasing thickness were generated and their structures optimized: with increasing thickness the slabs show regions having the bulk structure within the B and W crystals and the twinning energy converges to a constant value, as will be shown in the following. In a such situation, the disturbance due to the surface of the slab does not propagate to the 2352
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Figure 2. Optimized (100) contact twin viewed along (a) [010] and (b) [001], respectively. The vertical axis is parallel to τ*100. The bold blue segment is the trace of the geometrical plane where the black crystal has been reflected to generate the white one. BBS identifies the singular layer; W1 and W2 designate the two layers of water molecules within the thickness d200.
structure of the interface on which twin is composed and vice versa. The comparison between surface relaxation in untwinned and twinned slabs, shown in Figure 3, demonstrates that the surface relaxation is strictly the same for an untwinned slab and a twinned one with the same number of layers d200: this validates the calculation to obtain the twinning energy. The optimized twin structure shows a singular layer separating the black from the white crystals. This singular layer, BBS, located on the side of the B crystal (Figure 2), has a new disposition of the water molecules not related to the diad axis. Then, adjacent to the singular layer, on the side of the B crystal there is a type A layer, while on the side of the W one there is a layer of type B. The two crystals experience a relative bulk translation Tx, Ty, Tz: the first two components lay in the OCP parallel to τ3 and τ2, while the third is perpendicular to the 100 plane. Atomic translations t in the two-dimensional phase, BBS, can be qualitatively appreciated looking at the orientation of the SO4 tetrahedra and of the water molecules (Figure 2). 3.1. Water Molecules at the Twin Interface. The freedom of the water molecules in the BBS layer is much higher than that of both calcium and sulfate groups. In the layers A and B close to BBS the molecules of water experience small rotations and some fluctuations of the bond angle (Figure 2). The bond angle of the water experiences independent variations: the optimal HOH angle is ∼100.9°, while it is 104.3° in the bulk; OH bond distances experience minor variations. The orientation of the four molecules of water in the transition layer is such that two (W11 and W12) mimic the stacking sequence in the W crystal while the other two (W21 and W22) the stacking in the B one. Although the force field used has the limitations of being empirical, it is worth reporting some numerical values. 3.2. The Components of the Atomic Relaxations Perpendicular to the Twin Interface. In direction Tz the atomic positions oscillate over several d100 layers on the two sides of the plane of reflection, and therefore the perturbation of the twin interface propagates far from the geometrical interface. This is illustrated in Figure 4 where the mean oscillations of layers of the named particles and the mean oscillation (averaged over the displacements of all atoms in the layers) of the layers’ thickness in either crystal are reported. Comparing Figure 4 with Figure 3, we see that the particles at the interface separating B from W crystals
are rather constrained and the amplitude of their damped oscillation is by far less than at the two surfaces of the slab. The mean of displacements over an increasing number of particles smoothes the resultant amplitude of the oscillations of the layers. This is illustrated in Figure 5 where we report the mean oscillations of the water oxygen resolved in the components of the two nontranslational equivalents oxygen atoms in the slice. The tz(H2O) component of the translation of the oxygen atoms are such that the distance between layers of atoms of type W1 (Figure 5) shows the wider relaxation displacements on the side of the B crystal while the oxygen atoms of type W2 show symmetrical displacements. At the B/W interface, the local distortion of the coordination generates forces conjugated with the normal displacements. To explain the origin of the atomic surface relaxation, Allan28 (see also refs 12 and 29) proposed a model based on the minimization of the crystal elastic energy. In the harmonic approximation, the forces and displacements are related to the dynamical matrix (Djk) determining the bulk phonon spectrum: ω2 uj = Σk Djk uk, where ω is the angular frequency and u is the displacement. Imposing the boundary condition that the elastic forces vanish away from the surface, the solution of the equations for the relaxation displacements is a superposition of oscillations having complex wave vectors k ¼ 6 0 such to make the frequency values ω2(k) = 0. The model helps to elucidate which components of the potential are essential to catch the frozen displacements of planes 100 of atoms of both at the surface of the slab and at the interface of the twin we are dealing with. Thus, the relaxation at the interface between B and W crystals can be thought of as an unstable surface configuration frozen and stabilized, during growth, by a stacking fault: in case a twin develops, the instability is recorded by the singular layer of lower 2D symmetry. The minimization of energy gives atomic positions but does not allow this insight. 3.3. The Atomic Shifts Far from the Interface and at the Interface Level. The absolute shifts, measured in Å, in the 010 plane are described by T x ¼ xn j½001j; T y ¼ yn j½010j where xn and yn represent the fractional coordinates of the atom (n) referring to the modulus of the vectors [001] = 6.5233 Å and [010] = 15.2199 Å, respectively. 2353
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Figure 3. Variation of the distance d200 between Ca sublattice planes, through the slab. Layers 1 and 64 are at the opposite surfaces of the slab. The value of d200 in the bulk is 2.563 Å.
Figure 4. Frozen oscillations, about the bulk d200 = 2.563 Å, of the thickness of the layers as obtained from the atomic positions (of Ca, S, O(S) = oxygen of the sulfate group, O(W) = oxygen of the water molecule and H), along with the derived mean thickness of the B/W interface. The line of symmetry of the figures between the layers numbered 3233 locates the geometrical interface. 2354
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Table 1. The Atomic Shifts Far from the Interface (Bulk) and at the Interface Levela xn
atom
yn
tx (Å)
ty (Å)
Ca
bulk
0.1763
0.3436
0.0313
0.0533
S
interface bulk
0.1715 0.1763
0.3401 0.3436
0.0320
0.0198
interface
0.1812
0.3423 0.2714
0.1157
O2 (water)
bulk
0.1763
0.3436
interface
0.1347
0.3360
a
xn and yn are fractional coordinates of the vectors [001] and [010], respectively, while tx and ty represent the components of the vector t = Tinterface Tbulk.
The discussion presented relies on the calculated atomic positions of the optimal twin structure. They depend on the quality of the force field used which, however good, is empirical: it follows that our arguments pretend to give a sound physical picture although not quantitatively correct.
Figure 5. Thickness oscillation along Tz: (a) of the planes of the oxygen atoms of W1 type water; (b) of the planes of the oxygen atoms of W2 type water.
Shifts obviously vary not only from an atom to another but also when going from the interface level to the layers far from the interface (bulk). Results are illustrated in Table 1 that shows the components of the difference t = Tinterface Tbulk. Further, it is worth outlining that the tetrahedral oxygen atoms experience the same translations of the sulfur atom; the same applies to the hydrogen atoms linked to O2 (water) in the bulk. Thus, the components Tx and Ty are the macroscopic translation of the W crystal with respect to the B one. As a consequence, the chains of CaSO4 groups parallel to (010) in the BB layer are not, strictly speaking, mirror images of the chains in the layer Aw. From Table 1 it ensues that the more mobile atom is the oxygen of the water, O2, at the interface. The amplitude of the movements at the interface decays sharply within two layers. The translation of S and of O2 (water) are accompanied by the constrained translation of the oxygen and hydrogen atoms respectively linked to them: to this translation a rotation of the molecules is superimposed. Indeed, the SO4 tetrahedra in the singular layer (BBS) are oriented slightly differently than in the volume. The plane of water molecules (type W1), which are contained in the layer closest to the W region, is rotated by 177.8° degrees in respect to the plane of the corresponding molecules in the bulk of the B region; no further rotation is necessary to superpose the two molecules. In the layer closest to the B crystal, the W2 molecules experience only small rotations in respect to the position of the corresponding ones in the bulk.
4. THE FORMATION OF (100) TWINS The work of formation of twins determines the lowest value of the affinity of crystallization at which twinning can occur. If affinity values higher than this limit can occur in nature or be obtained in experiments, twinning occurs by 2D or 3D nucleation, and at different length scale from stacking faults to bicrystal.18,30,31 The higher the probability of formation of twins, the higher the adhesion energy of the W on the B crystal. This is related to the work of formation of the B/W interface.18 In the case of 2D nucleation, the edge energy of the twinned nucleus, which is somewhat higher than that of an untwinned one,18 should be accounted for. With our calculation procedure (see Bruno et al.31 for more details), we obtain directly the interface energy γbw (eq 1). From the Dupre relationship (eq 2) and the calculated surface energy, γb, of the untwinned crystal the adhesion energy, βbw is obtained as well.14,15 Eb Ebw S
ð1Þ
γbw ¼ 2γb βbw
ð2Þ
γbw ¼
In eq 1, Eb and Ebw are the energy of an untwinned and twinned crystal slabs limited by 100 planes, having the same surface configuration and number of layers d200. The surface energy γb of the stable (100) surface, corresponding to the profile of the B/W interface (γb = γ100 = 705.7 erg cm2, corresponding to 0.7057 J m2)14,15 has been used for calculating βbw. The numerical values depend on the model used, shortly recollected in the following. The area S of the two-dimensional mesh is the same for both B and W crystals. As in the calculation, the translation vectors parallel to the 100 plane are fixed and periodic boundary conditions are imposed, the 2D parameters at the surface are the same as in the bulk, and the energy converges to a minimum when the atomic coordinates are optimized. The potential function implemented in GULP have been determined by Adam17 and used previously14,15 to calculate the surface energy of gypsum. The good agreement between calculated equilibrium morphology of 2355
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The value at convergence is γbw =13.55 erg cm2. The value of γbw is of uppermost importance as it modifies the * of a 2D nucleus,30 given in eq 3. In this work of formation ΔG2D formula: - a is the surface occupied by a CaSO4 3 2H2O growth unit on the (100) face, that is, one-half of the (100) 2D mesh area; - constants cl are related to the shape of the 2D nucleus which is made by l edges having edge energies Fl. - Δμ = kBT ln β is the thermodynamical supersaturation; kB is the Boltzmann’s constant, T the absolute temperature and β the supersaturation ratio; - Δμ0 = aγbw represents the threshold to be overcome for 2D twin-nucleation to occur at the 100 interface
ΔG2D ¼
½∑Fl cl 2 ½∑Fl cl 2 ¼ 4ðΔμ aγbw Þ 4ðΔμ Δμ0 Þ
ð3Þ
We accept that twinning by 2D nucleation can occur if the condition Δμ > Δμ0 = aγbw is satisfied for a reasonable value of Δμ. From a = 1/2 c0 b0 = 47.6326 1016 cm2 and γbw = 13.55 erg cm2, we obtain Δμ > 6.45 1014 erg. 2D nucleation can occur if the supersaturation of the reaction: Ca2þ þ SO4 2 þ 2H2 O f CaSO4 3 2H2 O
ð4Þ
is Δμ = kBT ln (Q/Keq) > 6.45 1014 erg, where Q (< Keq) is the actual activity product of the reaction (4) and Keq the equilibrium constant of reaction (4); here, we follow the conventional crystal growth jargon designating the affinity of crystallization “thermodynamical supersaturation” and symbolized Δμ. Taking unit activity of the solvent and considering Q = m(Ca2þ)2 we obtain the concentration at which 2D nucleation occurs as a function of the supersaturation: Δμ 1=2 mðCa2þ Þ ¼ Keq exp ð5Þ 2kT
Figure 6. (a) Calculated interface energy (γbw) from eq 1, as a function of the number of layers. (b) Calculated adhesion energy from eq 2. Dotted lines are a guide for the eye.
To calculate ΔG*2D: - Let32 Keq = 104.6 at 298.15 K, - Mean edge energy is estimated considering that a 2D nucleus on the (100) face is limited mainly by steps (of thickness d200 = 2.563 Å) showing the surface of the (010) face. To do this, the athermal surface energy (γ010 = 432 erg cm2) was used, as calculated in a previous work,15
Figure 7. Work of formation, by 2D nucleation, of a twin on the (100) face of gypsum, as a function of supersaturation ratio β = (mCa2þ/meq,Ca2þ). 2356
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Crystal Growth & Design hence obtaining a very reasonable value: = γ010 d200 = 1.11 105 erg cm1 - Finally, we assume a nucleus with the reasonable size and shape of 10•[001] 4•[010] unit cells; with these values the activation energy for 2D twinned nucleation is represented in Figure 7. The values obtained give an upper estimate of the order of magnitude of the activation energy because the edge energy of the 2D nucleus, in aqueous solution at room temperature, is lower than that calculated in a vacuum at 0 K.
5. CONCLUSIONS The (100) face, even if it has stepped character, appears on the equilibrium morphology of gypsum14,15 and, as we found in the case of the kinked (00.1) face of calcite,31 it can host 2D nuclei in fault position. In this work, we show that the relations between B and W lattices allow us to identify 100 as the original composition plane and to build the twin. The optimization of crystal geometry gives the structure of the interface and the relative displacements of the B and W crystals. It allows us also to assess that the structure of the B/W interface is a parent of the structure of the surface layer, but the damped oscillatory rumpling is less pronounced. A similar behavior was recently found when dealing with the twin laws of calcite.31 We could build a twin on a less stable surface structure: indeed, the most stable face profile corresponds to a different legal way to divide the crystal along the 100 plane, and it has two more molecules of water on the 2D surface cell. Therefore, the surface on which we compose the twin has its characteristic lifetime during the growth of the (100) face, for the growth occurs by particle addition. The pattern of the layers’ expansion (see for example Figure 1) indicates that the surface takes an unstable structure that can evolve toward that of the underlying bulk crystal or, if particles deposit in anomalous sites, can host a stacking fault and eventually a twinned crystal. The picture we draw essentially agrees with the general ideas exposed in ref 18. The conclusions by Simon20,21 who states that the “swallow tail” twin, geometrically related to the 100 twin law, may only form through a penetration mechanism having either 010 or 120 as original composition planes are not correct. Finally, the formation of this twin does not produce interface dislocations as the damped rumpling and shifts in the interface layers of W and B crystals release the energy associated with the structural discontinuity. Because of the coherence of the W and B lattices, a residual interface stress remains as indicated, for instance, by the deformations of bond angles of the water molecules in the interface. ’ AUTHOR INFORMATION Corresponding Author
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(2) Friedel, G. Lec-ons de Cristallographie; Berger-Levraut: Paris, France, 1926. (3) Cahn, R. W. Adv. Phys. 1954, 3, 363–445. (4) Fleming, S. D.; Parkinson, G. M.; Rohl, A. L. J. Cryst. Growth 1997, 178, 402–409. (5) Ostapovets, A. Comput. Mater. Sci. 2010, 49, 882–887. (6) Nogaret, T.; Curtin, W. A.; Yasi, J. A.; Acta Mater. 2010, 58, 4332–4343. (7) Serra, A.; Bacon, D. J. Phil. Mag. 2010, 90, 845–861. (8) Wang, J.; Hirth, J. P.; Tome, C. N. Acta Mater. 2009, 57, 5521–5530. (9) Wang, J.; Hoagland, R. G.; Hirth, J. P.; Scr. Mater. 2009, 61, 903–906. (10) Christian, J. W. The Theory of Transformations in Metals and Alloys; Pergamon Press: Oxford, 1981. (11) Khachaturian, A. G.; Theory of Structural Transformations in Solids; John Wiley & Sons: New York, 1983. (12) Sutton, A. P.; Balluffi, R. W. Interfaces in Crystalline Materials; Clarendon Press: Oxford, 2006. (13) De Jong, W. F.; Bouman, J. Z. Krystallogr. 1938, 100, 275–276. (14) Massaro, F. R.; Rubbo, M.; Aquilano, D. Cryst. Growth Des. 2010, 10, 2870–2878. (15) Massaro, F. R.; Rubbo, M.; Aquilano, D. Cryst. Growth Des. 2011, DOI: 10.1021/cg101570c. (16) Boyens, J. C. A.; Icharam, V. V. H. Z. Krystallogr. 2002, 217, 9–10. (17) Adam, C. D. J. Solid State Chem. 2003, 174 (1), 141–151. (18) Kern, R. Bull. Soc. franc-. Miner. Crist. 1961, LXXXIV, 292–311. (19) Simon, B.; Kern, R. Acta Crystallogr. A 1963, 16, A144, 17(i).23. (20) Simon, B.; Bienfait, M. Acta Crystallogr. A 1965, 19, 750–756. (21) Simon, B. Contribution a l’etude de la formation des macles de croissance. These docteur es-sciences physiques, Marseille, France, 1968. (22) Frey, M.; Monier, J. C. Bull. Soc. franc-. Miner. Crist. 1964, 87, 39–47. (23) Lacmann, F. These 3eme cycle, Paris, 1963. (24) Hartman, P. Z. Kristallogr. 1956, 107, 225–237. Hartman, P. Acta Crystallogr. 1958, 11, 459–464.Hartman, P. In Crystal Growth: An Introduction; Hartman, P., Ed.; North Holland Publishing Co.: Amsterdam, 1973; pp 367402. (25) Curien, H; Kern, R. Bull. Soc. Franc-. Miner. Crist. 1957, 80, 111–132. (26) Fleming, S.; Rohl, A. Z. Kristallogr. 2005, 220 (56), 580–584. (27) Gale, J. D. Faraday Trans. 1997, 93 (4), 629–637. (28) Allan, G. Prog. Surf. Sci. 1987, 25 (14), 43–56. (29) Houchmandzadeh, B.; Lajzerowicz, J; Salje, E J. Phys.: Condens. Matter 1992, 4, 9779–9794. (30) Mutaftschiev, B. The Atomistic Nature of Crystal Growth; Springer: Berlin, Germany, 2001; Kern, R; Metois, J. J.; Lelay, G. in Current Topics in Material Science; Kaldis, Ed.; North Holland Publishing Co.: Amsterdam, 1979; Vol. 3, pp 131419. (31) Bruno, M.; Massaro, F. R.; Rubbo, M.; Prencipe, M.; Aquilano, D. Cryst. Growth Des. 2010, 10, 3102–3109. (32) Krauskopf, K. B.; Bird, D. K. Introduction to Geochemistry; McGraw-Hill: Singapore, 1995; p 598.
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We like to offer special thanks to Professor Raymond Kern for fruitful discussions and criticisms. We also thank the anonymous reviewers for their helpful comments and Jeanne Griffin for improving the readability of the manuscript. ’ REFERENCES (1) Friedel, G. Etude sur les groupements cristallins; Saint-Etienne: France, 1904. 2357
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