Introduction

Hexagonal BN (hBN) lattices composed of alternating boron (B) and nitrogen (N) atoms can form a range of morphologies, including boron nitride nanotubes (BNNTs), planar hBNs, or closed BN cages, similar to their carbon counterparts of carbon nanotubes (CNTs), graphenes, or fullerenes, respectively [1, 2]. Recently, BNNTs have attracted great attention due to their intriguing properties that are distinct from CNTs, such as their wide band gap, high thermal stability, piezoelectricity, and neutron shielding capability [3, 4]. However, the progress of introducing BNNTs into real-world applications in various sectors such as composites, sensors, or electronics has been hampered by the limited access to BNNTs with high crystallinity and high purity. The recent advances in BNNT synthesis by laser and thermal plasma processes have achieved rapid growth of hBN phases from BN precursors and enabled the large-scale production of high-quality BNNTs [5,6,7,8,9]. However, the as-produced materials still contain various hBN impurities, including hBN sheets and shells, requiring additional purification steps. To date, extensive efforts have been made to obtain highly pure BNNTs at large quantities from synthesis to post-synthesis processes [10,11,12,13].

Surface functionalization could be exploited to achieve selective dispersion of BNNTs in various solvents [14, 15]. Covalent modification of the BNNT surface by attaching chemical moieties may not be effective for selective functionalization because BNNTs and other BN impurities, mainly hBN sheets, are chemically identical yet differ only in their morphology. Recently, Mapleback et al. have formulated stable BNNT dispersions coexisting with hBN by reacting BNNT/hBN with hydroxyl (·OH) radicals produced in an advanced oxidation process [16]. Physical adsorption of polymers could be an excellent alternative to differentiate BNNTs from hBN by exploring the effect of dimensionality and size of hBN structures on the polymer binding behavior [17,18,19,20]. A conjugate polymer poly(m-phenylenevinylene-co-2, 5-dioctoxy-p-phenylenevinylene) (PmPV) had shown potential to solubilize and effectively separate BNNTs from BN impurities by wrapping around the tubes through strong π–π interactions, while other BN particles and fibers with large diameters remained unreacted and thus precipitated [19]. Recently, Yu et al. reported that polyfluorenes, which have been largely explored for the chirality and diameter selection of CNTs [21,22,23,24,25,26], greatly improved isolation and dispersion of BNNTs by wrapping BNNTs selectively rather than other hBN or BNNT bundles [20].

As demonstrated above, the previous BNNT functionalization approaches have been largely adopted from the methods initially developed for CNT separation or sorting. However, it would be more desirable if one could fully exploit the unique surface characteristics of BNNTs originating from the mixed ionic and covalent bonding nature of a BN lattice. This may allow for more effective implementation of BNNT functionalization strategies [27,28,29]. One notable approach is to utilize the Lewis acid–base interaction: affinity of the electron-rich N (or phosphorus) atoms of functional moieties (i.e., Lewis bases) toward the electron-poor B atoms in BN lattices (i.e., Lewis acids) [30,31,32]. Recently, pyridine with Lewis base character was shown to serve as a good BNNT dispersant, effective in a wide range of solvents [33]. For instance, pyridine-contained poly(4-vinylpyridine) (P4VP) polymers have shown great promise to functionalize and disperse BNNTs whereby separating BNNTs from hBN sheets, though impurities less than 100 nm remained unseparated [34, 35]. However, the detailed BNNT dispersion mechanism is still elusive.

Molecular dynamics (MD) simulations can shed a light on the separation mechanism of BNNTs from other hBN morphologies by investigating the interaction strength and molecular details at the interface between various BN lattices and relevant dispersants. The previous MD simulations were largely devoted to studying the interfacial binding between CNTs/BNNTs and polymers wrapping around the nanotubes [36,37,38,39,40,41,42,43,44], yet there has been a lack of studies that attempt to examine the effect of adsorbent lattice morphology upon interaction with polymers. This study aims at a systematic investigation of binding energetics and adsorption characteristics of polymers interacting with a variety of BN lattices, including 12 different BNNTs of chirality and diameter varying from 0.4 to 4.1 nm, and 8 different hBN sheets of size ranging from 8 to 24 nm. Specifically, we focused on a 120-mer P4VP polymer to scrutinize the role of distinct electrostatic interaction with partially charged BN lattices. We also performed simulations for the same polymer chain interacting with CNTs and graphene sheets, for which electrostatic interactions with polymers are almost absent, thereby highlighting the role of the BN lattice’s partial ionic character. Our simulation results revealed prominent differences in P4VP conformations and interaction energetics, not only between the BNNT and hBN surfaces, but also between the BN materials and their carbon counterparts. We discuss that our calculation results on the relative stability among adsorption complexes can be ultimately exploited to search for the optimal binding energy window, ensuring BNNT solubilization with morphology selection. In addition, the analysis of the different energy components (i.e., polymer molecular energy and van der Waals and Coulomb interactions) explained the interplay between different energy contributions to determine the binding energy, thus providing mechanistic insights of designing the ideal polymeric dispersants for BNNT dispersion.

Materials and methods

Molecular models

Lattice models of BNNTs and hBNs are constructed with a B–N bond length of 1.446 Å, while CNT and graphene lattices are modeled with a C–C bond length of 1.418 Å. The nanotubes have a finite length of 25 nm, and their diameters vary from 0.4 to 4.1 nm. Both armchair (n, n) and zigzag (n, 0) structures are considered with n = 5, 10, 15, 20, 25, 30, and n = 5, 7, 10, 15, 20, 25, respectively. The hexagon-shape hBN and graphene sheets are modeled, and their lateral lengths are ranging from 8 to 24 nm (see Fig. 1). B and N atoms in the lattices are partially charged with + 0.4 e and − 0.4 e, respectively, while carbon atoms are neutral. The polymer model of P4VP is composed of 120 monomer units, and the total number of atoms and the corresponding molecular weight being 1802 and 12,619 g/mol, respectively. The length of P4VP is sufficiently long enough to highlight conformation changes upon adsorption, though the polymer end effect appears as the tube diameter increases (> 4 nm). For P4VP–BNNT/CNT/hBN/graphene complexes, a 120-mer P4VP was initially placed in parallel alignment with the axes of nanotubes, or the central line in the plane of hBN/graphene with a separation distance of ~ 7 Å.

Figure 1
figure 1

Molecular models of (a) P4VP polymer, (b) BNNT, and (c) hBN sheet. The P4VP polymer model is composed of 120 pyridine rings. The C, N, and H atoms are colored in gray, blue, and white, respectively, while the carbon backbone of the polymer is colored in red for the clarity of polymer conformation.

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Computational details

All MD simulations were implemented and carried out using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) code. The interatomic interactions of BNNT/hBN and CNT/graphene lattices were simulated using the Tersoff potential [45,46,47]. The OPLS-AA force field parameterizations [48] were used to calculate the intramolecular interactions of P4VP, as well as the intermolecular interactions between P4VP and BNNT/hBN (or CNT/graphene). We note that more accurate force field potentials, such as the interface force field (IFF) that were recently developed for prediction of interfacial energies in nanomaterials [40], are available for graphitic materials, but not yet developed for hBN materials. The combination of the OPLS-AA and Tersoff parameterizations allowed us to simulate both systems on equal footing and systematically compare them with one another. The Lennard–Jones coefficients employed for B, N, and C were \({\sigma }_{\mathrm{B}}=3.453 \) Å, \({\sigma }_{\mathrm{N}}=3.365 \) Å, \({\sigma }_{\mathrm{C}}=3.369 \) Å, \({\varepsilon }_{\mathrm{B}}=0.0949\) kcal/mol, \({\varepsilon }_{\mathrm{N}}=0.1448\) kcal/mol, and \({\varepsilon }_{\mathrm{C}}=0.06076\) kcal/mol. The cutoff distance of non-bonded van der Waals and Coulomb interactions was set to 12.5 Å. The partial charges of the polymer were computed using the charge equilibration (Qeq) method [49]. All simulations were performed in NVT ensemble at 300 K with Nose–Hoover thermostat, and consisted of 1 ns equilibrium runs with a time step of 0.1 fs, followed by 1 ns production runs with a time step of 1 fs. Both P4VP and BNNT/CNT/hBN/graphene lattices were allowed to fully relax during the entire simulation run. The simulation cell dimensions were 300 Å × 300 Å × 300 Å. Note that all the data reported here were obtained by averaging over each production run of three independent simulations. The effects of solvent were not considered (i.e., simulated in vacuum) as the main focus of this study is to understand the intrinsic interaction energetics between polymer and BN or C lattice surfaces. We note that the net binding energies could be affected by additional polymer–solvent or lattice–solvent interactions [40].

Results and discussion

Adsorption geometries of P4VP–BNNT complexes

We first examined the morphological aspects of the P4VP–BNNT complexes by considering a 120-mer P4VP polymer interacting with 12 different types of BNNTs in terms of chirality and diameters. The adsorption geometries of P4VP on the BNNT surfaces were obtained from MD simulations, and the corresponding snapshots at the end of the production runs are displayed in Fig. 2. P4VP helically wrapped around the BNNTs and their helical pitches varied with the tube diameter. We note that the polymer was initially aligned along the tube axis without a particular wrapping conformation and then allowed to freely interact with the BNNTs. It is remarkable to observe that P4VP developed such high selectivity of helical pitches depending on the tube diameter. Although it is generally expected that polymers adopt helical wrapping configuration on both nanotubes of CNTs and BNNTs, the origin of helical wrapping conformation, in particular, whether it is the global free energy minimum or a metastable state, is not clearly known yet. It is speculated that the structures of P4VP backbone and its side group and their interaction energetics with BNNTs together with the polymer intrinsic properties such as bending stiffness may greatly promote the spontaneous helical wrapping of P4VP on the BNNT surface. On the (15, 15) BNNT, the polymer chain formed a kink, which is likely a kinetically trapped intermediate conformation. P4VP exhibits a more regular helical structure on small-diameter BNNTs compared to large diameter BNNTs. Owing to the relatively short polymer length, the effect of the polymer ends has probably become more evident. Note that a 120-mer P4VP is not sufficiently long enough to make a full circle around the (25, 25) and (30, 30) BNNTs. Large-scale fluctuations of the nanotube backbones were noticeable for the small-diameter tubes of (5, 0), (7, 0), (10, 0), and (5, 5) (< ~ 1 nm), implying the amplitude of BNNT bending fluctuations strongly depends on the tube diameter [see Fig. 2(b)].

Figure 2
figure 2

Adsorption geometries of a 120-mer P4VP on the surface of BNNTs with various diameters and chirality: (a) side and (b) top views.

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The cross-sectional views of the complexes showed that tube diameter also influenced the orientation ordering of the pyridine side rings. Upon wrapping on the large diameters of BNNTs [e.g., (20, 0), (25, 0), (10, 10), (15, 15), (20, 20)], the plane of the pyridine rings oriented perpendicular to the tube surface. In contrast, the rings oriented more randomly for the smaller diameter tubes [e.g., (5, 0), (7, 0), (10, 0), and (5, 5)]. Blow-ups of the snapshots in Fig. 3 revealed that three pyridine rings (inside the yellow box) were collectively engaged in wrapping, in which two side rings “grabbed” the BNNT surfaces, while the middle ring formed a dihedral angle of 90° with the BNNT surface, whereby the N atom of pyridine pointed outward from the tube surface. Such orientation of the pyridine rings in P4VP are quite opposite to that of a single pyridine molecule upon its interaction with BNNTs. From the previous DFT calculation [32, 34], the most stable configuration of a pyridine–BNNT complex showed that the negatively charged N atom in pyridine pointed inward to the BNNT surface, thereby positioning itself 1.68 Å apart from the positively charged B site of the BNNT with a binding affinity of − 17.0 kcal/mol. The single pyridine interaction with BNNTs is dominantly governed by the strong electrostatic interaction between the electron lone pair of the N atom and the empty p-orbital of the B atom. On the other hand, the current study suggests that the adsorption configuration of macromolecular P4VP on BNNTs, rather, would be determined by the collective behavior of the long chain polymer through the complex interplay between the electrostatic and van der Waals interactions (i.e., strong π–π-stacking interactions between aromatic N-heterocycles and BNNT).

Figure 3
figure 3

Close-up views of P4VP polymers wrapping around (5, 0), (10, 10), and (20, 20) BNNTs, which are shown in (a), (b), and (c), respectively. The 3-monomer units in the yellow box repeat their arrangement while wrapping the nanotube surfaces.

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Benchmarking simulations were performed for P4VP–CNT complexes and their simulation snapshots are provided in Fig. 4. Like the P4VP–BNNT complexes, P4VP helically wrapped around CNTs, though formation of kinks and folding in the polymer backbones appeared more frequently compared to the P4VP–BNNT complexes. This observation implies that P4VP is in a better contact with the surface of BNNTs than CNTs. The most notable difference in polymer adsorption conformation was observed with the largest diameter tube of (30, 30), in which P4VP exhibited a random coil conformation on the CNT. The different morphology of P4VP could be originated from the stronger interaction of P4VP with BN than C lattices. A similar observation was made from the adsorption geometries of P4VP on planar hBNs and graphene sheets, as we will discuss below. Large-scale bending fluctuations of CNTs were also observed for the small-diameter tubes of (5, 0), (7, 0), (10, 0), and (5, 5) (< ~ 1 nm), similar to the case of BNNTs, indicating that the bending stiffness of CNTs and BNNTs is comparable as predicted from the elastic continuum model with ab initio parametrization [50]. It is worth noting that the bending stiffness of single-walled CNTs was estimated by measuring Brownian bending dynamics, which scales as the cube of tube diameters, regardless of chirality [51].

Figure 4
figure 4

Adsorption geometries of a 120-mer P4VP on the surface of CNTs with various diameters and chirality: (a) side and (b) top views.

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We now investigate the adsorption geometries of P4VP on planar hBN and graphene sheets with a lateral size ranging from 8 to 24 nm, presented in Figs. 5 and 6, respectively. For the small sized sheets (i.e., D ≤ 12 nm), P4VP formed random coil structures regardless of the atom type. Interestingly, when the sheet size was about 8 nm, the polymer was folded and wrapped around both side of the hBN, maximizing polymer coverage on the lattice surface, while the entire chain resided on one side of the sheet for the same size of graphene sheet. For the larger sheets (i.e., D ≥ 14 nm), P4VP adopted extended coil configurations on hBNs, thereby increasing polymer contacts with the lattice surface. In contrast, P4VP on graphene sheets showed crumpled coil structures, thereby increasing the intramolecular contact of P4VP (i.e., polymer chain self-interaction). It is interesting to find an analogy of polymer configuration in a “good” and “bad” solvent environment: hBN and graphene serve as a “good” and “poor” adsorbent lattice, leading P4VP to form an extended coil and compact globule state, respectively.

Figure 5
figure 5

Morphologies of P4VP adsorbed on hexagonal-shape hBN sheets with different sizes.

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Figure 6
figure 6

Morphologies of P4VP absorbed on hexagonal-shape graphene sheets with different sizes.

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As confirmed by the simulation snapshots, P4VP could adsorb on all the different types of adsorbent lattices considered in this study, though its adsorption conformation is greatly influenced by the atomic type (C- or BN-) and morphology (tubular or planar) of interacting lattices. P4VP helically wrapped around both nanotubes of BNNTs and CNTs, while they adsorbed on planar lattices of hBN/graphene in a random coil conformation. We have also calculated and plotted the radius of gyration \({R}_{\mathrm{g}}\) of P4VP with respect to the system size (i.e., the total number of BN or C atoms constructing finite-size lattices) in Fig. 7. Here,\({R}_{\mathrm{g}}\) is defined by \({R}_{\mathrm{g}}^{2}=\frac{1}{M}\sum_{i}{m}_{i}{\left({r}_{i}-{r}_{\mathrm{cm}}\right)}^{2}\), where M is the total mass of P4VP, mi and ri are the mass and position of ith constituent atom, respectively, and rcm is the center of mass position of P4VP. It was found that \({R}_{\mathrm{g}}\) of the adsorbed P4VP is always larger than that of a free P4VP being estimated as 15.9 Å [see its conformation in Fig. 7(a)]. When comparing between BN and C adsorbent lattices for the same size and dimensionality, P4VP was always more extended on the BN lattice rather than C lattice, thus resulting in a larger \({R}_{\mathrm{g}}\). As the size of the graphene sheet increases, the magnitude of \({R}_{\mathrm{g}}\) remains almost constant, while as the size of hBN sheets increases, it also increases due to further extension of the polymer fully exploring on the available lattice surface.

Figure 7
figure 7

(a) Conformation of a free 120-mer P4VP, (b) P4VP adsorption conformations on (25, 0) CNT and BNNT, calculated radius of gyration of P4VP adsorbed on (c) BNNT/hBN and (d) CNT/graphene.

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Binding energies of P4VP on the surface of BNNT/CNT/hBN/graphene

Binding energy between P4VP and BNNT \({(E}_{\mathrm{P}4\mathrm{VP}{-}\mathrm{BNNT}}^{\mathrm{b}})\) is estimated by the difference between potential energy of the P4VP–BNNT adsorption complex \({(E}_{\mathrm{P}4\mathrm{VP}{-}\mathrm{BNNT}})\) and the sum of potential energies of P4VP \({(E}_{\mathrm{P}4\mathrm{VP}})\) and BNNT \(({E}_{\mathrm{BNNT}})\) molecules as follows:

$${E}_{\mathrm{P}4\mathrm{VP}{-}\mathrm{BNNT}}^{\mathrm{b}}={E}_{\mathrm{P}4\mathrm{VP}{-}\mathrm{BNNT}}-\left({E}_{\mathrm{P}4\mathrm{VP}}+{E}_{\mathrm{BNNT}}\right).$$
(1)

Similarly, we have also calculated binding energies of P4VP interacting with CNTs, hBN, and graphene sheets, which are denoted by \({E}_{\mathrm{P}4\mathrm{VP}-\mathrm{CNT}}^{\mathrm{b}}\), \({E}_{\mathrm{P}4\mathrm{VP}-\mathrm{hBN}}^{\mathrm{b}},\) and \({E}_{\mathrm{P}4\mathrm{VP}-\mathrm{graphene}}^{\mathrm{b}}\), respectively.

In Fig. 8, we plotted binding energies of P4VP adsorbed on BNNTs and hBN sheets (a) and CNTs and graphenes (b) as a function of the total number of BN and C atoms, which were also marked with the corresponding chirality of BNNTs/CNTs and lateral size (nm) of hBN/graphene sheets. Such plots allow us to identify the effect of atom type and morphology of interacting hexagonal lattices on the adsorption strength. The results showed that binding energies are negative for all types of lattices, and thus, P4VP adsorption is always energetically favorable. Overall, binding of P4VP showed approximately two times stronger adsorption on the BNNT/hBN surfaces than on their carbon counterparts. The stronger adsorption of P4VP on the BNNTs may enhance contact between the polymer and the tube surface, resulting in the regular helical conformation of P4VP on the BNNTs with less kinks and folding as shown in Fig. 2. For both BNNTs and CNTs, binding energies increase with their diameter. A similar trend has been evidenced by previous studies on other polymer–CNT complex systems [24, 41]. For BNNTs, P4VP was found to be more stable on the zigzag tubes of (n, 0) than on the armchair tubes of (n, n), while for CNTs, the binding energies simply increase with diameter, independently of chirality. An energy bump was observed for the (15, 15) BNNT, which is likely associated with the kink formed on the polymer backbone (see Fig. 2), though the reason for the similar feature found on the (20, 0) BNNT is not clear. For the planar lattices of hBN and graphene, binding energies remained independent of the sheet size. However, there is an important difference: binding of P4VP on graphene is always energetically favorable over CNTs, while it becomes comparable on hBN and BNNTs as the diameter of BNNTs approaches greater than ~ 3 nm.

Figure 8
figure 8

Binding energies of a 120-mer P4VP polymer interacting with (a) BNNTs and hBN sheets and (b) CNTs and graphene sheets as a function of the number of BN or C. The (n, m) chirality of nanotubes and lateral size (nm) of planar lattices were denoted together with their number of atoms.

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The analysis on the binding stability of polymer-adsorbed complexes has been widely explored to understand the principles of the selectivity of adsorbent morphologies. For example, conjugated polymer poly[9, 9-dioctylfluorenyl-2,7-diyl] (PFO) has proven to selectively solubilize particular diameters of CNTs [22,23,24]. In this case, binding energy between PFO and CNTs serves as an indicator to assess the stability of PFO wrapping and thus CNT dispersion. As such, our calculations of binding energies can predict the ability of P4VP to disperse interacting adsorbent lattices. The stronger adsorption of P4VP on BNNTs as compared to CNTs suggests that P4VP could provide more stable dispersion for BNNTs than CNTs. Experimentally, it was reported that P4VP served as a good dispersant for both CNTs and BNNTs [34, 35, 52, 53]. The main purpose of P4VP treatment for CNTs was debundling of nanotubes [52, 53], suggesting that the calculated P4VP binding energies are sufficiently large enough to debundle and solubilize CNTs. For BNNTs, P4VP was employed to selectively disperse BNNTs (~ 5 nm) from the as-produced mixture containing hBN impurities [35]. It was reported that while P4VP greatly improved the dispersion stability of BNNTs over large hBN flakes, hBN impurities smaller than ~ 100 nm were also dispersed and stable in the solution.

According to our binding energy analysis, both BNNTs and hBN sheets interact with P4VP with a strong binding stability despite the differences in the adsorption conformation. This may present a challenge in the effective separation of BNNTs from hBN impurities when using P4VP. However, we note that in order to estimate the true dispersion stability of BNNTs or hBNs, interactions beyond the single polymer–BNNT/hBN level must be taken into account. Differences in the morphological structures between BNNT (e.g., few-layered) and hBN (e.g., flakes stacked with many layers) also provide differences in the surface area available for interaction with polymers (i.e., degree of functionalization), which ultimately determines the binding stability. For a given number of B and N atoms, debundled, few-layer BNNTs provide more surfaces available for interaction with P4VP over hBN flakes composed of many layers. This can make separation of BNNTs from large hBN flakes feasible. For small size, few-layer hBN flakes, there is no difference in terms of the layer number compared with BNNTs, rendering its effect negligible. Our simulation results predicted that the binding of P4VP on a single-layer hBN is more stable than the wrapping of P4VP on BNNT when the BNNT diameter is smaller than 3 nm. Thus, small few-layer hBN can have a comparable or better dispersion stability over BNNTs. This may explain the reason why large hBN flakes were successfully removed, yet small hBNs remained stable in the solution with BNNTs in the previous experimental study [35]. Our work improves the understanding of the purification mechanism, and provides insight into finding the optimal binding energy window that selectively prefers one lattice over another by fine tuning of the interplay between polymer–lattice binding affinity and morphology (e.g., number of layers and dimensionality) [54].

Contributions from van der Waals and Coulomb pair interactions and molecular energy

The potential energy of each system in Eq. (1) consists of different energy contributions, such as polymer molecular energies (\({E}_{\mathrm{sys}\_\mathrm{mol}}\)) including bond stretching, angle bending and torsion energies, non-bonded Coulomb (\({E}_{\mathrm{sys}\_\mathrm{coul}}\)) and van der Waals (\({E}_{\mathrm{sys}\_\mathrm{vdw}}\)) interaction energies, and the Tersoff lattice energies (\({E}_{\mathrm{sys}\_\mathrm{tersoff}}\)). For systems of P4VP–BNNT complex, P4VP, and BNNT, the total potential energies are represented by Eqs. (2), (3), and (4), respectively:

$${E}_{\mathrm{P}4\mathrm{VP}{-}\mathrm{BNNT}}={E}_{\mathrm{P}4\mathrm{VP}{-}\mathrm{BNNT}\_\mathrm{mol}}+{E}_{\mathrm{P}4\mathrm{VP}{-}\mathrm{BNNT}\_\mathrm{vdw}}+{E}_{\mathrm{P}4\mathrm{VP}{-}\mathrm{BNNT}\_\mathrm{coul}}+{E}_{\mathrm{BNNT}\_\mathrm{tersoff}}{^{\prime}},$$
(2)
$${E}_{\mathrm{P}4\mathrm{VP}}={E}_{\mathrm{P}4\mathrm{VP}\_\mathrm{mol}}+{E}_{\mathrm{P}4\mathrm{VP}\_\mathrm{vdw}}+{E}_{\mathrm{P}4\mathrm{VP}\_\mathrm{coul}},$$
(3)
$${E}_{\mathrm{BNNT}}={E}_{{\mathrm{BNNT}}_{\mathrm{tersoff}}.}$$
(4)

Here, \({E}_{\mathrm{P}4\mathrm{VP}-\mathrm{BNNT}\_\mathrm{vdw}}\) and \({E}_{\mathrm{P}4\mathrm{VP}-\mathrm{BNNT}\_\mathrm{coul}}\) are attributed to both the contributions of the intermolecular interaction between P4VP and BNNT and the intramolecular interaction of P4VP. \({E}_{\mathrm{BNNT}\_\mathrm{tersoff}}{^{\prime}}\) and \({E}_{\mathrm{BNNT}\_\mathrm{tersoff}}\) are BNNT lattice energies with and without polymer adsorption, respectively, which are calculated by the Tersoff potential parameterization [45,46,47]. Then, the binding energy in Eq. (1) can be represented by the following three contributions of polymer molecular energies (\(\Delta {E}_{\mathrm{mol}}\)), van der Waals interactions (\({\Delta E}_{\mathrm{vdw}}\)), and Coulomb interactions (\({\Delta E}_{\mathrm{coul}}\)):

$${E}_{\mathrm{P}4\mathrm{VP}{-}\mathrm{BNNT}}^{\mathrm{b}}=\Delta {E}_{\mathrm{mol}}+\Delta {E}_{\mathrm{vdw}}+\Delta {E}_{\mathrm{coul}},$$
(5)

where

$$\Delta {E}_{\mathrm{mol}}={E}_{\mathrm{P}4\mathrm{VP}{-}\mathrm{BNNT}\_\mathrm{mol}}-{E}_{\mathrm{P}4\mathrm{VP}\_\mathrm{mol}},$$
(6)
$$\Delta {E}_{\mathrm{vdw}}={E}_{\mathrm{P}4\mathrm{VP}{-}\mathrm{BNNT}\_\mathrm{vdw}}-{E}_{\mathrm{P}4\mathrm{VP}\_\mathrm{vdw}}+\left({E}_{\mathrm{BNNT}\_\mathrm{tersoff}}{^{\prime}}-{E}_{\mathrm{BNNT}\_\mathrm{tersoff}}\right),$$
(7)
$$\Delta {E}_{\mathrm{coul}}={E}_{\mathrm{P}4\mathrm{VP}{-}\mathrm{BNNT}\_\mathrm{coul}}-{E}_{\mathrm{P}4\mathrm{VP}\_\mathrm{coul}}.$$
(8)

Here, \(\Delta {E}_{\mathrm{mol}}\) measures the difference in the intramolecular bonding energies (i.e., bond stretching, angle bending, and torsion) between free and adsorbed polymers, which describes the energy cost due to polymer conformation change upon adsorption. Note that for \(\Delta {E}_{\mathrm{vdw}}\) and \(\Delta {E}_{\mathrm{coul}}\), the intermolecular interactions between P4VP and BNNT are the dominant contributor, while the changes in the Tersoff lattice energy \(({\Delta {E}_{\mathrm{tersoff}}=E}_{{\mathrm{BNNT}}_{\mathrm{tersoff}}}{^{\prime}}-{E}_{{\mathrm{BNNT}}_{\mathrm{tersoff}}})\) are negligible.

In Fig. 9, we plotted \(\Delta {E}_{\mathrm{mol}}\), \(\Delta {E}_{\mathrm{vdw}},\) and \(\Delta {E}_{\mathrm{coul}}\) for the interactions of P4VP with BNNT/hBN (left panel) and CNT/graphene (right panel). Firstly,\(\Delta {E}_{\mathrm{mol}}\) predicted that the molecular energy cost of P4VP upon adsorption would increase with diameter for both BNNTs and CNTs. As we observed from our simulation snapshots in Figs. 2(b) and Fig. 3, the orientations of the pyridine rings are more constrained on the large diameter tubes, leading to an energy penalty \((\Delta {E}_{\mathrm{mol}}>0)\). On the other hand, P4VP formed less stressed polymer structures on the smaller diameter tubes due to the larger rotational degrees of freedom for the pyridine rings and the smaller bending stiffness of the tube, thereby gaining energy \((\Delta {E}_{\mathrm{mol}}<0)\). Note that P4VP adsorption always costs molecular energy penalties (\(\Delta {E}_{\mathrm{mol}}>0)\) on both hBN and graphene. On the planar lattices, the pyridine rings have the rotational degrees of freedom available only over the half of the plane, thereby forcing them under more constraints when compared to free polymers or the polymers confined onto one-dimensional structures. We note that the entropic effects due to the restriction of P4VP’s rotational degrees of freedom upon adsorption have not been considered in this study. It was found that \(\Delta {E}_{\mathrm{mol}}\) remains approximately the same for hBN as the sheet size increases, yet it increases for graphene. Without considering the non-bonded interactions, the molecular energy of the crumpled and compact P4VP on graphene with a smaller \({R}_{\mathrm{g}}\) is higher than that of the extended P4VP on hBN.

Figure 9
figure 9

Energy contributions to binding energies: (a), (b) polymer molecular energy, (c), (d) van der Waals interactions, and (e), (f) Coulomb interactions, for BNNT/hBN (left panel) and CNT/graphene (right panel). The number of contacts between N atoms in P4VP and B atoms on BNNT/hBN lattice calculated within a cutoff of 3.5 Å are shown in the inset of (c). All energy values are plotted as a function of the number of BN or C atoms constructing the finite-size lattices.

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Our calculations showed that van der Waals interactions predominantly contributed to the resulting binding energy. The geometries (i.e., the curvature) of the adsorbent lattices have shown to play an important role in determining the van der Waals interaction strength. The large curvature of the small-diameter tubes lowered the number of contacts within the van der Waals contact distance along the circumferential direction, leading to weaker van der Waals interaction. This is supported by the number of contacts between N atoms in P4VP and B atoms on the BNNT/hBN lattice calculated within a cutoff of 3.5 Å, as presented in the inset of Fig. 9(c). The result indicated a strong correlation between the number of intermolecular contacts and the van der Waals interactions. The planar hBN sheets have zero curvature, thus the van der Waals interaction for hBN is not sensitive to the sheet size. However, the van der Waals interaction with graphene increased with the sheet size. It is speculated that this increment is associated with the intramolecular van der Waals gains due to the collapsed polymer conformation on graphene, which is balanced with the large polymer energy costs [Fig. 9(b)]. Consequently, binding energies on graphene remained independent with respect to the sheet size.

Due to the ionic characteristic of BN lattices, the contribution of Coulomb interactions is expected to be more significant for BNNT/hBN than for CNT/graphene, which is evidenced by the calculation of \(\Delta {E}_{\mathrm{coul}}\) [see Fig. 9(e) and (f)]. In particular, the electrostatic interaction was shown to be markedly strong for the large diameter BNNTs [see the red circle in Fig. 9(e)]. The electrostatic gains compensated for the high van der Waals costs of BNNTs are relative to hBN, thus improving the binding stability for the large diameter BNNTs and ultimately making their binding stability comparable to that of hBN. For CNT/graphene, the Coulomb interactions were found to be almost negligible, and the dominant energy competition occurred only between polymer molecular energy and van der Waals interactions to determine their binding energy.

Conclusion

In this study, we systematically investigated the adsorption conformations and binding energetics of P4VP interacting with nanotubes (BNNTs and CNTs) of 12 different diameters as well as hexagonal sheets (hBN and graphene) of 8 different sizes by performing all-atom MD simulations. The polymer adsorption conformation was found to be greatly influenced by the lattice curvature (i.e., tube diameter), morphology (i.e., tubular or planar), and atom type (BN or C). Strong adsorption energies between P4VP and BNNT led the polymer to form a tightly ordered helical wrapping structure around the BNNT. Notably, P4VP is shown to pick out a well-defined helical pitch that depends on diameter upon wrapping on BNNTs. P4VP adsorption was found to be less stable on small-diameter BNNTs (< 3 nm) than on hBN, though it becomes comparable when the diameter of BNNT is greater than ~ 3 nm due to the enhanced electrostatic interactions. This result of binding energy estimation implies that small, few-layer hBNs have better, or similar, dispersion stability compared with BNNTs. This is in qualitative agreement with the challenges encountered in the previous experimental study where small hBN flakes of less than ~ 100 nm were not removable. By considering their stacking morphologies (e.g., number of layers), the mechanism of selective dispersion can be better addressed, in which the total binding energies can be tuned by the interplay between morphological parameters and intrinsic binding affinity. A comparative study between BN and C lattices highlighted the role of strong electrostatic interactions for the BN systems. Finally, we analyzed the different energy contributions (i.e., polymer molecular energy and van der Waals and Coulomb interactions) and their delicate competition to determine the binding energy, which provided insight into the design of a polymer able to separate BNNTs from other hBN phases.