Research Article

Austin J Nanomed Nanotechnol.2014;2(2): 1013.

# Cluster model expanded to C-nanostructures: Fullerenes, tubes, graphenes and their buds

Francisco Torrens, Gloria Castellano

^{1}Institute for Molecular Science, University of Valencia, Spain

^{2}Faculty of Veterinary and Experimental Sciences, Valencia Catholic University, Spain

***Corresponding author: ** : Francisco Torrens, Institute for Molecular Science, University of Valencia, Building Institutes Paterna, P. O. Box 22085, E-46071 Valencia, Spain

**Received: **January 02, 2014; **Accepted: **January 23, 2014; **Published: **January 30, 2014

## Abstract

The existence of nanographene (GR) and GR-fullerene bud (GR-BUD) in cluster form is discussed in organic solvents. Theories are developed based on columnlet, bundlet and droplet models describing size-distribution functions. The phenomena present a unified explanation in columnlet model, in which free energy of GR involved in cluster comes from its volume, proportional to number of molecules n in cluster. Columnlet model enables describing distribution function of GR stacks by size. From purely geometrical considerations, columnlet (GR/GR-BUD), bundlet [single-wall carbon nanotube (SWNT) (CNT) (NT) and NT-fullerene bud (NT-BUD)] and droplet (fullerene) models predict dissimilar behaviours. Interaction-energy parameters of GR/GR-BUD are taken from C_{60}. An NT-BUD behaviour or further is expected. Solubility decays with temperature result smaller for GR/GR-BUD than SWNT/NT-BUD than C_{60}, in agreement with lesser numbers of units in clusters. Discrepancy between experimental data of the heat of solution of fullerenes, CNT/NT-BUDs and GR/GR-BUDs is ascribed to sharp concentration dependence of the heat of solution. Diffusion coefficient drops with temperature result greater for GR/GR-BUD than SWNT/NT-BUD than C_{60}, corresponding to lesser number of units in clusters. Aggregates (C_{60})_{13}, SWNT/NT-BUD_{7} and GR/GR-BUD_{3} are representative of droplet, bundlet and columnlet models.

**Keywords:** Solubility of graphene-fullerene bud; *Columnlet* cluster model; *Bundlet* cluster model; Droplet cluster model; Nanobud; Fullerene

## Introduction

Nanoparticles interest arose from shape-dependent physical properties of nanoscale materials [1,2]. Single-wall C-nanocones (SWNCs) (CNCs) allowed curved-structures nucleation/growth suggesting pentagon role that, introduced into nanographene (GR) via extraction of a 60° sector, forms a cone leaf. Pentagons in SWNC apex are analogues of single-wall C-nanotube (SWNT) (CNT) tip topology. Classes of positive-curvature CNCs [3-5] /Clar theory [6-10] were analyzed. Ends of SWNTs predicted electronic states related to GR topological defects [11]. Resonant peaks in density of states appeared in SWNTs [12] /multiple-wall CNTs (MWNTs) [13]. SWNCs with discrete opening angles θ = 19°, 39°, 60°, 85° and 113° in pyrolytic C were explained by cone wall model of wrapped GR sheets, where geometrical requirement for seamless connection accounted for semi-discrete character [14]. Total disclinations are multiples of 60° for P≥0 pentagons in SWNC apices. From symmetry/Euler theorem five SWNC types are obtained from continuous GR sheet for P = 1-5: sin(θ/2) = 1 - P/6, leading to flat discs and caped SWNTs matching to P = 0 and 6, respectively; most abundant SWNC (P = 5) is nanohorn (SWNH). Configurations exist for given cone angle depending on pentagon arrangement: *isolated pentagon rule* led to somers more stable than grouped ones [15]; others derived from ab initio calculations [16]. Functionalization of SWNCs with NH_{4}^{+} improved solubility [17] that was achieved by skeleton [18-20]/cone-end [21] functionalization and supramolecular p-p stackings [22-24] with pyrenes/porphyrins. An MNDO computation of BN substitutions in C_{60} showed analogous B_{30}N_{30}[25]. Substitution in C_{diamond} by alternating B/N provided BN-cubic [26]. BN-hexagonal (h) resembles C_{graphite} since fused planar six-membered B_{3}N_{3} rings; however, interlayer B-N exist. BN nanotubes were visualized [27-29]. BN-h was proposed [30]. BN nanocones were observed [31-33] /calculated [34-39]; most abundant ones present 240/300° disclinations. Junction BN/AlN [40] /BC_{2}N nanotubes [41] were computed. Other layered materials are: WS_{2}, etc. [42-46]. Pyrolytic nano B_{x}C_{y}N_{z} shows C/BN domains; compound provides materials useful as nanocomposites (NCs)/semiconductor devices enhanced towards oxidation [47-49]. CNTs are inert/difficult to integrate into NCs/electronics.

C-NanoBuds ™ (NT-BUDs, fullerene-functionalized SWNTs) were synthesized [50]; all are semiconductors [51]. GR sparked potential to be ingredient of devices (e.g., single molecule gas sensors, ballistic transistors, spintronic) [52]. It was called *mother of all graphitic forms* because it can be wrapped into fullerenes, rolled into CNTs and stacked into graphite [53]. It consists of hexagonal arrangement of C-atoms in two-dimensional (2D) honeycomb crystals. It differs from most conventional 3D materials. Basic GR is semimetal/zero-gap semiconductor. Zigzag-edges nature imposes localization of electron density with maximum at the border C-atoms, leading to formation of flat conduction/valence bands near Fermi level. Localization states are spin polarized and in case of ordering electron spin along zigzag edges, GR is established in anti/ferromagnetic phase. The former breaks GR sublatice symmetry that changes its band structure and opens a gap. GR/CNTs show third-order nonlinerity. Electronic properties of semiconductor monolayers are better than the bulk, spawning efforts to create functionalized monolayers of other bonded crystal structures. Higher carrier mobility is achieved via ultra-thin topologies but terminating monolayers with ligands for specific applications, ultra-thin materials are made more sensitive than the bulk for sensors. Solvent selection was analyzed [54-56]. Coronado group examined multifunctional hybrid nanocomposites based on CNTs/chemically modified GR [57-59]. Other 2D materials were analyzed [60-66]. Some GR-fullerene nanobuds (GR-BUDs) [67] are calculated magnetic [68].

In earlier publications SWNT [69-74] /(BC2N/BN-)SWNC [75-78] *bundlet*, GR *columnlet* [79] cluster models, Sc/GR clusters polarizability and GR_{1/2}-cation interactions [80] were presented. A class of phenomena accompanying solution behaviour is analyzed from a unique point of view taking into account cluster formation. Different structures with delocalized electrons in droplet/bundlet/columnlet models are examined. Based on droplet/bundlet models, GR/GR-BUD columnlet is examined. The aim of the present report is to perform a comparative study of fullerene, SWNT/NT-BUD and GR/GR-BUD. The following section describes the computational method. The next two sections present/discuss results. Finally, the last section summarizes our conclusions.

## Computational method

Aggregation changes thermodynamic parameters that displays phase equilibrium and changes solubility. Columnlet is valid when GR sheet number in cluster n << 1. In saturated solution, chemical potentials per sheet for dissolved substance, crystal and clusters match. Cluster free energy depends on its volume, proportional to cluster sheet number n [81]. Our model assumes that clusters present columnlet shape. Gibbs energy G_{n} for n-sized cluster is:

where G_{1} and G_{2} are responsible for contribution to Gibbs energy of molecules placed inside volume and on surface of cluster, respectively, and correspond to formation energies An and -B. The chemical potential μ_{n} of a cluster of size *n* is:

where *T* is absolute temperature and C_{n}, concentration of *n*-sized cluster. With (1) it results:

where G_{1} and G_{2} are expressed in temperature units. In saturated sheet solution, cluster-size distribution function is determined via equilibrium condition linking clusters of specified size with solid phase, which corresponds to equality between chemical potentials for sheets incorporated into clusters of any size and crystal, resulting in the expression for the distribution function in a saturated solution:

where A is equilibrium difference between sheet interaction energies with its surroundings in solid phase and cluster volume, B, similarly on cluster surface and g_{n}, statistical weight of *n*-sized cluster. One neglects g_{n}(*n,T*) dependences in comparison with exponential (4), which normalization:

requires A < 0, and C is solubility in relative units. As n << 1, normalization (5) results:

$C={\overline{g}}_{n}{\displaystyle {\int}_{n=1}^{\infty}n\mathrm{exp}\left(\frac{-An+B}{T}\right)dn}={C}_{0}{\displaystyle {\int}_{n=1}^{\infty}n\mathrm{exp}\left(\frac{-An+B}{T}\right)dn}\text{(6)}$where is g _{n} cluster statistical weight averaged over *n* that makes major contribution to integral (6) and C_{0}, sheet molar fraction. The *A, B* and C_{0} were taken from C_{60} in hexane/toluene/CS_{2} (A = 320K, B = 970K, C_{0} = 5•10^{-8}). Correction takes into account packing efficiencies of C_{60}/sheet:

where ηsph = π/3(2)^{1/2} and η_{cyl} = π/2(3)^{1/2} are spheres (facecentred cubic, FCC) and cylinder packing efficiencies, respectively. Distribution-function dependences on concentration/temperature lead to sheet thermodynamic/kinetic parameters. For unsaturated solution, distribution function is obtained by clusters equilibrium condition. From Eq. (4) distribution function vs. concentration is:

where λ depends on concentration and is determined by normalization condition: $C={C}_{0}{\displaystyle {\int}_{n=1}^{\infty}n{\lambda}^{n}\mathrm{exp}\left(\frac{-An+B}{T}\right)dn}\text{(9)}$

where C_{0} defines absolute concentration: C_{0} = 10^{-4}mol•L^{-1} is found requiring saturation in Eq. (9). The formation energy of *n*-sized cluster results:

Using the distribution function one obtains the heat of solution per mole of dissolved sheet:

$H=\frac{{\displaystyle {\sum}_{n=1}^{\infty}{E}_{n}{f}_{n}\left(C\right)}}{{\displaystyle {\sum}_{n=1}^{\infty}n{f}_{n}\left(C\right)}}{N}_{A}=\frac{{\displaystyle {\sum}_{n=1}^{\infty}n\left(An-B\right){\lambda}^{n}\mathrm{exp}\left[\left(-An+B\right)/T\right]}}{{\displaystyle {\sum}_{n=1}^{\infty}n{\lambda}^{n}\mathrm{exp}\left[\left(-An+B\right)/T\right]}}{N}_{A}\text{(11)}$where *N _{A}* is the Avogadro number and λ depends on solution total concentration by normalization condition (9). The solute diffusion coefficient results:

where D_{0} is the diffusion coefficient of a unit that was taken equal to that of C_{60} in toluene D_{0} = 10^{-9}m^{2}•s^{-1}; Eqs. (1)-(12) are modelled in a home-built program available from authors. A *droplet* cluster model of C_{60} is proposed following modified Eqs. (1')-(12'):

A *bundlet* cluster model of SWNT and NT-BUD is proposed following customized Eqs. (1'')-(12''):

## Calculation results

The equilibrium difference between the Gibbs free energies of
interaction of an SWNT with its surroundings in solid phase, and
cluster volume or on surface (cf. Figure. 1) shows that on going from
C_{60} (droplet) to SWNT (*bundlet*) the minimum is less marked (68%
of C_{60}), causing a lesser number of units in SWNT (n_{min} ≈ 2) than in
C_{60} clusters (≈8) and a longer abscissa in C_{60} (n_{abs} ≈ 28) than in SWNT
(≈9). Thinner NT-BUD bundles (bundlet) result less stable while
wider ones appear more stable than SWNT packages. The minimum
of NT-BUD appears 55% of C_{60}. The minimum of GR (columnlet,
67%) is similar to SWNT but with fewer units (≈1) and shorter
abscissa (n_{abs} ≈ 3). Shorter GR-BUD stackings (columnlet) result less
stable while longer ones appear more stable than GR columns. The
minimum of GR-BUD (49% of C_{60}) is alike NT-BUD.

**Figure 1:**C

_{60}/SWNT/NT-BUD/GR/GR-BUD interaction energy with surroundings in cluster volume/surface.

Figure 1:C_{60}/SWNT/NT-BUD/GR/GR-BUD interaction energy with surroundings in cluster volume/surface.

The solubility of SWNT/NT-BUD and GR/GR-BUD vs.temperature (*cf.* Figure. 2) shows a solubility decay because of cluster formation. At T ≈ 260K, C_{60}-FCC presents a phase transition to simple cubic (SC). The solubility drops with temperature result less marked for SWNT/NT-BUD and even GR/GR-BUD, in agreement with lesser numbers of units in clusters (Figure. 1). At *T* = 260K from C_{60} (droplet) to SWNT, NT-BUD (bundlet), GR and GR-BUD (columnlet) the solubility (Figure. 2) decreases to 3%, 2%, 0.3% and 0.1% of C_{60}, respectively.

**Figure 2:**Temperature dependence of solubility of C

_{60}/SWNT/NT-BUD/GR/ GR-BUD.

Figure 2:Temperature dependence of solubility of C_{60}/SWNT/NT-BUD/GR/ GR-BUD.

The cluster distribution function by size in CS_{2}, calculated for saturation concentration at solvent temperature *T* = 298.15K (*cf.* Figure. 3), shows that on going from C_{60} (droplet) to SWNT (bundlet) the maximum aggregate size decays from n_{max} &asymp 8 to &asymp2 and spreading is narrowed, in agreement with lesser number of units in clusters (Figure. 1). The dispersal of NT-BUDs (bundlet) is somewhat enlarged to wider bundles with respect to SWNT. The dissemination of GRs (columnlet) is strongly narrowed in concordance with the fewest units (n_{max} &asymp 1). The scattering of GR-BUDs (columnlet) is rather increased to longer stacks with regard to GR.

**Figure 3:**Cluster distribution saturated in CS

_{2}at 298.15K of C

_{60}/SWNT/NTBUD/ GR/GR-BUD.

Figure 3:Cluster distribution saturated in CS_{2}at 298.15K of C_{60}/SWNT/NTBUD/ GR/GR-BUD.

The concentration dependence for the heat of solution in toluene, benzene and CS_{2} calculated at solvent temperature *T* = 298.15K (*cf.* Figure. 4) shows that for C_{60} (droplet), on going from *C* < 0.1% of saturated (<*n*> &asymp 1) to *C* = 15% (<*n*> &asymp 7) the heat of solution decays by 73%. In turn, for SWNT (bundlet) the heat of solution rises by 54% in the same range in agreement with a lesser number of units in clusters (Figs. 1 and 3). Moreover, in NT-BUD (bundlet), GR and GR-BUD (columnlet) the heat of solution increases by 98%, 392% and 680%, respectively. The discrepancy between experimental data for the heat of solution of fullerenes, CNT/NT-BUDs and GR/GRBUDs s ascribed to the sharp concentration dependence for the heat of solution.

**Figure 4:**Heat of solution vs. concentration of C

_{60}/SWNT/NT-BUD/GR/GRBUD in toluene/benzene/CS

_{2}at 298.15K.

Figure 4:Heat of solution vs. concentration of C_{60}/SWNT/NT-BUD/GR/GRBUD in toluene/benzene/CS_{2}at 298.15K.

In the temperature dependence for the heat of solution in toluene, benzene and CS_{2} calculated for saturation concentration (*cf.* Figure. 5) C_{60} results are plotted for *T* > 260K after FCC/SC transition. For C_{60} (droplet) on going from *T* = 260K to *T* = 400K the heat of solution rises 2.7kJ·mol^{-1}. The heat of solution of SWNT (bundlet) increases 10.4kJ·mol-1 in the same range and becomes endergonic. For NTBUD (bundlet) the heat of solution augments 11.3kJ·mol^{-1} similarly endergonic. Moreover, GR and GR-BUD (columnlet) heats of solution enlarge 82 and 76kJ·mol^{-1}, respectively, likewise endergonic.

**Figure 5:**Heat of solution

*vs.*temperature of C

_{60}/SWNT/NT-BUD/GR/GRBUD in toluene/benzene/CS

_{2}for saturation.

Figure 5:Heat of solutionvs.temperature of C_{60}/SWNT/NT-BUD/GR/GRBUD in toluene/benzene/CS_{2}for saturation.

The diffusion coefficient vs. concentration in toluene at *T* = 298.15K (*cf*. Figure. 6) shows that cluster formation, close to saturation, decreases the diffusion coefficients by 56%, 69%, 73%, 95% and 97% for C_{60} (droplet), SWNT, NT-BUD (bundlet), GR and GR-BUD (columnlet), respectively, compared with (C_{60})_{1}. Diffusion coefficients of SWNT, NT-BUD, GR and GR-BUD decay by 29%, 37%, 88% and 93% contrasted with (C_{60})_{n} in agreement with lesser number of units in clusters.

**Figure 6:**Diffusion coefficient

*vs.*concentration of C

_{60}/SWNT/NT-BUD/GR/GR-BUD in toluene at 298.15K.

Figure 6:Diffusion coefficientvs.concentration of C_{60}/SWNT/NT-BUD/GR/GR-BUD in toluene at 298.15K.

## Discussion

For a long time and, because of the fact that some theoretical works established that the long-range crystalline order was impossible in a strictly 2D material, GR was considered as only a reference structure that is found in the origin of other type of stable structures with which one is more familiarized, e.g., CNTs, fullerenes or simply C_{graphite} itself [82]. Not by the fact that it was considered as a hypothetical material, it left of being in the mind of some researchers. The first and practically unique intentions of obtaining GR were based on chemical exfoliation where C_{graphite}, adequately intercalated, could present considerably separated GR sheets. The compound so formed resulted, however, of scarce interest.

From purely geometrical considerations the columnlet (GR/GRBUD), bundlet (SWNT/NT-BUD) and droplet (fullerene) cluster models predict different behaviours. (1) Spheres (droplet) pack in three dimensions (3D), while cylinders (bundlet) do equivalently to circles in 2D and sheets (columnlet) stack against 1D. The C_{60} clusters can be greater than SWNT/NT-BUD bundles than GR/GR-BUD stacks, because of C_{60} > SWNT/NT-BUD > GR/GR-BUD additional curvatures and dimensions. (2) The co-ordination numbers are: CN_{spheres} = 12, CN_{cylinders} = 6 and CN_{sheets} = 2. As surface affects energy [Eqs. (10), (10') and (10'')] on changing the number of atoms the clusters should present discontinuities in areas and stabilities, especially about the corresponding co-ordination numbers. Consequently a discontinuity is expected between closed (C_{60})_{13} and open (C_{60})_{14} (droplet); the same happens between stopped up SWNT/NT-BUD_{7} and ajar SWNT/NT-BUD_{8} (bundlet), and connecting congested GR/GR-BUD_{3} and unlocked GR/GR-BUD_{4} (columnlet), which different sizes are in concordance with C_{60} > SWNT/NT-BUD > GR/GR-BUD extra curvatures and dimensions. (3) Smaller clusters are of the least consideration in columnlet, bundlet and droplet models because all three approaches are valid when n » 1 unit. (4) It has not escaped our notice that aggregates near (C_{60})_{13}, SWNT/NT-BUD7 and GR/GRBUD_{3} could be representative of the droplet, bundlet and columnlet models, respectively, with different sizes in agreement with C_{60} > SWNT/NT-BUD > GR/GR-BUD added curvatures and dimensions. However, all our calculations were performed with the whole size distribution. (5) Because spherical (2D-curved) C_{60} is more reactive than cylindrical (1D-bended) SWNTs than sheet (straight) GR, a greater dispersion in the results of C_{60} cluster size (n ≠ 13) is expected, with regard to lesser reactive SWNTs that are waited more uniform in SWNT_{n} (n ~ 7) and least reactive GRs that are anticipated more consistent in GR_{n} (n ≈ 3). In addition C_{60} is more soluble than SWNTs than GR in organic solvents. (6) The GR is strongly influenced by the materials it comes into contact with, whether solid, liquid or gas. (7) Strictly 2D (single layer) GR is rarely pristine since it contains a number of impurities, C or foreign ad-atoms, vacancies, etc., and the finite crystalline size and support-related effects, e.g., conformality, wrinkles, etc. For a group of potential applications the presence of those defects provides interesting properties related to the spin phenomena, for advanced nanoelectronics and to make possible the attachment of specific molecules. (8) In the investigation at the atomic scale of the impact that atomic defects have on the structural, electronic and magnetic properties of GR layers grown on different materials the pure bidimensionality of GR gives to these defects a critical role. (9) The study of the coupling of GR with its local environment is absolutely critical to be able to integrate it in tomorrow's electronic devices. (10) The atomically thin single-crystal membranes offer ample scope for fundamental research and new technologies, whereas the observed corrugations in the third dimension may provide subtle reasons for the stability of 2D crystals.

The microscopic dimensional considerations above present macroscopic effects. In a linear chain, for a linear model the square fluctuation is proportional to the distance because the amplitude of the fluctuation rises with the squared remoteness. It is because of thiscircumstance that one should attribute the origin of the qualitative distinction that exists between the solid and liquid states. In a liquid,the coherence is conserved at only long distances while naturally the neighbouring atoms preferably form the arrangements of minimalenergy. In the 1D case, the difference between solid and liquid will be only quantitative and instead of a well-determined fusion point there will be a continuum passing.

## Conclusions

From the discussion of the present results the following conclusions can be drawn.

- The nanoworld structural diversity is a consequence of its quantum nature. Several criteria reduced the analysis to a manageable number of magnitudes,
*viz.*closeness, curvature, dimension and efficiency. Our non-computationally intensive approach,*i.e.*, object clustering plus propertyprediction, assessed reliability. Type, dimensions and producer selection of fullerenes, nanotubes and graphenes must be chosen to ensure the transfer of methods developed between laboratories. Our interaction energy parameters for nanobuds are taken from C_{60}. For nanotube bud an C_{60}/tube intermediate behaviour was expected. However, nanotubebud properties result closer to tubes. Thinner nanotube-bud bundles appear less stable but wider ones are more stable than tube packages. - Association energy parameters of nanographene are obtained from C
_{60}. A nanotube-bud behaviour or further was expected. For C_{graphene}nanobud an C_{60}/C_{graphene}in-between behaviour as anticipated. Notwithstanding, nanobud features appear closer to C_{graphene}. Shorter nanobud stacks result less stable but longer ones appear more stable than C_{graphene}columns. The solubility decays with temperature result smaller for C_{graphene}/bud than nanotube/bud than C_{60}, in agreement with lesser numbers of units in clusters. The discrepancy between experimental data for the heat of solution of fullerenes, nanotubes, graphenes and their buds is ascribed to the sharp concentration dependence for the heat of solution. The diffusion coefficient drops with temperature result greater for C_{graphene}/bud than nanotube/bud than C_{60}, corresponding to lesser number of units in aggregates. - Some systems are dominated by the isolated pentagon rule, some others are not. Further work will explore similar nanostructures nature: generalization to systems more complex;
*e.g.*, a way of bypassing weak homonuclear bonding exists in closed B_{x}N_{x}(involving replacement of 5-membered by 4-ring B_{2}N_{2}with heteroatom alternation), BN/AlN tubes/heterojunctions, silicene, germanene and carbine. The C-nanostructures are more controllable while heterostructures present richer behaviour, especially for transtition-metal compounds, showing lubricant and electronic uses.

## Acknowledgments

The authors want to dedicate this manuscript to Dr. Luis Serrano- Andrés, who was greatly interested in this research and would have loved to see its conclusion.

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