Membrane Transport

Electrolyte and Electrode Separation

The electrodes in electrochemical cells are physically separated, and electrolyte occupies the space between the two electrodes. The manner in which this separation is implemented can vary and has significant impact on transport. Let’s inspect the space between electrodes for three electrochemical systems to highlight the differences. The first system we consider is the electrorefining of copper as discussed earlier in this chapter. In this system, there is no physical separator between the electrodes, and the space between them is filled with electrolyte. Since the electrolyte does not conduct electrons, the electrodes are insulated from each other as they must be. Current flows in solution by the transport of ions as we have discussed throughout this chapter (Figure 4.18a).

**Figure 4.18** (a) Transport across a simple electrode gap filled with electrolyte. (b) Porous inert separator filled with electrolyte. (c) Solid membrane representing a separate phase.

The second system is a lithium-ion battery. For battery operation, it is critical to minimize losses in order to get as much power as possible from the battery. Lithium ions are shuttled between the two electrodes during charge and discharge. In order to minimize losses related to transport, we want the distance between electrodes to be as small as possible, say 25 μm. At the same time, the two electrodes cannot touch each other. We cannot reliably construct a large area cell with the two electrodes so close together without having them touch if we only have liquid electrolyte between the electrodes. Instead, a porous separator made from a nonconductive material (e.g., polymer) is used (Figure 4.18b). The electrolyte fills the pores and enables ionic transport, while the polymer separator material prevents contact between the two electrodes. Movement through the pores restricts transport; therefore, transport properties, conductivity, and diffusivity must be adjusted to account for volume fraction filled by the inert polymer and for the tortuous path through the separator. These adjustments are discussed in detail in Chapter 5.

Finally, the third type of system we consider is one that uses a membrane between the electrodes. An important example of a membrane-based system is the chlor-alkali process used to produce chlorine gas and caustic. Many fuel-cell systems also use membranes. In contrast to a porous separator film, a membrane has no open porosity. Rather, the molecular structure of the membrane permits selective transport of one or more species through the membrane while blocking other species. For example, the chlor-alkali membrane selectively allows cations to move through the membrane, but excludes anions. This feature is critical to an efficient chlor-alkali process (Chapter 14).

Ion Transport Through Membranes

When used as the separator in an electrochemical process, the membrane itself serves as an electrolyte to allow ionic current to flow between the electrodes. It represents an additional phase in our electrochemical system, and there is selective partitioning of species between the bulk electrolyte and separator. In other words, the concentration of a species in the membrane is different from, but related to, the concentration of that species in the bulk electrolyte at the interface between the membrane and the electrolyte. We will only consider situations where the interfacial compositions are in equilibrium and can be described by a thermodynamic relationship. A partition or distribution coefficient is used to provide the required relationship as described below.

The partition coefficient is simply the ratio of the concentration of species i in phase α divided by its concentration in phase β (see Figure 4.18c):

(4.66)

These values are usually determined experimentally and are different for different species. In general, as you might expect, concentrations in the nonporous membrane are lower than those in the bulk.

The concentration difference between the values on either side of the membrane provides a driving force for transport. However, the potential field and charge interactions are also important for the transport of ions. Thus, ionic transport through the membrane can be quite complex and are not usually adequately described by the Nernst–Planck equation. The relationships needed to accurately describe transport in cation-exchange membranes that are commonly used in fuel cells are presented in Chapter 9.

Transport of Gases Through Membrane’s Thin Liquid Films

An approach similar to the partition coefficient mentioned previously can be used to describe the transport of gases in films. Such gas-phase transport is much simpler than ion transport since charge interactions are not important. However, transport of gases through membranes can result in important losses of efficiency. For example, hydrogen gas in a fuel cell may transport across the membrane to the cathode and react directly with oxygen instead of reacting separately on the anode to produce power. For gases, equilibrium at the interface between the gas and the membrane is expressed with Henry’s law:

(4.67)

where x and y are the mole fractions in the membrane and gas phases, p is the total pressure, and H_i is Henry’s law constant. As defined, 1/H_i describes the solubility of gas i in the membrane. A common physical situation is solution and diffusion, where a gas dissolves into and then diffuses across the membrane. To illustrate, we describe the transport of a gas species i. On the right side of the membrane the partial pressure is p_i, and on the left side it is zero. Equation 4.10 is applied to the membrane, which at steady state reduces to

(4.68)

where z is the position in the membrane in the “thickness direction.” Integration gives

(4.69)

As for boundary conditions, we express the concentration of the gas in the membrane with Henry’s law to yield

and

where δ is the membrane thickness, c is the total molar concentration or molar density of the membrane, and we have made use of the fact that x_i = c_i/c. We can now determine the constants A and B to yield the following expression for the concentration as a function of position

(4.70)

The flux is determined with Fick’s law:

(4.71)

The rate of transport of gas through the membrane will depend both on its solubility (1/H_i) and the diffusivity of the species (D_i). Often, these properties are combined into a quantity called the permeability,

(4.72)

We also see that the rate of transport depends linearly on the difference in partial pressure of gas across the membrane, as expected. When the species must first go into solution and then diffuse across the membrane, these processes are combined and described as permeation.

Closure

Transport in electrochemical systems by molecular diffusion and convection was introduced. Distinctive to electrochemical systems, the electric field plays a role in transport of ions. The Nernst–Planck equation was presented and several important simplifications were examined. Two important circumstances are the binary electrolyte and the case of an excess supporting electrolyte. For all intents and purposes, we treat the electrolyte as electrically neutral, which has the effect of coupling ion transport. For a binary electrolyte, there are three transport properties: electrical conductivity, diffusivity of the salt, and the transference number. Additionally, the significance of the current distribution was presented, which is influenced by geometry, kinetics, and mass transfer. The Wagner number is a dimensionless ratio of the ohmic and kinetic resistances. This dimensionless group provides a good starting point for assessing the uniformity of a current distribution. For transport of gases in liquids and membranes, the solubility and diffusivity can be combined to define the permeability; this quantity plays an important role in fuel cells and gas transport in membranes.

Notes

¹Mohanta, S. and Fahidy, T.Z. (1977) J. Appl. Electrochem., 7, 235.²Sigrist, L. Dossenbach, O., and Ibl, N. (1979) Int. J. Heat Mass Transfer., 22, 1393.

Problems

4.1. Oxygen diffuses through a stagnant film as shown in the figure. At the electrode, oxygen is reduced to form water. Estimate the limiting current density, that is, when the concentration of oxygen at the electrode goes to zero, corresponding to the maximum flux of oxygen. The following data are provided: thickness of the film 5 μm, the diffusivity of oxygen is 2.1 × 10⁻¹⁰ m²·s⁻¹, and the concentration of oxygen at the film surface in contact with gas is 3 mol·m⁻³.

4.2. For a binary electrolyte, show that the electric field can be eliminated and Equation 4.19 results.

4.3. At the positive electrode of a lead–acid battery, the reaction is

If the electrolyte is treated as a binary system consisting of H⁺ and , show that at the surface the current is given by
What issues could arise from the formation of solid lead sulfate on the surface?

4.4. A porous film separates two solutions. The left side contains 2 M sulfuric acid, whereas on the right side there is 2 M Na₂SO₄. Discuss the rate of transport of sulfate ions across the membrane and the final equilibrium state.

4.5. The purpose of this problem is to compare the limiting current for the electrorefining of copper both with and without supporting electrolyte. Assume that copper is reduced at the left electrode (x = 0) and oxidized at the right electrode (x = L). You should also assume steady state and no convection. Finally, assume that the current efficiency for both copper oxidation and reduction are 100%. (Note that the assumption of no convection leads to very low values of the limiting current since transport by diffusion is slow over the 5 cm cell gap.)

Derive an expression for the limiting current as a function of the diffusivity, the initial concentration of copper (which is also the average concentration), and the cell gap, L.
Derive an analogous expression for the limiting current in the presence of a supporting electrolyte.
How do the two expressions compare? Why is the limiting current lower in the presence of a supporting electrolyte?
For the binary system, derive an expression for the concentration profile as a function of current for values below the limiting current. What is the sign of the current in this expression?

4.6. One type of Li-ion battery includes a graphite negative electrode and manganese dioxide spinel positive electrode, described by the following reactions (written in the discharge direction):

The electrolyte consists of an organic solvent that contains a binary LiPF₆ lithium salt, where the anion is . Assume a one-dimensional cell with the cathode located at x = 0 and the anode located at x = L. The following properties are known:

For our purposes here, we assume that the electrodes are flat surfaces, and that they are separated by an electrolyte-containing separator that is 25 μm thick. The initial concentration of LiPF₆ in the electrolyte is 1.0 M. The parameters given in the question are for transport in the separator. The same expression derived in the text for the concentration also applies to this situation, Equation 4.39.

In which direction does the current in solution flow during discharge? Is this positive or negative relative to the x-direction?
Where is the concentration highest, at x = 0 or x = L?
What is the concentration difference across the separator if a cell with a 2 cm × 5 cm electrode is operated at a current of 10 mA? Is this difference significant?
Briefly describe how you would calculate the cell potential for a constant current discharge of this cell if the current is known and the concentration variation is known. The potential that would be measured between the current collectors of the cathode and anode during discharge. You do not need to include all the equations that you would use. The important thing is that you know and are able to identify the factors that contribute to the measured cell potential, and that you know the process by which you might determine their values.

4.7. A potential step experiment is conducted on a solution of 0.5 M K₃Fe(CN)₆, and 0.5 M Na₂CO₃. The potential is large enough so that the reduction of ferricyanide is under diffusion control. Using the data provided, estimate the diffusion coefficient of the ferricyanide.

Time (s)	Current density (A·m⁻²)
1	731
1.7	564
2.8	438
4.6	340
7.7	263
12.9	201
21.5	156
35.9	122
59.9	94
100	72

4.8. Platinum is used as a catalyst for oxygen reduction in low-temperature, acid fuel cells. At high potentials, the platinum is unstable and can dissolve (see Problem 2.17), although the concentration of Pt²⁺ is quite small. The electrolyte conductivity is 10 S·m⁻¹, and the current density, carried by protons, is 100 A·m⁻². Assuming that the temperature is 80 °C and the Pt²⁺ is transported over a distance of 20 μm, is it reasonable to neglect migration when analyzing the transport of platinum ions? Why or why not?

4.9. Calculate the diffusivity, transference number, and conductivity of a solution of 0.05 M KOH at room temperature.

4.10. Estimate the conductivity of a 0.1 M solution of CuSO₄ at 25 °C.

4.11. An electrochemical process is planned where there is flow between two parallel electrodes. In order to properly design the system, an empirical correlation for the mass-transfer coefficient is sought. An aqueous solution containing 0.05 M K₄Fe(CN)₆, 0.1 M, M K₃Fe(CN)₆, and 0.5 M Na₂CO₃ is circulated between the electrodes. The reactions at the electrodes are

If the potential difference between electrodes is increased slowly from zero, sketch the current–voltage relationship that would result. Include on the graph, the equilibrium potential for the reaction, the open-circuit potential, the limiting current, and the decomposition of water.
For the conditions provided, which electrode would you expect to reach the limiting current first?
Show how to calculate a mass-transfer coefficient, k_c, from these limiting current data.

4.12. For the system described in Problem 4.11,

Use the data provided for I_lim versus flow rate to develop a correlation for dimensionless mass-transfer coefficient, Sh. The Re should be defined with the hydraulic diameter, . At 298 K, the diffusivity of the ferri/ferrocyanide is 7.2 × 10^-10 m²·s⁻¹. Treat the electrodes as 0.1 m × 0.1 m squares separated by a distance of 1 cm. Use density as 998 kg·m⁻³ and a viscosity of 0.001 Pa·s. Based on the literature, the following correlations are suggested.
What is the importance of the sodium carbonate in this experiment?
Justify the need for the additional dimensionless factor .

Flow rate [cm³s⁻¹]	Limiting current [A]
0.96	0.15
2.06	0.19
3.69	0.22
6.41	0.27
9.16	0.30
11.89	0.34
14.62	0.36
17.35	0.39
20.22	0.41
22.83	0.44
25.54	0.46
28.56	0.50

4.13. For the Cu electrorefining problem (see Illustration 4.7),

Calculate the mass-transfer coefficient, k_c [m·s⁻¹], based on natural convection using the height of the electrode as the characteristic length.
It is desired to increase the mass-transfer coefficient by a factor of 10 and thereby raise the limiting current. If forced convection is used, what fluid velocity and Re are required? The following correlations for mass transfer between parallel planes are available. The distance between the electrodes is 3 cm. Assume that the electrodes are square.Note that in contrast to the correlation for natural convection, the Sh and Re here are based on the equivalent diameter .
For this arrangement, sparging with air at a rate of 2 L/min per square meter of electrode area results in a mass-transfer coefficient on the order of 2 × 10⁻⁵ m·s⁻¹. Calculate the superficial velocity of the air and compare with the velocity obtained in part (b). Why might air sparging be preferred over forced convection?

4.14. Correlations for mass-transfer coefficients for full-sized, commercial, electrowinning cells under the actual operating conditions can be difficult to measure.

What could be some of the challenges with measuring these mass-transfer coefficients?
It has been suggested (J. Electrochem. Soc., 121, 867 (1974)) that k_c can be estimated by codeposition of a trace element that is more noble. For instance, in the electrowinning of Ni (U^θ = 0.26 V), Ag (U^θ = 0.80 V) could be used. The idea is that because the equilibrium potential for the more noble material is higher, the limiting current will be reached sooner. After a period of deposition, the electrode composition is analyzed to determine the local and average rates of mass transfer. Would the mass-transfer coefficient for Ni be the same as for Ag? If not, how would you propose correcting the measured value?

4.15. A porous flow-through electrode has been suggested for the reduction of bromine in a Zn–Br battery.

Calculate the limiting current if the bulk concentration of bromine is 5.5 mM using the correlation

The Re is based on the diameter of the carbon particles, d_p, that make up the porous electrode.

4.16. Chlorine gas is being evolved on the anode of a cell. L = 0.5 m, W = 0.5 m, and h = 0.03 m. The electrode is operating at a uniform current density of 1000 A·m⁻².

Using the geometry from Figure 4.7, relate the superficial velocity of the gas between electrodes as a function of x, distance from bottom of the electrode.
If the bubbles have a diameter of 3 mm, estimate the void fraction of gas, ε, between the electrodes. Use a surface tension of 0.072 J·m⁻² and a density of 997 kg·m⁻³. Assume the bubbles travel with a terminal velocity given by
Compare estimates for the mass-transfer coefficient from Equations 4.37 and 4.38. Use a diffusivity of 1 × 10⁻⁹ m⁻²·s⁻¹. The viscosity of the solution is 0.00089 Pa·s.

4.17. The Hull cell is used to assess the ability of a plating bath to deposit coatings uniformly; this is known as the “throwing power” of the bath. The cell shown below has two electrodes that are not parallel. The other sides of the cell are insulating. The electrodeposition of a metal is measured on the long side (cathode), and the more uniform the coating, the better the throwing power.

Sketch the potential and current distribution assuming a primary current distribution.

4.18. Please comment on the effect that moving the counter electrode farther away from the working electrode has on the primary and secondary current distributions. Also, name at least one advantage and one disadvantage of doing so.

4.19. Electropolishing is an electrochemical process to improve the surface finish of a metal whereby rough spots are removed anodically. Sometimes this leveling process is characterized as acting on macroroughness or microroughness. Assume that there is a mass-transfer boundary layer on the surface.

Propose a mechanism for the leveling of macroroughness, specifically where the thickness of the boundary layer is small compared to variations in the roughness.
When silver is electropolished in a cyanide bath, CN⁻ ions are needed at the surface to form Ag(CN)₂⁻. Transport of the cyanide ion to the surface may limit the rate of dissolution of silver. Sketch how the current density for anodic dissolution of silver changes with the anode potential.
With microroughness, the variation in thickness of the surface is less than that of the boundary layer. How would the diffusion of the cyanide ion to the surface affect the electropolishing process?

4.20. Derive Equations 4.64 and 4.65.

4.21. Suppose that you have an electrolysis tank that contains 10 pairs of electrodes. The tank is 75 cm long and the gap between electrodes is 4 cm. The tank is 26 cm wide, and the width of the electrodes is 20 cm. The distance between the electrodes and each side of the tank is 3 cm. The temperature is 25 °C.

The tank is used for the electrorefining of copper (i_o = 0.01 A·m⁻², α_a = 1.5, α_c = 0.5) at an average current density of 250 A·m⁻². The composition of the solution is 0.7 M CuSO₄ and 1 M H₂SO₄. The following properties are known:

Would you expect the secondary current distribution to be uniform or nonuniform? Please support your response quantitatively.

4.22. Use a charge balance on a differential control volume to show that at steady state.

4.23. Permeation of vanadium across an ionomer separator of a redox flow battery should be kept small for high efficiency. In contrast to a simple porous material, for an ionomer membrane separator there can be partitioning between the bulk solution and the ionomer membrane (see Figure 4.18c). Consider the case when the current is zero.

If this equilibrium is represented by a partition coefficient, , what is the expression for the molar flux across the separator analogous to Fick’s law?
The product of solubility and diffusivity is called permeability, here equal to DK. For a permeability of 4 × 10⁻¹¹ m²·s⁻¹, estimate the steady-state flux of vanadium across a membrane that is 100 μm thick. On one side, the concentration of V is 1 M, whereas on the other side assume the concentration is zero.
This flux results in a loss of current efficiency for the cell. If the valence state of vanadium changes by one, what current density does this flux represent?

4.24. It is possible to obtain both the diffusivity and the solubility of a diffusing species from one experiment. The experiment involves establishing the concentration on one side of the membrane and measuring the total amount of solute that is transported across the membrane as a function of time. Assume that at the start of the experiment, there is no solute in the membrane.

Figure depicting a total amount (mole or g)-time (s) plot.

Time [s]	Total flux [mol·m⁻²]
45.9	0.0
90.0	0.0
136.9	0.0
179.3	0.5
226.0	1.1
270.9	2.8
315.8	4.4
359.0	6.7
403.5	9.4
446.6	11.9
489.6	14.7
536.9	17.7

Sketch concentration across the membrane as a function of time. Your sketch should include the initial concentration, and the pseudo-steady-state profile, that is, when the flux becomes constant.
Using the data provided, determine the permeability.
The time lag can be estimated, and the diffusivity calculated from the formula . Using these data, estimate both the diffusion coefficient and the partition coefficient for the solute. The concentration on one side is 1 M, and the thickness of the membrane is 400 μm.

4.25. The relative permeability of oxygen and hydrogen, , in a membrane is measured to be 0.4, where H_i is Henry’s law constant, representing the equilibrium between the gas phase and the ionomer defined in Equation 4.67. If air is on one side of the membrane and pure hydrogen on the other, both at atmospheric pressure, and the thickness of the membrane is L,

Estimate the relative rate of hydrogen permeation and oxygen permeation across the membrane film
If there is a catalyst in the membrane where oxygen and hydrogen react instantaneously to form water, sketch the concentration of hydrogen and oxygen across the film.
Derive an expression that describes at what location in the membrane this reaction occurs? How will your answer change if the air is replaced with pure oxygen?

4.26. Calculate the Wagner number for the Zn and Ni electrodes discussed in Section 3.4 (i = 4640 A·m⁻²). Comment on the values that you obtain and the relative magnitude of the kinetic and ohmic resistances that you calculated for this system. Are the values consistent with your expectations? Why or why not?CopycopyHighlighthighlightAdd NotenoteGet Linklink