E Giglio

K Tőkési

R D Dubois

Relaxation dynamics of charge patches formed inside an insulating capillary by ion impact

Keywords: capillary, insulator, charge relaxation, surface charge dynamics, multipole expansion

The dynamics of the charge distribution produced at the inner surface of an insulating capillary by low energy ion impact are investigated theoretically in the case when the entrance, exit and outer capillary surface are grounded. Starting with the surface continuity equation, that describes the charge dynamics at the inner surface, we deduced an analytical solution of the continuity equation in the form of a linear combination of surface charge moments that satisfy the corresponding boundary conditions. We determined the relaxation rate of each moment, which is given as a function of the dimensions and electrical properties of the capillary. We found an approximate expression of the time evolution of the electric potential and field generated by a charge patch during and after beam irradiation. The time evolution of the total charge is also described in detail. Our findings are illustrated with examples.

Introduction

Under charged particle impact, insulator surfaces accumulate electric charges. The latter generate an electric field that, if sufficiently strong, prevents the following beam particles from hitting the insulator surface. Ion beam transmission through insulating capillaries and their surprising guiding power has been intensively studied since its discovery by Stolterfoht et al. [1] for nano-capillaries and by Ikeda et al. [2] for macroscopic glass capillaries. The injected ions are guided by the selforganized formation of charge patches at the inner capillary walls. As a result, slow ions can pass through insulator capillaries, even when the geometrical conditions do not allow it. Experimental and theoretical studies of this topic have been summarized in three review articles [3][4][5]. While the guiding of ions by charged patches is qualitatively understood, the complex nature of the electric conduction in such insulators makes quantitative predictions still a challenging task. Indeed, for a given ion beam, the guiding is entirely determined by the dynamics of the charge patches at the surface, which in turn depend (i) on the electrical properties of insulators under ion beam irradiation and (ii) on the position and geometry of grounded electrodes. In order to get some insights into the discharge dynamics of formerly created charged patches, a theoretical study of the charge relaxation in a glass tube is performed. For a capillary, tilted with respect to the beam axis, the beam generates one or several charge patches at various places at the inner capillary surface wall. If the injected beam is blocked, the deposited charge patches relax due to conductivity of the insulator. We propose in this work to study the charge relaxation of injected charges patches during and after beam irradiation in the case of a glass tube capillary of large aspect ratio and where the entrance, the outlet and the outer surface are grounded. We give an analytical solution of the continuity equation in the form of a linear combination of surface charge moments that satisfy the corresponding boundary conditions. We deduce from the boundary conditions a general expression of the electric potential in the capillary as well as the guiding electric field. We also give the discharge of the total accumulated charge in the capillary. Finally, we illustrate the finding for the glass tube used in an experimental setup that will be described in a subsequent paper. We would like to mention that while the development in the manuscript has been done for a cylindrical shaped capillary, it can also be extended to conical shaped capillaries, which is another interesting class of capillaries.

Surface charge dynamics

We consider a cylindrically shaped glass capillary of length H, with an inner radius 𝑅 1 , an outer radius of 𝑅 2 and which has a dielectric constant 𝜀 𝑟 (see Fig. 1). The entrance and the outer surface of the capillary are electrically grounded. The underlying model considers two interfaces namely the inner glass-vacuum interface of the capillary and the outer glass-metal interface. We assume here that each projectile of charge +𝑞 that hits the surface, injects exactly 𝑞 holes at the impact point, which are immediately trapped by hole-centers at the inner surface. We thus implicitly assume that the secondary electrons, emitted from the impact point, fall back to the impact point except for 𝑞 electrons, which are picked up by the projectile [6]. In alkali glasses, the mobility of holes, electrons and anions is negligible compared to the mobility of alkali cations and the bulk conductivity 𝜅 𝑏 in those glasses is dominated by the mobile cations. We further assume 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 ≪ 𝜏 𝑏 -1 = 𝜅 𝑏 𝜀 0 𝜀 𝑟 . This implies that the charge accumulates only at the inner surface and not in the bulk and can thus be represented by a surface charge density 𝜎 [7]. Consequently, the mobile ions are driven by a divergence-free (solenoidal) electric field ∇ ⃗ ⃗ ⋅ 𝐸 ⃗ = 0 through the glass bulk. They can however accumulate or deplete at the interfaces. The dynamics of the surface charge density at the inner interface is given by the surface continuity equation,

𝜕𝜎(𝑠 ,𝑡) 𝜕𝑡 = -𝜅 𝑏 𝐸 𝑟 (𝑠 , 𝑡) - 𝜕 𝜕𝑧 [𝜅 𝑠 𝐸 𝑧 (𝑠 , 𝑡)] - 1 𝑅 1 𝜕 𝜕𝜃 [𝜅 𝑠 𝐸 𝜃 (𝑠 , 𝑡)] + γ(𝑠 , 𝑡)𝐼 inj (1)
where all quantities are taken at the inner surface, 𝑠 = (𝑅 1 , 𝜃, 𝑧). The first right hand term, which is proportional to the bulk conductivity κb, 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 𝜃 . The surface current is proportional to the surface conductivity 𝜅 𝑠 . The last term represents the injected charge per unit time and unit area, with 𝐼 inj being the beam current and 𝛾 the hole injection cross-section. The electric field 𝐸 ⃗ (𝑟 , 𝑡) = -∇ ⃗ ⃗ 𝑉(𝑟 , 𝑡) that guides the ions and drives the deposited surface charge density 𝜎(𝑠 , 𝑡) away from the production region is deduced from the potential where 𝜎 𝑝 stands for the bounded (polarization) surface charges at the inner glass-vacuum interface (𝑆 1 )and 𝜎 𝑓 for the induced free charges at the conducting outer surface (𝑆 2 ). The induced bound polarization charges 𝜎 𝑝 at the inner interface are obtained by satisfying the jump condition of the electric field through the interface separating two dielectric media, with 𝑅 1 + and 𝑅 1 -being the respectively the outer and inner region at the interface 𝑆 1 ,

𝑉(𝑟 , 𝑡) = ∫ (𝜎 + 𝜎 𝑝 )𝑑𝑠 1 4𝜋𝜀 0 |𝑟 -𝑠 1 | 𝑆 1 + ∫ 𝜎 𝑓 𝑑𝑠 2 4𝜋𝜀 0 |𝑟 -𝑠 2 | 𝑆 2 (2)
𝜀 𝑟 𝐸 𝑟 (𝑟 = 𝑅 1 + ) -𝐸 𝑟 (𝑟 = 𝑅 1 -) = 𝜎 𝜖 0 . (3)
The induced free surface charges 𝜎 𝑓 at the outer glass-metal interface surrounding the capillary (also referred to as image charges) are deduced by satisfying the boundary conditions, i.e. the potential is zero at the outer surface,

𝑉(𝑟 = 𝑅 2 ) = 0 . ( 4
)

Multipole expansion

To solve the continuity equation (1), we use a multipole expansion of the quantities involved in (1).

The electric field being divergence-free means that the potential satisfies the Laplace equation with the boundary conditions given by ( 3) and (4). Also, the entrance and outlet being grounded, the potential is zero at the entrance, 𝑉(𝑟, 𝜃, 0) = 0 and at the outlet 𝑉(𝑟, 𝜃, 𝐻) = 0. The latter conditions allow for a simple expansion of the potential by discretizing the wave number 𝑘 𝑛 = 𝑛 𝜋 𝐻 ⁄ , with 𝑛 > 0. The symmetry axis of the beam and of the tilted capillary define the x0z plane. The potential is assumed symmetric with respect to the x0z plane, so that 𝑉(𝑟, 𝜃, 𝑧) = 𝑉(𝑟, -𝜃, 𝑧). The expansion of the potential on harmonic functions differs depending on the domain. The function 𝑉 -represents the

R 1 R 2 S 2 ε r ε r S 1 S 2 E + E - S 1 V - V +
potential for the inner domain (0 ≤ 𝑟 ≤ 𝑅 1 ) and 𝑉 + the potential for the domain (𝑅

Where we introduced the auxiliary function ℎ 𝑚𝑛 (𝑟) defined as

ℎ 𝑚𝑛 (𝑟) = 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 )
, 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 order 𝑚. Similar, we expand the surface density and injection cross section defined on the interface 𝑆 1 , using the same basis as for the potential. Indeed, the grounded inlet and outlet imply that the surface density and source term are also zero at 𝑧 = 0 and 𝑧 = 𝐻, 𝜎(𝜃, 𝑧, 𝑡) = ∑ 𝜎 𝑚,𝑛 (𝑡) cos(𝑚𝜃) sin(𝑘 𝑛 𝑧)

𝑚,𝑛 (5c)
𝛾(𝜃, 𝑧, 𝑡) = ∑ 𝛾 𝑚,𝑛 (𝑡) cos(𝑚𝜃) sin(𝑘 𝑛 𝑧) 𝑚,𝑛 (5d)
Injecting equations (5a -5d) into (1) yields an equation for each moment (m,n) of the expansion,

𝜕𝜎 𝑚,𝑛 𝜕𝑡 = 𝑣 𝑚,𝑛 (𝜅 𝑏 ℎ 𝑚𝑛 ′ (𝑅 1 ) -𝜅 𝑠 ( 𝑚 2 𝑅 1 2 + 𝑘 𝑛 2 )) + 𝛾 𝑚,𝑛 𝐼 inj (6)
The primed quantities in (6) (𝑡). Finally, the dynamics of the surface charge density can be obtained by injecting (10) into the surface charge expansion (5c).

In a nutshell, we have solved equation (1) in the general case of a cylindrical capillary with grounded outer surface (including the entrance and the outlet). The dimensions of the capillary are explicitly taken into account by the factors 𝑏 𝑚,𝑛 and 𝑐 𝑚,𝑛 . The solution is given as a linear combination of time dependent surface charge moments 𝜎 𝑚,𝑛 (𝑡). From the above charge dynamics equation (10), we can evaluate the time evolution of the electric field as well as the time evolution of the total accumulated charge 𝑄 in the capillary.

Inner electric potential and field

The guiding power of insulating capillaries is due to the charge patch formation that guides the ions. A charge patch at the inner surface generates an electric field inside the capillary that deflects/guides the injected ion beam. Using relation (7) that links the surface charge moment to the potential moments, we obtain the time-dependent potential inside the capillary as a linear combination of the time-dependent surface charge moments 𝜎 𝑚,𝑛 (𝑡),

𝑉 -(𝑟, 𝜃, 𝑧, 𝑡) = ∑ 𝜎 𝑚,𝑛 (𝑡) 𝜀 0 𝑎 𝑚,𝑛 𝐼 𝑚 (𝑘 𝑛 𝑟) 𝐼 𝑚 (𝑘 𝑛 𝑅 1 ) cos(𝑚𝜃) sin(𝑘 𝑛 𝑧) 𝑚,𝑛 , 0 ≤ 𝑟 ≤ 𝑅 1 , ( 11
)
with the dimensionless coefficients 𝑎 𝑚,𝑛 defined by (12). The latter give the "intensity" of the potential field generated by each surface charge moment.

𝑎 𝑚,𝑛 = 1 𝜀 𝑟 ℎ 𝑚𝑛 ′ (𝑅 1 ) - 𝐼 𝑚 ′ (𝑘 𝑛 𝑅 1 ) 𝐼 𝑚 (𝑘 𝑛 𝑅 2 ) (12)
The time-dependent inner electric field is simply given by the gradient of the inner potential 𝐸 - ⃗⃗⃗⃗⃗ (𝑟 , 𝑡) = -∇ ⃗ ⃗ 𝑉 -. For example, let 𝑠 𝑝 = (𝑟 = 𝑅 1 -, 𝜃 = 0, 𝑧 = 𝑧 𝑝 ) be vector pointing at the center of a charge patch located at the inner surface. giving the intensity of the inner electric field for each surface charge moment. Again, the dimensions and dielectric constant of the capillary are "hidden" in the coefficients 𝑎 𝑚,𝑛 and 𝑎′ 𝑚,𝑛 .

Total charge

We may now turn to the evaluation of the total accumulated charge in the capillary. Integrating the surface charge density over the surface 𝑆 1 , yields the time evolution of the total charge,

𝑄(𝑡) = ∫ 𝑑𝑧 𝐻 0 ∫ 𝜎(𝜃, 𝑧, 𝑡)𝑅 1 𝑑𝜃 = 2𝜋𝑅 1 𝐻 ∑ 𝜎 0,𝑛 (𝑡) 𝑛 1 -cos(𝑛𝜋) 𝜋 𝑛>0 𝜋 -𝜋 = 4𝑅 1 𝐻 ∑ 𝜎 0,𝑛 (𝑡) 𝑛 𝑛>0, odd (14a)
Using the explicit expression of the surface charge moments 𝜎 0,𝑛 (𝑡) yields the general expression of the time evolution of the total charge in the capillary in the presence of a source term.

𝑄(𝑡) = ∑ 𝑞 0,𝑛 (0) 𝑛 exp (-𝑡 ( 1 𝜏 𝑏 + 𝑐 0,𝑛 𝜏 𝑠 )) 𝑛>0, odd + 𝐼 inj ∫ Γ 0,𝑛 (𝑡 ′ ) 𝑛 exp (-𝑡′ ( 1 𝜏 𝑏 + 𝑐 0,𝑛 𝜏 𝑠 )) 𝑡 0 𝑑𝑡′ (14b)
For simplicity, we introduced in (14b) the initial charge moment 𝑞 0,𝑛 (0) = 4𝑅 1 𝐻 𝜎 0,𝑛 (0) and probability Γ 0,𝑛 (𝑡) = 4𝑅 1 𝐻 𝛾 0,𝑛 (𝑡). Only angular moment 𝑚 = 0 and wave numbers 𝑘 𝑛 with odd integer 𝑛 contribute to the total charge. The selection rule on the integer 𝑛 is due to our boundary condition, i.e. the outlet is grounded. Relaxing the constraint would imply a continuously varying 𝑘 and remove the selection rule as 𝑘𝐻/𝜋 would not necessarily be an integer any more. We also note that the weight of the charge moments decrease inversely proportional with n. For smooth patches the contribution of the higher modes to the total charge becomes negligible.

Application to a borosilicate glass tube

We illustrate the theoretical development with an example. In the following we consider the case of a borosilicate capillary with the dimension like the one used in our experimental setup. The borosilicate capillary is H=43 mm long with an inner radius 𝑅 1 = 0.29 mm, an outer radius of 𝑅 2 = 2.75 mm and has a dielectric constant 𝜀 𝑟 = 4.6. Note the large difference in the ratios 𝑅 1 𝐻 ⁄ ≪ 𝑅 2 𝐻 ⁄ . We assume in the following that the variation of the charge distribution along the z-axis is sufficiently smooth within a step of 1 mm, so that the expansion in (10a-10d) can be limited to the integer 𝑁 = Int ( 𝐻 1 ) = 43 and only wave numbers, 𝑘 𝑛≤𝑁 , are considered. Further we assume that the charge distribution of a patch is well described by the first three angular moments, (𝑚 ≤ 2), namely the monopole, the dipole and the quadrupole angular moment. Note that the angular octupole moment (𝑚 = 3) can be added easily, if needed. The factors 𝑎 𝑚,𝑛 𝑏 𝑚,𝑛 and 𝑐 𝑚,𝑛 and can now be evaluated and a selection of the factors for 𝑛 ≤ 𝑁 and 𝑚 ≤ 2 is given in the table 1. Additionally, the relative variation of the factors as a function of 𝑛 < 𝑁 is shown in figure 2. The electric field generated by the monopole moments 𝜎 0,𝑛≤𝑁 is weak compared to the one generated by the dipole 𝜎 1,𝑛≤𝑁 and quadrupole moments 𝜎 2,𝑛≤𝑁 , as can be seen from the factors 𝑎 𝑚,𝑛 in table 1. Indeed, for a given integer 𝑛, the coefficients 𝑎 0,𝑛≤𝑁 of monopole term are at least 5 times smaller than those of the dipole and quadrupole term, 𝑎 0,𝑛≤𝑁 ≪ 𝑎 1,𝑛<𝑁 ≃ 𝑎 2,𝑛≤𝑁 so that the monopole terms the dynamics of the dipole and quadrupole moments can be approached by

𝜎 1,𝑛≤𝑁 (𝑡) = 𝜎 1,𝑛≤𝑁 (0)𝑒 -𝑡/𝜏 1 + 𝐼 inj ∫ 𝛾 0,𝑛≤𝑁 (𝑡 ′ )𝑒 -(𝑡 ′ -𝑡)/𝜏 1 𝑡 0 𝑑𝑡 ′ (16a) 𝜎 2,𝑛≤𝑁 (𝑡) = 𝜎 2,𝑛≤𝑁 (0)𝑒 -𝑡/𝜏 2 + 𝐼 inj ∫ 𝛾 2,𝑛≤𝑁 (𝑡 ′ )𝑒 -(𝑡 ′ -𝑡)/𝜏 2 𝑡 0 𝑑𝑡 ′ (16b)
Injecting (16a) and (16b) into (15) gives an expression that approaches the time-evolution of the inner electric field,

𝐸 𝑟 (𝑅 1 -, 0, 𝑧 𝑝 , 𝑡) ≃ 𝐸 1 𝑒 - 𝑡 𝜏 1 + 𝐸 2 𝑒 - 𝑡 𝜏 2 + 𝐼 inj 𝜀 0 ∫ 𝐽 1 (𝑡 ′ )𝑒 -(𝑡 ′ -𝑡)/𝜏 1 𝑑𝑡 ′ 𝑡 0 + 𝐼 inj 𝜀 0 ∫ 𝐽 2 (𝑡 ′ )𝑒 -(𝑡 ′ -𝑡)/𝜏 2 𝑡 0 𝑑𝑡 ′ (17)
where

𝐸 1 (𝑧 𝑝 ) = ∑ 𝑎′ 1,𝑛 𝜎 1,𝑛 (0) sin(𝑘 𝑛 𝑧 𝑝 ) 𝜀 0 ⁄ 𝑁 𝑛>0 (18a) 𝐸 2 (𝑧 𝑝 ) = ∑ 𝑎′ 2,𝑛 𝜎 2,𝑛 (0) sin(𝑘 𝑛 𝑧 𝑝 ) 𝜀 0 (18b) ⁄ 𝑁 𝑛>0
are the amplitudes of the dipole and quadrupole moment of the electric field at point 𝑧 = 𝑧 𝑝 and

𝐽 1 (𝑧 𝑝 , 𝑡) = ∑ 𝑎′ 1,𝑛 𝛾 1,𝑛 𝑁 𝑛>0 (𝑡) (18c) 𝐽 2 (𝑧 𝑝 , 𝑡) = ∑ 𝑎′ 2,𝑛 𝛾 2,𝑛 𝑁 𝑛>0 (𝑡) (18d)
the dipole and quadrupole injection cross section at the observation point 𝑧 = 𝑧 𝑝 . In the case where no beam is injected, 𝐽 1 (𝑡) = 𝐽 2 (𝑡) = 0, the relaxation of the inner electric field simplifies to a sum of two exponential functions, corresponding to the dipole and quadrupole contribution.

𝐸 𝑟 (𝑅 1 -, 0, 𝑧 𝑝 , 𝑡) ≃ 𝐸 1 𝑒 - 𝑡 𝜏 1 + 𝐸 2 𝑒 - 𝑡 𝜏 2 (19)
Expression ( 19) may be used to adjust the time-evolution of the force that deflects the ion beam. The advantage of using (19) as a fitting function of experimental data, is that it provides valuable information about the electrical conductivities of the insulating material.

We turn now to the expression of the time evolution of the total charge in the capillary. We note in figure 2 that the factors 𝑏 0,𝑛≤𝑁 ≃ 1 are close to unity, while we see in figure 3 that the factors 𝑐 0,𝑛 are monotonically increasing with 𝑛, starting with 𝑐 0,0 = 0. This differs from the 𝑐 1,𝑛 and 𝑐 2,𝑛 which are relatively constant. Also, the factors 𝑐 0,𝑛≤𝑁 of the monopole moments are much lower than the factors 𝑐 1,𝑛≤𝑁 of the dipole moment. The reason is that, for 𝑚 = 0, one of the two surface relaxation channels becomes inactive, namely the one along the angular direction. The bulk relaxation channel dominates the surface relaxation channel for all modes 𝑛 ≤ 𝑁 if the following condition is satisfied,

𝑐 0,𝑛≤ 𝑁 ≪ 𝜏 𝑠 𝜏 𝑏 𝑦𝑖𝑒𝑙𝑑𝑠 → 𝑏 0,𝑛≤𝑁 𝜏 𝑏 + 𝑐 0,𝑛≤𝑁 𝜏 𝑠 ≃ 1 𝜏 𝑏 (20)
In that case, the time evolution of the total charge reads

𝑄(𝑡) ≃ 𝑄(0) exp (- 𝑡 𝜏 𝑏 ) + ∫ 𝐼 dep (𝑡 ′ ) exp ( 𝑡′ -𝑡 𝜏 𝑏 ) 𝑡 0 𝑑𝑡 ′ (21)
where we introduced the total initial charge 𝑄(0) = ∑ 𝑞 0,𝑛 (0) 𝑛 ⁄ 𝑛>0, odd and the deposited current 𝐼 dep (𝑡) defined as the difference between the injected and transmitted current,

𝐼 inj -𝐼 trans (𝑡) = 𝐼 dep (𝑡) = 𝐼 inj ∫ 𝑑𝑧 𝐻 0 ∫ 𝛾(𝜃, 𝑧, 𝑡)𝑅 1 𝑑𝜃 𝜋 -𝜋 (22)
For borosilicate glass, the typical bulk conductivity is about 𝜅 𝑏 ∼ 1.5 × 10 -13 S/m at room temperature [8,9] yielding a bulk relaxation rate of about, 𝜏 𝑏 -1 = 3.7 mHz. The surface conductivity in borosilicate is not a well-known quantity and depends strongly on the adsorbed impurities. At room temperature, it is typically less than 10 -15 S in dry environment so that the surface relaxation rate is typically lower than 630 mHz (𝜏 𝑠 -1 ≤ 630 mHz). Surface conductivity of Pyrex ® borosilicate glass plates have been recently measured by Gruber et al. [9], who found a surface conductivity of about 𝜅 𝑠 ∼ 10 -16 S. Assuming thus a bulk conductivity of 𝜅 𝑏 ∼ 1.5 × 10 -13 S/m and surface conductivity 𝜅 𝑠 ∼ 10 -16 S, (yielding a surface relaxation rate of 𝜏 𝑠 -1 = 63 mHz) we see from table 1 that the condition (20) is only reasonably fulfilled for 𝑛 ≤ 𝑛 𝑐 = 10 . 𝑐 0,𝑛 𝑐 = 0.008 ≪ 𝜏 𝑠 𝜏 𝑏 = 0.06 (23a)

For 𝑛 ≥ 𝑛 𝑐 the condition holds less and less, and the surface relaxation channel participates in a nonnegligible way to the decay of the charge moments 𝑞 0,𝑛>𝑛 𝑐 . The trend of the factors 𝑐 0,𝑛≤𝑁 is shown in figure 2. Note that the factors 𝑐 0,𝑛 increase almost linearly for 𝑛 𝑐 < 𝑛 < 𝑁 with the slope 𝛽 = 0.002, 𝑐 0,𝑛≥𝑛 𝑐 ≃ 𝛽(𝑛 -𝑛 𝑐 ) (23b)

We will in the following try to give a general expression for decay of the initially accumulated charge in the absence of source term (no injected beam). Without losing any generality we can rewrite the decay of the accumulated charge using an auxiliary function 𝑄 𝑐 (𝑡), which corresponds to the part of the charge that decays not only via the bulk channel but also via a non-negligible surface channel,

𝑄
At this point, we have to make an assumption about the initial occupation numbers of the moments 𝑞 0,𝑛 (0) . We give here the example, where the initial occupation numbers of the charge moments decrease inversely proportional with ≥ 𝑛 𝑐 , 𝑞 0,𝑛≥𝑛 𝑐 (0) ≃ 𝑞 0,𝑛 𝑐 (0) 𝑛 (26)

In this particular case, the auxiliary function can be approached by . Noting that 𝑄 𝑐 (0) = 𝑞 0,𝑛 𝑐 (0) 𝜙(1, 𝑛 𝑐 , 2), the auxiliary function may be expressed by

𝑄 𝑐 ( 𝑡𝛽 𝜏 𝑠 ) ≃ 𝑞 0,𝑛 𝑐 (0) ∑ 𝑒 -𝑡𝛽(𝑛-𝑛 𝑐 ) 𝜏 𝑠 ⁄ 𝑛 2 = 𝑞 0,𝑛 𝑐 (0) ∑ (𝑒 -𝑡𝛽 𝜏 𝑠 ⁄ ) 𝑛 (𝑛 + 𝑛 𝑐 ) 2 (27)
𝑄 𝑐 ( 𝑡𝛽 𝜏 𝑠 ) ≃ 𝑄 𝑐 (0) 𝜙(𝑒 -𝑡𝛽 𝜏 𝑠 ⁄ , 𝑛 𝑐 , 2) 𝜙(1, 𝑛 𝑐 , 2) (28)
We obtain finally an approximated expression for the decay of the total charge, 𝑄(𝑡) ≃ 𝑒 -𝑡 𝜏 𝑏 ⁄ [(𝑄(0) -𝑄 𝑐 (0)) + 𝑄 𝑐 (0) 𝜙(𝑒 -𝑡𝛽 𝜏 𝑠 ⁄ , 𝑛 𝑐 , 2)

𝜙(1, 𝑛 𝑐 , 2) ] (29)

The quantity 𝑄(0) -𝑄 𝑐 (0) is the charge that decays purely via the bulk channel to the outer surface and 𝑄 𝑐 (0) the amount of charge located near the entrance and outlet and that decays via both channels, surface and bulk. In figure 4 we illustrate the influence of the surface decay channel on the total charge decay by showing normalized discharge curves for three different surface charge rates 𝜏 𝑠 -1 , ranging from 0 to 630 mHz. We used 𝛽 = 0.002, 𝑛 𝑐 = 10, and 𝜏 𝑏 -1 = 3.7 mHz. The percentage of the amount of charge that decays via both channels was fixed to (𝑄 𝑐 (0) 𝑄(0) ⁄ = 20%). For 𝜏 𝑠 -1 = 0 mHz, the surface channel is deactivated and all the charge decays via the bulk channel. With increasing 𝜏 𝑠 -1 , the initial decay is accelerated by the additional surface channel but tends then asymptotically to the bulk decay rate as the charge located near the grounded entrance and exit is removed. The effect becomes non-negligible for 𝜏 𝑠 -1 ≥ 630 mHz. The auxiliary function 𝑄 𝑐 (𝑡) defined by (25) depends on charge occupation numbers 𝑞 0,𝑛 (0). For a different law than (26), the auxiliary function 𝑄 𝑐 (𝑡) will be modified accordingly. In this example, we used a glass capillary with the ratios 𝑅 1 𝐻 ⁄ ≃ 4 × 10 -3 ≪ 𝑅 2 𝐻 ⁄ ≃ 5 × 10 -2 . Choosing a capillary with an outer radius 𝑅 2 of the order of 𝑅 1 would significantly reduce the factors 𝑐 𝑚,𝑛 which are associated with the surface relaxation channel. Indeed, taking for example 𝑅 2 = 0.3 mm, would reduce by 70% the dipole and by 30% the quadrupole factors associated with the surface channel while increasing in (lower proportion) the factors 𝑏 𝑚,𝑛 associated with the bulk channel. The development can also be applied to cylindrical shaped nano-capillaries in silicon sheets.

The idea was here to propose a general method for predicting the decay of the total charge as a function of the initial surface charge distribution. Expression (29) may be used, for example, to fit the decay of the total charge of a cylindrical capillary that was initially charged by a tilted ion beam as a non-negligible part of the charge is located near the grounded entrance. The charge decay deviates from a purely exponential decrease, putting boundaries on the value of the surface conductivity of the insulator.

Conclusion:

We solved the surface charge continuity equation in the case of an insulating capillary tube with the outer surface, entrance and exit grounded. The analytical solution is given as a linear combination of surface charge moments that satisfy the corresponding boundary conditions. Each surface charge moment decays exponentially with a relaxation rate depending on its mode (m,kn) and on the dimensions of the capillary. The analytical expression of the relaxation rate of each surface charge moment is given explicitly. Each relaxation rate includes contributions of the three possible relaxation channels, namely through the bulk, along the angular and along the axial direction at the surface. We showed that, depending on the mode (m,kn), the relative importance of the three channels varies. We deduced from the boundary conditions the time evolution of the electric potential and electric field due to the charge patch produced by the injected ions at the inner capillary surface. The equations given in the text are of general scope and can be used to evaluate the charge dynamics in insulating capillaries of different dimensions and even in nano-capillaries of cylindrical shape. The method developed in the paper can also be extended to conical shaped capillaries, following the same approach. We illustrated our findings by an example that shows in particular how to use our solutions for the electric field and for the total charge to fit experimental data, allowing an easier interpretation of the experimental results.

Figure 1 :

 1 Figure 1: Scheme of the capillary. The outer surface S2 is grounded. The insulating inner surface S1 carries the accumulated surface charge density 𝝈. The induced polarization charges 𝝈 𝒑 and free charges 𝝈 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 current through the bulk to the outer grounded surface (red arrows in right figure).

Figure 2 :

 2 Figure 2: Relative variation of the factors as a function of n

Figure 3 :

 3 Figure 3: Dependence of the factors 𝒄 𝟎,𝒏 on the integer n

  in (27) has been rearranged to reveal the definition of the Lerch transcendent function ϕ(𝑥, 𝑛 𝑐 , 2) = ∑ 𝑥 𝑛 (𝑛 + 𝑛 𝑐 )2 ⁄ ∞ 0

Figure 4 :

 4 Figure 4: Discharge curves Q(t) for three different surface relaxation rates namely 𝛕 𝐬 = 𝟎 (grey full line), 𝛕 𝐬 = 𝟔𝟑 mHz (dashed black line), 𝛕 𝐬 = 𝟔𝟑𝟎 mHz (black full line). The percentage of the amount of charge that decays via both channels was fixed to 20% (𝐐 𝐜 (𝟎) 𝐐(𝟎) ⁄ = 𝟎. 𝟐).

  1 ≤ 𝑟 ≤ 𝑅 2 ), with the continuity condition 𝑉 -(𝑅 1 , 𝜃, 𝑧) = 𝑉 + (𝑅 1 , 𝜃, 𝑧) , 𝑉 -(𝑟, 𝜃, 𝑧, 𝑡) = ∑ 𝑣 𝑚,𝑛 (𝑡) 𝐼 𝑚 (𝑘 𝑛 𝑟) 𝐼 𝑚 (𝑘 𝑛 𝑅 1 ) cos(𝑚𝜃) sin(𝑘 𝑛 𝑧)

	𝑚,𝑛	, 0 ≤ 𝑟 ≤ 𝑅 1 (5a)
	𝑉 + (𝑟, 𝜃, 𝑧, 𝑡) = ∑ 𝑣 𝑚,𝑛 (𝑡)ℎ 𝑚𝑛 (𝑟) cos(𝑚𝜃) sin(𝑘 𝑛 𝑧) 𝑚,𝑛	, 𝑅 1 ≤ 𝑟 ≤ 𝑅 2

  The factors 𝑏 𝑚,𝑛 and 𝑐 𝑚,𝑛 give respectively the strength of the bulk relaxation rate and surface relaxation rate for each moment 𝜎 𝑚,𝑛 . They depend on the dimensions of the capillary, and more precisely on the ratios 𝑅 1 𝐻 ⁄ and 𝑅 2 𝐻 ⁄ , as well as its dielectric constant. The general solution of equation (13) is given by the expression 𝜎 𝑚,𝑛 (𝑡) = 𝜎 𝑚,𝑛 (0) exp (-𝑡 (

			𝑐 𝑚,𝑛 =	-𝜀 𝑟 ℎ 𝑚𝑛 1 𝑅 1 ′ (𝑅 1 ) -(𝑚 2 + 𝑘 𝑛 2 𝑅 1 2 ) 𝐼 𝑚 𝐼 𝑚 (𝑘 𝑛 𝑅 2 ) ′ (𝑘 𝑛 𝑅 1 )	(9b)
				𝑏 𝑚,𝑛 𝜏 𝑏	+	𝑐 𝑚,𝑛 𝜏 𝑠	)) + 𝐼 inj ∫ 𝛾 𝑚,𝑛 (𝑡 ′ ) exp ((𝑡 ′ -𝑡) ( 𝑏 𝑚,𝑛 𝜏 𝑏 𝑡 0	+	𝑐 𝑚,𝑛 𝜏 𝑠	))	𝑑𝑡 ′ (10)
	Each moment has its proper decay rate	𝑏 𝑚,𝑛 𝜏 𝑏	+	𝑐 𝑚,𝑛 𝜏 𝑠	and source term 𝐼 inj 𝛾 𝑚,𝑛
	𝐼 𝑚 ′ (𝑘 𝑛 𝑟) =	𝑘 𝑛 2	( 𝐼 𝑚-1 (𝑘 𝑛 𝑟) + 𝐼 𝑚+1 (𝑘 𝑛 𝑟)) and 𝐾 𝑚 ′ (𝑘 𝑛 𝑅 1 ) = -	𝑘 𝑛 2	(𝐾 𝑚-1 (𝑘 𝑛 𝑟) + 𝐾(𝑘 𝑛 𝑟)) . Using
	the boundary condition (3),
			𝑣 𝑚,𝑛 (𝜀 𝑟 ℎ 𝑚𝑛 ′ (𝑅 1 ) -	𝐼 𝑚 ′ (𝑘 𝑛 𝑅 1 ) 𝐼 𝑚 (𝑘 𝑛 𝑅 2 )	) = -	𝜎 𝑚,𝑛 𝜀 0	(7)
	allows eliminating the potential moments 𝑣 𝑚,𝑛 in (6), giving a first order differential equation for the
	surface charge moments 𝜎 𝑚,𝑛 (𝑡)
			𝜕𝜎 𝑚,𝑛 𝜕𝑡	= -𝜎 𝑚,𝑛 (𝑡) (	𝑏 𝑚,𝑛 𝜏 𝑏	+	𝑐 𝑚,𝑛 𝜏 𝑠	) + 𝛾 𝑚,𝑛 (𝑡)𝐼 inj (8)
	where we have introduced the bulk relaxation rate 𝜏 𝑏 -1 = 𝜅 𝑠 𝜀 0 𝑅 1 and the dimensionless factors 𝑏 𝑚,𝑛 and 𝑐 𝑚,𝑛 defined by 𝜅 𝑏 𝜀 0 𝜀 𝑟	and the surface relaxation rate 𝜏 𝑠 -1 =
			𝑏 𝑚,𝑛 =	ℎ′ 𝑚𝑛 (𝑅 1 ) ℎ′ 𝑚𝑛 (𝑅 1 ) -𝜀 𝑟 𝐼 𝑚 (𝑘 𝑛 𝑅 2 ) -1 𝐼 𝑚 ′ (𝑘 𝑛 𝑅 1 )	(9a)

are first order derivatives with respect to the radial coordinate r, i.e.

  The time-dependent inner electric field normal to the surface (radial component) at position 𝑠 𝑝 is given by,

	𝐸 𝑟 -(𝑠 𝑝 ) = ∑ ∑ 𝑎′ 𝑚,𝑛 𝑛>0 𝑚≥0	𝜎 𝑚,𝑛 (𝑡) 𝜀 0	sin(𝑘 𝑛 𝑧 𝑝 ) ,	𝑟 < 𝑅 1 (13)
	with the dimensionless coefficients	𝑎 ′	𝑚,𝑛 = 𝑎 𝑚,𝑛	𝐼 𝑚 ′ (𝑘 𝑛 𝑅 1 ) 𝐼 𝑚 (𝑘 𝑛 𝑅 1 )

Table 1 :

 1 A selection of factors 𝒃 𝒎,𝒏 and 𝒄 𝒎,𝒏 , giving respectively the strength of the bulk and surface relaxation rate for each moment 𝝈 𝒎≤𝑵,𝒏≤𝟐 . The coefficients 𝒂′ 𝒎,𝒏 give the ratio between the surface charge moment and the associated inner electric field

	n	𝒂′ 𝟎,𝒏 𝒂′ 𝟏,𝒏 𝒂′ 𝟐,𝒏	𝒃 𝟎,𝒏	𝒃 𝟏,𝒏	𝒃 𝟐,𝒏	𝒄 𝟎,𝒏	𝒄 𝟏,𝒏	𝒄 𝟐,𝒏
	40 0.03	0.151 0.173 0.97	0.847 0.826 0.061	0.18	0.362
	30 0.02	0.158 0.175 0.98	0.841 0.824 0.042	0.176 0.359
	20 0.01	0.165 0.177 0.988	0.834 0.823 0.023	0.174 0.358
	10 0.004 0.173 0.178 0.996	0.826 0.821 0.008	0.175 0.357

5 0.001 0.175 0.178 0.999 0.824 0.821 0.002 0.176 0.357 1 10 -5 0.176 0.178 0.9999 0.823 0.821 0.0001 0.176 0.357

  Table1and figure2show that the factors 𝑏 1,𝑛≤𝑁 , 𝑐 1,𝑛≤𝑁 , 𝑏 2,𝑛≤𝑁 and 𝑐 2,𝑛≤𝑁 vary by less than 3% when n goes from 1 to N=43, so that they can be considered constant over that range. Hence, introducing the dipole and quadrupole relaxation rates,

	𝐸 𝑟 (𝑅 1 -, 0, 𝑧 𝑝 , 𝑡) ≃ ∑ [𝑎′ 1,𝑛 𝑁 𝑛>0	𝜎 1,𝑛 (𝑡) 𝜀 0	+ 𝑎′ 2,𝑛	𝜎 2,𝑛 (𝑡) 𝜀 0	]	sin(𝑘 𝑛 𝑧 𝑝 ) (15)
	1 𝜏 1	=	0.83 𝜏 𝑏	+	0.17 𝜏 𝑠
	1 𝜏 2	=	0.83 𝜏 𝑏	+	0.36 𝜏 𝑠

Acknowledgements

This work was supported by PICS N° 245 358 Hongrie 2018 founding, by Hubert Curien (PHC) Balaton 2018 program, project number 40301VK and by the Bilateral relationships between France and Hungary in Science and Technology (S&T) under the project number 2017-2.2.5-TÉT-FR-2017-00008.


Ali Nadar
email: ali.nadar@eurecom.fr

Jérôme Härri
email: haerri@eurecom.fr

An AI-as-a-Service Platform for an Artificial Intelligence of Things (AIoT)

Keywords: AI-as-a-Service, AI-of-Things, Machine learning, Ontology, Semantic, Driving Simulator, CARLA

With decentralized Machine Learning (ML) strategies and modern edge Tensor Processing Unit (TPUs), smart devices are no longer only consumers but also producers of Artificial Intelligence (AI), transforming an Internet of Things (IoT) into a global and decentralized Artificial Intelligence of Things (AIoT). With the availability of a large amount of AI actors comes not only the challenge to discover and network with them, but also the potential to use their AI capabilities as a service. However, the heterogeneity of the AI actors, their AI capabilities, AI contextual environment, mobility or even the AI characteristic available or sought requires not only a robust IoT architecture but also flexible AI semantics. In this paper, we present an AI-as-a-Service platform assisting AI consumers to identify existing AI tailored to their needs among the AIoT. We describe the architecture, the APIs, the message flows and AI semantics to identify the most appropriate AI workers when and where needed to efficiently generate AI models from distributed vehicles. As a proof-of-concept, we select an application scenario demonstrating the trainability/changeability of AI models between vehicles according to their context using the CARLA driving simulator.

I. INTRODUCTION

With advances in sensing, communication, and networking, autonomous vehicles and unmanned aerial vehicles (UAVs) are expected to play a vital role in a variety of Internet-of-Things (IoT) areas, including Industry 4.0, logistics, transportation, and public safety. Embedded with their multitude of sensors, these IoT actors are expected to be a significant source of data, which can be digested by state-of-art AI models to either improve the operation of the IoT or as functions supporting the IoT actors themselves. The inefficiency to share large amount of data in the IoT coupled with the significant energy requirement to train AI models on IoT infrastructures and the availability of modern Tensor Processing Units (TPUs) enabled an alternative strategy to distribute the AI tasks directly to the IoT actors.

Providing AI-as-a-Service (AIaaS) yet faces various challenges. First, coordination would be required for decentralized AI mechanisms to identify and cherry-pick the most appropriate AIoT entities to cooperatively perform AI tasks. Second, AI knowledge needs to be named and compared for a match between demand and supply, and AI ontologies should be defined for AI knowledge apparently named differently to be inferred actually as similar. Finally, trust and traceability should be enforced to guarantee the service outcome. In this paper, we present an AIaaS platform that allows users to play as AI actors by accessing and utilizing ML models without the need for heavy investment in infrastructure and expertise in producing such models.

Due to the diversity of user's environment, the available contexts may change over time. It is challenging to anticipate a complete set of contexts while we design a context aware system. In this paper, we propose a context modeling approach which can dynamically handle various context types and values. More specifically, our middleware platform is contextaware embedded system designed and implemented to deploy context management system for providing services to vehicles depending to their contextual information. We use ontologies to enhance the meaning of a user's context values and automatically identify the relations among different context values. Our contributions are fourfold: first we provide an overview of potential challenges to address AIaaS in AIoT; second we propose a centralized AI-as-a-Service (AIaaS) architecture called Infrastructure Assisted Knowledge Management (IAKM); third we propose a Map-based ML ontology for the metadata of AI-models and our approach for proximity metrics; finally, we conduct a tangible experiment showcasing the integration of our IAKM platform with CARLA driving simulator. The platform is available as open-source under an Apache 2.0 license and is developed as a multi-docker container platform.

The rest of this paper is organized as follows: Section III introduces AIoT, whereas Section IV, we present the IAKM platform and its components. We describe in Section V our proposed ontology and the similarity search mechanism, whereas in Section VI, we demonstrate an experiment for IAKM-CARLA implementation and integration scenario, in Section VII we summarize and conclude the paper.

II. LITERATURE SURVEY

A literature survey on the benefits and challenges of AIdriven applications in IoT domains has been performed, and the means of creating/consuming knowledge on decentralised learning for distributing knowledge, as well as on Mapbased ML ontology for vehicular systems and AI proximity metric have been proposed. In AIoT architectures literature, in [START_REF] Zhang | Empowering things with intelligence: A survey of the progress, challenges, and opportunities in artificial intelligence of things[END_REF] Zhang and Tao made a comprehensive survey of AIoT, covering AIoT computing architectures; AI technologies for empowering IoT with perceiving, learning, reasoning, and behaving abilities; promising AIoT applications; and the challenges and opportunities facing AIoT research. In [START_REF] Deveaux | On the networking of knowledge in vehicular networks[END_REF], an architecture for an envisioned Knowledge IoT Platform has been proposed, such proposal needs more elaboration for components architecture, technology selection as well as the deception of the followed methodology to manage the messages processing between different components of Edge/Client. Nevertheless, we noticed based on our survey that the concept of AI-driven applications and AI knowledge interchangeability remains on the conceptual/survey level and propose an algorithm to measure the relatedness of two terms using the knowledge base of BDpedia by computing the semantic relatedness of two terms as proximity rather than distance, as in similar ontology based approaches. Searching for data using a pure SPARQL query requires a full schematic knowledge about the knowledge graph, including nodes, properties and their relationships. Although lots of efforts have been devoted to the graph similarity search, [START_REF] Zhang | Sapper: Subgraph indexing and approximate matching in large graphs[END_REF]- [START_REF] Yang | Schemaless and structureless graph querying[END_REF], they suffer from various drawbacks, where most of them focus on the structure similarity, such as SAPPER [START_REF] Khan | Towards proximity pattern mining in large graphs[END_REF], kGPM [START_REF] Cheng | Top-k graph pattern matching over large graphs[END_REF] and Ness [START_REF] Khan | Towards proximity pattern mining in large graphs[END_REF]. SAPPER [START_REF] Khan | Towards proximity pattern mining in large graphs[END_REF] investigates the problem of approximate subgraph search allowing some edges unmatched. It does not support the vertex/edge label substitution. kGPM [START_REF] Cheng | Top-k graph pattern matching over large graphs[END_REF] proposes a graph pattern query, which allows a path to match an edge. However, it restricts that the vertex/edge labels specified in the query graph q should be exactly matched. Exploiting the neighborhood-based similarity measure, Ness [START_REF] Khan | Towards proximity pattern mining in large graphs[END_REF] and NeMa [START_REF] Khan | Nema: Fast graph search with label similarity[END_REF] try to identify the top-k approximate matches of a query graph q. Generally speaking, most of these methods concentrate on the graph structure similarity without considering the semantic similarity. However, in RDF graphs, two graph patterns may have large structural dissimilarity, such as Fig. 1 (a), (b) and(c), but they describe the identical semantic meaning for roundabout.

III. ARTIFICIAL INTELLIGENCE OF THINGS (AIOT)

Artificial Intelligence of Things (AIoT) relies on a serviceoriented IoT architecture to support decentralized large-scale dynamic AI management. In an AIoT, every component produces or consumes AI-based knowledge. It is therefore likely that an AI required by a particular service already exists in the AIoT and can be transferred rather than being recreated if it may be discovered.

Applying decentralized AI functions to IoT is however not straightforward due to peculiar AI requirements. Considering a risk assessment AI model for the roundabout scenario as described by Duncan et al. in [START_REF] Deveaux | A knowledge networking approach for ai-driven roundabout risk assessment[END_REF] and summarily illustrated in Fig. 2, AIoT resources are defined not only in the spatial domain (where) but also in the temporal domain (when). Red vehicles train as long as they remain in the roundabout, blue vehicles may train as soon as they enter the roundabout, and other grey vehicles are simply not suited for training, either because they never access the roundabout or do not have the required capabilities. AIoT capabilities must therefore be identified in space and time and match with the requirement of AI models. Accordingly, the success of AIoT will depend on services functions capable, among others, to solve the following challenges:

• Naming -what kind of AI model is it ?

• Context -where shall AI models be inferred/trained ? • Timing -when shall AI models be inferred or trained?

• Processing -how can AI models be trained ? Would these challenges be solved, an AIoT has the potential to move the AI paradigm from traditional AI-as-a-Duty (i.e. I create the AI I need) toward AI-as-a-Service (i.e. I create AI for someone else).

IV. INFRASTRUCTURE-ASSISTED KNOWLEDGE MANAGEMENT (IAKM)

We present in this section the Infrastructure-Assisted Knowledge Management (IAKM) platform, as a centralized implementation of AIaaS. The IAKM architecture consists of a server and a client entity as depicted on Fig. 3. IAKM server is accessed by An AI actor (consumer/producer) through an IAKM agent over a RESTful AIaaS-APIs. The IAKM agent may be located either on a vehicle or integrated in an external entity requiring IAKM services.

A. IAKM server

The IAKM server corresponds to the backend processing of the AIaaS framework, it consists of a message passing For data storage, it has a RDF knowledge graph database to store the metadata of AI models as well as a file database to store byte-codes AI models. Their roles are described below:

• Message Passing -provides publish/subscribe mechanisms to exchange information either between the client and the server or between different servers.

• Databases -provides a dual-storage approach, utilizing file format storage database like 'MongoDB' for AI models and graph format storage database like 'GraphDB' for context storage in RDF triples • Knowledge Management -identifies AI models available locally according to the AI engine proximity metric, or coordinate the localization and predictive caching of most appropriate AI models. • Data Access Management -maps the AI model metadata to its byte-codes based on the KM requests, also handles the transforms of graph-based queries into SPARQL/Lucene query. The IAKM server may be located either in the Cloud or at a 5G MEC or even on edge devices, and will adopt the semantic context corresponding to its location or scope (i.e. to optimize AI model caching).

B. IAKM Agent

IAKM agent could communicate with many AI actors, it manages user request in threefold; first it exposes AIaaS-API to receive request content. Second, using the semantic functions described in section V-C, it interprets the graphbased query to validate, analyse and align the query with the structure of IAKM ontology. Finally, it communicates with IAKM server microservices using publish/subscribe messaging pattern. The described AIaaS-API as depicted on Fig. 3 is an HTTP RESTful service, all requests must be in JSON/RDFformat that describes the Actor/AI context. The proposed endpoints are as following:

C. AI actor

In our proposed architecture, AI actor acts as a client for IAKM platform, it could be a consumer, producer or even as an AI worker with ML model training task. The actor communicates with IAKM agent using RESTful API requests, the request should include a graph-based query that contextually describes the needed AI, the map context and the capabilities of an AI actor.

V. MAP-BASED ML ONTOLOGY AND SEMANTIC

In fact, AIaaS framework enables AI actors to experiment with AI for various purposes without a large initial investment, these actors could belong to a various distributed and heterogeneous systems. In vehicular systems domain, every stakeholder has its own representations of knowledge, concepts and proprieties. Therefore the data interpretation, interoperability/interchangeability between the different AI actors becomes challenging due the semantics ambiguity and non-shared vocabularies. Therefore, a common knowledge representation method is necessary to harmonize this diverse data by providing a common framework for interpretation. In this section, we focus on ontology/semantic activities; first we define our domain of consideration; second we design an ontology for IAKM platform; finally we describe our approach to implement the AI semantic similarity search.

A. Knowledge Domains

Without loss of generalities, we define in this work an IAKM ontology that includes terms and naming related to the application of machine learning in vehicular systems. For this purpose, we describe the main domains/concepts required to create such ontology: "Model" is the conceptual class for all instance of AI models,"Framework" is the machine learning framework required to manage AI models. "Algorithm" is a set of mathematical processes or techniques by which an AI system conducts its tasks. "Dependency" describes the required inputs and the expected output for AI model. "Dictionary" is the acronyms, nicknames and phrases for which an interpretation is required. "MLType" identifies the type of machine learning (supervised, unsupervised, reinforcement learning ...). "Actor" (respresents the IAKM client, it could be) is all types of AI actors who created and improved the available resources to facilitate the use and application of Machine Learning the information of actors managing AI models. "Map" describes the contextual topology on which an AI application (model) could be used.

B. Ontology

Braga et al. [START_REF] Braga | A machine learning domain ontology to populate knowledge base to support intelligent agents working in autonomous systems domains of the internet infrastructure[END_REF] proposed an ontology adapted to Machine Learning, which provides a natural structuring of ML parameter. Zhao et al. [START_REF] Zhao | Core ontologies for safe autonomous driving[END_REF] presented a core ontology for autonomous driving. Inspiring by their contributions, in our study as depicted on Fig. 4 we build an IAKM ontology that combines the two ontological aspects. On top of it, we define 3 additional entities which are directly related to our IAKM platform; Actor to store actor capabilities and information as well as to register/unregister the availability of AI workers according to their context, Model to bind entities and store model metadata and Map to describe the map context correspond to the AI model.

C. Semantic Search

AI semantics represent a critical function of AIaaS, and needs to remain flexible enough to identify the most proximate AI, when the request is either incomplete or an exact AI model is not available. Compared to IoT or Named Data Networking (NDN) semantics, it is highly unlikely that a globally agreed way to describe an AI model or its usability context could be defined, considering that any AIoT component may produce any model under any condition. However, ontologies and knowledge graphs are designed not only to describe content but also to capture and infer relationships between data elements. In the IAKM, semantic search mechanism is a data search technique which uses the intent and contextual meaning behind a search query to deliver more relevant results. In this work, we present our methodology for searching the best fitted AI model according to its metadata stored in graph database and therefore find the unique identifier of AI data as stored in file database. In Fig. 5, we describe our approach for AI semantic search in IAKM platform According to Fig. 5, the graph-based query passes through multiple functions across IAKM component before being landed at the SPARQL query endpoint. we describe below the processing functions as the following: -In our approach, for storing/retrieving AI metadata using query-based graph, every node in the graph should be an instance of a class included in IAKM ontology concepts, in other RDF term every instance should has "rdf:type" predicate to one class within Model, Algorithm, Framework, Dependency, MLType and Actor, otherwise query graph must be rejected. We except the Dictionary class as it is not instantiable and considered as a static class that contains all acronyms definitions.

2) Context Map Alignment Function (CMAF): -In automotive industries, car manufacturers provide HD-maps, which may have their own structure, describing the road map, road structure, and spatial composition based on a existing road centerline concept. In such heterogeneous map description vocabularies context, the IAKM platform uses semantic matching to find the structural commonalities between the user's map context expression and the IAKM ontology definitions. To this end, IAKM platform presents a semantic matching function called Context-Map Alignment Function (CMAF) that uses Natural Language Processing (NLP) to align large language expressions with IAKM ontology. For this purpose, we use NLP-as-a-Service powered by OpenAI API with the large model gpt-3.5-turbo, this model could semantically select the closest defined ontology concept to the unknown requested map context expression.

semM atch(gpt, x, nodes) = node if x ∈ Roads ∅ if x / ∈ Roads
The function "semMatch" takes as parameters; 'gpt' for openAI model, 'x' for user unknown expression and 'nodes' for list of all map context sub-classes of IAKM ontology. For instance; CMAF could ask openAI with this message: "in one word, in roads context, which one is closer and could mean 'circular intersection': Lane, Junction, Intersection, Roundabout, Turn, RoadSegment. In this case the function matches the term "circular intersection" with the ontology class "Roundabout", which has semantically the same meaning. In contrast, for tricky expressions in road context, such as 'circular computer', the function catchs the unrelatedness by answering: "None, it's not a perfect match for a 'circular computer' in a roads context"

3) Literals Processing Function (LPF): -This function helps to make inference-like process by finding the common representation between the RDF global-KG attributes with query graph attributes (RDF literals). In our study, we use the exact match for attributes similarity as we keep LPF in conceptual level.

4) SPARQL & Lucene:

-In IAKM server,we propose a similarity search technique using SPARQL query with Lucene connector to determine how relevant a given stored AI model is to a consumer query. Lucene uses the Boolean model to first narrow down the documents that need to be scored based on the use of Boolean logic in the query specification. GraphDB Lucene engine is a Full-Text Search (FTS) using native Lucene queries. Technically, our proposed similarity search is in 2 dimensions; first, transforming the graph triplets into SPARQL query constraints in order to find a similar representation in global IAKM RDF graph. Second, transforming the query literal attributes into FTS terms in order to find the most relevant AI metadata available in IAKM RDF graph by leveraging graphDB Lucene connector.

VI. IAKM IMPLEMENTATION WITH CARLA

Having established the theoretical outlines for IAKM architecture, map-based ML ontology and semantic search mechanism, our focus now shifts to presenting a clear experiment that highlights the IAKM platform's capabilities. Our approach involves leveraging the capabilities of the CARLA driving simulator to animate the interaction between IAKM and AI actors within the virtual environment when an AI actor is symbolically represented using a CARLA vehicle. CARLA is an open-source simulator for autonomous driving research, it has been developed from the ground up to support development, training, and validation of autonomous driving systems. In the following subsections we describe the implementation steps for the experiment; First, we devise a ML control model that caters to different types of road junctions, then we delve into the requests processing among CARLA vehicles acting as AI actors, adopting consumer/producer roles and the IAKM server, and finally we explain the results of experimenting with the proposed traffic scenarios.

A. ML model for dynamic decision control

To ensure the realism of our experiment in simulating AI exchange between CARLA vehicles and the IAKM platform, it is imperative to utilize an authentic machine learning model. Without loss of generalities, in this work we introduce a simplified ML model designed to make decision controls for vehicles while nearing a complex traffic situations like intersection, roundabout, etc.. Thus, we opted for the Random Forest Classifier (RFC) due to its robust ensemble learning capabilities. Ensemble methods, such as RFC, are known for their ability to combine multiple weak learners to enhance overall model performance. This is particularly advantageous in our context of vehicle decision control, where the system can benefit from the collective intelligence of diverse decision trees within the random forest.

The proposed ML model aims to avoid collision, as shown in Fig. 6, we identify a Collision-Spot in road way-points where the paths of the two vehicles intersect, and then we define a control area (in red) where the ego vehicle must exercise caution, specifically by according priority to incoming vehicles. Building upon this identification, we can now precisely define four key features that form the foundation of our model: vehicle within the red control area. While ego is in control area, data collection is performed at each time step to capture the 4 input features and the control tag as output label for RFC model, we pick up the label according to the mapping table using the pair of (throttle & break) of actual dynamic control. Once ego vehicle leaves the control area, ego vehicle trains its ML model using the collected dataset where the combination of features helps the model learn patterns and make accurate predictions. For the categorical output label, we identify 15 dynamic control labels as showing in

B. IAKM integration with CARLA driving simulator

The designed ML model serves as a key integration for the systematic exchange process between the AI actors (CARLA vehicles) and the backend server (IAKM). In Fig. 9a, [veh-A] is considered as human-driven control and already has a non trained RFC ML model for driving control, while approaching a roundabout inside the control area, it collects the dataset (4 features, 1 label), then by exiting the control area it trains the model with the dataset, and as AI producer it pushes the model to IAKM server according to its context as depicted in (Fig. 7). At another point in time, [veh-B] upon nearing a road junction, it recognizes the complexity and lacks an ML model to aid in decision-making, it acts as AI consumer and asks IAKM server for AI model according to its context. At IAKM server, upon request receipt, it activates a similarity search mechanism to locate the closest model, noticing that in our experiment we consider 70% as the similarity threshold for SPARQL/Lucene search in graphDB for context closeness of consumer's graph-based query. Upon receiving a response from the server, the vehicle dynamically adapts its behavior: if the server provides a high-scored (bigger than 70%) machine learning model, the vehicle incorporates it for decision-making; otherwise, it employs a waiting strategy as a benchmark for control, wherein it remains stationary until the surrounding environment is devoid of other vehicles, ensuring a safe and opportune moment to cross the junction, as shown in Fig. 9b [veh-C] remains in a halted state until the blue zone is clear of incoming vehicles, ensuring a safe and obstructionfree environment before proceeding, and consequently leading to a traffic queue formation behind it.

Semantically speaking and according to our defined ontology, as a producer [veh-A] produces his trained ML model along with its context as a graph-based query as illustrated in Fig. 7, this graph/model should be stored in IAKM server at graphDB/fileDB respectively, the graph describes the semantic context that a vehicle as a consumer should fit in order to receive and consume the relevant ML model. To construct the graph, the producer instantiates all entities in IAKM ontology as following; for [MapContext], according to the producer vocabulary, he describes his map context as round intersection, for [algorithm] the ML type is "RandomForest", for [framework] the package is "sklearn", and for [dependency] context the model has four features and one output label as shown in Fig. 6, and finally it connects all instances to the core model instance, the latter has "control" attribute for its [type] and "NIL" for its [unique-ID] (UID). We note at this stage that model UID is marked as "NIL" in order to be assigned to graph in IAKM server level after the model storage in fileDB. Once producer sends his model/graph-based query, the query should be successfully processed in semantic functions of IAKM agent. For the case of [veh-A], the MapContext is described with round intersection which is not defined in IAKM ontology. For this reason, CMAF makes a processing in map alignment with IAKM ontology by replacing the unknown round intersection entity with the defined roundabout entity in IAKM ontology. Finally, IAKM server receives the producer message, and save consequently its ML model in fileDB and query graph at its RDF graphDB. At a later stage, an AI consumer [veh-B] while approaching the T-intersection, it realises the complexity of the situation where no local AI model is available to make a control decision. Therefore, it generates graph-based query by defining the map-context as "junction", and in similar way for other entities like in producer case but with more generic to simplify the request and to emphasis on similarity search process to find best fitted AI model. Likewise, at the IAKM agent level, the query graph passes through the semantic functions where QVF validates its structure, CMAF finds the structure conformity of "junction" with IAKM ontology and for LPF function we skip it as we keep the module at a conceptual level and we will define and implement it in future work. Once IAKM server receives the query, the DataAccess manager activates the similarity search process using SPARQL engine with Lucene embedded to determine how relevant a stored AI models is to a consumer graph query. Lucene uses the Boolean model to first narrow down the documents that need to be scored based on the use of Boolean logic in the query specification. As illustrated in Table. III, we apply the similarity search in 2 phases; in first we make sure that query structure is aligned with the proposed ontology, and second we applied Lucene connector functionality to find the most relevant AI metadata. we see more detailed description in section VI-C.

C. Experimentation

As we previously mentioned, we use CARLA simulator to incorporate a traffic scenario integrated with IAKM platform where a producer [veh-A] and a consumer [veh-B] as illustrated in Fig. 9a could precept their surrounding environments. We start the simulation by spawning a considerable number of auto-piloted vehicles, then we spawn [veh-A] in autopilot mode before the red control area of the "round intersection" in order to collect a full featured dataset for its dynamic control during his existence in the red control area, upon leaving the designated area [veh-A] initiates ML model training and subsequently pushes the trained model to the IAKM server specifying "round intersection" as the map context in a graphbased query. In a later step, we spawn [veh-B] without any AI model, also in autopilot mode at a random position before the control area of "junction", then by approaching this "junction", it detects the complexity of such situation and sends request to IAKM server to get the best fitted AI model and indicating "junction" as map context in its graph-based query.

On IAKM server, the DataAccess service activates the query transformation from RDF graph format to SPARQL query format. As depicted in table.III the similarity search approach is bi-dimensional:

• SPARQL constraints; by adding graph instances and its relations as SPARQL constraints to ensure that all resulting models adhere to the graph structure of the query. • Lucene FTS; by adding all attributes of query graph into the Lucene FTS field as a chain of (key:value) pairs In SPARQL constraints, logically speaking and as we defined in our ontology for MapContext, every roundabout is a junction, but not every junction is a roundabout. Accordingly, inference might be very complicated in SQL/NoSQL database. However being a semantic database, graphDB leverages OWL2-RL rulesets for advanced inferences, allowing it to implicitly identify relations through graph reasoning. In the proposed IAKM platform, the similarity result may yield zero or more pairs of "model-id" and "scores", if bigger than zero, IAKM server selects the best scored model-id and finds its associated AI model in fileDB in order to send it to the consumer, otherwise if zero, it means the query context does not match, then it replies with NIL response to the consumer as shown in Fig. 9b.

VII. FROM THEORY TO APPLICATION: IAKM KEY

CONTRIBUTIONS AND CONCLUSIONS We introduced in this paper the potentials of AI-as-a-Service for an AI-of-Things (AIoT) and presented an architecture for the Infrastructure Assisted Knowledge Management (IAKM) platform as a centralized implementation of an AIaaS framework, wherein we designed an integrated mapbased ML ontology tailored to accommodate diverse AI-driven applications within vehicular domains. To address semantic nuances, our research introduced a framework outlining the methodology for the storage and retrieval of AI models based on contextual information and leveraging an integrated strategy, we employed the SPARQL/Lucene engine for similarity search mechanisms while relying on the semantic alignment prowess of the OpenAI large language model to harmonize diverse context vocabularies. Additionally, we proposed the implementation of a RESTful API to enable seamless and decoupled communication between AI actors and IAKM agents. Furthermore, to validate the effectiveness of our approach, we conducted a real experiment using the CARLA driving simulator, specifically focusing on machine learning support in an AI-driven junction risk assessment scenario. The platform is provided as open-source, the full code and video for the IAKM implementation and its integration with CARLA in a concrete junction management scenario is available on EURECOM gitlab: gitlab.eurecom.fr/cats/carla-iakm

Figure 1 :

 1 Figure 1: Diverse semantic graph structures

Figure 2 :

 2 Figure 2: Space & time AIoT resources for AI functions.

  capability

Figure 3 :

 3 Figure 3: IAKM actor/server architecture for AIaaS

Figure 4 :

 4 Figure 4: Proposed IAKM Ontology

Figure 5 :

 5 Figure 5: Semantic processing & Similarity search

Figure 6 :

 6 Figure 6: AI model for intersection management

Figure 7 :

 7 Figure 7: graph-based query to store AI metadata for veh-A (Producer)

Figure 8 :

 8 Figure 8: graph-based query to find AI metadata for veh-B (Consumer)

  Figure 9: CARLA-IAKM scenarios

Table I :

 I AIoT resources for AI functions

			status(t)	AI role(t)
	A	able to use	asking to use AI-model	AI consumer
	B,C	able to train	training AI-model	AI producer
	D	able to train	ready for training AI-model	AI producer
	E	not-able to train	No computation resources	-
	F	not-able to train	unfitted vehicle class	-
	G	able to train	can train in roundabout	-
	H	able to train	exited roundabout	-

•

  Register -a client (AI actor) announces his availability to train AI model for a given context. It sends a registration request indicated by PUSH method and http://agenthost:agent-port/actor available train/{context} as url • Unregister -a client (AI actor) removes his availability for a given context, either for not more fitting his previous context or for saving resources.

	It sends a remove registration request indicated
	by DELETE method and http://agent-host:agent-
	port/actor available train/{context} as url
	• Find -while approaching a complex situation, an AI
	actor (i.e. Consumer) asks for an AI model accord-
	ing to a giving context. It sends a search request
	indicated by GET method and http://agent-host:agent-
	port/ai model/{context} as url.
	• Store -a client (Producer) already owning a trained
	AI model for a given context, it sends a save request
	indicated by PUSH method using http://agent-host:agent-
	port/ai model/{context} as url and the AI model byte-
	codes in the message body.

Table . II

 .

	Dynamic Control
	Throttle	Break	Label
	0.0 < t ≤ 0.1	0	t1
	0.1 < t ≤ 0.2	0	t2
	.....	.....	.....
	0.6 < t ≤ 0.7	0	t7
	0	0	n
	0	0.0 < b ≤ 0.1	b1
	0	0.1 < b ≤ 0.2	b2
	.....	.....	.....
	0	0.6 < b ≤ 0.7	b7
	* RFC model has 15 labels: 7/7 levels for throttle/break and neutral

Table II :

 II Mapping control(throttle,break) to model target(label)

Table III :

 III Similarity Search SPARQL/Lucene Full query PREFIX luc: http://ontotext.com/connectors/lucene PREFIX luc-index: http://ontotext.com/connectors/lucene/instance PREFIX iakm: http://example.org/IAKM.owl SELECT ?model ?score { ?search a luc-index:iakm-lucene-connector; luc:query "E2"; luc:entities ?model . { E1 } }

	Query Level	Expression
		?model rdf:type iakm:model .
		type : control
	E2	TTC-ego : float
	Lucene Search	TTC-other : float
	(attributes)	speed : float
		distance : float
		label : float

?map rdf:type iakm:junction . E1 ?dependency rdf:type iakm:dependency . SPARQL Search ?inputs rdf:type iakm:inputs . (instances ?outputs rdf:type iakm:outputs . & ?model iakm:hasMap ?map . relationships) ?model iakm:hasDependency ?dependency . ?dependency iakm:hasInputs ?inputs . ?dependency iakm:hasOutputs ?outputs .