IV – SUBMODELS
IV.1 – Thermodynamics and kinetics; IV.2 – Gas transport; IV.3 – Water adsorption and transport in PCP; IV.4 – Charge transport; IV.5 – References.
IV.1 – Thermodynamics and kinetics
The mathematical description of water recombination reaction:
2 2 (g) ) ACP ( ) PCP ( O H O H 2 + + − (reac. IV.1.1)
starts from the equilibrium condition and all the relationships that we will write about the electrochemical kinetics must be coherent with the description at the equilibrium. We have already shown the genesis of a difference of electric potential between ACP and PCP in par. II.2 by simple qualitative considerations, now we are going to enter in details.
Consider the CM as a closed system at equilibrium, it leads to uniform conditions of temperature, pressure and phase compositions. The equilibrium is described by the Gibbs energy of the system, in particular the balance of electrochemical potentials of species participating in the reaction is (Bard and Faulkner, 2003):
eq ) g ( w eq ) ACP ( O eq ) PCP ( H ~ ~ ~ 2µ + +µ −2 =µ (eq. IV.1.1)
In particular, it is possible to explode each term:
(
)
(
+ + + +)
+(
− +(
−)
−)
= eq ACP eq ) ACP ( O g 0 ) ACP ( O eq PCP eq ) PCP ( H g 0 ) PCP ( H R Tlna FV R Tlna 2FV 2µ µ 2 2( )
eq ) g ( w g 0 ) g ( w +R Tlna =µ
(eq. IV.1.2)in which the terms aieq represent the activities of species compared with standard
conditions (marked with the superscript 0). By combining together the standard chemical potentials
µ
i0 and arranging the equation, we obtain the absolute difference of(
)
− − = − = − + eq ) ACP ( O 2 eq ) PCP ( H eq ) g ( w g 0 CM eq PCP eq ACP eq abs 2 a a a ln F 2 T R F 2 G V V V ∆ ∆ (eq. IV.1.3)in which we call ∆G0CM the standard Gibbs free energy of (reac. IV.1.1), i.e.:
0 ) ACP ( O 0 ) PCP ( H 0 ) g ( w 0 CM 2 2 G =µ − µ + −µ − ∆ (eq. IV.1.4)
Now, for species in solid phases (i.e. H+ and O-2) activities are equal to 1 because they are invariable with external conditions1 (it is the same consideration that we usually do for activities of solid species in the equilibrium of solubility, for example), for water in gas phase the activity can be expressed as the ratio between the partial pressure at the equilibrium pweq and the partial pressure in standard conditions pw0 = 1.013·105Pa.
We could apply the criterion of (eq. IV.1.1) also to the global cell reaction obtaining the Nernst law of equilibrium as expressed in (eq. I.3.11). Note that also (eq. IV.1.3) represents the Nernst law, it is applied to (reac. IV.1.1) instead of the global reaction of the whole cell. In this way, we have mathematically shown that the water recombination reaction that occurs in the CM follows the laws of electrochemical equilibrium starting from elementary thermodynamic definitions.
It is possible now to investigate about the kinetics. We have already said in par. II.2 that the rate of water recombination is dependent on the value of the difference of potential VACP - VPCP compared with the same difference in equilibrium conditions and
we have also shown the reason. In particular, if the overpotential η, defined as:
(
ACP PCP)
eq abs V V V − − =∆ η (eq. IV.1.5)1 We will see in par. IV.3 that the concentration of protonic defects in PCP is a variable; it leads to think that the activity aH
+
(PCP) is not always equal to 1. But we will also demonstrate that protonic defects are not exactly protons that participate to water recombination reaction, they are the way used by PCP to adsorb water in solid state. It could be misleading, but for the electrochemical reaction the activity of protons is not a variable that we can change as, for example, the partial pressure of water in gas phase; on the other hand, the concentration of protonic defects is a variable (and changes the conductivity of PCP) but it represents only the measure of the concentration of adsorbed water in PCP (note that adsorbed water is not involved directly in (reac. IV.1.1)). The misunderstanding comes from the fact that protons (that participate to the reaction) and protonic defects (i.e. water adsorbed) are quite similar and can be confused; this situation would not happen if there was another substance instead of water (that is the product of the reaction) that was adsorbed and that affected the conductivity of PCP.
is positive, (reac. IV.1.1) flows from left to right because it is easier for protons and oxygen ions to react according to the fact that the difference of electric potential between ACP and PCP is lower than at equilibrium. The difference of potential between ACP and PCP is a contribution of energy that shall be won such as the thermal-chemical activation energy and it is valid for both direct and reverse reactions. In other words, the overpotential enters in the kinetic constants and affects the rate of the reaction. We do not want to illustrate all the details concerning electrochemical kinetics, for a comprehensive explanation see Bard and Faulkner (2001).
We assume that the kinetics can be represented by the expression:
(
)
− − − = α η α η T R F 1 exp T R F exp i i g g 0 s (eq. IV.1.6)in which is is the current per unit of length of TPB (i.e. A/mTPB in SI), i0 is the exchange
current (i.e. the kinetic parameter, close to the definition of kinetic constant in chemical
reactions) and α the transfer coefficient (a parameter that affects the symmetry of the relationship is(η)). Note that if is > 0 water is produced and vice versa; in particular,
considering protons and oxygen ions as current and considering that their recombination yields water, is represents the flow of water that leaves the reaction site per unit of
length of TPB (i.e. mol/(mTPB·s) in SI) multiplied by 2F (remember that water “adsorbs”
2 charges per molecule according to (reac. IV.1.1)). Note that the kinetics is consistent with equilibrium: when η = 0 the water production is zero.
(eq. IV.1.6) is a macro-kinetic description that is valid under the conditions:
1. the rate limiting step of the mechanism is the transfer of a single charged species; 2. no mass-transfer effects.
Point 1 means that the mechanism of (reac. IV.1.1) is composed of some elementary steps but only one is the rate determining one and, in particular, it involves the transfer of a single charged species; for example, we could imagine that the rate determining step could be the transfer of a proton on ACP or of OH- (i.e. an oxygen ion previously combined with a proton) from ACP to PCP. It is a reasonable assumption because several electrochemical reactions follow this kind of behaviour. The second assumption is a simplification, in this way there is not a direct dependence of is on concentrations of
reagents or products. Then, we also assume α = 0.5, i.e. the curve is(η) is
partial pressure of water, electric potentials, etc.), it is a kinetic parameter that depends only on temperature.
This is the simplest form that we can assume and this assumption is reasonable because we do not know anything about the kinetics of water recombination. Specific investigations are needed in this direction. Thus, also other electrochemical macro-kinetics or micro-macro-kinetics forms could be considered in the model of the CM.
All the considerations that we have done are valid and coherent even if we do not know the standard Gibbs free energy of water recombination as in our case; so, it is an unknown value but we do not need it (i.e. we do not need the knowledge of the value
∆V0). When only CM is modeled, absolute voltages are not of interest, it is important only the overpotential, i.e. the relative gap of electric potentials from equilibrium. When the cell is in equilibrium with a constant external partial pressure of water pwex, we can
assume as reference ∆Veq = 0 (or another generic constant value): in this situation in the CM ACP and PCP have ideally the same potentials and there is not reaction (it means that also VACP and VPCP are not absolute potentials but referred to this condition, i.e. we
have set a reference value for electric potentials2). When external circuit is closed, water recombination may occur in each point of the CM yielding local values of partial pressure of water pw > pwex. So, we calculate the local condition of equilibrium as:
− = ex w w g eq p p ln F 2 T R V
∆
(eq. IV.1.7)and again is by (eq. IV.1.6) with η defined as (eq. IV.1.5) by using the relative ∆Veq
instead of the absolute ∆Veqabs in (eq. IV.1.3), i.e.:
(
ACP PCP)
eq V V V − − =∆ η (eq. IV.1.8)Note that the equilibrium is respected (i.e. when pw = pwex reaction stops if no
overpotential is applied). Obviously, the relationships are still valid if the cell works in the direction of water consumption.
This last consideration could seem quite obscure, but the use of (eq. IV.1.7) is
equivalent to use (eq. IV.1.3) in which we subtract ∆Veq,exabs (that is a constant value if
pwex is fixed) from both sides of the equation as:
(
)
− − = − + eq ) ACP ( O 2 eq ) PCP ( H 0 ex w g 0 CM ex , eq abs 2 a a p p ln F 2 T R F 2 G V ∆ ∆ (eq. IV.1.9)that in (eq. IV.1.3) yields the relative difference of potential at local equilibrium ∆Veq:
(
)
− ⇒ − − = − − + ex , eq abs eq ) ACP ( O 2 eq ) PCP ( H 0 w g 0 CM ex , eq abs eq abs V a a p p ln F 2 T R F 2 G V V 2∆
∆
∆
∆
− = ⇒ ex w w g eq p p ln F 2 T R V∆
(eq. IV.1.10)Concluding, in the model of the CM we will use (eq. IV.1.6) as kinetic law coupled with (eq. IV.1.8) and (eq. IV.1.7) instead of the system given by (eq. IV.1.6), (eq. IV.1.5) and (eq. IV.1.3) that needs the knowledge of ∆G0CM.
IV.2 – Gas transport
In par. II.5 we said that we expect to describe the transfer of gaseous species in the porous media by the resultant of a combination of driving forces, i.e. gradient of total pressure, that produces a viscous non-separative flow, and gradients of partial pressure of components, that produce separative flows (i.e. mixing). All the description must take into account the flow region according to Knudsen number: continuum (Kn < 0.1), transition (0.1 < Kn < 10) or Knudsen region (Kn > 10)3. We neglect the surface diffusion of gaseous substances (i.e. the transport along the solid surfaces of physical adsorbed molecules) and assume the gas phase as an ideal gas (this assumption is reasonable because temperature is high and pressure is not too high).
3
The limits among regions are reasonable but other values are sometimes used in literature; these limits are the same used by Kenney et al. (2009).
The mathematical description of the mass transfer in gas phase in porous media can be done by using two different models: the mean transport pore model (MTPM), that assumes that the porous media can be visualized as cylindrical capillaries (with a distribution of their diameters around a mean value), or the Dusty Gas model (DGM), that visualizes the porous structure as giant spherical molecules (dust particles) kept in their fixed position by external forces. Both of them yield relationships for fluxes4 as a function of total pressure and composition of the gas mixture. The second model is closer to the reality of the CM and to our approach to describe the porous media (i.e. a packing of quite spherical particles), so we use it for a binary mixture of water (suffix: w) and nitrogen (suffix: B).
In the DGM the movements of gaseous components in pores are described by using the kinetic theory of gases. According to the temperatures, pressures and the mean dimensions of pore of the CM, we expect to be in transition region; in particular, at 873K for the range of total pressure 0.435-17.03atm gas transport is in transition region5, it is a reasonable range for our case. However, we will always verify Knudsen numbers after the solution of the model. In the DGM for transition region non-separative fluxes are due to gradient of total pressure with permeability as proportional factor (i.e. Darcy law is used) while diffusive fluxes are due to gradients of partial pressure of components and the proportionality factor is a diffusion coefficient that combines the ordinary diffusion coefficient and the Knudsen diffusion coefficient. The expressions of the fluxes of water and nitrogen in gas phase (respectively Nw,g and NB,g,
mol/(m2·s) in SI), according to Arnošt and Schneider (1995), are:
P D B 1 T R x x T R P N D x D x 1 D 1 N K,app w g w w g g , B app wB w app wB w app , K w g , w ∇ + − ∇ − = − + (eq. IV.2.1) P D B 1 T R x 1 x T R P N D x 1 D x D 1 N K,app B g w w g g , w app wB w app wB w app , K B g , B ∇ + − − ∇ + − = + (eq. IV.2.2)
in which P is the total pressure and xw the molar fraction of water in gas phase.
Diffusion coefficients, both ordinary and Knudsen, are apparent, i.e. they are corrected
4 As explained in par. II.7, fluxes are referred to the nominal surface, they are not the effective fluxes inside the pores but the apparent fluxes normalized on the global section of the media.
5 This calculus is made by using (eq. IV.2.13); for d
by the porosity and the tortuosity of the gas phase τg to take into account respectively
that the media is not a continuum and that pores are not straight paths6; it yields:
wB g fin g app wB D D τ φ = (eq. IV.2.3) K i g fin g app , K i D D τ φ = (eq. IV.2.4)
in which DwB is the ordinary diffusion coefficient of the mixture water-nitrogen (see
sec. V.4.3 for its estimation) and DiK are the Knudsen diffusion coefficients calculated
as (Arnošt and Schneider, 1995):
i g p K i M 2 T R d 3 4 D
π
= (eq. IV.2.5)in which dp is the mean pore diameter (see par. III.6 for its estimation)7 and Mi the
molecular weights of the substances (in kg/mol with other values in SI to obtain diffusivity in SI).
The permeability B is defined as (Arnošt and Schneider, 1995):
µ
τ
φ
P 32 d B 2 p g fin g = (eq. IV.2.6)so corrected by porosity and tortuosity, where
µ
is the dynamic viscosity of the mixture (see sec. V.4.3).Fluxes can be written in explicit forms by rearranging (eq. IV.2.1-2) yielding:
P x Nw,g =
α
w∇ w −β
w∇ (eq. IV.2.7) P x NB,g =−α
B∇ w −β
B∇ (eq. IV.2.8)6 We use a single value of tortuosity of gas phase to correct both ordinary and Knudsen diffusion coefficients, in general two values of τg should be used (i.e. τgK and τgo) because the mechanisms of transport are different (Kast and Hohenthanner, 2000). It is difficult to estimate tortuosity factors, so we neglect this distinction; moreover DGM does not take into account this matter.
7
In par. III.6 we talked about the mean hydraulic radius Rh and we said that dp = 4Rh; the mean radius of pores (called <r> by Arnošt and Schneider, 1995) is equal to dp/2, i.e. 2Rh, do not make confusion.
in which proportional factors
α
i andβ
i are functions of the variables xw and P (so (eq.IV.2.7-8) are not linear expressions):
(
)
(
)
2 app wB w w app wB w app , K B pp a wB w app , K w app , K B g w D x 1 x D x D 1 D x 1 D 1 D T R P − − + ⋅ − + ⋅ − =α
(eq. IV.2.9)(
)
(
)
2 app wB w w app wB w app , K B pp a wB w app , K w app wB w app , K B app , K w w app , K B app wB w w g w D x 1 x D x D 1 D x 1 D 1 D x D 1 D B 1 x D B 1 D x 1 x T R 1 − − + ⋅ − + + + + + − =β
(eq. IV.2.10)(
)
(
)
2 app wB w w app wB w app , K w pp a wB w app , K B app , K w g B D x 1 x D x 1 D 1 D x D 1 D T R P − − − + ⋅ + ⋅ − =α
(eq. IV.2.11)(
)
(
)
(
)
2 app wB w w app wB w app , K w pp a wB w app , K B app wB w app , K w app , K B w app , K w app wB w w g B D x 1 x D x 1 D 1 D x D 1 D x 1 D 1 D B 1 x 1 D B 1 D x 1 x T R 1 − − − + ⋅ + − + + − + + − =β
(eq. IV.2.12)To check the region of flow, Knudsen numbers8 for water and nitrogen shall be calculated according to Kenney et al. (2009) as:
p 2 i , m B p i , MFP i d P d 2 T d Kn ⋅ ⋅ = =
π
κ
λ
(eq. IV.2.13)in which
κ
B is the Boltzmann constant and dm,i are the mean diameters of gaseousmolecules (see sec. V.4.3);
λ
MFPi are the mean free paths of the gas molecules.The last parameter that closes the relationships mentioned above is the tortuosity of gas phase
τ
g. We need a consistent estimation of it. Now, the porous media is made by a
8
We use simplified expressions for Knudsen numbers in gas mixtures because their roles are marginal (we use them only to check the region of flow). For more details see Schneider (1978).
packing of spheres, so also Darcy law in Blake-Kozeny form (Mauri, 2005) can be used to describe the viscous (non-separative) component of flow Nvis:
(
)
P 1 150 D T R P N 2 fin g 3 fin g 2 p g vis ∇ − − =φ
φ
µ
(eq. IV.2.14) instead of: P T R B N gvis =− ∇ (eq. IV.2.15)
by using the DGM approach. Comparing the last two equations, by using (eq. IV.2.6) to calculate B and (eq. III.6.8-9) to write the mean particle diameter Dp as a function of the
mean pore diameter dp, we obtain:
083 . 2 32 6 150 4 2 2 g = ⋅ ⋅ =
τ
(eq. IV.2.16)So, for a porous media made of a packing of spheres the tortuosity factor is equal to 2.083, other porous media with different structures (e.g. a composite of fibers, solid with gas phase similar to a foam, etc.) will have different values of tortuosity. It is clear that (eq. IV.2.16) is a rough but coherent estimation of the tortuosity factor, specific measurements of permeability could be performed to check this value for the CM.
IV.3 –Water adsorption and transport in PCP
In par. II.6 we talked about the behaviour of some proton-conducting materials, in particular based on the perovskite-structure ABO3 (with A+2, B+4)9, to adsorb and
transfer water in solid phase; we did not say anything about the mechanism of transport in solid phase because to talk about a mechanism means that we are implying a law of transport. In this paragraph we will enter in details and we will show that water
9
The perovskite-cell has a cubic structure with B in the center of the cube, oxygens in the center of each face and A in the corners.
(
••)
) PCP ( O V(
x)
) PCP ( O O(
•)
) PCP ( O OH(
•)
) PCP ( O OHtransport through PCP is a diffusion and not another kind of transport (e.g. migration or a combination of diffusion and migration).
Let us take a piece of proton-conducting material surrounded by a wet gas phase, at the PCP-gas phase interface water is adsorbed to create protonic defects (Kreuer, 2003) as:
+ •• + O•(PCP) x ) PCP ( O ) PCP ( O ) g ( 2O V O 2OH H (reac. IV.3.1)
in Kröger-Vink notation and, in particular, OOx represents oxygen ion in the lattice,
VO oxide ion vacancy10 and OHO is a protonic defect. In other words, (reac. IV.3.1)
means that a water molecule reacts with a vacancy and an oxygen in the lattice to produce two protonic defects, i.e. the oxygen of water occupies the vacant site and gives a proton to the oxygen in the lattice (fig. IV.1).
fig. IV.1 – Representation of water adsorption in PCP.
Protonic defects can also be transported through the PCP, so entering in the bulk of the solid phase. Imagine a layer of dry proton-conducting material with one side exposed to wet gas phase (in particular we impose a constant adsorption flux of water) as in fig. IV.2a. Theoretically, within the PCP three species can be transported, i.e. OOx, VO and
OHO; we can assume the PCP as a continuum and call their concentrations in solid
phase CVO, CO and COH. We can also refer these concentrations per unit of
perovskite-cell, cVO, cO and cOH, as:
10 Oxide ion vacancies are due to the presence of a dopant (the material conductor is doped to achieve particular wished properties). When ABO3 is doped with an amount S of a trivalent ion D, AB1-SDSO3-S/2 is formed (e.g. in the central membrane BCY is used, i.e. BaCeO3 doped with yttria; when the dopant level is S the chemical formula is BaCe1-SYSO3-S/2). So, initially (i.e. without adsorption of water) the number of oxide ion vacancies is equal to S/2 per unit of perovskite-cell.
H H H H
a) b)
fig. IV.2 – a) Representation of a case study of water adsorption in PCP; b) situation if DOH > DO after a few seconds.
δ i i
C
c = (eq. IV.3.1)
in which δ is the density of perovskite-cells per unit volume (i.e. the number of moles of cells per cubic meter, see sec. V.4.5).
Now, because the total number of oxygen sites is constant and in particular for the perovskite-cell is:
3 c c
cVO + O + OH = (eq. IV.3.2)
the three transports are not independent. If we know two of the three concentrations the other one is assigned; it means that only two concentrations are independent variables, we arbitrary choose CO and COH. According to the fact that they are charged species
inside a solid phase, the general transport equations are (Bard and Faulkner, 2001):
PCP OH OH g OH OH OH D C V T R F C D N =− ∇ + ∇ (eq. IV.3.3) PCP O O g O O O D C V T R F 2 C D N =− ∇ + ∇ (eq. IV.3.4)
in which diffusion and migration are considered (we do not consider a convection contribution because species are in solid phase). Note that migration terms take into account the absolute charges of species, i.e. -1 for protonic defects and -2 for oxygen
H2O PCP H2O PCP + + + + + + + - - - - - - -
ions in the lattice (so vacancies have not got a charge, they are hollow sites). Within the PCP mass balances are equal to:
OH OH N t C ⋅ −∇ = ∂ ∂ (eq. IV.3.5) O O N t C ⋅ −∇ = ∂ ∂ (eq. IV.3.6)
The continuity of electric potential is described by the Poisson law:
0 r c PCP 2 V
ε
ε
ρ
− = ∇ (eq. IV.3.7)in which ε0 and εr are respectively the permittivity of the void and the relative
permittivity of PCP, ρc is the charge density that can be written as:
O OH
0
c = ρ −F⋅C −2F⋅C
ρ (eq. IV.3.8)
where ρ0 is the charge density due to the positive ions (i.e. A, B, D) of the
perovskite-structure, so it is a constant property of the material calculated as:
(
)
δρ0 = 6−S ⋅F⋅ (eq. IV.3.9)
11
The system made by (eq. IV.3.5-7) describes the global phenomenon of the transport in the PCP, coupled with (eq. IV.3-4) and (eq. IV.3.8-9). If we apply these equations to a layer of dry proton-conducting material with one side exposed to wet gas phase as in fig. IV.2a, it is possible to follow the concentration profiles versus time.
Imagine that DOH > DO (if it was DO > DOH it would be the same but changing
considerations for OH with O), this means that protonic defects go from left to right faster than oxygen ions in lattice from right to left. After a few seconds, this produces a unbalance of charges (remember that at the starting time the whole PCP was dry and electroneutral) that yields an internal electric field (fig. IV.2b). The intensity of the
11 The value 6 – S comes from the sum of the total positive charges per unit of perovskite-cell, i.e.:
(
1 S)
3 S 6 S 41
effects of the electric field to species depends on the product ε0εr: if it is high, electric
potentials are low, so migration contributions have a little effect on transports that are due to only diffusion; it leads to a distribution of charges in the phase (i.e. ρc ≠ 0). In the
other case, a strong electric field is created and it is opposed to the charge separation, i.e. it accelerates the flux of oxygen ions in lattice from right to left and brakes the flow of protonic defects from left to right: this leads to retain the electroneutrality (ρc = 0).
The second situation is more realistic because the void permittivity is very low (8.854·10-12F/m) and it is confirmed by simulations (not reported in this document), i.e. by the resolution of the system (eq. IV.3.5-7) by using as parameters the properties of the PCP used in the CM and realistic working conditions.
Thus, the transport of protonic defects through the PCP retains the electroneutrality, i.e. the fluxes of protonic defects, oxygen ions of the lattice and vacancies are exactly compensated. This means that the “constraint” of electroneutrality leads to balance the flows of species in PCP, so we can describe the three transports with a single equation of transport, i.e. a law of diffusion in which all the diffusivities DOH, DVO and DO are
equal. In particular, this kind of diffusion represents not only the transport of protonic defects but the transport of adsorbed water through the PCP. Due to the electroneutrality of the process (and to the constraint in (eq. IV.3.2), i.e. the constant number of oxygen sites), for two protonic defects that flow from left to right there are a vacancy and an oxygen of the lattice that flow from right to left as in fig. IV.3: this is a net transport of water.
fig. IV.3 – Net transport of water through PCP. H H H H
Thus, we have demonstrated that protonic defects are the way used by perovskite to store adsorbed water, in particular we can define the concentration of adsorbed water in PCP as: 2 C C OH PCP , w = (eq. IV.3.10)
just considering that one molecule of water produces two protonic defects.
We have also seen that transport of water within the PCP is a process that retains the electroneutrality of the material and can be described by the Fick law of diffusion as:
PCP , w PCP , w PCP , w D C N =− ∇ (eq. IV.3.11)
in which Dw,PCP is the diffusion coefficient of water in PCP (see sec. V.4.5 for its
estimation). Note that (eq. IV.3.11) is referred to a situation in which the system is made of only PCP, if we want to apply the relationship to a porous composite structure as the CM we must use an apparent diffusivity (see sec. V.4.5 for details)12.
After these considerations, it should be clear that also the adsorption of water is a process that retains the electroneutrality, i.e. as described in (reac. IV.3.1)13. Under these conditions, Kreuer (2003) suggested that equilibrium condition for (reac. IV.3.1) can be written as:
w O VO 2 OH p c c c K = (eq. IV.3.12)
in which K is the thermodynamic constant of equilibrium for water adsorption in PCP (see sec. V.4.5 for its estimation). The concentrations of oxide ion vacancies cVO and of
oxygen ions in the lattice cO can be expressed as a function of cOH by applying the
condition of electroneutrality ρc = 0 (i.e. by using (eq. IV.3.8-9)) and the constraint on
the total number of oxygen sites in (eq. IV.3.2); it yields:
12 It shall be noted that in the model of CM (chap. V) we will apply only (eq. IV.3.11) to describe the transport of water in PCP (it is the final equation), not all the relationships above.
13
Note that water is a neutral molecule, so it is reasonable that its adsorption does not affect the neutrality of the PCP.
2 S c 3 c OH O + − = (eq. IV.3.13) 2 c S c OH VO − = (eq. IV.3.14)
By substituting (eq. IV.3.13-14) into (eq. IV.3.12) and rearranging the equation, we obtain the expression of the concentration of protonic defects at the equilibrium as a function of partial pressure of water (i.e. an external condition), dopant level (i.e. a property of the material) and constant of equilibrium (i.e. a property of the system that depends on temperature) as in Kreuer (2003):
(
)
4 Kp S 4 S 24 S Kp S Kp 6 Kp 9 Kp Kp 3 c w 2 2 w w w w w eq OH − − + + − − = (eq. IV.3.15)Note that at saturation, i.e. when Kpw → +∞, (eq. IV.3.15) gives the maximum
concentration of protonic defects (i.e. the maximum concentration of water adsorbed):
S
cOHsat = (eq. IV.3.16)
At the same result we can arrive by (eq. IV.3.14) considering that at saturation cVO = 0.
Concerning the kinetics of water adsorption, according to the fact that a specific kinetic expression has not been found yet in literature for this phenomenon, we suggest an elementary kinetic expression for (reac. IV.3.1) consistent with equilibrium conditions: 2 OH d OH OH w d 2 OH d O VO w d ads c K k 2 S c 3 2 c S p k c K k c c p k v − + − − = − = (eq. IV.3.17)
in which vads is the rate of adsorption (i.e. moles of water adsorbed per unit area of PCP
per second) and kd is the kinetic constant; note that (eq. IV.3.13-14) have already been
substituted. The expression is coherent with equilibrium condition: when vads = 0, i.e. at
equilibrium, (eq. IV.3.17) gives (eq. IV.3.12).
In the model of the CM in chap. V we will use (eq. IV.3.11) for water transport in PCP and (eq. IV.3.17) for the rate of water adsorption.
IV.4 – Charge transport
The last submodel that we need concerns the transport of charges through ACP and PCP, i.e. the transport law of flows of protons and oxygen ions.
It is reasonable to assume that the transfer of charges in solid phases is due to migration, i.e. charges move according to a gradient of electric potential following the Ohm law:
V
i=−σ∇ (eq. IV.4.1)
In particular, considering that charges are protons (charge equal to +1) and oxygen ions (charge equal to -2) respectively in PCP and ACP, it is possible to write the currents in terms of molar flows as:
PCP PCP H V F N + =−σ ∇ (eq. IV.4.2) ACP ACP O V F 2 N −2 = ∇ σ (eq. IV.4.3)
Obviously, apparent conductivities must be used if these equations are applied to a porous composite media instead of a compact single-phase material (remember (eq. II.7.3) and see sec. V.4.4 for details).
The genesis of this kind of transport, i.e. a migration, is simple: to move through a solid media, charges have to win a resistance, so they spend their electric contribution of energy represented by the electric potential of the phase. Migration implies that a charge can move to the adjacent site if the charge that occupies it moves to the next site and so on as in a chain14. This model of behaviour is reasonable for solid phases and coherent with our description of ACP-PCP interface (par. II.2).
Now, for ACP this kind of behaviour is obvious: oxygen ions migrate through the ACP passing (and occupying) from a vacancy to another one if submitted to a gradient of electric potential. For protons in PCP it is the same but misunderstandings could be generated because it is easy to confuse transport of protons (i.e. a transport of charges) with transport of protonic defects (i.e. transport of water in PCP, an electroneutral
14 If a charge moved on the next site and the previous charge did not move on the hollow site, an internal electric field would be generated and it would force the charge to occupy the site. It is similar as we have seen in par. IV.3.
process). The transport of protonic defects, as explained in the previous paragraph, represents the transfer of adsorbed water through the PCP due to diffusion (i.e. the driving force is the gradient of concentration of adsorbed water) while the transport of charge is the transfer of protons from a protonic defect to another one due to migration (i.e. the driving force is the gradient of electric potential). Fig. IV.4 should clarify the concepts: in a) transport of water is represented, i.e. it is a net transport of water (bolt) without any transfer of charge within the phase; in b) transport of protons is shown, i.e. the migration process in which protons (not protonic defects) move from a protonic defect to the next one as in chain. In the last process, a net current is passing through the phase while in the first one it is not like that (i.e. there is only a net transport of water).
fig. IV.4 – Schematic representation of: a) transport of water (diffusion) b) transport of protons (migration) in PCP.
Now, because the concentration of protonic defects is not constant in PCP and it depends on the water transport, also the electric conductivity of PCP is not constant; this means that the local concentration of adsorbed water (i.e. the local concentration of
protonic defects cOH) affects the local conductivity of the material σPCP, in particular
conductivity increases if concentration of adsorbed water increases (Kreuer, 2003)15. It is reasonable to assume a linear relationship between conductivity of PCP and concentration of protonic defects, i.e. the same linear relationship that is shown in (eq. IV.3.3-4) to describe the contribution of migration. It yields:
15 Kreuer (2003) and also other authors (e.g. Katahira et al., 2000) found a relationship between conductivity and partial pressure of water when water adsorption is in equilibrium; according to par. IV.3, it is possible to link partial pressure of water with concentration of protonic defects to make a direct relationship between concentration of protonic defects and conductivity.
H H H H a) b) H H H H I
S cOH sat PCP PCP σ σ = (eq. IV.4.4)
in which σPCPsat is the conductivity at the saturation limit, i.e. when the concentration of
protonic defects is maximum and equal to S as in (eq. IV.3.16); this situation happens when Kpw → +∞.
Concluding, in the model of the CM we will take into account this phenomenon, i.e. the influence of partial pressure of water on conductivity of PCP, by using (eq. IV.4.4) and we will use (eq. IV.4.2-3) to describe the transports of protons and oxygen ions but using apparent conductivities instead of the conductivities of pure compact materials.
IV.5 – References
Arnošt D., Schneider P., “Dynamic transport of multicomponent mixtures of gases in
porous solids”, Chem. Eng. J., 57, pp. 91-99; 1995.
Bard A.J., Faulkner L.R., “Electrochemical Methods: Fundamentals and Applicationsm
2nd ed.”, John Wiley & Sons, pp. 60-62, pp. 87-132, p. 138, New York; 2001.
Kast W., Hohenthanner C.R., “Mass transfer within the gas-phase of porous media”,
Int. J. Heat and Mass Transfer”, 43, pp. 807-823; 2000.
Katahira K., Kohchi Y., Shimura T., Iwahara H., “Protonic conduction in
Zr-substituited BaCeO3”, Solid State Ionics, 138, pp. 91-98; 2000.
Kenney B., Valdmanis M., Baker C., Pharoah J.G., Karan K., “Computation of TPB
length, surface area and pore size from numerical reconstruction of composite solid oxide fuel cell electrodes”, J. Power Sources, 189, pp. 1051-1059; 2009.
Kreuer K.D., “Proton-conducting oxides”, Annu. Rev. Mater. Res., 33, pp. 333-359; 2003.
Mauri R., “Elementi di fenomeni di trasporto”, Ed. Plus – Pisa University Press, p. 68, Pisa; 2005.
Schneider P., “Multicomponent isothermal diffusion and forced flow of gases in
