Ring Connectivity: Measuring Network Connectivity in Network

Langmuir 2007, 23, 1123-1130

1123

Ring Connectivity: Measuring Network Connectivity in Network Covalent Solids Surendra K. Jain* and Keith E. Gubbins Center for High Performance Simulation and Department of Chemical and Biomolecular Engineering, North Carolina State UniVersity at Raleigh, Box 7905, Raleigh, North Carolina 27695-7905 ReceiVed December 15, 2005 In atomistic models of amorphous materials, ring statistics provide a measure of medium-range order. However, while ring statistics tell us the number of rings present in the model, they do not give us any information about the arrangement of rings, e.g., whether the rings are clustered and how big the cluster is. In this work we present a method to calculate the ring connectivity, or clustering, of rings. We first calculate the rings present in the model using the shortest path criteria of Franzblau and then find the rings that are connected together and group them into clusters. We apply our method to a set of models of disordered carbons, obtained using a reverse Monte Carlo procedure developed in a recent work. We found that in these carbon models the five-, six-, and seven-membered rings are connected together, forming clusters. After isolating the clusters, we found that they resemble defective graphene segments twisted in a complex way. The clusters give further insight about the arrangement of carbon atoms in microporous carbons at a larger length scale. Moreover, the method can be applied to any network covalent solid that contains rings and thus gives information about the ring connectivity present in such materials.

1. Introduction Amorphous materials are difficult to characterize because they do not have long-range order. In such materials information on short-range order can be obtained from the bond angle distributions, bond length distributions, and neighbor distributions. In many amorphous solids, ring statistics have become a generally accepted measure of medium-range order. There exist many methods for counting the rings. The shortest path (SP) criterion of Franzblau1 is one of the most widely used. The purpose of the SP criterion is to include only rings which have no shortcuts (paths which shorten a ring). Thus, if R is a shortest path ring, then R contains the shortest path between each pair of atoms which are part of the ring. In this work we are concerned with disordered carbons. Thus, in our case the rings contain only carbon atoms. The hydrogen atoms in the model are not present in a ring, since a hydrogen atom can have only one neighbor, and to be a part of the ring an atom needs to have at least two neighbors. By using the ring criterion, we can count how many n-carbon-membered rings are present in a graph (or an atomistic model in our case). Both crystalline and disordered networks can be characterized in terms of their ring distribution by SP analysis. n-fold rings are also used to describe the connectivity in the models. Large numbers of rings imply more connectivity. The structure of n-fold rings and their distribution have not been determined experimentally in an amorphous material. Rino et al.2 performed molecular dynamics simulations of vitreous SiO2 and studied the ring statistics and distribution of interatomic distances and bond angles in the rings. Apart from the ring statistics, many works have attempted to get more information about the connectivity in the computer-generated models of network solids. Godwin et al.3 simulated the synthesis of a range of hydrogenated amorphous carbons having different compositions * To whom correspondence should be addressed. E-mail: Skjain2@ unity.ncsu.edu. Phone: (919) 513-2051. Fax: (919) 513-2470. (1) Franzblau, D. S. Phys. ReV. B 1991, 44, 4925. (2) Rino, J. P.; Ebbsjo¨, I.; Kalia, R. K.; Nakano, A.; Vashishta, P. Phys. ReV. B 1993, 47, 3053. (3) Godwin, P. D.; Horsfiled, A. P.; Stoneham, A. M.; Bull, S. J.; Ford, I. J.; Harker, A. H.; Pettifor, D. G.; Sutton, A. P. Phys. ReV. B 1996, 54, 15785.

and formed under different processing conditions using a tight binding model. They found that there is clustering of 3-fold- and 4-fold-coordinated atoms, with the 4-fold-coordinated atoms providing cross-linking between these planes. The clustering was seen from the snapshots, with no quantitative information about the clusters. They investigated the correlation among five-, six-, and seven-membered rings in the samples by looking at whether rings share at least two adjacent members. In this way they were able to tell if two rings are connected, but the method does not give any further quantitative information about the possible presence of any clusters of rings. Bilek et al.4 carried out MD simulations of the structure of hydrogenated amorphous carbon at two densities (2.0 and 2.9 g/cm3). They found that the low-density structure is dominated by sp2 carbon and shows a strong tendency to form graphite-like planes. They tried to quantify the planarity of the graphite-like planes by summing the bond lengths projected onto the axis normal to the planes and comparing this with the sum of bond lengths projected onto the two in-plane directions. However, they did not find any information about the size of the graphitic planes or how the rings are arranged in the graphitic plane. O’Malley et al.5 developed a model of a glassy carbon using the reverse Monte Carlo (RMC) method. They characterized the graphitic short-range order in their model by a network definition of nearest neighbors. They assumed that an atom is regarded as an ith nearest neighbor of another atom if and only if the shortest path joining the two atoms via the network of bonds involves i bonds. They also adopted a criterion to identify the disordered region in their models. They assumed that an atom is regarded as in an ordered local neighborhood if three hexagonal rings pass through the atom, as in graphite. All other atoms are regarded as being in a disordered neighborhood. They also found that there are a small number of atoms in their model that are part of two six-membered rings and one five-membered ring, which is the same local environment as that of a carbon atom in C60 (4) Bilek, M. M. M.; McKenzie, D. R.; McCulloch, D. G.; Goringe, C. M. Phys. ReV. B 2000, 62, 3071. (5) O’Malley, B.; Snook, I.; McCulloch, D. Phys. ReV. B 1998, 57, 14148.

10.1021/la0534017 CCC: $37.00 © 2007 American Chemical Society Published on Web 12/21/2006

1124 Langmuir, Vol. 23, No. 3, 2007

fullerenes. Again for highly disordered carbon structures, significantly different from graphitic order, their result is not of much help. In these materials, carbon atoms may be arranged in the form of chains connecting two small graphene segments. In this case information about the size of the graphene segments and how they are arranged is useful. All the works mentioned above give information about individual rings or correlation between two rings. They do not give information about how the rings are arranged in the model, e.g., whether the rings are grouped together in a cluster or how large the cluster is. In this work we present an algorithm to get an estimate of the connectivity of the rings and clustering of the rings. The ring connectivity gives further information about the arrangement of atoms on a larger length scale and enhances our understanding of the structure. It can also be used as a measure of the extent of graphitization in the case of porous carbons. The ring connectivity method is a form of cluster algorithm. In the normal cluster algorithm, atoms are assumed to be a part of a cluster depending on a distance criterion. For example, Powles et al.6 have implemented a nearest neighbor distance criterion, which is a clustering algorithm, for identifying the atoms in a drop. Their method starts by picking an atom i. All atoms that satisfy r(ij) < r(cl), where r(cl) is the critical atom separation, are defined to be in the same cluster as i. Each such atom j is added to the cluster and is used in the same way as i to identify further members of the cluster. When the first cluster is complete, an atom outside the cluster is picked and the process repeated to generate a second cluster and so on. The whole process partitions the complete set of atoms into mutually exclusive clusters. In the ring connectivity method we find clusters of rings. It divides the set of atoms into mutually exclusive clusters of rings. There might be some atoms which are not part of any cluster (since they do not belong to any ring). In carbon structures, it is believed that the properties observed are due to the aromatic clusters formed by 5-, 6-, and 7-fold sp2-bonded rings.7 Nongraphitizing carbons could be described as composed of small microdomains of graphene sheets.8 Recent high-resolution electron microscopy studies by Harris et al.9 have found that microporous carbons are disordered and isotropic and are made up of tightly curved individual carbon layers. They tried to explain the experimental results by concluding that the curved and closed structures result from the presence of fullerenelike elements, in which both pentagons and heptagons are distributed randomly throughout a hexagonal network to produce the observed curvature. Much work has been done to build molecular models of carbons starting from perfect graphene segments10 or graphene segments with defects.11 In a more recent work, Smith et al.12 developed a model for the structure of a nongraphitizing carbon by randomly incorporating between 0.5% and 4% nonhexagonal rings into an extended sp2 carbon sheet. Disorder in the connectivity was incorporated by randomly adding a five- or seven-membered ring. These five- and seven-membered rings lead to curvature in the overall sheet. We find clusters of five-, six-, and seven-membered rings in our models. However, it should be noted that we can include any n-membered ring to be a part of the cluster, depending on the (6) Powles, J. G.; Fowler, R. F.; Evans, W. A. B. Phy. Lett. 1983, 98A, 421. (7) Frauenheim, Th.; Blaudeck, P.; Stephan, U.; Jungnickel, G. Phys. ReV. B 1993, 48, 4823. (8) Franklin, R. E. Proc. R. Soc. London, Ser. A 1951, 209, 196. (9) Harris, P. J. F.; Burian, A.; Duber, S. Philos. Mag. Lett. 2000, 80, 381. (10) Thomson, K. T.; Gubbins, K. E. Langmuir 2000, 16, 5761. (11) Acharya, M.; Strano, M. S.; Mathews, J. P.; Billinge, J. L.; Petknov, V.; Subramoney, S.; Foley, H. C. Philos. Mag. B 1999, 79, 1499. (12) Smith, M. A.; Foley, H. C.; Lobo, R. F. Carbon 2004, 42, 2041.

Jain and Gubbins

Figure 1. Snapshots of fullerenes.

system we are interested in. In the models for disordered carbons that we developed recently using a modified version of RMC.13

2. Algorithm The algorithm for finding clusters of rings is as follows. (1) The ring statistics, the number of three-, four-, five-, six-, seven-, eight-, etc.-membered rings present in the model, are calculated using the algorithm suggested by Franzblau. Periodic boundary conditions are used while the ring statistics and also the ring connectivity are tabulated. (2) Each ring is numbered, and the carbon atoms which are members of each ring are stored. Note that in computing the ring connectivity only five-, six-, and seven-membered rings are considered. (13) Jain, S. K.; Pellenq, R. J.-M.; Pikunic, J.; Gubbins, K. E. Langmuir 2006, 22, 9942.

Ring ConnectiVity in Network CoValent Solids

Langmuir, Vol. 23, No. 3, 2007 1125

Figure 3. Snapshots of graphene segments (pink and blue refer to different clusters). Gray color is used for the carbon atoms which are not part of any cluster. Table 1. Ring Connectivity Results for the Fullerenes

sample (a) C70 (b) C92 (c) C100

Figure 2. Snapshots of carbon nanotubes.

(3) We then start with a ring i and loop through all the member carbon atoms in the ring to check whether any carbon atom is also a member of another ring. If a carbon atom is found that is also a member of another ring, j, we store ring i, give it a cluster number, and go to ring j. (4) We repeat the same procedure for ring j that we adopted for ring i. We loop through all the carbon members of ring j and check whether any carbon atom is also a member of a third ring, k. If we find such a carbon atom, we store ring j, give it the same cluster number as ring i, and go to the third ring. (5) This procedure is continued until all the rings that are connected and belong to the same cluster as ring i have been found. Care is taken not to count a ring twice. (6) Another ring outside this cluster is then chosen, and the process is repeated to find further clusters of rings. (7) At the end of this process all rings have been identified and grouped in separate clusters (collection of rings that are

no. of no. of no. of total sevensixfivetotal no. of membered membered membered no. of rings in C atoms rings rings rings the cluster 37 48 52

12 12 12

25 36 40

0 0 0

70 92 100

connected together). The ring clusters are isolated from each other (that is, the individual clusters are not connected by any five-, six-, or seven-membered rings; they may be connected by chains of carbon atoms, however). The rings (and hence the atoms present in the rings) belong to only one cluster. There is an advantage of our ring clustering method over the normal clustering algorithm. In the normal clustering algorithm, clusters are found by observing the neighbors of an atom belonging to a cluster. In the case of disordered carbons, if there are two graphene segments connected with chains of carbon atoms, then the normal cluster algorithm, with the nearest neighbor criteria, will identify both graphene segments and the carbon chain as one cluster. However, we are interested in the individual graphene segments and want to discriminate between different graphene segments and the carbon chains. Our ring clustering algorithm gives us this information, and from this we can obtain the statistics of the graphene segments, including the rings present in the graphene segment, the size of the segment, and the nature of the segment.

3. Results 3.1. Fullerenes, Nanotubes, and Graphene Segments. As an example of the algorithm and the clusters that it counts, we apply our ring connectivity method to isolated fullerenes (Figure 1), nanotubes (Figure 2), and graphene segments (Figure 3). In all the cases we found that our method was able to identify all the rings that are connected and grouped them into one cluster. We found that the fullerene and nanotube were each identified as a cluster. We also calculated the number of rings (five-, six-, and seven-member rings) and the number of carbon atoms present in each cluster. Our method was able to report correctly the

1126 Langmuir, Vol. 23, No. 3, 2007

Jain and Gubbins

Table 2. Ring Connectivity Results for the Nanotubes no. of no. of no. of total no. of fivesixseventotal rings in membered membered membered no. of the cluster rings rings rings C atoms

sample a (6 × 6) b (8 × 8) c (10 × 10)

48 64 80

0 0 0

48 64 80

0 0 0

120 160 200

Table 3. Ring Connectivity Results for the Graphene Segments

sample

total no. of rings in each graphene sheet

no. of graphene sheets

no. of six-membered rings

total no. of C atoms

a b c d

3 3 0 3

1 2 0 2

3 3 0 3

13 13 0 13

Table 4. Specifications of the Different Carbon Models Obtained from RMC Simulation

sample CS400 CS1000 CS1000a

fraction of fraction of fraction of fraction of total C atoms C atoms C atoms C atoms no. of having one having two having three having four C atoms C neighbor C neighbors C neighbors C neighbors 827 1160 566

0.0326 0.0069 0.0265

0.4341 0.2052 0.3816

0.5284 0.7793 0.5883

0 0.0078 0.0035

Figure 4. Snapshot of the CS400 model obtained from RMC simulation. Different color codes indicate different graphene segments.

Table 5. Ring Statistics for the SP Rings Present in Different Carbon Models

sample

no. of no. of no. of no. of no. of threefourfivesixsevenmembered membered membered membered membered rings rings rings rings rings

CS400 CS1000 CS1000a

0 0 0

0 1 0

26 21 16

58 166 58

21 83 25

Table 6. Cluster Statistics for CS400 Showing the Number of Graphene Segments, the Number of Rings, and the Number of Carbon Atoms Present in Each Segment no. of no. of total sevensixno. of total membered membered no. of fivecluster no. of C atoms rings rings rings membered rings no. 1 2 3 4

57 13 8 4

12 3 1 1

33 5 6 3

12 5 1 0

196 46 29 17

number of atoms as well as the number of rings in each cluster. We also calculated the same for graphene sheets. We show the results of our algorithm in Table 1 (for the fullerenes), Table 2 (for the nanotubes), and Table 3 (for the graphene segments). In Figure 3 we label the different clusters with different colors. In Figure 3a we find a cluster of three rings. In Figure 3b we find two isolated clusters, one in each graphene segment. In Figure 3c we do not find any cluster as there are no rings present, and in Figure 3d we find two isolated clusters, one in each graphene segment. Here we are counting clusters of rings and not clusters of atoms. Thus, in Figure 3d the two carbon atoms that connect the graphene segments are not counted as being part of a cluster. However, in the normal clustering algorithm based on nearest neighbor criteria, the two graphene segments and the carbon chains connecting them would have been counted as a single cluster. Note that, in our definition of a cluster, we consider only those atoms as part of a cluster if the atom belongs to a five-, six-, or seven-membered ring. Some of the edge atoms in a graphene segment are not part of any ring and can be thought

Figure 5. Snapshot of the largest graphene segment (graphene segment containing the largest number of rings) present in the CS400 model.

of as terminating the rings; thus, they are not considered to be a part of a cluster. 3.2. Disordered Carbon Models. After checking our code with the standard and regular clusters, we calculated the ring connectivity in models of disordered carbons obtained using an RMC protocol that we have developed recently.11 This procedure11 has been used to develop molecular models of three saccharosebased carbons. These samples were named CS400, CS1000,14 and CS1000a;15 here CS ) carbon from saccharose, 400 and 1000 refer to the temperature (°C) of heat treatment, and a ) activated. We developed the models of these samples by two routes. In the first route, the models are built of only carbon (14) Pikunic, J.; Clinard, C.; Cohaut, N.; Gubbins, K. E.; Guet, J. M.; Pellenq, R. J.-M.; Rannou, I.; Rouzoud, J. N. Langmuir 2003, 19, 8565. (15) Jain, S. K.; Pikunic, J. P.; Pellenq, R. J.-M.; Gubbins, K. E. Adsorption 2005, 11, 355.


Langmuir, Vol. 23, No. 3, 2007 1127

Figure 6. Snapshot of the CS1000 model obtained from RMC simulation. Table 7. Cluster Statistics for CS1000 Showing the Number of Graphene Segments, the Number of Rings, and the Number of Carbon Atoms Present in Each Segment

cluster no.

total no. of rings

no. of fivemembered rings

no. of sixmembered rings

no. of sevenmembered rings


1

269

21

166

82

899

Figure 7. Snapshot of the largest graphene segment (graphene segment containing the largest number of rings) present in the CS1000 model.

Table 8. Cluster Statistics for CS1000a Showing the Number of Graphene Segments, the Number of Rings, and the Number of Carbon Atoms Present in Each Segment no. of no. of total sevensixno. of total membered membered no. of fivecluster no. of C atoms rings rings rings membered rings no. 1 2 3 4

72 10 5 5

10 3 1 1

41 6 2 3

21 1 2 1

234 35 22 22

atoms and other heteroatoms such as H, O, are ignored, and in the second route we built models by explicitly including hydrogen along with carbon atoms. In the results shown below we consider only those atoms as part of a cluster if the atom belongs to a five-, six-, or sevenmembered ring. We call the cluster a graphene segment. We report data for clusters containing more than three such rings. There may be some clusters present containing one, two, or three rings, which are not reported here. We also report the number of carbon atoms that are not part of any five-, six-, or sevenmembered ring and are thus not a member of any cluster. These carbon atoms may be thought of as forming small chains that connect or cross-link between different clusters. A few of them can be thought of as carbon atoms terminating the clusters, i.e., the edge atoms which we do not count as a member of the clusters. Some rings cross the boundaries of the computational cell and are closed in the neighboring periodic cells. This should be kept in mind when the results are compared with the structures shown in the figures. In the snapshots different color codes are used for different graphene segments (or clusters). 3.2.1. Porous Carbon Models without Hydrogen. The number of carbon atoms, the neighbor distribution, and the ring statistics of the carbon models for the three samples, obtained using route 1, are reported in Tables 4 and 5.

Figure 8. Snapshot of the CS1000a model obtained from RMC simulation. Different color codes indicate different graphene segments.

3.2.1.1. CS400. The number of carbon atoms not present in any five-, six-, or seven-membered ring, and thus not present in any graphene segment, is 429 (51.87%). From Table 6 we see that there is a large graphene segment with 57 rings and one with 13 rings, and a large percentage of carbon atoms are not part of any segment. This shows that this carbon sample has a few clusters, and most of the carbon atoms form chains which are considered as disordered segments. In addition, CS400 has a significant number of small segments with three rings or less (not shown in Table 6). This can be expected since the number of carbon atoms having three neighbors is less than that of the other carbons. Carbon atoms with three neighbors can be a part of more than one ring and hence lead to ring connectivity. In Figure 4 we show a snapshot of the CS400 model. The different color code in the snapshot represents the different graphene

1128 Langmuir, Vol. 23, No. 3, 2007

Jain and Gubbins


Figure 9. Snapshot of the largest graphene segment (graphene segment containing the largest number of rings) present in the CS1000a model. Table 9. Specifications of the Different Carbon Models Obtained from RMC Simulation total total fraction of fraction of fraction of fraction of no. of no. of C atoms C atoms C atoms C atoms C H having one having two having three having four sample atoms atoms C neighbor C neighbors C neighbors C neighbors CS400 827 CS1000 1160 CS1000a 566

438 174 52

0.0592 0.0198 0.0424

0.5115 0.3371 0.3940

0.4244 0.6284 0.5583

0.0024 0.0147 0.0035

Table 10. Ring Statistics for the SP Rings Present in Different Carbon Models

sample

no. of no. of no. of no. of no. of threefourfivesixsevenmembered membered membered membered membered rings rings rings rings rings

CS400 CS1000 CS1000a

0 1 1

0 0 1

8 9 7

19 65 58

14 44 23

Table 11. Cluster Statistics for CS400 Showing the Number of Graphene Segments, the Number of Rings, and the Number of Carbon Atoms Present in Each Segment no. of no. of total sevensixno. of total membered membered no. of fivecluster no. of C atoms rings rings rings membered rings no. 1 2

6 5

2 2

2 2

2 1

24 21

segments. In Figure 5 we show the largest graphene segment present in the CS400 model. 3.2.1.2. CS1000. The number of carbon atoms not present in any five-, six-, or seven-membered ring, and thus not present in any graphene segment, is 254 (21.89%). From Table 7, we see that the model has one very large graphene segment that is twisted. We show a snapshot of the CS1000 model in Figure 6. In this case all the colors refer to the same graphene segment (considering periodic boundary conditions), and this can be seen from the fact that the different (color) segments extend to the edge of the box.

Figure 11. Snapshot of the largest graphene segment (graphene segment containing the largest number of rings) present in the CS400 model. Table 12. Cluster Statistics for CS1000 Showing the Number of Graphene Segments, the Number of Rings, and the Number of Carbon Atoms Present in Each Segment

cluster no.

total no. of rings





1 2 3 4 5 6

67 22 16 12 11 6

7 1 1 1 1 0

36 10 8 6 6 4

24 11 7 5 4 2

250 89 58 46 43 28

In Figure 7 we show the largest graphene segment present in the CS1000 model. The amount of carbon atoms not part of any ring is small compared to that of CS400. This is again due to the comparatively high density of CS1000. The number of carbon atoms having three neighbors, which contribute to the connectivity of the rings, is higher in this case. 3.2.1.3. CS1000a. CS1000a is prepared by activation of CS1000 and thus has a much larger pore volume. The number of carbon atoms not present in any ring, and thus not present in any graphene segment, is 216 (38.16%). Table 8 shows that CS1000a contains a gaphene segment with 72 rings. In Figure 8 we show the snapshot


Langmuir, Vol. 23, No. 3, 2007 1129


Figure 14. Snapshot of the CS1000a model obtained from RMC simulation. Different color codes indicate different graphene segments.

Figure 13. Snapshot of the largest graphene segment (graphene segment containing the largest number of rings) present in the CS1000 model. Table 13. Cluster Statistics for CS1000a Showing the Number of Graphene Segments, the Number of Rings, and the Number of Carbon Atoms Present in Each Segment

cluster no.

total no. of rings





1 2 3 4 5 6

27 21 11 7 6 4

2 2 1 2 0 0

19 16 5 2 3 3

6 3 5 3 3 1

94 78 41 30 26 18

of the CS1000a model. Different color codes represent different graphene segments present in the model. Figure 9 shows thelargest graphene segment present in the CS1000a model. The amount of carbon atoms not present in any segment is less than that for CS400, but more than that for CS1000. This is again reflected by the fact that the number of carbon atoms having three neighbors is the largest in CS1000. Due to the high density of CS1000, the number of bonds is also high and results in a high connectivity. We have shown results for graphene segments containing more than three rings. However, there are also smaller graphene segments containing three, two, or one ring. CS400 has the largest number of these smaller graphene segments among the three

Figure 15. Snapshot of the largest graphene segment (graphene segment containing the largest number of rings) present in the CS1000a model.

samples, and CS1000 has the lowest. In CS1000, apart from the large graphene segments there is only one smaller graphene segment containing one ring. Thus, CS1000 consists of one large graphene segment, whereas CS400 and CS1000a contain many discrete graphene segments. 3.2.2. Porous Carbon Models with Hydrogen. The number of carbon atoms, the number of hydrogen atoms, the neighbor distribution, and the ring statistics of the carbon models for the three samples, obtained using route 2, are reported in Tables 9 and 10. From Tables 9 and 4 we see that inclusion of hydrogen results in a decrease in the fraction of carbon atoms having three carbon neighbors. This is to be expected, as the presence of hydrogen reduces the number of carbon-carbon bonds. The number of rings also decreases (see Tables 5 and 10), and this is consistent with the fact that the number of carbon atom having three neighbors decreases. Moreover, hydrogen inhibits ring formation since hydrogen is singly coordinated. The amount of hydrogen present in CS1000a is much smaller than that for the other two samples. Below we show results of the effect of hydrogen on the connectivity of rings.

1130 Langmuir, Vol. 23, No. 3, 2007

3.2.2.1. CS400. The number of carbon atoms not present in any ring, and thus not present in any graphene segment, is 508 (61.427%). This percentage is higher than that of the model obtained without taking hydrogen into account. The presence of hydrogen reduces the size of the graphene segments since hydrogen terminates the segments. Table 11 shows the statistics of the graphene segments present in the model. From the snapshot (Figure 10), we see that there are a few small graphene segments and most of the carbon atoms are arranged in a chain fashion. In Figure 11 we show the largest graphene segment present in the CS400 model. 3.2.2.2. CS1000. The number of carbon atoms not present in any ring, and thus not present in any graphene segment, is 484 (41.72%). Here again the number of carbon atoms not part of any segment is increased upon adding hydrogen. Moreover, in the case of the model obtained without hydrogen, we found one large graphene segment. However, upon adding hydrogen, the single large graphene segment of 269 rings found for CS1000 without hydrogen gives way to 1 graphene segment with 67 rings and many other medium-sized graphene segments (see Table 12). Figure 12 shows the snapshot of the CS1000 model, and Figure 13 shows the largest graphene segment present in the model. 3.2.2.3. CS1000a. The number of carbon atoms not present in any ring, and thus not present in any graphene segment, is 217 (38.34%). The number of carbon atoms not part of any graphene segment is almost the same as that of the model obtained without hydrogen. This is because the percentage of hydrogen in CS1000a is much smaller than that for CS1000, and even less than that for CS400. There is a large graphene segment of 72 rings in the model obtained without hydrogen, which now gives way to graphene segments of 27 and 21 rings and other small graphene segments (see Table 13). The snapshot of the CS1000a model is shown in Figure 14, and Figure 15 shows the largest graphene segment present in the model. Thus, we found that hydrogen reduces the connectivity in the network, which was evident from the fact that the size of the grahene segments was reduced upon adding hydrogen for all three samples.

Jain and Gubbins

4. Conclusion We have presented a method to calculate the connectivity of rings. The ring connectivity method is important to identify the connectivity in model materials, such as carbons. The ring clustering algorithm is different from the normal clustering algorithm based on the nearest neighbor criterion. In the clustering algorithm based on the nearest neighbor criterion, the atoms connected together are counted as a cluster. This method when applied to disordered carbons counts the graphene segments and the carbon chains connecting the graphene segments as one cluster. The ring clustering algorithm counts only the rings that are connected together and groups them into a cluster and thus can distinguish between the graphene segments and the carbon chains. We have applied our method to calculate the connectivity for models of disordered carbons and found that they are made up of graphene segments containing five-, six-, and seven-membered rings, which are arranged in a complex way. We also found that the size of the graphene segments varies between different samples. As expected, upon inclusion of hydrogen the connectivity in the models was reduced. The size of the graphene segments depends on the density of the sample, the amount of order in the system, and also the amount of hydrogen present in the sample. Thus, ring connectivity gives us an idea of the size of the graphene segments which are present in the model, and also the network connectivity, and can be used to differentiate between different models of carbon. Finally, the ring connectivity method is not limited to porous carbons. It can be applied to any covalent network solid which contains rings and can give valuable information about the connectivity and size of the clusters in those models. Acknowledgment. We thank the Department of Energy (Grant No. DE-FGO2-98ER14847) for support of this research. LA0534017

Ring Connectivity: Measuring Network Connectivity in Network

Recommend Documents