Francisco Torrens; Gloria Castellano

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

¹Institute for Molecular Science, University of Valencia, Spain

²Faculty 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 C₆₀. An NT-BUD behaviour or further is expected. Solubility decays with temperature result smaller for GR/GR-BUD than SWNT/NT-BUD than C₆₀, 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 C₆₀, corresponding to lesser number of units in clusters. Aggregates (C₆₀)₁₃, SWNT/NT-BUD₇ and GR/GR-BUD₃ 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 NH₄⁺ 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 C₆₀ showed analogous B₃₀N₃₀[25]. Substitution in C_diamond by alternating B/N provided BN-cubic [26]. BN-hexagonal (h) resembles C_graphite since fused planar six-membered B₃N₃ 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] /BC₂N nanotubes [41] were computed. Other layered materials are: WS₂, etc. [42-46]. Pyrolytic nano B_xC_yN_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:

G_{n} = G_{1} n - G_{2} (1)

where G₁ and G₂ 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:

μ_{n} = G_{n} + T \ln C_{n} (2)

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

μ_{n} = G_{1} n - G_{2} + T \ln C_{n} (3)

where G₁ and G₂ 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:

f (n) = g_{n} \exp (\frac{- A n + B}{T}) (4)

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:

\sum_{n = 1}^{\infty} f (n) n = C (5)

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

C = {\bar{g}}_{n} \int_{n = 1}^{\infty} n \exp (\frac{- A n + B}{T}) d n = C_{0} \int_{n = 1}^{\infty} n \exp (\frac{- A n + B}{T}) d n (6)

where is g _n cluster statistical weight averaged over n that makes major contribution to integral (6) and C₀, sheet molar fraction. The A, B and C₀ were taken from C₆₀ in hexane/toluene/CS₂ (A = 320K, B = 970K, C₀ = 5•10^-8). Correction takes into account packing efficiencies of C₆₀/sheet:

A' = \frac{A}{η_{sph}}

and

B' = \frac{B}{η_{sph}}

(sheet)

A'' = \frac{η_{cyl}}{η_{sph}} A

and

B'' = \frac{η_{cyl}}{η_{sph}} B (7)

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:

f_{n} (C) = λ^{n} \exp (\frac{- A n + B}{T}) (8)

where λ depends on concentration and is determined by normalization condition: $C = C_{0} \int_{n = 1}^{\infty} n λ^{n} \exp (\frac{- A n + B}{T}) d n (9)$

where C₀ defines absolute concentration: C₀ = 10^-4mol•L^-1 is found requiring saturation in Eq. (9). The formation energy of n-sized cluster results:

E_{n} = n (A n - B) (10)

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

H = \frac{\sum_{n = 1}^{\infty} E_{n} f_{n} (C)}{\sum_{n = 1}^{\infty} n f_{n} (C)} N_{A} = \frac{\sum_{n = 1}^{\infty} n (A n - B) λ^{n} \exp [(- A n + B) / T]}{\sum_{n = 1}^{\infty} n λ^{n} \exp [(- A n + B) / T]} N_{A} (11)

where N_A is the Avogadro number and λ depends on solution total concentration by normalization condition (9). The solute diffusion coefficient results:

D = D_{0} \frac{\int_{n = 1}^{\infty} n λ^{n - 1} \exp [(- A n + B) / T] d n}{\int_{n = 1}^{\infty} n^{2} λ^{n - 1} \exp [(- A n + B) / T] d n} (12)

where D₀ is the diffusion coefficient of a unit that was taken equal to that of C₆₀ in toluene D₀ = 10^-9m²•s^-1; Eqs. (1)-(12) are modelled in a home-built program available from authors. A droplet cluster model of C₆₀ is proposed following modified Eqs. (1')-(12'):

G_{n} = G_{1} n - G_{2} n^{2 / 3} (1)

μ_{n} = G_{1} n - G_{2} n^{2 / 3} + T \ln C_{n} (3)

f (n) = g_{n} \exp (\frac{- A n + B n^{2 / 3}}{T}) (4)

C = {\bar{g}}_{n} \int_{n = 1}^{\infty} n \exp (\frac{- A n + B n^{2 / 3}}{T}) d n = C_{0} \int_{n = 1}^{\infty} n \exp (\frac{- A n + B n^{2 / 3}}{T}) d n (6)

f_{n} (C) = λ^{n} \exp (\frac{- A n + B n^{2 / 3}}{T}) (8)

f_{n} (C) = λ^{n} \exp (\frac{- A n + B n^{2 / 3}}{T}) (8)

C = C_{0} \int_{n = 1}^{\infty} n λ^{n} \exp (\frac{- A n + B n^{2 / 3}}{T}) d n (9)

E_{n} = n (A n - B n^{2 / 3}) (10)

H = \frac{\sum_{n = 1}^{\infty} E_{n} f_{n} (C)}{\sum_{n = 1}^{\infty} n f_{n} (C)} N_{A} = \frac{\sum_{n = 1}^{\infty} n (A n - B n^{2 / 3}) λ^{n} \exp [(- A n + B n^{2 / 3}) / T]}{\sum_{n = 1}^{\infty} n λ^{n} \exp [(- A n + B n^{2 / 3}) / T]} N_{A} (11)

D = D_{0} \frac{\int_{n = 1}^{\infty} n^{5 / 3} λ^{n - 1} \exp [(- A n + B n^{2 / 3}) / T] d n}{\int_{n = 1}^{\infty} n^{2} λ^{n - 1} \exp [(- A n + B n^{2 / 3}) / T] d n} (12)

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

G_{n} = G_{1} n - G_{2} n^{1 / 2} (1)

μ_{n} = G_{1} n - G_{2} n^{1 / 2} + T \ln C_{n} (3)

f (n) = g_{n} \exp (\frac{- A n + B n^{1 / 2}}{T}) (4)

C = {\bar{g}}_{n} \int_{n = 1}^{\infty} n \exp (\frac{- A n + B n^{1 / 2}}{T}) d n = C_{0} \int_{n = 1}^{\infty} n \exp (\frac{- A n + B n^{1 / 2}}{T}) d n (6)

f_{n} (C) = λ^{n} \exp (\frac{- A n + B n^{1 / 2}}{T}) (8)

C = C_{0} \int_{n = 1}^{\infty} n λ^{n} \exp (\frac{- A n + B n^{1 / 2}}{T}) d n (9)

E_{n} = n (A n - B n^{1 / 2}) (10)

H = \frac{\sum_{n = 1}^{\infty} E_{n} f_{n} (C)}{\sum_{n = 1}^{\infty} n f_{n} (C)} N_{A} = \frac{\sum_{n = 1}^{\infty} n (A n - B n^{1 / 2}) λ^{n} \exp [(- A n + B n^{1 / 2}) / T]}{\sum_{n = 1}^{\infty} n λ^{n} \exp [(- A n + B n^{1 / 2}) / T]} N_{A} (11)

D = D_{0} \frac{\int_{n = 1}^{\infty} n^{3 / 2} λ^{n - 1} \exp [(- A n + B n^{1 / 2}) / T] d n}{\int_{n = 1}^{\infty} n^{2} λ^{n - 1} \exp [(- A n + B n^{1 / 2}) / T] d n} (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 C₆₀ (droplet) to SWNT (bundlet) the minimum is less marked (68% of C₆₀), causing a lesser number of units in SWNT (n_min ≈ 2) than in C₆₀ clusters (≈8) and a longer abscissa in C₆₀ (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 C₆₀. 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 C₆₀) is alike NT-BUD.

Figure 1: C₆₀/SWNT/NT-BUD/GR/GR-BUD interaction energy with surroundings in cluster volume/surface.



Figure 1:  C₆₀/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, C₆₀-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 C₆₀ (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 C₆₀, respectively.

Figure 2: Temperature dependence of solubility of C₆₀/SWNT/NT-BUD/GR/ GR-BUD.



Figure 2:  Temperature dependence of solubility of C₆₀/SWNT/NT-BUD/GR/
GR-BUD.

The cluster distribution function by size in CS₂, calculated for saturation concentration at solvent temperature T = 298.15K (cf. Figure. 3), shows that on going from C₆₀ (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 CS₂ at 298.15K of C₆₀/SWNT/NTBUD/ GR/GR-BUD.



Figure 3:  Cluster distribution saturated in CS₂ at 298.15K of C₆₀/SWNT/NTBUD/ GR/GR-BUD.

The concentration dependence for the heat of solution in toluene, benzene and CS₂ calculated at solvent temperature T = 298.15K (cf. Figure. 4) shows that for C₆₀ (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.

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

Abstract

Introduction

Computational method

Calculation results

Discussion

Conclusions

Acknowledgments

References