Permeation of Fullerenes Through Graphynes: Theoretical Design of

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Permeation of Fullerenes Through Graphynes: Theoretical Design of Nanomechanical Oscillators Anto James, and Rotti Srinivasamurthy Swathi J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b00992 • Publication Date (Web): 01 Apr 2019 Downloaded from http://pubs.acs.org on April 1, 2019

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The Journal of Physical Chemistry

Permeation of Fullerenes Through Graphynes: Theoretical Design of Nanomechanical Oscillators Anto James and Rotti Srinivasamurthy Swathi* School of Chemistry, Indian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM), Vithura, Thiruvananthapuram 695551, India

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The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ABSTRACT

Carbon allotropes such as fullerenes, carbon nanotubes (CNTs) and graphene have shown great promise for device applications in recent times. Nanomechanical oscillators based on CNTs are well explored theoretically as well as experimentally. Investigations on the motion of C 60 through CNTs have revealed oscillator frequencies of up to 74 GHz. However, the absence of mature technology in the terahertz regime has resulted in the so-called terahertz gap. Based on an analysis of the permeation of fullerenes through the nanopores of graphynes, herein, we report the theoretical design of nanomechanical oscillators in the 0.1-0.5 THz regime. The design strategy involves employing electronic structure methods as well as atomistic model potentials to probe the permeation process of a set of fullerenes, namely, C 20 , C 42 , C 50 , C 60 , C 70 and C 84 through the triangular and the rhombus-like nanopores of γ−GY-N (N=4-6) and r−GY-N (N=4-5), respectively. Considering the results from the electronic structure methods as a benchmark, we adopt a fitting procedure to extract the optimal values of the parameters in the atomistic model potentials that could be useful for researchers performing force field calculations on fullerenegraphyne systems. Our findings indicate that a discrete atomistic potential of the improved Lennard-Jones type can describe the permeation process leading to the oscillatory response with reasonable accuracy at a computational cost much lower than the electronic structure calculations.


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INTRODUCTION

Design of high-quality devices that can exploit the ‘plenty of room at the bottom’ is currently one of the hotly pursued research areas.1 Nanomechanical systems such as molecular motors2, molecular switches3, rollers4, wheels5, shuttles6 and chemical or physical sensors7 are being explored for more than a decade, to fabricate miniaturized and efficient devices. An important family of such devices, namely nanomechanical oscillators have found applications in quartz crystal microbalances8, non-contacting atomic force microscopy9, molecular electronics10, photodetectors11, optical filters and nano-antennae12. Carbon allotropes of reduced dimensions, namely fullerenes, carbon nanotubes (CNTs) and graphene are well known for their unique properties such as high mechanical and thermal stability, elasticity, electrical conductivity, flexibility etc., and are currently being explored for the design of devices. Nanomechanical oscillators based on CNTs are well explored both experimentally and theoretically.13-25 Pioneering work in CNT-based oscillators was carried out independently by Cumings and Zettl as well as by Yu et al. in the year 2000.13-14 Cumings employed multi-walled carbon nanotubes (MWCNTs) to fabricate a nanoscale linear bearing possessing ultra-low friction. The core nanotubes in MWCNTs can perform telescopic extension and retraction motion along the uncapped end of the outer CNT. Zheng and Jiang further proposed that MWCNTs can be exploited to devise nanomechanical oscillators of frequencies up to several gigahertz.15 To obtain a sustained oscillation with less damping, the difference between the radii of core and outer nanotubes must be around 3.4 Å and the outer CNT should be uncapped at both ends. Oscillation frequencies reported are ~38 GHz.1617

Zheng and co-workers have shown that the oscillator frequency decreases with the increase in

the length of the core nanotube.15, 18 However, Rivera and co-workers predicted that MWCNTs with equal lengths for inner and outer tubes could also sustain oscillations for longer time.23 An


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oscillation frequency of up to 74 GHz is achieved in case of the motion of C 60 inside a CNT.19-20 The highest oscillator frequency reported among the neutral oscillators is for the carbon nanotorusC 60 complex, with a frequency of around 150 GHz.26 Besides investigations on the motion of fullerenes and CNTs through CNTs for nano-oscillator applications, the oscillatory motion of a variety of atoms and ions through CNTs has also been studied. The oscillatory motion of carbon19, 26 and neon27 atoms, as well as chloride ion27-28 through CNTs are recently reported. The frequencies of oscillation are found to be as high as 137 GHz and 250 GHz for the neon atom and the chloride ion, respectively. The frequencies depend on many factors such as initial velocity, initial axial position, length of the CNT and so on. Atoms are also found to show an escaping tendency from the nanotube during the oscillation.28 Potassium ion bound in a CNT is predicted to show an oscillation frequency of up to 400 GHz.29 The ion can induce a strong dielectric response and polarization inside the nanotube. Nanomechanical oscillators based on boron nitride nanotubes have also been predicted to exhibit similar oscillatory response.30-31 The oscillator frequency for CNT-based nano-oscillators is limited up to 150 GHz except for the ion-based oscillators and the frequency varies with damping.29 Terahertz frequency has become important in recent times for many applications ranging from molecular detection and bio-imaging to security and communication systems.32-33 However, the absence of mature technology in this frequency region has resulted in a gap, namely the terahertz gap. The gap lies in the region 0.3 to 30 THz of the electromagnetic spectrum and researchers are attempting to develop devices for the detection and generation of waves in this region.11, 34 Another class of nanomechanical oscillators involves the rattling motion of the oscillatory moieties adsorbed on various substrates. The oscillatory motion of fullerenes over graphene is one of the important examples in this direction. Adsorption of fullerenes on graphene can happen via


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chemisorption or physisorption.35-44 The attachment of fullerenes to the graphene sheet by way of covalent bonds results in the formation of fullerene-graphene nanobuds.35,

45-50

While in

physisorption, the binding happens due to the weak van der Waals interactions.51-52 Followed by the binding of fullerenes on graphene, a charge transfer can result from fullerenes to graphene.37 Neek-Amal and co-workers have predicted the existence of an oscillatory motion similar to that of a simple harmonic oscillator for fullerenes adsorbed on graphene.53 According to Ghavanloo and Fazelzadeh, such oscillatory behavior of spherical fullerenes (C 60 , C 240 and C 540 ) can be exploited to devise nano-oscillators akin to the CNT oscillators.40 Further, from the analytical calculations based on the Lennard-Jones potential and Newtonian mechanics, the authors have reported the frequencies of oscillation to be in the range of terahertz and the frequency of oscillation is found to increase with an increase in the size of the fullerene. However, similar analytical calculations by Rekhviashvili et al. suggest that the oscillatory motion of fullerene adsorbed on epitaxial graphene that lies on a copper surface is heavily damped and aperiodic.54 Our group has recently investigated the rattling motion of alkali and alkaline earth metal ions through the pores of graphynes, the latest members in the family of synthetic allotropes of carbon possessing sp and sp2 hybridized carbon atoms.55-56 In addition to the investigations on the oscillatory behavior of various species in the vicinity of CNTs and two-dimensional carbon membranes, interesting oscillatory motion involving various molecular complexes is also reported in the literature.57-59 The oscillatory motion in such complexes can arise due to the tunneling as well as the passage over the barrier of the oscillating moiety. Some of the pioneering reports in the field of molecular rattles are the theoretical studies on the passage of protons as well as ions through small aromatic rings such as benzene and cyclopentadienyl anion, cyclononatetraenyl anion etc.57-58 Potential energy profiles for the motion of proton or ions through these aromatic rings are of the nature of a


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double well. Cyclononatetraenyl-lithium complex is shown to behave as a molecular oscillator with a frequency of 8.2 THz. Satpati and co-workers have proposed a molecular inversion oscillator based on a B 12 cluster.59 The cluster possesses a bowl-shaped structure with a threemembered ring of boron atoms standing above a nine-membered ring of boron atoms. The threemembered ring is predicted to tunnel through the larger ring resulting in an inversion of the bowl. Carbon membranes are increasingly becoming popular for achieving selectivity in the adsorption and permeation of various chemical species.60-66 Nanoporous graphene membranes are relatively difficult to synthesize, and many of the existing methods of synthesis do not give good control over the porosities. Graphynes (GYs) have recently emerged as an interesting class of carbon membranes with well-defined intrinsic pores for potential applications in water desalination, molecular adsorption, gas sensing, energy storage, catalysis and optoelectronics.67-70 GYs are designed by replacing the C-C bonds in graphene selectively with acetylenic carbon linker chains in specific patterns. According to the pattern in which the acetylenic linkers are introduced, GYs are classified into various groups, namely α, β, γ, rhombic GYs and so on. Based on the number of acetylenic groups (N) in the linkers, GYs are named as GY-N. GYs were first proposed theoretically by Baughman and co-workers in 1987.71 Successful syntheses of various members of γ−graphynes (γ-GYs) and rhombic−graphynes (r−GYs) as well as other analogues provided further impetus to the research on these carbon membranes.72-79 In this study, we report the theoretical design of nanomechanical oscillators based on the permeation of fullerenes through various members of the graphyne family. The permeation of a set of fullerenes, namely C 20 , C 42 , C 50 , C 60 , C 70 and C 84 through γ−GY-N (N=4-6) and r−GY-N (N=4-5) is probed by employing electronic structure calculations as well as atomistic model potentials. In some selected fullerenegraphyne complexes, we report nanomechanical oscillatory response with frequencies in the range


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of 0.1 to 0.5 THz, which belongs to the terahertz gap. One of the unique aspects of this study is the demonstration of the oscillatory response from smaller fullerenes (C 20 , C 42 and C 50 ), while most of the previous findings have focused on C 60 and C 70 . Ozmaian et al. have recently studied the diffusion and self-assembly of C 60 on γ−GYs (ranging from γ−GY-1 to γ−GY-5) using molecular dynamics simulations with an adaptive intermolecular reactive empirical bond order potential.80 While the focus of this study is on the binding configurations of C 60 at various sites of GYs and the role of the attractive interactions between proximal fullerenes in governing their mobilities etc., we attempt to analyze the permeation of various fullerenes through the pores of γ−GYs as well as r−GYs in great detail with the objective of retrieving functional nanomechanical oscillatory response. Furthermore, by a careful comparison of the potential energy profiles obtained from the electronic structure calculations and the atomistic potentials of the LennardJones and the improved Lennard-Jones types, we arrive at the optimal parameters of the model potentials that could be useful for researchers performing force field calculations on fullerenegraphyne systems.

THEORETICAL AND COMPUTATIONAL METHODOLOGIES Electronic Structure Calculations: Electronic structure calculations are performed using density functional theory (DFT) with the M06-2X81 and B97D82 functionals and a triple-𝜁𝜁 basis set, 6311G(d,p) using the Gaussian 09 suite83. M06-2X is a Minnesota functional which describes the medium-range electron correlations reasonably well, while B97D is a dispersion-corrected functional used for accounting for the long-range dispersive forces accurately.84-85 The M06-2X functional employs meta-generalized gradient approximation and incorporates double the amount of non-local exchange. B97D is a DFT-D functional employing generalized gradient


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approximation and including atom-pairwise dispersion corrections empirically with R-6 terms.8182

Both functionals have earlier been employed to describe the non-covalent interactions in carbon-

based systems and are shown to yield accurate results.41-42 Various forms of GYs are represented by the annulenic model systems.55-56 The potential energy curves for the permeation of fullerenes through GYs are obtained by placing fullerenes at various vertical positions from the pore centers of GYs and calculating the single point electronic energies. The interaction energies of the fullerene-graphyne complexes at various vertical positions are evaluated using 𝐸𝐸𝑖𝑖𝑖𝑖𝑖𝑖 = 𝐸𝐸𝑠𝑠𝑠𝑠 (𝑐𝑐𝑐𝑐𝑚𝑚𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝) − 𝐸𝐸𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 − 𝐸𝐸𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔ℎ𝑦𝑦𝑦𝑦𝑦𝑦

where 𝐸𝐸𝑠𝑠𝑠𝑠 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) refers to the single point energies of the fullerene-graphyne complexes at

various vertical positions and 𝐸𝐸𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 and 𝐸𝐸𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔ℎ𝑦𝑦𝑦𝑦𝑦𝑦 are the energies of the optimized geometries of isolated fullerene and graphyne systems, respectively. The energy barriers for the permeation of fullerenes through the pores of GYs are evaluated using 𝐸𝐸𝑏𝑏 = 𝐸𝐸𝑠𝑠𝑠𝑠 (𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) − 𝐸𝐸𝑠𝑠𝑠𝑠 (𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑐𝑐𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜)

where 𝐸𝐸𝑠𝑠𝑠𝑠 (𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) and 𝐸𝐸𝑠𝑠𝑠𝑠 (𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) are the single point

energies of the minimum energy configurations and the tranisition states obtained during the potential energy scans. The binding energies of the various fullerene-graphyne complexes are

obtained by performing full geometry optimizations, starting with the geometries of the minimum energy configurations of the potential energy scans. The binding energies are evaluated using 𝐸𝐸𝐵𝐵𝐵𝐵 = 𝐸𝐸𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 + 𝐸𝐸𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔ℎ𝑦𝑦𝑦𝑦𝑦𝑦 − 𝐸𝐸𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

where 𝐸𝐸𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 refers to the energies of the optimized fullerene-graphyne complexes. The adsorption heights of fullerenes in various complexes are evaluated as the distances between the centres of mass of fullerenes and GYs in the optimized geometries of the fullerene-graphyne complexes.


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Structural Analysis: The molecular diameters of fullerenes and the pore sizes of GYs are evaluated using pywindow, a python package recently developed by Miklitz and Jelfs.86 In pywindow, the average diameter of a molecule is defined as the average distance between the center of mass and the outer edge of the van der Waals sphere of atoms. Similarly, the intrinsic pore diameter is defined as the distance from the center of mass of the molecule to the inner edge of the van der Waals sphere of the closest atom. Atomistic Potential Calculations: The atomistic potential calculations to investigate the fullerene-graphyne interactions are carried out by employing the Lennard-Jones87-88 (LJ) and the improved Lennard-Jones89-90 (ILJ) potentials. All the calculations are implemented in python. The ILJ potential is a modified form of the LJ potential, initially proposed by Maitland and Smith in 1973 and further modified by Pirani and co-workers in 2004 with a simplified expression for the coefficient of the repulsive part of the potential.89, 91 The energy of interaction between the ith atom in fullerene and the jth atom in graphyne using the atomistic potentials are given by 12

and

𝑟𝑟𝑚𝑚 𝑉𝑉𝐿𝐿𝐿𝐿 �𝑟𝑟𝑖𝑖𝑖𝑖 � = 𝜀𝜀 �� 𝑟𝑟𝑖𝑖𝑖𝑖

𝑛𝑛�𝑟𝑟𝑖𝑖𝑖𝑖 �

𝑚𝑚

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𝑟𝑟𝑚𝑚 − 2� � � 𝑟𝑟𝑖𝑖𝑖𝑖

𝑟𝑟𝑚𝑚 𝑉𝑉𝐼𝐼𝐼𝐼𝐼𝐼 �𝑟𝑟𝑖𝑖𝑖𝑖 � = 𝜀𝜀 � � � 𝑛𝑛�𝑟𝑟𝑖𝑖𝑖𝑖 � − 𝑚𝑚 𝑟𝑟𝑖𝑖𝑖𝑖

𝑛𝑛�𝑟𝑟𝑖𝑖𝑖𝑖 �

𝑚𝑚

𝑟𝑟𝑚𝑚 − � � � 𝑛𝑛�𝑟𝑟𝑖𝑖𝑖𝑖 � − 𝑚𝑚 𝑟𝑟𝑖𝑖𝑖𝑖

where 𝑉𝑉𝐿𝐿𝐿𝐿 and 𝑉𝑉𝐼𝐼𝐼𝐼𝐼𝐼 are the interaction energies for LJ and ILJ potentials, respectively. In the above, ε is the potential well depth and 𝑟𝑟𝑚𝑚 is the distance at which the potential energy reaches the

minimum for a LJ type of interaction. For interactions between disimmilar atoms, ε and 𝑟𝑟𝑚𝑚 are

obtained by the Lorentz-Berthelot mixing rules92 as:

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𝜀𝜀 = �𝜀𝜀𝑎𝑎 𝜀𝜀𝑏𝑏 and 𝑟𝑟𝑚𝑚 = 2 (𝑟𝑟𝑚𝑚𝑚𝑚 + 𝑟𝑟𝑚𝑚𝑚𝑚 ) ACS Paragon Plus Environment

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where a and b refer to two dissimilar atoms. For all neutral-neutral systems, the term m in 𝑉𝑉𝐼𝐼𝐼𝐼𝐼𝐼 has a value of 6.93 The exponent of the first term 𝑛𝑛(𝑟𝑟𝑖𝑖𝑖𝑖 ) in 𝑉𝑉𝐼𝐼𝐼𝐼𝐼𝐼 depends on the distance between the

two interacting atoms as

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𝑟𝑟𝑚𝑚 𝑛𝑛�𝑟𝑟𝑖𝑖𝑖𝑖 � = 𝛽𝛽 + 4 � � 𝑟𝑟𝑖𝑖𝑖𝑖

where β is a dimensionless constant. 𝑛𝑛�𝑟𝑟𝑖𝑖𝑖𝑖 � modifies the repulsive part of the LJ potential. Both

𝑛𝑛(𝑟𝑟𝑖𝑖𝑖𝑖 ) and β are indicative of the hardness of the potential of the two interacting moieties. The total

interaction energies in various fullerene-graphyne complexes are then evaluated as the sums of all the pairwise interactions using the LJ and the ILJ pair potentials. Bartolomei and co-workers have recently investigated the permeation of water molecules93 and gases94 (He, Ne, CH 4 ) through GYs

using the dispersion-corrected second-order Møller–Plesset perturbation theory (MP2C) method and compared the results with those obtained using the LJ and ILJ pair potentials. We have adopted a similar approach here for analyzing the permeation of fullerenes through GYs.

RESULTS AND DISCUSSION With the objective of eliciting functional nanomechanical oscillatory response from the fullerenegraphyne complexes, we probe the permeation of a set of fullerenes, namely C 20 , C 42 , C 50 , C 60 , C 70 and C 84 through the nanopores of various γ−GYs (γ−GY-N; N=4-6) as well as r−GYs (r−GY-N; N=4-5). The optimized geometries of the annulenic model compounds of GYs obtained at the B97D/6-311G(d,p) level of theory are shown in Figure 1. The initial geometries of fullerenes are taken from an online database provided as part of electronic supporting information to a monograph written by Tománek95 and geometry optimizations are performed at the B97D/6311G(d,p) level. In case of fullerenes possessing multiple isomers, the isomers with the highest


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symmetry are considered. The optimized geometries of various fullerenes are depicted in Figure 2. All the graphyne model compounds and fullerenes are optimized considering their molecular symmetries (see Table 1 for the molecular symmetries). Permeation of a species through a membrane mainly depends on two structural factors, the molecular diameter of the permeating species and the pore size of the membrane. The average molecular diameters of fullerenes and the intrinsic pore diameters of GYs are therefore analyzed using a recently reported pywindow python library86, and their numerical values are given in Table 1.

Figure 1. The optimized geometries of various annulenic model compounds of graphynes.


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Figure 2. The optimized geometries of various fullerenes.


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Table 1. Structural properties of fullerenes and graphynes.

Fullerene

Average molecular diameter (Å)

Graphyne

Pore size (Å)

C 20 (D 3d )

6.85

γ−GY-4 (D 3h )

5.25

C 42 (D 3 )

8.81

γ−GY-5 (D 3h )

6.71

C 50 (D 5h )

9.37

γ−GY-6 (D 3h )

8.21

C 60 (Ih )

10.02

r−GY-4 (D 2h )

8.91

C 70 (D 5h )

10.58

r−GY-5 (D 2h )

11.35

C 84 (D 6h )

11.34

Permeation Barriers from Electronic Structure Calculations: We now analyze the potential energy profiles for the permeation of the fullerenes through various forms of GYs. Considering both GYs and fullerenes as rigid structures (as found on geometry optimization), interaction energies (𝐸𝐸𝑖𝑖𝑖𝑖𝑖𝑖 ; section on Theoretical and Computational Methodologies) are evaluated at various

vertical positions of the fullerenes with respect to GYs using the B97D and M06-2X functionals to obtain the scans. The energy profiles for the passage of various fullerenes through various γ−GYs and r−GYs are shown in Figures 3 and 4. As can be seen from Table 1, the pore size of γ−GY-4 is smaller than the molecular diameters of all the fullerenes and therefore one would expect non-zero energy barriers for the permeation of all the fullerenes through γ−GY-4. In tune with this, we find that the permeation of C 20 through γ−GY-4 is associated with an energy barrier

of ~56 kcal/mol (Figure 3). For all the other fullerenes, the permeation barriers through γ−GY-4 are much higher and the energy scans for these systems are not presented. On the other extreme, the pore size of r−GY-5 is larger than the molecular diameters of all the fullerenes and therefore the permeation process is expected to be facile. Indeed, the energy scans do indicate this (Figures


13


3 and 4). Thus, it is clear that, both γ−GY-4 and r−GY-5 are not appropriate substrates for achieving selectivity in the permeation of various fullerenes through the nanopores. On the other hand, by employing γ−GY-5, γ−GY-6 and r−GY-4, it is possible to achieve selective permeation. C 20 can pass through γ−GY-5 by overcoming the tiny barrier of ~2 kcal/mol if it approaches the substrate with an initial kinetic energy, while none of the other fullerenes can permeate through γ−GY-5 (Figure 3). The barriers for the passage of C 42 , C 50 , C 60 , C 70 and C 84 through γ−GY-5 are very high. In contrast, C 20 permeates through γ−GY-6 in a facile manner, while C 42 requires a tiny initial energy for permeation. The motion of C 50 , C 60 , C 70 and C 84 through γ−GY-6 involves higher energy costs. In the case of r−GY-4, the transmission of C 20 and C 42 involves zero energy barrier while the transmission barrier for C 50 is small (~7 kcal/mol). Again, the permeation of C 60 , C 70 and C 84 through r−GY-4 is difficult because of large energy barriers (Figure 4). Thus, our analysis indicates that γ−GY5, γ−GY-6 and r−GY-4 membranes could be very useful to isolate smaller fullerenes from complex fullerene mixtures in carbon soot. Similarly, r−GY-5 membranes can be used to separate the fullerenes investigated herein from the larger ones such as C 240 and C 540 that are also frequently encountered.40 The energy profile for the permeation of C 84 through r−GY-5 indicates that C 84 can just about pass through, implying that all the higher order fullerenes will face significant energy barriers for permeation. The computed energy profiles for the permeation of fullerenes reveal a significant difference between interaction energy values computed using the B97D and the M06-2X functionals of DFT. The energy values for the various fullerene-graphyne complexes obtained using M06-2X are larger compared to those obtained using B97D. In the long-range (at farther vertical distances on either side of GYs), the interaction energies die out faster for M06-2X when compared to B97D. This variation can be attributed to the fact that the M06-2X is a mid-range functional while B97D is a long-range functional and the


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difference in the treatment of long-range dispersion interactions by both the functionals accounts for the differential nature of energy profiles computed using the two approaches. Interestingly, the computed energy barriers for permeation using both the functionals are in reasonable agreement with each other, since the barriers are primarily dictated by the atoms in the vicinity of the nanopores. In order to further establish that the energetics of the permeation process is dictated by the nanopore configuration and is not strongly affected by the size of the annulenic model systems employed to represent various GYs, we have calculated the potential energy profiles for the permeation of C 20 through γ−GY-5 and γ−GY-6 by considering larger annulenic model compounds, C 162 H 18 and C 186 H 18 , respectively than what we have considered earlier (C 48 H 12 and C 54 H 12 ). The permeation profiles for C 20 through the larger annulenic model compounds of γ−GY-5 and γ−GY-6 are in reasonable agreement with the smaller ones (Figure S1). Therefore, the inferences drawn from molecular model systems can be extrapolated to the corresponding carbon membranes. Such an approach of representing the carbon membranes by the corresponding annulenic model systems has earlier been adopted by us as well as other groups.36, 42, 55-56, 93


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Figure 3. Potential energy profiles for the permeation of C 20 , C 42 and C 50 through various graphynes computed using the B97D and the M06-2X functionals with the 6-311G(d,p) basis set.


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Figure 4. Potential energy profiles for the permeation of C 60 , C 70 and C 84 through various graphynes computed using the B97D and the M06-2X functionals with the 6-311G(d,p) basis set. Permeation Barriers from Atomistic Potentials: Atomistic potentials and empirical force fields can describe molecular interactions at lower computational costs than the first-principles and semiempirical approaches. One of the challenges in materials science research is to propose force field parameters that can enable modeling complex systems such as fullerene-graphyne complexes that are otherwise not easily amenable to first-principles calculations. With the reports on successful synthesis of various forms of GYs in the last five years or so72-74, research on the development of accurate force fields for describing interactions involving GYs has begun.93-94, 96-97 In the context of modeling fullerene-graphyne interactions, it is all the more important to develop force fields since accurate electronic structure calculations on such systems are fairly expensive. Therefore, herein, we have attempted to propose atomistic potential parameters for the LJ and the ILJ potentials (section on Theoretical and Computational Methodologies) that can be useful starting


17


points for force field calculations on fullerene-graphyne systems. The ε and 𝑟𝑟𝑚𝑚 values depend on

the chemical nature of the interacting atoms and their environment. There are numerous reports in the literature employing different values of ε and 𝑟𝑟𝑚𝑚 parameters for carbon-based materials such as fullerenes, CNTs and graphene.40,

54, 98-101

Similarly, the 𝛽𝛽 parameter in the ILJ potential

depends on the chemical nature and the hardness of the interacting species94 and it is evaluated to be in the range of 8.5 ± 2.5 for neutral systems.97 Bartolomei et al. have recently evaluated 𝛽𝛽 as 7.5 and 6.5 for graphdiyne (𝛾𝛾-GY-2) using second-order dispersion-corrected Møller–Plesset

perturbation (MP2C) calculations while studying the permeation of gases96 and water molecules93, respectively. For the interaction of noble gases with graphene, 𝛽𝛽 is evaluated as 8.5.94

Since the fullerene-graphyne complexes have so far not been investigated in the literature, we set

out to arrive at the optimal LJ and the ILJ parameters for describing the intermolecular interactions in such complexes. Considering the results from the electronic structure calculations reported in the previous section as a benchmark, herein, we adopt a fitting procedure to fit the potential energy scans for permeation (computed at the DFT level) with the LJ and ILJ functional forms thereby extracting the parameters of interest. The non-covalent interactions between GYs and fullerenes comprise two types of interactions based on the chemical constituents, carbon-carbon interactions and carbon-hydrogen interactions. The number of carbon-hydrogen interactions is much smaller in comparison with the carbon-carbon interactions. Further, since the hydrogen atoms in the model compounds are located at the terminal positions, the contribution of carbon-hydrogen interactions to the total interaction energies is expected to be very small. Thus, for simplicity of the fitting process, ε and 𝑟𝑟𝑚𝑚 parameters for the carbon-hydrogen interactions are adopted from the

literature.101 Thus, ε and 𝑟𝑟𝑚𝑚 for the carbon-carbon interactions and β are used as fit parameters to fit the energy scans obtained at the B97D/6-311G(d,p) level with the total interaction energies


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calculated using the ILJ potential. A flow chart depicting the fitting procedure is shown in Figure 5. The fitting procedure is explained below: 1. Initial values of ε, 𝑟𝑟𝑚𝑚 for carbon-carbon interactions and 𝛽𝛽 are chosen to be ε = 0.15 kcal/mol, 𝑟𝑟𝑚𝑚 = 3 Å and 𝛽𝛽 = 7.5.93, 101

2. Keeping the value of 𝛽𝛽 fixed (𝛽𝛽 = 7.5), ε and 𝑟𝑟𝑚𝑚 are used as fit parameters to reproduce the energy profile for the permeation of C 20 through γ−GY-4.

3. The ε, 𝑟𝑟𝑚𝑚 values obtained in the previous step along with 𝛽𝛽=7.5 do not however yield good

fits for other fullerene-graphyne complexes. Major deviations are noted in the description of the short-range repulsive region of the potential. This effect is more pronounced in the

fullerene-graphyne complexes possessing large repulsions for the approach of fullerenes towards the GYs. As a result, for further fitting, the DFT scans with very large interaction energies in the short-range repulsion region are considered. Therefore, using the revised values of ε and 𝑟𝑟𝑚𝑚 (obtained after step 2), 𝛽𝛽 is tuned over a range of values such that the energy scan for the complex of C 42 with γ−GY-5 is reproduced. Also, since the 𝛽𝛽

parameter is introduced in the ILJ potential to account for the limitations of the LJ potential, 𝛽𝛽 is varied initially to get a good fit. Using the new value of 𝛽𝛽, we again fit for ε and 𝑟𝑟𝑚𝑚

for the accurate description of the scan for the complex of C 20 with γ−GY-4. This procedure is followed few times such that eventually the energy scans for the complexes of C 20 and C 42 with γ−GY-4 and γ−GY-5, respectively are reasonably fitted.

4. The obtained fit parameters (ε, 𝑟𝑟𝑚𝑚 and β obtained after step 3) for the ILJ potential are then

employed to check for their accuracy in describing the scans for the other complexes. If a


19


major deviation is observed, the empirical parameters can be fitted further considering the new set of DFT scans. However, in our case, the ε, 𝑟𝑟𝑚𝑚 , 𝛽𝛽 parameters obtained by fitting the

scans for the complexes of C 20 and C 42 with γ−GY-4 and γ−GY-5, respectively were accurate enough to reproduce the data for the other fullerene-graphyne complexes.

Figure 5. Illustration of the fitting procedure for obtaining the optimal parameters of the improved Lennard-Jones potential. On employing the fitting procedure, the ε, 𝑟𝑟𝑚𝑚 and 𝛽𝛽 parameters obtained are 0.09 kcal/mol, 3.91 Å and 6.0, respectively. The ε and 𝑟𝑟𝑚𝑚 parameters obtained from the ILJ potential are used for the

LJ potential as well in order to have a better comparison. The atomistic potential energy scans thus

obtained for the passage of fullerenes through GYs are shown in Figures 6 and 7 and Figure S2. A comparison of the permeation barriers, binding positions and interaction energies at the potential


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minima for various fullerene-graphyne complexes obtained from DFT scans and atomistic potential calculations is shown in Figures S3, S4 and S5. As evident from the atomistic scans, the LJ potential heavily overestimates the repulsive region of the interaction. Modification of the repulsive coefficient, 𝑛𝑛�𝑟𝑟𝑖𝑖𝑖𝑖 � in the ILJ potential makes it a much softer core potential compared to the LJ potential. The estimated interaction energies (at the minima) from the ILJ and the LJ

potentials are very close, implying that the description of the attractive regime by both the potentials is similar. As evident from the figures, the binding positions and the interaction energies at the potential minima estimated by the fitted LJ and ILJ force fields are in agreement with the values estimated from the B97D functional (Figure S4). We note that the fitting procedure is employed only for the B97D functional in view of its accuracy when compared to the M06-2X functional.


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Figure 6. Potential energy profiles for the permeation of C 20 , C 42 and C 50 through various graphynes obtained using the Lennard-Jones and the improved Lennard-Jones atomistic potentials.


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Figure 7. Potential energy profiles for the permeation of C 60 , C 70 and C 84 through various graphynes obtained using the Lennard-Jones and the improved Lennard-Jones atomistic potentials.

Geometry Optimization of Fullerene-Graphyne Complexes: We further performed full geometry optimization of the fullerene-graphyne complexes using B97D and M06-2X functionals of DFT. The minimum energy structures of the fullerene-graphyne complexes in energy scans are considered as the initial structures for geometry optimizations. The veracity of the structures to be true minima on the potential energy surfaces is verified by the frequency analysis of the optimized complexes. The optimized geometries and the binding energies of various fullerene-graphyne complexes obtained using the B97D functional are shown in Figures 8 and 9. The triangular pores of γ−GY-5 and γ−GY-6 are ideal for accommodating C 20 and C 42 , respectively into their pockets (Figure 8). The annulenic model compounds for r−GYs in the optimized geometries of the


23


complexes are observed to be considerably distorted from planar configurations (Figure 9). This can be due to the decrease in the number of sp2 carbons, and thus Young’s modulus. r−GYs have earlier been shown to be much more flexible compared to γ−GYs and graphene.67, 102 As a result, the model compounds of r−GYs act like molecular tweezers or buckycatchers to wrap around fullerenes in order to maximize the interactions.41, 103-104 However, the distortions from planarity will be less pronounced in extended graphyne sheets compared to the model compounds due to extensive covalent bonding in two dimensions. Further, temperature plays a crucial role in governing the dynamics of fullerenes on GYs. The thermal effects are best revealed using molecular dynamics simulations. The permeation of C 60 through γ−GYs has earlier been analysed by Ozmaian and co-workers.80 The authors have found that at higher temperatures, C 60 bound on the hexagonal pores of γ−GYs can escape from the GY surface. However, when it is bound on the triangular pores (which is the case in the current study), desorption rarely occurs. It would be interesting to perform detailed simulations on the various fullerene-GY complexes reported here and we are currently investigating these aspects.


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Figure 8. The optimized geometries and binding energies of the complexes of C 20 and C 42 with γ−graphynes.


25


Figure 9. The optimized geometries and binding energies of the complexes of fullerenes with r−graphynes.

Nanomechanical Oscillatory Motion: Molecules that are non-covalently adsorbed on surfaces can have surface vibrations in the physisorption well. Theoretical studies of Ghavanloo and Fazelzadeh have predicted the frequencies of oscillation for the out-of-plane motion of spherical fullerenes on the graphene surface to be in the terahertz region.40 Ozmaian and co-workers have


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reported an effective out-of-plane vibrational motion of fullerene in the normal direction of graphyne sheet during molecular dynamics simulations of a single fullerene over graphynes.80 In view of the above, the optimized complexes of the fullerenes with the GYs are further analyzed to probe the vibrational motion of fullerenes on graphyne surfaces. The objective herein is to find the appropriate complexes that can lead to an interesting oscillatory motion of fullerenes. The frequency data obtained from the optimized geometries of the fullerene-graphyne complexes (Table 2) showed that such oscillatory motion could be observed in the complexes of C 20 and C 42 with γ−GY-5 and γ−GY-6, respectively with oscillator frequencies in the terahertz regime (0.3 – 0.5 THz). The timescales of oscillation are found to be in the picosecond regime. Videos showing the nanomechanical oscillatory motion in these non-covalent complexes are presented in Supporting Information. In the previous studies on molecular rattles, the potential energy profiles for permeation of the oscillating moieties are found to possess double well nature with very low permeation barriers.55, 58-59 Atomistic and DFT-based scans for the permeation of C 20 and C 42 through γ−GY-5 and γ−GY-6, respectively also reveal a double well nature with low barriers (Figures 3 and 6), validating the current findings.

Vibrational analysis of the optimized geometries of the complexes of fullerenes with various r−GYs shows that similar oscillatory motion can be obtained in r−GY-4−C 20 , r−GY-4−C 50 , r−GY-5−C 60 , r−GY-5−C 70 and r−GY-5−C 84 complexes. Among these, the energy scans for the permeation of C 50 and C 84 through r−GY-4 and r−GY-5, respectively show very low permeation barriers (Figures 3 and 4). The potential energy profiles also indicate that r−GY-4 allows facile passage of C 20 (Figure S2). Similar is the case for the motion of C 60 and C 70 through r−GY-5 (Figure 4). Thus, vibrational analysis of the optimized complexes and energy scans reveal oscillatory motion for the passage of C 50 and C 84 through r−GY-4 and r−GY-5, respectively. In


27


contrast, the oscillatory behavior obtained from the optimized geometries of the r−GY-4−C 20 , r−GY-5−C 60 and r−GY-5−C 70 complexes can also be a consequence of distortions in the geometries of the model compounds of r−GYs. To investigate this aspect further, we evaluated potential energy scans for fullerene permeation through the distorted model compounds of r−GYs as observed in their optimized geometries (buckycatcher configurations) and the resultant profiles are shown in Figure S6. For comparison, we have also shown the energy profiles obtained with planar configurations of r−GYs. The potential energy scans indeed show a double well nature. The curves are asymmetric because of the asymmetry in the geometries due to distortion. Therefore, the rattling motion in the r−GY-4−C 20 , r−GY-5−C 60 and r−GY-5−C 70 complexes might not be observable on graphyne sheets since the sheets are planar due to conformational constraints of extensive covalent bonding. However, the annulenic subunits of such r−GYs are potential buckycatchers. Thus far, the analysis of the oscillatory motion is based on the optimized geometries of the fullerene-graphyne complexes. We now estimate the oscillator frequencies from potential energy scans. Assuming that the rattling motion of fullerenes through GYs can be represented by a simple harmonic oscillator near the minima of the double well, we further evaluated the corresponding frequencies of oscillation from the atomistic as well as the DFT potential energy scans.58 For DFT scans, the force constants, k are calculated by fitting the interaction energy near to the bottom of the well with the function 1 𝑉𝑉 = 𝑘𝑘𝑧𝑧 2 + 𝐶𝐶 2

where z denotes the vertical positions of fullerenes from the pore centers of GYs. For the atomistic potentials, the force constants are evaluated by taking the second derivatives of the potential at the minima of the interaction energies:


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𝜕𝜕 2 𝑉𝑉 𝑘𝑘 = � 2 � 𝜕𝜕𝑧𝑧 𝑅𝑅

0

where 𝑅𝑅0 refers to the minima positions in the energy scans. Second derivatives are evaluated

numerically by applying the limit definition of derivatives. Vibrational frequencies, 𝜈𝜈 and time

periods of oscillation, T are then evaluated using 1

𝑘𝑘

𝜇𝜇

𝑣𝑣 = 2𝜋𝜋 �𝜇𝜇 and 𝑇𝑇 = 2𝜋𝜋�𝑘𝑘

where µ is the reduced mass of graphyne and fullerene in a given complex. The frequencies and timescales of oscillation thus evaluated for the rattling motion of the various oscillator complexes are given in Table 2. Therefore, vibrational analysis based on the DFT optimized structures and the potential energy scans suggests that the oscillation frequencies for the out-of-plane motion of fullerenes over graphynes are in the range of 0.1 to 0.5 THz. The timescale of oscillatory motion is of the order of few picoseconds. Indeed, Ozmaian and co-workers had earlier reported a similar timescale for the motion of C 60 over various γ−GYs.80 We believe that the nanomechanical oscillatory response from various fullerene-graphyne complexes reported herein would soon be realized in experiments.


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Table 2. Frequencies and time periods corresponding to the rattling motion of fullerenes through graphynes. Graphyne

γ−GY-5

γ−GY-6

r−GY-4

r−GY-5

Fullerene

C 20

C 42

C 50

C 84

Method

Frequency (THz)†

Time of oscillation (ps)†

DFT optimization

0.30 (0.35)

3.33 (2.86)

DFT PES scan

0.43 (0.39)

2.32 (2.57)

Atomistic PES scan

0.45 (0.48)

2.22 (2.08)

DFT optimization

0.35 (0.46)

2.86 (2.17)

DFT PES scan

0.39 (0.47)

2.56 (2.11)

Atomistic PES scan

0.46 (0.67)

2.17 (1.49)

DFT optimization

0.41 (0.33)

2.44 (3.03)

DFT PES scan

0.50 (0.57)

2.00 (1.76)

Atomistic PES scan

0.49 (0.59)

2.04 (1.69)

DFT optimization

0.28 (0.10)

3.57 (10.48)

DFT PES scan

0.32 (0.30)

3.12 (3.37)

Atomistic PES scan

0.15 (0.12)

6.66 (8.33)

†

Given values correspond to the B97D functional of DFT and the ILJ atomistic potential. The values in parentheses are from the M06-2X functional of DFT and the LJ atomistic potential. PES stands for potential energy surface. CONCLUSIONS In conclusion, the current study explores the possibility of graphynes as potential substrates for nanomechanical devices. By employing a set of fullerenes (C 20 , C 42 , C 50 , C 60 , C 70 and C 84 ) that are carefully chosen based on their comparable sizes to the triangular and rhombic pores of γ−GYs and r−GYs respectively as oscillators, we report the frequencies for the rattling motion of fullerenes through the nanoporous carbon membranes to be in the THz regime. The observed


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oscillator frequencies are one of the highest frequencies reported for carbon-based oscillators in the region of the terahertz gap. The computed permeation barriers indicate that γ−GY-5, γ−GY6 and r−GY-4 could be interesting substrates for achieving the separation of various fullerenes. The permeation barriers from the electronic structure calculations are reproduced well by an atomistic model potential of the improved Lennard-Jones type. The LJ potential of course overestimates the barriers, in agreement with previous findings. The parameters of the ILJ potential reported herein could be useful for force field calculations on fullerene-graphyne complexes. Furthermore, our findings indicate that annulenic model compounds of rhombic forms of GYs are potential buckycatchers. ASSOCIATED CONTENT Supporting Information Supplementary figures and videos showing the nano-oscillatory motion are given in the Supporting Information. AUTHOR INFORMATION Corresponding Author *Email: [email protected] Notes The authors declare no competing financial interest. ACKNOWLEDGMENT The authors thank IISER-TVM for the computational facilities. R.S.S. acknowledges the Kerala State Council for Science, Technology and Environment (KSCSTE) for financial support of this


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AUTHOR BIOGRAPHY

R S Swathi obtained her PhD from Indian Institute of Science, Bangalore under the guidance of Prof. K L Sebastian in the area of theoretical chemistry. Subsequently, she joined as an Assistant Professor in School of Chemistry, IISER-TVM. Her research group employs analytical as well as computational approaches for modeling the interactions of atoms, ions and molecules with carbonbased and metal-based nanostructures. She is a recipient of the Young Scientist Awards from Indian National Science Academy, New Delhi, National Academy of Sciences, Allahabad and Kerala State Council for Science, Technology and Environment. She was an Young Associate of the Indian Academy of Sciences, Bangalore. Swathi has also been awarded the Distinguished Lectureship Award of the Chemical Society of Japan for her work in the area of theoretical chemistry. She is currently on the Editorial Advisory Board of the journal, ACS Applied Materials & Interfaces.


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Permeation of Fullerenes Through Graphynes: Theoretical Design of

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