Over the past decade, the interactions of low-energy ions with the insulating capillaries have been extensively studied. Unlike the phenomena occurring in the metal capillaries,^[1] ions can be transmitted along the insulating capillary axis with their initial charge state and incident energy even when the capillaries are tilted by a larger angle with respect to the incident ion-beam direction.^[2–5] This phenomenon is called ion guiding, which is attributed to the self-organized formation of charge patches on the capillary internal wall.^[2,6]

The guiding process can be qualitatively described as follows: Ions entering into the capillary tilted by an angle hit the wall and, in turn, deposit their charges at the collision points so that a charge patch is generated near the entrance region (hereafter referred as to entrance charge patch). When the entrance charge patch is sufficient to produce a strong repulsive electric field, subsequent ions will be deflected at large distances from the capillary wall. As a result, the close collisions and charge exchanges between ions and atoms on the capillary wall are inhibited. Then, additional small charge patches are formed further inside the capillary.^[6–12] Owing to the capillary conductivities, the entrance charge patch is extended and the small charge patches are strongly weakened so that the electric field within the capillary keeps unchanged to maintain ion guiding.^[8,11,12]

Experimentally, a large number of researchers have studied the guiding effect using different insulating capillaries^{[2–6,13–16]} accompanied by negative ions,^[17–19] molecular ions,^[20,21] and electrons,^[22,23] as well as various positive ions. Through analysis of various experimental results, many outstanding achievements are acquired. For instance, the observed oscillations of the ion emission angle provide evidence of sequential charge patches formed on the capillary internal wall.^[6,24] Particularly, it is found that the guided transmission are not dependent of the ion mass but primarily dependent of the projectile charge-to-energy ratio.^[25,26] Recent experiments reported that 1-MeV proton microbeam could be guided through a single cylindrical Teflon macrocapillary,^[27] which still needs further interpretations.

Motivated by the desire to understand the guiding mechanisms, theoretical investigations by means of Monte Carlo simulations.^[8–12,28] and analytic calculations^[29] have been performed in several research groups. These theoretical studies successfully provide more detailed information about ion guiding. It is demonstrated that most of the deposited charges are located near the entrance region.^[9–12] To establish ion-guiding equilibrium in simulations, the transport of charges deposited on the capillary wall should be invoked. However, there exist two different models of the charge transport used in previous simulations, namely the drift model^[8,11,12] introducing a nonlinear effect on the charge migration in an electric field, and the diffusion model^[9,10] adopting an unbiased random walk to describe the spread and depletion of the deposited charges. Here, it should be pointed out that some general properties of ion guiding can be reproduced by either of these two models. Therefore, a deeper understanding of guiding process within the capillary is still needed. A review on both the experimental measurements and theoretical description of the capillary guiding can be found in Refs. [30] and [31].

The present work is primarily intended to simulate the transmission of ions guided through a nanocapillary and to compare the simulated results with those reported previously. The program is written in MATLAB language which can simplify the procedure codes and improve the developing efficiency greatly. Section 2 describes the simulation methods, inevitably part of which have considerable overlap with those in previous work.^[9–11] However, a more justified treatment of the surface-charge decay, which was indicated in early literature,^[32–34] is implemented. Section 3 on the simulated results marks the time evolution of the ion transmission, the ion-mass effects on guided transmission, and the electric fields within the nanocapillary. Among the most important finding is that the maximum of the repulsive field is independent of not only the tilt angle but also the ion energy. The summary of the present work is taken up in the final section.

The simulations were designed to explore ion guiding through a cylindrical nanocapillary in polyethylene terephthalate (PET) materials. The nanocapillary has a diameter of 100 nm and a length of 10 μm, whose geometric illustration is shown in Fig. 1 to indicate the coordinate directions.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) The geometric illustration of PET nanocapillary with d = 100 nm in diameter and L = 10 μm in length (not to scale). The coordinates y and z are indicated in the graph, and the direction of x axis is perpendicular to the paper and points to it (not to show). The ions enter into the capillary with tilt angle ψ relative to the capillary axis, i.e., z axis.

The simulation methods are very similar to those used in Refs. [9]–[12] except for the charge transport caused by the capillary conductivities, so that the overlapped part are outlined briefly. The capillary wall becomes charged due to the deposition of incident ions but without considering the addition charge-up via the secondary electron emission induced by the ion impact with the capillary wall. The trajectories of ions under the function of electric fields produced by the charged wall, are determined by solving Newton’s equation of motion. Note that the electric fields are weakened by the image charge of the deposited charges.^[35,36] For PET materials with relative permittivity ε ≃ 3.2,^[37] the electric fields are reduced by a factor of (1 + ε)/2 ≃ 2. Here, it should be pointed out that since we use a single nanocapillary to simulate ion guiding, influences of the capillary neighbors are neglected. Nevertheless, this neglect is justified when the number of capillaries per unit area in foils is not very high.^[38]

Due to the capillary conductivities, the charges deposited on the capillary wall will undergo spread and decay, which is key to the guiding process. Since the surface conductivity is much larger than the bulk,^[9] only the spread along the capillary surface is taken into account in our simulations. In addition, compared to the drift mechanism, the effect of the diffusion on the charge transport is rather minor.^[32] Hence, in our simulations, the diffusion is not invoked.

The spread of deposited charges is considered in terms of the built-in electric field-driven migrations. It is assumed that the charges reaching the golden layer on the front and back surface of the capillary foil will disappear immediately. The carrier mobility governing the charge migration is found to exponentially depend on the square root of the electric field,^[39] which is similar to the conductivity modeled by Poole–Frenkel mechanism.^[40] To date, it is ambiguous for us whether these two dependencies are exactly within the same framework. Therefore, to avoid the possible inconsistencies, we directly quote the empirical expression for the mobility introduced in Ref. [39], namely,

It is obtained from Ref. [41] that the mobility μ₀ of electrons in Mylar (PET) foil is in the neighborhood of 1 nm²/(V·s). Unlike the treatment of the ion deposition in Ref. [11], we assume that the ion deposited on the capillary wall would become an ion-carrier determining the position of the deposited ion. Accordingly, the mobility for ions is estimated to be smaller than the value of 1 nm²/(V·s) as a result of the heavy mass. However, as pointed out in Ref. [11], the mobilities in insulating materials have large uncertainties spanning more than eight orders of magnitude since they are in close relation to the treatment of the internal surface of the hole, such as the humidity, cleanliness.^[37,42] Thus, it is unlikely to directly use these mobilities in the simulations. Our methods of determining the mobility is to test the calculations with values varying from 1 nm²/V·s to 10⁻⁴ nm²/V·s, and then find the optimal one by comparing the calculated results to the previous experiments of the capillary guiding. After some tests, it is found that the μ₀ value of 10⁻⁴ nm²/V·s is appropriate for Ne⁷⁺-ion guiding. The test results are presented in Appendix A.

Then, combining the field E with Eq. (1) gives the distance of the ion migration on the capillary surface at the time interval of two incident ions^[11]

Equation (3) is used to determine the positions of charges deposited on the capillary wall. Only when these charges reach the grounded layer, would they be neutralized suddenly. We expect that the number of deposited charges should decrease as they are moving on the capillary wall (see Fig. 2).

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) The deposited charge Q(t) is changed into Q (t + Δt) after a time Δt, and the position is governed by Eq. (3).

In previous simulations,^[10,11] the decay of the deposited charge Q(t) is assumed to follow an exponential law to describe the charge diffusion into the capillary bulk. However, as mentioned in Refs. [9]–[12] the bulk conductivity could be neglected compared to the surface conductivity. That is to say, the bulk discharge cannot have a significant impact on guided transmission. Moreover, the guiding experiments^[43] exactly claim the charge decay not to be exponential with time. Fortunately, it has been achieved in early studies^[32–34] that the charge decay on the insulating surface obeys the following hyperbolic formula:

Since both equations (3) and (4) are valid within the drift mechanism, one may argue that they give rise to the same charge decay. In order to explain this, we simulate the decay of a uniformly charged square plate in area of 1 μm² by using Eq. (3). One side of the plate is grounded. In such simulation, however, the exponential term in Eq. (3) is cancelled (Note that equation (4) was obtained without considering the nonlinear effect). The simulated data, fitted by hyperbolic and exponential functions, is shown in Fig. 3. It is clearly seen that the time evolutions of charges described by Eqs. (3) and (4) are not the same. Besides, it is found from simulations that the charge decay determined by Eq. (3) decreases rapidly at the beginning, and then becomes very slow. This implies that only the charges near the grounded side have a rapid depletion, while those charges far away from the grounded side are difficult to discharge in the way of Eq. (3). Hence, we adopt both of Eqs. (3) and (4) in the simulations. Here, it should be added that from our test calculations the guided transmission cannot be obtained in absence of Eq. (4) (See Appendix A).

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) The decay of a uniformly charged plate simulated by using Eq. (3) without the exponential term. The plate area is 1 μm2.

The selection of ions and the beam currents, to some extent, is subjective. However, one of the selection criteria is that the ion and the current should be readily available from previous experiments for sake of assessing the precision of our simulations. Besides, on the premise of saving the computing time, the selection is best not to introduce extra approximations and simplifications. Therefore, Ne⁷⁺ ions with current of 200 pA/mm² is selected, which leads to that the number of Ne⁷⁺ ions injected into the capillary per second is the equivalent of about 1 with consideration of dielectric screening. To make the simulations more factual, the divergence of incident beam is set to be 0.3° full width at half maximum (FWHM) estimated in previous experiments.^[20,44] Here, it is added that we can not explain from first principles why the values of the parameters μ₀ and t₀ should, respectively, be 10⁻⁴ nm²/V·s and 1000 s but simply take them as empirical fact that turns out to agree with the experimental observations reported before.

In summary, the present simulations capture the essential features of the ion transmission guided through the insulating nanocapillary. The simulations indicate that the ions undergo oscillatory trajectories to the exit of the capillary, which result from the several charge patches sequentially formed on the capillary internal surface. Moreover, it needs a longer time for lower-energy ions to get guiding equilibrium as a result of more oscillations, in agreement with the experiments.^[45]

On the other hand, the simulations provide direct evidence that the transmission of ions guided through the insulating capillary is nearly independent of ion mass. As expected by the semiempirical scaling law,^[25,26] the key controlling parameter of ion guiding is the ratio of charge to energy. This implies that the insulating capillary has pretty promising future in manipulating various charged particles regardless of the mass.

Most importantly, the field component E_y is found to be nearly unchanged by varying the ion energy as well as the tilt angle. Hence, to guide ions with high transversal energy caused by the increasing of the tilt angle or incident energy, the effective length of the field E_y is extended in a self-organized manner.