Influence of Point Defects on the Free-Radical Scavenging Capability

Apr 2, 2010 - They have been proposed as nanopipets,(1) nanotweezers,(2) nanocontainers,(3) and field emission devices,(4-56) as well as for chemical ...
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J. Phys. Chem. C 2010, 114, 8302–8308

Influence of Point Defects on the Free-Radical Scavenging Capability of Single-Walled Carbon Nanotubes Annia Galano,*,† Misaela Francisco-Marquez,† and Ana Martı´nez‡ Departamento de Quı´mica, UniVersidad Auto´noma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Col. Vicentina, Iztapalapa, C. P. 09340, Me´xico D. F., Me´xico, and Departamento de Materia Condensada y Criogenia, Instituto de InVestigaciones en Materiales, UniVersidad Nacional Auto´noma de Me´xico, Circuito Exterior S. N., Ciudad UniVersitaria, CP 04510, Me´xico D. F., Mexico ReceiVed: February 20, 2010; ReVised Manuscript ReceiVed: March 20, 2010

The effect of point defects on the free-radical scavenging activity of armchair and zigzag single-walled carbon nanotubes (SWCNTs), through a radical adduct formation mechanism, has been studied using density functional theory calculations. SWCNTs with different vacancy (V), adatom (AA), and Stone-Wales (SW) defects have been considered, as well as their pristine partners. All the studied reactions were found to be significantly exothermic and exergonic, which supports their viability. The presence of point defects in the carbon lattice of SWCNTs is predicted to increase their free-radical scavenging activity. The AA and V point defects, involving C atoms with dangling bonds, are expected to cause a larger increase on the SWCNTs’ reactivity toward free radicals than the SW and vacancy defects without C atoms with dangling bonds. The studied Stone-Wales point defect shows the largest site-dependent effect on the free-radical scavenging activity of SWCNTs. The presence of nonpolar environments is not expected to change the proposed trends. Characteristic infrared bands in the 3300 and 900-1100 cm-1 regions have been assigned to the νO-H and νC-O vibrations of the OH radical adducts. Introduction Carbon nanotubes (CNTs) have attracted great deal of attention in the last two decades. This is probably due to their amazing and diverse properties. They have been proposed as nanopipets,1 nanotweezers,2 nanocontainers,3 and field emission devices,4–56 as well as for chemical detection,7,8 gas storage,9–11 membrane separation,12,13 and drug delivery and diagnostic,14 among many other applications. Very recently, they have also been proposed as efficient free-radical scavengers.15–20 However, the production of CNTs does not yield only one kind of welldefined molecule. On the contrary, they preferentially aggregate into bundles of different characteristics. The obtained mixtures widely vary in length; diameter; helicity; and kind, location, and number of defects.21–23 It has been shown that structural defects can play a central role in the modification of the physical24–36 and chemical37–46 properties of single-walled carbon nanotubes (SWCNTs) and even in the toxicity of CNTs.47,48 Point defects on the sidewalls of SWCNTs have been described and identified,49–51 and even high-quality single-walled carbon nanotubes (SWCNTs) have been confirmed to present one defect per 4 µm on average.50 Point defects can be classified in three classes: vacancy (V), adatom (AA), and Stone-Wales (SW) defects; and any other can be formed by a combination of two or more of them.33 Some questions have been recently raised regarding the influence of point defects on the reactivity of SWCNTs toward free radicals and, therefore, on their freeradical scavenging activity,20 and that is the main aim of the present work. For that purpose, the reactions of hydroxyl radicals with finite fragments of (4,4) armchair and (7,0) zigzag * To whom correspondence should be addressed. E-mail: agalano@ prodigy.net.mx. † Universidad Auto´noma Metropolitana-Iztapalapa. ‡ Universidad Nacional Auto´noma de Me´xico.

SWCNTs, with different kinds of point defects, have been modeled using density functional theory (DFT). We have investigated the thermochemistry of the reactions and compared it with that of their pristine partners to predict if the presence of the studied point defects increases or decreases the free-radical scavenging ability of SWCNTs. Different sites of reaction have been studied, and the most reactive sites have been identified. The role of the nature of the point defects on the viability of the •OH scavenging activity of the tubes has also been analyzed. In addition, the infrared (IR) bands related to the formed radical adducts have been assigned. Computational Details Electronic structure calculations have been performed with the Gaussian 0352 package of programs. Full geometry optimizations and frequency calculations were carried out for all the stationary points using the B3LYP functional and the 3-21G basis set. No symmetry constraints have been imposed in the geometry optimizations. The energies of all the stationary points were improved by single-point calculations at the B3LYP/6311+G(d) level of theory. Thermodynamic corrections at 298.15 K were included in the calculation of relative energies. Spinrestricted calculations were used for closed-shell systems and unrestricted ones for open-shell systems. Spin contamination was checked for all the studied radical adducts. Even though it was found to be unusually large for DFT calculation, in all the cases, the deviations from the correct value (〈S2〉 ) 0.75) were lower than 10% and 3% before (〈S2〉 e 0.82) and after (〈S2〉 e 0.77) annihilation of the first spin contaminant. Local minima were identified by the absence of imaginary frequencies. It seems worthwhile to emphasize on the fact that any theoretical model aiming to make predictions concerning practical applications must be analyzed in terms of Gibbs free energies, which implies the necessity of performing frequency calculations, which are

10.1021/jp101544u  2010 American Chemical Society Published on Web 04/02/2010

Influence of Point Defects on SWCNTs

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Figure 1. Studied SWCNT fragments and acronyms used in this work. White ) H atoms, gray ) C atoms in nondefective zones, and dark gray ) C atoms in the point defects.

Figure 2. Schematic representation of the different sites of reaction.

particularly expensive. Accordingly, it is important to include frequency calculations at least at a modest level of theory. The stationary points were first modeled in gas phase (vacuum), and solvent effects were included a posteriori by single-point calculations using a polarizable continuum model, specifically, the integral-equation formalism (IEF-PCM)53–56 at the B3LYP/6-311+G(d) level of theory, with benzene as the solvent for mimicking nonpolar environments. Polar environments were not included because pure carbon nanotubes are not expected to be soluble in such media. The solvent cage effects have been included in the calculations of the Gibbs free energies of reaction according to the corrections proposed by Okuno,57 taking into account the free volume theory.58 These corrections are in good agreement with those independently obtained by Ardura et al.59 and have been successfully used by other authors.60–63 This correction is important because the packing effects of the solvent reduce the entropy loss associated with any addition reaction. Results and Discussion SWCNTs are well-described as graphene sheets rolled up along a wrapping vector (Ch ) na1 + ma2) with characteristic chiral indices (n,m). In the present work, finite fragments of (4,4) armchair and (7,0) zigzag SWCNTs, about 11 Å long, have been studied. The dangling bonds at the ends of the SWCNTs have been saturated by hydrogen atoms to avoid unwanted distortions. The addition reactions of OH radicals to

the walls of the tubes have been modeled for different defective fragments as well as for their pristine partners. Three different kinds of defects have been taken into account: vacancy (V), adatom (AA), and Stone-Wales (SW) defects, which can be formed by rotating a C-C bond by 90°. (Figure 1). Reactions involving all the sites in the defective region of the tubes have been considered. These sites have been numbered to facilitate discussion according to the schematic representation shown in Figure 2. Accordingly the acronyms Pn, with n ) {1,2,...7}, will be used for referring to the products of reaction formed by •OH addition to site n. Because of the symmetry of the studied defects, there are several sites that are chemically equivalent; therefore, these sites have been labeled with the same number and one letter (i.e., 1a, 2a, etc.) Some of the modeled species evolved during geometry optimizations toward different structures than those initially intended. The geometry optimization of the (7,0) fragment with the AA defect lead to a structure with a three-membered ring including atoms 1, 2, and 2a. The 2-2a bond distance was found to be equal to 1.52 Å. Therefore, although the AA defect in the armchair fragment corresponds to two neighbor heptagons, in the zigzag fragment, it corresponds to two hexagons and a triangle, which include the C-C bond connecting the hexagons and a third C atom (1) out of the cylindrical network formed by all the atoms (Figure 3). The geometry optimizations of all the other modeled defective tubes lead to the intended structures. The addition products to sites 1 and 4 on the (7,0)-AA SWCNT

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Galano et al. TABLE 1: C-O Bond Distance (Å) in the Addition Products Involving (4,4) SWCNTs and Frequencies (cm-1) of Selected Vibrational Modes, from Calculations in Gas Phase pristine V1

AA

V2

Figure 3. Deviations from the intended (7,0)-AA structures arising from full geometry optimizations. C atoms are in gray, H atoms in white, and O atoms in red.

SW

site

d(C-O)

νO-H

1 2 3 4 5 6 7 1 2 3 4 1 2 3 4 1 2 3 4 5

1.467 1.467 1.399 1.457 1.462 1.466 1.468 1.354 1.386 1.456 1.460 1.460 1.450 1.469 1.470 1.454 1.489 1.448 1.464 1.474 1.467

3309 3318 3339 3325 3310 3313 3294 3280 3364 3310 3318 2990 3324 3320 3312 3344 3317 3317 3325 3302 3318

νC-O 934, 945, 978, 928, 931, 931, 882, 994, 959, 872, 926, 921, 968, 887, 923, 929, 880, 937, 935, 925, 940,

965, 978, 999, 1055 947, 971, 988, 998, 1059 1014, 1052 975, 994, 1006, 1029, 1053 939, 962, 974, 1004, 1066 938, 950, 970 910, 990, 994, 1018 1017, 1036, 1055 968, 1024, 1052, 1085, 1102 899, 977, 1044 962, 968 971, 990 994, 1060, 1073, 1080, 1088 907, 962, 982, 997 930, 981 992, 996, 1004, 1021 893 988, 1006, 1029, 1067 950, 977, 994, 1039 930, 949 948, 959, 994, 1045

TABLE 2: C-O Bond Distance (Å) in the Addition Products Involving (7,0) SWCNTs and Frequencies (cm-1) of Selected Vibrational Modes, from Calculations in Gas Phase pristine V1

AA

V2 Figure 4. Deviations from the intended structure of the adduct formed by the OH addition to site 2 on (4,4)-V1. Atoms highlighted in yellow are those involved in the V1 defective region. Atoms highlighted in green are outside the defective zone in the reactant and become part of the V1 region in the adduct. Atoms highlighted in pink are involved in the defective zone of the isolated reactant and out of it in the adduct.

also show the same deformation than that of the isolated SWCNTs, while the adduct formed at site 3 evolved to a structure with the •OH attached to site 1 and with no bond between atoms 1 and 2a (Figure 3). The adduct formed by the •OH addition to site 2 on the (4,4)-V1 SWCNT also deviates from the intended structure after geometry optimization. In this case, as the •OH approaches the reaction site and the C-O bond is formed, the bond 2-3 breaks while a new bond is formed between C atoms at sites 3 and 7 (Figure 4). Such a transformation causes the atoms labeled as 8, 9, and 10 (highlighted in green in Figure 4), which are outside the defective zone in the reactant SWCNT, to become part of the V1 region in the adduct. On the other hand, atoms 4a, 5a, and 6a (highlighted in rose color in Figure 4), which are involved in the defective zone of the isolated reactant, get out of it in the adduct. As a consequence of the described reorganization, in product 2, the radical is attached to a site that is chemically closer to the

SW

site

d(C-O)

νO-H

νC-O

1 2 3 4 5 6 7 1 2 3 4 1 2 3 4 1 2 3 4 5

1.473 1.457 1.472 1.430 1.473 1.464 1.475 1.373 1.377 1.447 1.353 1.471 1.462 1.445 1.453 1.485 1.447 1.476 1.469 1.460 1.466

3310 3322 3320 3333 3341 3323 3301 3325 3324 3349 3319 3323 3330 3332 3254 3313 3335 3305 3315 3332 3317

915, 927, 966, 970, 1016 927, 940, 986, 1015, 1051, 1062 913, 923, 985, 1021 984, 1006, 1037, 1078 913, 938, 959 919, 923, 968, 976 879, 910, 946, 962 997, 1050 1044, 1057, 1089, 1104 908, 949, 967 1114, 1123, 1101, 1096 913, 959, 984 952, 956, 1045 891, 959, 1007, 1049, 1093 932, 942, 952, 966, 985 905, 910 917, 998, 1009, 1020, 1025, 1063 913, 920, 939, 947 908, 940, 960, 970 972, 1006, 1011, 1065 906, 928, 981

original site 7 than to the original site 2. No other adducts deviate from the intended structure after geometry optimization. The most relevant geometrical feature in the adducts formed through •OH additions on the walls of the modeled SWCNTs is the distance of the new formed bond, d(C-O). Their values are reported in Tables 1 and 2 for the (4,4) and (7,0) SWCNTs, respectively. For all the defective structures, several addition products have shorter C-O distances than that of the corresponding pristine tube, indicating stronger bonds. The presence of V1 and AA defects on the walls of SWCNTs implicates C atoms with dangling bonds. They correspond to sites 7 and 1 for the V1 and AA structures, respectively. Therefore, the C-O bonds corresponding to •OH additions to these sites are expected to be stronger than those arising from additions to other sites on the SWCNT walls. They were, in fact, the shortest C-O bonds among all the studied ones. One exception can be noticed, though, the C-O bond distance in product P3 of the (7,0)-AA

Influence of Point Defects on SWCNTs

J. Phys. Chem. C, Vol. 114, No. 18, 2010 8305 TABLE 3: Enthalpies (∆H) of the •OH + SWCNT Reactions, in kcal/mol, Computed at 298.15 K, in Gas Phase and in Benzene gas pristine V1

Figure 5. Intramolecular interaction in the product of OH addition to site 4 in the studied (4,4)-AA SWCNT. The distance is in Å. AA

SWCNT (Figure 3). The very short distance of the C-O bond in this case can be explained by the deformation of the carbon network previously described and shown in Figure 3. This particular product still has a C atom with a dangling bond, and this is precisely the C atom bonded to the OH group. Accordingly, its C-O bond is expected to be particularly strong. The C-O bond length in product P2 of the (4,4)-V1 SWCNT is also short. This can be explained by the geometry evolution of this adduct described above (Figure 4) The frequencies of the infrared (IR) bands corresponding to the O-H and C-O stretching vibrations of the products of •OH additions to the modeled SWCNTs are also included in Tables 1 and 2. The reported frequencies have been scaled using a scaling factor of 0.9627, as recommended by Irikura el al.64 The IR band corresponding to the νO-H vibration appears around 3300 cm-1 for all the studied products, except for the (4,4)AA-P4. For this particular product, this band appears at a frequency about 300 cm-1 lower, that is, at ∼3000 cm-1. This is caused by the intramolecular interaction between the H atom in the OH group and the C atom with a dangling bond in the defective zone (site 1) found in this particular adduct (Figure 5). This interaction weakens the O-H bond, causing the lower shift in the corresponding frequency. The C-O stretching vibrations are characterized by an IR region rather than by a unique band. In all the studied cases, more than one IR band was found involving this motion and all of them in the ∼900 to 1100 cm-1 region. The splitting of the νC-O band arises from the coupling between this vibration and those corresponding to the C-C stretching modes of the carbon atoms in the SWCNT walls. Therefore, multiple bands in this region, or a wide band, are expected to be observed for the addition products of oxygenated radicals to SWCNTs. All the studied reactions were found to be exothermic and exergonic at room temperature (Tables 3 and 4). The values of the calculated enthalpies and Gibbs free energies of reaction are, in all the cases, large enough to overcome any inaccuracy of the used methodology. To help visualize the effects of the studied defects on the thermochemistry of the •OH addition processes to SWCNTs, the variations of ∆H and ∆G of reaction, with respect to those involving nondefective SWCNTs, have been plotted in Figures 6 and 7. For the OH addition to both (4,4) and (7,0) SWCNT fragments, the presence of a nonpolar environment was found to have only a minor effect on the enthalpies of reaction. The moderate increase in exergonicity, on the other hand, can be attributed to the cage effect of the solvent. The values of both ∆H and ∆G of reaction were found to be systematically more negative for the •OH additions to the (7,0) SWCNT than for those taking place at equivalent sites of reaction on the (4,4) SWCNT. It should be noticed here that, in the present study,

V2

SW

benzene

site

(4,4)

(7,0)

(4,4)

(7,0)

1 2 3 4 5 6 7 1 2 3 4 1 2 3 4 1 2 3 4 5

-35.86 -36.91 -70.94 -50.43 -43.60 -43.74 -28.40 -86.22 -99.58 -35.16 -37.33 -33.91 -48.32 -32.03 -32.74 -41.96 -29.42 -59.13 -43.57 -44.59 -41.67

-59.62 -73.04 -56.82 -87.22 -56.83 -74.89 -55.42 -130.99 -119.37 -81.14 -117.72 -66.69 -65.19 -66.90 -71.01 -54.14 -69.72 -57.19 -44.61 -66.94 -45.35

-35.17 -36.00 -70.07 -49.42 -42.60 -42.81 -27.70 -84.71 -98.41 -34.46 -36.19 -32.65 -47.56 -31.11 -32.09 -40.87 -28.59 -58.05 -42.81 -43.71 -40.82

-58.94 -71.66 -54.99 -84.92 -57.43 -78.08 -53.54 -121.72 -117.47 -75.55 -116.82 -65.43 -63.95 -65.60 -69.35 -52.63 -68.97 -56.11 -44.13 -64.49 -46.32

TABLE 4: Gibbs Free Energies (∆G) of the •OH + SWCNT Reactions, in kcal/mol, Computed at 298.15 K gas pristine V1

AA

V2

SW

benzene

Site

(4,4)

(7,0)

(4,4)

(7,0)

1 2 3 4 5 6 7 1 2 3 4 1 2 3 4 1 2 3 4 5

-25.64 -26.24 -60.42 -39.51 -33.07 -32.89 -18.13 -75.19 -88.62 -24.83 -27.01 -23.02 -37.54 -21.01 -22.16 -31.47 -20.72 -48.76 -33.37 -34.30 -31.47

-47.11 -62.56 -46.62 -76.81 -46.50 -64.28 -45.24 -119.83 -107.88 -69.65 -107.28 -56.01 -55.06 -57.12 -61.23 -44.26 -58.62 -45.39 -32.81 -55.57 -33.87

-29.38 -29.76 -63.99 -42.93 -36.50 -36.39 -21.87 -78.11 -91.88 -28.55 -30.30 -26.19 -41.21 -24.51 -25.95 -34.81 -24.32 -52.10 -37.04 -37.85 -35.06

-50.86 -65.61 -49.23 -78.94 -51.53 -71.91 -47.79 -114.99 -110.42 -65.15 -110.80 -59.17 -58.25 -60.24 -64.00 -47.18 -62.30 -48.74 -36.75 -57.55 -39.27

the modeled SWCNTs correspond to ultrashort fragments, and therefore, none of the studied SWCNT fragments have actual metallic character. For tubes with length . diameter, the abovementioned trend, between armchair and zigzag SWCNTs, might change. As expected, due to the presence of C atoms with dangling bonds in the defective zone, the most exothermic and exergonic reactions are those involving OH radical additions to sites 1 and 7 in structures AA and V1, respectively. In addition, the geometrical characteristics described above for adducts (7,0)AA-P3 and (4,4)-V1-P2 cause a significant increase in the exothermicity and exergonicity of the formation of these products, with respect to the equivalent reactions involving the corresponding pristine tubes. In general, according to Figures 6 and 7, the adatom and V1 vacancy point defects are expected

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Figure 6. Variation in the enthalpies of reactions involving defective tubes, in kcal/mol, with respect to those involving the corresponding pristine SWCNTs.

Figure 7. Variation in the Gibbs free energies of reactions involving defective tubes, in kcal/mol, with respect to those involving the corresponding pristine SWCNTs.

to cause a larger increase on the free-radical scavenging activity of SWCNTs than the Stone-Wales and V2 vacancy defects. For the (7,0)-AA SWCNT, the •OH additions to all the reaction sites in the point defect were found to be more thermodynamically favored than the equivalent process, involving the pristine (7,0) nanotube. For the (4,4)-AA SWCNT, on the other hand, only the •OH addition to the C atom with a dangling bond (site 1) was found to have an increase on viability with respect to the pristine (4,4) SWCNT. This suggests that the effect of adatom point defects on the reactivity of SWCNTs toward •OH, and probably toward other oxygenated radicals, is larger for the armchair carbon nanotubes than for the zigzag ones. However, because, in both defective structures, there is at least one site of reaction that increases the exothermicity and exergonicity of the free-radical scavenging process, it can be stated that adatom point defects increase the free-radical scavenging activity of SWCNTs. Significant variations in the exothermicity and exergonicity of the •OH addition reactions were also found for the SWCNTs

with V1 vacancy defects. In addition to the process involving the C atom with a dangling bond (site 7, P7), •OH additions to other sites of reaction are predicted to have more negative ∆H and ∆G than those involving C atoms in nondefective regions. For the (4,4)-V1 fragment, these reactions sites are 2, 3, 4, and 5. The viability of the •OH additions to sites 1 and 6, on the other hand, is predicted to be equally and less thermodynamically viable than that of the pristine tube. However, they still are significantly exothermic and exergonic. For the (7,0)-V1 SWCNT, sites 1, 3, and 5 increase the exothermicity and exergonicity of the studied reaction with respect to the pristine sites. The ∆H and ∆G values of the reactions involving sites 2, 4, and 6 are slightly less negative than the corresponding pristine nanotubes, but negative enough to be thermodynamically viable. Comparing the results for the reactions involving C atoms in vacancy point defect regions, it stands out that not all vacancy defects have the same effect on the thermochemistry of the •OH + SWCNT reactions. According to our results, the presence of V2 point defects influences the free-radical scavenging activity

Influence of Point Defects on SWCNTs of SWCNTs to a less extent than the presence of V1 point defects. It seems to be a logical finding because there is one C atom with a dangling bond in the V1 region, while there is none in the V2 one. Apparently, such a C atom not only has a higher reactivity itself but also influences the reactivity of the other atoms in the point defect region. For the (7,0)-V2 SWCNT, modest increases in exothermicity and exergonicity were found for sites 1, 2, and 3, while the •OH addition to site 4 is predicted to have less negative values of ∆H and ∆G than the reactions involving pristine tubes. In the case of the (4,4)-V2 SWCNT, sites 2 and 3 are predicted to be less reactive and sites 1 and 4 more reactive than sites at nondefective regions. It seems worthwhile to emphasize that, beyond their relative reactivity, all the reaction sites are proposed to efficiently trap OH radicals. The studied Stone-Wales point defect shows the largest sitedependent effect on the free-radical scavenging activity of SWCNTs. For OH additions to site 1 (Figure 2), the reaction becomes more exothermic and more exergonic than that of the pristine tube for the (7,0)-SW SWCNT, whereas it becomes less exothermic and less exergonic for the (4,4)-SW SWCNT. For OH additions to site 2, the values of ∆H and ∆G are significantly more negative for the (4,4)-SW fragment than for the pristine one, whereas the variations are almost negligible, but positive, for the (7,0)-SW fragment. Sites 3 and 5 show just the opposite trend than site 1; that is, the reaction becomes less exothermic and less exergonic than that of the pristine tube for the (7,0)-SW SWCNT and more exothermic and more exergonic for the (4,4)-SW SWCNT. •OH additions to site 4 are the only processes that have the same behavior, regardless of the helicity of the tube. In this particular case, both kinds of defective SWCNTs showed an increase in their reactivity toward the free radical. As in all the other studied defective structures, the •OH additions to all the sites in the SW defective region are thermodynamically feasible. Conclusions Because chemical reactions are assumed to take place at the most reactive sites, it can be concluded that the presence of point defects in the carbon lattice of SWCNTs would increase their free-radical scavenging activity. The adatom and V1 vacancy point defects are expected to cause a larger increase on reactivity than the Stone-Wales and V2 vacancy defects. This is explained by the fact that there is one C atom with a dangling bond in the AA and in V1 regions. These atoms seem to influence the other atoms in the defective region as well. Different vacancy defects were found to affect in a different way the thermochemistry of the •OH + SWCNT reactions. The studied Stone-Wales point defect shows the largest sitedependent effect on the free-radical scavenging activity of SWCNTs. The presence of nonpolar environments is not expected to change the proposed trends. In summary, it is not imperative to obtain pristine SWCNTs for achieving good freeradical scavenging activity. Carbon nanotubes with point defects are also expected to have such activity, probably to a larger extent than the pristine tubes. Acknowledgment. This study was made possible due to funding from the Consejo Nacional de Ciencia y Tecnologı´a (CONACyT), as well as resources provided by the Instituto de Investigaciones en Materiales IIM, UNAM. The work was carried out, using a KanBalam supercomputer, provided by DGSCA, UNAM, and the facilities at Laboratorio de Superco´mputo y Visualizacio´n en Paralelo of UAM Iztapalapa. A.M. is grateful for financial support from DGAPA-UNAM-Me´xico.

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