Asymptotic Analysis of Coagulation–Fragmentation Equations of Carbon Nanotube Clusters
- Francisco Torrens^{1}Email author and
- Gloria Castellano^{2}
Received: 17 May 2007
Accepted: 6 June 2007
Published: 27 June 2007
Abstract
The possibility of the existence of single-wall carbon nanotubes (SWNTs) in organic solvents in the form of clusters is discussed. A theory is developed based on abundlet model for clusters describing the distribution function of clusters by size. The phenomena have a unified explanation in the framework of the bundlet model of a cluster, in accordance with which the free energy of an SWNT involved in a cluster is combined from two components: a volume one, proportional to the number of moleculesn in a cluster, and a surface one, proportional ton ^{1/2}. During the latter stage of the fusion process, the dynamics were governed mainly by the displacement of the volume of liquid around the fusion site between the fused clusters. The same order of magnitude for the average cluster-fusion velocity is deduced if the fusion process starts with several fusion sites. Based on a simple kinetic model and starting from the initial state of pure monomers, micellization of rod-like aggregates at high critical micelle concentration occurs in three separated stages. A convenient relation is obtained for <n> at transient stage. At equilibrium, another relation determines dimensionless binding energy α. A relation with surface dilatational viscosity is obtained.
Keywords
Introduction
Among the unusual properties of fullerene solutions should be mentioned the nonmonotonic temperature dependence of solubility of fullerenes [1] and the nonlinear concentration dependence of the third-order nonlinear optical susceptibility [2]. The solvatochromic effect [3, 4] is exhibited in a sharp alteration in the spectrum of the optical absorption of C_{70}, dissolved in a mixture of organic solvents, of a result of a slight change in the solvent content. The peculiarities in the behaviour of fullerenes in solutions are attributable to the phenomenon, predicted theoretically and revealed in experiments [5, 6], of the formation of clusters. It was examined the decrease in pyridine-soluble material observed after soaking coals in solvents, which is due to an increase in cross-linking associated with the formation of ionic domains or clusters, similar to those observed in ionomers [7]. It is not possible to extract C_{60-70} from a solution in toluene to water and from a dispersion in water to toluene [8]. Upon contact with water, under a variety of conditions, C_{60} spontaneously forms a stable aggregate with nanoscale dimensions [9]. Water itself might form a donor–acceptor complex with C_{60} leading to a weakly charged colloid [10–12]. C_{60}, dissolved in water via complexation with cyclodextrin_{8}, was extracted to toluene [13, 14]. In C_{60} incorporated into artificial lipid membranes, it was not extracted to toluene, but the extraction became possible once the vesicle was destructed by adding KCl [15]. Addition of KCl was also required to extract poly(vinylpyrrolidone)-solubilized C_{60–70} to toluene [16].
An assembly of randomly packed spheres can represent certain features of the geometry of simple liquids [17]. Models of randomly packed hard spheres exhibited some features of the properties of simple liquids [18]. Using a new series acceleration method, the virial expansion for the pressure of hard discs and hard spheres was found to be a monotonically increasing function of the number density ρ and diverged at the density of closest packing with the critical exponent γ = 1 [19]. The general problem of open packing of spheres is difficult, since the answers depend on the assumptions about the local connectivity. At the purely mathematical level the only thing that counts is that there is continuity from one sphere to the next. From the engineering viewpoint of the stability of a pile of dust particles or a rime of ice crystals, each particle must be in contact with several other particles but not with crystallographic regularity. An open packing of spheres must be regular, at least in two dimensions, and preference is given to arrangements that are related to (4;2)-connected three-dimensional (3D) nets. The problem of stability is difficult because it involves chemical bonding. From the viewpoint of simple ionic bonding, any open packing, in general, is not electrostatically stable with respect to a more compact one. Material encapsulated during synthesis can promote stability of open frameworks, but removal of the encapsulated material should result in collapse of the framework as the minimum of electrostatic energy is favoured. From the viewpoint of ionic plus covalent bonding, open structures can persist metastably if bonds remain unbroken. The safest approach, in considering nets with extremely low density, is to look first at all theoretical possibilities, irrespective of chemical implications, and then to look at the complex topochemical possibilities. Low-density sphere packings were invented for a continuous, locally symmetric arrangement, in which each line joining the points of contact of successive spheres passes through the centres of the spheres. The most open packing has 94.4% void space. The line-centre restriction is critical to mechanical stability of a sphere packing, but is not necessary for chemical stability. Replacement of one sphere by a triangle of three spheres is an important technique for creating new packings. Relaxation of the stability criterion allows invention of sphere packings of even lower density, including ones with 95.5 and 95.8% void space. In earlier publications the bundlet model for clusters of SWNTs was presented [20–22]. The aim of the present report is to perform a comparative study of the properties of fullerenes (droplet model) and SWNTs (bundlet model). A wide class of phenomena accompanying the behaviour of SWNTs in solutions is analyzed from a unique point of view, taking into account the tendency of SWNTs to cluster formation in solutions. Based on the droplet model of C_{60-70}, the bundlet model of SWNTs and droplet model of single-wall carbon nanoholes (SWNHs) are proposed.
Computational Method
Solubility of Single-wall Carbon Nanotubes
where λ is determined by the total concentration of formed solution via normalization condition (Eq. 9).
Transfer Phenomena in Single-wall Carbon Nanotube Solutions
Equation 26 differs from the estimate by a factor (−An + Bn ^{1/2})/T ≫ 1. Under conditions favourable to cluster formation the thermal diffusion mechanism, associated with SWNT aggregation in solution, proves much more efficient as compared with the more general mechanism.
Fractal Structures in Single-wall Carbon Nanotube Solutions
Equation 46 calculated for D = 2.08, α = 2, and γ_{0} = 10^{−7} showed that the dependence agrees quite well with experiment. The calculated dependence almost coincides with the calculation result within the simplified model with D = 2.
Real-space Imaging of Nucleation and Growth in Colloidal Crystallization
where R is the radius, γ, the surface tension, Δμ, the difference between the liquid and solid chemical potentials, and N, the number density of particles in the crystallite [31]. The size of the critical nucleus is R _{ c } = 2γ/(Δ μ N), corresponding to the maximum of ΔG (Eq. 47). The radius of gyration R _{ g } of crystallites was related to the number of particles n within each crystallite as with the fractal dimension D = 2.35 ± 0.15 for all values of packing volume fraction η; the fractal behaviour presumably reflects the roughness of their surfaces. The interfacial tension between the crystal and fluid phases is a key parameter that controls the nucleation process, yet γ is difficult to calculate and to measure experimentally, but one can directly measure γ by examining the statistics of the smallest nuclei. For R ≪ R _{ c } the surface term in Eq. 47 dominates the free energy of the crystallites, and one expects the number of crystallites to be n _{cry}(A) ∝ exp[− A γ/(kT)] where A is the surface area, which one approximates by an ellipsoid. The (r _{0} is the particle radius = 1.26 μm) and may decrease slightly with increasing η values. The γ value is in reasonable agreement with density functional calculations for hard spheres and Lennard-Jones systems. The measurement of a low value of γ is consistent with the observed rough surfaces of the crystallites; this may reflect the effects of the softer potential due to the weak charges of our particles. Approximating the critical nucleus as an ellipsoid with , one obtains A _{ c } = 880 μm^{2} , and .
Dimensional Analysis for the Early and Later Stages of Fusion-site Expansion
The two stages of cluster fusion, a fast early and a slower later stage, were detected also in vesicle fusion. During the former the fusion site opened rapidly: the expansion velocity of the rim of the site was . The fusion pore opens up to micrometres within a hundred microseconds. One would relate this time τ_{early} to fast relaxation of membrane tension. The tension of the clusters achieved before fusion was in the stretching regime of the membrane. The τ_{early} should be primarily governed by the relaxation of membrane stretching. Viscous dissipation can be associated with two contributions: in-plane dilatational shear as the fusion site expands and intermonolayer slip among the leaflets of the multilayer membrane in the fusion-site zone. The latter is negligible for fusion-site diameter L larger than half a micrometre. The τ_{early} ∼ η_{ s }/σ, where η_{ s } is the surface dilatational viscosity coefficient of the membrane with units [bulk viscosity coeffcient] × [membrane thickness] [32]. For membrane tensions close to the tension of rupture one obtains τ_{early} ∼ η_{ s }/σ ∼ 100μs, in agreement with experiment . During the later stage of the fusion process the site expansion velocity slowed down by two orders of magnitude. The dynamics was governed by the displacement of volume ΔV of fluid around the fusion site between the fused clusters. The restoring force was related to the bending elasticity of the membrane. Decay time τ_{late} ∼ ηΔV/κ where η is the bulk viscosity coefficient of the solvent, ΔV ∼ R ^{3}, and κ, the bending elasticity modulus of the membrane . For a cluster size of R = 20 μm one obtains τ_{late} ∼ 100 s, which is the time scale measured for complete fusion-site opening. When two clusters fuse at several contact points and form some fusion sites, the coalescence of these fusion sites can lead to small, contact-zone clusters. Consider three fusion sites, which expand and touch each other in such a way that they enclose a roughly triangular segment of the contact zone. If the three sites are circular and have grown up to a diameter L _{1}, the enclosed contact-zone segment will form a contact-zone cluster of radius , as follows from geometric considerations. The coalescence of these several sites can lead to small contact-zone clusters encapsulating solvent. One expects that these clusters be interconnected by thin tethers, because pinching the membrane off completely would require additional energy. The fusion-induced cluster formation resembles the membrane processes during cell division, when one looks at them in a time-reversed manner. During the initial stages of the division process, the cell accumulates membrane in the form of small vesicles that define the division plane and transform into two adjacent cell membranes. From dimensional analysis it is found an appropriate time scale τ for the later stage of the expansion of the fusion site. The driving force for this expansion is provided by membrane tension σ, whereas the hydrodynamic-Stokes friction is governed by solution viscosity coefficient η. The system is characterized by two well-separated length scales: the membrane thickness l and a typical cluster size R. It is chosen R = (R _{1} + R _{2})/2 where R _{1} and R _{2} are the radii of the two clusters before they were brought into contact. The only time scale, which one can obtain from a combination of the four variables σ, η, l and R, is given by τ = (ηR/σ)f(l/R) with the dimensionless function f(l/R). Because l ≪ R one can replace f(l/R) by f(0) and ignore corrections or order (l/R). Let v (in m s^{−1}) be the average site expansion velocity for a single site. The same order of magnitude for the average expansion velocity is deduced if one assumes that the fusion process startes with N > 1 fusion sites. The fusion sites would grow until they start to touch and coalesce. They would then create a coalesced site of diameter L if each site had grown up to L/N ^{1/2}, which implies an average expansion velocity , still of the same order of magnitude even if N were as large as 10.
Description of the Asymptotic Coagulation–Fragmentation Equations
where α kT is the monomer–monomer binding energy [33]. As it is considered the Becker–Döring model, it is taken into account reactions only between monomers and other clusters. The expression for the binding energy is suitable for aggregates of certain kinds of lipids, when these form rod-like clusters. The molecules of these lipids typically have a hydrophilic head and a hydrophobic tail so, in aqueous solution, they spontaneously arrange themselves so that tails are away from the surrounding water, and heads, in contact with it. Depending on the shape of the particular molecule, they can form spherical aggregates with tails pointing inwards and heads pointing outwards, or form lipid bilayers, where lipid molecules form a double layer with heads on the external surface and tails on the inside. Clusters formed by lipids in aqueous solution are called micelles, and the process by which they form is called micellization. To determine the time scale, one needs a measure of the kinetic coefficient of the d decay reaction, which was set equal to one. A convenient relation could be an equation, which in dimensional units is . In case the self-similar size distribution is not reached during the intermediate phase, another way to determine d is to study the equilibration era and compare the experimental size distribution with the numerical solution of the model. By combining τ_{early} ∼ η_{ s }/σ with it is obtained . The original software used in the investigation is available from the authors.
Calculation Results
Packing-efficiency correction factors and numerical values of the parameters determining the interaction energy^{a}
Molecule | Packing efficiency | η-correction factor | A′ (K) | B′ (K) | σ′ (K) |
---|---|---|---|---|---|
C_{60}-face-centred cubic^{b}, SWNH^{c}, SWNT^{d} | 0.74048 | 1.0 | 320 | 970 | 647 |
SWNH^{c} η-correction | 0.82565 | 1.11501 | 357 | 1082 | 721 |
SWNT^{d} η-correction | 0.90690 | 1.22474 | 392 | 1188 | 792 |
Conclusions
- 1.
For a cluster nature of SWNT solubility to be completely established direct experimental exploration is necessary. The measurements of infrared absorption spectra of an SWNT solution, involving concentrations at various temperatures and a constant optical path length, can be conceived; the dependence will indicate the presence of clusters in solution. According to Raoult’s law, the saturation vapour pressure of a solvent above a solution differs from that above a pure solvent by a value proportional to solute-particle concentration. The solvent vapour flow will enable the dependence of solute particle concentration. If the dependence is nonlinear it will indicate the existence of clusters in solution.
- 2.
The C_{60}aggregation in benzene is reversible and (C_{60})_{ n }exhibits a loose structure. The experimental results confirm the variation of cluster-size distribution. The structure of (C_{60})_{ n }changes from compact spherelike to larger and looser clusters. The formation of fullerene-SWNT clusters is rapid (∼10^{−6} s), while their growth process, slow (∼ 10^{6} s). The key for the explanation of process nature is found, what makes thinking that the clustersheath is filled with numerous pores. The establishment of themembranous character of growth process in clusters allows explaining the high experimental data dispersion.
- 3.
During the latter stage of the fusion process the site expansion velocity slowed down by two orders of magnitude. The dynamics were governed mainly by the displacement of the volume of liquid around the fusion site between the fused clusters, which is confirmed by dimensional analysis. The same order of magnitude for the average cluster-fusion velocity is deduced if the fusion process starts with several fusion sites, even if there were as much as 10 sites.
- 4.
Based on a simple kinetic model and starting from the initial state of pure monomers, micellization of rod-like aggregates at high critical micelle concentration occurs in three separated stages. In the first era many small clusters are produced while the number of monomers decreases sharply. During the second era aggregates are increasing steadily in size, and their distribution approaches a self-similar solution of the diffusion equation. Before the continuum limit can be realized the average size of the nuclei becomes comparable to its equilibrium value, and a simple mean-field Fokker–Planck equation describes the final era until the equilibrium distribution is reached. A continuum size distribution does not describe micellization until the third era has started; during the first two eras the effects of discreteness dominate the dynamics. To validate the theory by an experiment, it would be important to measure the average cluster size as a function of time. To determine the time scale one needs a measure of kinetic coefficientd of decay reaction. A convenient relation could be an equation, which in dimensional units is . In case the self-similar size distribution is not reached during the intermediate phase, another way to determined is to study the equilibration era and compare the experimentally obtained size distribution with the numerical solution of the model. At equilibrium , which determines dimensionless binding energy α.
- 5.
where , the surface tension, and v = V/M, the molecular volume .
Declarations
Acknowledgements
The authors acknowledge financial support from the Spanish MEC DGCyT (Project No. CTQ2004-07768-C02-01/BQU), EU (Program FEDER) and Generalitat Valenciana (DGEUI INF01-051, INFRA03-047, OCYT GRUPOS03-173 and COMP06-147).
Authors’ Affiliations
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