Chapter 16. Mass Transport in Electrolytic Systems

Transport of ions or charged species is encountered in a wide variety of processes. For example, sodium chloride is ionized in water and exists as sodium and chloride ion pair. Thus the diffusion of sodium chloride in reality involves the diffusion of positively charged sodium ions and negatively charged chloride ions. Such systems are of importance in electrochemical reaction engineering and in electric field–assisted separations. Examples can be found, for example, in electro-winning of metals, fuel cells, batteries, and so on. Equally important is the transport of charged species in charged membranes or solids carrying a net surface charge. An example of such a system is the ion-exchange membrane, which is widely used in water purification, metal recovery by electrodialysis, and selectivity modulation of chemical reactors. Transport of charged species in biological membranes is another example of transport across charged membranes. In addition, a number of techniques to separate large molecules such as DNA or protein involve application of an electric field. Electrophoresis is such an example. The ionic transport effects are also important in the study of surface properties of colloids.

The key feature in the model for transport in such systems is the charge migration due to the electric field. The constitutive equation for diffusion in such systems the classical Nernst-Planck equation, which is discussed in this chapter together with some application examples. Further complexity in models for these systems arise due to the requirement that the electric field has to be computed simultaneously with the concentration field. We explore these situations in detail in this chapter. This chapter therefore provides the background for modeling electrochemical reactors and electric field–based separation processes. These are taken up in some additional details in later chapters (24 and 30).

υ_i = μ_iE

Note that the velocity υ_i is defined for a system with zero total velocity, that is, in a frame of reference moving with a mixture velocity. The previous equation also applies to a positive charge. A negative charge moves in the opposite direction to the field. The two cases can be reconciled by introducing z_i, the valency of the ion. Hence the previous equation can be written as

Here z_i is positive for cations and negative for anions.

The unit for electric field is N/C as it is force on a Coulomb of charge and hence the unit of mobility is m.C/N s or, equivalently m²/V s.

The electric field can be represented as the negative of the gradient of a scalar potential:

The φ is the electric potential with units of J/C same as 1 V.

The mobility is related to diffusivity by the Einstein equation:

where D_i is the diffusion coefficient for the ion. Here F is the Faraday constant, which has a numerical value of 96485 C/mole and represents the charge on one mole of electrons.

Typical values of diffusion coefficients of the various common ions are shown in Table 16.1. Note the relatively high value of H⁺ (protons), which is somewhat inconsistent with its size. The higher value is due to molecular interaction with the water molecules, the so-called Grotthuss effect.

Table 16.1 Diffusion Coefficients of Common Ions in Water at 25°C

Ion	H+	Na+	K++	Ca++	OH–	Cl–	SO4−−
D × 109m2/s	9.313	1.334	1.957	0.7920	5.860	2.032	1.065

The mobility relation can be combined with the Fick’s law of diffusion, leading to the Nernst-Planck equation discussed next.

Finally, expressing the electric field in terms of the potential gradient as per Equation 16.2 we have

Migration flux due to electric field = –D_iz_iC_i(F/R_gT)∇φ

The total flux of a charged species is then obtained by combining Fick’s law, which gives the diffusion due to the concentration gradient, and the flux due to mobility under the electric field. The x-component of this flux, for example, is therefore

If there is a net superimposed system velocity of υ, the contribution of the convective term is also added to obtain the combined flux:

This equation is known as the Nernst-Planck equation. The vector representation is

The three terms on the right-hand side can be identified as diffusion, migration, and convection, respectively. The migration term is the key to transport in charged systems and does not appear in the flux model for uncharged species.

It is common to express the fluxes in terms of electric current in electrochemistry. Flux of a charged species produces a current. The total current density in an electrolyte solution is related to the fluxes by the following equation:

Also, due to conservation requirements the divergence of the current density is zero:

This provides an additional relation between the fluxes and is similar in status to the determinancy condition used in diffusion type of problems.

Note that both the concentration field and the electric potential field have to be computed simultaneously to calculate the flux of each ionic species. The additional equations needed to compute the potential field are presented in Section 16.3.

The following discussion shows why the charge neutrality holds in general and the region where it does not hold.

The electric field is related to charge density ρ_c by Maxwell’s equation:

∇⋅E=ρc∊

The ∊ is the dielectric permittivity of the medium. The electric field can be expressed as the gradient of the electric potential (Equation 16.2). Hence the equation for the potential field is Poisson’s equation, given as

The electrial charge density ρ_c is given as

Using these definitions in Equation 16.10 we have

The permittivity is the product of the free space permittivity and the dielectric constant of the system. The value of ∊₀ (the free space permittivity) is 8.8 × 10^–12 C/V.m and the dielectric constant for water is 80.2. Hence the value of the parameter F/∊ is equal to 1.3592 × 10¹⁶ V.cm/gmol.eq for water. Hence the coefficient on the right-hand side of Equation 16.12 is large. This implies that the left-hand side is nearly zero and hence that the charge density (Equation 16.12) will be nearly zero in most cases. This provides the basis for the charge neutrality relation widely used in electrochemical reaction engineering.

The charge neutrality condition is valid everywhere except near the solid (electrode) surface where there is a thin layer (of the order of 1 to 10 nm) over which the charge density is non-zero. This thin region is known as the electrical double layer or the Debye layer. The thickness of the Debye layer, denoted as λ, can be calculated by the following equation:

where C_∞ is the concentration of either the cation or anion just outside the Debye layer where electroneutrality holds.

Electroneutrality does not hold in the Debye region but holds everywhere else to a first approximation. However, often electroneutrality is assumed to hold all the way to the electrode surface and the effects due to a double layer are incorporated indirectly into the boundary condition at the electrode.

The last term is taken as zero due to electroneutrality and is dropped from further discussion. We define the conductivity of the solution by the following equation:

κ=F∑izi2μiCi

We can also write this in terms of the diffusivity using the Einstein relation as

κ=F2RgT(∑izi2DiCi)

Both expressions are completely equivalent. Using this expression for the conductivity, Equation 16.14 can be written as

Rearranging this we get the following general expression for the electric field:

The three terms on the right-hand side can be interpreted as an Ohmic term with the actual current, the potential caused by diffusion, and the streaming potential caused by the bulk flow of the charges. The last term is generally zero whenever electroneutrality holds but may not be zero in some cases, for example, transport in charged membranes. Here the electroneutrality applies only if the immobile charge of the membrane is also included. Thus the term Σz_ic_i is not zero for mobile charges (charges transported across the charged membranes) and this term contributes to the streaming potential.

Consider Equation 16.15. If there are no concentration gradients in the system, the current is simply related to the electric potential gradient and Ohm’s law is applicable:

Using the current continuity condition in Equation 16.7 we find that

∇²φ = 0

Hence the Laplace equation holds for the potential field. As seen from the previous discussion, this equation for electric potential is valid only under certain conditions such as constant electrical conductivity of the medium and no concentration gradients in the system (e.g., bulk liquid outside the diffusion boundary layers). In such cases, it is appropriate to use the Laplace equation to calculate the potential field, but such calculations may be inaccurate near the concentration boundary layer (diffusion film) near the electrodes. In general Equation 16.16 is appliable for finding the potential.

ti=zi2μici∑izi2μici

Differences in transport number arise from differences in electrical mobility. For example, in a solution of sodium chloride, less than half of the current is carried by the positively charged sodium cations and more than half is carried by the negatively charged chloride anions because the chloride ions are able to move faster, that is, chloride ions have higher mobility/diffusivity than sodium cations (see Table 16.1 for values). The sum of the transport numbers for all of the ions in a solution equals unity.

–∇ · N _i + R_i = 0

where R_i is the rate of production of species i by homogeneous reaction in the liquid. Note that the reaction on the electrode surface is not included here and will appear as the wall boundary condition.

The reaction at the electrode surface is modeled as a Robin boundary condition at the wall. Simply stated this is the balance of the flux to the surface to the reaction rate for each species. The surface concentration of a reacting species at the electrode is determined by the kinetics of the reaction. There is discussion on the kinetic models for electrode reactions in Chapter 24 where we show that the kinetics of the reaction is usually modeled by the Butler-Volmer kinetics or by a simple Tafel model. In general, rate is a function of the overpotential (potential at the surface minus the equilibrium value). Thus rate can be varied by changing the potential of the electrode. Rate is also a function of the concentration and temperature.

If the surface reaction is rapid, the concentration of the reacting species is zero at the electrode. This condition determines the maximum current in the system, also known as the limiting current. This concept is again similar to a catalytic heterogeneous reaction; the concentration of a limiting reactant is nearly zero for a fast reaction at a surface.

With this background material in hand, we show some illustrative examples for some prototypical problems.

We assume that the electrolyte is fully ionized and diffuses as M+ and X– ions. The fluxes are given by the Nernst-Planck equation:

J(M+)=−D+dCMdx−z+D+(F/RgT)CMdφdx

A similar equation holds for X–:

J(X−)=−D−dCXdx−z−D−(F/RgT)CXdφdx

A univalent electrolyte is considered for further discussion; z₊ = 1 and z_– = –1 is used further on. The electroneutrality implies that

C_M = C_X = say, C

where C is either C_M or C_X.

Also, if no net current is drawn across the membrane the two fluxes are equal:

J(M+) = J(X–) = J_salt

This helps us to eliminate the potential term. Hence the following expression for dφ/dx can be obtained by combining the two flux equations with the requirements of electroneutrality and equal fluxes for both cations and anions:

Integrating across the system gives an expression for the potential:

This potential is known as the diffusion potential and arises whenever the diffusion coefficients of the cation and anion are not equal.

Using this back in the flux expression, we can now calculate the flux of either of the species. The result can be expressed in a Fick’s law type of equation as

J(M+)=J(X−)=J(salt)=−DeffdCdx

where the effective diffusion coefficient turns out to be

which is known as the ambipolar diffusion coefficient. The expression can be written in a simple form for the case of equal charges, z₊ = –z_–, as

Deff=2D+D−D++D−

or

2Deff=1D++1D−

Thus the effective diffusivity is a harmonic mean of the positive and negative ion diffusivities. The concept of ambipolar diffusion is illustrated in Figure 16.1.

In Figure 16.1 the direction of the transport is from left to right and both H⁺ and Cl^– move to the right. The faster moving H⁺ ions set up an electric field in the opposite direction to its motion (to the left). The electric field slows the hydrogen ions down and these ions are pushed back somewhat by the field. The overall result is that H⁺ and Cl^– diffuse at the same rate and have the same (ambipolar) diffusivity.

The problem analyzed is that of a membrane exposed to two different salt concentrations on either side (x = 0 and x = L). The transport of the salt in ionized form as cations and anions take place across the membrane. Also we assume there is no current in the system. (The current will be non-zero if some electrode reactions are taking place downstream and upstream of the membrane, for example, in electrodialysis; a modified analysis is needed for such cases.) The flux of the salt across the system is to be calculated. External mass transport resistance near the diffusion film on either side of the membrane is neglected for simplicity since we want to focus on the transport rate in the membrane itself. A univalent electrolyte is considered for simplicity.

The concentrations at the interfaces (x = 0 and L) are different at the liquid side and the membrane side. This is first calculated from thermodynamic considerations.

Since the charge of the membrane is fixed and does not penetrate into the solution there is a concentration jump/fall at the solution–membrane interface. Hence the concentrations at the membrane surface are not equal to the bulk concentrations and are to be calculated by the following procedure. This discussion uses a univalent electrolyte of type MX with z₊ = 1 and z_– = –1, but the concepts can be extended to other types of electrolytes.

The ionic products of the cations and anions are fixed. They should be the same on the bulk side and the membrane side of the interface. Hence

C_+,b × C_–,b = C_+,0 × C_–,0

where b represents the bulk values and 0 refers to values at x = 0⁺, the membrane side values.

Electroneutrality on the membrane side requires that C_+,0 = C_–,0 + C_–,m for a negatively charged membrane carrying a charge of C_–,m. Electroneutrality in the bulk requires C_+,b = C_–,b. This is illustrated in Figure 16.2.

Now, equating the ionic products on either side we get a quadratic equation for C_–,0, the solution of which results in the boxed equation for C_–,0 shown in Figure 16.2. This equation is often called the Donnan equilibrium conditions. The value of C_+,0 can then be computed using the electroneutrality on the membrane side of the interface. A similar equation can be derived for C_–,L as well.

Having obtained the concentrations on the membrane side of each of the boundaries, we now look at the transport model.

Hence

and

The electroneutrality condition, as stated earlier, is

C₊ = C_– + C_–m

Differentiating this with respect to x we get

dC+dx=dC−dx

Here C_–m is taken as a constant, that is, there is constant fixed charge across the membrane.

The current in the system is given as F (N₊ – N_–) for a univalent binary electrolyte. We analyze the case here where the net current is zero. This means that the fluxes are equal. Hence

N₊ = N_– = N_salt

Subtracting the two Nernst-Planck equations and using these two conditions you will be able to show that the potential field is given by the following equation:

This can be now backsubstituted into either of the two Nernst-Plank equations to get the flux of each ion, which is also the flux of the salt transported across the system. This results in the following expression for the salt flux in terms of the concentration gradient of the anions:

Integrating this across the system, keeping N_salt constant, yields an expression for the flux across the membrane:

The following limiting cases can be identified:

• No charge on the membrane: In this case C_–m = 0 and the model reverts to the case of an uncharged membrane. Only the first term on the right-hand side of Equation 16.25 remains and the term with all the diffusion coefficients can be identified as the ambipolar diffusion coefficient.

• Cations and anions have the same diffusion coefficient: In this case again the uncharged membrane model applies (with Donnan correction at the end points) since the potential across the membrane can now be shown to be equal to zero.

• Highly charged membrane: The limit of C_–m tending to infinity is now taken. It can be shown that the mass transfer of anions is the rate limiting step:

  Nsalt=−D−dC−dx

  The Donnan relation simplifies to C_– equals C0,b2/C−m for this case. Hence the previous equation can be expressed in terms of the bulk concentrations of the anions as

  Nsalt=D−L(C0,b2−CL,b2)C−m(16.26)

We assume quasi-steady state conditions and use the concept of a diffusion film near the cathode. We also neglect convection effects and use the low mass flux approximation: J_A = N_A. The concentration of the salt is denoted as C_Mb in the bulk liquid and C_Ms at the cathode surface. Recall that the film model flux is described by the following relation in the absence of migration effects:

J(M+)=DMδf(CMb−CMs)=kL(CMb−CMs)

Here δ_f is the film thickness. Since M is a charged species, the migration contribution to the flux has to be included and the goal of this section is to study the effect of migration and reexamine the enhancement in transport rate due to migration. A schematic for this problem is presented in Figure 16.3.

The model starts with the Nernst-Planck equation, which is applied for both species X^– and M⁺. The procedure is similar to that presented in the previous two sections with the difference that there is a net current in the system. Thus the determinancy condition has to be expressed differently here.

Since there is no flux of species X^– (denoted by subscript X) we can equate J_X to zero:

Note that z_i is taken as –1 in the Nernst-Planck equation. The electric field can be solved from the previous equation:

Charge equality means C_M = C. (M here means the species M⁺.) Hence the electric field can also be expressed as

dφdx=RgTF1CMdCMdx

For M⁺ we have from the Nernst-Planck equation

JM=−DMdCMdx−DMFRgTCMdφdx

Note the minus sign on the second term of the right-hand side since z_i = +1 now.

Using the expression for the electric field given by Equation 16.28 and simplifying we find

The simple Fick’s law would have produced the above result but without the factor of two. Hence the effect of the electric field is to enhance the transport by a factor of two over that given by the simple Fick’s law.

Since J_M is constant in the diffusion film, we find the concentration profile is linear. Also, the concentration profiles for both species M and X are the same. Thus there is a concentration gradient for species X in the film as well, but this does not imply that X is diffusing toward the cathode. The diffusive flux of X caused by the favorable concentration gradient is balanced by the migration flux of X in the opposite direction. The net flux of X is therefore zero! The transport processes of diffusion and migration for both cations and anions near a cathode are schematically explained in Figure 16.4.

The electric potential field can be computed by integration of Equation 16.28, keeping the concentration gradient constant. The result is

ϕ(0)−ϕ(δf)=RgTFln(CX(0)CX(δf))

The surface concentration of M will depend on the rate of electrode reaction. Hence the transport rate of M in the film is coupled with a kinetic model for the electrode reaction. Commonly used kinetic models are shown in Chapter 24 where we demonstrate examples of the combined model to find the rate of electrode reaction. The emphasis here is to show the analogy with mass transfer followed by the heterogeneous reaction studied in Section 6.4. A limiting case is examined next.

For a fast reaction the concentration of M at the electrode is nearly zero. Hence C_Ms is set to zero and the process is limited by mass transfer rate to the electrode. The corresponding current in the system can be calculated as

iL=2FDMδfCMb

and is called the limiting current.

Note that the factor of 2 arises due to the enhancement in diffusional mass transfer due to migration. This factor is applicable when there are no other electrolytes in the system. The role of a second electrolyte is discussed briefly in the following paragraph.

The enhancement factor of two (over Fick’s value) in transport rate is valid if there are no additional ions in the system. If there is another salt of composition NX then the effect of the migration term will be smaller than one and in the limiting case there will be no enhancement and Fick’s law simply holds. This happens because the electroneutrality now requires X = M + N. Since N is not reacting, the overall contribution of migration is reduced. Thus the rate of mass transport can be calculated based on the diffusional considerations alone for cases where there is an excess of a “supporting” electrolyte.

16.2 What is the relation between R_gT/F and k_BT/e?

16.3 Calculate the value of R_gT/F at 298 K.

16.4 Verify that the Einstein relation given by Equation 16.3 is dimensionally consistent.

16.5 Find the mobility of H⁺, OH^– ions from the diffusivity data given in Table 16.1.

16.6 Define the conductivity of an electrolyte solution and the transference number and give equations to calculate these.

16.7 When does Ohm’s law apply in an electrolyte solution? When does it not apply?

16.8 What is the Debye length and what is its physical significance?

16.9 What is meant by the term limiting current in an electrochemical reactor?

16.10 Explain briefly the concept of ambipolar diffusion.

16.11 When does an electric potential develop for transport of ions across an uncharged membrane?

16.12 What is the direction of the electric field in an uncharged membrane?

16.13 State the basis of the Donnan equilibrium relation.

16.14 When is the potential across a charged membrane equal to zero?

16.15 What is the enhancement in mass transfer for a cation diffusing and reacting at a cathode?

16.16 When should the enhancement due to migration effects be applied for film diffusion toward an electrode?

16.2 Conductivity of a salt solution. Calculate the value conductivity of 0.1 M and 1 M solutions of NaCl. Also calculate the transference numbers and indicate the current carried by the cation and anion separately.

16.3 Conditions for Laplace equation to hold. Verify that the divergence of the current is equal to zero. Hence show that the Laplace equation holds for the potential field for cases with no concentration gradients.

16.4 Poisson equation for the electric potential. When there is a concentration gradient in the system show that the potential gradient is composed of two terms: an Ohm’s law contribution and a diffusional contribution. Here we assume there is no contribution of the superimposed velocity v. State the equation for the current. Now take the divergence of the current and show that the following Poisson equation holds for the potential field:

∇⋅(κ∇ϕ)=−F∑izi∇⋅(Di∇ci)

Show the similarity to mass transport with generation with variable diffusivity. What assumption is implicit in the above equation?

16.5 Induced potential due to difference in mobilities. A membrane separates two bulk solutions of 0.5 M NaCl and 0.1 M NaCl. Find the potential difference developed across the membrane and the flux of NaCl across the membrane.

16.6 Diffusion potential calculations. Calculate the diffusion potential for an uncharged membrane with a concentration of 0.5 M on one side and 0.1 M on the other side for the solutions of (1) CuSO₄, (2) MgCl₂, and (3) KCl. Also calculate the ambipolar diffusion coefficient for these systems.

16.7 Transport rate across an uncharged membrane. Consider an uncharged membrane of thickness 100 μm with concentrations of 1 M HCl on one side and 0.1 M HCl on the other side. Assume that the HCl is completely ionized. (1) Find the flux of the HCl in the system. (2) Find the diffusion potential generated in the system. (3) Which side of the membrane is at a higher potential? (4) Find the diffusion flux and migration flux of Cl^– and H⁺ ions and tabulate these. Comment on the results.

16.8 Diffusion across a disk for a mixture of two salts. Two solutions are separated by a porous sintered disk (1 mm thick) that permits diffusion across the disk. On one side we have a mixture of 1 M HCl and 1 M BaCl₂ while on the other side we have pure water. Both salts are completely ionized and diffuse as H⁺, Cl^–, and Ba⁺⁺ across the disk. It is required to find the flux across the system. Assume zero current flow across the membrane. Set up the model to compute the fluxes. Based on the fluxes find the effective diffusivity of these ions across the disk. Use the following value of ionic diffusivity for Ba ions: 0.85 × 10^–9m²/s. For the other ions use the values in Table 16.1.

16.9 Diffusivity of a weakly ionizing acid solution. Diffusion of weakly ionized acids such as acetic acid is an interesting and complex problem in diffusion. The diffusion occurs by transport of the ionized species CH₃COO^– as well as by the unionized acid CH₃COOH. The diffusion of acetate ion is affected by H-ions due to electroneutrality while that for unionized is constant. For acetate ions an ambipolar diffusion coefficient may be used. Set up a model to compute the flux based on this model. Express the results in terms of the total concentration of acetic acid on either side of a region. What is the dependency of flux on concentration? Is flux linear in acetic acid concentration?

16.10 Concentration jump at a charged membrane interface. A surface of a charged membrane is in contact with a 0.25 M NaCl solution. The membrane carries a fixed anion concentration of 0.05 M. Find the concentrations of sodium and chloride ions on the membrane side of the interface.

16.11 Transport rate across a charged membrane. Consider a charged membrane of thickness 100 μm with concentrations of 1 M HCl on one side and 0.1 M HCl on the other side. The membrane has a negative charge. Assume this generates a constant electric field of 1 N/C. Find the flux of HCl in the system using the model for transport across a charged membrane given in the text.

16.12 Transport across a charged membrane for a mixture of two electrolytes. Extend the analysis for a system for transport in a charged membrane for the case of a mixture of two electrolyte salts with a common anion (a mixture of NaCl and KCl for example). Derive a formula for the membrane potential when one side of the membrane is exposed to concentrations of C₁(0) for KCl and C₂(0) for NaCl while the other side at L is exposed to C₁(L) and C₂(L). Calculate the potential for membrane that is 100 μm thick with the concentrations of 0.1 and 0.01 M on either side.

16.13 Copper deposition at an electrode and the limiting current. Copper is deposited at a cathode from solution with a bulk concentration of 0.5 M at a rate of 3.0 g/m²s. Find the surface concentration of Cu⁺⁺ at the cathode if the mass transfer coefficient from the bulk to the surface is 1 × 10^–4 m/s. Find the current density. Determine the maximum rate of deposition.

16.14 Effect of supporting electrolyte in copper deposition. Consider a system with a “supporting” electrolyte, for example, CuSO₄ and a second salt Na₂SO₄ that serves as the supporting electrolyte. The system now consists of Cu⁺⁺, SO4−−, and Na⁺. Assume that only Cu⁺⁺ can react at the cathode. Hence the flux of Cu⁺⁺ should be assumed to be non-zero while that of the other species is zero. Develop an equation to find the diffusion potential developed in this system and the factor by which Cu⁺⁺ flux to the cathode is enhanced.