Solute Transport
7.7 Transient solute transport to an interface for electrolyte solutionselectrolyte solutions
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 in𝑇−, until steady state is reached. In steady state, the profile in𝑇− is symmetric across the midplane with a single maximum value at that position which is around 6% larger than𝑡−. For the two cations the profiles in transport number are asymmetric during the lead-up to steady state as well as in steady state.
7.7 Transient solute transport to an interface for
Transient solute transport to an interface for electrolyte solutions 191 change upward to flow away from the surface. Thus, as function of time𝑡, the flux starts off infinitely high, and then drops down, ultimately to zero, because the concentration profiles extend further and further away, becoming more and more shallow. The concentration change at each distance𝑥away from the surface, as a fraction of the applied step changeΔ𝑐, is given by erf
𝑥/
2
√
𝐷 𝑡 , where erf is the Error function. Note that penetration theory often uses the flux averaged from time zero to time𝑡. This averaged flux is equal to twice the ‘instantaneous’ flux (i.e., the flux at any timet) that is given by Eq. (7.57).
When instead of a step change in concentration, a step change in boundary flux𝐽is applied from time𝑡=0 onward, the solution for the surface concentration vs. time is given by Sand’s equation
Δ𝑐=±2𝐽
√︂
𝑡 𝜋 𝐷
. (7.58)
In electrochemical processes, we often analyze the situation of a step-change in current 𝐽ch from time𝑡=0 onward (with current zero before𝑡=0). As explained above, together with information on the transport numbers𝑇𝑖 (or𝜆for a 1:1 salt, and then𝜆=𝑇+−𝑇−), we can convert this information into a concentration gradient at the surface, see Eq. (7.41). And results from §7.1 can be rewritten such that they apply to a binary salt solution, by making the replacements𝐽 → −𝐷 𝜕 𝑐/𝜕 𝑥|∗ and𝐷 → 𝐷hm. Concentrationc is the monovalent equivalent (m.e.) salt concentration.
For a 1:1 salt and a perfectly selective interface, i.e., an interface where only one type of ion reacts or moves across (the counterion, abbreviated as ‘ct’), we can use Eq. (7.43), and combination with Eq. (7.58) then results in
Δ𝑐=±𝐽
ch
𝐷ct
√︂
𝐷hm
𝜋
𝑡 (7.59)
which shows that the dynamics of the change in surface concentration depend on the diffusion coefficients of both ions, also the one that is completely blocked from the interface.
If we start at a concentration𝑐∞, which is the unvarying bulk concentration, and if we remove counterions from solution through the interface for a fixed current (and𝜆=1), then the surface concentration drops, first fast then slower, and it reaches zero at the transition time, which we can calculate after rearranging Eq. (7.59) to
𝑡=
𝜋 𝐷2
ct𝑐2∞
/ 𝐷hm𝐽2
ch
. (7.60)
At this time, the salt concentration at the interface reaches zero, and the voltage across the solution phase diverges. Measurement of this transition time can be helpful to establish ion
192 Solute Transport
0 0.2 0.4 0.6 0.8
0 0.5 1 1.5 2
film layer of finite thickness
current density i
transition time t
semi-infinite
0 0.2 0.4 0.6 0.8 1
0 1 2 3 4 5
surface concentration c
time t B)
A)
c/x=0.5
c/x=0.75
Fig. 7.6:Sand’s equation for the transition time for diffusion to a surface. a) Qualitative illustration of the transition time for a semi-infinite layer of electrolyte solution (orange dots), as considered by Eq. (7.61), compared with the case of a film layer of finite thickness, according to Eq. (28) in Van Soestbergenet al.(2010) (blue dots). For currents below the limiting current, in this second case (blue dots), we never reach a surface concentration of zero. For high enough currents, the two calculations give the same result, with the transition time dependent on current to the power -2. b) The surface concentration in a model with diffusion and dispersion with a variable refreshment time 𝜏, all as function of time ˜𝑡. Only when the flux is high enough (resulting in a high enough value of𝜕𝑐˜/𝜕𝑥˜) will the concentration at the surface drop to zero at some point. All calculations are based on a 1:1 salt, equal𝐷’s and current efficiency𝜆=1.
transport numbers when the measured time for the voltage to diverge is longer than predicted by Eq. (7.60).
Indeed, when the surface is not perfectly selective for counterions, and thus the transport numbers are not either 0 or 1, and when we have a 1:1 salt with both ions having the same diffusion coefficients,𝐷=𝐷
hm, then we can combine Eq. (7.44) with Eq. (7.59), and arrive for the relationship between transition time and transport numbers at
𝑡=𝜋 𝐷 𝑐2∞/( (𝑇+−𝑇−) 𝐽
ch)2 . (7.61)
Eq. (7.61) for the transition time predicts that at some time the concentration will always reach zero and the current will diverge, see the orange dots in Fig. 7.6A. This relates to the assumption that we have an unstirred bulk solution that extends to far away. It is, however, more realistic to consider a film layer with a certain thickness, or assume some dispersion.
Then for currents that are too low, the concentration will never hit zero and the voltage will not diverge, see the blue dots in Fig. 7.6A. So in the latter case there is a minimum current
Transient solute transport to an interface for electrolyte solutions 193 required to reach zero concentration after some time. For the standard film layer model with thickness𝛿, Eq. (28) in Van Soestbergen et al., Phys. Rev. E81, 021503 (2010) is used, which as function of𝛿(for a 1:1 salt, equal ion diffusion coefficients, and for𝑇
ct=1), provides the relationship between transition time and current, see the blue dots in Fig. 7.6A.
Next we consider the case where in addition to diffusion and electromigration towards the surface we have dispersion described by the refreshment approach. Here we analyze the case that𝜏is a decreasing function with distance to the surface, see §7.1.4. For this model we can rewrite the salt balance, Eq. (7.29), to
𝜕𝑐˜
𝜕𝑡˜
= 𝜕2𝑐˜
𝜕𝑥˜2 + (1−𝑐˜)𝑥˜ (7.62) where ˜𝑐=𝑐/𝑐∞and ˜𝑡=𝑡·√︁3
𝐷 𝛾2. For a given gradient𝜕𝑐˜/𝜕𝑥˜at the surface –proportional to the applied current,𝐽
ch, and also dependent on the counterion transport number,𝑇
ct– we can analyze how ˜𝑐at the surface drops with time ˜𝑡, see Fig. 7.6B. Results here show that we do not reach a zero concentration at the surface when the current, thus𝜕𝑐/𝜕˜ 𝑥˜, is too low, but for a higher value of the current we do reach zero concentration at the surface after some time. When that happens, the voltage will steeply increase. If the aim is to avoid reaching this limiting situation, and when current is fixed, then we must reduce the mass transfer resistance, i.e., stir more (increase𝛾), or make the surface more leaky for coions, because that reduces𝑇ct.
This finalizes our discussion of diffusion and electromigration to a selective surface, such as an electrode or membrane, for steady state and for the dynamic approach to steady state, for arbitrary binary salt solutions (i.e., solutions with two ions that can have different valencies and/or diffusion coefficients). We also showed that when we have more than two ions, the situation can become drastically different, and for instance we can reach a limiting current in one of the ions, but not in other ion types.
Sand’s equation with background salt. Note that we presented here results of concentration changes in time for an electrolyte with one type of anion and one type of cation, and we include all electric field effects. Instead, the classical analysis of the Sand equation and the transition time (e.g., Bard & Faulkner (1980) p. 252) is based on the absence of electric fields because of addition of indifferent background electrolyte, or because the species moving to the interface is neutral and therefore only experiences a diffusional force. This is why in that literature there is effectively a factor 2 difference with our equations, i.e., their results make use of𝐽
ch = −𝐷
ct 𝜕 𝑐/𝜕 𝑥|∗. In addition,
194 Solute Transport
because of neglecting electromigration, results in these literature sources never have an influence of the diffusion coefficient of the coion, or a dependence on transport numbers. In an electrochemical experiment indeed the electric field can be cancelled by adding ‘indifferent electrolyte’, but of course this is not generally advisable in an electrochemical process, for instance for water desalination with membranes.