Journal of Crystal Growth 343 (2012) 77–85

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Journal of Crystal Growth

0022-02

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jcrysgro

Simulating vapour growth morphology of crystalline urea using modifiedattachment energy model

M.K. Singh a,n, Arup Banerjee b, P.K. Gupta a

a Laser Materials Development and Devices Division, Raja Ramanna Centre for Advanced Technology, Indore 452013, Indiab BARC Training School at Raja Ramanna Centre for Advanced Technology (RRCAT), Indore 452013, India

a r t i c l e i n f o

Article history:

Received 3 June 2011

Received in revised form

11 January 2012

Accepted 16 January 2012

Communicated by M. Uwahathe orientation. The molecular orientations in a slice of thickness dhkl may differ from each other and

Available online 24 January 2012

Keywords:

A1. Attachment energy

A1. Growth morphology

A1. Hartman–Perdok

A1. Molecular layer

A1. Urea crystal

48/$ - see front matter & 2012 Elsevier B.V. A

016/j.jcrysgro.2012.01.032

esponding author. Tel.: þ91 731 2488677.

ail addresses: mksingh@rrcat.gov.in, iasingh@

a b s t r a c t

We report a computational model to simulate vapour growth morphology of urea crystal by

considering molecular anisotropy and surface relaxation of different crystal faces. It has been argued

that the faces’ growth occurs through the adsorption of molecular layers rather than a slice of thickness

dhkl. The molecular layer is a 2-D periodic arrangement of molecules in which each molecule has same

depend on crystallographic orientation of the slice. The discussed approach has been employed to

simulate vapour growth shape of crystalline urea by calculating attachment energy of molecular layers

using Hartee–Fock and density functional theories. The calculated growth morphology is in good

agreement with the vapour grown shape of urea crystal. The role of thermal and growth kinetics

affecting the vapour growth morphology has been discussed. The observed polar growth morphology of

urea crystal has also been discussed particularly in the context of different atomic environments of

(111) and (�1�1�1) faces.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

The study and manipulation of the forms of crystals haveattracted immense interest of artists and academicians since theBronze Age [1]. Understanding the correlation between thecrystallisation forms and its internal structure has motivatedthe scientists long before the crystal structure could be deter-mined from diffraction data. This knowledge base has enabled thecrystal grower to manipulate the crystallisation processes toobtain a desired shape of a crystal. The forms of the crystals areof great relevance in many industrial processes and have a directimpact on the separation efficiency, the stability of crystallinechemicals, the bioavailability and effective drugs delivery.

The growth theory of Hartman and Perdok (HP) aims tocompute shape of crystals [2–5]. In 1955, a significant contribu-tion was made by HP [2,3] and later modified by Hartman andBennema [4,5], who developed the idea of strong bonds in theirperiodic bond chain (PBC) theory, connected nets and rougheningto predict the external shape of a crystal. A series of contributionsby Hartman and Bennema provided the attachment energy (AE)model, which was designed to realise the significance of non-bonded energies in morphology prediction. The development of

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gmail.com (M.K. Singh).

the AE model was an important milestone in the prediction ofcrystal morphology. The HP method provides a systematic way togenerate stable growth planes of a crystal. It removes one of theassumptions of the attachment energy method, namely thegrowth planes are always ideal and flat. The prediction of growthshape involves the calculation of attachment energy of slices,which requires the determination of the lattice and slice energies.The energy released per mole on the formation of a newelementary growth layer, called a slice, with a thickness of dhkl,is the slice energy, Ehkl

slab, while the energy released per mole on theattachment of a slice on an existing crystal face is the attachmentenergy, Ehkl

AE . The lattice energy, Elatt, is the energy calculated bysumming all nonbonded interactions for the entire perfect crystal.HP introduced a proportionality relation between the growth rateof a flat face and its AE [2,3]. According to their hypothesis, faceswith the smallest attachment energies grow slowly compared tothe faces with higher attachment energies. Therefore, the faceswith the lowest AE play an important role in crystal morphology.The attachment energy model often provides good predictions forcrystals that are grown from the vapour phase [6,7]. However,this model does not take into account molecular anisotropy,surface scaling factor [8] and relaxation of the habit faces onthe growth morphology. It is established that the surface relaxa-tion plays an important role in calculating growth morphologiesas the atomic structure at the outermost boundary controls thecrystal growth mechanism [9,10]. The attachment energy model

Fig. 1. Schematic diagram showing two distinct molecular layers ‘A’ and ‘B’ on

(001) form of the urea crystal.

M.K. Singh et al. / Journal of Crystal Growth 343 (2012) 77–8578

also failed to take into account the influence of the driving forcethrough which assembly of crystal faces occurs [11,12], which iswidely thought to be a major factor affecting crystal shape. It isnot surprising that the attachment energy model should fail,because only AE of the slice, with a thickness of dhkl, is taken ashabit controlling energetics without considering the effect ofmolecular anisotropy in the crystal face. It is thus logical todevelop a computational model based on the modified attach-ment energies that takes into account molecular anisotropy andsurface relaxation to calculate growth morphology and that isprecisely what we have attempted in this paper. We note herethat in a series of papers by Strom [13] a graph theoretic approachhas been utilised to develop the modern version of the HP model,which indirectly takes into account surface relaxation on growthmorphology. In addition the PBC approach has also been used tostudy the influence of additives such as antifreeze proteins on thehabit of crystals [14].

We have used a periodic ab initio method implemented inCRYSTAL09 to calculate habit controlling energetics [15]. How-ever, we are aware that previously CRYSTAL95 had beenemployed to obtain atomic charges in the urea crystal and thento calculate vapour growth morphology of urea crystal [16]. Thecomputed atomic charges were then used to calculate lattice andattachment energies using a semi-empirical force field method.We have utilised a first principles method primarily because asemi-empirical force field method has the following inherentlimitations: (i) calculation of partial charges is not unique anddepends on the method used [17], (ii) the partial charges differ forbulk and slice structure and hence the atomic charges obtainedfrom bulk crystal structure cannot be used for slice and attach-ment energy calculation and (iii) the short-range potential is wellstudied for bulk but the applicability is limited when applied tosurfaces [18]. Keeping these factors in mind we use a firstprinciples method for studying crystal growth morphology.

Several authors [10(b)],[19–22] have reported the failure of HPmodel to predict vapour growth morphology of urea crystal. Weanticipate that the predictive power of the HP model is limitedprimarily because the molecular orientation has not been con-sidered to calculate growth morphology. The role of molecularorientation on AE of growth units was studied by Singh et al.[10(a)], Bisker-Leib and Doherty [22] and Liu and Bennema [8].Bisker-Leib and Doherty have calculated modified attachmentenergies of a reference urea molecule, M1, summing all theinteractions outside a stoichiometric slice up to a limiting radiusand then repeating the calculation for the second urea moleculeM2. The morphology obtained from their modified attachmentenergies of the molecules showed an improvement over theresults of earlier calculations. The methodology of their calcula-tion thus takes into account molecular anisotropy in the faces ofurea crystal. Liu and Bennema [8] formulated what is known asthe interfacial structure analysis theory and predicted the shapeof urea crystals in aqueous solutions. They have utilised theconcept of effective growth unit, which in turn depends onmolecular orientation of growth units to calculate growth shapeof urea crystal. They were the first to model in a systematic waythe shape of crystals grown from solution by performing complexmolecular dynamic simulations. They have carried out moleculardynamic simulation to produce the genuine interfacial structurein different crystallographic orientations. However, in thisapproach, calculation of various quantities within affordablesimulation time is a formidable task. However, these authors[8,20] have considered only experimentally observed faces tocalculate the growth morphology of the urea crystal.

In the HP model, growth layers are usually assumed to havethickness dhkl or less [4,5]. To investigate the effect of molecularorientation on growth morphology, the growth layers can be

further partitioned into molecular layers. Accordingly, a slice ofthickness dhkl may comprise more than one distinct molecularlayer depending upon the molecular anisotropy in the slice. Themolecular layer can be considered as a 2-D periodic slice in whichevery molecule has the same orientation. We hypothesize thatcrystallisation occurs through the adsorption of molecular layersrather than slices of dhkl thickness to the crystal surfaces. Theassumption is justified as during crystal growth, growth units areadsorbed on the crystal surfaces. In the case of molecular crystal,the growth unit can be monomer or dimer of the molecule. In thepresent paper we assume that the growth units are monomers.The AE of each distinct molecular layer in a slice need not beequal and it is determined by the relative orientation of themolecules. To illustrate the concept of molecular layers, we chosethe (001) face of urea crystal as shown in Fig. 1, in which a slice isdivided into two distinct molecular layers, namely ‘A’ and ‘B’. It isclear from Fig. 1 that the molecular orientation of layer ‘A’ isdifferent from that of layer ‘B’. The growth of this face proceedsthrough the successive attachment of molecular layers ‘A’ and ‘B’.The facet’s growth is rate limited by the adsorption of themolecular layer that has the minimum attachment energy [7].We have calculated the growth shape of urea crystal by calculat-ing relaxed and un-relaxed attachment energies of molecularlayers. The calculated relaxed growth shape is in good agreementwith the experimental morphology.

2. Methodology

The following methodology has been adopted to constructslices of different crystallographic orientations. The slices werecreated from the bulk crystal structure, each limited by an idealsurface that is parallel to the given plane (hkl). An atomic layer isdefined by a set of atoms with the same z coordinate, withreference to the new Cartesian frame. The thickness of the 2-Dslice is determined by the number of atomic layers. The chargeneutrality of the slice is maintained during the process of slicegeneration. The slice must possess the minimum energy config-uration. If the slice has dipole moment, attention must be paid tothe surface termination, because slices can possess a net dipoleperpendicular to the surface. The stability of the slice depends onits orientation, (hkl), number of atomic layers, correct surfacetermination and relaxation. The number of atomic layers in a sliceis in turn determined by convergence of the surface energy and tomaintain the stoichiometry, we have taken all atomic layers in aslice of thickness dhkl. Often more than one surface termination ispossible for a slice. To identify the correct surface termination, theconfiguration of slice must possess minimum (i) total energy and(ii) surface energy. In most cases these conditions are satisfied atthe same configuration. The crystal surface that is taken as the

M.K. Singh et al. / Journal of Crystal Growth 343 (2012) 77–85 79

starting point is not simply a straight cut from a zero temperaturecrystal structure, but the dismembered molecules are recombinedto make them complete. Naturally, the resulting configuration isnot favoured energetically because, in molecular crystals, themolecular topology must be preserved and only intermolecularbonds are cut during the surface creation. The topology ofdismembered molecules must be restored in order to obtainstable surface configuration. We have calculated required displa-cements of the fragmented molecules using structural informa-tion like bond lengths, bond angles and dihedral angles to carryout molecular restoration of the dismembered molecules. Finally,the surface configuration thus obtained for each face is subjectedto structural optimisation to obtain relaxed structure. This isnecessary because the ideal crystal surface may undergo relaxa-tion, without loss of translational symmetry, or exhibit partialreconstruction. We wish to mention that during geometry opti-misation of the slices, molecules are treated to be rigid. In Table 1we list all the parameters that are required for constructing thedifferent slices.

3. Computational method

All the calculations have been carried out using the CRYSTAL09code, an ab initio program based on linear combination of atomic

Table 1The parameter required to create slices of different crystallographic orientations of

urea crystal. These parameters are needed to uniquely determine the stable slices

structure. The table lists the number of atomic layers in one dhkl thick slices, label

of surface atoms and label of the molecule that needs restoration of integrity in

the faces.

Crystal

face

Number of

atomic layers

in a dhkl thick

slice

Label of surface

atom

Label of the cut

urea molecule

whose integrity

needs to

be restored

(100) 14 3(O), 5(N),

10(H), 12(H)

M2

(110) 7 2(C), 7(N) M2

(101) 16 4(O) None

(001) 10 2(C), 10(H) M2

(111) 13 5(N) M1

Fig. 2. (a) Dependence of different proportions of BSSE correction on lattice energy

correction for crystalline urea, with HF and DFT-B3LYP methods and (b) urea molecule

orbitals (LCAO) for the treatment of periodic systems. The 6-21Gmolecular all-electron basis set has been employed. Crystallineorbitals are represented as linear combinations of Bloch functionsand are evaluated over a regular three-dimensional mesh inreciprocal space. Each Bloch function is built from atom-centredatomic orbitals, which are contractions (linear combinations withconstant coefficient) of Gaussian-type functions (GTF), each GTFbeing the product of a Gaussian times a real solid sphericalharmonic. The level of accuracy in evaluating the Coulomb andexchange series is controlled by five thresholds [15], for whichvalues 10�10, 10�10, 10�10, 10�10 and 10�20 were used. Thedensity functional exchange-correlation contributions are evalu-ated by numerical integration over the cell volume. The SCF cycleterminates when the root mean square (RMS) of the change ineigenvalues from two subsequent cycles is less than 10�10

hartree or the change in the absolute value of the total energy isless than 10�9 hartree. The shrinking factors along the reciprocallattice vectors are set to 2, 2 and 2, corresponding to 8 reciprocalspace points of the irreducible Brillouin zone at which the Hamilto-nian matrix is diagonalized. To calculate the relaxed structure ofbulk crystal and slices of different orientations, we start with theexperimental crystal structure [23] as an initial guess. The structuraloptimisations have been carried out by computing analytical energygradients with respect to atomic fractional coordinates in the unitcell, and then standard conjugated gradient routines are used toobtain the lowest energy configuration. Geometry convergence istested on the RMS and absolute value of the largest component ofgradients and estimated displacements. The thresholds for themaximum force, the RMS force, the maximum atomic displace-ment and the RMS atomic displacement on all atoms have been setto 0.00045 a.u., 0.00030 a.u., 0.00180 a.u. and 0.00120 a.u., respec-tively. The optimisation is successfully completed when all the fourconditions set by the above thresholds are simultaneously satis-fied. The symmetry is maintained during the process of structuralrelaxation.

We employed Hartree–Fock (HF) and density functional theory(DFT) methods with hybrid exchange-correlation functional(B3LYP) to obtain the optimised structure and energetics of themolecular layers. The hybrid functional is chosen because of itssuccess in predicting the properties of molecular crystals [24,25].An important issue connected with the calculation of latticeenergy and AE of molecular layers is the basis-set superpositionerror (BSSE). To define the BSSE we consider a system (AB) formedby two interacting fragments A and B. The stabilisation energy of

of urea crystal as a function number of neighbouring atoms included in the CP

surrounded by 89 neighbouring ghost atoms in urea crystal.

M.K. Singh et al. / Journal of Crystal Growth 343 (2012) 77–8580

the system (AB) can be calculated by

DEðABÞ ¼ EðABÞ�EðAÞ�EðBÞ: ð1Þ

There is an error associated to such an approximation becausewe are forced to use finite basis sets, which leads to approximate

Table 2Experimental and calculated lattice parameters and lattice energy of urea crystal

obtained from HF and DFT-B3LYP methods at 0 K. The % deviation in lattice

parameters and lattice energy clearly shows that the result obtained from the HF

method is in close agreement with experimental results.

Experimental lattice

parameters [23] and

sublimation enthalpy

[27] (extrapolated at

zero temperature)

Hartree–Fock DFT-B3LYP

dev. (%) dev.

(%)

A (A) 5.661 5.332 5.8 5.187 8.4

C (A) 4.712 4.630 1.7 4.545 3.5

Lattice energy

(kcal/mole )

�22.2 �21.0 5.4 �29.3 32.0

Table 3Un-relaxed and relaxed attachment energy of molecular layers ‘A’ and ‘B’ on different

Hamiltonian Crystal face

(hkl)

Attachment energy of molecular layers (kcal/mole

Un-relaxed structure

Layer ‘A’ BSSE correction Layer ‘B’ BSSE co

50% 100% 50%

HF (100) �4.7 �3.8 �4.7

(110) �5.5 �2.9 �3.4

(101) �24.0 �21.4 �6.8

(001) �25.7 �22.9 �5.4

(111) �21.5 �19.2 �5.3

DFT-B3LYP (100) �6.1 �4.8 �6.1

(110) �8.1 �4.3 �4.5

(101) �29.4 �26.2 �9.4

(001) �29.9 �26.7 �6.9

(111) �26.0 �23.2 �7.3

Fig. 3. Optimised structure of (001) face of the urea crystal of (a) 3dhkl, (c) 2ð1=2ÞdðhklÞ an

layers ‘A’ and ‘B’, respectively.

energies of each of the fragments. However the calculation ofE(AB) is more accurate than that E(A) or E(B) as it has been obtainedusing a different basis set. As a consequence, E(A) and E(B) areunderestimated compared to E(AB), and the interaction energy isoverestimated. It is precisely this mismatch of E(AB) against E(A) andE(B) that is known as the basis set superposition error. It affects thelattice energy and AE more than the binding energy of moleculeswhen the local basis set is incomplete. The counterpoise (CP)correction [26] is an a posteriori procedure to correct for BSSE. It isbased on the improved calculation of the fragments using all thebasis functions of the complex to obtain the energy of eachfragment, so as to calculate fragments and complex at the samelevel of theory. The way this is implemented in the CRYSTAL09 codeis by supplementing the adopted basis set of the isolated moleculewith the basis functions of an increasing number of ghost atomsbelonging to the surrounding array of molecules that would bepresent in the crystal or slices. For BSSE correction to the lattice andattachment energies, the monomer energy is calculated by placing89 ghost atoms in a cluster surrounding the monomer at the atomicpositions obtained from the optimised crystal structure at the samecomputational level. We restrict up to 89 ghost atoms since thelattice energies of crystals thus obtained are well converged and arein good agreement with experimental enthalpy of sublimation [27].

faces of urea crystal obtained using HF and DFT-B3LYP methods.

)

Relaxed structure

rrection Layer ‘A’ BSSE correction Layer ‘B’ BSSE correction

100% 50% 100% 50% 100%

�3.8 �6.4 �5.3 �6.4 �5.3

�1.6 �4.9 �2.6 �4.5 �2.1

�5.8 �18.7 �16.7 �8.7 �7.5

�4.6 �22.9 �20.4 �6.9 �5.9

�4.5 �18.4 �16.4 �6.8 �5.8

�4.8 �8.7 �7.1 �8.7 �7.1

�2.1 �7.2 �3.8 �6.0 �2.8

�8.0 �22.9 �20.5 �12.0 �10.4

�7.3 �26.8 �23.9 �9.6 �8.2

�6.2 �22.3 �19.9 �9.4 �7.0

d (d) 2dhkl thick slices while (b) and (e) show the optimised structure of molecular

M.K. Singh et al. / Journal of Crystal Growth 343 (2012) 77–85 81

4. Results and discussion

As discussed in the previous section the BSSE leads to anoverestimation of the interaction energies of interacting system.On the other hand, when 100% BSSE correction is considered,interaction energies are often underestimated [28]. Kim et al. [29]have studied the effect of different proportions of BSSE correctionon the binding energies of weakly bound p-complexes and foundthat 50% BSSE correction yields reasonably accurate interactionenergies, especially with moderate basis sets. We have appliedthis idea to compute the lattice energy of urea crystal and findthat it is useful to employ a 50% BSSE correction when comparingthe theoretically evaluated quantities such as lattice energy andenthalpy with the corresponding experimental data. It should bementioned that the 50% BSSE correction cannot be rigorouslyjustified. However, in the particular case of calculations for largemolecular complexes with moderate basis sets, the 50% schemementioned above often yields good results because the ordinaryBSSE tends to cancel the lack of dispersive interactions in smalland moderate basis sets. Fig. 2 shows (a) the dependence ofdifferent proportions of BSSE correction on the lattice energy ofurea crystal as a function number of neighbouring ghost atomsincluded in CP correction for crystalline urea, with HF and DFT-B3LYP using 6-21G basis set and (b) a urea molecule surroundedby 89 neighbouring ghost atoms in the urea crystal. Our calcula-tion reveals that the CP correction of urea crystal converges to alimiting value of about 9.1 kcal/mole, with the largest calculation

Fig. 4. Same as Fig. 3 but for (100) face; the molecular orientations in layer ‘A’ and

‘B’ are the same and AE of these layers are equal.

Fig. 5. Same as Fig. 3 b

including 89 ghost atoms. For completeness, we note here thatinclusion of 30 ghost atoms in the calculation is enough to reach avalue of 8.5 kcal/mole, which represents about 93% of the entireCP correction. Thus, proper consideration of the BSSE correctiondecreases lattice energy considerably at all levels of the theory.The experimental and calculated lattice parameters and latticeenergy of urea crystal obtained from HF and DFT-B3LYP calcula-tions are presented in Table 2. It is evident from Table 2 that thecalculated lattice energies and lattice parameters obtained usingthe HF method are in agreement with the experimental enthalpyof sublimation [27]. The % deviation in lattice parameters andlattice energy clearly shows that the result obtained from the HFmethod is in closer agreement with the experimental result thanthe result obtained using the DFT-B3LYP method. We have alsoconsidered relatively larger basis sets (6-31G(d, p) and TZP) tostudy the effect of BSSE on lattice energies of urea crystal and analmost negligible dependence on the basis set used has beenfound, although the BSSE correction decreases with the basis setsize. However, use of larger basis sets leads to a dramatic increasein the computational time. Keeping this in mind, we employ 50%BSSE correction and a moderate sized basis set to calculate latticeenergy and AE of molecular layers on various faces of urea crystal.

Having discussed the role of BSSE on lattice energy, we now turnour attention to the calculation of vapour growth morphology of ureacrystal. The growth shape of urea crystal is well studied [8(b)–(d)],[10,16,19–22] due to its comparatively simpler molecular structure.The semi-empirical force-field method has been employed tocalculate habit controlling energetics for which formal charges ateach atomic site are derived either from semi-empirical or from ab

initio calculations for the bulk crystal. Accurate determination of theatomic charges is needed to calculate the Coulombic contribution tothe intermolecular interaction energy. Docherty et al. [16] have usedthe HF method using a periodic boundary calculation to deriveatomic charges in a 3D unit cell. Boek et al. [19] have calculatedcharge densities from the HF method and decompose them intomulti-pole moments to calculate the Coulombic contribution to theintermolecular interaction energy. However Docherty et al. and Boeket al. have not considered (101) and (200) faces and also neglectedthe effect of structural relaxation of the faces to calculate growthshape. The growth morphology predicted by Docherty et al. showedthe appearance of (001), (110) and (111) faces. Moreover, the shapecalculated by Docherty et al. using the attachment energy model isclaimed to be in agreement with experimental observations. Ashleyet al. [20] have considered all the above mentioned faces to calculategrowth shape of urea crystal but their predicted morphologyshowed the high morphological importance of (101) and (200) facesbut these faces are completely absent in the experimental morphol-ogy. Apart from this, their calculation also underestimated themorphological importance of (111) face. Engkvist et al. [21] have

ut for (101) face.

M.K. Singh et al. / Journal of Crystal Growth 343 (2012) 77–8582

employed accurate anisotropic atom–atom intermolecular poten-tials, including the distributed-multipole electrostatic force-fieldmodel to calculate AE but their predicted morphology showed theappearance of (200) and (101) faces. Bisker-Leib and Doherty [22]have considered molecular orientation but their predicted shape didnot correctly reproduce experimentally observed shape of ureacrystal. There is a disagreement in the literature on the preferredprocess to select the faces for the attachment energy model.Hartman and Bennema [5] and Boek et al. [19] carried out a detailedPBC analysis to identify the strong interactions between moleculesforming the crystal, followed by the attachment energy calculationof flat faces. In Ref. [6] possible forms were obtained using unit cellparameters and space group symmetry. In earlier investigationsdevoted to the morphology of urea crystal [18,19], an assumptionhas been made that the (200) and (101) faces are not morphologi-cally important and thus they do not appear in the experimentalmorphology. Note that adding this assumption to the attachmentenergy/equilibrium model is out of line with the basic premise thatthe face selection process should be dictated by the calculation ofrelative attachment/surface free energies. Keeping this in mind werelax this assumption in our present calculation because these facescorrespond to large inter-planer spacing and due considerations arerequired to calculate growth shape.

We now proceed with the discussion of the results of AE ofmolecular layers on different faces of urea crystal. In order to

Fig. 7. Same as Fig. 3

Fig. 6. Same as Fig. 3 but for (110) face.

predict the growth shape of a crystal, we start with selection of aset of candidate planes. The accurate prediction of crystal shaperequires correct selection of slow growing planes, which areimportant in crystal growth. To this end, we consider five low-index planes, namely (001), (100), (110), (101) and (111), forcalculating AE of molecular layers on these faces. We define theattachment energies of molecular layers ‘A’ and ‘B’ to the crystalface, ðEðhklÞ

attðlayerÞÞ, as

EðhklÞattðlayerAÞ ¼

1

2Etotal

3dðhklÞ�Etotal

212dðhklÞ�Etotal

12dðhklÞlayerA

� �, ð2Þ

EðhklÞattðlayerBÞ ¼

1

2Etotal

212dðhklÞ�Etotal

2dðhklÞ�Etotal

12dðhklÞlayerB

� �: ð3Þ

The factor 1/2 is placed in Eqs. 2 and 3 because a molecular

layer has two surfaces [30]. Etotal3dðhklÞ

, Etotal21

2dðhklÞand Etotal

2dðhklÞare the total

energies of slices with thickness 3d(hkl), 2ð1=2ÞdðhklÞand 2d(hkl),

respectively. Etotal12dðhklÞlayerA

and Etotal12dðhklÞlayerB

are the total energies of the

molecular layers ‘A’ and ‘B’, respectively. In Fig. 3(a)–(e) we showthe growth process of (001) face. The growth of (001) faceproceeds by the attachment of molecular layer ‘B’ (shown inFig. 3(e)) upon the slice of thickness 2d(hkl) (Fig. 3(d)). The resulting

slice of thickness 2ð1=2Þdð001Þ is shown in Fig. 3(c). The molecular

layer ‘A’, shown in Fig. 3(b), is now attached to the slice as shownin Fig. 3(c). Fig. 3(a) shows the slice of thickness 3d(001). Thusthrough the successive deposition of layer ‘A’ and ‘B’, the growth of(001) face occurs. In Table 3 we list the attachment energies ofmolecular layers ‘A’ and ‘B’ on different faces of urea crystal forun-relaxed and relaxed structures using 50% and 100% BSSEcorrections. Figs. 4(a)–(e), 5(a)–(e), 6(a)–(e) and 7(a)–(e) showthe growth step of (100), (101), (110) and (111) faces, respectively,through the attachment of molecular layers ‘B’ and ‘A’. It is evidentfrom Table 3 that the AE of molecular layer ‘A’ is more than that ofthe layer ‘B’ and the growth is determined by the adsorption oflayer ‘B’ except for (100) face, in which AE of layer ‘A’ is equal tolayer ‘B’ since the molecular orientations in these layers are thesame. The molecular layer having lowest AE is chosen to be ratedetermining, as the growth is assumed to be controlled by theslowest growing layer [6]. At this point we wish to note that thereexists a parallelism between molecular layer considered in thispaper and the PBC in Refs. [4,13,14]. For example, both layers ‘A’and ‘B’ in (001), (101), (110) and (111) faces have differentmolecular orientations and according to the classification madeby the PBC theory, they are characterised as F(flat)-faces. On the

but for (111) face.

M.K. Singh et al. / Journal of Crystal Growth 343 (2012) 77–85 83

other hand, molecular layers in (100) face have the same orienta-tions and they are classified as S(step)-faces.

The predicted growth shapes of urea crystal, obtained byemploying 50% BSSE correction are presented in Fig. 8(a) and(d) for un-relaxed and relaxed structures from HF and DFT-B3LYPmethods, respectively. The relaxed growth morphologiesobtained using HF and DFT methods are in excellent agreementwith the experimental morphology of urea crystal obtained fromvapour phase [16,31]. Both calculated and experimentallyobserved shapes of urea crystal show the appearance of (110),(001) and (111) faces. In Fig. 9 (a)–(d), we show the predictedshape of urea crystal when 100% BSSE correction has been appliedto obtain AE of molecular layers. A comparison with a shapedrawn from a scanning electron micrograph of urea crystallisedfrom the vapour phase [16], as shown in Fig. 10, reveals that thereis an excellent agreement between our simulated shape obtainedusing 50% BSSE correction (shown in Fig. 8) and the experimentalshape. The un-relaxed shape shown in Fig. 8 shows a slightappearance of (100) face but it disappears when structuralrelaxation is allowed. It is clear from Table 3 that the structuralrelaxation considerably changes the attachment energies of themolecular layers and hence the shapes. Using the results pre-sented in Table 3, the aspect ratio (AS¼R001/R110) is found to be1.54 and 1.60 for HF and DFT-B3LYP methods, respectively.We note here that our results for the aspect ratio are in excellentagreement with the corresponding experimental value of 1.5[16,31]. The quantitative agreement between experimental andcalculated shapes of urea crystal, where energetics has beenobtained using 50% BSSE correction, is presented in Table 4. Itlists the measured and calculated percentage form area andpercentage deviation in the experimental and calculated shapesof urea crystal. It reveals an excellent agreement between our

Fig. 10. Experimental growth shape from vapour, reproduced from Docherty

et al. [16].

Fig. 9. Same as Fig. 8 but habit controlling energetics obtained using 100% BSSE

correction.

Fig. 8. The predicted vapour growth morphology of crystalline urea: (a) and (c)

un-relaxed and (b) and (d) relaxed structures using 50% BSSE correction obtained

from HF and DFT-B3LYP methods.

calculated relaxed morphologies using the DFT method and theexperimental results for vapour grown crystal [16]. Figs. 9 and10show that when 100% BSSE correction has been employed tocalculate AE of molecular layers the shape obtained does notcorrespond to experimental shape. The calculated aspect ratio

Table 4Measured and calculated percentage form area and percentage deviation in experimental and calculated morphology obtained using 50% BSSE correction of urea crystal.

Forms in the

urea crystal

Data from

experimental

morphology [16]

Data from our calculated morphologies

Hartree–Fock DFT-B3LYP

Un-relaxed Relaxed Un-relaxed Relaxed

Form area (%) Form area (%) Dev. (%) Form area (%) Dev. (%) Form area (%) Dev. Form area (%) Dev. (%)

(110) 70.3 66.4 5.5 68.4 2.6 66.3 5.6 70.1 0.3

(001) 18.3 20.1 10.1 19.9 8.5 21.5 17.2 19.5 6.7

(111) 11.5 10.7 7.0 11.7 1.8 7.5 70.38 10.4 9.6

(100) 0.0 2.8 100 0.0 0.0 4.7 100.0 0.00 0.00

(101) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

M.K. Singh et al. / Journal of Crystal Growth 343 (2012) 77–8584

from HF and DFT-B3LYP methods is 2.8 and 2.9, respectively.We wish to mention that the discussed model is currently beingapplied to other molecular crystals like b-succinic acid anda-resorcinol crystals to calculate growth shape. The details willbe published in due course of time.

It is interesting to re-examine the effects of thermal andgrowth kinetics on vapour growth morphology of molecularcrystal since growth morphology can be modulated through theirmanipulations. The attachment energy model correlates thegrowth rate of the face to the energies in the crystal structurebut does not include the effect of temperature, growth mechan-ism or driving forces for crystallisation. In reality, the growth rateis determined by the step free energy, since both birth-and-spread and spiral growth models have this quantity, apart fromthe supersaturation, as their key parameter. Temperature mainlyshows up in roughening of crystal faces, usually resulting inrelatively high growth rates depending on the growth orientation(hkl). Although we are aware that the accurate understanding ofgrowth mechanism needs step energy calculation and not attach-ment energy, the attachment energy method often gives reason-able results [7]. This can be explained by considering that smallattachment energies usually mean large slice energies. For iso-tropic connected net, that is a net in which the bonds are more orless equal in all directions, large slice energies result in large stepenergies leading to slow growth. An anisotropic connected netconsists of a very strong and a very weak PBC; that is, it isconnected very strongly in one direction and very weakly inanother direction. Hartman and Bennema [5] showed, by studyingtheories of crystal growth mechanisms such as spiral growth and2-D nucleation mechanism, that relative growth rates of faces arein principle proportional to attachment energy of the face. It hasbeen shown by Hartman and Bennema that at very low super-saturation such as the case of vapour grown crystal, the crystalgrowth morphology is independent of the supersaturation. Forlarge attachment energies and particularly at high supersatura-tion the morphology is isometric. In between, the habit tends tobe non-isometric. When the supersaturation is very high, thehabit is hardly affected by changing supersaturation.

We close this section by mentioning that the polar habit ofvapour-grown urea crystal further poses a problem for attach-ment energy model: the (111) faces are observed, but not theirFreidel opposite (�1�1�1) faces; the usual calculation of AE isnecessarily equal for the two faces. Two explanations have beenproposed to substantiate for polar shape [16,20]. Docherty et al.[16] suggested that the polarisation of the (111) and (�1�1�1)crystal surfaces would be sufficiently different to each other,which causes anomalous growth rate of (111) and (�1�1�1)faces. George et al. [20] proposed that the anomalous growth of(111) and (�1�1�1) faces arises from the difference in the

relaxation of the surfaces. In light of these studies, we have alsomade an attempt to calculate AE of molecular layers in (111) and(�1�1�1) faces of urea crystal before and after structuralrelaxation of the faces but could not find any difference in theirAE. We anticipate that the attachment energy formalism couldnot predict the polar morphology of urea crystal from vapourphase. To solve this problem, we have already started to examinethe growth mechanism of (111) and (�1�1�1) faces of ureacrystal using the surface docking method. We expect that theanomalous growths of (111) and (�1�1�1) faces of urea crystalare fundamental characteristics of the growth of polar crystalsupon which relaxation may be superimposed. In the case ofsolution growth, pronounced differences in growth rate of theabove faces are associated with differences in the solid–liquidinteractions at the interface and external growth parameters.

5. Summary and conclusion

In conclusion, we studied the vapour growth shape of crystal-line urea using a modified HP model by performing periodicab initio total energy calculation to evaluate habit controllingenergetics of molecular layers. The roles of molecular orientationand surface relaxation on vapour growth morphology of ureacrystal were studied. The calculated relaxed shape of urea crystalobtained using our approach discussed in this paper reproducedall experimentally observed forms and excellent agreement of therelative forms area.

Acknowledgement

The authors are grateful for the support and motivationreceived from Dr. V.K. Wadhawan. Discussions with S.K. Sharmaand Dr. V.S. Tiwari are gratefully acknowledged. The referees forthis paper are also thanked for their helpful comments. All thecalculations were performed on the Brahma and Nalanda Linuxclusters at our centre.

Appendix A. Supporting material

Supplementary data associated with this article can be foundin the online version at doi:10.1016/j.jcrysgro.2012.01.032.

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