Encapsulation of organic molecules in carbon nanotubes: role of the van der Waals interactions

In this section we will review the main methods which have been developed recently to treat the question of weak and vdW interaction at the microscopic level. We therefore present here the essential approaches related to DFT calculations. Other approaches, such as quantum chemistry methods, are very important for the description of vdW interactions. However, these methods are rather expensive with respect to the computational cost and, consequently, they have not been applied as often to describe molecular encapsulation in CNTs. For more general approaches, the reader is referred to review articles from Grimme or Dobson for example [21–23]. As is well known, vdW energy is associated to charge fluctuations in the material due to the coupling with the vacuum electromagnetic field, giving rise to non-permanent dipoles. Those non-permanent dipoles interact together yielding an attractive interaction energy, namely the vdW energy. Consequently, this is a field interaction not directly related with an overlap of the spatial electronic densities. This constitutes the main problem for the DFT treatment based on spatial electronic density approximations. For example, local density approximation (LDA) or generalized gradient approximation (GGA) fail totally to describe such interactions [19, 20]. Another way of considering the problem lies in the exponential decay of the DFT calculated interactions, due to the exponential behaviour of the wave functions overlaps, while in the mean time, the vdW interaction is rather long range. As a consequence, one can notice that it is really difficult to find a functional able to handle in the mean time short-range and covalent interactions on the one hand, and long-range and weak interactions on the other.

Consequently, in recent years, many approaches have been developed to overcome this difficulty in the frame of DFT, either as an intrinsic contribution or as an added external potential. Here we will present the main kinds of approximations, ranging from semi-empirical model, to full RPA implementation in DFT, through intermolecular perturbation theory. We will then discuss some applications of these different approaches, especially in the case of encapsulated molecules in CNTs.

Many works have been devoted to the modelling of systems ruled by weak and vdW interactions, using semi-empirical approaches. The very first popular model was the Lennard-Jones potential:

$$\begin{equation} u(r) = 4\varepsilon \Big[-\Big(\frac{\sigma}{r}\Big)^6 + \Big(\frac{\sigma}{r}\Big)^{12}\Big], \end{equation} \tag{ 1 }$$

where the parameters ε and σ are chosen empirically to fit the considered material. The r⁻⁶ term represents the vdW interaction as traditionally obtained from dipole–dipole interactions. The r⁻¹² term is more arbitrary and is chosen to describe the repulsion at short range, whose origin is mainly the Pauli exclusion principle. This potential describes the interaction energy between two atoms at a distance r. To obtain the total binding energy of the system, one has to sum all the pair energies of the two subsystems, taking into account the geometry of the system through the interatomic distance. This empirical method has been successful in providing a unified and consistent description of properties which depend on weak chemical interactions and vdW dipolar interactions. But even though this potential is widely used and has given numerous interesting results, it presents a strong disadvantage: the parameters have to be adjusted for each considered material, and it cannot take into account structural modifications of this material at the atomic level. For example, the deformation of a molecule during an adsorption process, which obviously will affect the interatomic potential, cannot be described correctly through this potential. Moreover, the arbitrarity of the r⁻¹² term often has to be compensated by the dipolar term, leading to a bad estimation of the r⁻⁶ term, the pure vdW energy [24, 25]. This feature can be seen in the table represented in figure 1, where the coefficient A and B, which we usually call C₆ and C₁₂, are related to equation (1) through C₆ = 4εσ⁶ and C₁₂ = 4εσ¹².

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It clearly shows that according to the considered system, the coefficients can be different for around 30%. However, this formalism has provided good results in the description of encapsulated molecules in nanotubes, as for example, C₆₀ phases in various CNTs (figure 2).

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This formalism was strongly improved in the work of Grimme et al [27–29] by substituting the C₁₂ coefficients by a damping function, which is much more efficient to describe the repulsive part of the interaction and in overall to cancel the vdW contribution at short distance. Namely, the DFT-D formalism [27] is based on a standard DFT calculation where a dispersion term of the form

$$\begin{equation} E_{\rm disp} = s_{6}\Sigma_{i=1}^{N_{\rm at}-1}\Sigma_{j=i+1}^{N_{\rm nat}}\frac{C^{ij}_6}{R_{ij}^{6}}f_{\rm dmp}(R_{ij}) \end{equation} \tag{ 2 }$$

is added to the total energy of the system. Here, N_at is the number of atoms in the system, $C_{6}^{ij}$ denotes the dispersion coefficient for atom pair (i, j), s₆ is a global scaling factor and R_ij is the interatomic distance. The damping function is used to avoid singularity at small distances and is given by

$$\begin{equation} f_{\rm dmp}(R) = \frac{1}{1+\rme^{-\alpha(R/R_{0}-1)}}, \end{equation} \tag{ 3 }$$

where R₀ is the sum of atomic vdW radii.

Even though there are no results yet on molecular encapsulation in CNTs, this method is really useful and is still being used extensively for molecular systems. For example, we have also used this damping function in combination with our vdW calculation to determine the benzene–gold equilibrium distance accurately [30].

A following improvement to these approaches is the method developed by Tkatchenko et al [31, 32]. In this formalism, the vdW energy is written as

$$\begin{equation} E_{\rm vdW} = -\case{1}{2}\Sigma_{A,B}f_{\rm damp}(R_{AB},R_{A}^{0},R_{B}^{0})C_{6AB}R_{AB}^{-6}, \end{equation} \tag{ 4 }$$

where R_AB is the distance between atoms A and B, C_6AB is the corresponding C₆ coefficient, $R_{A}^{0}$ and $R_{B}^{0}$ are the vdW radii. The $R_{AB}^{-6}$ singularity at small distances is eliminated by the short-ranged damping function $f_{\rm damp}(R_{AB},R_{A}^{0},R_{B}^{0})$. This approach also considers a Fermi-type damping function similar to the one proposed by Grimme et al. This approach is pretty similar to the one of Grimme et al, except that the C₆ coefficients are no longer adjusted but determined from referenced polarizabilities and ground state electronic densities obtained with DFT calculations. This takes into account the atomic environment. This method represents an important step in the non-empirical determination of C₆ coefficients for vdW forces.

The next step in the development from semi-empirical to first-principles methods to characterize accurately the weak and vdW interactions is the vdW-DF approach proposed by Langreth, Lundqvist and collaborators [33–36]. In this model, the vdW energy is described as

$$\begin{equation} E_{\rm vdW-DF} = E_{\rm GGA} - E_{{\rm GGA,c}} + E_{{\rm LDA,c}} + E_{\rm c}^{\rm nl}, \end{equation} \tag{ 5 }$$

where the LDA correlation energy E_LDA,c is substituted by the GGA correlation energy, E_GGA,c, and where the non-local correlation energy $E_{\rm c}^{\rm nl}$ is added. This non-local correlation energy is expressed as

$$\begin{equation} E_{\rm c}^{\rm nl} = \case{1}{2}\int \rmd^{3}r\rmd^{3}r'n({\bit r})\phi({\bit r},{\bit r'})n({\bit r'}), \end{equation} \tag{ 6 }$$

where φ(r, r') is a function depending on |r − r'|, the charge density n and its gradient at the positions r and r'|, respectively. This formulation of the vdW energy is not yet a functional in the present form, but it has been implemented self-consistently as a functional in the SIESTA code [37]. This self-consistent version of the Lundqvist and Langreth functional has been applied to the structural determination of a double-wall CNT. The non-self-consistent vdW-DF has been applied in general for sparse matter characterization, including molecules and CNTs [38] as illustrated in figure 3.

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Notice that this approach has been used by several groups for the study of aromatic molecule adsorption on noble metal surfaces for example, in combination with semi-empirical corrections to accurately determine the molecule–metal equilibrium distance [40].

The last further step in the treatment of weak and vdW interactions in DFT is the development of methodologies based on the adiabatic-connection fluctuation-dissipation (ACFDT) [41–43] used directly in the random phase approximation (RPA). In this frame, the correlation energy (including vdW interaction) is written as

$$\begin{equation} E_{\rm c} = \frac{1}{2\pi}\int_{0}^{\infty}\,\rmd\omega Tr{\ln[1-\chi^{\rm KS}(\rmi\omega)\nu] + \chi^{\rm KS}(\rmi\omega)\nu}, \end{equation} \tag{ 7 }$$

with χ^KS being the independent particle (non-interacting) Kohn–Sham (KS) response function, and ν being the Coulomb kernel. The exchange energy is calculated exactly using the Hartree–Fock expression. Calculations on graphene have been performed within this formalism, using the VASP code [44, 45].

Besides these developments, we can also cite the work of Silvestrelli et al [46] where the vdW energy is determined through the use of Wannier functions. Due to high computational costs, those last methods are rather prohibitive for the study of big systems like encapsulated molecules in nanotubes.

In this part we will now present our treatment of weak and vdW interactions, i.e. an intermolecular perturbation theory approach within the frame of DFT. Namely, the LCAO-S² + vdW model has been first developed for the study of rare gas dimers [47] and then to describe the interaction between two graphene planes [48, 49] and graphitic materials in general [50]. This approach is based on the LCAO-OO formalism. Details about this formalism can be found in [51–53].

As a starting point, we first study each isolated subsystem (for example a molecule and a nanotube, before encapsulation), in order to properly describe subsequently their interaction. The DFT computational scheme, as well as the theoretical foundations underlying our calculations—a very efficient DFT localized orbital molecular dynamics technique (FIREBALL)—have been described in full detail elsewhere [54–57], and here we will only summarize the main points.

We first analyse all the systems involved in this study by using a self-consistent version of the Harris–Foulkes LDA functional [58, 59], instead of the traditional KS functional based on the electronic density, where the KS potential is calculated by approximating the total charge by a superposition of spherical charges around each atom. The FIREBALL simulation package uses a localized optimized minimal basis set [60], and the self-consistency is achieved over the occupation numbers through the Harris functional [61]. Besides, the LDA exchange-correlation energy is calculated using the efficient multi-centre weighted exchange-correlation density approximation (McWEDA) [55, 56]. To these DFT calculations we add 'weak interactions', which can be seen as two opposite contributions. The first one, named 'weak chemical' interaction, is due to the small overlaps between electronic densities of the interacting subsystems. Therefore, this energy can be determined as an expansion of the wave functions and operators with respect to these overlaps. This expansion is based on a development in S² (since in the weak-interacting case the overlaps are really small) of the S^−1/2 term appearing in the Löwdin orthogonalization, which induces a shift of the occupied eigenenergies of each independent subsystem, and leads to a repulsion energy between them. This shift can be defined as follows:

$$\begin{eqnarray} &&\fl \nonumber \delta^S \varepsilon_n= -\!\sum_m\! \case{1}{2} \left[S_{nm}T_{mn}+ T_{nm}S_{mn} \right] + \case{1}{4}\!\sum_m | S_{nm}|^2(\varepsilon_n-\varepsilon_m ),\nonumber\\ &&\fl \delta^S \varepsilon_m=-\!\sum_n\! \case{1}{2} \left[ S_{mn}T_{nm} + T_{mn}S_{nm} \right] + \case{1}{4}\!\sum_n | S_{nm}|^2(\varepsilon_m-\varepsilon_n ) ,\nonumber\\ \end{eqnarray} \tag{ 8 }$$

where S_nm and T_mn are the overlapping and hopping integrals between eigenvectors m and n in the two interacting subsystems, for example the molecule and the nanotube. The effect of the hopping matrix elements, T_mn, can be calculated in a standard intermolecular second-order perturbation theory as

$$\begin{equation*} \nonumber\delta^T \varepsilon_n= \sum_m \frac{|T_{mn}|^2} {\varepsilon_n-\varepsilon_m}, \end{equation*}$$

$$\begin{equation} \delta^T \varepsilon_m= \sum_n \frac{|T_{mn}|^2} {\varepsilon_m-\varepsilon_n}. \end{equation} \tag{ 9 }$$

Thus, we obtain the following 'weak chemical' contribution to the interaction energy:

$$\begin{eqnarray} &&\fl E_{{\rm weak~chemical}}=2\!\!\sum_{n={\rm occ.}}(\delta^S\varepsilon_n+ \delta^T\varepsilon_n) + 2\!\!\sum_{m={\rm occ.}}(\delta^S\varepsilon_m+\delta^T\varepsilon_m), \nonumber\\ \end{eqnarray} \tag{ 10 }$$

where a multiplicative factor of 2 has been included to take into account the spin degeneracy and only filled states are considered. Typically, in graphitic or hydrocarbonated materials this contribution is repulsive.

The second contribution is the pure vdW interaction, which finds its origin in charge fluctuations in each subsystems, arising from oscillating dipoles whose interaction gives the attractive part of the cohesive energy. This interaction is treated in the dipolar approximation, and added in perturbation to the total energy of the system. In this frame, the vdW energy can be written as

$$\begin{equation} E^{\rm vdW}=4\sum_{i,j,\alpha,\beta}(J_{i,j,\alpha,\beta}^{\rm vdW})^2 \frac{n_i (1 - n_j) n_{\alpha} (1 - n_{\beta})} {(\overline{e}_i - \overline{e}_j + \overline{e}_{\alpha} - \overline{e}_{\beta})}. \end{equation} \tag{ 11 }$$

In this expression, n_i are the orbital occupation numbers (per spin)

$$\begin{equation} n_i = \int_{\rm occupied} \rho_i (\varepsilon) \,\rmd\varepsilon \end{equation} \tag{ 12 }$$

and

$$\begin{equation} \overline{e}_i = \int_{\rm occupied} \varepsilon \rho_i (\varepsilon) \,\rmd\varepsilon \Big/ \int_{\rm occupied} \rho_i (\varepsilon) \,\rmd\varepsilon \end{equation} \tag{ 13 }$$

$$\begin{equation} \overline{e}_j = \int_{\rm empty} \varepsilon \rho_j (\varepsilon) \,\rmd\varepsilon \Big/ \int_{\rm empty} \rho_j (\varepsilon) \,\rmd\varepsilon \end{equation} \tag{ 14 }$$

are average occupied and empty levels. The integral $J_{i,j,\alpha,\beta}^{\rm vdW}$ takes into account all the dipole–dipole excitations of the system. It is actually the result of the calculation of a four-centre Coulombic integral in the dipolar approximation and its corresponding expression is

$$\begin{eqnarray} &&\fl J_{i,j;\alpha,\beta}^{\rm vdW} = \frac{1}{R^3}( \langle i | x | j \rangle \langle \alpha | x' | \beta \rangle + \langle i | y | j \rangle \langle \alpha | y' | \beta \rangle\nonumber\\ &&- 2\langle i | z | j \rangle \langle \alpha | z' | \beta \rangle ) \end{eqnarray} \tag{ 15 }$$

R is the distance between the two atoms assumed in this expression along the z-axis, which depends on the different dipolar matrix elements in each atom. Of course, this approximation would not be valid anymore if we would like to evaluate vdW interaction at really short distances, but this is not the case for this work. Moreover, as it was discussed above, this $J_{i,j;\alpha,\beta}^{\rm vdW}$ represents the bare vdW interaction, without screening, as approximated in our model.

Using these expressions and the DFT band structures of each isolated subsystem obtained previously with FIREBALL, we can evaluate the intermolecular vdW energy. By combining this energy with the weak chemical repulsion obtained in the LCAO-S² approach, we can determine the binding energy of the full system and compare it accurately with experimental results and other theoretical determinations. The balance of the two contributions gives the equilibrium configuration of the system. As an illustration, energetic calculations using this formalism for different structures of graphite are presented in figure 4 [49]. This formalism has been used in the example of molecular encapsulation in nanotubes that we will present in the next section. We have also used some extensions of this formalism to study the case of metal organics interfaces [30, 62].

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As a last remark, it is worth noticing that a general trend nowadays in the development of formalisms which take into account the vdW energy is the screening of the interaction. Indeed, this dynamical effect is mostly neglected even though it can actually reduce the C₆ coefficients by up to a factor of 4. Going into detail on this topic is far beyond the scope of this review and moreover there is no relation yet with molecular encapsulation, but some approaches have been developed in [31, 63].

Since the discovery of nanotubes and their ability to encapsulate molecules in their inner space, many studies have been devoted to these hybrid systems. The most famous and common example is the peapod structure [64], where C₆₀ molecules are inserted in the nanotube, as illustrated in figure 5. This system is particularly interesting for the realization of true one-dimensional systems, were fullerenes are aligned inside the nanotube along one specific direction, due to the confinement inside the limited space of the nanotube. Consequently, the interaction between the fullerenes can be affected by the interaction with the nanotube. On another hand, the encapsulation of molecules inside the nanotube can lead to electronic modifications of the nanotube, as well as changes in its mechanical properties. Therefore, this hybrid system is of high interest for scientific and industrial applications.

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Several models have been considered to characterize these systems. As an illustration, we can mention the recent work of Tang and Yang [66] who described the encapsulation of C₆₀ molecules in SWCNT in solvent conditions by the mean of molecular dynamics simulations. In this approach, the vdW energy is included in the potentials used for the simulation and which have been fitted on reference materials earlier. Considering three different solvents, they have demonstrated the ability of inserting these molecules in the nanotube in good agreement with experimental results. In particular, they have shown that the insertion is not always possible due to a strong potential barrier at the entrance of the tube, stressing the important role of the solvent in molecular encapsulation in nanotube.

Coming back to first-principles methods, we have to say that many studies of encapsulated molecules in CNTs, and in particular peapods, are performed using standard DFT calculations without including weak and vdW forces. Although it seems to be a strong approximation, it can be justified in two ways: first, the emergence of methods able to treat this kind of interactions is very recent and, until they appeared, DFT calculations were considered as reference works even for weakly interacting systems. Second, in many cases, due to a compensation of errors, standard DFT can recover more or less the geometry of the system (with however a strong difference on the energy value), leading to a reasonable theoretical characterization of the peapod structure. This is the case for example of the work of Okada [67] where the energetics of C₆₀ molecule inserted in CNTs ranging from the smallest diameter, (10, 10) to the largest, (20, 20), has been determined using DFT. It has been revealed that there is an important deformation of the nanotube with the inclusion of the fullerenes, which modifies the tube cross section from circular to elliptical shape. The corresponding deformation energy is around 10 meV per atom. In addition, the author has considered the case of the C₆₀ dimerization in the nanotube, and found that this reaction is exothermic, the gain in energy being around 0.1 eV per C₆₀. However, the activation barrier for the dimerization is found to be 1.4 eV indicating that polymerization is unlikely to take place without an external added energy like irradiation. Consequently, this work is really interesting for gaining an insight in the understanding of fullerene polymerization inside a CNT. However, when we have a look at the binding energy, we can notice that the lack of vdW energy changes the quantitative aspects of this system dramatically. For example, in this work, Okada finds a binding energy of around 1 eV per C₆₀ molecule in a (12,12) nanotube.

Using the LCAO-S²+vdW approach, we have determined the binding energy of the encapsulated C₆₀ in a (12,12) CNT [50]. The inclusion of vdW energy yields a binding energy of 2.28 eV per molecule, as represented in figure 6, which is more than twice of what is obtained in standard DFT calculation.

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This particular example stresses quite well the fact that even if DFT calculations can yield the correct geometry for those weak-interacting systems, the energetics are really dominated by vdW interactions. Therefore, their inclusion in the formalism is a requisite to obtain an accurate energetic description of the system.

In the next example, we will see that, in addition, there are weak-interacting systems where not only the energetics but also the geometry of the system are ruled by vdW interactions. In those cases, not including vdW energy in the formalism will lead to an erroneous result. The system is the following: we consider a C₆₀ molecule encapsulated in a double-walled CNT (DWCNT) [68]. The inner tube is a (15,0) CNT meanwhile the outer tube is a (24,0), as represented in figure 7.

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This system has been chosen in agreement with the experiments reported by Khlobystov et al [69] where assembled molecular arrays of C₆₀ inside DWCNTs have been observed by means of high-resolution transmission electronic microscopy (HRTEM). The specificity of this system lies in the really small radius of the inner tube. Indeed, DFT calculations with or without inclusion of vdW forces show that C₆₀ molecular encapsulation in the SWCNT (15,0) is not allowed thermodynamically due to the strong repulsion with the tube walls. The idea is then to stabilize the system using a second nanotube, the (24,0) around the C₆₀@(15,0).

First of all, we can observe that the standard DFT-LDA calculation, performed with the FIREBALL code, yields a totally repulsive interaction, around 8.7 eV for the whole C₆₀ molecule at the minimum of interaction, that is to say, in the centre of the DWCNT [68]. This result means that either the C₆₀ molecule in the DWCNT(15,0)@(24,0) is not stable or the DFT-LDA formalism is unable to describe it properly since weak and vdW interactions are not considered. Now, we consider the evolution of the C₆₀ molecule binding energy, calculated with the LCAO-S² + vdW formalism, along the DWCNT axis. The purpose here is to observe the energy variation when moving the molecule inside and outside the tube. This binding energy is shown in figure 8.

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As we can observe, between 0 and 3.5 Å, there are oscillations of the binding energy, roughly between −1 and −0.5 eV/molecule. In this spatial range, the C₆₀ molecule is located inside the DWCNT, and the oscillations correspond to the tube corrugation associated with the loss of interacting atoms when the molecule is getting closer to the tube entrance. At 0 Å, the binding energy value is 668 meV for the whole C₆₀. Between 3.5 and 5 Å, we can see a potential barrier that the C₆₀ molecule has to overcome to go out of the DWCNT. From 5 Å, we observe an increase of the binding energy which is explained by the weak interaction between the C₆₀ molecule and the DWCNT entrance. This interaction presents an important energy minimum of almost 4 eV around 8.3 Å. This energy is much more important than in the DWCNT, because in this configuration the C₆₀ feels much less repulsion from the tube than in the confined configuration. Consequently, the attractive part is not so well balanced, and the binding energy becomes bigger. The barrier energy to be overcome by the C₆₀ molecule to go in the DWCNT is 3.426 eV in good agreement with previous work from Girifalco et al [24]. Farther from 8.3 Å, the binding energy begins to decrease, until there is no more interaction at very large distance. Experimentally, one of the proposed mechanisms entails that fullerenes enter nanotubes through defect openings which would then require less energy cost. Another mechanism would be (especially for DWCNTs) that fullerenes enter the tubes from their open ends after the DWCNTs are formed and for this a certain temperature is required. However, as far as we know the mechanism is not fully understood.

Consequently, in agreement with experiments the C₆₀ molecule accommodation is confirmed when and only when the vdW interactions are included. This is an indication that results based only on standard DFT methods (whether LDA or GGA) can be misleading to predict the energetics for these systems if vdW interactions are not properly included. Finally this result stresses the fundamental importance of these kind of interactions, not only as DFT corrections, but as fundamental for the correct description of the physics of the system.

To close this part, we mention here an experimental work as an example of peapod extensions, which is the study of metallofullerene encapsulations in CNTs [70]. The authors have achieved the encapsulation of metallofullerene like La₂@C₈₀ and Er_xSc_3−xN@C₈₀ (x = 0–3) and studied their structural properties by the mean of HRTEM and electron diffraction. Moreover, the thermal stability of these supramolecular assemblies has been characterized, which opens the way for the study of new electronic effects in such hybrid supramolecular structures. Similarly, on the theoretical side, we can mention the work of Ganji et al [71] who have considered the encapsulation of azafullerenes (nitrogen-doped fullerenes) on the basis of a DFT treatment. The obtained results are comparable to the one of standard peapods.

In this part, we will now consider examples of more general molecules encapsulated in CNTs. In order to remain in the all-carbon systems, i.e. graphitic materials, we can consider the inclusion of graphene nanoribbons in SWCNT (GNR@SWCNT) [72]. In this study, Mandal et al characterize theoretically the encapsulation of coronene and perylene based nanoribbons in zig-zag SWCNT of different sizes, ranging from the (14,0) to the (20,0). The computational methodology is a self-consistent charge density-functional tight-binding (SCC-DFTB) method, considering vdW interaction between nanoribbon and nanotube (on the basis of a set of parameters for H and C), based on the Slater–Kirkwood model. Graphene nanoribbons have been placed in the centre of the tubes in order to perform geometry optimization and energetics calculations. As a result, the nanoribbon encapsulation in a nanotube is possible, as an exothermic process, which opens many experimental potentialities. However, if most of the systems remain with an unchanged shape, the smaller tubes have to get adapted to the size of the encapsulated nanoribbon. This aspect is obviously related to the repulsive energy between the tube wall and the nanoribbon when the tube is too small, leading to an unfavourable energetic situation. The most stable coronene based structure, GNR(Coronene)@(18,0), presents a metallic character while the other coronene based structures are semiconducting. Regarding the perylene based nanoribbons, the hybrid encapsulated systems appear to be of great interest for solar cell applications, due to a lower charge recombination rate.

Another prototypical system of graphitic-like molecules as a π-conjugated organic system is the molecule of benzene [73]. Its encapsulation in a zig-zag CNT (n, 0), with n ranging from 10 to 18, has been studied theoretically using the vdW-DF approach [35]. In this work, the position and orientation of the molecule inside the nanotube have been explored, in order to find the optimal tube diameter for encapsulation. This diameter appears to be around 1 nm. The wall–molecule distance depends on this diameter, but in all the cases, the molecule tends to align its molecular plane with the SWCNT axis. The chirality of the tubes is shown to play no role, but the tube has an influence on the intermolecular distances for confined benzene arrays. Here as well the structure and energetics of the hybrid system is ruled by vdW interactions.

In the same manner, we can mention the nice experimental work of Fujihara et al [15] where they report novel GNR growth through the vapour-phase doping of coronene in CNTs. The process is schematically represented in figure 9. The system is then characterized by HRTEM and Raman spectroscopy, revealing the encapsulation of linearly condensed coronene oligomers, that is to say GNRs. Consequently, this work shows that a CNT can constitute the host of molecular polymerization of graphitic molecules in order to grow GNR.

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Similarly, in a theoretical study from Kim et al, the authors determine the energetics of a single acetylene (C₂H₂) molecule and the formation energy of acetylene polymerization inside the nanotube [74]. Using the PWSCF-DFT package, they calculate the activation energy barrier in polymerization of the acetylene molecules confined in a (5,5) nanotube, as represented in figure 10. The process is found to be exothermic since there is practically no activation energy barrier. As an application, a process is proposed to fabricate acetylene polymers in a CNT, without interchain coupling, helping to understand the intrinsic properties of the individual chains.

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Notice that in another theoretical work [75], the characterization of carbon chain molecules encapsulated in an SWCNT has been achieved using the DMOL program [76]. They show that polyynes and their dehydrogenated forms can be spontaneously encapsulated in an open-ended SWCNT. No polymerization is determined in this work, but the encapsulation reaction is also found to be exothermic, confirming the results found in the above cited work and showing that hydrogenocarbonated molecules encapsulation is an energetically favourable process.

As an extrapolation, we can remark that a specific graphitic molecule can be constituted by another nanotube inserted in the host nanotube. This system has been studied theoretically as a reference system using the LCAO-S² + vdW by calculating the inter-wall energy of a DWCNT like (4,4)@(10,10). The resulting cohesion energy is 0.92 eV Å⁻¹, which is totally comparable with the one obtained with the fullerene encapsulation in the same case [50].

To close this section, we will finish by presenting theoretical calculations on polythiophene encapsulation in CNTs. In a first study [77], terthiophene molecules are inserted in zig-zag CNTs with diameters ranging from 8 to 13 Å. Calculations have been performed using the SIESTA code without inclusion of vdW forces [78]. Here also the encapsulation process is shown to be exothermic as represented in figure 11 for a cleaned or hydrogenated opened SWCNT.

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Similar energy profiles are observed in both cases, indicating that electrostatic forces induced by the dangling bonds at the SWCNT termination are not determinant for the insertion process. This means again that vdW forces dominate the molecular encapsulation process in CNTs. It is observed then that the incorporation reaction is exothermic by 1.2 eV. Thus, it is energetically more favourable that the molecule enters into an uncapped nanotube than remains outside, exhibiting a barrierless insertion process. Since the CNT is slightly charged due to a curvature effect, the encapsulation of the terthiophene is driven by nanocapillarity forces, induced by the interaction between both electronic densities of the nanotube and the molecule. Moreover, band-structure calculations show that the electronic structure of the molecule is preserved through encapsulation.

The second work on polythiophene encapsulation in CNTs present a general scheme for predicting the electronic structure of vdW bound systems, using this example of encapsulation [79]. They have therefore proceeded to DFT calculations, using the QUANTUM-ESPRESSO package [80] and considering vdW forces through the use of the vdW-DF method in a post-GGA manner. Several SWCNTs of (n, 0) type have been considered, with n ranging from 10 to 22. Here as well, the exothermic character of the process is exhibited, and an electronic structure study is performed to show the negligible charge transfer between the tube and the molecule due to vdW interactions.

In the next sections, we will precisely consider those electronic aspects in the encapsulation process. We have seen many examples of structures and configurations, we will now consider some possible applications as consequences of these structural aspects, as magnetic, charge transfer and electronic transport, and optical properties of encapsulated molecules in CNTs.