Another quantity that must be computed, is the mean curvature $H$, which depends only on the surface vector $\vec{s}$. For an axisymmetric surface $\vec{s}$ depending in curvilinear coordinates on $(\eta, \phi)$ (valid for prolate and spherical coordinate systems), the mean curvature is defined as

$$2H = \frac{eG + gE}{EG} \qquad (A.14)$$

with

$$E = \left| \frac{\partial \vec{s}}{\partial \eta} \right|^2 = \left| h_{\xi} \frac{\partial \xi(\eta)}{\partial \eta} \vec{e}_{\xi} + h_{\eta} \vec{e}_{\eta} \right|^2 \qquad (A.15)$$

$$G = \left| \frac{\partial \vec{s}}{\partial \phi} \right|^2 = |h_{\phi} \vec{e}_{\phi}|^2 \qquad (A.16)$$

being the non-zero coefficients of the first fundamental form and

$$e = \vec{n} \cdot \frac{\partial^2 \vec{s}}{\partial \eta^2} \qquad (A.17)$$

$$g = \vec{n} \cdot \frac{\partial^2 \vec{s}}{\partial \phi^2} \qquad (A.18)$$

the non-zero coefficients of the second fundamental form<fnref n="4" />. The mean curvature depends thus on the first and second derivatives of $\xi(\eta)$ with respect to $\eta$ and are computed using a Gauss-Legendre interpolation scheme [18], which is an extremely precise algorithm. (However, for pointed tips, $H$ becomes near singular at the extremities, which is usually a source of numerical instabilities.)

$$\frac{\partial \vec{e}_j}{\partial q_j} = -\vec{e}_k \frac{1}{h_k} \frac{\partial h_j}{\partial q_k} - \vec{e}_i \frac{1}{h_i} \frac{\partial h_j}{\partial q_i} \qquad (A.19)$$

$$\frac{\partial \vec{e}_k}{\partial q_j} = \vec{e}_j \frac{1}{h_k} \frac{\partial h_j}{\partial q_k} \quad k \neq j \qquad (A.20)$$

the coefficients of second fundamental form become,

$$e = \vec{n} \cdot \left( \frac{\partial}{\partial \eta} \left( g_{\xi} \frac{\partial \xi}{\partial \eta} \vec{e}_{\xi} + h_{\eta} \vec{e}_{\eta} \right) \right) \qquad (A.21) \\ = \vec{n} \cdot \left( \left( \frac{\partial h_{\xi}}{\partial \eta} \frac{\partial \xi}{\partial \eta} + h_{\xi} \frac{\partial^2 \xi}{\partial \eta^2} - \frac{h_{\eta}}{h_{\xi}} \frac{\partial h_{\eta}}{\partial \xi} \right) \vec{e}_{\xi} + \left( \frac{\partial \xi}{\partial \eta} \frac{\partial h_{\eta}}{\partial \xi} + \frac{\partial h_{\eta}}{\partial \eta} \frac{\partial \xi}{\partial \eta} \right) \vec{e}_{\eta} \right)$$

$$g = h_{\phi} \vec{n} \cdot \frac{\partial \vec{e}_{\phi}}{\partial \phi} \qquad (A.22) \\ = h_{\phi} \vec{n} \cdot \left( - \frac{\vec{e}_{\xi}}{h_{\xi}} \frac{\partial h_{\phi}}{\partial \xi} - \frac{\vec{e}_{\eta}}{h_{\eta}} \frac{\partial h_{\phi}}{\partial \eta} \right)$$

<footnote n="4">⁴Using the relations for the unit vectors,</footnote>
## A.2 Time propagator (valid also for VCVPF and RCVPF)

The axisymmetric shape of a perfectly conducting droplet is propagated in time by the following scheme. Knowing the shape $\vec{s}(t)$ and velocity potential $\Psi(t)$ at time $t$, we propagate the surface to $\vec{s}'(t + dt)$. This is the predictor step. Then, we compute the pressure and velocity at this new intermediate surface $\vec{s}'(t + dt)$. Finally, taking the mean value of the pressure computed at $\vec{s}(t)$ and $\vec{s}'(t + dt)$, the corrector step calculates the new surface $\vec{s}(t + dt)$ and velocity potential $\Psi(t + dt)$. The 9 computing steps are summarized below:

1. $\vec{u}(t) = \vec{\nabla}_{\vec{s}(t)} \Psi_n(t)$

2. $H(t) \equiv H(\vec{s}(t))$, $\sigma(t) \equiv \sigma(\vec{s}(t))$

3. $\dot{\psi}(t) = 2H(t) + 2\sigma^2(t) - \vec{u}^2(t)/2$ (c.f. 1.68)

4. $\vec{s}'(t + dt) = \vec{s}(t) + dt\vec{u}(t)$ (c.f. 1.28)

5. $\Psi'(t + dt) = \Psi(t) + dt\dot{\psi}(t)$

6. $\vec{u}'(t + dt) = \vec{\nabla}_{\vec{s}(t+dt)} \Psi_n(t + dt)$

7. $\dot{\psi}'(t + dt) = 2H'(t + dt) + 2\sigma'^2(t + dt) - \vec{u}'^2(t + dt)/2$

8. $\Psi(t + dt) = dt(\dot{\psi}(t) + \dot{\psi}'(t + dt))/2$

9. $\vec{s}(t + dt) = \vec{s}(t) + dt(\vec{u}(t) + \vec{u}'(t + dt))/2$

The dimensionless velocity increases exponentially in time during the first step of the deformation. We used thus also a time adaptive scheme, where the time step is reduced by a factor two as soon as $|\vec{u}|dt > \epsilon$, where $\epsilon \simeq 10^{-5}$ is a small arbitrary number. This second order time adaptive predictor-corrector-like scheme allows to propagate the shape of the droplet, while conserving the volume and energy of the droplet (inviscid liquid). Typically, the volume and energy are conserved within a relative error of less then $10^{-9}$ and $10^{-5}$ respectively during the whole deformation pathway, except for the moments where pointed tips (singularities) are formed and where the numerical method accumulated larger errors or simply fails. The accumulation of numerical errors is significant when pointed ends develop at the tip as the harmonic basis, used for solving the Laplace equation, is no longer adapted, meaning that a large number of basis functions are necessary to capture the near singular velocity potential at the tips mainly because the harmonic basis functions are not necessarily orthogonal on the droplet surface. The jet emission is modelled
by reducing instantaneously the drop charge. Usually, the charge is reduced by a given fraction just before the singularities develop, making the droplet subcritically charged. At that point, the pressure due to the surface tension dominates again and the tip formation is stopped and reversed (tips become flattened) and the calculation can continue.

### A.3 Electric field in the case of a non-equilibrium charge distribution

In the case the potential $V$ at the surface in not an equipotential, the technique based on A.8 is not applicable and the evaluation of the electric field needs to be adapted. The potential $V^-$ inside the drop and $V^+$ outside the drop are assumed to satisfy the Laplace equation and can therefore be expanded on harmonic functions $f_n$ and $g_n$ having the appropriate behaviour at the origin and at infinity.

$$V^{-}(\xi, \eta) = \sum_{n}^{N} a_{n}f_{n}(\xi, \eta) \quad \text{in } \Omega \qquad (A.23)$$

$$V^{+}(\xi, \eta) = \sum_{n}^{N} b_{n}g_{n}(\xi, \eta) \quad \text{in } \mathbb{R}^{3}\setminus\Omega \qquad (A.24)$$

The functions $f_n$ and $g_n$ were introduced in (A.1) (A.6) respectively. The expansion coefficients $\{a_n\}$ and $\{b_n\}$ are obtained by solving the boundary conditions equations at the $i \in [1, N_i]$ grid points $\vec{s}_i$ of the droplet interface.

$$V^{-}(\vec{s}_i) - V^{+}(\vec{s}_i) = 0 \quad \text{on } \partial\Omega \qquad (A.25)$$

$$\vec{n}_i \cdot (\varepsilon_r \vec{\nabla} V^{-}(\vec{s}_i) - \vec{\nabla} V^{+}(\vec{s}_i)) = \frac{\sigma}{\varepsilon_0} \quad \text{on } \partial\Omega \qquad (A.26)$$

Injecting (A.23, A.24) into the above boundary equations yields a $N_i \times N$ system of equations for the coefficients. Once the system is solved using a matrix inversion techniques like SVD<fnref n="5" /> and the coefficients determined, the electric field inside and outside the drop are readily computed. Note that for $n \le N = 150$ and $i \le N_i = 200$, this technique is CPU efficient compared to the cost of the other functions in the numerical code. The method however loses its precision as soon as the matrices $f_{ni} \equiv f_n(\vec{s}_i)$ or $g_{ni} \equiv g_n(\vec{s}_i)$ become singular and a more robust technique like BEM<fnref n="6" />, introduced in section 1.5.4, should be used.

<footnote n="5">⁵Singular Value Decomposition, for over-determined system of equations with more equations than unknowns $N_i \ge N$, gives the solution is the sense of a least square method.</footnote>

<footnote n="6">⁶Boundary element method</footnote>
## A.4 Numerical method for the evaluation of $p_μ$ in VCVPF

The VCVPF model is only useful if one can compute the viscous pressure $p_μ$, which enters the normal stress term. We propose here a numerical method to compute the viscous pressure in VCVPF. Making use of the fact that the Laplace operator is separable in prolate spheroid coordinate system ($\xi \ge 1, -1 \le \eta \le 1, 0 \le \varphi \le 2\pi$), we get a simple expression for the harmonic functions $f_m$ that satisfy the Laplace equation $\nabla^2 f_m = 0$. As we want $f_m(\xi, \eta) = f_m(\xi, -\eta)$ to be axisymmetric and even and defined at the origin $f(1, 0) < \infty$, we have

$$f_m(\xi, \eta) = P_{2m}(\xi)P_{2m}(\eta) \qquad (A.27)$$

where $P_m()$ are Legendre polynomials of order $m$. Let $\{f_m\}$ be a basis of harmonic function. We may now expand the viscous pressure on the harmonic basis $f_m$,

$$p_{\mu}(\xi, \eta) = \sum_{m=1}^{M} b_m f_m(\xi, \eta) \qquad (A.28)$$

The expansion coefficients $\{b_m\}$ are deduced using the boundary condition (1.64),

$$\sum_{m=1}^{M} b_m \underbrace{f_m(\vec{s}_i)u_n(\vec{s}_i)ds_i}_{F_{mi}} = \underbrace{u_{\tau}(\vec{s}_i)(2\mu D_{\tau n}(\vec{s}_i)ds_i)}_{d_i} - \underbrace{u_{\tau}(\vec{s}_i)\sigma_i(\vec{s}_i)E_{\tau}(\vec{s}_i)ds_i}_{e_i} \qquad (A.29)$$

where $\vec{s}_i \le N$ is an ensemble of $N$ points at the surface. Equation (A.29) is a system of $N$ equations and $M$ unknowns of the form

$$\sum_{m=1}^{M} b_m F_{mi} = d_i - e_i, \quad 1 \le i \le N \qquad (A.30)$$

If $N \ge M$ the system is overdetermined and can be solved using a singular value decomposition (SVD), which finds the coefficient $\{b_m\}$ as least squares solutions.
## A.5 Derivation of the RCVPF model

We define the symmetric tensors $K$ and $\Lambda$, standing respectively for the irrotational and rotational contribution of the basis vectors to the strain-rate tensor:

$$ K = \frac{1}{2} (\vec{\nabla} \vec{\kappa} + (\vec{\nabla} \vec{\kappa})^t) \qquad (A.31) $$

$$ \Lambda = \frac{1}{2} (\vec{\nabla} \vec{\lambda} + (\vec{\nabla} \vec{\lambda})^t) \qquad (A.32) $$

$$ D = \sum_{k=1}^{L} a_k K \qquad (A.33) $$

$$ W = \sum_{\lambda=1}^{L} b_{\lambda} \Lambda \qquad (A.34) $$

We remind that $K$ and $D$ are divergence-free tensors, unlike $\Lambda$ and $W$.

The time evolution of the irrotational velocity field $\vec{v}$ in the RCVPF model is obtained by projecting the Navier-Stokes equation 1.12 on the (irrotational) basis vector $\vec{k}$.

$$ \begin{aligned} \rho \left( \vec{k} \cdot \frac{d\vec{u}}{dt} \right)_{\Omega} &= \left( \vec{k} \cdot (\vec{\nabla} \cdot T) \right)_{\Omega} && (A.36) \\ &= -(K : T)_{\Omega} + \int_{\partial\Omega} \kappa_n T_{nn} + \kappa_{\tau} T_{\tau n} ds \\ &= -2\mu (K : (D + W))_{\Omega} + \int_{\partial\Omega} \kappa_n (T_{nn} - M_{nn}) ds \\ &\quad + \int_{\partial\Omega} \kappa_{\tau} (T_{\tau n} - M_{\tau n}) ds , \end{aligned} $$

where $(\cdot)_{\Omega}$ means integration over the drop volume $\Omega$. Injecting the boundary conditions (1.95) and (1.96) and noting that $^7$

$$ (K : D)_{\Omega} = \int_{\partial\Omega} (\kappa_n D_{nn} + \kappa_{\tau} D_{\tau n}) ds \qquad (A.37) $$

$$ (K : W)_{\Omega} = \int_{\partial\Omega} (w_n K_{nn} + w_{\tau} K_{\tau n}) ds \qquad (A.38) $$

$$ \left( \vec{\kappa} \cdot \frac{d\vec{u}}{dt} \right)_{\Omega} = \int_{\partial\Omega} \psi_k \frac{d\vec{u}}{dt} \cdot \vec{n} \, ds \qquad (A.39) $$

<footnote>$^7$Equation A.38 results from the commutativity of the double dot product $K : W = W : K$. The latter property yields also $(\vec{v}(\vec{\nabla} \cdot W))_{\Omega} = (\vec{v} \cdot \nabla^2 \vec{w})_{\Omega} = \int_{\partial\Omega} (\vec{v} \cdot W - \vec{w} \cdot D) \vec{n} ds$, where we recognize the Greens vector identity.</footnote>
one has

$$ \rho \int_{\partial\Omega} \psi_k \frac{d\vec{u}}{dt} \cdot \vec{n} \, ds = \int_{\partial\Omega} (\kappa_n (-P_s + P_c) + \kappa_{\tau}(\sigma E_{\tau})) ds \\ -2\mu \int_{\partial\Omega} (\kappa_n D_{nn} + \kappa_{\tau} D_{\tau n}) ds \\ -2\mu \int_{\partial\Omega} (w_n K_{nn} + w_{\tau} K_{\tau n}) ds \qquad (A.40) $$

We deduce from (A.40) the traction and dissipation power of the RCVPF model, by summing over $k$,

$$ \frac{dE_{\text{kin}}}{dt} = \rho \sum_{k}^{L} a_k \int_{\partial\Omega} \psi_k \frac{d\vec{u}}{dt} \cdot \vec{n} \, ds \qquad (A.41) $$

$$ = \rho \int_{\partial\Omega} \Psi \frac{d\vec{u}}{dt} \cdot \vec{n} \, ds \qquad (A.42) $$

$$ = \int_{\partial\Omega} (v_n (-P_s + P_c) + v_{\tau}(\sigma E_{\tau})) ds \\ -2\mu \int_{\partial\Omega} (v_n D_{nn} + v_{\tau} D_{\tau n}) ds \\ -2\mu \int_{\partial\Omega} (w_n D_{nn} + w_{\tau} D_{\tau n}) ds \qquad (A.43) $$

## A.6 Numerical method of RCVPF

We propose the following scheme to solve numerically the RCVPF equations. We assumes that the basis function $f_l(\vec{s}) = 0$ on the boundary so that the vectors $\vec{\kappa}$ and $\vec{\lambda}$ are orthogonal in the drop volume. The surface integrals (A.6) and equations (A.40) and (A.43) involving $\lambda_n$ and $w_n$ thus vanish and the coefficients $b(t+\Delta t)$ are found by satisfying adiabatically by the shear stress boundary condition. Using the expansion of the velocity field (1.90), we can re-write equation(A.40) and (1.103) in a matrix form,

$$ \left( \begin{matrix} \int \phi_{k'} \kappa_n ds & \underbrace{\int \phi_{k'} \lambda_n ds}_{=0} \\ \int \kappa'_{\tau} K_{\tau n} ds & \int \kappa'_{\tau} \Lambda_{\tau n} ds \end{matrix} \right) \left( \begin{matrix} a_k(t + \Delta t) \\ b_{\lambda}(t + \Delta t) \end{matrix} \right) = \left( \begin{matrix} \int_{\partial\Omega} \phi_{k'} u_n ds + \Delta t f_{k'} \\ \int \kappa'_{\tau} \sigma E_{\tau} ds / 2\mu \end{matrix} \right) \qquad (A.44) $$

which is resolved using a matrix inversion technique like SVD. The computed $a_k(t+\Delta t)$ and $b_l(t+\Delta t)$ give the velocity vector $\vec{u} = \vec{v} + \vec{w}$ at the surface at time $t+\Delta t$.
Using the predictor-corrector propagation scheme (A.2), the surface deformation is computed. Here we assumed that $\lambda_n$ is zero at the surface (Orthogonal Helmholtz decomposition). The right hand terms are all taken at time $t$, with $f_k$ containing the traction and the dissipation power as well as the non-linear advection term of the rotational and irrotational field,

$$f_{k'} = \int_{\partial\Omega} \kappa'_{n} \left( P_s + P_c - \frac{v^2}{2} \right) ds - 2\mu \int_{\partial\Omega} (\kappa'_{n} D_{nn} + \underbrace{w_n K'_{nn}}_{0}) ds \\ +2\mu \int_{\partial\Omega} (\kappa'_{\tau} W_{\tau n} - w_{\tau} K'_{\tau n}) ds + (\vec{\kappa}' \cdot (\vec{v} \wedge (\vec{\nabla} \wedge \vec{w})))_{\Omega} \quad (A.45)$$

We will in the following neglect the last (advection) term containing the rotation field $\vec{w}$, which is usually small compared to the irrotational vector field $\vec{v}$. The term $f_k$ differs from the right hand term of (A.40). Indeed, the shear stress boundary condition was not used in $f_k$ because it is expressed by the system of equations (A.44) and thus automatically fulfilled by the solution. So equations (A.40) and (A.44) are equivalent. Note that the coefficients $b_l(t)$ are not actually propagated in time. They satisfy adiabatically the boundary condition while itself propagates in time.
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# Chapter 2

## Guiding and focusing power of
insulating capillaries

### Acknowledgments

This work was funded by the French National Research Agency (ANR) in the P2N 2012 program (ANR-12-NANO-008) for the PELIICAEN project. A big thank to Hongqiang ZANG of the School of Nuclear Science and Technology, Lanzhou University, for making the capillary setup again fully operational after several months of shut down and for the early successful measurements presented in figure 2.13. An enormous thanks to Stéphane GUILLOUS (CIMAP) for helping to improve the experimental setup and for co-designed the capillary holder. Stéphane also participated in the generation of numerous experimental data. The author also thanks Amine CASSIMI and Benoit GERVAIS (CIMAP) for the numerous fruitful discussions that helped to clarify the insights. A special thank to Jimmy RANGAMA for his helpful simulations with ®SIMION and help with setting up FASTER for data acquisition. I am indebted to Emmanuel GARDES (CIMAP) for clean cutting the home-maid capillaries and for the fabrication of the emittance meter with the FIB. A big thank also to Sylvain GIRARD and Claire FEIERSTEIN for engineering the capillary puller that allowed us the pull highly symmetric tapered capillaries. Many thanks to our local group of mechanics, Thierry BEEN, Patrick GUINEMENT and Jeam-Marc RAMILLON for the machining of the capillary holder and maintenance of the setup.

**Collaborations** Finally I want to thank Karoly TÖKÉSI and Gyula NAGY from Debrecen (H) and Robert D. DUBOIS from Rolla University (USA) for our fruitful collaboration.
**Introduction**

When low energetic ions hit an insulating surface, charge carriers are injected at the impact points and trapped by defects. As a result, charge accumulates in the dielectric, generating an electric field that, if sufficiently strong, deviates the following beam particles, preventing them to hit the insulator surface. This basic property led to the discovery that slow ions can pass through insulator capillaries (having an aspect ratio, length over diameter, larger than 50) without any change, neither in the ion charge state nor in the ion kinetic energy, even when the geometrical conditions do not allow it. The phenomenon is called charged particle guiding by insulating capillaries and it has become an intensively studied field since its discovery by the pioneering work by N. Stoltherfoht for polyethylene terephthalate (PET or Mylar) nano-capillaries [17] and T. Ikeda for glass macro-capillaries [22], more than 10 years ago. The first simulations showing the guiding effect were

<figure><img src="image_46.png" /><figcaption>Figure 2.1: Examples of nano and macro-capillaries</figcaption></figure>

developed by K. Schiessl et al. [24, 25]. They developed a classical trajectory transport theory of this self-organized guiding process. The guiding electric field was deduced from the accumulated charge in the capillary and the induced image charge. For the charge relaxation channel, they mainly considered surface and bulk diffusion. Despite ignoring the charge relaxation channel due to electrical conductivity, they found a relative good agreement of the simulated transmission coefficients with experimental data.

M. B. Sahana et al. observed Ne<sup>+7</sup> ions being transmitted through well-ordered,
<figure><img src="image_47.png" /><figcaption>Figure 2.2: Scheme of the transmission of an ion beam, tilted with respect to the capillary axis, through insulating nano-capillaries. Extracted from [24]</figcaption></figure>

parallel SiO₂ nanocapillaries with a decreasing transmitted intensity up to a factor 100, when increasing the capillary tilt angles up to 4°. In particular, they found narrower angular distribution in comparison to PET nano-capillaries [17].

In 2007, A.R. Milosavljevic *et al.* reported the guiding of low-energy electrons by highly ordered Al₂O₃ nanocapillaries with large aspect ratio (140 nm diameter and 15 µm length). The guiding of the electron beam was observed for tilt angles up to 12°. The experimental results clearly showed the existence of the guiding effect, as found for highly charged ions. The energy losses suggest that electrons undergo multiple elastic and inelastic scattering within the capillaries before being transmitted or lost inside the capillary. Despite the considerable energy losses, it has been found that the inelastically as well as the elastically scattered electrons are guided through the capillaries.

P. Skog *et al.* gave evidence that 7 keV Ne⁺⁷ ions are guided through nanocapillaries by sequentially formed charge patches [32]. A. Cassimi *et al.* showed that 230 keV Xe²³⁺ ions are guided through glass macro-capillaries, in agreement with the picture of charge patches created sequentially along the capillary and thus deflecting the beam until a stationary state is reached [37]. Later it was shown that the Xe²³⁺ ions show no evidence of energy loss or change in the charge state of the ion, changing except for the production of a small neutral fraction. The latter was determined to be due to ions that had become neutralized at the interface to form atoms rather than due to photon emission [44].

N. Stolterfoht presented in 2013 simulations that indicate that non-linear conductivity to transport the deposited charges along the capillary surface must be taken into account to describe properly the observed transmission in nano-capillaries [48, 49]. Remarkably, he found that if the ion beam is tilted with respect to the capillary axis, ion trajectories are found to be focused, enhanc-
ing the ion guiding, a phenomenon that will be discussed in more detail in the
manuscript. Later the same model was used to simulate the guiding of 4.5-keV
Ar<sup>7+</sup> ion guiding through a conical glass macrocapillary. For the latter, the sur-
face conductivity was ignored and the charge dynamics modeled by assuming
that the injected charges fade away at a rate depending on the bulk conduc-
tivity and thus temperature of the insulator [53]. Recently, evidence of a 1 MeV
proton microbeam guided through insulating macrocapillary was given by Nagy
et al. in [68]. Other advances in the field are described in this non-exhaustive list
[31, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 46, 47, 51, 62, 64, 67], where the guiding
power of insulating nano-capillaries as well as straight, curved and tapered macro-
capillaries in various conditions is demonstrated. Detailed reviews about guiding
properties of macro- and nano-capillaries are given in [50] and [59]. A recent review
including curved teflon macro-capillaries is presented by T. M. Kojima in [63].

<figure><img src="image_48.png" /><figcaption>Figure 2.3: Scheme of the transmission of an ion beam, tilted with respect to the capillary axis, through an insulating tapered (conical) macro-capillaries, by sequentially formed charge patches.</figcaption></figure>

While the guiding of ions through glass-capillaries due to charged patches is
qualitatively understood, the complex nature of the electric conduction in such
insulators makes quantitative theoretical predictions still a challenging task. In-
deed, for a given ion beam, the guiding is entirely controlled by the dynamics
of the charge patches in the dielectric, which in turn depends (a) on the electri-
cal properties of insulator, (b) on the possible presence of electrodes that affect
the electric field inside the capillary and (c) on the charge transfer at the inter-
faces. Some theoretical studies, simulating the ion trajectories through capillaries
have already been presented in [24, 25, 48, 49, 50]. But they suffer mainly by a
simplified description of the charge dynamics of the patches and are limited to a
relative low number of trajectories due to relatively inefficient algorithms. As a
result, they may not have access to the long time (asymptotic) behaviour of the
transmission, mainly dominated by the typical long relaxation time of the charge
patches in insulators. Also, they do not account for near-laying grounded or float-
ing electrodes that might modify the electric field inside the insulators with low
dielectric constant, affecting the trajectories of the ions and thus the charge patch
formation.

In order to get some insights into the relaxation dynamics in glass capillaries, the author developed a numerical code, labelled *InCa4D*, that simulates the ion transport through insulating capillaries. The code is based on the explicit description of the charge accumulation and charge dynamics at the insulator interfaces. Much work was put into developing a CPU efficient numerical code for the calculation of the ion trajectories and charge dynamics in the capillary wall, so that the asymptotic behaviour could be analysed. The model can simulate capillaries having rotational symmetry, such as tubes and conical shaped capillaries. The development of the model started in 2013 and was since developed incrementally to include more and more features, that will be addressed in later sections. One original motivation for studying the transport of ions through insulating capillaries, was to prove that conical glass capillaries can focus an injected beam, similar to electrostatic lenses (Einzel Lenz). While some authors imagined such process, it was not jet demonstrated neither experimentally nor theoretically. Another motivation for the development of *InCa4D* was the rich amount of experimental data available in the literature, awaiting for more detailed theoretical support. In return, the experimental data provided constraints to the model that the simulations had to fulfil. Using a feedback-loop, the model was refined in order to reproduce the experimental data. We foresee to make *InCa4D* freely available, for simulating with high accuracy the ion transport in insulating capillaries, in the spirit of SIMION®.

Regarding the history of capillaries in CIMAP: Capillaries were already studied experimentally in CIMAP since 2005 using a setup developed by A. Cassimi and L. Maunoury. Several colleagues collaborated since on the setup and their findings were published in [31, 37, 44]. In 2016, S. Guillous and E. Giglio, started to develop a capillary holder specially designed to allow the observation of the radial focusing effect in conical capillaries. The development of the capillary holder is discussed in the manuscript in section 2.7. The symmetry constraints on the capillary holder as well as the additional features integrated in the design, were suggested by simulations. Indeed, the holder and the numerical code were development hand in hand so as to make comparison between simulated and measured results as meaningful as possible.
## 2.1 Electric conductivity in glasses

In a first step, the author will discuss in some detail the electrical conductivity in insulating material (mainly ionic glasses) and at the insulator-vacuum and insulator-conductor interfaces. This will allow to introduce the assumptions used in the theoretical model underlying the numerical code *IncA4D*.

### 2.1.1 Bulk conductivity

<figure><img src="image_49.png" /><figcaption>Figure 2.4: Borosilicate glassy network structure</figcaption></figure>

In amorphous insulators with a large band gap, the electrical conductivity is entirely due to extrinsic charge carriers. In glasses, the electric conductivity is mainly based on the transport of impurity ions through the glassy network. The mobility of electrons and holes in those glasses is usually too small to contribute significantly to the electric conductivity [6] and will be neglected in this study. For example, for ®PYREX borosilicate glass, the dominating charge carriers are alkali ions, mostly Na⁺ ions. The following discussion however should hold for most ionic glasses. In borosilicate glass, the sodium cation is mainly trapped in the vicinity of a negatively charged non-bridging oxygen (NBO) or tetrahedral BO₄ (B⁴) unit and migrates via a thermally activated hopping mechanisms from one trap to another [2, 4, 5, 7, 8, 9, 13, 39, 52]. The NBO and B⁴ anions however are firmly linked to the glassy network and are considered immobile. The conductivity
in borosilicate glass follows the Arrhenius law, where $E_a$ is the activation energy of the alkali cation, $k_B$ the Boltzmann constant, and $T$ the temperature,

$$ \kappa = A \exp \left( \frac{-E_a}{k_B T} \right) \qquad (2.1) $$

For ®Pyrex 7740, we have typically $A = 10^{-3.24}$ S/m and $E_a = 0.42$ eV.

**Weak electrolyte:** In a material, whose electrical conduction results from the migration of a unique charge carrier species, the electric bulk conductivity $\kappa_b$ may be expressed as

$$ \kappa_b = qn\mu \qquad (2.2) $$

with $q$ being the charge for the charge species, $n$ the effective charge carrier concentration and $\mu$ its mobility. The latter is related to the ion diffusivity $D$ through Einstein's relation

$$ D = \frac{\mu k_B T}{q} . \qquad (2.3) $$

Expression 2.2 allows for two different interpretations, assuming that either all the ions move with the same low mobility (strong electrolytes), or only a small fraction of impurity ions move with a higher mobility (weak electrolytes). According to J.J. Muray [5] and J.L. Souquet [9, 13, 39], alkali glasses behave like a weak electrolyte<fnref n="1" />. Only those $Na^+$ ions that are thermally excited from a regular deep trap to a weak interstitial trap, where two ions share the same NBO or $B^4$ unit, can be considered mobile, with a non-negligible mobility $\mu^+$. Let $n^+$ be the density of thermally activated $Na^+$, the conductivity of a weak electrolyte with one dominant charge carrier may thus be written as,

$$ \kappa_b = e \mu^+ n^+ . \qquad (2.4) $$

Assuming a typical sodium ion density for borosilicate of about $n_0^+ \simeq 3 \times 10^{22}$ $Na^+$ ions/cm³, a bulk conductivity of $\kappa_b \simeq 10^{-15}$ S/cm and a charge carrier mobility

<footnote n="1">¹At room temperature and higher, the electrical conduction of electrically neutral borosilicate glasses is based on the transport of $Na^+$ ions through the glassy network. In its regular site, the cation is bond by a Coulomb interaction preferably to a negatively charged $BO_4^-$ borate group. With enough thermal energy, the cation can leave the regular site and occupy for a short time an interstitial site formed by two alkali cations sharing the same counter-ion. Such a defect has a higher mobility and is responsible for the alkali transport in silicates, as the lower activation energy of the weakly bound $Na^+$ allows it more easily to jump to a nearby regular site. The density of defect pairs depends on the concentration of $Na^+$ ions, their bond strength to the network in their regular cationic site and the thermal energy. The short lifetime of about $10^{-10}$ s of these defects indicates that only a small number of alkali ions participate to the conductivity at a given time, making borosilicate a weak electrolyte.</footnote>
of about $\mu^+ \simeq 4 \times 10^{-10} \text{ cm}^2\text{V}^{-1}\text{s}^{-1}$ at room temperature [6, 5], about $n^+ = \kappa_b/(q\mu) \simeq 5 \times 10^{13} \text{ ions/cm}^3$ are thermally activated at given time and contribute to the conductivity Compared to a typical density in borosilicate, mobile ions account for a fraction of $n^+/n_0^+ \simeq 10^{-9}$ at room temperature, making borosilicate indeed a weak electrolyte.

## Enhanced conductivity due to excess charge carriers:

In the case where a glass is positively charged, such that there are less negative counter-ions $n_0^-$ than alkali ions $n_0^+$, the ionic conductivity of the charged glass may be enhanced with respect to a neutral glass. As mentioned before, in borosilicate, the mobile ions are mainly those which share the same NBO or B⁴ unit. Now the excess density $\rho = e(n_0^+ - n_0^-)$ stands for those alkali ions that have lost their counter ion, and have to share a negative unit with another alkali, exactly like the thermally activated ones. Little data exists in the literature that describes such phenomenon in glass, mainly because it is negligible in most cases. For charged capillaries however, the charge stored in the glass may be high enough to enhance sensibly the bulk conductivity. Indeed, for glass-capillaries we are interested in, we typically store about 2 nC of charge in a volume of glass of $10^{-2} \text{ cm}^3$. This gives a density of $10^{12} \text{ ions/cm}^3$ that are activated by the excess charge. This is to be compared to the thermally activated sodium density of $\simeq 5 \times 10^{13} \text{ ions/cm}^3$ at room temperature<fnref n="2" />. Thus, for a weak electrolyte like glasses, the density of thermally activated ions does not necessarily dominates the density of excess alkali ions and the latter may contribute significantly to the conductivity. Especially, below room temperature. In the absence of any data regarding their mobility, we assume in the following that the mobility of the sodium ions having lost their counter ion equals the mobility $\mu^+$ of the thermally activated alkali. Within this hypothesis, the density of mobile ions is locally enhanced by the excess charge carriers and the ionic conductivity eventually reads

$$ \kappa = e\mu^+(n^+ + (n_0^+ - n_0^-)) \quad (2.5) $$

$$ = \kappa_b + \mu^+ \rho \quad (2.6) $$

While at higher temperature, the last term of equation (2.6) is certainly negligible compared to the bulk conductivity, it may no longer be dominated at low temperature and for high space charge in the insulator. For completeness, the enhancement of the conductivity by positive space charge is taken into account in this model.

<footnote n="2">²In contrast, in strong aqueous electrolytes, containing for example HCl (hydrochloric acid) or (NaOH (sodium hydroxide), where all the ions are active, the density of active ions would only marginally increase ($n_0^+ - n_0^-$) $\ll n_0^+$, so that the contribution of excess charges to the conductivity may be safely ignored.</footnote>
## 2.1.2 Surface conductivity

Usually, the surface conductivity of insulating surfaces, labelled $κ_s$, stands for the conductivity along the insulator-gas interface due to adsorbed impurities. The surface conductivity is expressed in Siemens (S). In the atmosphere for example, the surface conductivity of glass samples depends strongly on the humidity in the atmosphere, meaning that the adsorbed water ions are the main contributors to the anomalous large surface conductivity for glasses in the atmosphere. In near vacuum, the humidity is clearly low and the surface conductivity decreases be several orders of magnitude with respect to the humide atmosphere. It was found by Gruber et al. [43] that the surface conductivity in Borosilicate glass follows an Arrhenius type law, with a typical value of $κ_s = 10^{-16}$ S at room temperature.

## 2.1.3 Non-linear conductivity

The conductivity may be written $κ_b = qnμ$, where the separation of carrier density $n$ and mobility $μ$ is artificial; carrier density and mobility are strongly correlated and may depend on the spatial and energy distribution of the charge carriers. One distinguishes shallow and deep (regular) traps. In its regular site, the cation is bond by a Coulomb interaction preferably to a negatively charged $BO_4^-$ borate group. With enough thermal energy, the cation can leave the regular site and occupy for a short time an interstitial site formed by two alkali cations sharing the same counter-ion [9, 13, 39]. Such an interstitial defect has a higher mobility and is responsible for the alkali transport in silicates, as the lower activation energy of the weakly bound $Na^+$ allows it more easily to jump to a nearby regular site. The ability for a carrier to escape the regular site are influenced by the temperature and by the applied electric field $E$. In the thermally assisted hopping model, the rate of escape

$$ \Omega_0(T) = \nu \exp \left( -\frac{\Delta H}{k_B T} \right) $$

depends on the attempts frequency $ν$ and the average trap depth $ΔH$. In the presence of an electric field, the probability of escape is the sum of the probability in both directions, with the field and against the field.

$$ \Omega(E, T) = \Omega_0 2 \sinh \left( \frac{eEa}{2k_B T} \right), $$

with $a \approx 8$ nm being the average anomalously large distance between interstitial traps [9]. $Ω$ can be linked to the carrier mobility through the drift velocity,

$$ a\Omega = \mu E $$
From the definition of the current density $j = \kappa_b E = n\mu E = na\Omega$ one deduces the carrier mobility,

$$ \mu = \frac{qa^2\nu}{k_b T} \exp\left(-\frac{\Delta H}{k_B T}\right) \sinhc\left(\frac{qEa}{2k_B T}\right) \quad (2.7) $$

$$ = \mu_0 \sinhc(E/E_c) \quad (2.8) $$

where sinhс() the hyperbolic sinus cardinal and $E_c = \frac{2k_B T}{qa} \approx 8$ kV/mm, the critical field well below the dielectric strength field of 14 kV/mm in pyrex. For

$$ E \ll E_c \quad (2.9) $$

we recover the field independent mobility $\mu_0$. For ionic conductivity, the thermally assisted hopping model (TAHM) is here preferred to the Poole-Frenkel model, $\mu \propto \exp(\sqrt{E/E_c^{PF}})$ [1], which rather describes the electronic conduction of holes and electrons in polymers like Teflon or Mylar [48]. For macro-capillaries, like the one we studied here, the condition 2.9 is usually fulfilled and we will consider thus in the following that the conductivity is field independent, except if stated otherwise.

### 2.1.4 Interface charge carrier injection

#### Inner surface

When an insulating surface is irradiated by a positively charged beam, the ions may be implanted into, and/or electrons are removed from the surface, depending on the nature, impact angle and energy of the ions. For grazing impact angles, like the one considered here, cations preferably injects holes into the solid in the vicinity of the impact. We consider the case where a low energy ion beam is injected into a borosilicate capillary. When slow grazing Ar$^{q+}$ ion hit the inner surface of the dielectric, electrons are ejected from (or holes injected into) the surface at the impact point. These holes are quickly trapped near the impact point, mostly by NBO's, which act as hole traps. While usually q ejected electron are captured by the Ar$^+$ ion, the others will fall back on the surface and trapped by electron traps at the interface and eventually recombine with the holes. As a result, the net deposited charge per impact is q holes captured by q NBO units. This analyses can be generalized to others simply or multiply charged ions. Indeed, in the case of a Xe$^{23+}$ ion beam at 10 keV, Cassimi et al. [31] found only neutrals and ions of charge state of +23 in the transmitted beam, giving the indirect evidence that the ions hitting the surface have all been neutralized, by capturing 23 electrons. Of course, the number of electrons ejected from the impact point generally exceed the charge state of the ion. But those electrons, which are not captured by the ion,
fall back on the surface. We assume in the following that the non-captured electrons fall back to the impact point from where they were emitted<fnref n="3" />.

As a result, the deposited charge per ion at the impact point is equal to the ion charge, independent of the charge state of the surface and energy of the ion. The number of injected holes $N_h$ per unit surface $ds$ and unit time $dt$ at an inner surface point $\vec{r}$, namely $\gamma^h(\vec{r})$, is thus given by

$$e \frac{d^2 N_h}{ds dt} \equiv \gamma^h(\vec{r}) = \vec{\eta}(\vec{r}) \cdot \vec{\gamma}(\vec{r}) \quad , \qquad (2.10)$$

where $\vec{\gamma}$ is the incoming beam current density, $\vec{\eta}$ the vector normal to the surface and $e$ the elementary charge. For the NBO units that have captured a hole, the charge of the near lying Na⁺ ion is no longer compensated. The alkali ions will start to repel each other and move away, driven by the electric field, which is generated by the uncompensated Na⁺ ions. This way, in alkali glasses like borosilicate, the positive charge injected by beam ions are not removed because of the migration of electrons or holes, but rather because of the migration of uncompensated alkali ions away from the impact point. Possibly, the migration of electrons and holes in those glasses may happen on a much longer time scale.

## Outer surface

**Stray electrons:** At the outer glass-vacuum interface, no ionic charge transfer takes place. However, potential low energetic stray electrons, that are attracted by the positive charge stored in the capillary walls, may be injected at the outer interface, where they are quickly trapped by electron centres. A reliable expression for the injection rate of stray electron per unit surface may be taken from [62], where a detailed study on the injection rate of stray electrons was proposed. The authors assumed that the stray electrons are mostly produced by the beam ions that hit metal parts in the chamber. The intensity of stray electrons is thus taken proportional to the injected ion beam current $I$. This proportionality factor, $\lambda_s \ge 0$, depends on the shielding of the capillary’s outer surface from secondary electron sources and on the potential of the capillary. Naturally, a positively charged capillary attracts more electrons than a neutral one. The injection rate must thus also depend on the charge of the capillary, or alternatively on its Coulomb potential $U$. In the study presented in [62], the authors found that, for a given beam current $I$ and a given shielding quantified by the value $\lambda_s$, the injection rate increases linearly for low capillary potentials but tends asymptotically to $I\lambda_s$ above a threshold potential $U_c$, where all the stray electrons, that bypass the shielding, are injected

<footnote n="3">³This assumption will be relaxed in a later development of the model and numerical code so that secondary electrons emitted from the impact point are transported through the capillary and injected at new impact point, redistributing the injected charge in the capillary.</footnote>
into the outer surface. The rate of injected stray electron per unit surface at an outer surface point $\vec{r}$ is thus simply given by

$$ \gamma_{\text{str}}^{e}(\vec{r}) = I \lambda_{s} \text{erf} \left( \frac{U(\vec{r})}{U_{c}} \right) . \qquad (2.11) $$

Note that stray electrons may represent a major neutralizing channel, if the capillary is not sufficiently shielded from secondary electrons sources. Indeed, in the setup presented in [62], the shielding was obviously incomplete and $\lambda_s$ was measured to be about 0.95. The study showed that the neutralization channel due to stray electrons was so strong in that setup that the asymptotic potential of the capillary stayed well below the extraction potential of the ion source for all injected beam intensities.

**Interface conductivity:** In the presence of an electric field, mobile cations are field driven until they reach an interface. If the interface is blocking for the ionic charge carrier so that no exchange of ions can take place, they accumulate near the interface. This is certainly true for the glass-vacuum interface but also for many electrode-glass interfaces. Typically, electrodes consisting of a silver or carbon paste layer, as has been used in many capillary setups [43, 37], can be considered as quasi blocking for the $Na^+$ cations. As a result, there is accumulation or depletion of ions near the interfaces, leading to the formation of space-charge layers. The voltage drops rapidly in these layers, which implies a huge electrical interface polarization of the material and a drop of electric field in the bulk. For completely blocking glass-electrode interfaces, the charge migration would vanish asymptotically. That it does however not vanish but tends instead towards a steady-state current, as is observed in metal-insulator-metal resistivity measurements, is due to a surface process that removes sodium ions from the interface to provide for continuity of the current [15]. A possible mechanism, proposed by Rohatgi [3], is some sort of surface recombination of $Na^+$ ions with surface electron traps, replenished by dawning electrons from the metal electrode to the surface of the glass. The latter phenomenon may explain the observed glass-conductor interface conductivity. Interface conductivity may differ by several orders of magnitude from the bulk conductivity of glass and depends mainly on the type of contact. If the interface conductivity is higher than the bulk conductivity of the dielectric, then the discharge current is controlled by the bulk conductivity. Otherwise the discharge current is dominated by the interface conductivity. A detailed description of the insulator-metal interface conductivity may be found in [19]. For each charge carrier, one distinguishes at the glass-conductor interface between ohmic and blocking type of contact.

Let us assume that the glass-electrode interface is blocking for the ionic charge carriers and only electrons are allowed to pass the interface. This is certainly
the case if the electrode is a graphite layer, which cannot supply or receive alkali ions. If the inner surface is positively charged by ion beam impacts, the outer surface is negatively biased (cathode), favouring electron injection through the conductor-glass interface into the insulator, with the injection rate depending again on the type of contact. In the extreme case where the contact is also blocking for electrons, the interface conductivity would be zero. On the other extreme, for an ohmic contact, electrons flow readily from the electrode into the insulator where they are quickly trapped by electron traps at the glass interface. As the mobility of trapped excess electrons is negligibly low, they accumulate there. The electron flow through the interface stops once the thermodynamic equilibrium at the interface is attained. There is thus a reservoir of filled electron centres at the interface that re-trap the Na⁺ cations that reach the interface, giving rise to a neutral pair. The recombining rate of a sodium ion with an electron centre (NBO, B⁴ ...) at the interface corresponds thus, in the case of an ohmic contact, to the rate at which the ions approach the interface and is thus controlled by the bulk conductivity.

For simplicity, we will in the following assume that the interface contact is ohmic. Within this assumption, the interface current density $\gamma_{int}$ is proportional to the normal component of the electric field $E_n(\vec{r})$ at the glass-conductor interface, with the proportionality factor $\kappa_b$ standing for the ionic bulk conductivity.

$$ \gamma_{int}(\vec{r}) = \kappa_b E_n(\vec{r}) \quad . \qquad (2.12) $$

If the type of contact is known at the interface, $\kappa_b$ in equation 2.12 may be replaced by the corresponding expression.

## 2.2 Charge dynamics in the capillary walls

In the capillary field, several groups have developed a numerical code in order to support theoretically the observed guiding power in insulating capillaries [24, 25, 48, 49, 53]. In all cases, the classical trajectory of the projectiles are deduced from the electric field generated by the accumulated charges in the capillary wall. The time-dependent ion transmission depends of course on the time evolution of the guiding electric field and thus on the charge dynamics of the excess charge carriers in the insulating capillary. Thus much attention must be paid to the description of the charge dynamics in capillaries in order to obtain reliable simulations that are accurate and predictive. In the previous works, in order to keep the CPU cost viable, simplified models for the charge dynamics were proposed. The relaxation channels that were considered in the description of the charge dynamics in [24, 25, 53] were usually described phenomenologically, by retaining mainly the characteristic time related to the considered channels. While this approach captures the main effect of the channel on the charge dynamics, it
lacks the refinement of an explicit field dependent charge dynamics. In [48, 49], N. Stolterfoht included a relaxation channel where the injected charges migrate along the surface with a velocity that depends non-linearly on the electric field. This was a refinement over the previous approach, which allowed him to explain the guiding power in nano-capillaries. In this work, the author will go a step further and give a detailed description of the charge dynamics in insulating capillaries by considering explicitly the charge current associated with each relaxation channel. The resulting model is assumed to be applicable in most situation, may it be for nano- or macro capillaries, even if here, the description mainly emphasizes on glass macro-capillaries.

### 2.2.1 Excess charge carrier density

The time-evolution of the density of the charge carriers in the insulator is described by the continuity equation. Although the mobility of negatively charged point defects like NBO, B⁴ or other filled electron centres is negligible low, the density $n_0^-$ of those negative units varies in time due to hole injection at the inner surface and possible electron injection at the outer surface,

$$e \frac{\partial n_0^-}{\partial t} = \vec{\nabla} \cdot (\gamma^h - \gamma_{\text{str}}^e - \gamma_{\text{int}}^e) \vec{\eta} \quad . \qquad (2.13)$$

Here e stands for the elementary charge. The density $n_0^+$ of Na⁺ ions, on contrary, varies because a part of them have a mobility high enough to be field driven,

$$e \frac{\partial n_0^+}{\partial t} + \vec{\nabla} \cdot \vec{j} = 0 \quad , \qquad (2.14)$$

where $\vec{j}$ stands for the ionic current density. As we assumed that no ionic charge transfer takes place at the interfaces, no source term is added. Eventually, the excess charge density, which is defined as

$$\rho = e(n_0^+ - n_0^-) \quad , \qquad (2.15)$$

evolves in time accordingly and reads

$$\frac{\partial \rho}{\partial t} + \vec{\nabla} \cdot \vec{j} = -\vec{\nabla} \cdot (\gamma^h - \gamma_{\text{str}}^e - \gamma_{\text{int}}^e) \vec{\eta} \quad . \qquad (2.16)$$

Inside the dielectric, $\vec{j}$ has two main contributions, one proportional to the electric field, $\vec{E} = -\vec{\nabla}V$, and one proportional to the charge density concentration gradient $\vec{\nabla}\rho$.

$$\vec{j} = \kappa \vec{E} + D \vec{\nabla} \rho \quad , \qquad (2.17)$$
with $κ$ and $D$ being respectively the alkali conductivity and diffusivity. The diffusion term may be neglected here as it is quickly dominated by the electric field term<fnref n="4" />. As discussed before, in weak electrolyte like glasses, the density of thermally activated ions does not necessarily dominates the density of excess alkali ions and the latter may contribute significantly to the conductivity. As a result, the ionic bulk conductivity $κ_b = eμn^+$ is locally enhanced by the excess charge carriers and the current density eventually reads

$$ \vec{j} = \kappa \vec{E} = e\mu(n^+ + (n_0^+ - n_0^-))\vec{E} \quad (2.18) $$

$$ = (\kappa_b + \mu\rho)\vec{E} \quad . \quad (2.19) $$

In a sense, this is similar to adding salt into deionized water.

### 2.2.2 Surface charge approximation

In alkali glasses like borosilicate and soda lime glasses, the mobility of holes, electrons and anions is negligible compared to the mobility of alkali cations and the bulk conductivity $κ$ in those glasses is dominated by the mobile alkali cations. Also, electrons and holes that are injected at the interfaces are quickly trapped by electron traps or hole traps at the interface. We assume thus that the rate $τ_h^{-1}$ at which the injected holes migrate into the bulk is negligible compared to the charge relaxation rate due to bulk conductivity, $τ_h^{-1} \ll τ_b^{-1} = \frac{\kappa}{\varepsilon_0\varepsilon_r}$. This implies that the electrons and holes that are injected at the interfaces accumulates only at the inner and outer surface and not in the bulk. For example, if a hole migrates into the bulk, the mobile alkali ions will rearrange around the hole at the rate $τ_b^{-1}$, so that the hole is balanced ("neutralized") by a deficit of mobile ions. As a result, for each hole that migrates into the bulk, an alkali ion will migrate "rapidly" to the interface, leaving a zero net charge in the bulk. The injected charge can thus be represented by a surface charge density. Consequently, the mobile charge carriers in the insulator are driven by a divergence-free electric field $\nabla \cdot \vec{E} = 0$ through the glass bulk. If the electric field is directed from the inner to the outer surface, field-driven Na⁺ cations tend to deplete the inner interface and accumulate at the blocking outer interfaces $S_2$, so that the excess charge density $ρ$ is mainly non-zero at the interfaces. Further away from the interfaces, the excess charge density $ρ$ tends to zero. These trends are depicted in figure 2.6. Because the excess charge density $ρ$ is mainly non-zero at the interfaces, we propose to replace the charge

<footnote n="4">⁴Using the Einstein relation, linking the diffusivity to the charge carrier mobility, $D = \frac{\mu k_B T}{e}$, we deduce from the dimensionless analysis that the diffusion term is negligible if $k_B T \ll eU$ where $U$ is the potential in the capillary. At room temperature, $T = 300$ K, this condition is fulfilled for $U \gg 0.24$ V.</footnote>
<figure><img src="image_50.png" /><figcaption>Figure 2.5: Negative $n_0^-$, positive $n_0^+$ and excess charge carrier distribution $\rho$ in the glass wall following charge injection at the interface. Because the mobile alkali ions are field driven from the inner $S_1$ to the outer $S_2$ interface by the electric field $E$, they deplete at the inner and accumulate at the outer interface. The negative immobile anion charge density $n_0^-$ depletes at the inner interface because of hole injections by the projectiles at the inner surface. At the outer interface the accumulation of negative charges accounts for a possible injection of electrons by either stray electrons or by a grounded electrode. The excess charge density $\rho$ is simply the sum of the positive and negative charge densities and is mainly non-zero at the interfaces.</figcaption></figure>

distribution $\rho$ by surface charge densities $\sigma_1$ and $\sigma_2$ at the inner and outer surface respectively. Using the cylindrical coordinates system $(r, \theta, z)$, the inner and outer surface of an axisymmetric capillary are given by the function $R_1(z)$ and $R_2(z)$, respectively. The surface charge densities at the two interfaces may then be defined by

$$ \int_{R_1(z)}^{R_2(z)} r \rho(r, \theta, z) dr \approx \sigma_1(\theta, z) \frac{ds_1}{dz} + \sigma_2(\theta, z) \frac{ds_2}{dz} \quad (2.20) $$

where $ds_i = \sqrt{R_i^2(z) + R_i'^2(z)} dz$ is the surface element at the interface $i$.

### 2.2.3 Surface charge dynamics at the interfaces

Motivated by the above discussion, we assume here that no excess charges are stored inside the bulk $\rho = 0$ and that the electric field is therefore divergence-free inside the capillary walls

$$ \nabla \cdot \vec{E} = 0 . \quad (2.21) $$
The time evolution of the charge density in the capillary (2.16) is replaced by the time evolution of the surface charge density at the interfaces ($S_1$) and ($S_2$) and best described by the surface continuity equation [60, 61]. At the inner interface ($S_1$), the time evolution of $\sigma_1$ is given by

$$ \frac{\partial \sigma_1}{\partial t} + \vec{\nabla}_s \cdot ((\kappa_s + \mu \sigma_1) \vec{E}_1) = -\kappa E_{n,1}^{+} + \gamma^h \quad (2.22) $$

where the operators

$$ \vec{\nabla}_s = \vec{\nabla} - \vec{n} (\vec{n} \cdot \vec{\nabla}) , \quad \vec{\nabla}_s \cdot \vec{T} = \vec{n} \cdot (\vec{\nabla} \wedge (\vec{n} \wedge \vec{T})) $$

are respectively the surface gradient and the surface divergence, with $\vec{T}$ being an arbitrary vector tangent to the surface. The source term $\gamma^h$ stands for the injected holes by ion impacts and $\kappa E_{n,1}^+$ for the current density of mobile charge carriers that are field-driven from the inner interface into the bulk. The term $\kappa E_{n,1}^-$ is not present because no mobile charge carriers are injected from the vacuum at the inner surface. On the outer interface ($S_2$) the dynamics of the surface charge $\sigma_2$ becomes,

$$ \frac{\partial \sigma_2}{\partial t} + \vec{\nabla}_s \cdot ((\kappa_s + \mu \sigma_2) \vec{E}_2) = \kappa E_{n,2}^{-} + \gamma_{\text{str}}^e + \gamma_{\text{int}}^e \quad (2.23) $$

The source terms $\gamma_{\text{str}}^e + \gamma_{\text{int}}^e$ stand for the injected electrons at the outer surface and $\kappa E_{n,2}^{-}$ for the mobile charge carriers that are field driven typically from the bulk to the outer interface where they accumulate. The second left hand term in both equations gives the flow of mobile charges along the respective interface.

Limiting the charge dynamics in the capillary to both interfaces allows reducing by one (spatial) dimension the charge dynamics equation (2.16) and thus the CPU cost without sacrificing the scope of the model presented above. The coupling is due the electric field $\vec{E}$, which is a functional of two excess surface charge distribution $\sigma_{1,2}$. That the electric field needs to be evaluated only at the two interfaces helps to reduce even more the CPU cost for the charge dynamics in the dielectric.

### 2.2.4 Evaluation of the electric field at the interfaces

#### Polarization charges in dielectrics

We consider the general case where the capillary of relative dielectric constant $\varepsilon_r$ is surrounded by an axisymmetric conducting (metal) surface ($S_3$), sharing the same symmetry axis then the capillary. The profile of the ($S_3$) metal-vacuum interface is defined by $R_3(z)$ with $R_3(z) > R_2(z)$. If no conducting surface is present for a given $z$, then $R_3(z)$ is pushed to infinity. We have eventually to
<figure><img src="image_51.png" /><figcaption>Figure 2.6: Scheme of the capillary wall with the 3 interfaces, namely $S_1$ and $S_2$ standing for the inner and outer capillary glass-vacuum interfaces and $S_3$ for the metal-vacuum interface of the metal layer surrounding the capillary. The red charges are the polarization charges appearing around the positive charges in the bulk and at the glass-vacuum interfaces. The blue surface charges at the vacuum-metal interface $S_3$ are free charges that accommodate to satisfy the Dirichlet condition of a constant potential $V_3$ at the surface of metal layer.</figcaption></figure>

deal with three interfaces, two vacuum-dielectric interfaces, ($S_1$) and ($S_2$) and one metal-vacuum interface ($S_3$). In the presence of dielectrics and metal surfaces, some attention must be paid to the evaluation of the electric field $\vec{E}$ generated by the deposited charges in the capillary wall. Indeed, polarization surface charges induced at the glass-vacuum interfaces as well as free surface charges induced at the metal-vacuum interface affect the electric field generated by the deposited charges. A systematic error in the evaluation of the field induces discrepancies in the trajectories of the injected ions and consequently on the source term $\gamma^h$, which in turn controls the dynamics of the deposited charges in the capillary walls. Induced surface charges need thus to be computed accurately in order to avoid a systematic and accumulative errors in the simulations.

Let $\vec{r}_1$, $\vec{r}_2$ and $\vec{r}_3$ be the position vectors pointing respectively at the inner $S_1$, outer surface $S_2$ and metal surface $S_3$. Following the discussion in section 2.2.2, the accumulated charge is represented by the surface charge densities $\sigma_{1,2}$ defined on the interfaces $S_1$ and $S_2$. At both vacuum-glass interfaces $S_1$ and $S_2$ appear polarization surface charges $\tilde{\sigma}_{1,2}$. At the metal-vacuum interface, $S_3$, free surface charges $\tilde{\sigma}_3$ appear so as to maintain an equipotential on $S_3$. A schematic
representation of the interfaces in given in 2.6. Note that in the literature, these surface polarization charges are often referred to as image charges. The electric potential V at arbitrary point $\vec{r}$, due to the accumulated charge distributions $\sigma_{1,2}$ and the three induced surface charge densities $\tilde{\sigma}_{i=1,2,3}$ at the three interfaces $S_{i=1,2,3}$, is eventually given by

$$V(\vec{r}) = \int_{S_1} \frac{\sigma_1 ds_1}{4\pi\epsilon_0|\vec{r} - \vec{r}_1|} + \int_{S_2} \frac{\sigma_2 ds_2}{4\pi\epsilon_0|\vec{r} - \vec{r}_2|} + \sum_{i=1}^{3} \int_{S_i} \frac{\tilde{\sigma}_i d^2 s_i}{4\pi\epsilon_0|\vec{r} - \vec{r}_i|} \quad (2.24)$$

with $ds_i$ being the surface element. The polarization charges $\tilde{\sigma}_1$ and $\tilde{\sigma}_2$ at the inner and outer surface of the dielectric are deduced from the boundary conditions of the normal component of the electric field across both interfaces. In the presence of free surface charges $\sigma_{1,2}$, one has,

$$(S_1): \quad -\epsilon_r \left. \frac{\partial V}{\partial \vec{n}} \right|_{\vec{n}=r_1^+} + \left. \frac{\partial V}{\partial \vec{n}} \right|_{\vec{n}=r_1^-} = \frac{\sigma_1}{\epsilon_0} \quad (2.25)$$

$$(S_2): \quad - \left. \frac{\partial V}{\partial \vec{n}} \right|_{\vec{n}=r_2^+} + \epsilon_r \left. \frac{\partial V}{\partial \vec{n}} \right|_{\vec{n}=r_2^-} = \frac{\sigma_2}{\epsilon_0} \quad (2.26)$$

The surface density $\tilde{\sigma}_3$ of free charges at the metal layer is deduced from a Dirichlet boundary condition, i.e. the potential is an equipotential at the metal interface.

$$(S_3): \quad V(\vec{r}_3) = V_3 \quad . \quad (2.27)$$

In the case where the value of the equipotential $V_3$ of the electrode is not imposed but floating, we use the constraint that the total initial charge of a floating isolated electrode does not vary in time,

$$\int \sigma_3 ds_3 = Q_3 \quad .$$

If for example the electrode was initially neutral, then $Q_3 = 0$. With this supplementary condition, the value of the equipotential $V_3$ is found.

### Matrix representation of the surface integrals

In order to compute the surface densities $\tilde{\sigma}_i$, one needs to evaluate the electric potential and the normal component of the electric field at the interfaces. It is convenient, for the sake of a compact notation, to cast the surface integrals, found in (2.24), into matrix forms,

$$\int_{S_i} \frac{\tilde{\sigma}_i ds_i}{4\pi\epsilon_0|\vec{r}_j - \vec{r}_i|} \equiv K_{ji}\tilde{\sigma}_i \quad , \quad (2.28)$$
where the indexes i = {1, 2, 3} and j = {1, 2, 3} stand for the three interfaces. Similar, the normal derivatives of the surface integrals taken at the interfaces may be expressed in matrix form,

$$ \left. \frac{\partial}{\partial \vec{n}} \int_{S_i} \frac{\tilde{\sigma}_i ds_i}{4\pi\epsilon_0 |\vec{r} - \vec{r}_i|} \right|_{r=r_j} \equiv N_{ji} \tilde{\sigma}_i . \qquad (2.29) $$

Details about $K_{ji}$ and $N_{ji}$ are given in the appendix. Because of the discontinuity of the normal component of the electric field through an interface, we must distinguish between the inner (-) and outer (+) field in the case where $i = j$. Eventually, we obtain for each boundary condition (2.25), (2.26) and (2.27) a matrix equation. These equations are coupled by the induced surface charges $\tilde{\sigma}_{1,2,3}$ at the interfaces, yielding a system of matrix equations of the form,

$$ \begin{bmatrix} N_{11}^{-} - \epsilon_{r} N_{11}^{+} & \beta N_{12} & \beta N_{13} \\ -\beta N_{21} & \epsilon_{r} N_{22}^{-} - N_{22}^{+} & -\beta N_{23} \\ K_{31} & K_{32} & K_{33} \end{bmatrix} \times \begin{bmatrix} \tilde{\sigma}_{1} \\ \tilde{\sigma}_{2} \\ \tilde{\sigma}_{3} \end{bmatrix} = \begin{bmatrix} (\text{RHT}_1) \\ (\text{RHT}_2) \\ V_3 - V_\rho \end{bmatrix} \qquad (2.30) $$

with $\beta = (1 - \epsilon_r)$. The right hand terms $(\text{RHT}_1)$ and $(\text{RTH}_2)$ stand for the electric field at the interfaces $S_1$ and $S_2$ evaluated using the known quantities $\sigma_{1,2}$

$$ \begin{aligned} (\text{RHT}_1) &= (\epsilon_0^{-1} + \epsilon_r N_{11}^{+} - N_{11}^{-})\sigma_1 - \beta N_{12}\sigma_2 \\ (\text{RHT}_2) &= \beta N_{21}\sigma_1 + (\epsilon_0^{-1} - \epsilon_r N_{22}^{-} + N_{22}^{+})\sigma_2 \end{aligned} \qquad (2.31) $$

At the interface $S_3$, the known quantity at the right hand side of the matrix system is given by the difference between the equipotential $V_3$ of the metal layer and the Coulomb potential $V_\rho(\vec{r}_3)$,

$$ (S_3) : V_3 - V_\rho(\vec{r}_3) . \qquad (2.32) $$

The polarization charges $\tilde{\sigma}_{1,2}$ as well as the free charges on the metal layer $\tilde{\sigma}_3$ are readily computed by solving the system by matrix inversion. Finally, the potential at an arbitrary point is now evaluated using equation (2.24), from which the electric field $\vec{E} = -\nabla V$ is deduced. More detail about the evaluation of the electric field and potential are given in appendix 2.11.3. The influence of the polarization charge on the electric field is illustrated by an example in appendix 2.11.4.

### 2.2.5 Image charge potential of projectile

The image charge potential of projectile of charge $q$ becomes non-negligibly when the projectile approaches the inner glass-vacuum interface within a nanometer scale
distance, so that the ion can induce enough polarization charges at the interface S₁ to affect its trajectory. In order to avoid that the image charge force diverges, the projectile's charge q is screened below a distance dscr = 10⁻⁹ m from the surface. For micro-capillaries with an inner radius much larger the the screening distance dscr, we preferably use the analytic expression of the image charge potential, in the adiabatic approximation, obtained in the case of an infinite plane interface between a dielectric of dielectric constant εr and the vacuum.

$$V_{\text{im}}(r, \theta, z) = \frac{q}{8\pi\varepsilon_0} \frac{\varepsilon_r - 1}{\varepsilon_r + 1} \frac{1}{R_1(z) - r} \text{erf} \left( \frac{R_1(z) - r}{d_{\text{scr}}} \right) \quad (2.33)$$

However for a cylindrical surface with a radius in the nano-meter range, this expression may over-estimate the image charge force. The image force for a cylindrical interface may then be expressed more accurately by a (slowly converging) series of Bessel terms [21, 54].

## 2.3 Equation of motion of the projectile

For low enough ion beams intensities I < 1 pA, the coulomb interaction between the ions of the beam can be neglected and the trajectories regarded as independent one from another. The equation of motion (EOM) of a particle of charge q and mass m at the position $\vec{r}_p$ interacting with a charged capillary of dielectric constant εr may be written

$$\frac{d^2 \vec{r}_p}{dt^2} = -\frac{q}{m} \nabla (V[\rho, \tilde{\sigma}_{1,2,3}] + V_{\text{im}}[d(\vec{r}_p, S_1)]) \Big|_{\vec{r}=\vec{r}_p}, \quad (2.34)$$

where the potential $V[\rho, \tilde{\sigma}_{1,2,3}]$ is given by (2.24) and where $V_{\text{im}}$ stands for the non-dissipative part of the image charge potential of the projectile. Eventually, the trajectory of the ion is computed on the fly by evaluating the electric field $-\nabla(V + V_{\text{im}})$ at the projectile position using equation (2.24). The propagation is based on a 4th order time adaptive Predictor-Corrector scheme [29].

## 2.4 Charge relaxation rates

In the following section, we will show how to extract the charge relaxation rate for insulating capillaries like those used in [43, 51, 53]. A detailed study of the charge relaxation dynamics of charge patches and their associated relaxation time scales is given in Giglio et al. [67]. I will give here a short summary of the main ideas presented in the paper. We consider a cylindrically shaped glass capillary (tube) of length H, with an inner radius R₁, an outer radius of R₂ and which has
a dielectric constant $\epsilon_r$ (see Fig. 2.7). The insulating capillary tube of symmetry axis $Oz$ has the outer surface, entrance and exit grounded. We assume that the electric charge is injected by the ion beam at the inner interface and that the charges accumulate only at the interface but not in the bulk. As a result, the electric field is divergence-free and the electric potential $V$ satisfies the Laplace equation everywhere. The underlying model considers thus merely two interfaces namely the inner glass-vacuum interface $S_1$ of the capillary and the outer glass-metal interface $S_2$. The charge dynamics taking place at the inner interface $S_1$ is expressed in the cylindrical coordinate system by the equation,

$$ \frac{\partial \sigma}{\partial t} = -\kappa_b E_r - \frac{\partial}{\partial z} (\kappa_s E_z) - \frac{1}{R} \frac{\partial}{\partial \theta} (\kappa_s E_{\theta}) + \gamma I_{\text{inj}} \quad (2.35) $$

The first right hand term, which is proportional to the bulk conductivity $\kappa_b$, stands for the mobile charge carriers that are driven from the inner to the grounded outer surface by the radial component $E_r$ of the electric field. The second and third right hand terms stand for the charge carriers that are field driven at the inner interface respectively along the symmetry axis by the tangential field $E_z$ and along the angular direction by the tangential field $E_{\theta}$. The surface current is proportional

<figure><img src="image_52.png" /><figcaption>Figure 2.7: Scheme of the surface current in the capillary. The outer surface $S_2$ is grounded. The insulating inner surface $S_1$ carries the accumulated surface charge density $\sigma$. The induced polarization charges $p$ and free charges $f$ are also shown. The tilted injected beam (large red arrow) deposits a charge patch near the entrance, which will relax in time due to a surface current along the angular direction (red arrows in left figure) a surface current along the axial direction and a current through the bulk to the outer grounded surface (red arrows in right figure).</figcaption></figure>
to the surface conductivity $\kappa_s$. The last term represents the injected charge per unit time and unit area, with $I_{\text{inj}}$ being the beam current and $\gamma$ the hole-injection cross-section. To keep the math simple, we omitted here the part of the surface conductivity due to the excess charges $\mu\sigma$. The latter can however be added without larger difficulties in principle. Extracting the charge relaxation rate of the injected patch via a dimensionless analysis of (2.35) is not straightforward as it would yield three different characteristic relaxation rates related to the three independent relaxation channels,

$$ \tau_b^{-1} = \frac{\kappa_b}{\varepsilon_0\varepsilon_r} \qquad (2.36) $$

$$ \tau_s^{-1} = \frac{\kappa_s}{\varepsilon_0 R_1} \qquad (2.37) $$

$$ \tau_z^{-1} = \frac{\kappa_s}{\varepsilon_0 H} \qquad (2.38) $$

and we have no clear way to combine them to get the total relaxation rate. Here, we present a simple method that allows us to extract the total charge relaxation rate of an injected charge patch located at the $S_1$ interface. The charge relaxation dynamics will be obtained in the case where the capillary potential satisfies the following boundary and symmetry conditions:

1. The outer surface is grounded: $V(S_2) = 0$

2. The entrance and exit are grounded: $V(z = 0) = V(z = H) = 0$

3. The potential is assumed having the $xOz$ plane symmetry so that: $V(\theta) = V(-\theta)$

4. If we label $V^-$ the potential in the inner domain $r \le R_1$ and $V^+$ the potential in the bulk domain $R_1 \le r \le R_2$, then the continuity of the potential implies $V^-(R_1, \theta, z) = V^+(R_1, \theta, z)$

Equation (2.35) is solved by using the multipole expansion technique. The potentials are expanded on a harmonic basis which satisfying the Laplace equation as well as the imposed symmetry and boundary conditions:

$$ V^{-}(r, \theta, z, t) = \sum_{mn} v_{mn}(t) \frac{I_{m}(k_{n}r)}{I_{m}(k_{n}R_{1})} \cos(m\theta) \sin(k_{n}z), \quad r \le R_{1} \qquad (2.39) $$

$$ V^{+}(r, \theta, z, t) = \sum_{mn} v_{mn}(t) h_{mn}(r) \cos(m\theta) \sin(k_n z) , \quad R_1 < r \le R_2 \qquad (2.40) $$
where we introduced the auxiliary function $h_{mn}(r)$ defined as

$$h_{mn}(r) = \frac{I_m(k_n r) K_m(k_n R_2) - I_m(k_n R_2) K_m(k_n r)}{I_m(k_n R_1) K_m(k_n R_2) - I_m(k_n R_2) K_m(k_n R_1)} \quad (2.41)$$

in order to simplify the notations, with $I_m(k_n r)$ and $K_m(k_n r)$ being modified Bessel functions of the first and second kind of order $m$. Here $m \ge 0$ gives the angular moment of the distribution and $k_n = \frac{n\pi}{H}$ the wave number of the mode $n \ge 1$. Note that $h_{mn}(R_1) = 1$ and $h_{mn}(R_2) = 0$ satisfy the imposed boundary conditions (5) and (1). Similar, we expand the surface charge density and injection cross section defined on the interface $S_1$ onto a basis set that satisfies also the $xOy$ plane symmetry and zero boundary conditions at the endpoints. The latter is justified by the fact that the grounded inlet and outlet imposes that the free surface density and source term are also zero at $z = 0$ and $z = H$,

$$\sigma(\theta, z, t) = \sum_{mn} \sigma_{mn}(t) \cos(m\theta) \sin(k_n z) \quad (2.42)$$

$$\gamma(\theta, z, t) = \sum_{mn} \gamma_{mn}(t) \cos(m\theta) \sin(k_n z) \quad (2.43)$$

Using the jump condition of the electric field through an interface separating to dielectrics, which relates the free surface charge density $\sigma$ to the electric potentials $V^+$ and $V^-$

$$\epsilon_r \left. \frac{\partial V^+}{\partial r} \right|_{(S_1)} - \left. \frac{\partial V^-}{\partial r} \right|_{(S_1)} = \frac{\sigma}{\epsilon_0} \quad (2.44)$$

allows to express $v_{mn}(t)$ as a function of $\sigma_{mn}(t)$,

$$v_{mn}(t) = \frac{\sigma_{mn}(t)}{\epsilon_0} (\epsilon_r h'_{mn}(R_1) - h'_{mn}(R_2)) \quad (2.45)$$

The prime symbol stands for the partial derivatives with respect to the radial coordinate $r$. Injecting (2.45) into the expansions (2.39 - 2.43) into 2.35 yields a differential equation (uncoupled) for each surface charge moment $\sigma_{mn}$ of the form,

$$\frac{\partial \sigma_{mn}(t)}{\partial t} = -\frac{1}{\tau_{mn}} \sigma_{mn}(t) + \gamma_{mn}(t) I_{\text{inj}} \quad (2.46)$$

where $\tau_{mn}^{-1}$ is the charge relaxation rate associated with the mode $(m, n)$. The rate $\tau_{mn}^{-1}$ is given as the weighted sum of the surface $\tau_s^{-1} = \frac{\kappa_s}{\epsilon_0 R_1}$ and bulk rate $\tau_b^{-1} = \frac{\kappa_b}{\epsilon_0 \epsilon_r}$,

$$\frac{1}{\tau_{m,n}} = \frac{b_{m,n}}{\tau_b} + \frac{c_{m,n}}{\tau_s}, \quad (2.47)$$
<figure><img src="image_53.png" /><figcaption>Figure 2.8: Weight factors $b_{mn}$ and $c_{mn}$ for $m \le 2$ and $1 \le n \le 40$. Left: all but $c_{0n}$ vary by less than 4% with $n$, so that they can be considered constant over the range $n < 40$. Right: $c_{0n}$ varies asymptotically linear with $n$, starting from $c_{01} \simeq 10^{-3}$.</figcaption></figure>

where the positive coefficients $b_{m,n}$ and $c_{m,n}$ give the strength of the bulk and surface rate respectively,

$$b_{mn} = \frac{h'_{mn}(R_1)}{h'_{mn}(R_1) + \varepsilon_r^{-1} \frac{I'_{mn}(R_1)}{I_{mn}(R_2)}} \quad , \quad c_{mn} = \frac{(m^2 + k_n^2 R_1^2)/(\varepsilon_r R_1)}{h'_{mn}(R_1) - \varepsilon_r^{-1} \frac{I'_{mn}(R_1)}{I_{mn}(R_2)}} \quad .$$

The coefficients are time independent and need to be calculated only once for all. Equation (2.47) is a remarkable result as it shows that each relaxation rate $(\tau_{m,n})^{-1}$ includes contributions of the three possible relaxation channels, namely through the bulk, along the angular and along the axial direction at the surface and that the weight of each contribution varies with $(m, n)$. Both coefficients $b_{mn}$ and $c_{mn}$ depend on the dimensions of the capillary and more precisely on the ratios $R_1/H$ and $R_2/H$ as well as the dielectric constant $\varepsilon_r$ of the insulator. Usually, $b_{mn}$ is close to unit, while $c_{mn}$ scales as $(m^2/(k_n R_1) + k_n R_1)$ with $m$ and $n$. The trends of both coefficients with $m$ and $n$ are shown in Fig. 2.8 and discussed in detail in [67]. From (2.46) we deduce the time evolution of the accumulated charge distribution and the interface as well as the important result that each surface charge moment $\sigma_{mn}(t)$ decays exponentially with the relaxation rate $(\tau_{mn})^{-1}$

$$\sigma_{mn}(t) = \underbrace{\sigma_{mn}(0) \exp\left(-\frac{t}{\tau_{mn}}\right)}_{\text{discharge}} + \underbrace{I_{\text{inj}} \int_{0}^{t} \gamma_{mn}(t') \exp\left(\frac{t' - t}{\tau_{mn}}\right) dt'}_{\text{charge}} \quad . \quad (2.48)$$
Eventually, the analytical solution of (2.35) is given as a linear combination of surface charge moments that satisfy the given boundary conditions.

$$ \sigma(\theta, z, t) = \sum_{m,n} \sigma_{mn}(t) \cos(m\theta) \sin(k_n, z) . \qquad (2.49) $$

Let us consider now the interesting case where the accumulated charge is stationary, $\partial_t\sigma_{mn}(t) = 0$. In that case we have that the accumulated charge varies adiabatically as

$$ \sigma_{mn}(t) = I_{\text{inj}}\tau_{mn}\gamma_{mn}(t) . \qquad (2.50) $$

Using (2.45) and (2.39) the guiding electric potential can be evaluated. This may be a helpful tool to estimate if the guiding electric field is sufficient or not to guide the injected ion beam through the capillary. In conclusion, the advantage of the above multipole analyses is that we obtain an excellent estimate of the accumulated charge at the inner interface of the capillary as a function of the dimensions of the capillary beam parameters. Note that the method can be easily extended to account for an insulator-vacuum interface at $S_2$ instead and a grounded interface at $S_3$ surrounding $S_2$. Work for this more general case is underway. We give below a quick example of how to use equation (2.47) for the interpretation of experimental data.

## Examples

Consider for example a borosilicate glass tube with a length of 43 mm, an inner radius of $R_1 = 0.29$ mm and outer radius of $R_2 = 0.275$ mm, with the outer surface electrically grounded, like the one used in a theory-experiment comparison found in [69]. The values of the coefficients $b_{m,n}$ and $c_{m,n}$ for this particular capillary are published in [69]. For borosilicate at room temperature, one has typically $\kappa_b = 10^{-13}$ S/m and $\varepsilon_r = 4.6$, yielding $\tau_b^{-1} = 2.4$ mHz. The surface conductivity $\kappa_s$ is not a well known quantity as it depends strongly on the adsorbed ions at the surface. We assume here a value of $\kappa_s = 10^{-16}$ S as measured by [43], yielding $\tau_s^{-1} \simeq 40$ mHz. Let us consider a small charge patch of lengths $d = 4$ mm located several mm behind the entrance of the capillary, that has been injected by a tilted ion beam. We are interested in its charge relaxation rate of the accumulated charge once the ion beam stops, $\gamma_{mn} = 0$. As suggested by (2.47), the rate is given as a weighted mean value with the weights depending on the geometry of the capillary and the considered angular moment of the charge patch. Here, we limit our expansion (2.49) to the first two angular moments, namely $m = 0$ and $m = 1$. The dipole part ($m = 1$) of the accumulated charge distribution is the first non-zero angular moment and is mainly responsible for the deflection of the ion beam by the charge patch, so it is interesting to consider its relaxation rate. The
monopole moment $m = 0$ characterizes rather the charge relaxation of the total charge. Noting that the first $n \le 10$ are sufficient to describe the $d = 4$ mm charge patch, ($d \cdot k_{n=10} \approx \pi$), we limit our expansion to the modes $n \le 10$. As shown in Fig. 2.8, the coefficients $b_{mn}$ and $c_{mn}$ depend only weakly on $n$. We may thus get an excellent guess of the monopole and dipole relaxation considering only $n = 10$.

$$ \tau_{0,10}^{-1} \approx \frac{0.99}{\tau_b} + \frac{0.008}{\tau_s} \\ \approx 0.99 \times 2.4 + 0.008 \times 40 \approx 2.8 \text{ mHz} $$

$$ \tau_{1,10}^{-1} \approx \frac{0.826}{\tau_b} + \frac{0.176}{\tau_s} \\ \approx 0.826 \times 2.4 + 0.176 \times 40 \approx 7 \text{ mHz} $$

First, we note that the relaxation rate of the dipole is 2.5 times higher than the monopole one. Unsurprisingly, for the monopole part, the contribution of the surface channels are negligible with respect to the bulk channel. Indeed, only the surface channel along the symmetry axis contributes slightly, but most of the charges relax via the bulk channel. The dipole part on the other hand activates an additional surface channel, namely the one along the angular direction, which also becomes the dominant one. Measuring the relaxation of the total charge $\tau_{0,n}^{-1}$ in a capillary and the relaxation of the dipole part $\tau_{1,n}^{-1}$ of a charge patch yields thus complementary pieces of information that allows to determine unambiguously the inner surface and bulk conductivity of the insulating capillary!

## Dimensionless analysis

Instead of performing a dimensionless analysis of (2.35), we use instead (2.46) which is equivalent to (2.35) but has the advantage to define clearly the characteristic relaxation times $\tau_{mn}^{-1}$. Introducing the extraction potential $U_s$ of the ion source and the characteristic surface $S = 2\pi HR_1$, and capacity $C = 2\pi\epsilon_0\epsilon_r H/\log(R_2/R_1)$ of the capillary, we can define the characteristic charging time

$$ \tau_c = \frac{CU_s}{I_{\text{inj}}} \qquad (2.51) $$

and the following dimensionless quantities

$$ \tilde{t} = t/\tau_c \qquad (2.52) $$

$$ \tilde{\sigma}_{mn} = \frac{S_0}{CU_s}\sigma_{mn} \qquad (2.53) $$

$$ \tilde{\gamma}_{mn} = S\gamma_{mn} \qquad (2.54) $$
We can now re-write equation (2.46) in a dimensionless form,

$$ \frac{\partial \tilde{\sigma}_{mn}(t)}{\partial \tilde{t}} = - \frac{\tau_c}{\tau_{mn}} \tilde{\sigma}_{mn}(t) + \tilde{\gamma}_{mn}(t) \qquad (2.55) $$

The charge dynamics at the inner capillary surface is thus entirely controlled by the ratio

$$ \frac{\tau_c}{\tau_{mn}} = \underbrace{\frac{U_s}{I_{\text{inj}}}}_{\text{beam}} \times \underbrace{\frac{2\pi H}{\log\left(\frac{R_2}{R_1}\right)}}_{\text{length scale}} \times \underbrace{\left(b_{mn} \kappa_b + \frac{c_{mn}}{R_1} \kappa_s\right)}_{\text{conduct. & B.C}} \qquad (2.56) $$

The latter can be decomposed into a product of three terms. The first tells us that the charge dynamics scales with the fraction $U_s/I_{\text{inj}}$, that characterizes the injected beam. Thus doubling the injected intensity and the extraction potential simultaneously will lead to the same charge dynamics, with the same asymptotic dimensionless charge distribution, and thus ion transmission. The second term is merely a scale factor depending on the dimensions of the capillary. The third term regroups all the properties of the capillary, including the conductivities $\kappa_b$ and $\kappa_s$, the dielectric constant $\varepsilon_r$ and boundary conditions (B.C.) imposed to the electric field. Finally the ratio depends on the considered mode $(m, n)$. For our capillary in the above example, we may calculate the monopole $m = 0$ and dipole term $m = 1$ for $n = N = 10$

$$ m = 0: \quad \frac{\tau_c}{\tau_{0N}} = \frac{U_s}{I_{\text{inj}}} \times 0.73[\text{m}] \times \left( 0.99\kappa_b + \frac{52}{[\text{m}]}\kappa_s \right) \qquad (2.57) $$

$$ m = 1: \quad \frac{\tau_c}{\tau_{1N}} = \frac{U_s}{I_{\text{inj}}} \times 0.73[\text{m}] \times \left( 0.826 \kappa_b + \frac{926}{[\text{m}]} \kappa_s \right) \qquad (2.58) $$

where $\kappa_s$ is expressed in Siemens (S), $\kappa_b$ in (S/m) and the length in (m). For this particular capillary, we conclude that if $\kappa_s[S] \ll 10^{-3}\kappa_b[S/m]$, the surface conductivity does not affect the charge dynamics of both the dipole and the monopole angular distribution of the charge patch. If $\kappa_s[S] \sim 10^{-3}\kappa_b[S/m]$, then the surface conductivity does affect the dipole but still not the monopole distribution. For borosilicate glass (Pyrex), we are in the latter case, meaning that the surface conductivity affects the charge relaxation of the dipole part of the patch and thus the guiding properties of the capillary and may not be ignored in the modelling. The surface channel however affects only marginally the monopole part.
## 2.5 Simulations

### 2.5.1 Numerical model: InCa4D

The numerical model InCa4D<fnref n="5" /> is based on the model presented in section 2.2.2. The surface charge dynamics at the inner and outer interfaces $S_1$ and $S_2$ of the capillary are evaluated using the surface continuity equations,

$$ \frac{\partial \sigma_1}{\partial t} = -\kappa_b E_n^+ - \vec{\nabla}_s \cdot \left( (\kappa_s + \mu \sigma_1) \vec{E}_\tau \right) + \gamma^h, \quad \text{on } S_1 \qquad (2.59) $$

$$ \frac{\partial \sigma_2}{\partial t} = \kappa_b E_n^- - \vec{\nabla}_s \cdot \left( (\kappa_s + \mu \sigma_2) \vec{E}_\tau \right) + \gamma_{\text{str}}^e + \gamma_{\text{int}}^e, \quad \text{on } S_2 \quad . \qquad (2.60) $$

The sources terms $\gamma^h$ stands for the injected holes at the inner surface due to the ion beam and $\gamma_{\text{str}}^e$ for the injected electrons at the outer surface. In our simulations, $\gamma_{\text{str}}^e$ was set to zero, as the capillary holder in our setup screens the capillary from stray electrons. The entrance of the capillary is usually covered by a conducting layer and grounded. In our simulations, the conducting layer covered the first 11 mm of the outer surface $S_2$<fnref n="6" />, allowing electrons to be injected through the glass-conductor interface and neutralize the charge accumulated within the covered area. This is the only charge depletion channel we considered. The electron injection through the glass-conductor interface is described by the term $\gamma_{\text{int}}^e = -\kappa_b E_n(R_2^+) H(z_g - z)$, where $H$ is the Heaviside step function and $z_g$ the length of the outer conducting layer. The current density along the interface is proportional to the enhanced surface conductivity $\kappa_s + \mu\sigma$, while the current density through the bulk is proportional to the bulk conductivity $\kappa_b$ of the capillary. The diffusion term was always ignored. The model has thus three free parameter, $\kappa_b$, $\kappa_s$ and $\mu$ which need to be provided. Usually the charge mobility $\mu$ of the injected charged defects at the surface was set equal to the charge mobility of thermally activated "labile" Na⁺ ions in borosilicate, $\mu = 4 \times 10^{-14} \text{ m}^2/\text{V/s}$ [5], if not said otherwise. For such low mobilities, contribution of excess charge carriers to the surface conductivity $\kappa_s$ is negligible<fnref n="7" />.

<footnote n="5">⁵Label "InCa4D" stands for "Ions through Capillaries" and the 4D stands for the 3 dimensions xyz and time.</footnote>

<footnote n="6">⁶Our capillary holder has an 11 mm long cylindrical hole matching the outer diameter of about 1.5 mm of the capillary. The capillary was inserted into that hole in order to keep it in place.</footnote>

<footnote n="7">⁷The charge mobility of the injected charges at the surface is not a well known quantity. We assume that it is similar to mobility of interstitial pair defects (two alkali ions share one non-bridging oxygen anion NBO). The simulated insulators in this work are usually borosilicate (Pyrex) capillaries and the voltage of the source typically $U_s < 3$ kV. For borosilicate, a mobility of $\mu = 4 \times 10^{-14} \text{ m}^2/\text{V/s}$ at room temperature was found by [5, 6]. Assuming a typical value for $\sigma_0 = \frac{\epsilon_0 \epsilon_r U_s}{R_2 - R_1} < 2.5 \times 10^{-4} \text{ C/m}^2$, the additional surface conductivity $\mu\sigma_0 < 10^{-17} \text{ S}$, is sufficiently</footnote>
Additionally to the two glass-vacuum interfaces of the capillary, the code accounts for a third interface $S_3$, namely the interface of a metal cylinder surrounding the capillary. The electric potential of the latter is floating or kept at a fix potential and stands for the inner cylindrical electrode of the capacitor found in the experimental setup, see Fig. 2.19. This way, the boundary conditions of the electric potential in the model are the same than those in the experimental setup. The dielectric constant $\epsilon_r = 4.6$ of the borosilicate glass is also taken into account by computing explicitly the polarisation charges $\tilde{\sigma}_1$ and $\tilde{\sigma}_2$ at the inner and outer glass-vacuum interfaces. The induced surface charges $\tilde{\sigma}_3$ at the metal cylinder interface are calculated using (2.30) which satisfies the appropriate boundary conditions at $S_3$. Finally, the divergence-free electric field is deduced from the electric potential $V$ (2.24). Because the charges are computed only at the three interfaces, our approach is CPU cost efficient while keeping a realistic description of the charge transport in capillaries. In addition, as the boundary conditions at the 3 interfaces match the experimental ones, the numerical code is expected to reproduce with high accuracy the experimental electric field in the capillary and make reliable quantitative predictions. For example, long simulations, indented to study the radial focusing effect (typical $10^7$ trajectories), as presented in this manuscript, take about 36 hours on a modern (2018) CPU core.

## 2.5.2 Features of the Numerical model InCa4D

The numerical code simulates the 3D classical trajectories though an insulating capillary, where the guiding electric field is computed from the time-depending surface charges at the interfaces. The code has been developed over several years while adding small incremental improvements. By 2019, the code has the following remarkable features:

1. **Transmission of neutrals:** Slow ions that hit the inner insulating surface are generally neutralized by the ejected electrons. A fraction of the scattered neutrals are transmitted through the capillary and can be detected for example by a micro channel plate (MCP) coupled to Position Sensitive Detector (PSD). The distribution of neutrals on the PSD is particularly useful for aligning the ion beam with the capillary axis, as shown in the upper panel of figure 2.9a. Indeed, when both are aligned, the distribution of neutrals yields a perfectly uniform circle on the PSD. We implemented a function in the code that follows the neutralized ions allowing to compute the energy loss of the neutrals as well as their distribution on the PSD. The function

<footnote>small that the enhanced surface conductivity $\kappa_s + \mu\sigma$ is well approximated by $\kappa_s \sim 10^{-16}$ for the capillary we use</footnote>
<figure><img src="image_54.png" /><figcaption>(a) Left: Ring of Ar neutrals on the PSD, 270 mm behind the capillary outlet, before fine-tuning the alignment of the capillary axis with the beam axis. Spot of neutrals is off-centered and the ring is not uniformly intense. The transmitted ions Ar⁺ were deflected upwards. Right: Uniformly intense ring of neutral with centered spot after rotating the capillary by the 0.12 degree (azimuthal angle). The vertical black line shows the alignment of the ion beam spot with the neutral spot before and after fine-tuning. Beam intensity was below 100 fA and the capillary was heated to 70° C to avoid charge accumulation and guiding.</figcaption></figure>

<figure><img src="image_55.png" /><figcaption>(b) Black points give the simulated distribution of transmitted neutral Ar atoms on the PSD. Red spot gives the transmitted Ar ions. Bulk conductivity was put to $10^{-11}$ S/m to avoid charge accumulation at the interface. In the right panel, the tilt angle is 0° and the ion spot lies on the origin of the PSD. The neutral distribution is a circle, with a notable absence of neutrals for a radius of less than 2 mm, due to the tapered shape of the capillaryª. In the left panel, the tilt angle is -0.15°. The distribution of neutral is no longer symmetric, with a higher intensity opposite to the tilted direction of the capillary. This property allows to align the capillary with the beam axis within an uncertainty of about 0.05°.</figcaption></figure>

<footnote>ªR/H × 270 mm ≈ 2 mm, with R = 0.4 mm and H = 50 mm being the inner radius and length of the conical part of the capillary</footnote>
is based on the following algorithm: At the impact point, a scattering ve-
locity vector is chosen randomly from a data base that was generated in
advance using SRIM<fnref n="8" />. The latter software was diverted to recorded in a file
the backscattering velocity vector of a large number of impacting ions. We
recorded the scattering velocities, for example in the case of 3 keV Ar atoms<fnref n="9" />
impacting a fused quartz surface at grazing angle, but also for a 1 Mev H (or
proton) colliding with a Teflon and fused quartz surface. We recorded files
for grazing angles varying between 0.2° and 2°, allowing for a dependency
of the backscattered velocity vectors on the impact angles. With this addi-
tional feature, we could compute the transmitted neutrals as a function of
the tilt angle of the capillary. We give in the lower part of figure 2.9b the
simulated distribution of transmitted neutral atoms through a conical capil-
lary, projected on the PSD. For small tilt angles around zero, we found that
usually 1 neutral Ar is transmitted for 10 ions. The nice agreement between
observed and simulated distributions on the PSD, gives confidence in the
chosen algorithm. It also shows that aligning the capillary using neutrals is
relevant [64].

2. **Non-linear electrical conductivity:** In general, the de-trapping rate and thus the charge mobility $\mu \equiv \mu(E)$ depends on the local electric field $E$. As a result, the bulk conductivity $\kappa_b(E) = \rho\mu(E)$ depends on the electric field too. Same for the surface conductivity $\kappa_s \equiv \kappa_s(E)$. We implemented two models that take both into account the enhanced de-trapping rate of the charge carriers by the electric field. The first (2.8) is the thermally activated hopping model (TAHM) that is preferred for the ionic conductors and the second is the Poole-Frenkel model [1] which is preferred for electronic conduction in polymers, although it was also used for silicates [28]. Field depended non-linear conductivities have been shown crucial for explaining the guiding power of nano-capillaries [49]. Here, we limited our study to glass macrocapillaries in which case the electric field stays usually well below the critical electric field of 10 kV/mm. The latter is the typical value for which the transition from ohmic to non-ohmic conduction happens in materials like fused silica, HDPE polymer and low alkali glasses [66].

3. **Secondary electrons generated inside the capillary:** We recently added the tracking of the secondary electrons (SE) that are produced by the impaction ions at the insulating surface. Until then, electrons that were not

<footnote n="8">⁸SRIM - The Stopping and Range of Ions in Matter. SRIM includes quick calculations which produce tables of stopping powers, range and straggling distributions for any ion at any energy in any elemental target. We used the TRIM module to generate the backscattering velocities with the "Monolayer Collision Step/ Surface sputtering" option.</footnote>

<footnote n="9">⁹The TRIM results may be less precise for ions below 10keV</footnote>
captured by the ion, were assumed to fall back to the impact point. Now we propagate the emitted secondary electrons by the electric field in the capillary until they hit again the inner surface of the capillary or escape the capillary through the inlet and outlet. The emitted electrons have a cosine angular distribution and the energy follows a Poisson-like distribution peaked around 2 eV. The number of emitted electrons $N_{SE}$ per impact is an adjustable parameter, taking integer values typically $N_{SE} \le 3$. The CPU cost only increases marginally because usually not more than 3 electrons are propagated for a short path (typically the inner diameter of the capillary) per ion impact. In figure 2.10 we show the trajectories of SE (in blue) in a charged glass capillary. It was found that, initially, the ejected electrons are re-absorbed at the opposite side. But with increasing accumulated charge in patch, the electric field makes the ejected SE u-turn, so that the SE are re-absorbed some mm away from the original emission point. The SE channel thus contribute significantly to redistribute the accumulated charge at the interface. Mainly, the accumulated charge is smeared out by SE, resulting in a modified generated electric field and ion trajectories.

<figure><img src="image_56.png" /><figcaption>Figure 2.10: Secondary electrons (SE) emitted by impacting 2.5 keV Ar⁺ ions at the insulator interface. The capillary is tilted by 1.2° with respect to the beam axis. The electrons are propagated via the electric field until they hit again the interface, where they are re-absorbed. Note that a large fraction of the SE fall back some mm away from the original emission point, contributing to smear out the accumulated charge at the interface.</figcaption></figure>

4. **S₃ interface electrode:** The potential of the cylindrical interface S₃ surrounding the capillary can be kept floating or fixed to an arbitrary potential V₃ (see section 2.2.4). When kept for example grounded V₃ = 0, the electrode
can play the role of the cylindrically symmetric capillary holder. When kept
floating but neutral (no accumulated charge) the electrode can measure by
influence the charge stored in the capillary. The latter feature will be dis-
cussed in the following section. In the case where the potential is fixed to
a non-zero value, the S<sub>3</sub> electrode can be used to simulated the behaviour
of an Einzel lens, as was discussed in our collaboration with Ikeda Tokido.
Here, the entrance and the exit of a conical insulating capillary are painted
by a conducting paint and grounded (black layer in Fig. 2.11). The S<sub>3</sub>
conical interface (red layer) was connected to a signal generator and a time-
tependent potential V<sub>bc</sub> was imposed. An Ar<sup>+</sup> ion beam, extracted by a
potential V<sub>s</sub> = 2.5 keV, was injected into the capillary. By increasing V<sub>bc</sub>
from V<sub>bc</sub>/V<sub>s</sub> = 0 to 0.6, the focusing point of the ion beam was brought from
infinity to the outlet of the capillary. At that point, the transmitted frac-
tion was maximal. For an even larger ratio V<sub>bc</sub>/V<sub>s</sub> = 0.8, the focusing point
was located inside the capillary, reducing the transmitted fraction. We con-
cluded that with this electrode configuration, the capillary setup behaves like
an Einzel lens and is able to focus the injected beam. However it is crucial
that the electrode setup has axisymmetry in order to avoid the appearance
of dipole moments in the electric field, that would otherwise deflect the beam
into the capillary wall. Note that the focusing of the injected beam by the
electrode S<sub>3</sub> is to be distinguished from the self-organized radial focusing
described in section 2.6.2.
<figure><img src="image_57.png" /><figcaption>Figure 2.11: The $S_3$ conducting interface (full red line) is connected to an external time dependent potential $V_{bc}$. The outer capillary surface is grounded near the entrance and near the outlet (black full lines). The inner surface (thin gray line) is the insulating glass-vacuum interface. The electrodes generate a field inside the capillary that focuses the trajectories like an Einzel lens</figcaption></figure>
## 2.6 Most significant simulated results

We will present in this section merely the three most significant results obtained with our InCa4D code. The first one explains why the transmitted intensity increases when the capillary glass tube is tilted with respect to the beam axis. The second one shows that it is possible to focus an injected ion beam through a tapered glass capillary, similar to an Einzel lens. The third shows the influence of secondary electrons (SE) on the guiding power in cylindrical glass capillaries.

### 2.6.1 Transverse self-organized focusing

#### The mechanism

One has to distinguish between two mechanisms responsible for the enhanced transmitted current density. They differ by the symmetry of the accumulated charged distribution in the capillary wall. The first is due to a *non* axisymmetric charge deposition. For example, an injected ion beam, tilted with respect to the symmetry axis of the capillary, creates one or more local charge patches that guide the ions. The patches generate an electric field, which is stronger the closer it is to the charge patches. As a result, trajectories close to the charge patches are more deviated than trajectories farther away, resulting into a compression of the beam in one dimension. This compression mechanism is illustrated in Fig. 2.12. The injected current is 30 pA. After only 300 pC have been stored at the inner capillary surface, the transmitted beam is compressed by a factor $j_{out}/j_{in} \simeq 4$ by the x-component of the electric field, as highlighted by the two red transmitted trajectories. The beam compression takes place rather quickly after beam injection (see inset of Fig. 2.12), as *transverse* fields become efficient to deflect the beam after only a relatively small amount of charge has been accumulated in those patches. Indeed, in our simulation (Fig. 2.12), the beam compression takes place for a potential inside the capillary as low as 14 V, which is well below the extraction potential of the ion source of 3 kV. In the following, we refer to this mechanism as the transverse field compression. Compression of an ion beam, injected into a straight capillary tilted with respect to the beam axis, was already noted by N. Stolterfoht in 2013 [48, 49], but not further analysed. A detailed discussion of the transverse focusing/compression is given in [64]. A nice example of transverse focusing is illustrated in Fig. 2.16 where a tilted beam is compressed in one direction due to sequential reflection by multiple charge patches in a straight capillary.

#### Explaining experimental results

The transverse self-organized focusing effect explains the enhancement of the transmitted intensity for non-zero tilt angles in the experimental data shown by Gruber
<figure><img src="image_58.png" /><figcaption>Figure 2.12: Illustration of the transverse field compression of an injected ion beam by non axisymmetric deposited charge patches in a tapered capillary of large aspect ratio: Full violet curve stands for the inner surface of an insulating capillary, black curve for its outer metallized and grounded surface. The green areas (darkk gray) give the accumulated charge at opposite sides at the inner surface in arbitrary units. Gray curves represent simulated trajectories of an Ar⁺ beam at 3 keV and tilted by 0.8° with respect to the capillary axis. The inset shows the time-evolution of the transmitted current density *j*out normalized with respect to the injected current density *j*in.</figcaption></figure>

et al. for tapered (conical) glass capillaries [51]. N. Stolterfoht et al. [53] succeeded to simulate the trend observed in [51], but gave no further convincing explanation as to why there was an enhancement. Note that an enhancement of the transmitted fraction for non-zero tilt angles can only be observed for tapered capillaries, which is coroborated by the data in [43] for a strait capillary.

During his visit in CIMAP, H. Zhang of the School of Nuclear Science and Technology, Lanzhou, China, repeated the measurements of the transmission of an ion beam through tapered capillaries as a function of the tilt angle. Injecting an Ar⁺ at 3 keV into a tapered borosilicate glass capillary at room temperature, he found that at 0.75° tilt angle, the transmission was maximal and about twice
<figure><img src="image_59.png" /><figcaption>Figure 2.13: Transmission of a 3 keV Ar⁺ ion beam of Iᵢₗⱼ = 0.5 pA through a conical glass capillary with outer grounded surface as a function of tilt angle. The black points are experimental data. The blue stars, the green crosses and red boxes are simulations with respectively κ₁ = 10⁻¹⁴ S/cm, κ₂ = 3 × 10⁻¹⁴ S/cm and κ₃ = 5 × 10⁻¹⁵ S/cm</figcaption></figure>

as high as when the beam was aligned with the capillary axis (zero tilt angle). The observed results are reported in Fig. 2.13 (black points). Our simulations confirmed the result that a maximal transmitted fraction is obtained for non-zero tilt angle. They also confirmed the trend that the angle, for which the maximum transmission is observed, is shifted towards lower angles with increasing bulk conductivity, as was observed in [51]. On the other hand, the highest value of the transmission increases with decreasing conductivity, because more charge patches are formed in the capillary, multiplying the deflections and thus the transverse focusing.

2.6.2 Radial self organized Focusing

The other focusing mechanism is due to an axisymmetric charge accumulation,
generating an axisymmetric potential in the capillary, which focuses the injected
beam through the capillary outlet. The beam is compressed by the radial com-
ponent of the electric field, similar to an electrostatic Einzel lens. We found in
a preliminary numerical investigation, presented in [67], that the potential has to
approach 70% of the extraction potential of the ion source in order to focus the
injected beam through the outlet of an insulating conical shaped capillary. The radial focusing and the transverse compression differ thus not only by the symmetry of the charge accumulation in the capillary wall, but also on the value of the self organized potential which is needed to became efficient. Summarizing: for radial focusing to be observable, two conditions must be fulfilled. The potential inside the capillary must (i) approach at least 70% of the extraction potential of the injected ions and (ii) have axial symmetry. In the following, we refer to this mechanism as the radial self-organized focusing.

The self-organized focusing power of a tapered glass capillaries is shown in Fig. 2.14. The injected Ar⁺ ions have an intensity of $I_{in} = 210$ fA and the beam is aligned with the capillary’s symmetry axis (zero tilt angle). The initial phase-space distribution of the injected ions are chosen such as to match the experimental divergence and emittance of the beam used in section 2.8.2 and measured in Fig. 2.22, namely a divergence of 0.1° half opening angle and a rms emittance of about 0.1 mm.mrad. The simulations were performed using a bulk conductivity of $1.8 \times 10^{-13}$ S/m and a surface conductivity of $2.6 \times 10^{-17}$ S. The snapshots give the trajectories of the injected ions for different amounts of charge stored in the capillary. Below each snapshot we show the corresponding potential at the inner surface. For the initially discharged capillary, only 5.3% of the injected ions are transmitted, corresponding to the geometrical transmission, corrected by the divergence of the beam. Most ions hit the inner surface and charge the capillary inner wall. With increasing charge stored in the capillary, $Q = 900$ pC, the trajectories are slightly focused by the electric potential in the capillary and the transmission doubles to 10%. The potential $U$ in the capillary attains, at his highest value, already 1100 V, which corresponds roughly to half of the extraction potential $U_s = 2300$ V of the ion source. Once about $Q = 1600$ pC are stored in the capillary, the electric potential inside the capillary is strong enough to focus the beam at the outlet. The transmission fraction $I_{out}/I_{in}$ reaches a maximum of 60% (which corresponds to a focusing factor of 11), resulting in a transmitted maximum current $I_{out}^{max}$ of 124 fA. The latter is obtained for a potential $U$ inside the capillary of about $0.75 \times U_s = 1725$ V at its apex. Without surprise, the potential has its maximum where most of the charges are deposited, in the region where the inner diameter of the capillary equals the diameter of the inlet aperture. From the relation $\nabla \cdot \vec{E} = 0$, it results that the focusing radial field $E_r$ is strong there, where the negative curvature (second derivative) of the potential along the z-axis is large,

$$E_r(R(z), z) = \frac{1}{R(z)} \int_{0}^{R(z)} r \frac{\partial^2 V}{\partial z^2} dr \quad (2.61)$$

In the middle panel of Fig. 2.14, the maximum of the focusing radial field is thus located around $z = 30$ mm. The accumulated charges in the capillary is
depleted at a rate dominated by the bulk conductivity, $\tau^{-1} \sim \frac{\kappa_b}{\epsilon_0\epsilon_r}$ yielding a charge relaxation rate $\tau^{-1}$ of 4.4 mHz (2.47) for the monopole mode, which is the only non-zero mode. Now, the ratio, injected current over relaxation rate, $I_{in}/\tau^{-1}$,

<figure><img src="image_60.png" /><figcaption>Figure 2.14: Each snapshot, identified by the amount of charge Q stored in the capillary wall, is divided into two panels. The upper panel shows the ion trajectories (gray lines) through the capillary. The radius $R(z)$ of the inner surface of the capillary is represented by a violet (dark grey) full line. The lower panel shows the Coulomb potential $U(z)$ at the inner capillary surface. The potential $U(z)$ is normalized with respect to the extraction potential $U_s = 2300$ V of the ions source. The green curve (labelled $I = 75$ fA), obtained for an injected current of 75 fA, has been added for comparison. For all the other curves, the injected current was 210 fA.</figcaption></figure>
directly controls the steepness of the self-organized potential along the z-axis and thus its curvatures. The higher the value of $I_{in}\tau$, the steeper the variation of the potential along the z-axis and consequently the stronger its focusing power. This trend is illustrated by the flatter but not necessarily lower green curve, which is obtained for an injected current of only 75 fA and yields a transmission factor of barely 24%. The trend will be discussed more in details in the text section. Note that the capillary behaves similar to an Einzel lens, where the focal point is moved from infinity towards the outlet of the lens with increasing potential of the central electrode and with decreasing distance between the central and outer grounded electrodes [16].

The charge depletion rate $\tau_{leak}^{-1}$ is the rate at which the capillary evacuates the accumulated charge and is not to be confused with the charge relaxation rate $\tau^{-1}$ defined in the paragraph above. The charge depletion rate $\tau_{leak}^{-1}$ was simulated in section 2.8.1. With our particular setup and capillary, we found a depletion rate of $\tau_{leak}^{-1} = 0.01$ mHz for the given conductivities. At maximum transmission ($Q = 1600$ pC), the leakage current is estimated to be about $Q/\tau_{leak} \simeq 16$ fA. The deposited charge per time unit, $I_{in} - I_{out} = 85$ fA, still dominates the leakage current and the capillary continues to charge. As a result, the focusing distance continues to shorten and the focusing point enters the capillary tip. Part of the trajectories emerging from the focus point inside the capillary are now intercepted by the capillary walls and the potential rises quickly near the outlet. If the potential inside the capillary exceeds the potential of the ion source, the transmission blocks. In the lower panel of Fig. 2.14 ($Q = 2000$ pC), the potential at the last 5 mm exceeds the source potential $U_s$ of 2300 V and the ions u-turn at this point. This Coulomb blocking is inevitable, except of course if the deposited current is balanced by the leakage current before the potential in the capillary exceeds the source potential. Only then, can a stable transmission be expected. Nevertheless, this theoretical study clearly shows that insulating tapered capillaries can be used to focus a low energetic ion beam.

Short tapered capillaries are another class of capillaries for which focusing was observed experimentally, as was reported by Chen et al. [45]. They observed that the transmitted fraction of a 90 keV $O^{6+}$ ion beam was enhanced by a factor 4 with respect to the geometrical transmission in a tapered glass capillary that was only 14 mm long (see profile of the capillary in Fig. 2.15). Our simulation confirm that even such capillaries, having an extremely short tapered part, allow radial focusing as is shown by the blue trajectories in Fig. 2.15. Our simulations predicted that the transmitted fraction increases from the initially 10% to 35%, which is close to the 40% observed in [45]. This is a promising result as it shows that focusing can also been observed in short tapered capillaries which are not near conical, opening self-organized radial focusing to a new class of macro-capillaries.
<figure><img src="image_61.png" /><figcaption>Figure 2.15: Simulated trajectories of 90 keV O$^{6+}$ ions showing radial focusing through the short glass capillary similar to that used in [45]. The red and the black curve give the profile of the inner and outer interface of the capillary. In our simulations, the outer surface was grounded.</figcaption></figure>

Recently, Maurya et al. [65] measured self-focusing in high current density ion beam transport through 7.5 mm short insulating glass capillaries having an outlet diameter ranging from 200 to 20 µm. For the 20 µ they found a focusing factor of more than 1000. Those studies indicate that focusing of beam ions by short tapered capillaries is seen as a promising tool for the producing of micro- and nano-beams.

### 2.6.3 Influence of Secondary Electrons on high intensity beam transmission

In an experimental study presented in [69], it was shown that a 1 keV Ar$^+$ beam, tilted by 5° deg, was transmitted through a cylindrical glass capillary for injected intensities as high as 50 pA. The borosilicate glass ($\epsilon_r = 4.6$) capillary used in the experiment was $H = 43$ mm long, with an inner radius of $R_1 = 0.29$ mm and outer radius of $R_2 = 2.75$ mm. The outer surface of the capillary was grounded. We performed simulations for a capillary having the exact above dimensions and properties. In the simulation, the initial positions and velocities of the Ar$^+$ ions are sampled such as to produce a spatially uniform beam with a divergence of 0.15° half opening angle and a root mean square (rms) emittance of 0.3 mm.mrad, so as to reproduce the estimated experimental beam parameters. At that time,
<figure><img src="image_62.png" /><figcaption>Figure 2.16: Ion trajectories through straight capillary tilted by 5° with respect to the ion beam axis after 480 pC have been injected by a 12 pA, 1 keV Ar⁺ ion beam. Top: projection on the xOz plane; bottom: projection on the yOz plane. Note the compression of the current density in the x direction. After 40 s of beam time, the injected beam is deflected multiple times by at least 7 charge patches (illustrated by the red curve) but without escaping the capillary. The SE channel was inactivated, N<sub>SE</sub> = 0.</figcaption></figure>

our numerical code did not track the secondary electron that are emitted when an Ar⁺ projectile hits the inner insulating surface of the capillary. We naively thought that its influence would be negligible anyway. So, this former version of *InCa4D* had two free parameters<sup>10</sup>, namely the bulk and surface conductivities. We took for the bulk conductivity of borosilicate at room temperature the typical value of κ<sub>b</sub> = 1.5 × 10<sup>-13</sup> S/m. The surface conductivity in borosilicate is not a well-known quantity and depends strongly on adsorbed impurities. At room temperature and in vacuum, it is typically less than κ<sub>s</sub> < 10<sup>-15</sup> S. Experimentally, it was found that ion beams having an intensity as low as 0.28 pA, had a transmitted fraction of about 1%. We found from our simulations, by varying the surface conductivity, that κ<sub>s</sub> must be smaller than 5 × 10<sup>-16</sup> S if a 1% transmission should be observed for I<sub>inj</sub> = 0.28 pA. Indeed, if κ<sub>s</sub> > 5 × 10<sup>-16</sup>, no transmission is observed in our simulations. Simulations also showed that taking κ<sub>s</sub> = 3 × 10<sup>-16</sup> S, yields a transmission fraction of about 5%, which is too high compared to the observed one. Thus we deduce from our simulation that 3 × 10<sup>-16</sup> < κ<sub>s</sub> < 5 × 10<sup>-16</sup> S. Consequently, we used κ<sub>s</sub> = 5 × 10<sup>-16</sup> S, which is in approximate agreement with the measured surface conductivities of Pyrex borosilicate glass [43]. We simulated the time-evolution of the transmitted beam fraction for I<sub>inj</sub> ranging from 0.28 to 50 pA. With the secondary electron production, (N<sub>SE</sub> = 0) ignored, no stable

<footnote n="10"><sup>10</sup>The charge mobility of the injected charge carriers are set to μ = 4 × 10<sup>-14</sup> m<sup>2</sup>/V/s</footnote>
transmission could be simulated for beam intensities $I_{\text{inj}} > 8$ pA. For example, we found that up to 8 patches are formed inside the capillary by a 12-pA beam (see Fig. 2.16), which results in a fluctuating transmission (see green curve labeled $N_{\text{SE}} = 0$ in figure 2.17). The latter finally stops because, with increasing Coulomb potential in the capillary, deflections are so strong that the beam is stuck in the capillary. The simulated results however conflicts with the experimental data shown in Fig. 7 of [69], where a stable transmitted fraction, for $I_{\text{inj}} = 12$ pA, is observed for at least 500 s

<figure><img src="image_63.png" /><figcaption>Figure 2.17: Simulated transmitted fraction of a 12 pA, 1 keV $Ar^+$ beam through the 5° tilted capillary tube as a function of time. The green curve stands for the simulation without secondary electrons ($N_{\text{SE}} = 0$). The red and black curves stand for the simulations with respectively 1 and 2 secondary electrons emitted per ion impact.</figcaption></figure>

To address this conflict, we updated our numerical model to track secondary electrons (SE) generated by ion impacts at the inner surface, see section 3 for more details. We expected the SE emission to smear out the charge patches and possibly avoid blocking of the transmission at higher intensities. Most SE were found to fall back to the surface some hundreds of µm away from the point they were emitted, see Fig. 2.18. This helped indeed to redistribute the charge of the injected patch in a way that weakens the deflecting force of the patch, giving rise to a lower number of patches (~4) and eventually to a stable transmission as seen in Fig. 2.17. The case $N_{\text{SE}} = 2$, gave the best agreement with our experimental data. Based on the measurments of SE yields for low energetic $He^+$ and $Ar^+$ projectiles
on SiO₂ films by [18], we estimate that $N_{SE} = 2$ is a reasonable value for the SE yield in the case of grazing 1 keV Ar⁺ ions on borosilicate glass capillary. And indeed, with $N_{SE} = 2$, we were able to simulate a stable transmission for injected current ranging from 0.26 to 50 pA.

<figure><img src="image_64.png" /><figcaption>Figure 2.18: Snapshots of Ar⁺ trajectories (black lines), projected on the x-axis, through the glass capillary at 4, 11 and 74 seconds after the beam is injected. The injected intensity was 12 pA and beam was tilted by 5° with respect to the capillary axis. The horizontal black lines at $x = \pm 0.29$ mm show the inner capillary surfaces. Above and below the surface lines, the red curves show the intensity (arbitrary units) of the positive charge patches at the inner surfaces. Note that for $t = 74$ seconds, the amount of charge on the lower surface at the entrance is due to surface charge relaxation along the angular direction. The blue dashed lines are the trajectories of secondary electrons emitted from the impact point. Here we used $N_{SE} = 2$.</figcaption></figure>

Figure 2.18 shows snapshots of the charge distribution and trajectories at various times for a 12 pA injected beam. The simulation shows that a few seconds after beam injection, (upper panel of Fig. 2.18) three charge patches have formed, two on the upper surface and one on the lower surface. The electric fields from these patches are sufficient to deflect the incoming beam and already a transmitted
fraction of 30% through the entire capillary is observed. There is also an indication
that an additional charge patch will soon be formed on the lower surface near the
exit of the capillary. The middle panel shows that at a later time (11 seconds) the
"guiding electric fields" have caused the beam to interact closer the capillary en-
trance, thus producing a new charge patch further downstream. Now, about 60%
of the injected beam is transmitted. After additional time, (bottom panel) except
for the charge patch closer to the capillary entrance, the previously formed patches
are now weaker while a fairly uniform charge layer has formed along the last quar-
ter of the capillary. This leads to a significant portion (80%) of the injected beam
being transmitted. The simulations provide new valuable insight into the guiding
process, since they also model and track the redistribution of the charge deposited
on the capillary surface. This redistribution is seen by the intensities opposite the
charge patches formed where the beam impacts the surface. For example, in the
upper panel of figure 2.18, the beam interacts with the upper surface at horizontal
positions of approximately 2 and 30 mm and with the lower surface at 15 mm.
However, in the last panel of Fig. 2.18, our simulations show that some of the
charge patch produced on the upper or lower surface during the first few seconds
has moved along the circumference with the result being less intense patches of
charge on the opposite surface at the same horizontal distances. This redistribu-
tion of charge is most obvious near the capillary entrance. Thus, our simulations
imply that a portion of the deposited (or created) charge rather quickly moves
away from its original location with much of it moving to the opposite side. This
weakens the electric field precisely in the regions which contribute to the guiding
process. The combined theoretical and experimental study put into evidence that
SE must be taken into account for the description of the guiding power of insulat-
ing capillaries. Without the SE channel, simulations cannot reproduce accurately
the experimental results.

## 2.7 Experimental Setup

We designed a new capillary holder with the aim to put into evidence the ra-
dial focusing of the injected beam by a conical capillary and to perform precise
measurements of the charge relaxation in capillaries. From the simulations (Fig
2.14), it became clear that the total charge in the capillary must approach 1600
pC in order to expect focusing. It is thus crucial that the charged capillary is
well screened from stray electrons which would otherwise neutralize the capillary,
as was demonstrated in Giglio et al. [62]. Simulations also suggested that the
emittance of the beam must be low enough so that a major part of the injected
beam can be focused to a diameter lower or equal to the diameter of the capillary’s
outlet. Also, in order to evaluate precisely the transmitted fraction *I*<sub>out</sub>/I<sub>inj</sub>, we
need to know accurately the injected intensity $I_{\text{inj}}$. Finally, the electric field in the capillary must have axial symmetry for focusing to be observable. The capillary holder should thus have the following features:

* Measure the charge stored in the capillary as a function of time.

* Measure the intensity of the injected beam.

* Measure the emittance of the injected beam.

* Screen the capillary from stray electrons.

* Allow heating the capillary up to 70°C on demand for faster discharge.

* Ensure axial symmetry around the capillary allowing for radial focusing.

* Allow alignment of the beam axis with the capillary symmetry axis.

<figure><img src="image_65.png" /><figcaption>Figure 2.19: Left: cylindrical box with "repeller". Top: front of the box with 3 inlets for the ion beam. Bottom: Inside the box showing the Faraday-Cup (FC), Capacitor surrounding the capillary (not visible) and deflection plate.</figcaption></figure>

In order to monitor the time-evolution of the total electric charge stored in the capillary, the capillary is surrounded by a cylindrical capacitor (Fig. 2.19 and Fig. 2.20). The latter imposes axisymmetric boundary conditions to the electric potential, as required. The inner cylinder of the capacitor is electrically connected to the upper plate facing a grounded plate. The outer cylinder of the capacitor
is grounded. In this configuration, the capacity of the inner cylinder is $C = 13 \pm 0.2$ pF. Due to total influence, the potential $V$ of the floating inner cylinder of the capacitor and upper plate rises with the charge stored in the capillary, $V = Q/C$. An electric field builds up between the two plates with increasing potential $V$, deflecting the transmitted beam that passes through the plates. For low angles, the deflection $\Delta Y$, measured on a position sensitive detector (PSD) located 270 mm downstream, is thus proportional to the charge stored in the capillary. The proportionality factor is deduced from calibration measurements and is equal to $88 \pm 3$ pC/mm for this setup. The whole acts as a perfect Coulomb-meter. Assuming an initially neutral capillary, the charge stored in the capillary is eventually given by $Q[\text{pC}] = 88 \times \Delta Y[\text{mm}]$. Monitoring the accumulated charge in the capillary allows to indicate if and when an equilibrium has been reached for which the leakage current balances the deposited current such that the charge stored in the capillary is stationary. It also indicates if and when the charge stored in the capillary approaches the critical charge for which the capillary focuses efficiently the beam through the outlet. Last but not least, it tells if a blocking of the transmission is due to Coulomb-blocking or due to a charge patch that deviates the beam into the capillary walls.

<figure><img src="image_66.png" /><figcaption>Figure 2.20: A tapered glass capillary (green surface) is surrounded by a cylindrical capacitor $C$, which is electrically connected to the upper plate of a pair of deflection plates. The lower deflection plate is grounded. Due to total influence, the potential $V = Q/C$ of the upper plate rises with the charge $Q$ in the capillary. The transmitted beam is deflected by the electric field between the plates and the deflection $Y$ monitored on a position sensitive detector. The whole is inserted in a grounded metal box that screens the capillary and deflection plates from stray electrons. The entrance of the capillary is grounded (generally behind a grounded collimator having a slightly lower diameter than the capillary to avoid the charge up of the entrance).</figcaption></figure>

Particular attention has been paid to screen the capillary, capacitor and the deflection plates from stray electrons. The latter would otherwise be attracted by the positive potential of the capillary and limit the rise of the potential in the
<figure><img src="image_67.png" /><figcaption>Figure 2.21: With a Focused Ion Beam (FIB) a grid of $\emptyset$ = 10 µm holes were "digged" into a $\emptyset$ = 2mm inox disc having a thickness of 14 µm. The disc was placed in the left inlet of the holder. The injected beam produced on the PSD located 270 mm downstream a grid figure from which we can deduce the divergence of the injected beam and its rms emittance. Here, the divergence has a half-opening angle of 0.1° deg and a rms emittance of 0.1 mm.mrad.</figcaption></figure>

capillary [62]. The capillary, capacitor and the deflection plates are therefore put into a larger grounded cylindrical box that screens them from outer stray electrons (Fig. 2.19). The cylindrical metal box has three circular inlet holes, one for the capillary (located on the symmetry axis), one for the Faraday cup (1 cm right) and one with a grid pattern (1 cm left) that together with the PSD allows measuring the emittance of the beam. It has also a 30 × 10 mm² large outlet to let pass the transmitted beam. Behind the outlet, we added a grid-less negatively biased electrode "repeller", to avoid that low energetic stray electrons can enter the box. The holder can be heated by a THERMOCOAX cable in order to increase the conductivity of the capillary, for example, to speed up the discharge of capillaries. The temperature is monitored inside the box by a thermocouple attached to the capacitor surrounding the capillary. The holder is mounted on a high precision rotatable and translational plate, allowing for an alignment of the capillary axis with the beam axis by less than 0.05°, by adjusting the elevation and azimuthal angles. Upstream, the beam is collimated by a plate with a 2 mm hole. The plate is connected to an ammeter and the intensity recorded, allowing in the later analysis to account for the fluctuation of the intensity of the source.
<figure><img src="image_68.png" /><figcaption>Figure 2.22: Photo of the home-made capillary puller</figcaption></figure>

<figure><img src="image_69.png" /><figcaption>Figure 2.23: Comparison between the experimental and simulated charge-discharge curve of our capillary for two different temperatures, as indicated in the legend. The bulk and surface conductivity which were used in the simulations are given.</figcaption></figure>
## 2.8 Remarkable experimental results

### 2.8.1 Depletion (discharge) rate

We want to determine the surface and bulk conductivity of a borosilicate (®Pyrex 7740) glass capillary we used in our experiments. Our capillary is home-made by pulling a locally heated borosilicate glass tube (Fig 2.21). Much attention was paid to get a axis symmetric straight capillary so as to ensure an axis symmetric charge deposition at zero tilt angle <sup>11</sup>. The profile of the tapered capillary is given by the pink curve in Fig.2.14. Once mounted in the holder, only the 11 first mm of the outer surface of the capillary were grounded. As our setup allows measuring the accumulated charge as a function of time, we performed charge-discharge measurements of our capillary at different temperatures. In a first step, the capillary is charged by a 2.3 keV 0.1 pA-beam until a charge Q₀ of ~ 1.5 nC has been accumulated in the capillary. Then, the holder is shifted by 1 cm, so as to present a 20µm hole to the beam. The transmitted beam passes through the deflector plates and monitors, via the ΔY deflection, the charge stored in the capillary. This way, we follow the discharge of the capillary in time. In Fig. 2.23, we show the observed charge-discharge curves for T =31° and 43° C. With this particular setup, an experimental depletion (leakage) rate of about τ<sub>leak</sub><sup>-1</sup> ≃ 0.01 mHz was found for T =31° C. For a higher temperature T =43° C, we found τ<sub>leak</sub><sup>-1</sup> ≃ 0.03 mHz. We want to emphasize that the observed depletion rates<sup>12</sup> depend on the capacity C of the cylindrical capacitor that surrounds the capillary and are thus setup dependent. In our setup, we had C = 13±0.2 pF. The charge-discharge curves were simulated with our code and compared to the experimental data. To ensure a one to one comparison between the simulations and the experiments, a cylindrical electrode of 13 pF, and with floating potential, surrounded also the capillary in the simulations. Simulations with a surface conductivity κ<sub>s</sub> = 2.6 × 10<sup>-17</sup> S and bulk conductivity κ<sub>b</sub> = 1.8 × 10<sup>-15</sup> S/cm, in agreement with the measurements in [43], were found to reproduce the experimental discharge curve for T =31° C. In the case of T =43° C, an agreement was found for κ<sub>s</sub> = 9 × 10<sup>-17</sup> S and bulk conductivity κ<sub>b</sub> = 6.4 × 10<sup>-15</sup> S/cm. The depletion rate is a function of the surface and bulk conductivity, τ<sub>leak</sub><sup>-1</sup> = f(κ<sub>b</sub>, κ<sub>s</sub>). For an experimental discharge curve at given temperature, we found that several couples (κ<sub>b</sub>, κ<sub>s</sub>) yield simulated discharge curves that fit all nicely with the experimental data. In figure 2.24 we

<footnote n="11"><sup>11</sup>Analyzing the shape of our pulled capillaries, we noticed that the symmetry axis of the tapered part of the pulled capillary was systematically rotated by a small arbitrary angle < 0.3 deg with respect to the symmetry axis of cylindrical part of the capillary. After fine tuning the capillary puller, we succeeded in forming a capillary whose tapered part deviated by less than 0.1 deg.</footnote>

<footnote n="12"><sup>12</sup>They are not to be mistaken with the charge relaxation rates defined by (2.47).</footnote>
<figure><img src="image_70.png" /><figcaption>Figure 2.24: For the three indicated temperature, the set of ($\kappa_b, \kappa_s$) that give a good agreement with the measured discharge rates are plotted in the graph. The black dashed lines indicate the bulk conductivity of Pyrex at given temperatures.</figcaption></figure>

give for $T = 25^\circ C$, $T = 31^\circ C$ and $T = 43^\circ C$ the couples of ($\kappa_b, \kappa_s$) that yield discharge curves that are in agreement with the respective observed data. For each temperature, the solutions lay on a line, as could be expected by analogy from (2.47). The horizontal black dashed lines in Fig. 2.24 give the value of $\kappa_b$ of Pyrex for the three indicated temperatures, as provided by the manufacturer$^{13}$. From the intersection of the dashed line with the simulated data for each respective temperature, we deduce a non-zero surface conductivity that agrees well with the measured surface conductivity in [43]. On the other hand, taking $\kappa_s = 0$ yields a bulk conductivity that is systematically twice as large as the expected bulk conductivity for the given temperature. We deduce thus from this experiment that the surface conductivity plays a non-negligible role in the charge depletion in our glass capillary, and cannot be ignored in the modeling. Finally, we note that for capillaries mounted in the newly designed holder, *InCa4D* allows making accurate predictions, giving much confidence in the simulations.

## 2.8.2 Experimental evidence of self-organized focusing

Our original setup allows monitoring the transmitted beam (intensity and shape of the spot via the PSD) and accumulated charge in the capillary simultaneously. We take advantage of those features to give experimental evidence of the self

<footnote n="13">13 http://valleydesign.com/Datasheets/Corning-Pyrex-7740.pdf</footnote>
<figure><img src="image_71.png" /><figcaption>Figure 2.25: Y-projection of the transmitted beam on the PSD as a function of time. Colours indicate the normalized counts per second per area of the beam spot. The full green curve gives the simulated time-evolution of the charge Q stored in the capillary for an input current of 160 fA. The right scale is the accumulated charge, using the relation $Q[nC] = 0.088 \times \Delta Y[mm]$, defined in the text.</figcaption></figure>

organized radial focusing of an injected beam by a tapered insulating capillary. We used our home-made borosilicate capillary, with the profile given by the pink curve in Fig. 2.14. The capillary is aligned with beam axis using the technique involving the neutrals (see Fig. 2.9a), ensuring that the capillary is aligned with an error lower than 0.1°. We injected a low intensity Ar$^+$ ion beam at 2.3 keV and the transmitted fraction is monitored on the PSD. Figure 2.25, shows the y-projection of the transmitted beam spot on the PSD as a function of time for an injected intensity of 160 ± 7 fA, measured with a Faraday cup. The colors indicate the normalized counts per second (cps) of the spot. The deflection of the transmitted beam translates into the charge stored in the capillary using the relation $Q(t)[pC] = 88 \times \Delta Y(t)[mm]$. Initially, the spot on the detector has a size of less than 1 mm and is deflected vertically with increasing accumulated charge. After 2 hours of charging, the beam spot starts to spread, indicating that the focusing point approaches the outlet of the capillary. Two hours later, the electric field in the capillary is strong enough to focus the beam at the outlet, resulting into a high transmitted fraction of 50%, which is indicated by the darker colors. The spot has now a size of 7 mm corresponding to a full opening angle of 1.5 degree, close to the full opening angle of the capillary of around $tan(1.05/45) = 1.4$ degree. At this point, the total deflection of 21 mm indicates that the capillary accumulated a total charge of about 1.8 nC. A high transmission of more than 80 fA is kept for about 4 hours before the transmission slowly fades away. During the high transmission, the beam spot continues to spread, while its center rises only slowly. The
<figure><img src="image_72.png" /><figcaption>Figure 2.26: The transmitted intensity $I_{out}$ through the glass capillary as a function of time, for 6 different injected intensities ranging from 75 to 210 fA. Top panel: Experimental data. The transmitted intensity passing through a transmission grid of 10% is measured on the PSD and expressed in counts per second (cps), as indicated by the right vertical scale. Assuming an efficiency of the detector of 40%, the left vertical scale gives the transmitted intensity in fA. Bottom panel: Theoretical transmitted current, simulated with our numerical code InCa4D using the parameters defined in section 2.7.</figcaption></figure>

latter indicates that a quasi balance between the deposited and leakage current is attained. We add in Fig. 2.25 the simulated time-evolution of the accumulated charge $Q(t)$ in the capillary (full green curve) for a same input current of 160 fA. From the slope of the curve between 4h and 8h, we deduce that the capillary continues to accumulate charge at a rate of 15 fA. This low accumulation rate explains the long stable transmission for over 4 hours. Still, after 13 hours, a charge of 2100 pC has been accumulated and the transmission stops, because of Coulomb block-
ing. Monitoring the accumulated charge in the capillary is crucial for analysing and clearly identifying the different stages of the transmission dynamics, especially from the perspective to stabilize the transmission.

In a second step, we studied the transmitted beam intensity $I_{\text{out}}$ for six different injected beam intensities $I_{\text{in}}$, ranging from 75 to 210 fA. The experimental results are given in the upper panel of Fig.2.26. Note the typical time-evolution of the transmitted fraction in the case of radial focusing, which differs strongly from the one in the case of transverse focusing (see inset of Fig.2.12). Only after at least 800 pC have been stored in the capillary walls, does the transmission increases rapidly from the initial geometrical fraction. We found that the focusing factor increases from 4 to 10, or alternatively, the transmission fraction $I_{\text{out}}/I_{\text{in}}$ increased from 20% to 50% with increasing beam intensity. For injected beam intensities less than 120 fA, the beam was transmitted for more than 24 hours and no Coulomb blocking was observed for the time of the experiment. In these cases, we know from the $\Delta Y$ deviation of the spot on the PSD that the charge in the capillary never exceeded 2100 pC, which corresponds to the critical value for which the potential in the capillary is expected to exceed the source potential. For currents above 120 fA, the charge accumulated in the capillary during the time of the experiment was sufficient to generate a potential that exceeds the source potential and the transmissions faded slowly away. In these cases, the leakage current was too small to compensate for the deposited charge per time unit. Increasing the conductivity of the capillary (for example by heating) will certainly allow for higher input currents without experiencing Coulomb blocking.

We simulated for the same intensity range the transmission and compared it to the experimental data (lower panel of Fig.2.26). Simulations corroborate the experimental findings that the transmission fraction increased with the intensity. The quantitative agreement is remarkable, giving great confidence in the predictions that can be calculated by our model. This was made possible by developing the capillary holder, while keeping in mind that the boundary conditions imposed on the electric field in the capillary and its holder must be matched by our the numerical code. The simulations also corroborate that, with our particular experimental setup, the injected current always dominates the leakage current, so that the high transmission cannot be maintained and the Coulomb blocking not avoided. The numerical code offers thus a valuable tool to test the effect of different beam or capillary parameters on the stability of the transmission. For example, preliminary simulated results show that a lower beam emittance allow for higher focusing factors $j_{\text{out}}/j_{\text{in}}$ and for stable transmissions. Detailed studies about the influence of the beam emittance on the transmission are underway.

The simulations also confirm the trend that the Coulomb potential along the z-axis varies steeper for higher injected intensities. The profile of the potential along
<figure><img src="image_73.png" /><figcaption>Figure 2.27: color online: Radial electric field $E_r(r = R(z), z)$ at the inner surface inside the capillary, taken at the apex of the transmission, for different injected current intensities. Negative values have a radial focusing effect while positive values have a defocusing radial effect. Note that the fast fluctuation of the field around 11 mm is due to the negative charge accumulation at the grounded outer electrode, which covers the first 11 mm of the outer capillary surface. It has only a negligible effect on the ion trajectories and is irrelevant here.</figcaption></figure>

the z-axis exhibits thus a larger negative curvature, which yields a stronger focusing radial electric field (see eq. 2.61). Now, for axisymmetric charge distributions at the capillary interfaces, the radial component of the electric field $E_r$ is strong mainly outside of the capillary interfaces and should be very weak inside, even if non-zero. It is thus interesting to show $E_r$ inside the capillary at the moment the transmission is maximal. In figure 2.27, the axisymmetric radial field $E_r$, taken at the inner surface $R(z)$ of the capillary, is given for different injected current intensities. Negative values have a radial focusing effect while positive values have a defocusing effect. The focusing part of the field between 26 and 40 mm always dominates the preceding defocusing part and increases with increasing current intensity. This explains why focusing power and thus transmission fractions increases with higher injected currents. While the intensity of the focusing part of the fields are indeed relatively weak, ($< 10$ V/mm), they are sufficient to focus a 2300 keV $Ar^+$ beam through the capillary outlet.

## 2.9 Conclusion

In a first step, we discussed the electric conductivity in ionic glasses and motivated in detail the surface charge approximation used in our description of the accumulated charge at the interfaces. We then presented the theoretical model (and its
assumptions) that describes the surface charge dynamics at the capillary interface. We detailed the evaluation of the electric field that guides the ions through the capillary and drives the charge dynamics at the interfaces. We considered the general case where a third metal electrode surrounding the capillary is present, as the latter allows for a better comparison with experimental data. We presented an original analysis of the charge relaxation rates of the charge patches at the inner surface of an insulating capillary tube with the outer surface grounded. We showed that the relaxation rate is a weighted sum of the bulk and surface relaxation rate, with the weights depending on the dimensions of the capillary as well as the imposed boundary conditions. We gave an example where we used the theoretical predictions to analyse experimental relaxation rates.

We presented our numerical code *InCA4D* and highlighted its features. *InCA4D* takes into account secondary electrons, neutralized projectiles, induced polarization charges at the three interfaces and non-linear conductivity. In the second part of the chapter, we presented simulated results. We showed that one must distinguish between transverse and radial self-organized focusing. The transverse focusing explains the enhanced transmission in tapered capillaries for non-zero tilt angles. We showed that tapered capillaries can focus an ion beam similar to an Einzel lens. Simulations showed that the injected beam is efficiently focused through the outlet if the self-organized potential of our capillary approaches 75% of the extraction potential of the ion source. For a given electrical conductivity of the capillary, the self-organized potential is steeper with increasing injected intensity, resulting in a higher focusing radial electric field. We could thus explain the observed trend that the transmitted fraction increases with the injected beam intensity.

We presented an original capillary holder, that allows monitoring the accumulated charge in the capillary. The holder also screens the capillary from stray electrons and allows measuring the intensity and emittance of the injected beam with the integrated Faraday-Cup and micro-hole grid, respectively. We gave for the first time clear experimental evidence of the radial self-organized focusing by a tapered glass capillary, for intensities ranging between 70 to 200 fA. We found that the apex of the transmission could not be sustained for long times (days). By monitoring the accumulated charge, we could deduce the leakage current at the apex of the transmitted fraction and found that it was too low, compared to the deposited current, to maintain a stable transmission. Nevertheless, for intensities below 120 fA, the transmission was found stable for almost 24 hours. We expect that our simulations will be of immense help to improve and stabilize the transmission by identifying the bottlenecks and test original solutions. Indeed, simulations confirmed recently (not further discussed in the manuscript) that the emittance of the injected beam plays a crucial role on the stability of the trans-
mitted beam and that the high transmission could be maintained by grounding for
example the tip of the capillary. Reliable simulations could be achieved by devel-
oping hand in hand the capillary holder and the numerical code, ensuring a one to
one comparison between experimental and simulated results. Finally, fitting the
experimental charge-discharge curves for our glass capillary, allowed to extract the
surface and bulk conductivity of our Pyrex glass capillary and showed that surface
conductivity should not be ignored in the description of the charge dynamics at
the interface.
## 2.10 Perspectives

Three topics linked to the transmission of charged particles through insulating capillaries are presented below. Some may may give rise to Phd works with a mixed theoretical and experimental profile.

### 2.10.1 Stabilization of the focused ion beam

<figure><img src="image_74.png" /><figcaption>Figure 2.28: Influence of the emittance $E$ on the stability of the transmission of a focused beam.</figcaption></figure>

In the manuscript, we have highlighted the focusing power of a conical insulating capillary. Preliminary simulations indicated that if the injected beam has a low emittance, the transmission is maintained for a long time, at least well beyond the simulated time, see red curve in Fig. 2.28. In the extreme case of a zero emittance beam, the simulations predict even a 100% transmittance with a long time stability. In reality, the emittance of the injected beam is never zero and the focusing "point" has a non-zero spatial extension. If the diameter of the focusing area is larger than the diameter of the capillary outlet, a part of the injected beam is always intercepted by the capillary wall near the outlet. As a result the potential of the capillary continues to rise and eventually exceeds the extraction potential of the ion source (black curve in Fig. 2.28). Indeed, the leakage current, ie the current flowing to ground, is usually too small to compensate for the charge deposited per unit of time. The goal here is to optimize the leakage current in order to stabilize the charge in the capillary and therefore the transmission. The charge transport in the dielectric can be controlled by the following parameters:
1. The conductivity of the capillary (by varying the composition of the glass and / or its temperature);

2. Profile and dimensions of the capillary

3. Adding grounded electrodes on the outer tip surface of the capillary.

We privilege the latter and estimate that a grounded electrode surrounding the exit of the capillary should stabilize the transmission as the deposited charge is now faster removed, keeping the exit at low (near ground) potential. The beneficial effect of adding electrodes on the capillary was tested with the InCa4D code. As shown in Fig. 2.29, the idea is validated by the numerical code. We used the

<figure><img src="image_75.png" /></figure>

Figure 2.29: Electric potential in the capillary, in the presence of a grounded electrode surrounding the last 3 mm of the capillary tip. Legend of the panels indicates the injected charge. Beam parameters were those used in Fig. 2.14 for better comparison. The abscissa coordinates are given in mm.
same parameters for the injected beam as in Fig. 2.14. The potential near the
outlet stays now near zero, even after 9 nC have been injected. The Coulomb
blocking is avoided, contrary to what has been observed in the last panel of Fig.
2.14. Comparing the potential curve in middle and last panel, we see that the
potential in the capillary stays almost constant, meaning that an equilibrium be-
tween the leakage and injected current has been found. In a near future, we want
to bring the experimental evidence that such a setup stabilized the transmission.
In a second step we want to reduce the outlet diameter of the capillary in order to
reach stable beams of nanometre size. Capillaries with nano-sized outlets diameter
can be be achieved with our capillary puller. The profile and dimensions of the
tapered capillary are then measured with a microscope and the profile injected
into the numerical code. A theoretical study will then evaluate its potential to
stabilize transmission. In addition to the theoretical interest, Focused Ion Beams
(FIB) have a great practical interest. Stable beams of nanometre size could find
an application in the future European IMPLATION project for single ion implan-
tation. Thus the focusing of ion beams by micrometre capillaries also responds to
technological and societal issues.
## 2.10.2 Transmission through curved insulating capillaries

<figure><img src="image_76.png" /><figcaption>Figure 2.30: Curved capillary. Extracted from [63]</figcaption></figure>

The guiding power of insulating capillaries has also been tested for curved capillaries. It has been shown experimentally by T. M. Kojima et. al. [42] that ions can pass through a 5 cm long Teflon capillary arc, curved by 27°. The transmission was found to be about 20%. A review is given in [63]. Improvements are certainly possible and a theoretical study of the problem could identify the bottlenecks that have to be lifted and propose workarounds. I propose thus to simulate ion transmission through curved insulating capillaries. The goal is to find the necessary conditions for a capillary to transmit with a transmittance of more than 50% an ion beam through a curved 90° capillary. We would like to demonstrate that such a tube can effectively guide low energy (< 1000 V/q) multi-charged ions, keeping their energy and initial charge and without applying an external electric or magnetic field. To date, we have a numerical code, written in cylindrical coordinates, capable of simulating the transmission through axially symmetric capillaries. This code is however not suitable for describing curved capillaries. I propose here to modify the existing code in order to describe a curved tube up to 90 degrees for a given radius of curvature. Part of the work therefore consists of writing a function capable of solving the Laplace equation of the electrical potential in toroidal coordinates system (η, ξ, φ), (see Fig. 2.31). The success of the method is ensured by the fact that the Laplace equation is separable in the toroidal coordinate system,

$$ \Phi(\eta, \xi, \phi) = \sqrt{\xi - \cos(\eta)} \sum_{n,m} a_{mn} P_{n-1/2}^{m}(\xi) e^{\pm i n \eta} e^{\pm i m \phi} \rightarrow \nabla^2 \Phi = 0 \quad (2.62) $$

allowing for a fast computation of the electric potential and field, with a cost similar to the model in cylindrical coordinates. For the inner surface, the Neumann conditions will be imposed, in order to satisfy the discontinuity of the normal field through an interface separating two different dielectrics. For the outer surface, the Dirichlet conditions will be imposed (the potential of the outer surface is
<figure><img src="image_77.png" /><figcaption>Figure 2.31: Toroidal coordinate system. Surfaces with constant $\xi > 1$ are toroids and $\varphi$ the polar angle in the xOy plane.</figcaption></figure>

grounded). The goal is to understand through simulations how the transmittance varies according to the radius of curvature and the length of the capillary. We will try to predict and optimize with our simulations the transmittance, with the goal to increase the length of the arc up to 90°, if possible. Then we will experimentally test the theoretical predictions with our experimental setup. Again, we manage to create a feedback loop between simulations and experimental data that might allow us to quickly converge towards the desired solution.
### 2.10.3 Transmission of injected electron beam

Transmission of electrons through insulating capillaries has been observed more than a decade ago in the pioneering work of Milosavljevic *et al.* [30]. Recently, the capillary community has renewed interest in electron transmission through insulating capillaries [57, 58]. Today, a relatively large experimental database is available. These data suggest that electron transmission through capillaries might be similar to that of ions. As with ions, electrons are guided through the capillary even if no geometric transmission is possible. And indeed, 500 eV electron transmission through nano-capillaries in PET has recently been observed. However a more detailed comparison shows divergences that are related to interaction mechanisms with insulators, which are different between ions and electrons. While the ions always deposit a positive charge at the point of impact, the sign of the charge deposited by the electrons depends on their velocity and their incident angle. Thus, the impact of a 500 eV electron on a Teflon or PET surface emits on average 4 secondary electrons, thus depositing on average 3 elementary positive charges at the point of impact, whereas for an electron of 20 eV, the charge deposited is negative [14]. This raises the question of how exactly an electric field, capable of guiding 500 eV electrons through a capillary, emerges with positive charge deposition at the point of impact. Of course, it should be noted that overall the charge deposited in the capillary is negative. Electrons can also penetrate the surface for a few nm before being backscattered by losing a fraction of their incident energy. The ion on the other hand is neutralized on the surface and is no longer guided by the electric field. Finally, electrons scattered near the surface are always subject to image charge and can be attracted back to the surface. It follows that the electron charge deposition mechanism differs greatly from that of the ions.

I propose to develop a model with a realistic description of the electron-dielectric interaction. The goal is to improve our understanding of the behavioural differences between ion and electron transmission. The focus will be on the description of the charge deposition and charge dynamics at the dielectric interfaces. It involves reconstructing the electric field map inside the capillary via simulations to explain the transmission of electrons through the insulating capillaries. The comparison between the theoretical predictions and the experimental data will make it possible to validate or not the initial hypotheses of the mechanisms of electron-insulating interaction as well as the dynamics of the electrons in the insulator.
## 2.11 Appendix : Numerical methods

### 2.11.1 Injected charge at the inner surface

When an ion hits an insulating surface, electrons are ejected from the impact point. Some are captured by the projectile ion while the other, if we consider a closed surface, are re-injected into the surface. We suppose here that the projectile is neutralized by the ejected electrons and that the remaining electrons fall back onto the impact point, meaning that the deposited charge is equal to the initial charge of the ion. We ignore hereby the possible capture of electrons by the insulator farer away from the impact point, avoiding the description of the dynamics of the ejected electrons.

The deposited surface charge density per unit time is computed using the following scheme. Let the inner surface of the capillary be mapped onto a two-dimensional grid $(\theta_i, z_j)$ of dimension $N_\theta \times N_z$ (see figure 2.32). Along $\theta$ we chose an uniform grid path $\Delta\theta = 2\pi/N_\theta$ and $\theta_i = (i-1)\Delta\theta$. Along the $z$ axis the points $z_j = L \frac{(1+x_j)}{2}$ are distributed according to the Gauss collocation points $x_j$, where $L$ is the length of the capillary and $\{x_j\}$ the $N_z$ zeros of the Legendre polynomial of order $N_z+1$. For an impact point at the inner surface coordinate $((R_1(z), \theta, z))$, the

<figure><img src="image_78.png" /><figcaption>Figure 2.32: Deposited charge on the 2d-grid, mapping the inner surface of the capillary. The charge is distributed on the grid using the 2d normalized gaussian function given in Eq. 2.63 (blue transparent ellipse), with the impact point (black circle) being the center of the gaussian distribution</figcaption></figure>

deposited charge $q_d$ is distributed over the grid points $(\theta_i, z_j)$ using the normalized 2d gaussian function

$$f(\theta_i, z_j, \theta, z) = \frac{\exp\left(-\frac{(\theta-\theta_i)^2}{\gamma\Delta\theta^2}\right) \exp\left(-\frac{(z-z_j)^2}{\Delta z^2}\right)}{\Delta\theta \Delta z R_1(z) \pi \sqrt{1 + R_1'(z)^2} \sqrt{\gamma}} \quad (2.63)$$

Choosing $\gamma = 1/\ln(2)$ and $\Delta z = L/N_z$, assures that the convolution function $f$ is well integrated over the grid to unit with a relative error less then $10^{-6}$. As the surface element $\Delta\theta \Delta z R_1(z) \sqrt{1 + R_1'(z)^2}$ varies with $z$, the deposited charge is smeared over an area which depends on the impact abscissa $z$. In the case of
tapered capillaries, this may vary over one order of magnitude. It does however not affect the charging up as long as the surface element is small compared to the size of the beam spot projected on the inner surface. Our micro-capillaries being about 60 mm long with an inlet diameter of about 1 mm, we typically use in our simulations $N_{\theta} = 64$ and $N_{z} = 500$, yielding a grid mesh which is about 100µm × 120µm at its loosest point. This grid size satisfies the above condition and gives a good compromise between the precision with which the density is represented on the grid and CPU cost for computing the Coulomb potential and forces. Eventually, the deposited surface charge distribution at the grid points $(\theta_i, z_j)$ during the time interval $\Delta t$ is given by

$$ \gamma^h(\theta_i, z_j)\Delta t = q_d \sum_{\alpha} f(\theta_i, z_j, \theta_{\alpha}, z_{\alpha}) , \qquad (2.64) $$

where the sum runs over all the particles $\alpha$ that hit the surface during the time interval $\Delta t$.

### 2.11.2 Accelerating the charging up

The code is aimed at describing not only the guiding power of weakly charged insulating capillaries, but also to explore the possible focusing effect of highly charged tapered capillaries and even the Coulomb blocking. Coulomb blocking of the ion beam by the potential of the capillary happens only if the potential $V$ of the capillary equals the extraction potential $U_0$, while the focusing effect may appear for potentials above 60% of $U_0$. Consequently, $Q_0 = \varepsilon_0 L_0 U_0$, gives the order of magnitude of the total charge, distributed on the capillary, necessary to block the beam. Taking typical values for $U_0 = 3$ keV and $L_0 = 0.001$ m, gives a characteristic charge $Q_0$ of the order of $3 \times 10^{-11}$ C. One needs thus to simulate of about $Q_0/q$ trajectories in order to deposit a charge that generates a potential field strong enough to Coulomb block the ions. In the case of singly charged ions, this means $Q_0/q = 10^8$ trajectories. This huge number would push the CPU time for 1 simulation to several weeks, making it unreasonable for systematic studies. Let us now assume that each simulated trajectory accounts for an ensemble of $n$ particles where each particle deposits $q$ charges. If the deposited charge per impact $q_d = nq$ is small enough as to influence the Coulomb potential $V_c$ only in a negligible way, $n$ can be adjusted in order to accelerate the charging up. For $n > 2000$, we found that the first trajectories were affected by the poorer sampling, that is, the coarse graining of the deposited charge per trajectory, which spuriously over-deflected the subsequent ions. No significant difference however was found for $n = 10, n = 100, n = 500$. To stay on the safe side, we used in all our simulations an acceleration factor of less than 500, $n \le 500$. This leads to
about $10^6$ trajectories per simulation, which is a feasible number of trajectories that one modern CPU can simulate in 24 hours.

We found in the above paragraph that the number of trajectories increases with the extraction potential. To avoid this, we may fix $N_t = 10^6$ and calculate the appropriate accelerating factor $n$. By choosing for $n$ the particular value $n = Q_0/(N_tq)$, the deposited charge per ion impact becomes $q_d = nq = Q_0/N_t$. Note that the latter is independent of the ion charge $q$. For example in the case of Ar$^+$ extracted by a potential of 3000 V, we get an acceleration factor $n = Q_0/(N_tq) \simeq 160$. The latter is small enough as not to alter in a significant way the charging up with respect to $n = 1$. We found indeed no difference in the computed observables for $n = 160$ or $n = 16$, validating the assumption. By adjusting $n$ so as to keep a reasonable number $N_t$ of trajectories, allows to keep the same CPU cost per simulation independent on the extraction potential.

### 2.11.3 Evaluation of the potential and electric field inside the capillary

Even with a time step adaptive predictor corrector scheme, which is conservative on the evaluation of the force along the trajectory, an average of 100 evaluation of the electric potential / field are necessary for each trajectory. It is therefore vital to have a efficient numerical scheme to evaluate the electric potential and field at an arbitrary point inside the capillary. We give here an overview of the numerical scheme used in our code. We assume that the shape of the capillary has cylindrical symmetry. Let us decompose the angular part of the surface charge density and the Coulomb potential on the first $M$ moments. We consider the case where the capillary axis may be tilted with respect to the beam axis. The two axis define then a plane of symmetry, fixing the xOz plane and the origin of the $\theta$ angle. The deposited surface charge density $\sigma(s(z), \phi, z) \equiv \sigma(\phi, z)$ is thus symmetric with respect to the xOz plane $\sigma(-\phi, z) = \sigma(\phi, z)$. Consequently, only the cosines moments are non-zero in the angular decomposition so that,

$$ \sigma(\theta, z, t) = \sum_{m=0}^{M} = \sigma_m(z, t) \cos(m\theta) \qquad (2.65) $$

$$ V(r, \theta, z, t) = \sum_{m=0}^{M} = V_m(r, z, t) \cos(m\theta) \qquad (2.66) $$

Note that if the capillary axis and the beam axis are aligned, only the moment $m = 0$ need to be considered, reducing by a factor $M$ the CPU cost. The components $V_m$ are defined in the rOz plane. Instead of calculating $V_m$ for an arbitrary point of the rOz plane, we limit the evaluation of $V_m$ on the grid points
<figure><img src="image_79.png" /><figcaption>Figure 2.33: 2d grid of the rOz plane mapping the interior of the capillary as well as the space behind the outlet of the capillary so that the trajectories can be computed until the multipoles of the electric field become negligible.</figcaption></figure>

of the 2d mesh that maps the inner region of the capillary as well as the space behind the outlet of the capillary so that the trajectories can be computed until the multipoles of the electric field become negligible (see figure 2.33). The grid has $N_i = 6$ points in the Or direction and $N_j = 2000$ points in the Oz direction, labelled $z_j$. Because the inner radius of the capillary $s(z)$ is varying with $z$, the interval between the radial points depends on $z$ and is proportional to $s(z)$. The radial distribution depends thus explicitly on the coordinate $z$ of the symmetry axis Oz and the radial points $r_{ij}$ of the mesh read,

$$r_{ij} = x_i s(z_j), \qquad (2.67)$$

where $x_i$ are the collocation points of a $N_i$ order Gauss Legendre interpolation scheme. In the Oz direction the $N_j = 2000$ points are distributed uniformly except at the capillary outlet where the density is increased in order to cope with the strong variation of the field. We chose to represent the surface charge moments $\sigma_m(z_k)$ on $N_k = 500$ collocation points. For a capillary of 70 mm length, this is in average 1 point every 0.14 mm, much smaller than the typical spatial variation of the surface charge density and Coulomb potential along the $z$ axis.

The surface densities are evaluated on the interface grid as discussed in the previous section. The $m \le M$ moment of the potential, $V_m$ are efficiently computed by a simple algebraic equation

$$V_m(r_i, z_k, t) = \sum_q S_{i,k,q}^m \sigma_m(z_q, t) \qquad (2.68)$$

where the matrix $S^m$ is the Coulomb kernel for the moment $m$ and depends only on the shape of the capillary. As a result, $S_{i,k,q}^m$ is independent of time and can be computed once for all at the beginning of the simulation or read from disk if stored before. Only the moments $\sigma_m(z, t)$ depend on time and need to be undated regularly. Eventually, the potential at an arbitrary point inside the capillary is
obtained, first by interpolating $V_m(\rho, z, t)$ on the 2D grid using $V_m(\rho_i, z_k, t)$ and then by summing over the moments (2.66).

The same method is used for the electric field. As the amount of deposited charge per hit is small, it is sufficient to update the potential moments $V_m(\rho_i, z_k, t)$ on the 2D grid, via equation (2.68), only every $n_{up}$ trajectories, reducing even more the CPU costs. In our simulation we took $M = 6$, limiting the decomposition of $\sigma$ and $V$ to the duodecapole moment. We found that $M = 6$ is indeed sufficient for most simulated cases, as long as the beam size is of the same order than the inner capillary diameter. The frequency with which $V$ was updated depends on the characteristic dimensionless ratio $\tau_c/\tau_r = \kappa_b L_0 U_0 / I$. We typically take values for $N_k = 2000$, $N_i = 6$ and $n_{up}$ about 300, meaning that the potential is only updated every 300 trajectories, making this scheme efficient if $N_i \times N_k \ll 100 \times n_{up}$.

## The S matrix

The Coulomb potential in cylindrical coordinates is given by

$$V(r, \phi, z, t) = \frac{1}{4\pi\varepsilon_0} \int_0^L \frac{d\Sigma'}{\sqrt{(r+s(z'))^2 + (z-z')^2}} \times \int_{-\pi}^{\pi} \frac{\sigma(s(z), \phi, z, t)d\phi'}{\sqrt{1-w \cos^2\left(\frac{\phi'-\phi}{2}\right)}} \quad (2.69)$$

with $w = \frac{4\rho s(z')}{(r+s(z'))^2+(z-z')^2}$ and where $s(z)$ defines the inner capillary surface, $d\Sigma' = s(z')\sqrt{1+(\frac{ds}{dz'})^2} dz'$ the surface element and $L$ the length of the axisymmetric capillary. Injecting (2.65) into (2.69) allows to define the moments of the potential as defined in Eq. 2.66,

$$V(r, \phi, z, t) = \sum_{m=0}^{M} \frac{1}{4\pi\varepsilon_0} \int_{0}^{L} \frac{\sigma_m(z', t)d\Sigma'}{\sqrt{(r+s(z'))^2 + (z-z')^2}} \times \underbrace{\int_{-\pi}^{\pi} \frac{\cos(m\phi')d\phi'}{\sqrt{1-w \cos^2\left(\frac{\phi'-\phi}{2}\right)}}}_{F_m(w) \cos(m\phi)} \quad (2.70)$$

$$= \sum_{m=0}^{M} \underbrace{\frac{1}{4\pi\varepsilon_0} \int_{0}^{L} \frac{\sigma_m(z', t) F_m(w) d\Sigma'}{\sqrt{(r+s(z'))^2 + (z-z')^2}} \cos(m\phi)}_{V_m(\rho, z, t)} \quad (2.71)$$

$$= \sum_{m=0}^{M} V_m(r, z, t) \cos(m\phi) \quad (2.71)$$
The integrating over $\phi'$ in eq. (2.70) is done analytically. As examples, the first 3 terms, calculated with Mathematica [20], are given below,

$$F_0(w) = 4K(w) \qquad (2.72)$$

$$F_1(w) = \frac{4(2E(w) + (w-2)K(w))}{w} \qquad (2.73)$$

$$F_2(w) = \frac{4(8(w-2)E(w) + (w-4)(3w-4)K(w))}{3w^2} \qquad (2.74)$$

where $K()$ and $E()$ are complete elliptic function of type 1 and 2. In a second step, the surface charge density moment $\sigma_m(z, t)$ is interpolated by a sum of cardinal functions, $g_k(z)$, based on Legendre polynomials [23, 12]

$$\sigma_m(z, t) \approx \sum_{k}^{N_k} \sigma_m(z_k, t) g_k(z) \quad , \qquad (2.75)$$

where $z_k$ are the $N_k$ collocation points of the Legendre polynomial of order $N_k+1$. Injecting (2.75) into (2.71) yields

$$V_m(r, z, t) = \sum_{k}^{N_k} \sigma_m(z_k, t) \times \qquad (2.76)$$

$$\int_0^L \frac{g_k(z')F_m(w) \, d\Sigma'}{4\pi\epsilon_0\sqrt{(r+s(z'))^2 + (z-z')^2}} = \sum_{k}^{N_k} \sigma_m(z_k, t)S_{m,k}(r, z) \qquad (2.77)$$

The matrix $S_{m,k}$ links the moment $m$ of the surface charge distribution to the moment $m$ of Coulomb potential. Indeed, because of the cylindrical symmetry of the capillary, the moments do not mix. The moments $V_m(r, z)$ as well as $S$ are defined in the xOz plane. Finally the potential is recomposed by summing over all the moments,

$$V(r, \theta, z, t) = \sum_{m=0}^{M} = V_m(r, z, t) \cos(m\theta) \quad . \qquad (2.78)$$

yielding the potential at an arbitrary point inside the capillary. Both interpolation schemes can be differentiated with respect to $r$ or $z$ to calculate the electric field at arbitrary points. Alternatively, similar to the $S$ matrices for potential, the $S$ matrices for the $r$ and $z$ components of the electric field can be pre-calculated. The electric field can then be interpolate using the same scheme than for the
potential. The letter scheme is used in our simulation to calculate the force at an arbitrary point inside the capillary to propagate the projectiles until they hit the inner surface or until they pass through the outlet. It is also used to calculate the electric field at the interfaces $S_1$, $S_2$ and $S_3$.

### 2.11.4 Induced surface charges

In order to illustrate the importance of accurately computing the induced surface charges at the interfaces, we propose to compute the electric field in the case of a single charge patch in a glass tube of dielectric constant $\varepsilon_r = 4.8$. The tube is 55 mm long and the inner and outer radii are respectively 0.43 mm ($S_1$) and 0.75 mm ($S_2$). The tube is surrounded by a grounded metal tube of 3 mm inner radius ($S_3$). We start with given free surface charge distributions $\sigma_1$ and $\sigma_2$ at the inner and outer surface of the glass tube, respectively. The latter were created by injecting a low intensity ion beam of 0.6 mm diameter into the glass tube. The beam was tilted by 0.8 deg with respect to the tube axis, until the beam was deviated by the electric field due to the charge patches. The right hand terms (2.31) of the matrices system 2.30 are evaluated and the induced surface charges at the 3 interfaces computed. Finally the normal and tangential components of the electric field at the inner and outer part of the $S_1$ and $S_2$ interfaces are evaluated. The symmetry axis of the tube and of the beam define a symmetry plane. Cuts, containing the symmetry plane, of the induced surface charge distributions at the 3 interfaces are given in the left panel of figure 2.34. As shown in the top left panel, the surface polarization charges $\tilde{\sigma}_1$ and $\tilde{\sigma}_2$ are clearly not negligible with respect to deposited screened charges $\sigma_1/\varepsilon_r$ and $\sigma_2/\varepsilon_r$ in the dielectric. The surface charges $\tilde{\sigma}_3$ at the $S_3$ interface ensure that the equipotential of the metal tube equals zero. All three induced surface charges contribute sensibly to the Coulomb field in the dielectric.

We compare the computed screened surface charges to an alternative method, where the deposited charges are simply screened by the factor $2/(1+\varepsilon_r)$. The latter corresponds to the screening of a charge lying at the glass-vacuum interface of an infinite dielectric plane. Similarly, in this alternative approach, the grounded metal tube is taken into account via image charges in the spirit of an infinite grounded metal plane. This approach has been used by several authors in their simulations [48, 49]. While the latter method is easily implemented in numerical codes, it leaves the self-consistent nature of the above described matrix method. The right top panel compares the screened charges obtained by both methods. The pink color stands for the surface charges at the inner surface. We note that using a the factor $2/(1+\varepsilon_r)$ to screen the surface charges is indeed a viable approximation. However, for the outer screened surface charges (green full and dotted line) the difference is dramatic and it is clear that the factor $2/(1+\varepsilon_r)$ approach is failing.
<figure><img src="image_80.png" /><figcaption>Figure 2.34: Full line; Force acting on the projectile due to the induced polarization charges on a cylindrical glass-vacuum interface as a function of the distance separating the projectile from the interface. Dotted line: plane interface. The dielectric constant of the glass was set to $ε_r = 4.8$</figcaption></figure>

The left bottom panel shows the inner $E_1^-$ and outer $E_1^+$ part of the electric field at the interface $S_1$. They control respectively the deflection of the beam ions at the inner surface and the migration of the charge carriers in the dielectric from the inner to the outer surface. Again, we compare the computed electric fields to an alternative method, where the deposited charges are screened by the factor $2/(1 + ε_r)$. We found in both cases a discrepancy of about 30% in the electric fields between both approaches for the same deposited charge distributions. The reason for the discrepancy is that the boundary conditions (2.27) and (2.25, 2.26) are not satisfied at the 3 interfaces in that simple alternative approach. Indeed, in the right bottom panel, we note that $V_3(\text{alt})$ is non-zero and does not meet the required boundary condition of $V_3 = 0$. Of course the discrepancy between the simple screening of the form $2/(1 + ε_r)$ and the exact method depends on the geometry of the interfaces $S_{1,2,3}$. The above example however clearly illustrates the sensitivity of the electric field on the induced polarization charges at the interfaces
and demonstrates the numerical advantages of the precise self-consistent matrix
approach.
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[22] "Production of a microbeam of slow highly charged ions with a tapered glass capillary", T. Ikeda, Y. Kanai, T. M. Kojima, Y. Iwai, T. Kambara, and Y
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[37] A.Cassimi, L.Maunoury, T.Muranaka, B.Huber, K.R.Dey, H.Lebius, D.Lelievre, J.M.Ramillon, T.Been, T.Ikeda, Y.Kanai, T.M.Kojima, Y.Iwai, Y.Yamazaki, H.Khemliche, N.Bundaleski, P.Roncin, Nucl.Instrum.Meth.in Phys.Res.B **267** (2009) 674.

[38] N Bundaleski, H. Khemliche, P. Rousseau, A. Cassimi, L. Maunoury and P. Roncin, J. Phys.: Conf. Ser. 163 (2009) 012091

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[43] "Temperature control of ion guiding through insulating capillaries", E. Gruber, G. Kowarik, F. Ladening, J. P. Waclawek, F. Aumayr, R. J. Bereczky, K. Tőkési, P. Gunacker, T. Schweigler, C. Lemell and J. Burgdörfer, Phys. Rev. A **86** (2012) 062901.

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[62] "Evolution of the electric potential of an insulator under charged particle impact", E. Giglio, S. Guillous, A. Cassimi, H. Q. Zhang, G. U. L. Nagy, and K. Tőkési, Phys. Rev. A 95, 030702(R) (2017)

[63] Takao. M. Kojima, "Ion guiding in macro-size insulating capillaries: straight, tapered, and curved shapes", Journal of Physics B Atomic Molecular and Optical Physics, 51(4), 042001, 2018 https://doi.org/10.1088/1361-6455/aa9eaf

[64] "Self-organized focusing power in insulating capillaries" E. Giglio, S. Guillous, A. Cassimi, Phys. Rev. A 98, 052704 (2018)

[65] Sanjeev Kumar Maurya, Sushanta Barman, Samit Paul and Sudeep Bhat-tacharjee, "Charge dissipation and self focusing limit in high current density ion beam transport through a micro glass capillary", Journal of Physics D: Applied Physics, Volume 52, Number 5 (2018)

[66] Priyanka Dash, Mengxue Yuan, Jun Gao, Eugene Furman and Michael T Lanagan, "High electric field conduction in low-alkali boroaluminosilicate glass", JOURNAL OF APPLIED PHYSICS 123, 054102 (2018)

[67] "Relaxation dynamics of charge patches formed inside an insulating capillary by ion impact", E. Giglio, K. Tőkési, R D Dubois, Nuc. Inst. Meth. B (2019) accepted, in press, DOI: 10.1016/j.nimb.2018.12.027

[68] "Transmission dynamics of 1 MeV H+ microbeam guided through an insulating macrocapillary", G. Nagy, E. Giglio, I. Rajta and K. Tőkési, Nuc. Inst. Meth. B (2019) accepted, in press, DOI: 10.1016/j.nimb.2018.11.045
[69] "Guiding of Low-Energy Ions through an Insulating Capillary: A Study of the Charge Deposition, Redistribution and Decay capillaries", R D Dubois K. Tőkési, E. Giglio, submitted to PRA

**List of publications:** All my papers are fully available on ResearchGate, in pre-prints format with identical content as the published ones. A list of publications over the last decade as well as this HDR thesis in pdf format can be found on my CIMAP webpage.
<figure><img src="image_81.png" /></figure>
# ABSTRACT

**Charged droplets at the Rayleigh limit.** We model and simulate the shape evolution of critically charged droplets, from the initial spherical shape to the charge emission and back to the spherical shape. The shape deformation is described using the viscous correction for viscous potential flow model, which is a potential flow approximation of the Navier-Stokes equation for incompressible Newtonian fluids. The simulated shapes are compared to snapshots of experimentally observed drop deformations. We highlight the influence of the viscosity and charge carrier mobility of the liquid on the shape evolution of droplets and discuss the observed trends. We give an explanation as to why the observed deformation pathways of positively and negatively charged pure water droplets differ.

**Ion transport through insulating capillaries.** Self-organized charged particle guiding by insulating capillaries is modeled and simulated by our code InCa4D. Additionally, an original experimental setup is presented that allows us to monitor the accumulated total charge in the capillary and to link the charge to the transmitted fraction of the beam. In a combined theoretical and experimental study, we give evidence that the self-organized electric potential in tapered glass capillaries has the strength to focus a low energy ion beam, similar to Einzel lenses. The experimental data are corroborated by our simulations, which allows a valuable and comprehensive insight into the dynamics of the self-organized Coulomb potential in irradiated insulating capillaries.

# RESUMÉ

**Gouttelettes à la limite de Rayleigh.** Lorsqu'une gouttelette chargée atteint une charge critique, elle devient instable et se déforme spontanément. Initialement sphérique, la gouttelette s'allonge, puis émet 2 jets de matières chargés. Ici, nous modélisons et simulons l'évolution de la forme d'une gouttelette chargée et comparons nos simulations aux résultats expérimentaux. La déformation de la gouttelette est décrite en utilisant le modèle d'écoulement potentiel visqueux avec correction visqueuse (VCVPF), qui est une approximation de l'équation de Navier-Stokes pour les fluides newtoniens incompressibles et faiblement visqueux. Nos simulations mettent en évidence l'influence de la viscosité et mobilité des charges du liquide sur l'évolution de la forme des gouttelettes. Nous explications ainsi pourquoi la déformation observée pour des gouttelettes d'eau pure chargée positivement et négativement, est différente.

**Transport d'ions à travers les capillaires isolants.** Le guidage auto-organisé des particules chargées par des capillaires isolants est modélisé, puis simulé par notre code InCa4D. Un dispositif expérimentale original, qui permet de mesurer la charge totale dans le capillaire et de relier cette dernière à la fraction transmise du faisceau, est présenté. Dans une étude théorique et expérimentale combinée, nous montrons que le potentiel électrique auto-organisé dans les capillaires coniques peuvent focaliser un faisceau d'ions de basse énergie, semblable à une lentille d'Einzel. Les données expérimentales sont corroborées par nos simulations, permettant ainsi une compréhension complete de la dynamique de charge à l'interface des isolants.