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Project supported by the Funds from the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant No. 45005). Z. L. Mišković thanks the Natural Sciences and Engineering Research Council of Canada for Finacial Support.

We investigate the interactions of charged particles with straight and bent single-walled carbon nanotubes (SWNTs) under channeling conditions in the presence of dynamic polarization of the valence electrons in carbon. This polarization is described by a cylindrical, two-fluid hydrodynamic model with the parameters taken from the recent modelling of several independent experiments on electron energy loss spectroscopy of carbon nano-structures. We use the hydrodynamic model to calculate the image potential for protons moving through four types of SWNTs at a speed of 3 atomic units. The image potential is then combined with the Doyle–Turner atomic potential to obtain the total potential in the bent carbon nanotubes. Using that potential, we also compute the spatial and angular distributions of protons channeled through the bent carbon nanotubes, and compare the results with the distributions obtained without taking into account the image potential.

Soon after the discovery of carbon nanotubes, several groups started investigating interactions of charged particles with nanotubes. A number of theoretical papers have focused on ion channeling with the main aim to explore the possibility of guiding ion beams with carbon nanotubes.^{[1–14]} However, the experimental studying of ion channeling through carbon nanotubes is still in its beginning.^{[15,16]} Zhu *et al.*^{[15]} obtained the first experimental data on ion channeling through nanotubes with He^{+} ions of the kinetic energy of 2 MeV and an array of the well-ordered multi-wall carbon nanotubes grown in a porous anodic aluminum oxide (Al_{2}O_{3}) membrane. The authors measured and compared the yields of ions transmitted through the bare Al_{2}O_{3} sample and those through the Al_{2}O_{3} sample including the nanotubes. Chai *et al.* obtained the first experimental data with electrons and nanotubes.^{[16]} They used the 300-keV electrons and studied their transport through the multi-wall carbon nanotubes of the length in a range of 0.7 μm–3.0 μm, embedded in a carbon fiber coating. Nevertheless, this field of research still remains attractive.^{[17,18]}

An ion in the MeV energy range will induce a strong dynamic polarization of the valence electrons in carbon atoms. This will give rise to a sizeable image force on the ion, as well as a considerable energy loss due to the collective, or plasma, excitations of those electrons.^{[15–17]} Calculations of the image force in Refs. [19]–[21] were based on a two-dimensional (2D), one-fluid hydrodynamic model, which treats all four valence electrons in carbon atoms as a single fluid of quasi-free charges that occupy the surface of a cylinder. It was shown that the dynamic polarization effect exerts a large influence in the angular distributions of protons channeled through a short and straight (11, 9) SWNT.^{[19–21]}

However, in the ion channeling experiments, there is always a question about the curvature of carbon nanotubes, because in reality carbon nanotubes are seldom perfectly straight. There are some studies of channeling through the bent carbon nanotubes, but without taking into account the image force.^{[22–24]} So, it is important to study the influence of dynamic polarization of carbon nanotubes when the nanotubes are curved. Therefore, in this paper, we extend our investigation of the spatial and angular distributions of the channeled ions^{[19–21]} to the cases in which the carbon nanotubes are not straight. We evaluate the dynamic image potential for ions by means of an extended two-fluid hydrodynamic model. Unlike our previous work, where valence electrons in carbon were treated as a quasi-free charged fluid with zero damping,^{[19–21]} in the present work we use an extended version of the hydrodynamic model, where *π* and *σ* electrons in the nanotube wall are treated as two charged fluids, each with its own restoring frequency and damping factor.^{[25]} Such an extension of the hydrodynamic model with suitably chosen parameters showed great success and versatility in the recent modelling of several independent experiments involving electron energy loss spectra (EELS) of different carbon nanostructures. Specifically, the extended two-fluid hydrodynamic model was used to explain the experimental spectra in high-energy EELS of single- and multi-layer graphene,^{[26]} single-walled carbon nanotubes,^{[27]} and individual C_{60} molecules,^{[28]} as well as the low-energy reflection EELS of single-layer graphene supported by a metal substrate.^{[29]} While the trajectories of charged particles in those studies were either traversing or obliquely incident on carbon nanostructures, the present work presents our first application of the extended hydrodynamic model under channeling conditions. We use the dynamic image potential from the extended hydrodynamic model in a Monte Carlo (MC) type of computation to obtain the spatial and angular distributions of protons moving at a speed of *v* = 3 a.u. through the bent (6, 4), (8, 6), (11, 9), and (15, 10) SWNTs. By comparing those results with the distributions obtained without the image potential, we are able to give a preliminary assessment of the influence of the image potential on the distribution of ions channeled through the bent carbon nanotubes.

We first present the basic theory used in our modeling of proton channeling, then we discuss the results for spatial and angular distributions, and we finally give our concluding remarks. Atomic units are used throughout unless otherwise explicitly stated.

We model an SWNT as an infinitesimally thin cylindrical shell with radius *R* and length *L*, and assume that the valence electrons in the ground state may be considered as an electron gas distributed uniformly over a cylindrical surface, with a number density per unit area *π* and three *σ* electrons per carbon atom, respectively.^{[17]} We use cylindrical coordinates *r**ρ*, *φ*, *z*) and assume that a charged particle with the charge *Q* moves within the nanotube, with its trajectory parallel to the nanotube axis *z*, such that the instantaneous position of particle is given by *r*_{0}(*t*) = (*ρ*_{0}, *φ*_{0}, *vt*), where *v* is the particle’s speed.

Following Ref. [30], one may express the self-energy, or the image potential *U*_{im}, experienced by a point-charge ion *Q* on the trajectory *r**r*_{0} (*t*) as

_{m}and K

_{m}are cylindrical Bessel functions of integer order

*m*. We have set

*φ*

_{0}= 0 and used the symmetry properties of the real and imaginary parts of the density response function of a carbon nanotube, which is given by

^{[30]}

^{[25]}

*s*,

_{i}*ω*, and

_{ir}*γ*are the acoustic speed, restoring frequency, and the damping rate in the

_{i}*i*-th fluid (where the index

*i*takes values

*σ*and

*π*), respectively. In the Thomas–Fermi–Dirac approximation for the 2D hydrodynamic model, we use the acoustic speed which is given in Ref. [27]. As regards the adjustable parameters in Eq. (

^{[31]}and the high-resolution reflection EEL spectra of an SL graphene supported by a metal substrate,

^{[29]}namely,

*ω*

_{σr}= 0.48,

*ω*

_{πr}= 0.15,

*γ*

_{σ}= 0.1, and

*γ*

_{π}= 0.09.

In the present paper we also compute spatial and angular distributions of protons channeled through a bent SWNT with and without including the image force. The initial proton velocity is chosen to be 3 a.u. (corresponds to the energy 0.223 MeV) and it is taken to be parallel to the *z* axis. Numerical calculation is performed in a Cartesian coordinate system with the *z* axis that coincides with the nanotube axis and with the origin that lies in its entrance transverse plane. We assume that the nanotube is sufficiently short, for the proton energy loss is neglected, but is long enough so that we can omit the effect of nanotube ends on the image potential. The nanotube length of *L* = 0.2 μm is considered to satisfy both these requirements.^{[19]}

For the short-ranged proton interaction with carbon atoms in the nanotube wall, we use the repulsive Doyle–Turner potential (averaged axially and azimuthally over the nanotube wall),^{[3,32,33]} which gives

*Z*

_{1}= 1 and

*Z*

_{2}= 6 are the atomic numbers of the proton and carbon, respectively,

*R*is the nanotube radius,

*l*= 0.144 nm is the nanotube atomic bond length,

^{[34]}

*ρ*

_{0}is the distance between the proton and nanotube axis, I

_{0}designates the modified Bessel function of the first kind of the 0-th order, and

*a*= (0.115,0.188,0.072,0.020) and

_{j}*b*= (0.547,0.989,1.982,5.656) are the fitting parameters.

_{j}^{[33]}

We assume that the nanotube is bent along the positive direction of the *y* axis. Its bending angle is *α* = 4 mrad. The channeling mechanism in bent carbon nanotubes may be considered in a frame of reference that rotates around a center with the bending radius *R*_{b} of a nanotube. This means that the new reference frame rotates in the *yz* plane along the circle of radius *R*_{b} = *L*/*α* that is the radius of curvature of the nanotube.^{[23,24]} This is a non-inertial reference frame in which one need to treat the nanotube as if it were straight and use the effective proton-nanotube interaction potential

*U*(

*ρ*) in a straght nanotube.

^{[4]}Here,

*ρ*= (

*x*

^{2}+

*y*

^{2})

^{1/2}is the insrtantaneous distance of the proton from the nanotube axis in the new frame and

*v*

_{0}is its initial tangential velocity. As a consequance of the rotation of the moving frame, a centrifugal force arises that affects the particle energy, which may be described through the effective potential (

^{[23,24]}

The dynamic polarization of the nanotube by the proton is treated using the 2D extended two-fluid^{[25,27]} hydrodynamic model of the carbon valence electrons, which is based on a jellium-like description of the nanotube ion cores. Therefore, our model for the attractive image potential given by Eq. (

The angular and spatial distributions of the channeled protons in the exit transverse plane are generated using a Monte Carlo computer simulation method.^{[19]} The initial proton position, *x*_{0} and *y*_{0}, is chosen randomly from a 2D uniform distribution with the condition *a*_{sc} = [9*π*^{2}/(128Z_{2})]^{1/3} *a*_{0} is the nanotube atom screening radius and *a*_{0} is the Bohr radius. The initial number of protons is approximately 10^{6}. The current proton position (*x*,*y*) during its motion through nanotube is obtained by the numerical solution of the proton equations of motion in the transverse plane.^{[19]} The proton motion is non-relativistic, and is treated classically. The continuum approximation for the atomic potential in Eq. (^{[32]} Lindhard also neglected the transverse correlations between the positions of the atomic strings, and he made the assumption of statistical equilibrium of ion trajectories in the transverse position plane.

In the case of a straight nanotube the critical angle for channeling is *U*_{sc} is the total interaction potential at the distance from the nanotube wall, equal to the screening radius *a*_{sc}.^{[3,6]} In the case of a bent nanotube this formula for critical angle has a small correction.^{[24]} For the nanotube types and the proton energy that are chosen in this paper the critical angle for channeling is around 10 mrad. Our choice for the bending angle of 4 mrad is on the same order of magnitude as the critical angle for channeling.

In addition, in the case of proton channeling through a nanotube it is important to stress two facts, which justify our use of the continuum approximation in Eq. (^{[6]}

In order to justify the zero-thickness approximation for the *σ* and *π* electrons, one should consider the following facts. Compared with the *π* electrons, the *σ* electrons are more localized within the hexagonal lattice of the nanotube wall, whereas the *π* electron orbitals are localized within a range of about 0.1 nm on that wall. On the other hand, typical distances of turning points from the nanotube wall in the trajectories of channeled protons are expected to be larger than the size of the *π* electron orbital. For example, in Fig. *π* electrons, the image force is expected to turn weaker, whereas the interaction of the proton with the nanotube wall near the turning point of its trajectory is dominated by a strong, short-ranged repulsive force. Therefore, the difference between the widths of the *σ* and *π* electron orbitals becomes irrelevant and they both may be described as having zero thickness for channeled protons. That is why our 2D hydrodynamic model of the *σ* and *π* electrons described as two superimposed, continuous fluids of zero thickness^{[31]} represents a good approximation for the dynamic polarization force on proton trajectories within the nanotube.

In this section we first analyze the total potentials in four different nanotube types^{[36,37]} in the presence of the image potential calculated from both the one-fluid and the extended two-fluid hydrodynamic models.^{[25,31]} We then present and discuss the effects of the image potential from the extended two-fluid model on the spatial and angular distributions of the protons channeled through both the straight and bent nanotubes. In our analysis we assign the fixed values to several important parameters: proton speed *v* = 3 a.u., nanotube length *L* = 0.2 μm and, in the case of bent nanotubes, bending angle *α* = 4 mrad.

Figure *U*_{im} from the extended two-fluid model, the repulsive potential *U*_{r} and the resulting total potential *U*_{t} depend on the proton position *y* in the *yz* plane from the center of a straight nanotube. Also shown are the results for the total potential *U*_{t1f} calculated with the image potential from the one-fluid model with zero damping. Vertical solid bars represent the nanotube walls. There are four types of nanotubes with the values of radius *R* of 0.346, 0.483, 0.689, and 0.865 nm for (6,4), (8,6), (11,9), and (15,10) SWNTs, respectively. Vertical dotted bars show the value *R* − *a*_{sc} for each nanotube. If the proton position reaches the outside of the region, *ρ* < *R* − *a*_{sc} during its motion through the nanotube, we treat that proton as dechanneled one and we do not follow its trajectory any more. In all cases in Fig. *U _{i}*(

*y*) =

*U*(−

_{i}*y*), (

*i*= im,r,t). This figure also shows the influence of the image potential

*U*

_{im}on the total potential

*U*

_{t}. The total potential

*U*

_{t}is effectively lower if the image potential

*U*

_{im}is taken into consideration and becomes attractive towards the nanotube wall. When comparing the total potential calculated with the one-fluid model (

*U*

_{t1f}) and that obtained from the extended two-fluid models (

*U*

_{t}), one notices that in the former case the potential well is generally deeper throughout the nanotube, especially in the narrower nanotubes, and its attractive part is steeper towards the nanotube walls, especially in the broader nanotubes.

Figure *U*_{t}, calculated with the image potential *U*_{im} from the extended two-fluid model, depends on the proton position *y* from the center for each of the four types of bent and straight nanotubes. Also shown are the results for the total potential *U*_{t1f} calculated with the image potential from the one-fluid model with zero damping for the bent nanotubes. In the cases of bent nanotubes the total potential is no longer cylindrically symmetric and we cannot take *U*_{t}(*y*) = *U*_{t}(−*y*). This behavior of the potential function is a result of the centrifugal force that acts on the incoming proton. As the proton distance from the bent nanotube center increases, the differences in the potential are great compared with the case of the straight nanotubes. This may be rationalized by inspecting the expression for *U*_{t}(*ρ*) in Eq. (*y*. As a result, *U*_{t}(*y*) is attractive in a relatively large region of each nanotube, and the image potential *U*_{im} makes the attractive region increasingly deeper as the nanotube diameter increases. This effect of the image potential in the bent nanotubes is more pronounced in the case of the image potential from the one-fluid model with zero damping, but differences in comparison with the case of the image potential from the extended two-fluid model diminish as the nanotube diameter increases.

From Figs. ^{[19–21]} Hence, we adopt the extended two-fluid model to obtain the effects of image potential in our MC simulations of proton distributions in the bent nanotubes as shown in Figs.

Figure *α* = 4 mrad in the cases of (a) (6, 4) SWNT, (b) (8, 6) SWNT, (c) (11, 9) SWNT, and (d) (15, 10) SWNT. In the upper panels the distributions are calculated from proton trajectories obtained using the solid curves from Fig. *y* axis (lying in the nanotube bending plane) and in the direction of the *x* axis (perpendicular to the bending plane). The broadening effect appears to increase with increasing nanotube diameter. It is interesting that, inside wider nanotubes, the broadening in the perpendicular direction occurs over a relatively narrow, crescent-shaped region along the outer wall of the nanotube, which lies farther away from the center of curvature.

Figure *α* = 4 mrad in the cases of (a) (6, 4) SWNT, (b) (8, 6) SWNT, (c) (11, 9) SWNT, and (d) (15, 10) SWNT. In the upper panels the distributions are calculated for proton trajectories using the solid curves from Fig. *θ _{y}* ≈ −4 mrad, corresponding to undeflected protons, which is accentuated by the presence of the image potential.

Figure *y* axis, corresponding to the cross section with *x* = 0 in Fig.

Figure *θ _{y}* axis, corresponding to the cross section with

*θ*= 0 in Fig.

_{x}*θ*≈ 4 mrad, corresponding to a deflection that doubles the bending angle

_{y}*α*= 4 mrad due to single hard collision with the nanotube wall, and the nanotube with the image potential also shows peaks at

*θ*≈ 0 and even

_{y}*θ*≈ −4 mrad. It is surprising that in the case of latter peak, such a narrow nanotube still favors the direction almost with no bending in the presence of the image potential.

_{y}Figures *α* = 4 mrad in the cases of (11, 9) SWNT nanotube with the lengths *L* = 0.5 μm and *L* = 1 μm, respectively. In Fig.

In Fig. *L* = 0.2 μm. The nanotube bending angle, *α*, is varied between 0 and 100 mrad. We can see that for bending angle *α* = *ψ*_{c} = 11 mrad (critical angle for channeling) about 90% of the initial number of protons remains channeled. For *α* = 2*ψ*_{c} about 60% of the initial number of protons remains, because the nanotube is very short and majority of protons make less than one oscillation. For a few times longer nanotube, if *α* > *ψ*_{c} all protons would be dechanneled. Thus, for the bending angle *α* = 4 mrad and the length *L* = 0.2 μm that are chosen in Figs.

Next, we provide the main reasons for choosing the values of proton energy and the nanotube lengths in this work.

Future developments in our work will be concerned with simulations for longer nanotubes that include the effects of energy loss of channeled protons and the angular uncertainty because of the proton collisions with the nanotube electrons in the presence of the dynamic polarization of the nanotube electrons.

In this work, we study the channeling of protons that move at a speed of 3 a.u. through four different types of straight and bent SWNTs, namely (6, 4), (8, 6), (11, 9), and (15, 10). The effects of dynamic polarization of the valence electrons in the nanotube wall are assessed by both the one-fluid hydrodynamic model and an extended two-fluid hydrodynamic model. According to those models, we calculate the image potential for protons and combine it with the Doyle–Turner repulsive potential to obtain the total potentials in the straight and bent carbon nanotubes respectively. We then calculate individual proton trajectories by solving the classical equation of motion in a total potential based on the image potential from the extended two-fluid model. Finally, a Monte Carlo (MC) code is used to compute the spatial and angular distributions of protons channeled through short and bent nanotubes respectively. In the MC code we use the parameters for the extended two-fluid hydrodynamic model, which were determined in recent applications in the modelling of several independent experiments on electron energy loss spectra of different carbon nanostructures.^{[31]} Therefore, compared with our previous studies of the image potential effects in ion channeling that were based on a one-fluid model with zero damping for a quasi-free electron gas in nanotubes, this work provides a complete and accurate description of the dynamic polarization effects.

The results show that the presence of the image potential in a bent nanotube broadens and deepens the attractive potential well, which happens in each nanotube due to the centrifugal force in a non-inertial frame of reference for proton motion. As a consequence, the spatial distributions of the channeled protons in the presence of the image potential are generally broader in the directions that are parallel and perpendicular to the bending plane of all four types of nanotubes. Angular distributions also exhibit broadening, but to a lesser extent than the spatial distributions, which is accompanied with a significant redistribution of the proton flux that follows the nanotube bending angle. Overall, there is a tendency for the image potential to increase both the proton yield in the spatial distributions and the proton flux in the angular distributions, in the peripheral regions of the nanotubes, away from the bending center.

All our findings indicate that the influence of the dynamic polarization on ion channeling through bent carbon nanotubes in the MeV energy range is strong and it should not be omitted in simulations. In particular, the presence of the image potential may diminish the bending efficiency of carbon nanotubes for charged particles that move in the range of speeds of several Bohr velocities.

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