Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.chem.msu.ru/rus/lab/phys/cryschem/liter/pap_6.pdf Дата изменения: Fri Oct 29 13:12:10 2010 Дата индексирования: Sat Feb 12 23:37:13 2011 Кодировка: Windows-1251

View Online

COMMUNICATION

www.rsc.org/crystengcomm | CrystEngComm

On the topology of layered motifs (H2O)
Alexandru Banaru and Yuri L. Slovokhotov*
Received 9th September 2009, Accepted 10th November 2009 First published as an Advance Article on the web 25th November 2009 DOI: 10.1039/b918793f

N

Downloaded by Lomonosov Moscow State University on 29 October 2010 Published on 25 November 2009 on http://pubs.rsc.org | doi:10.1039/B918793F

On the basis of Infantes-Motherwell classification of water clusters in organic hydrates combined with the theory of planigons, feasible topologies of water layers were derived. A value of the protic excess for water motifs was introduced. Its dependence on the nature of an organic molecule, and its impact on the topology of a water layer, were analyzed.

1. Introduction
Infantes and Motherwell1 introduced a convenient symbolism intended for finite and infinite water clusters that were realized in organic hydrates: discrete chains (D), continuous chains (C), rings (R), tapes (T) and layers (L). On the basis of data retrieved from the CSD2 they analyzed the abundance of respective graphs over pure water motifs1 and motifs extended by hydrogen bond donors and acceptors of an organic molecule.3 Afterwards, a search for the conditions of formation of a hydrate was performed and the distribution of water molecules by types of coordination was presented.4 Amongst various water clusters layered water motifs Lm1(n1)m2(n2).mk(nk), composed of m1-, m2-, ., mk-membered rings, which share nodes (oxygen atoms) with n1, n2, ., nk adjacent cycles1 present a particular interest in our opinion. Firstly, some of these layers can be unalterably transported into a 3D framework, e.g. layer L5(7), which occurs in hexahydrates of pinacol (CSD refcode PINOLH) and piperazine (PIPERH), can also be visualized in the structure of water ice XII.5 Secondly, water networks can not only be formed on a relatively flat surface, but also on other 2D surfaces. Platonic and Archemedean solids that are in fact cells of hydrate clathrates, are formed on spherical surfaces. For instance, layer L5(7) is made up of dodecahedron patches. On a cylindrical surface tubes are formed, e.g. the structure of 18-crown-6$CH3NH3F$6H2O (LUQJEM6) which contains water nanotubes of the topology L6(6), identical with those of carbon (single-wall graphene nanotubes). The most important challenge is to derive feasible motifs on arbitrary 2D manifolds. The current communication represents a deductive derivation (topological analysis) of feasible water layers Lm1(n1)m2(n2).mk(nk) according to the stoichiometry of a hydrate.

forming no more than 4 hydrogen bonds within a cluster (H2O)m. Two nets correspond to symbol L5(7), one to symbol L6(6) and one to L4(8). Any other symbol Lm(n) does not fit a (H2O)N layer. Symbols Lm1(n1)m2(n2).mk(nk) with k > 1 behave in a different way. Apparently, as any of respective motifs is periodic, it can be mapped as a certain regular net with equal cells divided into k polygons. In particular, the most widespread in organic hydrates layer L4(6)5(7)6(8) can be transposed into its topologic equivalent, shown in Fig. 2(a), with a hexagonal fundamental region. Taking into consideration that in this layer one 4-membered ring is accompanied by one 6-membered and two 5-membered rings, a symbol reproducing the stoichiometry would be L4(6)5(7)5(7)6(8). Naturally, there are other topologically non-equivalent nets L4(6)5(7)5(7)6(8) (Fig. 2(b)), as well as truly L4(6)5(7)6(8) (Fig. 2(c)), though none of them have been found in a crystal structure of a hydrate so far. For the regular nets composed of equal t-sided polygons the following equation from a well-known Euler formula (eqn (1)) was derived:7 t 1 1 1 ? bрF Ю (1) F А F, ю F, ю ю ::: ю 2 a1 a2 at where F is the number of t-sided polygons within a certain simply connected domain, t/2 is the number of ribs per one polygon, ai is the degree of node i (i ? 1, ., t), and b(F) ( F. This equation follows from the fact that the ratio of a number of polygons adjacent to a simply connected domain to the number of its interior polygons (F)

2. Regular partitions of a flat surface
There are merely 11 regular partitions of a flat surface into equal polygons (planigons), called Laves nets.7 Yet only four of them (Fig. 1) account for water layers, since the others have nodes with a degree more than 4, while H2O, strictly speaking, is capable of
Moscow State University, Chemistry Faculty, Leninskie Gory, 1, Moscow, 119991, Russia. E-mail: slov@phys.chem.msu.ru; Tel: +7 495 9395434

Fig. 1 Laves nets applicable for description of layered motifs (H2O)N.

1054 | CrystEngComm, 2010, 12, 1054-1056

This journal is ? The Royal Society of Chemistry 2010


View Online

Fig. 2 Instances of topologically non-equivalent layers L4(6)5(7)5(7)6(8) (a, b) and L4(6)5(7)6(8) (c). Laves planigons are marked in red.

Downloaded by Lomonosov Moscow State University on 29 October 2010 Published on 25 November 2009 on http://pubs.rsc.org | doi:10.1039/B918793F

decreases with increase of F.There are E ? F$t/2 edges, V ? F$(1/a1 +1/a2 + . +1/at) vertices, and F faces per a domain, with (V А E + F) being as many times smaller than F as needed. We have modified this formula (eqn (2)) for nets (H2O)N containing merely tri- and tetra-coordinated nodes. Taking into consideration that each polygon mi(ni) consists of (ni А mi) tetraand (2mi А ni) tri-coordinated nodes, we obtain: 0P k k k k k P P P P1 mi ni А mi 2 mi А ni Bi?1 C i?1 i?1 C ю i ?1 F ,k А F , i?1 ю F ,B @ A ? b рF Ю 2 4 3 (2) Designating F/N
k P i?1

Fig. 3 Distribution of 158 trihydrates, 90 tetrahydrates, and 33 hexahydrates (only organic; with 3D-coordinates determined; number of residues ? 2; Z0 ? 1) retrieved from the CSD.2 The construction of motifs (H2O)m was performed using the Mercury program8 for bonds H2O/ OH2 not longer than 3.04 A (in accordance with van der Waals radii 9 proposed by Bondi ).Dimers D2 with protons ordered (a) and disordered by means of an inversion centre (b).

mi hM ;

k P i?1

ni hN we obtain, that in the limit

M + N ? 12k

(3)

at that M # N. If k ? 1, four nets formally satisfy eqn (3): L6(6), L5(7), L4(8) and L3(9), the latter not existing, since a triangle has 6 adjacent polygons at most. It is not difficult to show that the layer L4(6)5(7)6(8) also satisfies eqn (3), if k ? 3.

the common position (Fig. 3(a)) and in the inversion centre (Fig. 3(b)). The latter case implies excess protons being disordered. For structures with no disorder p$n$Z0 is mostly integer. When Z0 ? 1, p$n becomes an integer. In many cases this value equals the number of H-acceptors of an organic molecule H-bound with the motif (H2O)m. As it was shown previously,4 just about 15% of hydrate water molecules do not saturate donor H-bonds. Our results show unequal distribution over dimensionalities of (H2O)m motif (Fig. 4). Low values of p$n, in general, accounting for layers and for the majority of tapes, entail practically complete saturation of excess protons, whereas higher values of p$n entail less saturation. If water motif is a layer containing merely tri- and tetra-coordinated atoms O, then: x$n ? 2j, j N (4)

3. Protic excess
The protic excess of water motif (H2O)m is the average number of hydrogen atoms per molecule H2O, which do not belong to the ribs of the Om graph, designated as p. Evidently, 0 # p # 2; p > 1 for discrete chains (D), p ? 1 for continuous chains (C), for tapes generally 1/2 < p < 1. Since a tri-coordinated atom O possesses, on average, 1/2 an excess of protons, and a tetra-coordinated atom O has no excess protons, each layer L containing merely tri- and tetracoordinated nodes has p # 1/2. Let the stoichiometry of a hydrate be Y$nH2O, where Y is the organic component. Then p$n$Z0 is the number of excess protons, and (2 А p)$n$Z0 is the number of binding protons, per asymmetric unit. If O atoms do not occupy specific positions (site symmetry higher than 1) and are not disordered, then p$n$Z0 and (2 А p)$n$Z0 are both integers or non-integers. Excess protons can not occupy specific positions, otherwise the adjacent O would occupy the same position (generally a rotational axis), therefore when p$n$Z0 is a noninteger, some excess protons are disordered. On the contrary, binding protons can not be disordered, otherwise adjacent atoms O would be also disordered. Consequently, when (2 А p)$n$Z0 is a non-integer, some binding protons occupy specific positions. In Fig. 3 water dimers D2 are shown, in which the only binding proton is located in
This journal is ? The Royal Society of Chemistry 2010

where x ? 2p denotes fraction of tri-coordinated nodes. One may utilize eqn (4) in order to derive the most popular layers (H2O)N in hydrates of different n. In monohydrates (n ? 1) water layers can not occur, because x < 1. In dihydrates there can be merely a layer, in which every node is tri-coordinated, i.e. L6(6). In trihydrates, there can be a layer with x ? 2/3, in tetrahydrates a layer with x ? 1/4 and L6(6), etc. Let us express the protic excess of a layer via topologic characteristics M and N. For that we should count the number of binding protons per a simply connected domain of cycles mi(ni), i ? 1, ., k, by dividing the number of its ribs by the number of its nodes (eqn (5)):

Fig. 4 Dimers D2 with protons ordered (a) and disordered by means of an inversion centre (b).

CrystEngComm, 2010, 12, 1054-1056 | 1055


View Online

2Аp? It follows that:

M =2 рN А M Ю=4 юр2M А N Ю=3

(5)

4. Conclusion
It was shown that integer-valued number of protons not belonging to a graph ON (excess protons), per asymmetric unit, facilitates the prediction of plausible symbols Lm1(n1)m2(n2).mk(nk). For layered hydrates Y$nH2O this value, as a rule, equals the number of H-acceptors of Y bound with water molecules. Thus, the topology of water layer is implicitly defined by the nature of organic compound. The values k k k P P P mi ; ni А yi
i ?1 i?1 i?1

M 2Аp ? N 4 А 5p and, taking into account eqn (3), p?1А 2k M А 2k

(6)

(7)

Dividing the arisen ratio by k, and designating M/k ? mi (the mean size of cycles mi), we obtain:
Downloaded by Lomonosov Moscow State University on 29 October 2010 Published on 25 November 2009 on http://pubs.rsc.org | doi:10.1039/B918793F

p?1А

2 mi А 2

(8)

where yi is the number of bi-coordinated nodes in cycle mi(ni), and k, are interrelated. Hence, the protic excess corresponds to the mean size of cycles mi.

Eqn (7) and eqn (8) may be utilized to predict water layers in hydrates Y$nH2O with different n. For instance, in trihydrates, as it was shown above, p ? 1/3, therefore M ? 5k. Layers L5(7), L4(6)6(8) and L4(6)5(7)6(8) satisfy this criterion, and, taking into account that H2O tends to form 4-, 5- and 6-membered rings, the mentioned layers are likely to be merely plausible, when k # 3. By means of the reverse speculation one may find that the mentioned layers are to be realized in hydrates with n divisible by 3. For layers Lm1(n1)m2(n2).mk(nk) containing not only tri- and tetra-coordinated nodes, but bi-coordinated nodes, too, it is not complicated to prove the validity of the following equations: M + N А Y ? 12k M 2Аp ? N А Y 4 А 5p where Y ?
k P i?1

Acknowledgements
The authors are thankful to A. Plugaru for her help in preparing this manuscript.

References
1 L. Infantes and S. Motherwell, CrystEngComm, 2002, 4, 454. 2 F. H. Allen and W. D. S. Motherwell, Acta Crystallogr., Sect. B: Struct. Sci., 2002, 58, 407. 3 L. Infantes, J. Chisholm and S. Motherwell, CrystEngComm, 2003, 5, 480. 4 L. Infantes, L. Fabian and W. D. S. Motherwell, CrystEngComm, 2007, 9, 65. 5 E. A. Zheligovskaya and G. G. Malenkov, Russ. Chem. Rev., 2006, 75, 57. 6 K. A. Udachin and Ya. Lipkowski, J. Struct. Chem., 2002, 43, 705. 7 B. N. Delone, Izv. Akad. Nauk SSSR Ser. Mat., 1959, 23, 365. 8 C. F. Macrae, P. R. Edgington, P. McCabe, E. Pidcock, G. P. Shields, R. Taylor, M. Towler and J. van de Streek, J. Appl. Crystallogr., 2006, 39, 453. 9 A. Bondi, J. Phys. Chem., 1964, 68, 441.

(30 ) (60)

yi

and yi is the number of bi-coordinated nodes in cycle mi(ni). However, the form of eqn (7) should not be revised.

1056 | CrystEngComm, 2010, 12, 1054-1056

This journal is ? The Royal Society of Chemistry 2010