Intermolecular Energy and Structure of

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Oct 26, 1979 - caractériser des facettes tres étroites de la familie. {020}, qui étaient visiblement sur Ie point de disparaïtre de la morphologie des dendrites.
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MORPHOLOGIE THEORIQUE DU COMPOSE Al3Ni

BENNEMA, P. & GILMER, G. M. (1973). Kinetics ofCrystal Growth. In Crystal Growth: An Introduction, edited by P. HARTMAN. Amsterdam: North-Holland. BIENFAIT, M. & KERN, R. (1964). Buil. Soc. Fr. Minéral. Cristallogr. 87,604-613. DONNAY, J. D. H. & HARKER, D. (1937). Am. Minéral. 22, 446^167. DOWTY, E. (1976). Am. Minéral. 61,448-457. EUSTATHOPOULOS, N. (1974). These de Docteur-ès-Sciences Physiques, Univ. de Grenoble. FELIUS, R. O. (1976). Structural Morphology ofRutile and Trirutile Type Crystals. Dutch Efficiency Bureau, Pijnacker. HARTMAN, P. (1965). Absorption et Croissance Cristalline, Colloque CNRS, Nancy. HARTMAN, P. (1973). Structure and Morphology. In Crystal Growth: An Introduction, edited by P. HARTMAN. Amsterdam: North-Holland. HARTMAN, P. (1978). Buil. Soc. Fr. Minéral. Crystallogr. 101, 195-201.

HAVINGA, De Kruif PAULING, Ithaca, Selected Element SMITHELLS Butterworth. TASSONI, D. (1978). These de Docteur-Ingénieur, Univ. de Grenoble. TASSONI, D., RIQUET, J. P. & DURAND, F. (1978a). Acta Cryst. A34, 55-60. TASSONI, D., RIQUET, J. P. & DURAND, F. (19780). J. Cryst. Growth, 44, 241-246. Thermochemical Tables (1971). 2nd ed. JANAF, NBS, Washington. TORRENS, I. M. (1972). Interatomic Potentials. New York: Academie Press. WULFF, G. (1901). Z. Kristallogr. Minéral. 34,449-530.

Acta Cryst. (1980). A36,428-432

Intermolecular Energy and Structure of Tetrathiafulvalene (TTF) Stacks from AtomAtom Potentials BY H. A. J. GOVERS AND C. G. DE KRUIF General Chemistry Laboratory, Chemical Thermodynamics Group, State University of Utrecht, Padualaan 8, Utrecht 2506, The Netherlands (Received 26 October 1979; accepted 3 December 1979)

Abstract

Introduction

The lattice energy of isolated, regular tetrathiafulvalene stacks was minimized for a longitudinal slip of the molecules relative to each other at constant intermolecular separation and transverse slip. The van der Waals and repulsive interactions were calculated from atom—atom potentials. A simple expression is presented for the electrostatic interaction in neutral and charged stacks. This electrostatic contribution was calculated from CNDO/2 atomic point charges. The latter contribution proved to be negligible for stacks built up from neutral molecules. For these stacks the minimum of the lattice energy is achieved at a slip of 0-1-0-2 A below the observed values. Eclipsed stacks, with zero slip, appeared to be only 3-35 kJ mol"1 less stable than slipped ones. In stacks built up from positively charged molecules the van der Waals and repulsive contributions are dominated completely by the electrostatic interaction. These stacks tend to a structure with infinite slip. 0567-7394/80/030428-05S01.00

The existence of segregated stacks of acceptor and donor molecules (Soos, 1974) is an important condition for high electrical conductivity and other one-dimensional properties of compounds like tetrathiafulvalenetetracyanoquinodimethane, TTF-TCNQ (Kistenmacher, Phillips & Cowan, 1974). In TTF compounds two different kinds of stacking are found. In the first (Kistenmacher, Phillips & Cowan, 1974; Cooper, Edmonds, Wudl & Coppens, 1974), the flat TTF molecules are slipped relative to each other along the longitudinal molecular axis by d = 1-6-1-7 A. In the second (Scott, La Placa, Torrance, Silverman & Welber, 1977; Wudl, Schafer, Walsh, Rupp, Di Salvo, Waszczak, Kaplan & Thomas, 1977), the molecules eclipse with d ~ 0-0 A. In both kinds the slip, e, in the transverse direction of the short molecular axis is about 0-0 A. A range of 3-3 to 3-8 A is observed for the intermolecular separation, R. A second condition is the existence of partial charge © 1980 International Union of Crystallography

D. TASSONI, J. P. RIQUET ET F. DURAND

dendrites appartiennent aux formes de plans {110}, {101} et (111}. Lors d'expériences de croissance en solution métallique (Tassoni, 1978) de fines dendrites Al3Ni naissent a partir d'une fine couche de liquide hypereutectique par un processus de solidification plus rapide que pour celui étudié par Tassoni, Riquet & Durand (1978i). Les observations au microscope èlectronique a balayage conjointement au traitement par geometrie descriptive (Tassoni, Riquet & Durand, 1978Z») ont permis de caractériser des facettes tres étroites de la familie {020}, qui étaient visiblement sur Ie point de disparaïtre de la morphologie des dendrites. La Fig. 4 est une photographie d'une de ces dendrites. Dans les mêmes expériences de croissance en solution métallique (Tassoni, 1978) nous avons obtenu la solidification monophasée de Al3Ni autour d'un germe de nickel polycristallin. Nous avons caractérisé différents cristaux obtenus; les seules facettes présentes appartiennent aux formes {110} et {101} dans eet ordre d'importance. Par conséquent nos diverses expériences donnent dans l'ordre d'importance morphologique les families de plans {110}, {101}, {111} et {020}. Nous n'avons pas observé d'autres facettes pour Al3Ni. Nous constatons que ces families se retrouvent dans Ie Tableau 6, dans l'ordre même observé pour leurs extensions relatives. Cependant les plans {002} n'ont jamais été observés, et les plans {020} et {111} n'ont été observés que sur des dendrites et ils ont disparu sur des cristaux de taille plus importante.

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postulat suivant lequel la vitesse de croissance est proportionnelle a l'énergie d'attachement nous admettons implicitement qu'un même cristal présente toujours la même morphologie quel que soit Ie milieu de croissance. Nos observations confirment cette idee du moins en ce qui concerne la phase vapeur et notre solution métallique. Si nos calculs étaient repris pour simuler la croissance a partir de la solution liquide, et en admettant que nous disposions d'un modèle adéquat de l'alliage liquide, les valeurs E,, Eldc et EA se trouveraient considérablement réduites. Toutefois il n'est pas évident que les valeurs relatives des rapports Ef/Eiül. s'en trouvent bouleversées ainsi que l'allure des diagrammes de la Fig. 2. 8. Conclusions

Dans Ie composé Al3Ni nous avons determine 14 families de chaïnes périodiques de liaison, qui permettent de définir 11 directions de plans réticulaires a caractère F. Nous constatons que celles-ci sont également les 11 premières families données par la classification de Donnay & Harker (1937). Afin de déterminer ceux de ces plans qui font partie de la morphologie des cristaux, nous avons effectué un calcul énergétique. En utilisant Ie potentiel de LennardJones et l'expression des constantes donnée par

Havinga (1972), nous avons évalué les energies potentielles de liaison dans la structure Al3Ni, et les energies des fixations successives des atomes sur les

différentes facettes. Le calcul permet de visualiser l'ordre de fixation énergétiquement favorable. Il montre que sur une facette a caractère F les premières fixations Les calculs du §4 ont été effectués avec p = In = 12. sont nettement plus difficiles que les suivantes, ce qui Afin de déterminer l'influence éventuelle de ce traduit que la croissance d'une facette F s'effectue paramètre, nous les avons repris avec p = 2n = 10 et 8. tranche par tranche. Par une construction reposant sur la valeur de Les valeurs des energies reportées dans Ie Tableau 6 diminuent lentement et de fac.on monotone avec p. Les l'énergie d'attachement, nous en déduisons que la écarts relatifs entre les valeurs des facettes demeurent morphologie théorique de Al3Ni devrait comporter les sensiblement les mêmes. En particulier Ie classement facettes {110}, {101}, {11 U, {020} et {002} dans suivant Eati n'est pas modifié. Dans la 'construction de l'ordre d'extension décroissante. Nous avons observé Felius' (1976) les facettes {020} et {002} se trouvent les quatre premières families sur des dendrites et des faiblement réduites sans toutefois disparaïtre. La cristaux massifs. Seule la familie {002} n'a jamais été morphologie de croissance ne s'en trouve done pas observée. modifiée. Ce travail a été possible grace a la bourse dont l'un Pour simplifier l'étude nous avons considéré que la croissance s'effectue a partir de la vapeur. Or nos de nous (DT) a disposé de la part de la Fundacion Gran observations portent sur des cristaux ou des dendrites Mariscal de Ayacucho, Caracas, Venezuela. formes a partir d'un alliage liquide. Malgré cette difference nous constatons que notre calcul permet de prévoir les quatre families de facettes les plus étendues References dans la morphologie. Par définition l'énergie d'attachement ne dépend que BARIN, L, KNACKE, O. & KUBASCHEWSKI, O. (1977). des interactions a l'état solide, et pas du tout du milieu Thermochemical Properties of Inorganic Substance, dans lequel s'effectue la croissance. En appliquant Ie Supplement. Berlin: Springer. 7. Discussion

H. A. J. GOVERS AND C. G. DE KRUIF

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transfer, p, from the electron-donor to the electron- molecules in the stack and rk(i are the interatomic acceptor molecule (Soos, 1974). From the viewpoint of distances, which in our approach were calculated from classical electrostatic or Madelung interaction (Metzger the molecular geometry of TTF and from the value of & Bloch, 1975) one can describe this in a uniform the molecular separation, R = 3-47 A, observed in model, in which all molecules bear identical charges TTF-TCNQ (Kistenmacher, Phillips & Cowan, 1974). plus or minus p, or in a Wigner chain, in which some As transverse slip we used the model value e = 0-00 A. molecules are neutral and others are completely The distance, b, between the centres of two neighbour charge-transferred, monovalent ions. In pure TTF, all molecules in the stack was calculated from the relation molecules are neutral and no charge transfer is b2 = R2 + d2. Therefore b varies with the longitudinal observed. In TTF-TCNQ, p = 0-59e (Metzger, 1977). slip, S, Fig. 1. It has been shown recently (Silverman, 1979a) that The parameters At(j, Bti], Ct(i in (2) depend only on the energy of isolated TTF dimers exhibits a minimum the six different types, tij, of interatomic pairs CC, CH, for the eclipsed geometry only. However, this quantum- CS, HH, HS and SS, which exist for the C, H and S mechanical calculation conflicts with a packing atoms of TTF. We used set l of Table l by Govers analysis (Silverman, 19796), which shows that only the (1978). These parameters determine the van der Waals, slipped stacking geometry of TTF in TTF-TCNQ and £vdw, and repulsive, Enp, contributions to the lattice in pure TTF is consistent with the close packing of hard energy and were used as previously, i.e. in combispheres having atomic van der Waals radii. nation with summation limits of about 5-5 A yielding The atom-atom potential method (Kitaigorodskii, 80% of the lattice energy (Govers, 1978). 1973) can be considered to be of intermediate sophistiThe parameters et and ej in (2) are the point charges cation between a close-packing analysis and a quan- on the atoms / and j. These determine the electrostatic tum-mechanical calculation. This method has been contribution, Etttctt, to the lattice energy. We used the shown to be useful for the calculation of the lattice CNDO/2 charge distributions for TTF° (neutral) and energies of TTF-TCNQ crystals (Govers, 1978;* TTF+ (monovalent) molecules from the sets l and 4, Sandman, Epstein, Chickos, Ketchum, Fu & Scheraga, respectively, of Table 3 by Epstein, Lipari, Sandman & 1979). Here, it is our aim to predict the observed TTF Nielsen (1976).* The total electrostatic contribution to stack structures via this approximation, with a one- the chain energy cannot be calculated via a simple dimensional chain model. As far as we know no simple lattice sum of contributions eieilrkli, which is caused by method exists for the calculation of the electrostatic bad convergence properties. We used the expression interaction in chains built up from large numbers, of the £e.ectr = ^el'ectr + (Ne/b)[\nN - In (\Z' + 1) - l ], (3) order of Avogadro's number, of charged molecules. Therefore, we will also derive an expression for this with interaction. This work precedes a more complete three-dimensional analysis via the atom-atom approxi(4) « = II«/«y mation (Silverman & Govers, 1980). Therefore we are / J interested only in a simple calculation with no variation

of R, e, atom-atom potential parameters and charge distribution models.

In (3) E'eiectr 's

a

direct sum of interactions e,ej/rk(J

between the atoms of a central molecule and those ofz' neighbouring molecules. It was calculated in the normal

way via (1). The quantity e in (3) and (4) is an

Method

The lattice energy, E, is considered to be a pairwise sum of the interatomic interactions, EklJ, between the n

* Note that the charge on H (2) of TTF+ (set 4, Table 3) should be 0-0814 mstead of 0-0184 (Epstein, 1978).

atoms, i, of a central molecule in the stack and the n' atoms, j, of the z surrounding molecules, k (Govers, 1978):

^tfixi^o*,). * i j

CD

with

£k,/rkt/) = -A w T« + Btu exP (~ctu r* + eJ«A«' (2) In (1) the factor ^ is introduced to avoid doublé counting of pair interactions, N is the number of * The corrected values (m kcal mol-' = 4 19 kJ mol-') of £^e'cctr m Table 2 of this reference are -0-04, -0-01, +0-58 and +0 01, and m Table 3 -O- 38, -1 • 78, -9 -09, -1 • 88 and -6 • 35.

Fig 1. Nearest neighbour molecules in a TTF stack; definition of intermolecular separation, R longitudinal slip, S, transverse slip, s, and distance between the molecular centres, b.

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INTERMOLECULAR ENERGY AND STRUCTURE OF TETRATHIAFULVALENE

intermolecular sum over the products of the charges on the atoms i and j. For neutral TTF° molecules this sum is zero. Thus only E^ectr remains in (3). The derivation of (3) is given in the Appendix. An overall error of Z. The same neglect of boundary effects is used in the further simplification of (A 2) in combination with the Euler theorem (Hyslop, 1959): m-l

X

(!/£') = hi(m- l ) - l n ( Z + 1),

which holds almost exactly if Z «^ Af and Af is Avogadro's number. Substitution of U 3) and U4) into U 2) then results in (3). We next have to show that (A 1) holds if Z > 30, for example. For a stack with b = 4-0 A, built up from flat centrosymmetrical molecules with a longitudinal diameter of about 8 A as observed in TTF, the largest percentage deviation, bk' — rt,tf, for a molecule at k' = m + 30 amounts to ±7%, when the molecules are arranged in a direct line with each other. Thus for a symmetrical molecule these deviations counterbalance each other to a large degree. Moreover, the molecules are often far from being in a direct line with each other and only a small fraction of all rk,y has this largest deviation. Numerical calculations for eclipsed TTF stacks with p = 0-59, in a uniform model, with a CNDO/2 charge distribution with R=b= 3-47 A, as described in the Method, and performed for various values of Z yield results as given in Table 2. These results, which were calculated via (3), show that already at Z = 5, the total electrostatic energy, £electr, is calculated with an uncertainty of < l • 5 %c-

(A3)

*'=Z+1

which holds within 4%c if Z, m > 30, and in combination with the Stirling approximation (Hyslop, 1959) ln[(A r -Z)!/(Z+ l)\]=NlnN-N, (A4)

References

COOPER, W. F., EDMONDS, J. W., WUDL, F. & COPPENS, P.

Table 2. Influence of the number of neighbours, Z, on the electrostatic chain energy, calculated by (3) for an eclipsed and charged TTF stack as specified in the text E

„Kt,(P = 0-59) = (0-59)2 £ elKtr (p = 1-0) is used. Both energies are in kJ mol"1. Z = \z'.

Note that even at Z = 1000, with a calculation of £*iectr extending to neighbours at 3470 A in both directions of the chain, ^tiectr amounts to only 13% of the total electrostatic contribution.

At Z = 30, our choice, this amount is 6-5%. 2 5 8 11 30 100 300

1000

£