Solute Transport
7.6 Electrolytic conduction across a planar channel
184 Solute Transport
0.9 1 1.1 1.2 1.3 1.4
0 0.2 0.4 0.6 0.8 1
ฯch/ฯch|Ohmic
Jch/Jch,LC fixed ๏ดrefreshment model
standard film model
Fig. 7.3:The voltage across a channel that has a selective interface on one side, as function of current density, according to the standard film model, and based on the fixed-๐ refreshment theory. For parameter settings see main text.
voltage versus current in the limit of low currents, that we obtain a a difference by a factor of two in the predicted value of๐
L, depending on the chosen model. The standard film model with fixed thickness would lead to the prediction of a twice lower ๐
Lthan when the same data of voltage vs. current are analysed using the constant-๐refreshment model.
Electrolytic conduction across a planar channel 185 and migration. The two electrodes are planar and parallel and the ๐ฅ-coordinate runs at right angles from the one to the other electrode. We describe the system as if the Li-metal dissolves from one electrode and deposits on the other. In reality in a Li+-ion battery, there is no dissolution, but instead Li+-ions desorb from and absorb in porous battery electrodes.
Thus, we apply a constant current ๐ผ leading to a flow of Li-ions from one to the other electrode. The total amount of anions in the gap in between the electrodes is conserved, i.e.,
โซ๐ฟ
0 [๐]d๐ฅ=๐ฟ ๐
0with๐ฟthe width of the gap.
In this theoretical calculation the transport numbers at the two surfaces are๐+=1 and ๐โ=0. (With porous battery electrodes on each side of the channel we will not have these ideal numbers.) Based on Eq. (7.43) we then know the gradients in concentration at the two surfaces, only dependent on the cation diffusion coefficient. In between the surfaces we use the salt balance, Eq. (7.29) (with๐ฃ
F=0 and no refreshment). This salt balance immediately makes clear that in the steady-state the concentration changes linearly. This means that at any position in the channel the concentration gradient will be the same as at the surfaces, thus independent of the anion diffusion coefficient. The anion diffusion coefficient only influences the dynamic part of this process, but not the final steady-state behaviour.
Next we present results for concentration profiles and transport numbers for๐ท+=๐ทโ, and thus the transference number of both ions is๐ก+=๐กโ=ยฝ. Results are presented in Fig. 7.4.
As discussed in the previous section (and by Newman, 1983), at the start of this process the fraction of the current carried by the two ions is described by their transference number,๐ก๐, but this is no longer the case when concentration gradients develop. When we reach steady state, the transport number of the cation is unity,๐+=1, at each position in the channel, and for the anion, the transport number has become zero,๐โ=0, everywhere. We show the changes in๐โwith position and time (๐+always given by 1โ๐โ) in Fig. 7.4B.
Across the midplane of the channel, concentration profiles are exactly โpoint-symmetricโ, with concentrations increasing on the anode, and decreasing on the cathode in an exact mirror-image, also when๐ท+โ ๐ทโ, like shown in Fig. 7.4A. Initially, we have concentration profiles โgrowingโ from the electrodes into solution, until they โmeetโ in the middle. In steady state, we arrive at a linear concentration profile, also when ๐ท+ โ ๐ทโ. At the midplane, concentrations remain the same as initially, also when๐ท+โ ๐ทโ.
For equal anion and cation diffusion coefficients, there is an analytical solution for๐(๐ฅ , ๐ก) given as Eq. (15) in Van Soestbergenet al. (2010). Here, however, we numerically solve the equations: the salt balance, the charge balance, and the boundary conditions for๐ ๐/๐ ๐ฅ vs. current๐ฝ
ch. We choose a current such that in steady state we just reach the limiting current, i.e., the concentration eventually reaches zero at the cathode, which is on the right in Fig. 7.4A. With that choice, the entire problem is defined.
186 Solute Transport
Anion transport number T-
x-coordinate
c/ c
0x-coordinate
time
time
Fig. 7.4:Electrolytic conduction, i.e., the development of concentration profiles across the gap between anode and cathode which are perfectly selective for cations (all ions monovalent; current density equal to the limiting current).
On the anode the salt concentration increases, and on the cathode it goes down, to finally reach a linearly decaying profile in salt concentration. The transport number of the anion, ๐โ, is plotted in Fig. 7.4B, and we note how it quickly drops from the initial value of๐โ=0.5 to values close to zero after some time. The profiles for the cation transport number are a mirror image (vertically mirrored across the horizontal line at๐=0.5), and starting at ๐+=0.5 they increase to ultimately reach๐+=1.0 after some time. Interestingly, these curves for the transport number are symmetric across the midplane of the channel, even though the concentration profiles are not, and neither is the potential profile symmetric. This potential profile decays from left to right, going down steeper and steeper. Still, the plots for the transport numbers vs. position (at various moments in time) remain perfectly symmetric (between left of the midplane, and right of the midplane). This is even the case when the two ions have different diffusion coefficients, which to us is not an immediately intuitive outcome. [With different diffusion coefficients only the โstarting valueโ of๐๐shifts. This can be easily derived because at time zero,๐๐=๐ก๐, and๐ก๐depends via Eq. (7.21) on the diffusion coefficients of the two ions.]
Next we implement dispersion as an additional effect, as also addressed by Newman (1983, p. 11). We implement dispersion by adding to the mass balance the refreshment effect,(๐
0โ๐) /๐, due to flow parallel to the surface, see ยง7.1.4. We assume that the parallel fluid velocity,๐ฃโฅ, has a parabolic profile, starting at zero at the walls, with a maximum in the middle of the channel. Thus at each point, the salt is mixed up with the average value, ๐0, with a frequency that has a maximum in the middle of the channel and decreases to zero
Electrolytic conduction across a planar channel 187 at the two electrodes.
The outcome is presented in Fig. 7.5A which shows that the higher the refreshment of fluid and ions, the closer the anion transport number remains to the maximum value, which is given by the transference number, which in this case is๐ก+=0.5. But in all cases, near the sides of the channel the transport number for the anions drops to zero, also for a high refreshment rate. The profiles remain left-to-right symmetric across the midplane. When we reduce the anion diffusion coefficient, the transport number for the anion decreases, but the profiles of transport number are still symmetric with the maximum for๐โin the center of the channel (and here is the minimum for๐+).
The voltage across the channel (analysed as a positive number) goes down if the refreshment rate (the degree of dispersion) goes up. In the steady state, without any dispersion this voltage becomes infinite because we work at the limiting current, while dispersion leads to a steady drop in voltage the more we mix, though after some point the further reduction in voltage becomes very minor. When we reduce the diffusion coefficient of the anion (for a certain degree of dispersion) the voltage increases (the resistance of the solution goes up), also in steady state. Thus, while without dispersion the anion diffusion coefficient does not play a role in determining the steady state profiles (and neither does it have an effect on the voltage across the channel), it does play a role when there is some mixing.
Effect of indifferent salt. A related topic is the effect of added inert, or โindifferentโ
salt. Inert salt is added in many electrochemical experiments to reduce voltages across a channel. We make several calculations extending on the example just discussed.
In the calculation we add an extra cation that is inert, while for charge balance more (of the same, inert) anions are added. [Note that a very different outcome would be obtained when a third ion (either anion or cation) is added that has zero mobility, i.e., an ion which is homogeneously distributed across the channel. That would actually be a model for flow of ions across a charged gel or membrane.] Theoretical analysis is not straightforward because we have many variables to consider: concentrations, diffusion coefficients, type and intensity of dispersion. . . Some key results we obtain are as follows:
1. In all calculations, the transport number for the anion remains symmetric around the midplane, behaving similarly as above, dropping to zero over time in a a calculation without dispersion, and remaining at higher values (except near the surfaces) when there is dispersion.
188 Solute Transport
0.24 0.245 0.25 0.255
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5
0 0.2 0.4 0.6 0.8 1
Anion transport number T-
x-coordinate
Increasing convection
Anion transport number T-
x-coordinate B time
A
Fig. 7.5:Anion transport number,๐โ, for electrolytic conduction of a cation across a channel between two electrodes. a) Steady state profiles at various degrees of refreshment (convective mixing), described by a parabolic profile for the flow velocity along the electrodes. With increasing refreshment, the anion transport numbers more closely approach the maximum value, which is equal to the anion transference number๐กโ=0.5. (Except for refreshment (convective mixing), this is the same calculation as Fig. 7.4.) b) Development in time of๐โin a three-ion system with an alternative, realistic, mixing model where salt and water are exchanged between positions equally far from the electrode that are on either side of the channel. In this case the transference number is๐กโ=0.25 while the transport number,๐โ, can be both larger and smaller than๐กโ, and exhibits very non-monotonic profiles.
Electrolytic conduction across a planar channel 189
2. Now with a second cation, the transport number profiles for the cations can become asymmetric.
3. In a small channel, when there is not much mixing, or none at all, adding salt reduces the electric field, which thus reduces the migration term for the reactive cation, and as a consequence โfor the same currentโ its concentration profile must become steeper, and thus the limiting current decreases, i.e., certain high values of current can no longer be realized. (Interestingly, this limiting current in the reactive cation is now arrived at without the voltage across the channel diverging.) Nevertheless, adding this indifferent salt, for any current that can be realized, we obtain a lower voltage.
4. For wider channels, with mixing, adding salt reduces the voltage across the channel for the same current.
An alternative dispersion model for a rotating electrode pair. Interestingly, the calculation with three ions leads to slight errors when dispersion is included, which was considered by adding the term ๐
0,๐โ๐๐/๐to each ionโs individual mass balance, where ๐0,๐is the initial concentration in the channel. This works fine, but an interesting effect is that the total amount of the two cations in the channel will no longer be equal to the initial value. This is due to concentration profiles no longer remaining point-symmetric across the midplane. This is fine if the model is meant to represent a channel flown through from a larger container. But otherwise, there are two alternatives. The first is that we do impose mass conservation in the channel, and thus have the convective outflow, which is the termโ๐๐/๐, slightly corrected by a factor๐ผ(the same across the channel, varying in time), such that this convective outflow exactly equals the total inflow, which is always ๐0,๐/๐. The second option is much easier. And this second option is representative of the circulation patterns that are possible between two electrodes when the inner electrode rotates (Newman, 1983). What we do in this second option, mathematically, is to have the fluid (and the ions inside any volume of fluid) at a distance๐ฟfrom the one electrode, mix with fluid located at the same distance๐ฟfrom the other electrode.
The mixing frequency is related to the velocity profile in the direction parallel to the surface,๐ฃโฅ. This velocity, as before, will be zero at the electrode surface, and now is also zero in the center of the channel; thus we assume a parabolic profile in one half of the channel, as well as in the other half, thereby representing the flow pattern presented on p. 7 in Newman (1983). This second model is easily implemented in our numerical code and then works flawlessly. Interestingly, with this corrected model, that keeps all
190 Solute Transport
salt in the channel (no exchange with an outside reservoir), the transport numbers, for instance for the anion, can be both larger and smaller than the transference number, see Fig. 7.5B for one particular example calculation (details here) in which all three ions have different diffusion coefficients. Fig. 7.5B shows interesting non-monotic and asymmetric profiles in๐โ, until steady state is reached. In steady state, the profile in๐โ is symmetric across the midplane with a single maximum value at that position which is around 6% larger than๐กโ. For the two cations the profiles in transport number are asymmetric during the lead-up to steady state as well as in steady state.