2010 International Conference on Solid Dielectrics, Potsdam, Germany, July 4-9, 2010
On the transport of potassium ions
through borosilicate glass
A combined experimental and theoretical study
M. Schäfer, K. Lange, N. Rohman, A Schlemmer, H. Staesche, B. Roling, K.-M. Weitzel
Fachbereich Chemie, Physikalische Chemie, Philipps Universität Marburg
35032 Marburg, Germany
[email protected]
back side electrode is connected to an electrical high vacuum
feed through and fed into a home-made electrometer amplifier.
The current detected at this backside electrode is subsequently
A/D converted and processed in a personal computer. The
kinetic energy of the ion beam in the laboratory frame is given
by the repeller potential UR applied to L1 which can be adjusted
between about 50 Volt and 3000 Volt. L2 is always set to -1500
Volt, L3 is used for refocusing the ion beam, L4 (defining two
apertures spaced by 26 mm) and L5 (ring electrode) are set to
ground potential, the distance between the L4 and L5 is 2 mm.
In general the setup provides the possibility to investigate the
time evolution of the current. That means switching either the
intensity or the kinetic energy of the ion beam leads to a
material characteristic response of the back side current.
However, all measurements reported in this work have been
performed under stationary equilibrium conditions. The
effective input current impinging onto the sample is measured
by replacing the sample by a blank metal plate.
The transport of potassium ions through potassium borosilicate
glass has been investigated experimentally by measuring the back
side current induced by ion bombardment of the front side. The
experimental data are compared to calculations employing the
Nernst-Planck-Poisson model. The analysis allows deriving the
bulk ion conductivity respectively diffusion coefficient of the
material. The role of interface transport is discussed.
Ion conductivity;
equation (key words)
electrodiffusion;
I.
Nernst-Planck-Poisson
INTRODUCTION
The ion conductivity of glassy material is the basis for
several technical applications in the field of energy conversion
and storage. The generally accepted model for ion transport in
ion conducting glasses implies the thermally activated hopping
of ions between different sites separated by an activation
barrier [1]. Currently established techniques for determining
ion conductivities include the impedance spectroscopy
[2],[3],[4], the radioactive tracer technique [5] and pulsed field
gradient NMR [6]. Here, we describe an alternative approach
based on measuring the macroscopic transport of ions through
the material of interest.
II.
The glass sample investigated (K2O * SiO2 * 2 B2O3) has
been prepared by heating a stoichiometric mixture of K2CO3,
B2O3 und SiO2 for 2 h to about 1100 °C. The molten mixture is
cooled by pouring the material into a cylindrical steel pot with
2 cm diameter. It is annealed at 433 °C for five hours and then
cooled at a rate of 0.5 °C per minute to room temperature.
Afterwards the sample is cut into slices of approximately 800
µm thickness. Finally the surface of the disk is thoroughly
polished using SiC to the desired thickness of 100 µm to 250
µm. The thickness d of the sample used was 175 µm, the
temperature 50°C.
EXPERIMENTAL APPROACH
The experimental setup constructed for this investigation
consists of an ion source for production of a c.w. potassium ion
beam, an electrostatic ion guide and a sample holder connected
to an electrometer amplifier. The entire setup shown in Fig. 1 is
housed in a high vacuum chamber operated at a pressure of
about 10-6 mbar. The ion source employed is based on a homemade thermionic emitter consisting of a mixture of potassium
aluminosilicate (synthetic Leucite) with molybdenum [7]. This
source provides a continuous ion current of up to 20 nA. The
ion beam is guided towards the sample, which is mounted in an
electrically insulating holder. The back side of the sample is
mechanically pressed onto a stainless steel electrode. The front
side of the sample is covered by a ring electrode with central
opening of 2 cm diameter. The actual area A bombarded by the
ion beam is also circular with a diameter of 1 cm, enforced by
apertures mounted directly in front of the sample. In the
experiment ions are either adsorbed to the surface and then
transported into and through the bulk material or implanted into
the sample and then transported through the bulk material. The
Figure 1. Scheme of the experimental setup.
978-1-4244-7944-3/10/$26.00 ©2010 IEEE
1
III.
r
∂n
∂J
=−
∂t
∂z
THEORETICAL MODEL
In order to gain deeper insight into the transport processes
leading to the experimentally observed ion current at the
backside electrode, we have performed numerical simulations
of the ion transport. More specifically, we have setup a model
based on the Nernst-Planck-Poisson (NPP) equations.
According to the Nernst-Planck (NP) theory, the flux J of ions
transported per unit time through an area of interest is given by
v
uv Ze ⎞
⎛ uv
J = − D ⎜ ∇n + n ∇φ
⎟ ,
k BT ⎠
⎝
(1)
.
⎛ n − n n Δz + n Δz φ − φ e ⎞
Ji,i +1 = −Di ⎜ i +1 i + i i +1 i +1 i i +1 i
⎟
⎜ Δz
Δzi,i +1 kT ⎟⎠
2Δzi,i +1
i,i +1
⎝
(J − J )
d
n i (t) = i +1,i i,i −1
dt
Δz i
As shown by Eq. (8), the potential in increment i depends
on the ion current in the increment itself as well as on the
potentials in the neighboring increments i+1 and i-1. This
feature leads to the self-consistency problem that was already
mentioned above. If we assume the sample initially is electroneutral, the potential is zero everywhere. Subsequently, the
implantation of ions for example in the first increment leads to
a modification of the potential there. At the same time, the
modification of the potential in the first increment implies that
the potential in the second increment has to be calculated again
(since the potential in the second increment depends on the
potential of the first increment). A recalculation of the potential
in the second increment implies that the potential in the first
increment has to be calculated again and so one. We solve this
self-consistency problem by subsequently calculating the
potential (8) for increments 1 to nz several times until the
potential converges to values that are self-consistent with the
ion distribution in the glass. Convergence can be enhanced if
we keep the time step small such that the ion distributions
changes only marginally between two subsequent time steps
and take the converged potential of the previous time step as
starting point for the calculation in the actual time step.
(3)
where ε 0 is the vacuum permittivity and ε is the dielectric
function of the borosilicate glass.
We assume the ion beam shines homogenously on a circular region of the glass sample. The sample is shaped like a disk
with diameter much larger than the sample thickness. Hence,
only transport perpendicular to the sample surface has to be
taken into account, transport in radial direction may be neglected. Since the emission of ions other than potassium (K+)
by the emitter is negligible [7] and the electronic conductivity
of the borosilicate glass is much smaller than the ion
conductivity, we may exclusively concentrate on the transport
of potassium ions, consequently Z = +1. Additionally, we
assume isotropic material properties such that the dielectric
function can be expressed by a dielectric constant ε ≡ ε r = 10 .
As a consequence of these approximations, the NPP equations
simplify to the expressions
,
(4)
⎡ ∂ 2 φ(z) ⎤
ε0 εr ⎢
= −e ( n(z) − n bg ) ,
2 ⎥
⎣ ∂z ⎦
(5)
(9)
where Δzi is the size of increment i and Δzi,i+1 = 0.5 (Δzi +
Δzi+1). In general we do not restrict all Δzi being identical. The
diffusion coefficient Di is assumed to be the same in the entire
bulk sample, Dbulk. In general, the first increment on the grid is
assigned a different diffusion coefficient reflecting the
transport through the interface layer. Under the conditions
relevant to the current work this effect is negligible. See section
V for further discussion
As the ions themselves give rise to an electric potential, the
ion distribution inside the glass and the potential must be
calculated in a self consistent manner. Ultimately, the electric
potential has to fulfill the Poisson equation
v
e ∂φ(z) ⎤ v
⎡ ∂n(z)
J = −D ⎢
+n
ez
kT ∂z ⎥⎦
⎣ ∂z
(7)
⎛ e (n i − n bg ) Δz i Δz i +1,i Δzi,i −1 + ε 0 ε r ( φi +1 Δz i,i −1 + φi −1 Δz i +1,i ) ⎞
φi = ⎜
⎟
⎜
⎟
ε 0 ε r (Δz i,i −1 + Δz i +1,i )
⎝
⎠
(8)
(2)
uv
uv
ε0 ∇ ⋅ ⎡⎣ε∇φ⎤⎦ = −e Z(n − n bg ) ,
(6)
A. Discretization
In order to solve Eqs. (4)-(6), the z-axis of the sample is
discretized into nz increments and an ion density ni as well as a
potential φi is assigned to each of the increments. The ion flux
between
the two adjacent increments i and i+1 is given by
v
J i,i +1 . The discrete NPP equation are thus given by
where D is the diffusion coefficient. The first term describes
the diffusion of the ions due to the ion-density gradient
uv
∇n while the second term introduces the migration of the ions
uv
under the influence of a potential gradient ∇φ . The ion density
n contains the potassium ions injected as well as the
background potassium ions, nbg, present in the sample prior to
ion bombardment. The combination of the two contributions to
transport is termed electrodiffusion throughout the rest of this
work. The ion charge is given by Ze, kB is Boltzmann’s
constant and T the temperature. The time dependence of the ion
distribution enters via Fick’s second law
uv v
∂n
= −∇ ⋅ J
∂t
.
B. Boundary conditions
To consistently calculate Eq. (8), we need to define how the
potential behaves at the boundary of the glass sample. The
backside of the glass sample is in contact with a metal
electrode which is put to ground level. Neglecting contact
2
potentials, we may assume that the potential at the rear end of
the sample is φrear = 0. The potential at the front end of the
sample is the surface potential which is ultimately defined by
the ion distribution at the surface and inside the bulk material.
As the ions in the first increment define the ion density at the
surface and electrodes in front of the sample are far away and
thus should hardly influence the potential just in front of the
sample, we may set φfront = φ1 .
C. Backside current
We assume that the ions are neutralized as soon as they
reach the backside of the sample. Therefore, the current
detected at the backside electrode is given by
( UR − φ1 ) I
UR
0
(12)
where I0 is the experimentally measured blind current. This
leads to an additional source term in the Nernst-Planck
equation
(U − φ )
∂n
= R 1 I0
∂t source AΔzeUR
.
I0,A
0
0
400
800
Repeller voltage [ V ]
1
Iback(I0,C)
Iback(I0,B)
Iback(I0,A)
200
400
600
800
In the following, we present the results of a NPP analysis as
outlined in section III. The following information has been
used as input to the calculation. The blind current is directly
taken from the experimental measurement assuming it is
homogenously distributed over the relevant bombardment area.
The background ion density of the material is nbg = 7 · 1021
ions per cubic centimeter as given by the glass composition.
The only adjustable parameter is the effective bulk diffusion
coefficient.
(13)
The continuous transport of ions from the emitter to the
glass together with the electrodiffusion of ions through the
glass leads to a dynamic equilibrium of the ion distribution and,
thus, to an equilibrium back side current after some seconds. In
the following, we compare such steady state currents to the
experimental data. A more elaborate description of the
theoretical approach will be presented elsewhere [8].
IV.
2
Evidently, the slope of the current – voltage curve is small
between 50 and about 200 Volt but increases significantly
beyond this voltage. In order to get a first picture of the
transport characteristics, we assume that the ions are adsorbed
to the surface of the dielectric material such that the surface
potential, Usurf, is identical to the repeller voltage. This situation
would apply, if there are enough surface sites available and no
decay process of surface charges is operative. Note, that this
assumption places an upper limit to Usurf. In this situation, the
voltage drop across the sample would also be given by UR,
since the backside is basically operated at ground potential. By
assuming the transported current is simply field driven, i.e.
assuming Ohm’s law, the slope of the current – voltage curve
can be translated to a conductivity σ. In Table 1 we present the
effective conductivity (σeff,exp = σ·d/A) derived for the region
between 600 and 800 Volt. For comparison, the specific
conductivity measured by impedance spectroscopy is 4 · 10-14
S/cm, compatible with the data in Table 1.
(11)
,
4
Figure 2. Experimental current – voltage curves. The inlay displays the
blind current without a sample.
Thus, the ion current reaching the sample surface is given
Iin =
I0,B
Repeller voltage [ V ]
D. Ion source
When the potassium ions leave the emitter, they are
accelerated toward the sample via the repeller voltage UR. As a
consequence, the ions reaching the sample surface charge the
surface and lead to a surface potential that is self consistently
calculated by Eq. (8) such that the effective kinetic energy of
the ions reaching the surface is reduced corresponding to
by
Experiment
I0,C
6
0
with the electron charge e and the sample area A.
E kin = e ( UR − φ1 ) .
2
8
0
(10)
I B = e A J nz,nz +1 ,
blind current [ nA ]
back side current [ nA ]
3
One of the first results of the theoretical analysis pertains to
the surface potential. The actual surface potential is in general
smaller than the repeller potential due to two physical effects.
First, the transport of ions through the interface and
consequently through the bulk reduces the surface ion density.
RESULTS
TABLE I.
The current induced by the transport of potassium ions
through borosilicate glass has been measured for three different
blind current functions (see inlay of Fig. 1) corresponding to
average input currents of about 1.5 nA (I0,A), 3 nA (I0,B) and 6
nA (I0,C). As shown in Fig. 2, for all three input currents the
backside current increases with increasing repeller voltage.
EFFECTIVE CONDUCTIVITY OF THE BOROSILICATE GLASS
Effective conductivities in S/cm
I0,A
3
-14
σeff,exp
1.65 · 10
σ eff,theo
1.78 · 10-14
I0,B
I0,C
-14
8.02 · 10-14
3.45 · 10-14
5.30 · 10-14
3.67 · 10
back side current [ nA ]
3
number of reports in the literature demonstrating an increase of
ionic conductivity upon ion implantation. Most of these
experiments have been performed at much higher kinetic
energies than used in this work and are consequently defect
related [9],[10]. In the current work, on the other hand, the
conductivity directly scales with the ion number density
implanted.
Numerical simulation
2
Iback(I0,C)
Iback(I0,B)
Iback(I0,A)
1
As mentioned above, our theoretical model includes the
possibility to take into account the effect of interface transport.
We have carried out a limited number of calculations
employing an interface diffusion coefficient several orders of
magnitude smaller than the bulk diffusion coefficient. This
introduces a new bottleneck to the transport process. Since
such a small diffusion coefficient is not able to explain the
entire range of experimental data we have to assume that above
a certain kinetic energy implantation of ions into the material is
effectively bypassing the interface layer. Consequently, the
transport of the ions through the bulk material is the rate
determining step. This model leads to smaller calculated slopes
in the current – voltage curves at low repeller voltages. We do
have clear evidence for this scenario from additional
investigations [11].
0
0
200
400
600
800
Repeller voltage [ V ]
Figure 3. Theoretical current – voltage curves.
This is intrinsically taken into account in our analysis.
Second, a certain fraction of ions deposited onto the surface
can move parallel to the surface towards the ring electrode.
Both effects lead to a lowering of the surface potential. The
difference between the repeller potential and the actual surface
potential defines the kinetic energy with which the ions hit the
surface. Very clearly, this difference increases with increasing
repeller voltage.
To summarize, we have described a new approach for
investigating the conductivity respectively the diffusion
coefficient of a borosilicate glass by means of measuring
currents induced by ion bombardment. We are confident that
the technique can be further improved in the near future. In
particular we will focus on the attempt to determine interface
diffusion coefficients directly as a bottleneck in a macroscopic
transport process. Work along this line is in progress in our
laboratory. The actual approach employed here intentionally
deviates from electro-neutrality. It seems rewarding to study
the influence of ion injection on conductivities under
conditions operative for ion batteries.
The results of our calculations are presented in Fig. 3.
Evidently, there is qualitative agreement between the calculated
and the experimental currents (Fig. 2) for high repeller
voltages, albeit not quantitative agreement. Differences are
observed for low repeller voltage. The only adjustable
parameter Dbulk has been set to 2.6 · 10-18 cm2/s, in good
agreement with the value derived from impedance
spectroscopy (4 · 10-19 cm2/s). Given the fact that different
data curves are modeled by only one adjustable parameter the
agreement might be considered gratifying. As in the case of the
experimental data, we can also extract effective specific
conductivities from the theoretical current - voltage curves. The
numbers derived are given in Table 1. Evidently, there is good
agreement between the experimental and theoretical analysis of
conductivities.
V.
VI. REFERENCES
K. Funke, Jump Relaxation in Solid Electrolytes, Prog. Solid State
Chem., vol. 22, pp. 111-195, (1993).
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of glassy structure on spatial extent of nonrandom ion hopping, Phys.
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structures, Solid State Ionics, vol. 177, pp. 1551-1557, (2006).
[4] J. C. Dyre, P. Maass, B. Roling, and D. L. Sidebottom, Fundamental
questions relating to ion conduction in disordered solids, Reports on
Progress in Physics, vol. 72, pp. 046501-(1-15), (2009).
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Metals, Z. Phys. Chem, vol. 223, pp. 1143-1160, (2009).
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Weitzel, Field effects in alkali ion emitters: Transition from Langmuir–
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[8] M. Schäfer and K.-M. Weitzel, unpublished (2010).
[9] T. Okura and K. Yamashita, Ionic conductivities of Na+-ion implanted
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[11] unpublished results of this laboratory (2010).
[1]
SUMMARY AND DISCUSSION
We have measured the transport of potassium ions through
a borosilicate glass induced by ion bombardment. The ion
currents measured are shown to be in qualitative agreement
with a theoretical model based on the Nernst-Planck-Poisson
equations. The experimental and theoretical analysis yields
effective conductivities which exhibit very good agreement
(Table 1). We further showed that the effective conductivity
increases almost linearly with increasing input ion current for
both types of analysis. This originates from the linear increase
of the surface potential with the input current. In fact, plotting
the calculated backside ion current as a function of the effective
surface potential leads to a situation where all curves basically
fall on top of each other. Ultimately, this implies that the
transport is dominated by migration. For the future, it might be
rewarding to search for a situation where migration and
diffusion contribute similarly to the transport. There are a
4