the (100) contact twin of gypsum

Published: April 07, 2011

r 2011 American Chemical Society 2351 dx.doi.org/10.1021/cg2000816 | Cryst. Growth Des. 2011, 11, 2351–2357

ARTICLE

pubs.acs.org/crystal

The (100) Contact Twin of GypsumM. Rubbo,* M. Bruno, and D. Aquilano

Dipartimento di Scienze Mineralogiche e Petrologiche dell’ Universit�a 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 andadhesion free energies were calculated using empirical potential functions. By minimizing the twin energy, the optimal interfaceconfiguration was obtained and carefully described. The layer expansion and in plane deformation occurring at the twin interface isanalyzed in terms of the atomic relaxation displacements. A comparison with previous studies on this twin has been made. Adiscussion on the formation of the (100) contact twin follows where we show that this twin can be formed by two-dimensionalnucleation.

1. INTRODUCTION

Regular associations1�3 of bi(poly)-crystals can be studiedfrom different and complementary perspectives. There are topo-logical, geometrical, and structural aspects. All of these areneeded in order to understand the cause of twinning and tocalculate the energy of its formation.

A study based on an energy balance was proposed in a paperon twinning by Fleming et al.4 In that work, the differencesbetween the attachment energies of twinned and untwinnedgrowth slices were calculated; a small difference of the attach-ment energies indicates that contact twins could form. We,instead, calculated the difference of energy between twinnedand untwinned slabs consisting of the same number of layers.The thickness of the slabs was progressively increased to approachthe asymptotic value of the twin interface energy; successively theadhesion energy of the two crystals forming the twinwas calculated.Then the structure of the interface between the two crystals makingthe twin could be analyzed, as will be described in the following.

We are interested in a particular class of contact twins, thosegrowing in a crystal/solution heterogeneous system. There arerelatively few studies on these twins, while many works focus onthe formation of twinned regions due to martensitic transforma-tions and to the growth mechanism of deformation twins (see,for instance, some recent works5�9 and the classical books byChristian,10 Khachaturian,11 Sutton and Balluffi12).

This study is limited to the (100) contact twin of gypsum andto the thermodynamic conditions of its formation. For sake of

convention, the De Jong and Bouman reference frame is used forgypsum twin nomenclature in the results;13 however, the calcula-tions, in this as in our previous works,14,15 are based on thestructure by Boyens and Icharam16 (S.G. C2/c) which is adoptedin the reference structure of Adam’s17 work on the optimizationof the force field for sulfates (see Figure 1).

A lucid exposition18 “Sur la formation des macles de crois-sance” guides us. In his work, Kern reviews the knowledge ongrowth twins and puts forward ideas later quantitatively exploitedby Simon;19�21 however, it was limited to the computationalpower available in the 60s of the last century. Today, the power ofsmall workstations allows us to have a look at structural detailshardly detectable with sophisticated instruments. However, Freyand Monier22 and Lacmann have carried out some remarkablecomputations.23 Obviously, experimental progress is needed andwelcome.

In our opinion, one key idea proposed to describe the topologyand genesis of twins is that a lattice plane exists on which a stackingfault of growth units may occur at some stage of the growth.24,25 Itwas named “original composition plane”: OCP.

This statement is an induction inspired by the works byStranski, Krastanov, Dankov, and van der Merwe on nucleationand epitaxy. The affinity of formation of a nucleus in faulty

Received: January 19, 2011Revised: March 21, 2011

2352 dx.doi.org/10.1021/cg2000816 |Cryst. Growth Des. 2011, 11, 2351–2357

Crystal Growth & Design ARTICLE

position should then be calculated to evaluate the probability of itoccurring. As the probability of nucleation of twinned anduntwinned edifices is independent, the production of twins andstacking faults implies a higher rate of entropy production.However, a two-dimensional nucleation of an edifice in faultyposition can occur on a face only if the face itself can form duringgrowth and if the affinity of crystallization is higher than a criticalvalue.18 This requires a careful analysis of the stability of equili-brium and growth form which should take into account surfacerelaxation, which is most important when nonsingular faces(S and K in the Hartman’s classification24) are considered, asin the case of gypsum, where some of the S forms can enter theequilibrium morphology.14,15

We will come back to other points proposed by Kern in the1961 work18 in a following section. For the time being, we aregoing to present our calculation.

2. BUILDING THE CONTACT TWINS

Making a slab composed of the two crystals to produce a twinrequires several steps. At first, there is evidence of a movementwhich is not a symmetry of the crystal. This requires specifying aminimum number of macroscopic geometric variables or degreesof freedom.12 Two degrees of freedom are needed to specify thenormal to the interface, while three degrees are needed to specifythe movement producing the twin, for example, the orientationof the rotation axis and the rotation angle. These movementsdetermine the boundary conditions far from the interface and therelationships 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 freedomare sufficient to define this rigid body displacement representedby a vector T, in the following. However, to determine the localstructure of the interface requires assessing the atomic move-ments which can be analyzed as the composition of the collectivetranslationT and of atomic deviations from it. A small, bold twillindicate atomic movements at the interface. It ensues, from thediscussion, that the measurements of the angular relationsbetween the faces of the two crystals making the twin are notsufficient to describe the bicrystal, and, in some cases, it may bemisleading.1

3. THE (100) CONTACT TWIN

In the case of the (100) contact twin, the normal to the plane isτ*100, and the symmetry operation we used is a reflection withrespect to a plane parallel to a (100) acting as OCP or, what isequivalent, a rotation byπ about τ*100 followed by inversion withrespect to a center at the intersection of the 2-fold axis with theplane. This twin is often also described by a π rotation about[001]; the geometric interface can be identified with a reflectionplane relating the two individuals before the translations T and t.

The reflection generates a slab, 2D periodic, limited in thedirection τ*100, made of two crystals. In the following the lowerhalf of the slab is named black (B) and the upper one white (W).The relationships between the B andW lattices are described by a2D coincidence site lattice (CSL) whose vectors are [001] � τ3and [010] � τ2. Parallel to the plane 010 through the B crystal,periodic bond chains (PBCs24) develop in the direction [301]and continue in theW one. The misalignment between the B andW PBCs is 2.28� so that Simon21 could work out a supercell ofmultiplicity 3 on the common 010 plane and hence a 3D CSLcould be identified. However, the lattice coincidence does nothave a structural counterpart. The strict alignment of the PBCswould certainly give rise to short-range repulsions of atoms withtheir images at the B/W interface and to a long-range elastic fieldwhose energy would diverge with increasing crystal size. Ob-viously, this does not occur as the PBCs [301] in the B and Wcrystals are translated one with respect to the other by atomicmovements occurring on both sides of the geometrical interface.

In order to determine these movements, a translation isinitially made by inspecting the structure and facing the twocrystals in a sensible way, in order to avoid evident repulsion (inthis work this is done using the program GIDS26). Successively,the energy minimization (here performed using the GULPprogram;27 for computational details see Massaro et al.14,15)refines the value of the three components of the rigid bodytranslation and determines the atomic movements in the inter-face t. The atomic displacements (t) occur in a small volumeextending 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 appreciatedwhen looking at the couples of water molecules symmetryrelated, through the 010 glide plane, in the A and B layers,respectively. Hence, in the d100 layers there are four watermolecules, repeated by translations and partitioned in twocouples: W11, W12 in the layer W1 and W21, W22 in the layerW2, related by the diad axis in the bulk. Because of the energydegeneration of the two A and B configurations, two variants ofthe 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, thecoplanar Ca atoms and SO4 groups are separated by the vector1/2 [τ2 þ τ3]; the particles in A are related by the inversion tothose in B layers.

Before energy minimization the sequence of layers in the Bcrystal up to the mirror plane (bold lines in Figure 2) is (....A-B-A-B)B and the image (B-A-B-A-.....)W, in theW crystal. In order todetermine the optimal structure (that shows the minimumenergy), twinned slabs of increasing thickness were generatedand their structures optimized: with increasing thickness theslabs show regions having the bulk structure within the B and Wcrystals and the twinning energy converges to a constant value, aswill be shown in the following. In a such situation, the dis-turbance due to the surface of the slab does not propagate to the

Figure 1. The (100) contact twin viewed along the [010] direction. Themain forms and directions of the black (B) and white (W) crystals areindicated. The reference frame is by De Jong and Bouman.13



structure of the interface on which twin is composed and viceversa. The comparison between surface relaxation in untwinnedand twinned slabs, shown in Figure 3, demonstrates that thesurface relaxation is strictly the same for an untwinned slab and atwinned one with the same number of layers d200: this validatesthe calculation to obtain the twinning energy.

The optimized twin structure shows a singular layer separatingthe black from the white crystals. This singular layer, BBS, locatedon the side of the B crystal (Figure 2), has a new disposition ofthe watermolecules not related to the diad axis. Then, adjacent tothe singular layer, on the side of the B crystal there is a type Alayer, while on the side of theWone there is a layer of type B. Thetwo crystals experience a relative bulk translation Tx, Ty, Tz: thefirst two components lay in the OCP parallel to τ3 and τ2, whilethe third is perpendicular to the 100 plane. Atomic translations tin the two-dimensional phase, BBS, can be qualitatively appre-ciated looking at the orientation of the SO4 tetrahedra and of thewater 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 ofboth calcium and sulfate groups. In the layers A and B close toBBS the molecules of water experience small rotations and somefluctuations of the bond angle (Figure 2). The bond angle of thewater experiences independent variations: the optimal HOHangle is ∼100.9�, while it is 104.3� in the bulk; OH bonddistances experience minor variations. The orientation of thefour molecules of water in the transition layer is such that two(W11 and W12) mimic the stacking sequence in the W crystalwhile the other two (W21 and W22) the stacking in the B one.Although the force field used has the limitations of beingempirical, it is worth reporting some numerical values.3.2. The Components of the Atomic Relaxations Perpen-

dicular to the Twin Interface. In direction Tz the atomicpositions oscillate over several d100 layers on the two sides ofthe plane of reflection, and therefore the perturbation of the twininterface propagates far from the geometrical interface. This isillustrated in Figure 4 where the mean oscillations of layers of thenamed particles and the mean oscillation (averaged over thedisplacements of all atoms in the layers) of the layers’ thickness ineither crystal are reported. Comparing Figure 4 with Figure 3, wesee that the particles at the interface separating B fromW crystals

are rather constrained and the amplitude of their dampedoscillation is by far less than at the two surfaces of the slab.The mean of displacements over an increasing number ofparticles smoothes the resultant amplitude of the oscillations ofthe layers. This is illustrated in Figure 5 where we report themean oscillations of the water oxygen resolved in the compo-nents of the two nontranslational equivalents oxygen atoms inthe slice. The tz(H2O) component of the translation of theoxygen atoms are such that the distance between layers of atomsof type W1 (Figure 5) shows the wider relaxation displacementson 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. Toexplain the origin of the atomic surface relaxation, Allan28 (see alsorefs 12 and 29) proposed amodel based on theminimization of thecrystal elastic energy. In the harmonic approximation, the forcesand displacements are related to the dynamical matrix (Djk)determining the bulk phonon spectrum: ω2 uj = Σk Djk uk, whereω is the angular frequency and u is the displacement. Imposing theboundary condition that the elastic forces vanish away from thesurface, the solution of the equations for the relaxation displace-ments is a superposition of oscillations having complex wavevectors k 6¼ 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 planes100 of atoms of both at the surface of the slab and at the interfaceof the twin we are dealing with. Thus, the relaxation at the interfacebetween B andW crystals can be thought of as an unstable surfaceconfiguration frozen and stabilized, during growth, by a stackingfault: in case a twin develops, the instability is recorded by thesingular layer of lower 2D symmetry. The minimization of energygives 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 010plane are described by

Tx ¼ xn � j½001�j;Ty ¼ yn � j½010�jwhere 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.

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 segmentis 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.



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 valueof 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) = oxygenof the sulfate group, O(W) = oxygen of the water molecule andH), along with the derived mean thickness of the B/W interface. The line of symmetry ofthe figures between the layers numbered 32�33 locates the geometrical interface.



Shifts obviously vary not only from an atom to another but alsowhen going from the interface level to the layers far from theinterface (bulk). Results are illustrated in Table 1 that shows thecomponents 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 sameapplies to the hydrogen atoms linked to O2 (water) in the bulk.Thus, the components Tx and Ty are the macroscopic translationof the W crystal with respect to the B one. As a consequence, thechains 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 ofthe movements at the interface decays sharply within two layers.The translation of S and of O2 (water) are accompanied by theconstrained translation of the oxygen and hydrogen atomsrespectively linked to them: to this translation a rotation of themolecules is superimposed.Indeed, the SO4 tetrahedra in the singular layer (BBS) are

oriented slightly differently than in the volume. The plane ofwater molecules (type W1), which are contained in the layerclosest to the W region, is rotated by 177.8� degrees in respect tothe plane of the corresponding molecules in the bulk of the Bregion; no further rotation is necessary to superpose the twomolecules. In the layer closest to the B crystal, the W2 moleculesexperience only small rotations in respect to the position of thecorresponding ones in the bulk.

The discussion presented relies on the calculated atomicpositions of the optimal twin structure. They depend on thequality of the force field used which, however good, is empirical:it follows that our arguments pretend to give a sound physicalpicture although not quantitatively correct.

4. THE FORMATION OF (100) TWINS

The work of formation of twins determines the lowest value ofthe affinity of crystallization at which twinning can occur. Ifaffinity values higher than this limit can occur in nature or beobtained in experiments, twinning occurs by 2D or 3D nuclea-tion, and at different length scale from stacking faults tobicrystal.18,30,31

The higher the probability of formation of twins, the higherthe adhesion energy of the W on the B crystal. This is related tothe work of formation of the B/W interface.18 In the case of 2Dnucleation, the edge energy of the twinned nucleus, which issomewhat higher than that of an untwinned one,18 should beaccounted for.

With our calculation procedure (see Bruno et al.31 for moredetails), we obtain directly the interface energy γbw (eq 1). Fromthe Dupr�e relationship (eq 2) and the calculated surface energy,γb, of the untwinned crystal the adhesion energy, βbw is obtainedas well.14,15

γbw ¼ Eb � EbwS

ð1Þ

γbw ¼ 2γb � βbw ð2ÞIn eq 1, Eb and Ebw are the energy of an untwinned and twinnedcrystal slabs limited by 100 planes, having the same surfaceconfiguration and number of layers d200. The surface energy γbof the stable (100) surface, corresponding to the profile of theB/W interface (γb = γ100 = 705.7 erg cm�2, corresponding to0.7057 J m�2)14,15 has been used for calculating βbw.

The numerical values depend on the model used, shortly recol-lected in the following. The area S of the two-dimensional mesh is thesame for both B and W crystals. As in the calculation, the translationvectors parallel to the 100 plane are fixed and periodic boundaryconditions are imposed, the 2Dparameters at the surface are the sameas in the bulk, and the energy converges to a minimum when theatomic coordinates are optimized. The potential function imple-mented in GULP have been determined by Adam17 and usedpreviously14,15 to calculate the surface energy of gypsum. Thegood agreement between calculated equilibrium morphology of

Figure 5. Thickness oscillation alongTz: (a) of the planes of the oxygenatoms of W1 type water; (b) of the planes of the oxygen atoms of W2

type water.

Table 1. The Atomic Shifts Far from the Interface (Bulk) andat the Interface Levela

atom xn yn tx (Å) ty (Å)

Ca bulk 0.1763 0.3436 �0.0313 �0.0533

interface 0.1715 0.3401

S bulk 0.1763 0.3436 0.0320 �0.0198

interface 0.1812 0.3423

O2 (water) bulk 0.1763 0.3436 �0.2714 �0.1157

interface 0.1347 0.3360a xn and yn are fractional coordinates of the vectors [001] and [010],respectively, while tx and ty represent the components of the vectort = Tinterface � Tbulk.



gypsum with that of Naica crystals grown in condition not too farfrom saturation indicates the good quality of the Adam’s potential.

The interface energy and the adhesion energies are reported inFigure 6 as a function of the number of layersmaking the twinnedcrystal.

The value at convergence is γbw =13.55 erg cm�2.The value of γbw is of uppermost importance as it modifies the

work of formation ΔG2D* of a 2D nucleus,30 given in eq 3. In thisformula:- a is the surface occupied by a CaSO4 3 2H2O growth unit onthe (100) face, that is, one-half of the (100) 2D�mesh area;

- constants cl are related to the shape of the 2D nucleus whichis made by l edges having edge energies Fl.

- Δμ = kBT� ln β is the thermodynamical supersaturation; kBis the Boltzmann’s constant, T the absolute temperature andβ the supersaturation ratio;

- Δμ0 = aγbw represents the threshold to be overcome for 2Dtwin-nucleation to occur at the 100 interface

ΔG�2D ¼ ½∑Flcl�2

4ðΔμ� aγbwÞ¼ ½∑Flcl�2

4ðΔμ�Δμ0Þð3Þ

We accept that twinning by 2D nucleation can occur if thecondition Δμ > Δμ0 = aγbw is satisfied for a reasonable value ofΔμ. From a = 1/2 c0 � b0 = 47.6326 � 10�16 cm2 and γbw =13.55 erg cm�2, we obtain Δμ > 6.45 � 10�14 erg.

2D nucleation can occur if the supersaturation of the reaction:

Ca2þ þ SO42� þ 2H2O f CaSO4 3 2H2O ð4Þ

isΔμ =�kBT ln (Q/Keq) > 6.45� 10�14 erg, whereQ (<Keq) isthe actual activity product of the reaction (4) and Keq theequilibrium constant of reaction (4); here, we follow the con-ventional crystal growth jargon designating the affinity of crystal-lization “thermodynamical supersaturation” and symbolizedΔμ.Taking unit activity of the solvent and considering Q =m(Ca2þ)�2 we obtain the concentration at which 2D nucleationoccurs as a function of the supersaturation:

mðCa2þÞ ¼ K�1=2eq exp

Δμ

2kT

� �ð5Þ

To calculate ΔG2D* :- Let32 Keq = 104.6 at 298.15 K,- Mean edge energy <F> is estimated considering that a 2Dnucleus on the (100) face is limited mainly by steps (ofthickness d200 = 2.563 Å) showing the surface of the (010)face. To do this, the athermal surface energy (γ010 = 432erg cm�2) 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,Ca

2þ).

Figure 6. (a) Calculated interface energy (γbw) from eq 1, as a functionof the number of layers. (b) Calculated adhesion energy from eq 2.Dotted lines are a guide for the eye.



hence obtaining a very reasonable value: <F > = γ010 �d200 = 1.11 � 10�5 erg cm�1

- Finally, we assume a nucleus with the reasonable size andshape of 10•[001]� 4•[010] unit cells; with these values theactivation energy for 2D twinned nucleation is representedin Figure 7.

The values obtained give an upper estimate of the order ofmagnitude of the activation energy because the edge energy ofthe 2D nucleus, in aqueous solution at room temperature, islower than that calculated in a vacuum at 0 K.

5. CONCLUSIONS

The (100) face, even if it has stepped character, appears on theequilibrium morphology of gypsum14,15 and, as we found in thecase of the kinked (00.1) face of calcite,31 it can host 2D nuclei infault position. In this work, we show that the relations between BandW lattices allow us to identify 100 as the original compositionplane and to build the twin. The optimization of crystal geometrygives the structure of the interface and the relative displacementsof the B and W crystals. It allows us also to assess that thestructure of the B/W interface is a parent of the structure of thesurface layer, but the damped oscillatory rumpling is lesspronounced. A similar behavior was recently found when dealingwith 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 todivide the crystal along the 100 plane, and it has two moremolecules of water on the 2D surface cell. Therefore, the surfaceon which we compose the twin has its characteristic lifetimeduring the growth of the (100) face, for the growth occurs byparticle addition. The pattern of the layers’ expansion (see forexample Figure 1) indicates that the surface takes an unstablestructure that can evolve toward that of the underlying bulkcrystal or, if particles deposit in anomalous sites, can host astacking fault and eventually a twinned crystal.

The picture we draw essentially agrees with the general ideasexposed in ref 18. The conclusions by Simon20,21 who states thatthe “swallow tail” twin, geometrically related to the 100 twin law,may only form through a penetration mechanism having either010 or 120 as original composition planes are not correct.

Finally, the formation of this twin does not produce interfacedislocations as the damped rumpling and shifts in the interface layersof W and B crystals release the energy associated with the structuraldiscontinuity. Because of the coherence of the W and B lattices, aresidual interface stress remains as indicated, for instance, by thedeformations of bond angles of the water molecules in the interface.

’AUTHOR INFORMATION

Corresponding Author*E-mail: [email protected].

’ACKNOWLEDGMENT

We like to offer special thanks to Professor Raymond Kern forfruitful discussions and criticisms. We also thank the anonymousreviewers for their helpful comments and Jeanne Griffin forimproving the readability of the manuscript.

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the (100) contact twin of gypsum

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