Download Bombardment Induced Potassium Ion Transport

Z. Phys. Chem. 226 (2012) 341–353 / DOI 10.1524/zpch.2012.0215
© by Oldenbourg Wissenschaftsverlag, München
Bombardment Induced Potassium Ion Transport
Through a Sodium Ion Conductor: Conductivities
and Diffusion Profiles
By Lisa Rossrucker, Pramod V. Menezes, Julia Zakel, Martin Schäfer,
Bernhard Roling, and Karl-Michael Weitzel∗
Philipps-Universität Marburg, Fachbereich Chemie, Hans-Meerwein-Straße, 35032 Marburg, Germany
Dedicated to Paul Heitjans on the occasion of his 65th birthday
(Received January 26, 2012; accepted in revised form March 15, 2012)
(Published online May 14, 2012)
Ion Conductivity / Temperature Dependence / Mixed Alkali Effect /
Concentration Dependence of Diffusion Coefficient / Tof-SIMS / Migration
Ion transport through a Ca30 glass, a sodium ion (Na+ ) conductor, has been induced by
bombardment with a potassium ion (K+ ) beam. The measurement of back side ion currents as
a function of the ion beam kinetic energy by means of the recently developed BIIT (bombardment
induced ion transport) approach allows determining the conductivity of the material. Measurement
of this conductivity as a function of the temperature allows deriving the activation energy for
ion transport as 0.99 eV ± 0.01 eV in perfect agreement with impedance spectroscopy. While the
conductivity as well as the activation energy clearly correspond to the bulk property, i.e. the
transport of Na+ , depth profiling of the glass sample after the BIIT experiment exhibits K+
profiles reaching up to 100 nm into the glass. Ultimately, modeling of the experimental data by
means of the Nernst–Planck–Poisson theory provides access to a quantitative understanding of the
conductivities and the diffusion profiles under the condition of competing Na+ /K+ ion transport.
1. Introduction
The transport of ions through solid electrolytes is a core ingredient to the fields
of energy conversion and energy storage [1–3]. A broad spectrum of methods for
measuring and modeling ion transport is available today, including impedance spectroscopy (IS) [4,5], nuclear magnetic resonance spectroscopy (NMR), in particular
pulsed field gradient NMR (PFG-NMR) [6–9] and tracer diffusion (TD) techniques [10,
11] on the experimental side, hopping models [4] including random free energy barrier models [12], molecular dynamics models [13,14] and macroscopic electrodiffusion
models [15] on the theoretical side.
While the focus was directed at ionic conductivities in the past, concentration profiles induced in solid state ion conductors have attracted increasing interest in recent
* Corresponding author. E-mail: [email protected]
342
years. Concentration gradients together with potential gradients form the basis for what
is usually referred to as electrodiffusion [16]. Very pronounced concentration profiles
can e.g. be generated by electro-poling, where a solid electrolyte is exposed to an electric field at elevated temperature leading to migration of the charge carrier [17,18]. This
in turn generates a depletion zone, in other words a characteristic concentration profile,
which can be literally frozen by decreasing the temperature to a value where migration
is effectively stopped. This electro-poling is e.g. often employed in the context of bioactive glasses [19,20]. It has also been demonstrated to facilitate non-linear properties
of materials [17,21–23]. Pronounced concentration profiles of a charge carrier further
occur in the tracer diffusion studies mentioned above. More generally, concentration
profiles must occur where molecules of a certain sort are deposited on a solid sample
into which these molecules can diffuse.
In extension of the techniques available for ionic conductivity measurements, we
have recently developed the bombardment induced ion transport (BIIT) technique,
where a solid electrolyte sample is connected to a backside electrode allowing for a current measurement. The sample is bombarded from the front side by a c.w. alkali ion
beam. Attachment of the ions to the front side of the sample causes a well defined surface potential and consequently a potential gradient over the material but at the same
time also a concentration gradient inside the material. Both form the basis of what we
term electrodiffusion. This induces a neutralization current at the backside electrode.
Measuring this backside current as a function of the temperature ultimately provides
access to the bulk activation energy for ion transport.
If we bombard e.g. a potassium ion (K+ ) conductor with a K+ ion beam the excess
charges are located at the surface of the sample. Here, ion conduction ultimately implies that one K+ ion entering the sample at the front side leads to another K+ ion being
discharged at the backside of the sample, which is detected by an electrometer. Consequently, the K+ ion profile remains flat everywhere inside the sample. A very different
situation is expected when we e.g. bombard a sodium (Na+ ) ion conductor with a K+
ion beam. Intuitively, one would expect that the effective conductivity measured should
still be dominated by the conductivity of the bulk Na+ . However, a finite amount of K+
ions should be present in the front layers of the sample and the effective conductivity of
the sample could be different from that of the pure Na+ ion conductor.
It is the aim of this work to present experimental data and a theoretical analysis of K+ ion transport into/through a Na+ ion conductor. We demonstrate that the
macroscopic effective conductivity is dominated by the bulk properties of the material. Ultimately, the number density of various relevant elements will be investigated by
means of a well established depth profiling technique. Here, we employ time-of-flight
secondary-ion-mass-spectrometry (TOF-SIMS) [24,25], which has been successfully
applied in a large number of depth profiling studies [26].
2. Experimental approach
The experimental setup used for the BIIT investigations consists of an ion source, an
electrostatic ion optics and a sample holder with detection electrode that is connected
to an electrometer amplifier. The entire setup is housed in a high vacuum chamber at
BIIT of Ca30
343
Fig. 1. Sketch of the sample holder (color online).
an operating pressure of 10−6 mbar. We have employed a home made synthetic leucite
ion source [27] that is heated such that an ion beam is thermionically emitted [28].
The ions are repelled by a positive potential UR applied to a repeller lens and accelerated towards the sample. A set of electrostatic lenses and flight tubes is used to guide
and focus the ion beam towards the sample. After leaving the ion optics, the ions pass
a 20 mm thick grounded tube with 10 mm central opening that is positioned 3 mm in
front of the sample. Both sides of the tube are covered with copper meshes such that
a homogeneous field is created in front of the sample surface and field penetrations are
avoided. Eventually, an ion current of 8.7 nA reaches the sample position. The ions are
deposited at the sample surface such that an electrostatic potential builds up at the sample surface whereas the backside of the sample is glued on a copper electrode. Thus,
a potential gradient develops that leads to the electrodiffusion of ions through the sample towards the backside electrode where the ion current is detected with a home made
electrometer amplifier. In order to keep contact potentials between backside electrode
and sample small, we use an excellently conducting epoxide glue (Loctite 3880) to attach the electrode to the sample. The ion current is A/D-converted and processed in
a personal computer.
The front side of the sample is masked by a metal ring with 8 mm inner diameter to define the bombarded area A. The electric contact between the glass and the
front side mask is desired to be “bad” to avoid errors in the conductivity measurement.
Therefore, the mask is not glued but just loosely pressed. The influence of the mask on
the conductivity measurements has been tested by detecting the backside current once
with grounded mask and the second time without any electric contact to the mask and
otherwise unchanged parameters. Both measurements showed the same ion current at
the backside electrode implying that there is no influence of the mask on the bulk ion
transport.
The sample holder (Fig. 1) is mounted on a temperature controlled heating device
that permits us to define the sample temperature to an accuracy of 0.1 K. The entire
device can be moved using an ultrasonic motor such that the sample is shifted off the
ion beam and the ion current reaching the sample surface can be detected in a blind
measurement.
The glass sample investigated was prepared by heating a stoichiometric mixture of
Na2 CO3 , CaCO3 and (NH4 )2 HPO4 . The two carbonates have been heated to 393 K to re-
344
move water and adsorbed gases. After pestling, the educts have been heated to 1423 K
for 1 h in a platinum pot. The molten mixture has been poured into a preheated steel
cylinder with inner diameter of 20 mm. Subsequently, the melt was kept 40 K below the
glass transition temperature (Tg = 603 K) for 10 h before cooling to room temperature.
The actual sample is obtained by cutting thin discs from the glass cylinder and polishing
down to the desired thickness, in the current case 1.219 mm. The final composition of
the glass is 25% Na2 O – 30% CaO – 45% P2 O3 . In the rest of the paper, we will refer to
this Na+ ion conducting glass as Ca30.
Depth profiles of the sample have been recorded by means of ToF-SIMS employing
a TOF.SIMS 5 device (IONTOF GmbH, Münster, Germany). As a primary ion source,
Bi+ has been used in high current bunched mode (pulsed target current: 0.1 pA). The
dual source column (DSC-S) sputter gun, has been operating with O+2 as the sputter
species (target current: 170 nA). The sputter crater dimension has been 300 × 300 μm2
and the Bi+ ions were scanned over an area of 100 × 100 μm2 for analysis. All depth
profiles were acquired in non-interlaced mode (1.0 s sputter time, 0.5 s pause) and a low
energy electron gun (20 eV) was used for charge compensation. The crater depth was
determined post-analysis with as Sloan Dektak 3ST surface profilometer.
3. Theory
The demonstration of the diffusion profiles in the second part of this manuscript is
accompanied by numerical simulations. For these simulations, we employ the theory
introduced in [15] and calculate the coupled set of Nernst–Planck and Poisson equations [29] for the given experimental setup.
The experimentally investigated glass is thin compared to its radial extension such
that radial transport can be neglected. The ion beam hits a circular area and shows
circular symmetry in a very good approximation. Hence, angular transport can be neglected. As a consequence, we may concentrate on the ion transport alongside the axis
perpendicular to the glass surface; the z-axis.
Prior to the bombardment, the only mobile ion species in the Ca30 glass is Na+ .
The initial concentration of the Na+ ions is 7.16 × 109 ions/μm3 given by the stoichiometry and the density of the glass. The initial K+ content is set to zero, neglecting
the small amount of impurity. However, once the bombardment is started K+ ions are
deposited on the glass surface and a potential gradient builds up. In this situation,
both ion species contribute to the electrodiffusion and the Nernst–Planck equations
need to be formulated for two mobile ion species. Ion species other than K+ and
Na+ are neglected. Generally, one expects that the diffusion coefficients of the two
ion species are different from each other connected to the difference in the ion sizes.
Moreover, the effective diffusion coefficient Dν (ν = Na+ or K+ ) may depend on the
local environment in particular the concentration of both charge carriers Na+ and K+ .
As we will demonstrate below the sum of the Na+ and K+ concentration is uniform
throughout the sample (except directly at the surface, see [27]), hence, it is sufficient
to discuss Dν as a function of one of the two species. The dielectric constant is assumed to be space and concentration independent. It is set to εr = 10 everywhere in the
glass.
BIIT of Ca30
345
Accounting for these approximations, the Nernst–Planck equation for the two ions
species reads
∂n ν
∂ϕ e
Jν = −Dν (n Na+ )
+ nν
,
(1)
∂z
∂z kB T
where ν is the ion species index, Jν is the ion flux densities, Dν (n Na+ ) is the diffusion
coefficients of the mobile ions that depends on the local concentrations of both ions.
The ion concentration is n ν whereas the electrostatic potential is given by φ. The charge
of the ion species is e and kB T is Boltzmann’s constant times the temperature. The
concentrations of both ions species also enter the Poisson equation that is then given by
ε0 εr
∂2ϕ
= −e n K+ + n Na+ − n 0Na+ ,
∂z 2
(2)
with the vacuum dielectric permittivity ε and εr the relative dielectric constant. Because
the Na+ ions that belong to the Ca30 glass are explicitly treated in Eq. (3), we also need
to take into account the immobile negative charge of the glass matrix. We assume that
initially the Na+ ions are homogeneously distributed inside the glass and the glass is
electrically neutral such that the immobile negative charge q− of the glass matrix can
be written in terms of the space and time independent constant n 0Na+ , q − = (−e) · n 0Na+ .
Finally, the time evolution of the ion distributions is given by Fick’s second law
∂Jν
∂
nν = −
,
∂t
∂z
(3)
We discretize the z-axis into space elements that may have different sizes Δz i and
rewrite the Eqs. (1)–(3)
n ν,i+1 − n ν,i
ϕi+1 − ϕi e
i+1
i+1
Jν,i
= −Dν,i
+ n i+1
,
(4)
ν,i
Δz i
Δz i kB T
⎛ en i
⎞
Δz i Δz i+1,i Δz i,i−1 + (ϕi+1 Δz i,i−1 + ϕi−1 Δz i+1,i )
⎜ε ε
⎟
ϕi = ⎝ 0 r
(5)
⎠,
(Δz i,i−1 + Δz i+1,i )
i+1
i
− Jν,i−1
Jν,i
∂
,
n ν,i = −
∂t
Δz i
(6)
where n i is n K+ ,i + n Na+ ,i − n 0Na+ is the excess ion density in the space element i and
Δz i,i+1 = 1/2(Δz i + Δz i+1 ). The ion flux density between the adjacent space elements
i+1
i+1
while Dν,i
and n i+1
= (n i Δz i+1 + n i+1 Δz i )/(Δz i+1 + Δz i )
i and i + 1 is given by Jν,i
i
are the diffusion coefficient and the ion density at the boundary of these two space
elements.
Please note that the electric potential (Eq. 5) depends on the exact charge distribution inside the glass as well as on the potential in the neighboring space elements.
Therefore, we calculate the potential recursively using the potential of the previous
recursion step as an input for the actual recursion step. The converged potential subsequently enters the Nernst–Planck equation (4) where the electrodiffusion of the carriers
346
Fig. 2. Backside current, Iback , as function of the applied repeller voltage. Symbols: experimental data; solid
line: linear regression.
is calculated. The so calculated ion fluxes enter Fick’s second law such that the ion
distribution is modified. In the following time step, the potential is calculated again to
match the modified ion distribution and so forth.
In order to solve equations (4)–(6), it is essential to use the correct boundary conditions for the potential. Due to the grounding of the backside electrode, we use ϕ(L) = 0,
where L is the thickness of the sample. The potential directly in front of the sample surface is defined by a linear potential gradient between the charged glass surface and the
grounded tube in front of the glass. A detailed discussion of the boundary conditions
can be found in [15].
4. Results
We have bombarded a 1.219 nm thick Ca30 glass with a K+ ion beam. The ion beam
has been focused such that the ion current obtained at the backside electrode is maximal. Under theses conditions the inhomogeneities of the ion beam are minimal and the
overall blind current is about 8.7 nA. The temperature of the sample has first been set to
326.2 ± 0.1 K. Under these conditions, the backside current, Iback , has been detected as
a function of the applied repeller voltage. The repeller voltage has always been ramped
slowly compared to the surface charging. Consequently the ions are rather gently attached to the surface; implantation of ions into the sample does not play a major role.
Please note, that although modifying U R we have kept the electric field in front of the
emitter constant. This is provided by appropriately adjusting the voltage applied to the
first electrostatic lens behind the repeller. The abstracting field in front of the emitter has
been 2000 V/cm.
As shown in Fig. 2, the obtained backside current increases linearly with the repeller voltage in a range between 10 and 600 V. A linear regression of these data points
yields a slope of 0.00195 nA/V and a small offset of 0.025 nA that is mainly caused
by the electronics of the setup. For voltages larger than 600 V, the observed backside
BIIT of Ca30
347
Fig. 3. Arrhenius plot of the temperature dependence of the conductivity. Black squares: experimental IS
data, red circles: BIIT data. The solid line represents in each case the result of a linear regression to the
experimental data (color online).
current is smaller than the values extrapolated by the linear regression – an indication
for the onset of the transition regime which is caused by the inhomogeneities of the
ion beam [27]. Using the slope of the linear regression, we may calculate the actual
conductance of the sample G = ΔIback /ΔUR = 1.95 × 10−12 S and conclude a specific
conductivity of σ = G L/A = 4.73 × 10−13 S/cm.
It is well established that the dominant transport mechanism in disordered solids is
given by thermally activated hopping. Because the experiment operates under conditions where the migration contribution to the conductivity strongly dominates over the
concentration gradient driven diffusion, the temperature dependence of the conductivity
can be described via
σ=
S0 −E
act
e kB T .
T
(7)
Here, σ is the conductivity, E act is the activation energy for the DC transport and S0 is
the pre-exponential factor. Consequently, E act can be detected by measuring σ at different sample temperatures. The slope of an Arrhenius plot where ln(σT ) is plotted as
a function of the inverse temperature then yields the activation energy.
Thus, we have measured the conductivity in a temperature range between 306.2 K
and 334.2 K. The corresponding Arrhenius plot is shown in Fig. 3, where the BIIT data
are shown as red dots. For comparison, IS measurements with the same glass batch have
been performed in a temperature range between 333 K and 423 K. The corresponding
data are presented as black squares. For both data sets, linear regressions have been performed showing that Eq. (7) is fulfilled in the investigated temperature range. Moreover,
the analysis of the slopes of the two linear regressions shows that the activation energy
determined by BIIT (0.99 eV ± 0.01 eV) and by IS (0.98 eV ± 0.02 eV) perfectly match.
Even the overall conductivity determined by the two methods agrees very well. Nevertheless, we observe a small difference. The conductivities determined by BIIT are about
15 percent smaller compared to the ones determined by IS.
348
Fig. 4. Depth profiles of the most relevant elements in the sample. Depth = 0 refers to the surface of the
glass (color online).
In principle, part of this difference may originate from the injection of K+ ions in
the Ca30 glass during the bombardment process. Prior to the bombardment, the only
mobile ion species is Na+ while the K+ content is negligible. With proceeding bombardment more and more K+ is deposited at the surface and subsequently transported
into the material. As already mentioned, we expect that this modification should affect
the average diffusion coefficient of the entire glass and therefore the conductivity of
the sample. We can demonstrate that even for an experiment operated for about a week
the decrease in the effective conductivity is only on the order of 5%. The major part of
the difference between the BIIT and the IS conductivity (approximately 10%) must be
attributed to remaining differences between the two methods. Given the typical uncertainties of conductivity measurements, including uncertainties in sample preparation,
this agreement may still be considered very satisfying.
Obviously, the alteration of conductivities of a sample could be avoided by using
Na+ for the ion bombardment. A similar experiment with a borosilicate glass [27] performed in our laboratory showed excellent agreement between IS and BIIT regarding
the absolute value of the conductivity.
Yet, in this manuscript we are also interested in the answer to the question how
the introduction of the K+ influences the structure and therefore the conduction properties of the Ca30 glass. Therefore, a long term measurement has been performed where
a 1.543 mm thick Ca30 sample has been bombarded for a period of 24 days with an
average ion current of 1.5 nA. The ion beam hitting the sample showed a Gaussian like
ion flux density profile. As a consequence, the ion beam intensity has been larger than
the average ion flux density 1.5 nA/0.785 cm2 in the center and lower in the outskirts
of the bombarded area. The repeller voltage has been set to 100 V and the temperature
was kept at 373 K during the entire bombardment time.
Subsequent to the bombardment, the sample has been analyzed by time-of-flight
secondary-ion-mass-spectrometry. Fig. 4 shows the first 150 nm of the depth profiling
in the centre of the bombarded area where the ion beam current has been largest. The
point z = 0 has been defined at the onset of the phosphorus and the oxygen profiles.
BIIT of Ca30
349
The relative signal of the Na+ ions is shown as blue dots whereas the K+ signal is presented by the red squares. The calcium content is given by the green triangles whereas
the orange diamonds and the light blue triangles yield the phosphorus and the oxygen
distribution, respectively. The Na+ and the calcium signals have been normalized to the
average value deep inside the material where no K+ is present. The K+ signal has been
normalized to the maximum value directly below the front surface of the glass. For better visibility, the oxygen and the phosphorus signal have been normalized such that their
average value is one third.
Analyzing the depth profiles, one observes a diffusion front about 100 nm below
the glass surface. Between this front and the surface of the glass, most of the Na+
has been replaced by K+ . However, the Na+ concentration is not completely zero but
between 10% and 20% of the Na+ remains in this zone. The sum of the K+ and the
Na+ distributions is close to one everywhere in the sample, except in the region of
the diffusion front itself where the sum is smaller than one. Interestingly, the calcium signal shows an opposite feature at the same position and is space independent
elsewhere. Phosphorus and oxygen exhibit a constant distribution through the entire
material. Note, that the rise of the phosphorus and oxygen signals defines the front
of the sample. The depth profiles clearly show that the composition of the glass has
been substantially modified in an extended zone below the surface by the BIIT experiment. This zone, on the other hand, accounts for approximately 0.008% of the sample
thickness.
In order to gain more insights into the transport process, numerical simulations on
the basis of the coupled set of the Nernst–Planck and the Poisson equation have been
performed to complement the experiment. To this end, Eqs. (4)–(6) were solved with
an input current of 2.12 nA corresponding to 1.4 times the average ion flux reaching the
sample, a typical value for the Gaussian beam profile in the centre of the bombarded
zone in the used experimental setup. In these calculations the only free parameters are
the diffusion coefficients of Na+ and K+ . All other quantities are self-consistently calculated or defined by the experimental setup.
The diffusion coefficient of Na+ in the pure Na+ glass at 373 K can be determined by matching the calculated conductivity of the glass to the experimentally measured conductivity. The comparison between experiment and theory yields
DNa+ = 1.03 × 10−15 cm2 s−1 . Test calculations with concentration independent diffusion coefficients show that the slope of the diffusion profile at the position of the
diffusion front depends on the ratio between K+ and Na+ diffusion coefficient. Therefore, the K+ diffusion coefficient in a Ca30 glass can be estimated. It is predicted to be
DK+ = 2 × 10−18 cm2 s−1 .
Generally, the simulations with homogeneous diffusion coefficient provide already
some of the features obtained in the diffusion profiles of K+ and Na+ in Fig. 4, such as
a diffusion front, and the slope of the diffusion profile at the diffusion front. However,
these simulations also predict that all Na+ should be replaced in the first nanometers
below the surface, which is contradicting the experimental observations where about
10%–20% of the Na+ remain in the diffusion zone. In order to resolve this discrepancy,
we allow the diffusion coefficients to be concentration dependent. Generally, one may
expect that both coefficients could be concentration dependent. Yet, since the diffusion
coefficient of Na+ is larger than the one of K+ , we expect most of the effect to originate
350
Fig. 5. Concentration dependence of the Na+ diffusion coefficient.
Fig. 6. Potassium and sodium depth profiles: Symbols: experimental data; solid lines: numerical simulation
(color online).
from the concentration dependence of the Na+ diffusion coefficient. Hence, we keep the
diffusion coefficient of K+ constant.
The functional form of the concentration variation of DNa+ (n Na+ ) used for the simulations is shown in Fig. 5. The form follows the sigmoid equation
DNa+ (n Na+ ) = 1.0 × 10−20 cm2 s−1
+ 1.03 × 10−15 cm2 s−1
1
(8)
,
n +
1 + 2.0 × 104 exp −12.5 Na
n0
Please note, that the first term of the sum in Eq. (8) reflects the limit of the Na+ diffusion coefficient in a Ca30 glass where all Na+ is replaced by K+ . The prefactor of
the sigmoid function is the diffusion coefficient of the Na+ in the pure Na+ glass. The
result of the simulations performed with these parameters is shown as red line (K+
BIIT of Ca30
351
profile) and blue line (Na+ profile) in Fig. 6 and compared to the experiment. We observe excellent overall agreement between experiment and theory. Note, that agreement
is not only observed for the position of the diffusion front but also for the slopes of
this front and even the slopes in the depletion region. A very detailed inspection of the
data between about 80 nm and 120 nm indicates that the experimental Na+ signal is
slightly smaller than the calculated one. Most likely this difference is connected to the
small excess of calcium in this region, a species not included in the calculations (see
Fig. 4)
5. Discussion and summary
We have investigated the transport of ions in a Na+ ion conductor by means of K+
ion bombardment by the BIIT approach. The measured conductivities are shown to
be in nice agreement with impedance spectroscopy. The activation energy is clearly
that of the bulk Na+ transport. The BIIT result is E A = 0.99 eV ± 0.01 eV, the IS value
is E A = 0.98 eV ± 0.02 eV. However, the absolute BIIT conductivity is about 15% below the IS values. The discrepancy is in part (up to 5%) due to the incorporation of
the K+ into the material, which decreases the diffusion coefficient. There appear to remain, however, differences between the current IS and BIIT measurement on the order
of 10%. Given the common view on the precision of conductivity measurement, these
numbers may be considered as very good agreement.
Inducing Na+ transport by bombardment with K+ inevitably creates the prospect of
+
a K zone in the sample. This K+ profile has been quantitatively analyzed by means
of TOF-SIMS. Basically the BIIT experiment has led to a depletion of Na+ down to
approximately 100 nm below the surface of the material. In this zone between 80%
and 90% of all Na+ ions have been replaced by K+ . This diffusion profile has been
quantitatively modeled by means of the Nernst–Planck–Poisson equation. Quantitative
agreement between experimental and calculated data can only be achieved by assuming that the diffusion coefficient of the Na+ is concentration dependent. Clearly, the
diffusion coefficient for K+ is orders of magnitude smaller than that for Na+ . As the
Na+ , which are more mobile than the K+ , move towards the detector, they leave temporary vacancies behind, which are immediately filled by K+ , in other words the Na+
are dragging the K+ behind them.
One major result of the current work is the evidence for concentration dependent
diffusion coefficients. A similar effect has been discussed by Furini et al. [30]. For the
transport of K+ through a KcsA ion channel, where agreement between experiments
and Nernst–Planck–Poisson theory was only obtained, if the K+ ions were assumed
to be slowed down inside the channel, effectively represented by a smaller diffusion
coefficient.
As mentioned earlier diffusion profiles of alkali ions are also operative in voltage
assisted ion exchange [31]. E.g., voltage assisted 18 O tracer incorporation into oxides
leads to electrodiffusion profiles qualitatively similar to the ones discussed in the current work [32]. That experiment again employs two electrodes to define the voltage
drop across the sample. In contrast, our current experiment may be considered as voltage assisted ion exchange without the anode.
352
The concentration profiles discussed in this work clearly demonstrate that two alkali ions coexist in certain regions of the sample. The ratio of number densities even
varies along the profile. This stimulates thoughts, whether the conductivities observed
here might be related to the elusive mixed alkali ion effect [33–35]. In fact studies
of the mixed alkali effect also involves the discussion of concentration dependent diffusion coefficients [36,37]. It appears possible that the current experiment eventually
contributes to the understanding of the mixed alkali effect. However, significantly more
studies are required before reaching there.
What are the prospects of the current approach and what are the limitations? We
wish to stress the astonishing accuracy of our BIIT experiments, in terms of absolute
conductivities and activation energies, which is now comparable to IS. So far the BIIT
approach works best for extremely small diffusion coefficients. However, this is not
a limitation of the concept, but a limitation of the implementation. Work is currently
underway to extend the range accessible to BIIT to significantly larger diffusion coefficients, as those relevant for e.g. ion batteries.
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