J. Phys. Chem. C 2009, 113, 3467–3470
3467
The Origin of the Auxetic Effect in Prismanes: Bowtie Structure and the Mechanical Properties of Biprismanes. Nir Pour,† Eli Altus,‡ Harold Basch,† and Shmaryahu Hoz*,† Department of Chemistry, Bar-Ilan UniVersity, Ramat-Gan, 52900, Israel, and Faculty of Mechanical Engineering, Technion, Israel Institute of Technology, Haifa, 32000, Israel ReceiVed: NoVember 6, 2008; ReVised Manuscript ReceiVed: December 29, 2008
Prismanes have been shown to exhibit negative Poisson’s ratio (auxetic behavior) for their internal rings and normal Poisson’s behavior for the terminal rings. Various [3], [4], [5], and [6] biprismanes were studied by using quantum mechanical ab initio calculations. The unusual mechanical behavior of prismanes upon stretching was traced to the relaxation of geometrical distortion existing in the ground state. The increase of the Poissonic effect on the ring size correlates well with the Thorpe-Ingold effect. In biprismanes where the CH units in the external rings are replaced by nitrogen, the effect is accentuated. Substitution of the hydrogens by fluorine atoms markedly reduces the effect, this being an interesting example of substituent effect on mechanical properties. This effect is attenuated in longer prismanes. Introduction A positive Poisson’s ratio is the measure of the extent by which the stretching of a rod induces its thinning and compression results in its thickening.1 A negative Poisson’s ratio, namely,2 an auxetic behavior,3 is observed when the lengthening is accompanied, counterintuitively, by a thickening of the rod and vice versa. In the course of our examination of the applicability of rules and concepts of mechanical engineering to the nano-molecular world4,5 we have discovered that prismanes display the auxetic effect.5 This family of compounds is the first, and at the moment the only, known molecular system to demonstrate such behavior. Previously, the auxetic effect was found to be a mechanical property of the material derived from the special design of a system. Examples of such materials are foams,6-8 polymers,9-12 metals,13 silicates,14-17 and zeolites.18,19 Various modeling approaches were taken to describe this phenomenon.7,19,20 A main motif in these systems is presented by the inverted honeycomb structure, a portion of which is shown in Figure 1 (for animation see ref 21). Stretching the structure along the X-axis by pulling the lines designated as “a” away from each other will cause lines “b” to move in opposite directions causing an increase of the width in the Y dimension. Results and Discussion Biprismanes possess a bow tie structure. Namely, the central (middle) ring is smaller than the terminal rings. This interesting structural feature can be seen for bi[3], [4], [5], and [6]prismanes reported in the literature22 (the number in brackets denote the ring size). In principle, such a bowtie structure could be a possible source for the auxetic effect in prismanes. However, there is a clear indication that for prismanes this is not the case since, contrary to the expectation that the auxetic effect will decrease as the rings become more and more distant from the edges, this was found not to be the case.5 Moreover, unlike the material based
Figure 1. Auxetic effect in a honeycomb structure.
system, in prismanes the external rings shrink upon stretching. Thus, the bowtie geometry cannot be the origin of this effect in prismanes. The reversal in the behavior between terminal and internal rings in prismanes is another interesting feature of these systems. Upon longitudinal deformation, while one set expands, the other contracts. In this paper, using biprismanes, we will explore the origin of the auxetic effect and show that it is coupled to the bowtie phenomenon and that it stems from the Thorpe-Ingold effect.23 This effect, which was discovered in ring-closure reactions, was suggested to emerge from the fact that expansion of one angle on a tetrahedral carbon will result in the contraction of the opposite angle. In an ideal prismane, the bonds around a carbon, in an internal ring, are in a geometry as shown in Chart 1, “a”. The colinearity of the interlayer bonds is a marked deviation from the ideal tetrahedral geometry and bears a significant energy cost. The way biprismanes cope with the distortion is by increasing the bond length of the external ring and reducing that of the internal one leading to the bowtie structure in biprismanes. According to the Thorpe-Ingold effect, the contraction of one angle in a tetrahedral arrangement will inflict a concomitant increase in the opposite angle (R and β respectively going from “b” to “c” in Chart 1). Since in prismanes, rings of different sizes have different bond angles (R), there will be a difference in the CHART 1
* Towhomcorrespondenceshouldbeaddressed.E-mail:
[email protected]. † Bar Ilan University. ‡ Technion.
10.1021/jp809791j CCC: $40.75 2009 American Chemical Society Published on Web 02/05/2009
3468 J. Phys. Chem. C, Vol. 113, No. 9, 2009
Pour et al.
TABLE 1: Terminal Ring Bond Length in Bi[n]prismane terminal ring bond length, Å ring size
CH
CF
N
average
[3] [4] [5] [6]
1.540 1.605 1.604 1.616
1.557 1.611 1.604 1.617
1.524 1.596 1.606 1.659
1.540 1.604 1.605 1.631
TABLE 2: Internal Ring Bond Length in Bi[n]prismane internal ring bond length, Å ring size
CH
CF
N
average
[3] [4] [5] [6]
1.493 1.525 1.501 1.490
1.515 1.551 1.532 1.520
1.442 1.470 1.436 1.417
1.483 1.515 1.490 1.476
TABLE 3: Difference between Terminal and Internal Ring Bond Lengths in Bi[n]prismane differences between external and internal bond length, Å ring size
CH
CF
N
average
[3] [4] [5] [6]
0.047 0.080 0.103 0.126
0.043 0.060 0.072 0.097
0.082 0.126 0.170 0.242
0.058 0.089 0.115 0.155
optimal β for each ring size. As a result of this, the distortion toward the collinearity of the ideal prismane will have a different price tag for each ring size. Thus, in [3]prismane, where R is (formally) 60°, β is relatively large and the distortion toward β ) 180° will be much smaller than that in [6]prismane where R ) 120°, forcing β to be much smaller and further away from the 180° of the ideal prismane. It is therefore expected that the differences between the external and internal ring bond lengths will be larger, increasing the bowtie structure, as the ring size increases. We focused on bi[3], [4], [5], and [6] prismanes and examined, in addition to the regular prismanes (CH) two structural variations: (a) the terminal hydrogen atoms replaced by fluorines (CF) and (b) the terminal CH units replaced by nitrogen atoms (N). This yields biprismanes with two allnitrogen terminal rings and an all-carbon central ring. The three structures are shown for bi[3]prismane in Chart 2. CHART 2
The equilibrium structures were computed by using ab initio calculations. In a preceding paper5 we have shown that a variety of computational levels yields the same trends. Therefore, in the present paper we employ the B3LYP/ 6-31G* level for the CH systems using the Gaussian03 program.24 In nitrogen capped
Figure 2. The average value for the terminal ring bond lengths for the optimal and the stretched prismanes.
prismanes and for the CF prismanes we used the B3LYP/631+G* level. Essentially, the same bond lengths ((0.001Å) were obtained by using the two basis sets (Tables S1 and S2, Supporting Information). The energies and the geometrical parameters are given in the Supporting Information (Tables S3-S6). Bond lengths of the external and internal rings and the differences between them are given in Tables 1-3. As can be seen in Table 3, the differences in bond length between the external and internal rings are always positiVe and increase as the ring size of the prismane increases from [3] to [6]prismanes. Thus, as expected on the basis of the Thorpe-Ingold effect, the bowtie geometry is more pronounced in [6]prismane than in [3]prismane. One of the structural changes we made was to replace the hydrogen atoms on the external rings by fluorine atoms, on the assumption that these electronegative atoms, withdrawing electron density from the interlayer bonds, will diminish the Thorpe-Ingold effect. Indeed, comparison of the CH column with the CF column in Table 3 shows that the substitution by fluorine atoms reduces significantly the bowtie phenomenon (the differences for the CF are much smaller than those of the CH prismanes). For the series of the N prismanes, one could expect the opposite behavior, namely, an inverted bowtie structure, since in general, the N-N bond is shorter than the C-C bond. Focusing on the [3]prismanes as an example, calculations at the B3LYP/6-31G* and B3LYP/6-31+G* levels showed that the C-C bond length in cyclopropane is 1.509 and 1.511 Å, respectively, and the N-N bond lengths in its aza analogue N3H3 are 1.464 and 1.462 Å, respectively, showing that the N-N bond is indeed significantly shorter than the C-C bond. However, not only did the N prismanes retain the bowtie structure but they also enhanced it, as can be seen from Table 3 for N where the differences between the internal and external ring bond lengths are almost twice as much as those of the CH prismane, for all prismane sizes. Thus, it seems that the inherent “need” of the biprismanes to maintain the bowtie geometry and thus aVoid as much as possible the collinear geometry of the ideal prismane oVercomes the natural tendency to comply with the “optimal” bond lengths.25 We would like to examine now the effect of the above structural variations on the Poisson’s ratio of the internal and external rings. For this purpose we stretched the prismanes along their long axis by 5%. The resulting bond lengths for the terminal and internal rings are given in Tables S2-S5 in the Supporting Information. In general, the structure as a function of the ring size remains similar to the original ones as shown in Figures 2 and 3 for the average values. The Poisson’s ratio, which represents the change in the ring bond length per unit stretching, was calculated by using eq 1.
The Origin of the Auxetic Effect in Prismanes
J. Phys. Chem. C, Vol. 113, No. 9, 2009 3469 TABLE 6: Internal Rings Poisson’s Ratio for a Series of CF [3]Prismanesa Poisson’s ratio a
Figure 3. The average value for the internal ring bond lengths for the optimal and the stretched prismanes.
TABLE 4: Poisson’s Ratio for the Terminal Rings of Biprismanes Poisson’s ratio ring size
CH
CF
N
average
[3] [4] [5] [6]
0.052 0.075 0.086 0.114
0.029 0.046 0.033 0.061
0.094 0.125 0.224 0.220
0.058 0.082 0.114 0.132
Poisson’s ratio ring size
CH
CF
N
average
[3] [4] [5] [6]
-0.064 -0.065 -0.105 -0.123
-0.006 0.004 -0.049 -0.068
-0.227 -0.136 -0.250 -0.214
-0.099 -0.066 -0.135 -0.135
∆A L ∆L A
tri
tetra
penta
hexa
-0.006
-0.057
-0.062
-0.062
-0.067
Calculated at the B3LYP/6-31G* level.
have calculated a series of [3]prismanes from bi- to heptaprismane. As can be seen in Table 6, which displays the Poisson’s ratios for the internal rings, the fluorine effect decreases as the prismanic rods become longer. Namely, the auxetic behavior increases as the internal rings are further away from the fluorines. Not surprisingly, the largest effect was found for the N prismanes. In these prismanes, as mentioned above, the N-N bond was very long (compared to aza-cyclopropane) in the optimal structure and therefore one would expect that when given the option, it will shrink to a length closer to its normal value as was indeed observed. Conclusions
TABLE 5: Poisson’s Ratio for the Internal Rings of Biprismanes
υ)-
bi
In conclusion, the bowtie structure in biprismanes results from the resistance of the system to distort itself to a structure such as “a” in Chart 1. It is obtained by expanding the terminal rings and contracting the internal ones. Because of the Thorpe-Ingold effect, the bowtie structure is more pronounced with the larger rings. Stretching the interlayer bonds weakens them rendering the angle constrains less important. As a result, the expanded outer rings relax and shrink resulting in a positive Poisson’s ratio. Simultaneously, the shrunk internal rings relax by expanding, exhibiting an auxetic behavior. Because of the Thorpe effect, the positive/negative Poisson’s ratio increases with the ring size.
(1)
where L is the length of the rod, ∆L is the change in the length of the rod, A is the ring bond length, ∆A is the change in the ring bond length, and υ is the Poisson’s ratio. The Poisson’s ratios for the terminal and the internal rings are given in Tables 4 and 5. We assume that stretching the interlayer bonds will have an effect similar to that of the fluorine atoms. Namely, weakening the bond will reduce the differences in bond length between the internal and external rings. This was indeed observed as can be seen from Tables S3-S6 in the Supporting Information. Moreover, since the larger perimeter prismanes, as we have mentioned above, are more constrained (β is more deformed upon going to the ideal prismane), partial release of this strain, caused by lnterlayer bond stretching, will induce a larger relaxation of the system that will result in a larger Poisson’s ratio (in absolute values) as the ring size increases. The terminal rings will tend to contract (positive Poisson’s ratio) and the internal rings will expand (auxetic). The data in Tables 4 and 5 confirm this prediction for the CH series. The absolute Poisson’s ratio values increase upon going from [3]- to [6]prismanes. In the CF series where the C-C bonds are weaker and the energy constraints are smaller to begin with, as expected, the Poisson’s ratio was smaller (in absolute values) for both the internal and the external rings, and in one case (bi[4]prismane), the effect was even nullified. This is a unique example whereby substituents alter the mechanical property of a molecular rod. However, this effect is likely to decrease as the rod becomes longer since the inductive effect of the electronegative substituents is attenuated as a function of the distance. To test this we
Acknowledgement. . This work was supported by the Israeli Science Foundation. Supporting Information Available: Tables of bond lengths and energies. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) CRC Handbook of Chemistry and Physics, 67th ed.; CRC: Boca Raton, FL, 1986-1987; p, F97. (2) Lakes, R. S. Science 1987, 235, 1038. (3) Evans, K. E.; Nkansah, M. A.; Hutchinson, I. J.; Rogers, S. C. Nature 1991, 353, 124. (4) Itzhaki, L.; Altus, E.; Basch, H.; Hoz, S. Angew. Chem., Int. Ed. 2005, 44, 7432. (5) Pour, N.; Itzhaki, L.; Hoz, B.; Altus, E.; Basch, H.; Hoz, S. Angew. Chem., Int. Ed. 2006, 45, 5981. (6) Evans, K. E.; Nkansah, M. A.; Hutchinson, I. J. Acta Metall. Mater. 1994, 2, 1289. (7) Choi, J. B.; Lakes, R. S. J. Compos. Mater. 1995, 29, 113. (8) Smith, C. W.; Grima, J. N.; Evans, K. E. Acta Mater. 2000, 48, 4349. (9) Evans, K. E.; Caddock, B. D. J. Phys. D, Appl. Phys 1989, 22, 1883. (10) (a) Alderson, A.; Evans, K. E. J. Mater. Sci. 1997, 32, 2797–2809. (b) Alderson, K. L.; Alderson, A.; Webber, R. S.; Evans, K. E. J. Mater. Sci. Lett. 1998, 17, 1415. (11) Baughman, R. H.; Galvao, D. S. Nature 1993, 365, 735. (12) Grima, J. N.; Williams, J. J.; Evans, K. E. Chem. Commun. 2005, 4065. (13) Baughman, R. H.; Shacklette, J. M.; Zakhidov, A. A.; Stafstrom, S. Nature 1998, 392, 362. (14) Yeganeh-Haeri, A.; Weidner, D. J.; Parise, J. B. Science 1992, 257, 650–652. (15) Alderson, A.; Alderson, K. L.; Evans, K. E.; Grima, J. N.; Williams, M. J. Metastable Nanocryst. Mater. 2005, 23, 55.
3470 J. Phys. Chem. C, Vol. 113, No. 9, 2009 (16) Grima, J. N.; Gatt, R.; Alderson, A.; Evans, K. E. Mater. Sci. Eng., A 2006, 423, 219. (17) Kimizuka, H.; Ogata, S.; Shibutani, Y. Phys. Status Solidi B 2007, 244, 900. (18) Grima, J. N.; Gatt, R.; Zammit, V.; Williams, J. J.; Evans, K. E.; Alderson, A.; Walton, R. I. J. Appl. Phys. 2007, 101, 0861021-3. (19) Williams, J. J.; Smith, C. W.; Evans, K. E.; Lethbridge, Z. A. D.; Walton, R. I. Chem. Mater. 2007, 19, 2423. (20) Grima, J. N.; Zammit, V.; Gatt, R.; Alderson, A.; Evans, K. E. Phys. Status Solidi B 2007, 244, 866. (21) http://silver.neep.wisc.edu/∼lakes/Poisson.html.
Pour et al. (22) (a) Minyaev, R. M.; Minkin, V. I.; Gribanova, T. N.; Starikov, A. G.; Hoffmann, R. J. Org. Chem. 2003, 68, 8588. (b) Edward, T. S.; Schaefer, F. H. J. Am. Chem. Soc. 1991, 113, 1915. (c) Feng-Ling, L.; Ling, P. THEOCHEM 2004, 710, 163. (23) (a) Beesley, R. M.; Ingold, C. K.; Thorpe, J. F. J. Chem. Soc. 1915, 107, 1080. (b) Ingold, C. K. J. Chem. Soc. 1921, 119, 305. (24) Gaussian 03, Revision B.04; Frisch, M. J.; et al. (see the Supporting Information for the complete reference). (25) This probably results from a relatively shallow potential surface of the N-N bond.
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