CationâÏ Interactions and Rattling Motion through Two-Dimensional

Article pubs.acs.org/JPCC

Cation−π Interactions and Rattling Motion through TwoDimensional Carbon Networks: Graphene vs Graphynes S. Chandra Shekar and R. S. Swathi* School of Chemistry, Indian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM), Kerala, India - 695016 ABSTRACT: We study the cation−π interactions of alkali and alkaline earth cations with various forms of graphyne and graphdiyne and compare the results with those of graphene. The objective of the work is to explore the role of sp hybridized carbons present in various graphynes in determining the stabilities of the cation−π complexes. The two-dimensional network materials, graphene and graphynes are represented by a series of model compounds and their superstructures. Systematic investigations using density functional theory with the dispersion-including M06-2X functional and a triple-ζ basis set reveal that all the cations have a stronger binding with graphynes than graphene. The binding strengths of various ions across the model systems follow the order: Be2+ > Mg2+ > Ca2+ > Li+ > Na+ > K+. Electrostatic potential and molecular orbital analyses are used to explain relative binding strengths at various active sites of model systems. Further, we investigated the passage of the cations through the pores of various graphynes by estimating the energy barriers and rates of diffusion. γ-GY and rhombic GY are found to exhibit selectivity for the passage of certain cations. This opens up the possibility of these novel carbon materials to act as excellent platforms for ion filters.

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and graphynes (GYs).36 These fundamental studies are aimed at developing novel systems that could have a broad range of applications in batteries,37,38 ion channels,39 purification of water,40−42 superconductivity43 etc. GYs are the latest additions to the family of one-atom thick carbon networks containing sp and sp2 hybridized carbon atoms.44−47 γ-GY (the most studied form of GY) and graphdiyne48 (GDY) contain hexagonal carbon rings cross-linked with acetylenic and diacetylenic linkages and are based on dehydrobenzoannulene (DBA-12 and DBA-18 respectively) frameworks.49 Apart from the γ- form, various other forms of GY, namely α-, β- and rhombic forms are also currently being studied for various applications.45 GYs offer a variety of uniformly distributed pores of various sizes and hence are rather attractive candidates for use in gas sensing and separation,50 filtration,41,51 hydrogen storage,52 etc. In an earlier article, we had investigated the binding of alkali metal ions (Li+, Na+, and K+) with γ-GY and GDY and studied the rattling motion (out-of-plane motion) of the ions through the pores using cluster models.36 In this article, we extend our studies to the α-, β- and rhombic forms of GY and compare our results with those obtained for graphene. Furthermore, the effect of the size of the clusters used to represent various GYs is studied by performing calculations on superstructures. We also investigate the binding of alkaline earth cations to various model systems. A comprehensive analysis of the energy barriers for the passage of alkali and alkaline earth metal ions through various pores is performed and the rates of diffusion of the ions through the pores are estimated. The objective of the study is primarily to explore

INTRODUCTION The role of noncovalent interactions is enormous in governing the structures, energetics, functions and other dynamical properties of various chemical and biological systems.1−6 Despite the fact that they are weak in comparison with the covalent interactions, they play a key role in stabilizing biopolymers like DNA, RNA, proteins, and various other supramolecular assemblies.6−9 Cation−π interactions are an important class of noncovalent interactions of extreme relevance in chemistry10−13 and biology.14,15 Metal ions such as Na+, K+, Mg2+, Ca2+, Zn2+, Fe2+, and Co2+ play a key role in regulating various biological processes16−20 such as electron transfer,21 membrane ion channels,22 functioning of metalloproteins,23 and metalloenzymes24 in cell metabolism and molecular recognition.25 The interaction of alkali metal ions, Li+, Na+, and K+ with the π cloud of benzene ring is well studied as a classic case for investigating cation−π interactions.14,26−28 The binding strengths of the ions with the benzene ring decrease in the order Li+ > Na+ > K+ as a consequence of reduced electrostatic attraction due to decrease in the charge density with an increase in the size of the cation. Further studies on the adsorption of ions on ring-fused systems have found that the ionic binding energies increase with the increase in the number of rings.29,30 The interactions of various metal ions with innumerable types of πsystems have been studied by Dougherty and co-workers,10,14 to explore their role, relevance and range in various chemical and biological processes. Understanding these interactions is also crucial for designing novel ionophores, artificial ion channels and functional materials.22,31−33 Recent efforts in the field of cation−π interactions are directed toward understanding the interactions of various cations with extended two-dimensional carbon networks, graphene,32,34,35 © 2015 American Chemical Society

Received: December 17, 2014 Revised: March 10, 2015 Published: March 26, 2015 8912

DOI: 10.1021/jp512593r J. Phys. Chem. C 2015, 119, 8912−8923

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The Journal of Physical Chemistry C the role played by sp and sp2 hybridized carbon atoms in GYs in driving the binding of ions and passage of ions through the pores. Passage through pores is very interesting from the point of view of designing ion selective filters.

ΔG‡ = GTS − Gmin .

The rates of diffusion of ions for finite barriers are subsequently estimated using58

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k=χ

METHODOLOGY Graphene and various forms of GYs studied in this article are represented by cluster-based model systems. The geometries of all the clusters and the cation−π complexes reported herein are optimized by performing density functional theory calculations using the dispersion-including density functional, M06-2X with a basis set of triple-ζ quality using Gaussian 09 suite of programs.53 The M06-2X functional describes medium range electron correlations reasonably well and is shown to describe cation−π interactions accurately.9,54,55 The 6-311G(d,p) basis set is used for all energy minimizations. The cation−π interaction energies are evaluated using

kbT −ΔG‡ / RT e h

where χ is the transmission coefficient considering the tunneling effects, computed from the well-known Wigner formula:59 χ=1−

2 1 ⎛ hv ⎞ ⎛ RT ⎞ ⎜ ⎟ ⎜1 + ⎟. 24 ⎝ kbT ⎠ ⎝ Eb ⎠

In the above, ν is the imaginary frequency and kb, h, and R are the Boltzmann constant, Planck’s constant, and gas constant, respectively. T is the temperature, whose value is taken to be 298 K.

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RESULTS AND DISCUSSION One of the major objectives of this study is to understand the nature and the strength of cation−π interactions of alkali (Li+,

E int = Ecomplex − Emodel system − E ion

where Ecomplex is the energy of the cation−π complex and Emodel system and Eion refer to the energies of the model system and the ion, respectively. All the reported interaction energies are corrected for basis set superposition errors using the counterpoise technique. Since the cation−π interaction energies are predominantly governed by the electrostatic component,56 possible binding sites for various ions in the vicinity of the model systems are predicted by locating the regions of most negative electrostatic potential (ESP). This is also followed up by Hückel molecular orbital analysis performed using a pythonbased program.57 In our analysis of the rattling motion of ions through the pores of various model systems, energy barriers are estimated using two methods. In method 1, the ions are kept at various vertical positions from the pore centers of the model systems and single point potential energy scans are performed with a step size of 0.3 Å. In cases where a finite barrier for the out-of-plane motion exists, the potential energy profile is a double well potential. The energy barrier can be estimated as the difference in energies of the transition state (E′ complex,TS ) and minimum energy (E′complex,min) geometries: ′ ′ . E b′ = Ecomplex,TS − Ecomplex,min

Figure 1. Optimized geometries of the various model systems of graphene, γ-GY, and GDY. “a”, “b”, “c”, and “d” represent various active sites in the model systems for the binding of cations.

In method 2, full geometry optimization of the transition state is performed by placing the ions at the pore centers and constraining the symmetry of the system. The barrier is estimated as the energy difference between the optimized geometries of the transition state (Ecomplex, TS) and the minimum energy (Ecomplex, min) configuration:

Na+, and K+) and alkaline earth cations (Be2+, Mg2+, and Ca2+) with members of the GY family and compare the results with those obtained for graphene. The difference in electronic structures of GYs and graphene arises from the fact that GYs are one-atom thick carbon networks with sp and sp2 hybridized carbon atoms, while graphene is a one-atom thick carbon network with sp2 hybridized carbon atoms. The current study also considers the out-of-plane diffusion of the cations through the pores of the carbon networks and estimates the energy barriers and rates for such a motion. In all our calculations, graphene and GYs are represented by cluster-based models. Further, the effect of the size of the cluster is studied by performing calculations on a series of model compounds (MCs) and their superstructures. The family of GYs is rich and diverse with various forms, namely, α-GY, β-GY, γ-GY, and rhombic GY. γ-GY is the most studied form of GY and is obtained by introducing one acetylenic unit between the hexagons of graphene. GDY can be considered

E b = Ecomplex,TS − Ecomplex,min .

We also estimated the rates of diffusion of ions through the pores of the carbon networks using transition state theory (TST).58 For this, we make use of the thermochemistry variables obtained by calculating the partition functions using nonimaginary vibrational frequencies. First, we calculate the Gibbs free energies of the complexes (Gcomplex) and the transition states (GTS) as implemented in Gaussian 09. Free energy is computed using G = Ecomplex + Gcorr

where Gcorr is the free energy correction to the energy of the complex (Ecomplex). We then calculate the free energy of activation using 8913


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Table 1. Interaction Energies (in kcal mol−1) for the Cation−π Complexes of Li+, Na+, K+, Be2+, Mg2+, and Ca2+ Ions with Various Model Systems of Graphene, γ-GY, and GDY interaction energies (in kcal mol−1) Li+

Na+

K+

Be2+

Mg2+

Ca2+

a

a b c d a b c d a b c d a b c d a b c d a b c d

coronene

C54H18

DBA-12

C66H18

DBA-18

C90H18

−44.95 −47.42 − − −32.40 −32.45 − − −26.12 −25.93 − − −278.04 −284.45 − − −159.89 −161.54 − − −117.30 −117.59 − −

−50.20 −50.33 −49.44 −51.38 −35.53 −35.33 −35.46 −35.82 −29.99 −28.56 −29.26 −29.15 −307.64 −308.64 −306.15 −311.74 −181.80 −182.60 −181.97 −183.94 −137.30 −136.86 −137.16 −137.85

−64.22 −44.60 − − −46.36 −29.86 − − −35.68 −35.68 − − −341.44 −276.47 − − −207.60 −153.49 − − −145.07 −145.07 − −

−66.55 −45.83 −48.81 − −48.62 −31.53 −36.04 − −38.07 −38.07 −31.00 − −361.20 −293.71 −304.66 − −225.76 −170.43 −182.28 − −161.82 −161.82 −140.02 −

−58.01 −42.78 − − −50.14 −28.29 − − −41.79 −41.79 − − −312.79 −278.83 − − −200.51 −155.00 − − −158.91 −158.91 − −

−59.78 −42.80 −45.37 − −50.80 −28.72 −33.00 − −42.31 −42.31 −28.37 − −324.64 −292.66 a − −209.45 −169.40 −182.17 − −171.78 −171.78 −139.45 −

For the binding of Be2+ at site ‘c’ of C90H18, optimized geometry could not be obtained.

systems depend on the charge as well as the size of the ions. The divalent alkaline earth cations have stronger binding than the monovalent alkali metal ions. Within the monovalent and the divalent ions, smaller cations bind the model systems stronger. From Table 1, it is evident that the interaction of various ions with coronene is stronger at the terminal hexagonal ring site, “b”. An exception to this is the case of K+, in which stronger complex is formed at site “a” owing to the large size of K+. In the case of DBA-12 and DBA-18, stronger interaction is observed for the ions binding at site “a”. This is essentially due to the strong electrostatic interaction between the cations and the electron rich regions arising from the acetylenic carbons of DBA-12 and DBA18. It is important to note that, in case of complexes of larger cations (K+ and Ca2+) with initial geometries of ions at site “b” of DBA-12 and DBA-18, the computed lowest energy structures reflect a switching of the ionic positions toward the site “a”. This is also thus reflected in the same numerical values of Eint for these complexes, as can be seen from Table 1. It is interesting to note that coronene and DBA-12 are isomers. The lowest energy cation−π interaction energies reflect stronger binding for all the cations with DBA-12 than coronene. This is essentially due to the higher electron density on DBA-12, which arises from the sp-hybridized acetylenic units. As an illustration, in Figure 2, we show the optimized geometries and the Eint values for the adsorption of Li+ at various active sites of coronene and DBA-12. The lowest energy cation−π interaction energies of Li+ are −47.42 and −64.22 kcal mol−1 for coronene and DBA-12 respectively, showing that Li+ binds γ-GY better than graphene. Since coronene and DBA-12 are isomers, any difference in interaction energies with Li+ has to be attributed to the difference in electronic structure arising from the presence of carbon atoms in varying states of hybridization. Among the various MCs, DBA-

as an extended member of the family of GYs with a structure consisting of two acetylenic units between the hexagons of graphene. We first analyze the interactions of various cations with γ-GY and GDY and compare the results with those for graphene. Subsequently, we study cation−π interactions of various ions with α-GY, β-GY and rhombic GY. Cation−π Interactions Involving Graphene, γ-GY, and GDY. Coronene (C24H12), DBA-12 (C24H12) and DBA-18 (C30H12) are chosen as the MCs for graphene, γ-GY and GDY respectively. To study the effect of the size of the model systems representing the carbon networks, we also consider the superstructures formed from coronene, DBA-12 and DBA-18, namely, C54H18, C66H18 and C90H18 respectively.60 C54H18 (circumcoronene) was used as a model system for graphene in earlier studies.61 Figure 1 shows the optimized geometries of various model systems of graphene, γ-GY and GDY. We refer to the MCs and the superstructures collectively as model systems. Figure 1 also shows the various possible active sites on the model systems for ion adsorption. In the MCs, we consider two different active sites, denoted as “a” and “b”. Similarly, we choose various sites in the superstructures: C54H18 (a, b, c, and d), C66H18 (a, b, and c) and C90H18 (a, b, and c). Now, the cations (Li+, Na+, K+, Be2+, Mg2+, and Ca2+) are placed at various active sites at a distance of ∼3 Å above the plane of the model systems and full geometry optimizations are performed. The interaction energies (Eint) and the positions of ions in the optimized geometries of various resultant cation−π complexes are reported in Tables 1 and 2 respectively. The complexes are predominantly stabilized by electrostatic interactions between the cation and the negative charge densities on the model systems. Across all the model systems, the metal ion affinity follows the order: Be2+ > Mg2+ > Ca2+ > Li+ > Na+ > K+. The affinities of the cations to the π8914


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DBA-18 toward the acetylenic groups is 0.69 Å for Li+ in the complex. As one goes from MCs to superstructures, the complexes are more stabilized due to dispersion interactions from a large number of surrounding atoms. Circumcoronene (C54H18) forms most stable complexes with the ions at the terminal rings, with the exception of K+ which prefers to bind at the center (site “a”). Similar feature is reported in an earlier study by Jean-Francois Gal et al., on the interaction of Li+ with effective ring fused aromatic systems.61 The difference in the values of Eint for ion binding at various active sites of C54H18 is rather small (∼2 kcal mol−1). Analogous to the MCs, the superstructures of DBA-12 and DBA-18 (C66H18 and C90H18) form stable complexes at ring site “a”. However, the superstructures now have two different hexagonal ring sites, “b” and “c” and the numerical values of Eint suggest that ion binding at site “c” is stronger than at site “b”. In the case of K+ and Ca2+, this is not true due to the switching of ionic position from site “b” to site “a”. Most of the other features, including the order of binding strengths with various ions remain the same as discussed earlier for the MCs. Figure 3 shows the optimized geometries and the Eint values for the binding of the Li+ ion at various sites of the superstructures of graphene, γ-GY, and GDY. Thus, our studies on various model systems reveal that the cation−π binding strengths for Li+, Be2+, and Mg2+ with various carbon networks follow the order: γ-GY > GDY > graphene, while for Na+, K+, and Ca2+, the order is GDY > γ-GY > graphene. It is rather interesting to note that the cations bind strongly with the members of the GY family than with graphene. This can be rationalized from the electrostatic potential (ESP) surfaces. Figure 4 shows the ESP surfaces of various MCs and superstructures computed at an isosurface value of 0.0004 eV Å−3. Red regions in ESP surfaces represent regions of high electron density, where ions can bind strongly. Model systems of γ-GY and GDY exhibit enhanced negative regions at the larger pores in comparison with the pores of graphene, thus explaining the higher cation−π binding strengths for those systems. Further, the presence of a small empty region of no density in the triangular pores of DBA-18 and C90H18 validates the previous result that smaller cations bind at offset positions from the centers of the rings toward the acetylenic units. Preferential binding of the cations at the terminal hexagonal rings in the model systems of graphene and at the triangular pore sites in GYs can also be understood from the molecular orbital analysis. We perform Hückel molecular orbital analysis using a python-based program.57 Figures 5 and 6 show the molecular energy levels and the HOMO and LUMO orbitals of the various model systems. From the HOMO and LUMO of coronene and circumcoronene, it is obvious that there is higher electron density at the terminal hexagons and hence most of the cations form the most stable complexes at those sites. Similar is the case with the HOMO and LUMO of DBA-12 and DBA-18 and their superstructures. The electron density is concentrated at the acetylenic carbons validating the stronger binding of the cations at the triangular pores in these materials. We now analyze the out-of-plane diffusion of the cations through the pores of graphene and GYs. Study of passage of atoms, ions and molecules through pores is interesting because of a myriad of applications ranging from gas sensing62 and separation,50,63,64 ion channels39 to DNA sequencing.65 Inplane and out-of-plane diffusion of Li+ in the vicinity of graphene and GYs has been investigated in the context of lithium ion batteries.36,38 In this study, we estimate the energy barriers for

Table 2. Positions (in Å) of Li+, Na+, K+, Be2+, Mg2+, and Ca2+ Ions from the Molecular Planes in the Minimum Energy Geometries of the Cation−π Complexes with Various Model Systems of Graphene, γ-GY, and GDY positions of ions (in Å) Li+

Na+

K+

Be2+

Mg2+

Ca2+

a b c d a b c d a b c d a b c d a b c d a b c d

coronene

C54H18

DBA-12

C66H18

DBA-18

C90H18

1.80 1.78 − − 2.28 2.28 − − 2.69 2.73 − − 1.21 1.22 − − 1.82 1.82 − − 2.21 2.22 − −

1.79 1.77 1.78 1.77 2.27 2.26 2.28 2.27 2.68 2.70 2.68 2.67 1.20 1.21 1.19 1.22 1.81 1.81 1.80 1.80 2.18 2.17 2.17 2.19

1.10 1.80 − − 1.70 2.33 − − 2.17 2.17 − − 0.0 1.29 − − 0.98 1.87 − − 1.60 1.60 − −

0.91 1.80 1.82 − 1.68 2.37 2.33 − 2.22 2.22 2.73 − 0.0 1.27 1.46 − 0.95 1.95 1.83 − 1.59 1.59 2.22 −

0.0 1.80 − − 0.0 2.34 − − 0.68 0.68 − − 0.0 1.27 − − 0.0 1.86 − − 0.0 0.0 − −

0.0 1.80 1.83 − 0.0 2.36 2.35 − 0.74 0.74 2.74 − 0.0 1.26 a − 0.0 1.95 1.82 − 0.0 0.0 2.20 −

For the binding of Be2+ at site “c” of C90H18, optimized geometry could not be obtained.

a

Figure 2. Optimized geometries and interaction energies for the binding of Li+ at various sites of coronene and DBA-12.

12 forms the most stable complexes with the smaller ions, Li+, Be2+, and Mg2+, while DBA-18 forms the most stable complexes with the larger ions, Na+, K+, and Ca2+. Because of the large cavity size and lower electron density of DBA-18 at the ring site “a”, smaller cations are displaced toward the sp hybridized carbons away from the center. The offset distance from the center of 8915


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Figure 3. Optimized geometries and interaction energies for the binding of Li+ at various sites of C54H18, C66H18, and C90H18.

triangular pores in the plane of the model systems (see Table 2). However, in the case of GDY, most of the ions prefer to bind in the triangular pores in the plane of the model systems in the lowest energy configurations (except for K+). When the cation−π complexes have minimum energy configurations on either sides of the molecular plane, one can think of the out-ofplane diffusion of the ion from one minimum to the other via a planar transition state, referred to as the rattling motion.36 The energy barriers for rattling are estimated using two methods (see the section on Methodology). In the first (method 1), the ions are located at various positions along the z-axis (molecular plane is the xy-plane) and single point energy calculations are performed. The potential energy for the motion of the ions is a double well potential in some cases, while it is a rather flat potential in others. From the resultant potential energy profiles (see Figure 7 for the motion of Li+ across the pores of various model systems of graphene, γ-GY, and GDY) energy barriers (E′b) are estimated as the difference in energies of the transition states and the minimum energy structures. In the second (method 2), the ions are positioned at the pore centers in the molecular planes and geometry optimizations are performed to determine the transition state geometries. The difference in the energies of the transition states and the minimum energy structures gives the barrier height (Eb). Table 3 shows the barrier heights obtained using both the methods for the passage of various ions through the pores of model systems of graphene, γGY, and GDY. The key difference in the numerical values of the barrier heights estimated using both the methods is that method 2 allows for a relaxation in the geometry of the model system in the transition state when the ion is placed at the center. In the

Figure 4. ESP surfaces for various model systems of graphene, γ-GY, and GDY.

the out-of-plane diffusion of various ions through the model systems of graphene and GYs. From the cation−π complexes investigated thus far, it is clear that all the ions bind at locations away from the molecular plane on either side of the model systems in the complexes of graphene and γ-GY. An exception to this is the case of Be2+, wherein the ion binds with γ-GY in the 8916


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Figure 5. Hückel molecular orbital energy levels, HOMO and LUMO, of coronene, DBA-12, and DBA-18.

Figure 6. Hückel molecular orbital energy levels, HOMO and LUMO, of C54H18, C66H18, and C90H18.

passage through graphene, while it is barrierless for passage through GDY (K+ has a tiny barrier). In the case of γ-GY, only Be2+ can execute rattling motion without a barrier. Passage through the pores of γ-GY for Li+ and Mg2+ is possible (Eb < 10 kcal mol−1) if these ions hit the substrate with high energies. However, passage of Na+, K+ and Ca2+ through γ-GY is very difficult. This suggests that γ-GY is selective to the passage of certain ions and can be a potential material for ion filters. It is rather interesting to compare the motion of Li+ through graphene, γ-GY and GDY. Because of the smaller sized pores, Eb for passage through graphene is very high (Eb ∼ 180.0 kcal mol−1). However, the barrier drops down considerably for γ-GY, yielding a value of ∼4.0 kcal mol−1. The motion of Li+ becomes

transition state, the model system undergoes a ring expansion to accommodate the ion thus decreasing the barrier. In contrast, the geometry of the model system is kept rigid in method 1. As a consequence, the values of energy barrier obtained using method 2 are always lower than those obtained from method 1.36 For a given cation, the energy barriers for the passage through the pores follow the order: graphene > γ-GY > GDY. From the numerical values of Eb, it is clear that the energy barriers for the rattling motion of ions through MCs and superstructures are very close. This is because the bottleneck in the rattling process is the passage through the pores and this does not depend on the size of the model system. The out-of-plane diffusion of the various alkali and alkaline earth cations is characterized by a large barrier for 8917


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Figure 7. Potential energy profiles for the passage of Li+ through the pores of the model systems of graphene, γ-GY and GDY.

Table 3. Energy Barriers (in kcal mol−1) Obtained Using Method 2 for the Rattling Motion of Various Ions through the Pores of the Model Systems of Graphene, γ-GY, and GDYa energy barriers (in kcal mol−1) Li+ Na+ K+ Be2+ Mg2+ Ca2+ a

coronene

C54H18

DBA-12

C66H18

DBA-18

C90H18

179.41 (247.73) b b 99.03 (161.44) b b

178.98 (245.89) b b 96.44 (158.85) b b

3.54 (4.35) 35.60 (56.07) 102.18 (196.72) 0.0 8.21 (16.12) 58.94 (129.97)

3.66 (4.44) 37.28 (61.11) 107.24 (216.28) 0.0 7.87 (17.09) 60.25 (135.00)

0.0 0.0 0.29 (0.95) 0.0 0.0 0.0

0.0 0.0 0.36 (1.83) 0.0 0.0 0.0

The values in parentheses are obtained using method 1. bThe energy barriers for these systems are larger than 200 kcal mol−1.

Figure 8. Optimized geometries of the various model systems of α-GY, β-GY, and rhombic GY. “a” and “b” represent various active sites in the model systems for the binding of cations.

barrierless for passage through GDY. The free motion of Li+ through the pores of GDY provides an avenue for exploring GDY as an electrode material in lithium ion batteries. In our earlier study, we had used DBA-12 and DBA-18 as MCs for γ-GY and GDY respectively.36 The current study also considers C66H18 and

C90H18 as model systems, yielding results that are in agreement with those obtained using DBA-12 and DBA-18. Further, this demonstrates the utility of cluster models in providing useful insights into various phenomena involving extended twodimensional carbon networks. Since triangular pores of γ-GY 8918


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Table 4. Interaction Energies (in kcal mol−1) for the Cation−π Complexes of Li+, Na+, K+, Be2+, Mg2+ and Ca2+ Ions with Various Model Systems of α-GY, β-GY and Rhombic GY interaction energies (in kcal mol−1) Li+ Na+ K+ Be2+ Mg2+ Ca2+

a b a b a b a b a b a b

α-hexagon

C84H12

β-hexagon

DA-12

C60H12

rhombus

C68H16

−49.42 − −40.10 − −36.24 − −297.27 − −170.94 − −127.32 −

−54.29 −52.21 −45.89 −43.62 −42.46 −39.76 −290.81 −291.00 −209.49 −207.61 −163.39 −157.98

−53.62 − −45.34 − −41.89 − −295.64 − −191.73 − −147.29 −

− −54.02 − −38.02 − −28.41 − −304.56 − −172.39 − −114.12

−53.21 −58.01 −45.47 −41.64 −42.48 −32.01 −281.69 −351.63 −193.37 −215.73 −150.66 −151.54

−62.57 − −45.71 − −34.27 − −296.57 − −200.95 − −137.78 −

−67.54 −65.82 −50.33 −48.61 −39.21 −37.20 −331.88 −345.23 −237.11 −228.12 −172.36 −163.30

Table 5. Positions (in Å) of Li+, Na+, K+, Be2+, Mg2+, and Ca2+ Ions from the Molecular Planes in the Minimum Energy Geometries of the Cation−π Complexes with Various Model Systems of α-GY, β-GY, and Rhombic GY positions of ions (in Å) C84H12 Li+ Na+ K+ Be2+ Mg2+ Ca2+

C60H12

C68H16

α-hexagon

a

b

β-hexagon

DA-12

a

b

rhombus

a

b

0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.30 0.0 0.0

0.0 0.0 0.0 1.32 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0

0.96 1.59 2.23 0.0 1.06 1.66

0.0 0.0 0.0 1.18 0.0 0.0

0.97 1.71 2.25 1.25 0.95 1.56

0.0 0.97 1.82 0.0 0.0 1.03

0.0 1.01 1.83 0.0 0.0 0.98

0.0 1.01 1.82 0.0 0.0 0.98

Figure 10. Optimized geometries and interaction energies for the binding of Li+ at various sites of the model systems of β-GY. Figure 9. Optimized geometries and interaction energies for the binding of Li+ at various sites of the model systems of α-GY and rhombic GY.

geometries of various model systems of α-GY, β-GY and rhombic GY that we have considered for our investigations. αHexagon (C20H8) and C84H12 are chosen as model systems for αGY. β-GY can be thought of as a hybrid structure formed from αand γ-GY except for the presence of benzene rings in γ-GY. Hence, we chose β-Hexagon (C24H12) and DA-12 (C12H6) as the MCs for β-GY and C60H12 as a superstructure. Rhombus (C16H8) and C68H16 represent the model systems for rhombic GY. As shown in Figure 8, all the MCs contain only one active site, while the superstructures have two active sites. As earlier, the ions are kept at various active sites and full geometry optimizations are performed. The energies of interaction and

provide a finite barrier (Eb ∼ 4.0 kcal mol−1) for the rattling motion of Li+, it is interesting to calculate the rate for this motion. The rate with which the ion diffuses through the pores can be evaluated using the methods of transition state theory as outlined in the Methodology section. The rate of out-of-plane diffusion of Li+ through γ-GY is found to be 17.3 ns−1. Cation−π Interactions Involving α-GY, β-GY, and Rhombic GY. In this section, we study the interaction of alkali and alkaline earth cations with other forms of GY, namely α-GY, β-GY, and rhombic GY. Figure 8 shows the optimized 8919


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Figure 11. ESP surfaces for various model systems of α-GY, β-GY, and rhombic GY.

Table 6. Energy Barriers (in kcal mol−1) Obtained Using Method 2 for the Rattling Motion of Various Ions through the Pores of the Model Systems of α-GY, β-GY, and Rhombic GYa energy barriers (in kcal mol−1) Li+ Na+ K+ Be2+ Mg2+ Ca2+ a

α-hexagon

C84H12

β-hexagon

C60H12 (a)

DA-12

C60H12 (b)

rhombus

C68H16

0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 1.34 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0

3.75 (4.70) 35.82 (56.5) 98.82 (175.9) 0.0 11.27 (17.27) 63.67 (132.57)

4.33 (4.37) 38.82 (62.99) 105.34 (219.17) 0.0 7.28 (18.21) 54.44 (138.91)

0.0 1.96 (2.99) 26.84 (44.03) 0.0 0.0 7.21 (10.90)

0.0 2.79 (5.03) 31.94 (53.17) 0.0 0.0 8.05 (13.40)

The values in parentheses are obtained using method 1.

Figure 12. Potential energy profiles for the passage of Li+ and Na+ through the pores of the model systems of β-GY and rhombic GY respectively.

(triangular pore). In superstructures, binding at site “a” is more preferred for C84H12 and C68H16 (Be2+ is an exception). This can be attributed to the dispersion interactions from the peripheral rings in C84H12 and C68H16. However, site “b” of C60H12 has higher electron density than site “a” and hence is preferred by smaller cations like Li+, Be2+, Mg2+, and Ca2+. In contrast, larger cations like Na+ and K+ prefer to bind at site “a”. Interestingly,

positions of ions corresponding to the various minimum energy cation−π complexes are reported in Table 4 and Table 5 respectively. Among the cations, Li+, Na+, Mg2+ and Ca2+ bind strongly with rhombic GY. K+ is found to have strong binding with α-GY and site “a” of β-GY. Preference of K+ for the large hexagonal pores of α-GY and β-GY is due to its large size. Also, owing to its small size, Be2+ prefers to bind at the site “b” of β-GY 8920


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Figure 13. Summary of our results showing the interaction energies for the binding of various alkali and alkaline earth cations with graphene and graphynes.

GYs. Thus, from our studies on the binding of cations with graphene and various GYs, stronger binding of the ions is observed with various GYs than graphene, an effect that can be attributed to the presence of sp-hybridized carbon atoms in GYs.

this picture is not clearly captured in our studies on MCs. For instance, Mg2+ and Ca2+ show stronger binding with β-Hexagon than DA-12. However, in the superstructure formed from βHexagon and DA-12 units (C60H12), preference for site “b” can be seen. Further, a large structural distortion is observed in some of the complexes of the ions with MCs. The complexes with superstructures however do not show such distortions. Thus, an analysis of the cation−π complexes with the superstructures representing GYs as is done here is very important. For illustration, we show the optimized geometries of the complexes of Li+ with the model systems of α-GY, β-GY and rhombic GY in Figures 9 and 10. For the binding of Li+ at various “a” sites of α-GY and β-GY, the positions of ion binding are not the ring centers, but are located at a distance of ∼1 Å from the centers. This is essentially due to the small size of Li+ and the large pore size of model systems, as can be visualized from the ESP surfaces shown in Figure 11. Regions of high electron density can be found to be displaced toward the carbon framework from the ring centers. Subsequently, the energy barriers for the passage of various ions through the pores of model systems of α-GY, β-GY, and rhombic GY are determined using the two methods mentioned earlier and the results are shown in Table 6. The rattling motion is barrierless for the passage of all the ions through α-Hexagon, βHexagon, C84H12 (a small barrier of ∼1 kcal mol−1 exists for Be2+) and the binding site “a” of C60H12 owing to the large pores in these systems. We find a nonzero barrier for the passage of all the ions through DA-12 and the binding site ‘b’ of C60H12 (an exception to this is Be2+). It is interesting to note that the passage through the hexagonal pores of β-GY is barrierless for all ions, while the triangular pores are selective to certain ions. The rate of out-of-plane diffusion of Li+ through the triangular pores of β-GY is 0.40 ns−1. However, the model systems of rhombic GY (rhombus and C68H16) exhibit selectivity for the passage of various cations. The motion of Li+, Be2+, and Mg2+ is found to be barrierless, while that of Na+ and Ca2+ has a small, but finite barrier (

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