Can single-walled carbon nanotube diameter be defined by catalyst particle diameter?

Abstract

The need of designing and controlling single-walled carbon nanotube (SWCNT) properties is a challenge in a growing nanomaterials-related industry. Recently, great progress has been made experimentally to selectively control SWCNT diameter and chirality. However, there is not yet a complete understanding of the synthesis process and there is a lack of mathematical models that explain nucleation and diameter selectivity of stable carbon allotropes. Here, in-situ analysis of chemical vapor deposition SWCNT synthesis confirms that the nanoparticle to nanotube diameter ratio varies with the catalyst particle size. It is found that the tube diameter is larger than that of the particle below a specific size (d_c ≈ 2nm) and above this value is smaller than particle diameters. To explain these observations, we develop a statistical mechanics based model that correlates possible energy states of a nascent tube with the catalyst particle size. This model incorporates the equilibrium distance between the nucleating SWCNT layer and the metal catalyst (e.g. Fe, Co, Ni) evaluated with density functional theory (DFT) calculations. The theoretical analysis explains and predicts the observed correlation between tube and solid particle diameters during growth of supported SWCNTs. This work also brings together previous observations related to the stability condition for SWCNT nucleation. Tests of the model against various published data sets and our own experimental results show good agreement, making it a promising tool for evaluating SWCNT synthesis processes.

Graphical Abstract

INTRODUCTION

Single-walled carbon nanotubes (SWCNTs) are among the most studied and promising carbon allotropes for flexible electronics and biomedical applications due to their mechanical stability and high electronic conductivity^1–3. The design and controlled synthesis of carbon allotropes (e.g. nanotubes, fullerenes) and many other 2D polymorph materials have received increasing attention of several industries due to their wide range of applications^1–5. Catalytic chemical vapor deposition (CVD) is currently the most widely used technique for growing carbon nanotubes^6–9. This technology has been especially important in the mass production of SWCNTs and achieving a better selectivity in diameter and chirality. The structure of catalyst nanoparticles, including size, composition, morphology and their evolution during the CVD process (e.g. interaction with adsorbate gases and substrate) play a critical role in the growth of SWCNTs¹⁰.

There is some consensus among experimentalists about the diameter-controlled synthesis of SWCNTs grown using a supported nanocatalyst: A uniform distribution of supported small solid catalyst particles is suggested to produce a homogeneous narrow nanotube’s diameter distribution^11–14. These works show a strong relationship between the solid nanocatalysts size and the tube diameter profile. The phase of the catalyst (i.e. solid or liquid) has been suggested as an important factor for reducing the variability of possible chiral structures^15,16, as well as that for diameter selectivity, with solid structures favoring such selectivity. Statistical analyses establishing the ratio between the diameters of catalyst particles and those of SWCNTs^17–19 together with observations regarding the growth mode, have given birth to additional understanding, such as the tangential vs. perpendicular growth classification²⁰. However, a model that may quantitatively explain the link between SWCNTs and catalyst particle diameters has not been successfully established limiting the advances of targeted synthesis for selective nanotube production and chirality control.

Earlier studies have highlighted the thermodynamics-driven nature of the SWCNTs nucleation process²¹ and its connection with chirality selectivity²². For all chiral angles χ, the energy scale variability associated with the SWCNT caps is small, compared to that of the SWCNT to catalyst interface²³. A high interfacial surface stress between metal and carbon tends to peel off the cap. This process creates an incipient nanotube nucleus that could be described as composed of a thin curved tubular nanoribbon interacting with the metal surface topped by a semi fullerene cap as shown in Figure 1. We postulate that the fact that the nascent tube has elastic properties, and therefore curvature energy, allows the SWCNT to reach a defined range of energetically stable states. We show that the probability of reaching these stable states may determine the diameter of the tube.

Figure 1.

Simulated structure of a nucleated cap system, (green/orange) growing over a nanocatalyst (blue) supported on an oxygen-rich insulating substrate (e.g. SiO₂, Al₂O₃). [Top-Right] Nucleated semi-fullerene cap. [Bottom-Right] Curved tubular nanoribbon (orange) at the SWCNT open-end interacting with the catalyst.

Here we have used high-resolution in-situ environmental transmission electron microscopy (ETEM) to determine the diameter relationship between Co metal catalyst particles and SWCNTs experimentally, and propose an innovative model based on the elastic energy of the rim nanostructure (orange region in Figure 1) at the open-end tube edge. Such model offers an understanding of the correlation between catalyst particle and nanotube diameter through the analysis of most probable energy states for the atoms in contact with the metal immediately after cap nucleation.

The statistical-mechanical based model is tested against our own data and that from the literature. The paper is structured as follows: First we introduce the basic concepts on which the model is developed, and then we provide a physical description and define the main equations, assumptions, and parameters. Finally, we apply the model and compare the prediction with experimental results.

THEORETICAL BACKGROUND AND MODEL DEVELOPMENT

We model a system conformed by an infinitesimal section of the incipient SWCNT wall referred here as the tubular nanoribbon (Figure 2). The nanoribbon is described as the outer edge of the nascent tube in contact with the metal particle after the nucleation occurs. This tubular nanoribbon has a single degree of freedom, the radial position on a curved surface (i.e. tube diameter). According to the dislocation theory²⁴, the presence of free radicals at the edge of the early sp² carbon structure allows the carbon atoms in contact with the surface to spread into new planes (edge dislocations) forming pentagons and hexagons until the complete cap is formed. This stepped spreading of the nascent cap²⁵ allows the system to find a local minimum in the carbon structure’s curvature energy, and therefore a stable tube diameter (d_r). The probability function f_d(d_r) of a tube having a certain diameter d_T can be evaluated with a statistical-mechanical model. We define the diameter distribution f_d as a probability function of generalized coordinates (p,q) such that the statistical equilibrium condition can be expressed mathematically using Equation 1.

Figure 2.

Cylindrical carbon nanoribbon interacting with a catalyst. The nanoribbon interacts with the catalytic surface and positions itself at an equilibrium average distance δ₀ at the most stable diameter d_T. For a stable particle with diameter d_p, the tube diameter d_T is constrained within a diameter range described in Eq. 5 as the extension of phase’s radial limits.

$${\sum }_i\left(\frac{d{f}_d}{d{p}_i}{\dot{p}}_i+\frac{d{f}_d}{d{q}_i}{\dot{q}}_i\right)=0$$ Eq. 1

This condition of statistical equilibrium dictates that the system evolves in a way that conserves the density of states and probability function f_d within a multi-dimensional space $(\Pi d{p}_{i}d{q}_i)$, called the extension of phase26. For our system, this condition is fulfilled defining the probability f_d as a function of energy and including only conservative forces. Classical mechanics theory assigns energy states for an atom, or group of atoms, in terms of potential (U) and kinetic energy (K), these can be used to describe the configuration of the atoms for a microstate. Additional constraints to the degrees of freedom on the atoms are due to the C-C bonds in the nuclei layer that restrict the motion of the tubular nanoribbon. For this reason, we assumed that the carbon atoms on the tubular nanoribbon have a negligible velocity and the only relevant potential energy describing the intermolecular interactions is the energy stored as strain energy within the system (Equation 2), thanks to the flexible nature of the material.

$$E_{strain}=\frac{YL{a}^3\pi }{6d_T}$$ Eq. 2

The strain energy (E_strain) is defined in terms of Young modulus (Y), tubular nanoribbon length (L), nanoribbon wall thickness (a), and tube diameter (d_T). A schematic illustration of these variables can be found in Figure S1 (Supporting Information). The curvature energy (E_c) is a function of Y, a, d_T and the carbon surface density ρ_s and is defined as the strain energy normalized by N, the total number of C atoms, and reduced to the expression in Equation 3.

$$E_c=\frac{E_{strain}}{N}=\frac{Y{a}^3}{6{\rho }_s{d}_T^2}=\frac{\alpha }{d_T^2}$$ Eq. 3

The parameter α is defined in Equation 3. It is usually assumed constant for a defect and impurity-free material like the nanoribbon. Kudin et al. have shown the relation of α to the flexural rigidity using the continuum shell approach²⁷. Equations 2–3 are based on previous works^18,28–31 addressing diameter stability and their derivation can be found in the first section of the Supplemental Information.

The tubular nanoribbon system in its nascent form may exist in many different configurations (i.e. radial position and orientation on top the particle). Each configuration has a unique corresponding energy state (i.e. microstate) associated with it. Thus, to evaluate these microstates we define them mathematically using a continuous function for the curvature energy according to the work-energy principle for elastic materials. The probability f_d(d_T) of a tube having a certain diameter d_T will be proportional to the curvature energy E_c associated to a corresponding microstate as described by Equation 4.

$$f_d\left(d_T\right)\propto e^{-\frac{E_c\left(d_T\right)}{k_{B}T}}$$ Eq. 4

According to equation 1, the system is at statistical equilibrium, this means that the probability distribution function (f_d) is independent of time. Strictly speaking, a finite number of systems cannot be distributed continuously in the phase space. But in the limit of an infinite number of states, we may approximate the discrete distribution to a continuous function²⁶. The extension of phase’s radial limits in Equation 5 accounts or all possible tube diameter (d_T) configurations and it is related to the nanocatalyst diameter (d_p) in the upper limit (d_up = d_p + δ₀).

$$d_0<d_T<d_p+{\delta }_0$$ Eq. 5

Here, d₀ is the minimum equilibrium distance between two graphene layers in a graphite structure at the absolute zero temperature (≈ 0.34 nm), and δ₀ is approximated by the equilibrium average distance between the surface of the metal catalyst and the carbon nanoribbon as shown in Figure 2.

For a nascent carbon cap supported by a metal particle, the interfacial stress will bend the carbon structure to find a stable curvature. This quasi-static process follows an intrinsic energetic path under the principle of least action. Therefore, we propose a probability function in the pseudo-canonical ensemble (Equation 6) with a phase function distributed according to the Boltzmann probability function. In principle, this can be better understood as an a priori probability.

$$f_d(d_T,T)=\frac{Boltzmann\hspace{0.17em}Energy\hspace{0.17em}Distribution}{Sum\hspace{0.17em}over\hspace{0.17em}all\hspace{0.17em}possible\hspace{0.17em}states}=\frac{e^{-E_c(d_T)/k_{B}T}}{{\int }_{d_0}^{d_p+{\delta }_0}{e}^{-E_c(x)/k_{B}T}dx}$$ Eq. 6

At fixed temperature, d_T is the only variable describing every possible microstate of the tubular nanoribbon. Therefore, we can use the expression for curvature energy found in Equation 3 to integrate the denominator in Equation 6. The limits in the integral shown in the denominator of Equation 6 will match the boundaries in Equation 5 to account for all possible configurations. The resultant expression given in Equation 7 is a function only of the particle diameter (d_p), temperature (T), and the additional parameters (α, d₀, δ₀). As such, it depends strongly on the carbon and nanocatalyst intrinsic properties, as well as on the tube/nanocatalyst interactions.

$$f_d(d_T,T)=\frac{e^{-\left(\frac{\alpha }{d_T^2{k}_{B}T}\right)}}{{\left[\sqrt{\frac{\pi \alpha }{k_{B}T}}\text{erf}\left(\frac{\sqrt{\alpha /k_{B}T}}{x}\right)+x{e}^{-\left(\frac{\alpha }{x^2{k}_{b}T}\right)}\right]}_{x_1=d_0}^{x_2=d_{up}}}$$ Eq. 7

From Equation 6, we have obtained a probability density distribution given by Equation 7. We note that this is an approximation because we have neglected the translational kinetic energy (K → 0) and considered only the potential energy associated with the curvature of the nascent carbon structure (U = E_c). This doesn’t mean that the total energy of the tubular nanoribbon is not affected by other contributing factors, like chirality, and the velocity of the catalytic particle (if it is in motion). But the current approximation helps us to concentrate on the effect of the stability of the carbon nanostructure interacting with the catalytic particle that is assumed to be a dominant factor to determine the tube diameter.

$$<d_T>\hspace{0.17em}=\underset{x_1=d_0}{\overset{x_2=d_p+{\delta }_0}{\int }}x\hspace{0.17em}{f}_d\left(x,d_p,T\right)dx$$ Eq. 8

The experimentally observed diameter is predicted by calculating the most probable value (< d_T >) according to the probability density distribution (f_d) given by Equation 7, once the properties of the catalyst and catalyst/carbon interactions (mainly reflected in the δ₀ parameter) are determined from first principles as shown below. The result can be obtained by solving the expression in Equation 8. Moreover, the standard deviation (σ) can be found using the definition in Equation 9 and the probability distribution obtained previously (f_d). The standard deviation provides a measure of the most probable region where SWCNTs can grow, and it depends on temperature T and the particle diameter (d_p) through the upper integration limit d_up.

$${\sigma }^2=Var\left(d_T\right)={\int }_{d_0}^{d_{up}}{x}^2{f}_d\left(x,T\right)dx-<d_T{>}^2$$ Eq. 9

We note that some other conditions may affect the proposed representation of SWCNT curvature stability and the model parameters. For example, intrinsic properties of the growing tube, such as chirality and defects may cause small changes of quantum origin on the physical properties of the tube³² (e.g. elasticity Young modulus). Additionally, the distribution of accessible diameters should be discrete based on the existence of known distances between covalently bonded carbon atoms^28–30. However, the use of a classical expression for the elastic energy and the subsequent statistical analysis are valid under the equipartition theorem and the ergodic hypothesis^26,33.

Moreover, as we demonstrate in this paper, the model can reproduce observed nanotube’s diameter distributions using exclusively the measured catalyst diameter distribution profile. Additional simplifications to the model and approximations for the probability distribution f_d and the most probable diameter < d_T > are discussed in the supplementary information. We observed that a numerical evaluation of Equation 8 is the most accurate method for solving the expression, especially for small particles (Figures S3 and S4). In the following section we use this probability function to determine the most probable tube diameter as a function of particle size where the model parameters are evaluated for sets of experimental synthesis data of SWCNTs grown on cobalt nanoparticles.

RESULTS AND DISCUSSION

A critical part of the model and the basis to understand nanotube formation and its relationship to the nanocatalyst properties relies on studying the stability of the nascent tube and the associated catalyst-nanoribbon interaction. A deeper analysis of the interfacial interaction between the metal surface and the graphene layer is crucial to comprehend the forces involved during the tube’s nucleation. We use DFT to quantify the interaction between some common metal catalysts and carbon structures. These calculations are necessary to find an adequate model parameter value for the distance between the nascent nanotube wall and the catalyst surface. Additionally, we use our experimental data to assess the model and analyze its capabilities and limitations.

The Strength of Graphene-Metal Interaction and Evaluation of Model Parameters

Interlayer adhesion energies obtained using spin polarized DFT calculations reveal the information about the strength of interaction between metal (and metal carbide) catalysts and graphene. Table 1 displays values obtained by DFT under periodic boundary conditions. The calculations were set minimizing any graphene curvature contribution that could arise due to the finite size of the model. The values for energy in previous works shown in Table 1 correspond to multiple metal facets or in the case for cobalt carbide is the closer approximation found to the infinite layer approximation for this system. Previous work has reported the existence of a relation between the exposed metal (or metal-carbide) catalyst facet and the graphene/catalyst adhesion energy³⁴.

Table 1.

Adhesion energy (E_adh), metal-carbon interaction energy (E_MC) and equilibrium distance (δ₀) between (100) slabs and graphene layer. The calculated values are compared to previous published energies and distances.

System	Eadh[eV/nm2]		EMC[meV/atom]		${\mathbf{\delta }}_{\mathbf{0}}^{\mathbf{\infty }}\left[\mathbf{\text{nm}}\right]$
	Calc	Prev.	Calc	Prev.	Calc	Prev.
Iron (Fe)	−2.81	−5.5535	−74.20	−14935	0.204	[0.211 – 0.232]35,36
Nickel (Ni)	−0.94	−9.1337	−24.77	[−19; −135]38–41	0.205	[0.204 – 0.280]38,40,42*43–45
Cobalt (Co)	−2.22	−7.6146	−55.49	[−24.7; −160]46–48	0.338	[0.205 – 0.340]47,48
Cobalt Carbide (Co2C)	−2.96	[−14.4; −26.5]34	−77.59	-	0.220	[0.180 – 0.309]34

^*Experimental Techniques (43–45)

$$\Delta E_{int}=E_{slab\&grap}-E_{slab}-E_{grap}$$ Eq. 10

The values for the interfacial interaction energy (∆E_int) were calculated using Equation 10, where E_slab&grap is the energy of each of the systems shown in the DFT section of the supplemental information, E_slab, and E_grap are the energies of the isolated slab and graphene respectively. The (100) facet was chosen for the metal and metal carbide structures because it facilitates (in general) the match with the graphene lattice structure and therefore helps to minimize the catalyst/graphene lattice mismatch when modeling graphene in the infinite layer approximation. Details of the calculations are given in the Methodology section and in the supplemental information.

The metal-carbon interaction energy (E_MC) and the adhesion energy (E_adh) were obtained dividing the interfacial interaction energy (∆E_int) by the number of carbon atoms and by the transverse area of the simulation box respectively. This value is crucial because it defines the region where multiple stable carbon allotropes start to form^49,50 (e.g. fullerene vs. nanotube). In other words, the metal-carbon strength of interaction is an indicator of the encapsulation-growth transition. As such, it can help us to determine the minimum tube diameter value (d_T) where deactivation of the particle due to encapsulation becomes possible and may be used to characterize the stability regions as discussed further in the stability analysis.

The equilibrium distance (δ₀) is a parameter of the model that defines the extension of phase upper boundary (d_up = d_p + δ₀) and therefore the number of accessible microstates. The values ${\delta }_0^{\infty }$ shown in Table 1 correspond to the infinite layer approximation, two periodic metal – graphene layers interacting in the interface, this method is especially good to obtain a measure of the non-bonding, van der Waals interaction. Usually, bulk carbon atoms in a graphene sheet have a coordination number of three; however, edge carbon atoms may have a reduced coordination number. For example, the ${\delta }_0^{\infty }$ equilibrium distance in the cobalt slab (0.338 nm) decreases to 0.190 nm, 0.179 nm and 0.161 nm by reducing the number of neighbor carbon atoms coordinating with a central bulk graphene atom to two, one and zero respectively. This decrease in the equilibrium distance is related to a stronger interaction due to free electrons available to form bonds at the edge of the tube (${\delta }_o^{edge}\approx [0.5-0.6]\hspace{0.17em}{\delta }_0^{\infty }$). The tubular nanoribbon region, on which the model is based, is composed of a bulk – edge combination of carbon atoms. For this reason, we assumed that δ₀ in the upper limit of the radial extension of phase (d_up) is approximately equal to the infinite layer value (${\delta }_0\approx {\delta }_0^{\infty }\approx 2{\delta }_0^{edge}$) for the entire diameter.

An interesting observation based on the work-energy principle equations (Equations 2 and 3) is that theα carbon surface density (ρ_s) has a role defining the curvature energy function, or more explicitly the parameter. We evaluated the surface carbon density dependency on chirality for SWCNTs with similar diameter. A quick analysis of the smallest repetitive section of different chiral tubes shown in Figure 3 indicates that the surface carbon density is almost independent of the chiral angle and has a constant average value of 38.2 atoms nm⁻².

Figure 3.

Surface carbon density (ρ_s) dependence on chiral angle. Estimated values (•) using periodic units of different chiral tubes.

Experimental Results

The diameters of approximately 22 particles and the tubes associated with them were measured from the high resolution ETEM images recorded during SWCNT growth (see methods for details). It was not always possible to take images with both the particle and associated nanotube to be clearly visible in the 2D projection as often the particles are embedded in the support. Therefore, only the images with clear connection between the two were used for the data analysis. A broad distribution of three types of relationship can be established from the in situ data: (a) tube diameter is smaller than the particle diameter (Figure 4a and Figure 4b); (b) tube diameter is almost the same as particle diameter (Figure 4c and Figure 4d); and (c) tube diameter is larger than the particle diameter (Figure 4e). This last observation is intriguing as it results in a different metal to carbon interaction than the first two.

Figure 4.

ETEM images of SWCNTs growing on Co particles. a and b) Tube diameter smaller than particle diameter, apparent perpendicular growth; c and d) Tube diameter very close to the particle diameter, apparent tangential growth; e) Tube diameter larger than particle diameter, apparent tangential growth; f) Observed correlation between tube diameter and particle diameter in this study.

SWCNT production using a catalyst-support combination results in a clear dependence on particle diameter as indicated in Figure 4f. For particles above ≈ 3 nm, no SWCNT growth was observed. The proposed model does not predict a maximum diameter for the SWCNT growth on active particles. However, previous studies have shown the importance of the nanotube elastic behavior to keep its cylindrical structure and the existence of a transition between the cylindrical vs. the collapsed structure as a function of tube diameter^51,52. The elastic collapse has not been explored in this study, but it has been reported to impede the growth of SWCNTs above particle size of 3 nm⁵³.

Figure 4f shows the experimental correlation between the tube and particle diameter. There is a clear change in behavior of this function with respect to the straight line in Figure 4f (where d_r = d_p). The tube diameter is larger than that of the particle below ≈ 2 nm and above this value is smaller than particle diameters. This change agrees with the reported transition from tangential (Figures 4c, d, and e) to perpendicular growth mode²⁰ (Figures 4a and b). Note that in the tangential growth mode the small nanoparticle has more flexibility to let the tube diameter accommodate on its surface. Such flexibility (which depends on intrinsic properties of the catalyst such as its cohesive energy reflected in its melting point) disappears in the perpendicular mode, where the fate of the nascent tube is more related to the strength of the catalyst-carbon interaction. Moreover, in the tangential growth mode, the contact area between particle facets and nascent tube wall is much larger (Figures 4 c, d, e); thus we expect that the interfacial energies between carbon and catalyst be dominant. In contrast, in the perpendicular growth mode, the tube sits over the larger particle (Figures 4 a and b) and the interfacial interactions are predominantly between the tube rim and the catalyst surface. In addition, we know that the curvature energy of SWCNTs increases as the tube diameter decreases¹⁸. Therefore, the tube diameters from the smaller particles may be relatively larger. Molecular dynamics simulations have shown that the tube diameter is also dependent on the metal-support interactions and tubes with diameters within a stable region will nucleate from larger particles as the metal-support adhesive energy increases⁵⁴. The tube diameter has also been reported to depend on the carbon availability. However, as the C₂H₂ pressure in our ETEM experiments was kept constant, the availability of carbon can only vary with particle size. In other words, for the data in our study, the tube diameter will increase with increasing particle size, which is generally true, except for the fact that there appears to be at a minimum (≈ 1.1 nm) and a maximum (≈ 2.8 nm) limit for tube diameters from particles of diameters varying from ≈0.85 nm to ≈3.14 nm.

Model Predictive Capability

Test of the model for the small particle range (d_p < 5nm)

We first attempt to replicate the trend observed in our experimental data relating the diameter of the particle to the tube (Figure 4f). The interfacial distance δ₀ found previously for cobalt is used to define the upper integration limit d_up. The parameter α as discussed previously can be estimated using Equation 3 from the work-energy theory for elastic materials presented in the first part of the supplemental information. This relation makes α exclusively dependent of tube properties like Young modulus (Y), the atomic surface density (ρ_s) and the wall thickness (a). Values of Young moduli obtained from prior atomistic studies are largely scattered varying from (0.95 TPa to 5.5 TPa) for Y and [0.06 nm to 0.69 nm] for wall thickness (a)^53,55–58. The uncertainty on the definition and estimated values for these properties can impact α greatly. For this reason, a convenient procedure for including the variations in the carbon nanoribbon properties is to numerically optimize the α value from the experimental data and compare with previous estimations.

Figure 5 shows the results of fitting the model to our experimental data. The optimized α_opt value for our SWCNT data using a cobalt catalyst is 1.15 eV nm² atom⁻¹. The inflection point (i.e. the point where the ratio $d_T/d_p$ is approximately one) in the diameter’s behavior can be observed and explained within the model. For very small particles (< 2 nm) the most probable diameter (solid line) is close to the upper limit $(d_T\to d_p+{\delta }_0)$ in the extension of phase. This is due to the small range of possible states that results from a very sharp narrow probability distribution (Figure S1). On the other hand, the probability distribution for large particles is broader and the average is expected to be in an intermediate value between both limits of the range [d₀, d_up]. Hence, the most probable tube diameter in small particles is larger than d_p, and the growth should be tangential to the particle (d_T > d_p), whereas the growth for large particles is expected to be perpendicular (d_T < d_p).

Figure 5.

Adjusting the parameter α to our SWCNT experimental data. (+) High-resolution TEM experimental Data (this work). (−) Most probable or average diameter and (--) standard deviation limits (±σ) obtained after α_opt has been found.

It is important to further analyze the meaning of the α parameter. For this reason, it is also necessary to consider previous evaluations of this parameter. For example, Gülseren et al reported a value of α = 0.0214 eV nm² atom⁻¹ obtained from ab-initio calculations³¹ ($E_c=\alpha /R^2=E_{CNT}-E_{grap}$). This value is equivalent to α = 0.0856 eV nm² atom⁻¹ in our curvature energy representation ($E_c=\alpha /d_T^2$). However, α_opt (1.15 eV nm²atom⁻¹) is one order of magnitude higher than the one predicted by Gülseren et al. This difference in α values can be related to the uncertainty in the evaluation of the SWCNT wall thickness where approximations also vary in orders of magnitude^53,55,58. In an atomic thin shell model, the wall thickness is considered to be the graphite inter-layer spacing (0.34 nm), Cai et al. demonstrated that this value also corresponds closely to the thickness of the SWCNT electron cloud⁵⁷. To test the accuracy of our α, we estimated the Young modulus (Y) using the optimized α_opt value. Thus, using the 2D approximation (bending tubular carbon nanoribbon) in Equation 3, the previously calculated surface carbon density ρ_s, α_opt, and the value of 0.34 nm for the tubular nanoribbon thickness, we estimate a value of 1.07 TPa for the Young’s modulus of SWCNTs, that is within the range of many model approximations^56,59,60 (0.97 TPa to 5.5 TPa) and very close to the few experimental values reported for SWCNTs^61–63 (1.20 TPa to 1.25 TPa).

Test of the model for large particle range (d_p > 5 nm)

Next we used the experimental set by Tibbetts¹⁸, that reports inner diameters of multi-walled carbon nanotubes (MWCNTs) for very large particles. Although the growth mechanism of MWCNTs is not yet clear, we assume that the inner tube structure nucleation occurs under similar conditions to the ones mentioned previously for SWCNTs. Figure 6 shows that the inner diameters in MWCNTs can also be adjusted to our model yielding an α_opt value of 2.063 eV nm² atom⁻¹ using δ₀ for iron (Table 1). To explain the difference between the α_opt for SWCNTs in small particles and MWCNTs in larger particles it should be noted that the inner diameter in a MWCNT may be affected by the presence of compressive/attractive forces due to the additional walls. This can cause an increase in the ability of storing potential elastic energy within the curvature of the tube. An increase in the bending momentum stored by the carbon atoms at the edge of the tube is expected to affect the flexural rigidity (bending stiffness) of the tube and therefore the α value in the inner tube structure of MWCNTs. The difference could also be attributed to the van der Waals forces between the inner nuclei and the concentric layers of a MWCNT. These additional forces can modify the mechanical properties of the inner tube. Using the same approximation utilized for the SWCNT thickness (0.34 nm) and the α_opt obtained for MWCNTs, a Young’s modulus of 1.93 TPa is estimated, that is in reasonable agreement with reported Young moduli for MWCNTs⁶⁴.

Figure 6.

Adjusting the parameter α to inner diameters in the MWCNTs data set from Tibbetts. (x) Data collected for iron particles from Tibbets¹⁸. (−) Most probable or average diameter and (--) standard deviation limits (±σ) obtained after α_opt has been found.

We remark that the range of particle diameters in the two sets of data shown in Figure 5 and 6 is very large. To compare and test the applicability of the α values, we evaluated both data sets with the α_opt obtained for the smallest tubes and the value reported from Gülseren³¹ (Supplementary Information, Figure S6). The model shows that even using a rough approximation of α from the literature or that previously obtained in the fitting of our data, the model predicts a good estimate of the inner diameter value in large particles. It is found that the predicted tube diameters are much more sensitive to the α value in the small tube range. For larger tubes, where the probability function is distributed over many possible configurations, the α dependence becomes weaker. We note that although α_opt was obtained for the previous described experimental sets of data, it could be applied to other metal catalysts because its calculation only involves tube properties. For this reason, the prediction of a SWCNT diameter distribution in a bed of iron nanoparticles is proposed as a final test for α_opt.

Prediction of a nanotube diameter distribution corresponding to a given catalyst size distribution (experimental data from supported CVD)

Another validation and possible application of this model is shown in Figure 7. The model was used to predict the SWCNT diameter distributions corresponding to catalyst diameter profiles measured on a support of Si-SiO₂ wafer by Zou et al¹¹. The predictions were obtained using the diameter profiles for the catalyst particles at different etching times to define the upper limit (d_up). This is because the range of the particle diameter distribution is reduced after exposed to a longer etching time. The experimental data was normalized using Equation 11 and the parameters used for the prediction (Equations 7 and 8) were the equilibrium distance δ₀ for Iron (Table 1) and the parameter α_opt (1.15 eV nm²/atom).

Figure 7.

SWCNT’s diameter distribution prediction using Iron catalyst particle profiles measured by Zou et al¹¹. Figure 7a-c correspond to the experimental tube diameter distribution profile (orange) vs the prediction of the model (blue). Predictions were based on experimental catalyst particle diameters for particles (a) without etching, and (b and c) after 10 s to 15 s of etching, respectively. The darkest colored regions show the overlap between the experimental and the theoretical descriptions.

$$Normaliztion=\frac{Frequency}{\#Measurements}$$ Eq 11

We observe that all the predicted profiles in Figure 7a–c are in good agreement with the reported experimental sets. The slight right shift in the distributions of the three examples could be attributed to the assumptions on the estimation of δ₀ and α_opt. The equilibrium distance approximation for the nanoribbon (${\delta }_0\approx {\delta }^{\infty }\approx 2{\delta }^{edge}$), as discussed before, will affect the probability distribution for small particles. An overestimation of δ₀ may cause this type of shift due to an increase in the upper limit (d_up) and the number of possible accessible microstates. However, for the δ₀ range found in this and previous studies (0.2 to 0.35 nm) the shift due exclusively to δ₀ will not account for the total difference. A lack of information in the inactive particles and non-growth events may be contributing to overestimate the amount of tubes from the particle distribution as well.

Stability Analysis

An important question relates to the probability of a nascent SWCNT cap to develop a stable tube growth or stop the growth and encapsulate the catalytic particle (stable fullerene). Early studies in carbon allotropes have established the relation between curvature energy and tube diameter^{18,28–31,65} as described in the theoretical background (i.e. Equations 2, 3 and Equations S1 to S6). Burgos et al, for example, showed the existence of an explicit relation between adhesion energy, curvature and nucleation⁴⁹, as expressed in Equation 12. Furthermore, the analysis of an approximated support-particle interaction showed that the support nature has a strong influence on the catalyst structure, therefore on the catalyst shape (i.e. curvature) and tube’s diameter^54,66. Note that this discussion relates only to systems where the reaction conditions favor a slow carbon supply rate on the catalyst surface (e.g. low pressure of precursor gas). Lowering the carbon supply flow will likely allow the carbon structures to form more slowly, this annealing process may favor evolution to low energetic configurations⁶⁵. In this section we analyze the stability conditions and locate the estimated tube diameter prediction within the growth stability limits.

$$\underset{Tube\hspace{0.17em}Curvature\hspace{0.17em}Energy}{\underbrace{E_{cT}}}\le \underset{Metal\hspace{0.17em}supported\hspace{0.17em}Fullerene\hspace{0.17em}Energy}{\underbracket{\underset{Fullerence\hspace{0.17em}Curvature\hspace{0.17em}Energy}{\underbrace{E_{cF}}}-\underset{Metal-Graphene\hspace{0.17em}interaction\hspace{0.17em}energy}{\underbrace{E_{MC}}}}}$$ Eq. 12

Comparing the curvature energy per atom of the tube (E_cT) with the one for a fullerene with similar diameter (E_cF) shows that the tube is always more stable. However, the attractive metal-graphene interaction (E_MC) may reduce the energy necessary to bend the carbon bonds and therefore the curvature energy per atom of the fullerene capsule (E_sF = E_cF − E_MC). Hafner et al. introduced the stability criteria graphically using empirically fitted functions to represent the curvature energy of tubes and fullerenes⁶⁵. Following Hafner’s work, Figure 8 shows that the energy difference between both states is in the order of meV.

Figure 8.

Stability analysis for the nucleation of carbon allotropes on metal catalysts. Empirically fitted functions for the curvature energy in SWCNTs (E_cT) and fullerenes (E_sF) and the critical diameter of transition (d_c) for a cobalt catalyst.

Figure 8 offers a better understanding of Equation 12. Here, we used the interfacial cobalt-carbon interaction energy (Table 1) to estimate the fullerene capsule energy (E_sF). For this reason, even if a tube is less stable for diameters approximately above d_c in a cobalt particle, the energy difference is extremely small, and in many cases, nanotubes are observed to grow with a nucleation probability proportional to $e^{\left(-E_{MC}\right)/k_{b}T}$. An interesting observation is that d_c corresponds to the point where the ratio between $d_T/d_p$ is approximately one for our experimental data, and the growth behavior changes from tangential to perpendicular. Therefore, we could use this value as a point of reference and merge the stability analysis with our model.

Curvature stability plays a role delineating the stable tube growth regions. These are tube diameter stable zones for a specific particle diameter with boundaries defined by the critical diameter (d_c) horizontal line and the upper boundary in the extension of phase (d_up = d_p + δ₀) shown as a dashed red line in Figure 9. The small particle zone ($d_p/d_c\ll 1$) has an upper limit in stability and it is the d_c line. This means that the tube is always trying to reduce its curvature energy making the transversal area as big as possible until < d_T > or a value energetically accessible is reached. We can also observe that most of the experimental data is close to d_up for this zone, this is due to the existence of a very low standard deviation, leading to a small region with a high probability of nucleating the tube (< d_T > ± σ). Above d_up the probability distribution f_d rapidly collapses to zero (f_d → 0) so wider tubes beyond this limit will not grow. In our model the accuracy of the d_up limit will depend on the δ₀ estimation that has an associated error as discussed in the model parameters section.

Figure 9.

Diameter stable regions during the growth process of SWCNTs and high probability zones within one standard deviation from the most probable diameter (< d_T > ±σ). (+) High-resolution TEM experimental Data for Co catalysts. (--) Upper limit for the radial extension (d_up). (∙∙∙) Transition critical diameter (d_c) between stable fullerene and tube allotropes. (−) Most probable or average diameter and standard deviation limits (±σ). In the purple region, the probability of finding a diameter d_T is ≈ 0.

For SWCNTs growing from large particles ($d_p/d_c\gg 1$), the upper limit in stability (Figure 9) will be the line corresponding to d_up. Figure 9 shows that the tube diameter (for d_p > d_c and d_p approximately smaller than 2.5 nm) is also slightly smaller than the particle diameter in the transition zone. This is because the tube will try to minimize its internal strain energy by reducing the transversal area until reaching the stable diameter region with a value close to d_c. That is why we see an inversion in the data trend between the two zones (i.e. the tube diameter is no longer bigger than the particle). This zone also is characterized by a co-existence between<1 fullerenes and tube allotropes. For carbon allotropes with a weak metal-carbon interaction (< 1 eV) and large transversal area, the curvature energy will favor nucleation, but a strong metal-carbon binding energy (1 eV to 2 eV) favors encapsulation in every event for particles with a diameter above the transition diameter (d_c). The nature of the metal catalyst will change the value for d_c. For example, iron, with a strong attractive interaction with carbon is expected to slightly decrease the value of d_c, therefore the probability of encapsulation will increase, and the transition between tangential and perpendicular growth may be below 2 nm. However, a better estimation should include the distribution of stable facets and the average metal-carbon interaction energy (E_MC) according to the percentage of covered area for the case of metal particles below the melting point.

CONCLUSIONS

Using a combination of experimental data, quantum mechanical calculations, and statistical mechanics we developed a model to describe the relationship between catalyst size and SWCNT diameter. Good agreement of the predicted and the measured tube diameters demonstrates that the curvature energy of the nascent carbon nanostructure is a major contributor to the process of carbon nanotube nucleation. The proposed model offers a simple description of the correlation between tube diameter and catalyst particle diameter for a tube growing on an approximately spherical solid active nanocatalyst. Real conditions may show that a particle, especially less crystalline particles, are not completely spherical and multiple curvatures can be present in the catalyst (i.e. a nonhomogeneous particle diameter ). For this scenario, our model could include those multiple curvatures ($k=1/d_p$) as individual events represented in the probability distribution function f_d.

The DFT calculations of the interlayer adhesion energies provide important information about the strength of interaction between common metal catalyst particles and graphene. These data are valuable for the evaluation of model parameters like the tube-particle equilibrium distance and to develop a discussion about the stability of SWCNT’s diameters. CVD data collected using ETEM provided an insight into the tube-particle diameter relationship and allowed determination of an “optimized” model parameter α. We show that the value obtained α_opt) is only dependent of intrinsic properties of a graphene-like structure such as surface carbon density, wall thickness, and Young modulus, and could be used in many systems independently of the catalyst selection. The predicted Young moduli obtained from the optimized parameter for SWCNTs and MWCNTs are in good agreement with experimental values.

We recognize that particle composition and surface stability may play an important role during the SWCNT nucleation vs. catalyst encapsulation dynamics. Magnin et al⁶⁷ recently highlighted a correlation between the carbon concentration in the particle and graphene/catalyst wetting properties, thus leading to tangential vs. perpendicular growth. This analysis is based in partially molten (liquid) particles where surface tension can deform the particle and wet the tube. However, the direct effect between the formation of solid carbide phases, the mode of growth (perpendicular vs tangential) and deactivation is not yet completely explained. Our own unpublished work suggests that metal particles for which many possible carbide structures may exist, multimodal surface composition distributions may be responsible of the change of growth mode. But the effects of these observations on the tube diameter-particle diameter are not yet well understood and possibly could be incorporated into future models.

Moreover, Bets et al⁶⁸ demonstrated the important and interesting nature of the tube edge interacting with the metal surface. The analysis of the rim-metal interaction shows the stability of specific edge configurations and the effect on the chirality selectivity. Nevertheless, the study suggests that the influence of the catalyst particle size on the nucleation probability has to be added separately. This is because the edge-catalyst interaction does not account for the stability of the tube’s wall internal elastic energy. It could be concluded that for the perpendicular mode of growth (usually larger catalyst particles with d_p > d_c), the tube edge/metal interaction may be especially important to define chirality/diameter. This interaction could help to reduce the number of possible states of similar curvature energy, this aspect is not accounted in our current model.

Finally, the central focus of our paper was on whether single-walled carbon nanotube diameters are defined by catalyst particle diameters. With respect to the specific correlation between tube diameters and particle diameters, various new insights are obtained. It is found that for small catalyst particle diameters, the tube diameter is larger than or equal to the particle diameter, thus favoring tangential tube growth, and the range of possible tube diameters for a given particle is rather narrow corresponding to a sharp tube diameter distribution function. We identified a critical diameter that separates regions of stable nanotube and stable fullerene. Such critical diameter depends on the intrinsic catalyst properties and on its interactions with carbon. For d_T larger than such critical diameter, the growth behavior changes and the tube diameter tend to become smaller than the particle diameter, leading to a perpendicular growth. This transition to the perpendicular mode of growth, coincides with much broader probability distribution functions allowing a wider range of possible tube diameters for a given particle size. Interestingly, the model has also proved useful for the prediction of inner diameters of MWCNTs as well as to reproduce nanotube’s diameter distributions using exclusively the catalyst diameter distribution profile for supported CVD particles. A more complete characterization of inactive and encapsulated particles may improve the agreement of the model with the experimental results.

The results showed that the careful estimation of model parameters like α_opt and the tube wall-catalyst surface distance δ₀ is a requirement for a good accuracy of the model. We have noticed, for example, the ability of δ₀ and α for shifting the predicted tube diameter profile based on experimental catalyst size distributions. Thus, changes in δ₀ increase or reduce the amount of possible states in the proposed continuous function approximation for the tube diameter probability distribution (f_d). A different approach for the estimation of these parameters can be further explored for each metal or alloy after a detailed study of the stable facets present in the nanoparticle at growth conditions and their interaction with graphene.

METHODOLOGY

In-situ CVD growth

A high-resolution environmental transmission electron microscope (ETEM), with differentially pumped column capable of tolerating up to 2000 Pa gas pressure, monochromated high intensity field-emission gun, image corrected, operated at 80 KV, was used for in-situ observation of SWCNT nucleation and growth process. Co-Mo/MgO catalyst/support was dispersed on micro-electro-mechanical system (MEMS) based heating chip that were then loaded on the heating holder and introduced in the ETEM column. It is important to note that although the nominal composition of catalyst was Co-Mo, the catalyst particles active for SWCNT growth did not contain any Mo, as explained in previous publications³⁴. The sample was first heated to 900 °C in flowing O₂, maintaining ≈ 3000 Pa gas pressure in the sample area. This pre-treatment was needed to clean the catalyst/support from any contaminants that may be present. The ETEM column was then evacuated to initial high vacuum condition while keeping the sample at 900 °C. It is important to note the temperature is as measured by the heating holder and may not be the exact temperature at the investigated zone in the sample. We did not try to measure the actual temperature as it was not critical for the current experiments. Approximately, 0.055 Pa of C₂H₂ was then introduced to the sample chamber and videos were recorded at a rate of 6 frames⁻¹. Atomic-resolution images were also recorded from the area under observation as well as from the regions that were not exposed to electron beam under reaction condition. Both the images extracted from the videos and recorded post-synthesis were used to measure the diameters of the carbon nanostructures and associated catalyst particle.

Density Functional Theory (DFT) Calculations

We perform DFT calculations for geometry relaxations and adhesion energy for graphene on metallic slabs (e.g Nickel, Cobalt, Iron) with [100] and [111] orientation. All metal-graphene systems were modeled using periodic boundary conditions (PBC) to recreate two infinite long flat surfaces at absolute zero temperature. The Monkhorst-Pack scheme⁶⁹ was used for Brillouin zone’s k-point sampling with a characteristic length (l_c) optimized for each metal. Table 2 shows the optimization results for l_c, K-points, and the energy cut off used in every system.

Table 2.

Metal slabs parameters for the modeled system.

Metal Slab System	Length (lc)	K-points-Mesh	Energy Cutoff (eV)
Nickel	40	9 x 16 x 2	700
Cobalt	60	25 x 15 x 4	600
Cobalt Carbide (Co2C)	50	10 x 12 x 3	700
Iron	40	16 x 5 x 2	700

The exchange-correlation functional given by the Perdew-Burke-Ernzerhof (PBE) approximation⁷⁰, and the Projector Augmented Wave Method^71,72 (PAW) was employed for calculating core-electron energies. The dispersion correction to the Kohn-Sham energy was implemented using the DFT-D2⁷³ based method of Steinmann and Corminboeuf⁷⁴ (DFT-dDsC). This Van-der Waals energy correction method has the special characteristic that the dispersion coefficients and damping function are charge-density dependent⁷⁵.

The interfacial interaction energies and equilibrium distances (${\delta }_0^{\infty }$) were calculated using a [100] metal slab with lattice vectors optimized for the graphene-metal system. The approach for the [111] structures was different, the metal slab lattice was kept fixed and the graphene layerȦwas rotated until reducing the mismatchȦ with the periodic image. We constructed a vacuum space of 10 $\dot{\text{A}}$ for every slab and it was reduced to 7 $\dot{\text{A}}$ after the graphene was coupled to the metal system.

Figure 11 shows the test for the [100] structures in the graphene-metal slab system, we changed the lattice parameters and observed the variation in the system energy. We searched within the 0.7 – 1.2 range for the ratio (a₀/a_c) to ensure an energy minimum of the initial system and reduce the graphene’s curvature (lateral stress). The maximum change of the graphene lattice constant (a_c = 0.246 nm) after the relaxation of the system (Table S2) was a measure of the mismatch of graphene with respect the relaxed graphene periodic unit. The length of the lattice vectors and a measure of the mismatch caused by them can be found in the DFT calculation section of the supporting information.

Figure 11.

Test for optimizing the initial lattice vectors a₀, b₀ in the [100] system. The 0.7 – 1.2 range was tested for the ratio a₀/b₀, where a_c corresponds to the graphene lattice vector (0.246 nm). The iron system was optimized against 2a_c for a better fit.

Figure 10.

Metal – graphene interfacial systems for Cobalt carbide (Co₂C), Nickel, and Iron. Periodic boundaries were defined in all systems after relaxing a primitive cell with similar graphene’s lattice constant.

NOMENCLATURE

T

Temperature

P

Pressure

$\dot{\rho }$

Material flow at the open reactor system

k_B

Boltzmann constant

f_D

Probability distribution function

p

generalized momentum coordinate

q

generalized position coordinate

d_p

Particle diameter

d_T

Tube diameter

d_c

Transition critical diameter

U

Potential energy

K

Kinetic energy

E_c

Curvature energy

M

Flat 2D sheet momentum

Y

Young modulus

I

Moment of Inertia

R

Curvature radius

L

Flat 2D sheet length

α

Flat 2D sheet thickness

N

Number of atoms in the sheet

ρ_s

Surface atomic density

σ_a

Nanoribbon cylindrical length

d₀

Equilibrium distance between two graphene layers at absolute zero temperature (T = 0K).

δ₀

Equilibrium distance between metal catalyst and graphene walls

${\delta }_o^{edge}$

Equilibrium distance between metal catalyst and carbon atoms at the tube edges

${\delta }_0^{\infty }$

Equilibrium distance between an infinite metal catalyst slab and an infinite graphene layer

α

Material parameter

α_opt

Optimized material parameter

x₁,x₂

Integration limits corresponding to the extension of phase boundaries

l_c

Characteristic length for Brillouin zone’s k-point sampling using Monkhorst-Pack scheme

E_adh

Adhesion metal-graphene energy

∆E_int

Interfacial metal-graphene energy

E_slap&grap

Energy for the interacting slab-graphene system.

E_slap

Energy for the metal catalyst slab using PBC

E_grap

Energy for the graphene layer using PBC

E_MC

Interfacial metal – carbon interaction energy

E_cF

Curvature energy for a fullerene

E_sF

Supported curvature energy for a fullerene

E_cT

Curvature energy for a tube