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Multicomponent diffusion in a sublattice of an ionic crystal
Solid State Ionics 180 (2009) 1133–1138
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Solid State Ionics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / s s i
Multicomponent diffusion in a sublattice of an ionic crystal E. Gardés a,⁎, B. Poumellec b a b
Deutsches GeoForschungsZentrum, Section 3.3, Telegrafenberg, 14473 Potsdam, Germany Université Paris Sud 11, ICMMO, UMR CNRS-UPS 8182, Equipe PCES, 91405 Orsay Cedex, France
a r t i c l e
i n f o
Article history: Received 23 March 2009 Accepted 4 June 2009 Keywords: Multicomponent diffusion Chemical diffusion Tracer diffusion Self-diffusion Nernst field Diffusion potential Nernst–Planck equation Ionic crystal Aliovalent ion
a b s t r a c t We report a phenomenological description of multicomponent diffusion in ionic crystals with no restriction on the number and electric charge of species. The diffusion of ions is assumed to occur via a vacancy mechanism in one sublattice while the species of the other sublattices are immobile. Non-equilibrium thermodynamics is used to derive the entropy production and flux expressions during diffusion. Two relations on the fluxes, the zero net ion flux and the zero net electric current, are derived from the conservation of sites and electric charge. It is assumed that the internal electric field and chemical potential gradient of vacancies that rise from the movement of the species are such that they ensure the fulfillment of these two relations. General and simplified generalized Fick's laws are derived in the two cases of isovalent and non-isovalent diffusing ions. It is shown that neglecting the chemical potential gradient of vacancies implies a relation between the phenomenological coefficients when the ions are not isovalent. The consequence is that the off-diagonal phenomenological coefficients cannot be generally set to zero and particularly that the use of Nernst–Planck equation is not appropriate for describing the diffusion of non-isovalent ions in a sublattice of an ionic crystal. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Diffusion in ionic crystals differs from that in metals and alloys since electrostatic forces are important and ions are confined to their own sublattice (see, e.g., [1–4]). In ionic crystals, the diffusion of species gives rise to an internal electric field that exerts a strong force on ions. Furthermore, if species are immobile in at least one of the sublattices then the diffusion of the other species occurs without sublattice drift [5]. Even if much less numerous than for metals, there are several descriptions of the diffusion in ionic solids [5–11]. However, most of them are limited to the diffusion of two or three ions with identical valences. A generalization to the diffusion of any number of ions, and especially when they do not carry the same electric charge, is missing. We report here a phenomenological description of the diffusion of any number of ions carrying any electric charge in the sublattice of an ionic crystal containing Schottky defects. This description is done in the framework of non-equilibrium thermodynamics [12]. The entropy production is established as a function of independent conjugated fluxes and forces. Therefore the Onsager's reciprocal relations apply to the phenomenological coefficients that relate the fluxes linearly to the forces. Two important relations between the fluxes, the zero net ion flux and the zero net electric current, are derived from the conservation of sites and electric charge. Depending on the electric charges of the species, these two relations can be identical or independent so ⁎ Corresponding author. E-mail address: [email protected] (E. Gardés). 0167-2738/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2009.06.004
the treatment of the diffusion of isovalent ions is treated separately from that of non-isovalent ions. In both cases, fluxes are expressed in the form of the generalized Fick's law and the elements of the chemical diffusion matrix [D] are given. Some further (strongly) simplifying approximations are introduced in order to relate the Dij to tracer diffusion coefficients D⁎i and examples for simple cases are finally given and discussed. 2. Source of entropy and fluxes One considers an ionic crystal containing Schottky defects. It is assumed that species flows are non viscous, temperature is uniform, no magnetic field is applied and mechanical equilibrium is reached. Thus, the rate of entropy production per unit volume σ, when diffusion is one-directional, is given by [12] −Tσ =
X n
Xs
+
j = 1
Xm
k = 1
j
i = 1
J k Ak :
Ji; j
Aci; j AcV; j − Fi; j + JV; j − FV; j Ax Ax ð1Þ
The first term on the right hand side is the contribution of diffusion. This is the sum of the products of conjugated fluxes and thermodynamic forces of the ions and vacancies from the s sublattices of the crystal. J is the species flux, µ the molar chemical potential, F the external force exerted on one mole of species and n the number of different ions in the sublattice. The first subscript i or V refers to ions or vacancies and the second subscript j refers to the sublattice. The last term on the right hand side of entropy production is the contribution of the formation of
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Schottky defects. This is the sum of the products of the reaction rate J k and the chemical affinity Ak of the m different Schottky defects. For instance, these defects can be molecular vacancies, that involve vacancies of all of the sublattices in a proportion respective to that of the P crystal stoichiometry Ff sj = 1 mj Vj ;or defects involving vacancies of a smaller number of sublattices, what causes departure from crystal stoichiometry. It should be noted that, even at equilibrium, the chemical potential of the vacancies cannot be set to zero since the equilibrium condition is Ak = 0 but the chemical affinity is the sum of the different chemical potential of the vacancies of the Schottky defect, weighted by the stoichiometric coefficients [5,6]. As the species can be electrically charged, one has to take the electrostatic forces into account ð2Þ
where zi,j is the number of electric charge of ion i in the sublattice j, φ the electrostatic potential and F the Faraday's number. Thus, the thermodynamical forces are the electrochemical potential gradients. The species are described with their formal electric charges so there is no electric charge and therefore no electrostatic force on vacancies. It is assumed that there is no coupling between the species movements of the different sublattices and that the ions and vacancies of only one sublattice, say the first one, are diffusing while the ions and vacancies from the other sublattices are immobile and form a rigid framework. Therefore, Schottky defects creation is impossible since, obviously, it couples several sublattices J k = 0;
Aci; j A ci; j ui; j ; =− Ax At
ðk = 1; …; mÞ:
ð3Þ
As mechanical equilibrium is considered achieved, according to Prigogine's theorem [12], fluxes can be expressed relatively to any arbitrary velocity and thus relatively to that of the framework of sublattices in which species are immobile. If, for simplicity, that framework is assumed immobile in the laboratory frame, fluxes are expressed
the conservation law for vacancies, that is identical to the previous one because there is no vacancy production (Eq. (3)) AcV; j A cV; j uV; j ; =− Ax At
A Xn1 Ji;1 + JV;1 = 0: i = 1 Ax As the fluxes vanish outside the diffusion zone, it follows from the previous relation Xn
J i = 1 i;1 1
Ji; j = ci; j ui; j and JV; j = cV; j uV; j ;
−Tσ =
Ji; j = 0 and JV; j = 0;
ð j = 2; …; sÞ:
ð5Þ
Substituting Eqs. (2), (3) and (5), the rate of entropy production becomes Xn
1
i =
J 1 i;1
Aci;1 AcV;1 Au + JV;1 + zi;1 F : Ax Ax Ax
ð6Þ
It should be remarked that, due to mechanical equilibrium, the forces are only related by Gibbs–Duhem relation X n
Xs j = 1
j
i =
c 1 i; j
Aci; j AcV; j Au + cV; j + zi; j F = 0; Ax Ax Ax
ð7Þ
Xn
1
i =
J 1 i;1
Aci;1 AcV;1 Au − + zi;1 F : Ax Ax Ax
Acj;1 AcV;1 1 Xn1 Au − + zj;1 F ; Lij j = 1 T Ax Ax Ax
Lij = Lji : 3. Further relations between fluxes
The movement of ions with various mobilities can give rise to local electric charge excess. An electric field thus develops, even if the crystal is initially neutral and in the absence of an applied field, and opposes to this electric charge excess. This internal electric field is the Nernst field, at the origin of the drift term in the Nernst–Planck equation commonly used for describing the diffusive fluxes of electrically charged particles (e.g. [2]). Because of the strength of coulombic interactions, a small departure from electroneutrality produces an electric field strong enough for modifying the fluxes [13,14]. Therefore, departure from electroneutrality is much smaller than any concentration and it is thus reasonable to assume that the electric field is such that electroneutrality is fulfilled everywhere and at any time Xs
Xn
Xn
Xn
+ cV;1 = constant in time and space:
ð8Þ
ð12Þ
ð13Þ
or, since the species of the sublattices 2 to s are immobile
1
ð11Þ
and the phenomenological coefficients follow the Onsager's reciprocal relations [12]
thus the forces acting on the species of the first sublattice are independent between themselves. Because the species of the first sublattice are confined to diffuse in their own sublattice and the other species are immobile, the number of sites in the first sublattice remains constant c i = 1 i;1
ð10Þ
We are now left with n1 independent fluxes and n1 independent forces in the first sublattice. Therefore, under the usual assumption that fluxes are linearly dependent on forces, fluxes are expressed
ð4Þ
where c is the molar concentration and u the velocity in the laboratory frame. Thus, the reference frame of the sublattices in which ions are immobile coincides with the laboratory frame. It is follows from that definition that the species fluxes in these sublattices vanish
+ JV;1 = 0:
Thus, there is no drift of the first sublattice relatively to the others (the opposite would have been unacceptable). As the flux of vacancies is opposite to the sum of the fluxes of the ions, it can be eliminated from the entropy source
Ji;1 =−
−Tσ =
ð9Þ
and the definition of the fluxes (Eq. (4)), it is found that
Au ; Ax
Fi; j =− zi; j F
Starting from the time derivative of this expression, using the conservation law for ions
j = 1
c z i = 1 i;j i;j j
c z i = 1 i;1 i;1 1
= 0;
= constant in time and space:
ð14Þ
E. Gardés, B. Poumellec / Solid State Ionics 180 (2009) 1133–1138
Starting from the time derivative of this expression, using the conservation law (Eq. (9)) and the definition of the fluxes (Eq. (4)), it is found that there is zero net electric current in the sublattice Xn
1
i =
ð15Þ
z J = 0: 1 i;1 i;1
The movement of ions with various mobilities can also result in vacancy accumulation. This provokes gradients in the chemical potential of the vacancies, even if they were initially in equilibrium. Analogously with the negative retroaction of the electric field on electric charge excess, we assume ∂µV,1/∂x opposes to vacancy wind. Two phenomena support this assumption. The movement of species with different volumes induces an internal stress gradient (see, e.g., [15]). It is expected to be the highest in the vicinity of vacancy, where the ions repulse each other because of the missing ion with opposite electric charge. Thus, the internal stress gradient due to the movement of vacancies, implicitly included in ∂µV,1/∂x, opposes to vacancy accumulation. The internal stress gradient due to the movement of ions is neglected in comparison. Another phenomenon that will oppose to vacancy flux is the interaction of the vacancies of the first sublattice with those of the other sublattices, then forming immobile complexes (see, e.g., [2]). Thus, we assume that ∂µV,1/∂x ensures that the flux of vacancies is small and negligible compared to the fluxes of ions ð16Þ
JV;1 = 0:
It follows from Eq. (10) that there is zero net ion flux in the sublattice Xn
1
i =
ð17Þ
J = 0: 1 i;1
1135
The fluxes, expressed versus the ∂µj/∂x only, are then
Ji = −
1 Xn j = T
Pn 1
Lij −
Pn k = 1 Lik l = 1 Llj Pn k;l = 1 Lkl
!
Acj ; ði = 1; …; nÞ: Ax
The contributions of ∂φ/∂x and ∂µV/∂x in Eq. (19) cannot be distinguished. The reason is that the fulfillment of the zero net electric current relation also ensures the fulfillment of zero net ion flux in the case of isovalent ions. Thus, the same result would have been obtained in a description where the chemical potential gradient of vacancies is assumed to be zero, or a description without considering vacancies at all (e.g. [8,16]). However, ∂µV/∂x cannot be generally set to zero, especially in the case of the diffusion of non-isovalent ions as will be seen later. In order to find the expression of the chemical diffusion matrix [D], chemical potential gradients can be related to concentration gradients by: Xn Acj Acj Aci = : i = 1 Ac Ax Ax i
ð21Þ
The sum is limited to the n ions of the sublattice since the concentration gradients of the other species are zero. When the chemical potential of the ions has the form 0 cj = cj + RTln γj cj ; where µ0j is a standard value, R the gas constant and γj the activity coefficient, the chemical potential gradients can be written
Thus, there is no lattice distortion. The velocities of the sublattices are all identical and zero. The conservation law (Eq. (9)) implies
Xn Acj /ij + δij Aci = RT ; i = 1 Ax ci Ax
A Xn1 c = 0; i = 1 i;1 At
where δij is the Kronecker delta and ϕij supports non ideality
and, because of Eq. (8), one obtains Xn 1 c = constant in time and space: i = 1 i;1
/ij =
4. Diffusion of isovalent ions
ci Aγj : γ j Aci
Xn − 1 Ac Acn i =− : i = 1 Ax Ax
When all the ions have the same number of electric charge z, the fluxes (Eq. (12)) simplify to
Jn =−
Acj 1 Xn Au AcV − + zF ; L ij j = 1 T Ax Ax Ax
i; j = 1 P n
Lij
Acj Ax
i; j = 1 Lij
:
Xn −
1 j = 1
Dij
Acj ; Ax
ði = 1; …; n − 1Þ;
Xn − 1
J; i = 1 i
ð25aÞ ð25bÞ
where the coefficients of the chemical diffusion matrix [D] are
where the subscript 1 that refers to the first sublattice is dropped for clarity, here and in the following equations. Using the zero net electric current relation (Eq. (15)), or similarly the zero net ion flux relation (Eq. (17)), one obtains a relation between the forces Pn
ð24Þ
It is thus possible to express the first n-1 independent fluxes as a function of the first n-1 independent concentration gradients in the generalized Fick's law form, with the last flux given by Eq. (17) Ji = −
Au AcV − =− Ax Ax
ð23Þ
Because the sum of the concentrations of the n ions is a constant (Eq. (18)), only n-1 concentration gradients are independent
4.1. General expressions
zF
ð22Þ
ð18Þ
The zero net ion flux and zero net electric current relations impose two constraints on the fluxes but they are identical when all the ions have the same valence. Thus, two cases will be developed in the following: the isovalent case, where all the ions have the same valence and n-1 fluxes are independent, and the non-isovalent case, where at least one ion has a different valence from the others and only n-2 fluxes are independent.
Ji = −
ð20Þ
ð19Þ
Dij = R
Xn k = 1
/kj + δkj / + δkn − kn cj cn
! Lik −
! Pn l;p = 1 Lil Lpk P ; ði; j = 1; …; n −1Þ: n l;p = 1 Llp
ð26Þ 4.2. Simplifications 4.2.1. Approximations It may be assumed that the geometric correlations between the jumps of the ions are negligible compared to the correlations that
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derive from the zero net ion flux and zero net electric current conditions (Eqs. (15) and (17)). Therefore, if there are no supplementary constraints on the movement of the ions, as assumed in the present description, the off-diagonal phenomenological coefficients can be set to zero ði ≠ j with i; j = 1; …; nÞ:
Lij = 0;
ð27Þ
The diagonal phenomenological coefficients are then proportional to the self-diffusion coefficients or to the tracer diffusion coefficients D⁎i if correlation factors are neglected in experimental measurements [3,5] Lii =
ci D4 i ; ði = 1; …; nÞ: R
ð28Þ
A subsequent consistent approximation is to neglect the chemical interactions between the ions, that is to assume the mixture of the ions in the sublattice is ideal /ij = 0;
ði; j = 1; …; nÞ:
ð29Þ
Therefore, the elements of the chemical diffusion matrix (Eq. (26)) simplify considerably to 0
1 4 − D4n c D i j B C Dij = D4 A: i @δij − P n 4 c D k = 1 k k
ð30Þ
It has to be remarked that this expression, because of the above strongly simplifying approximations, can be inappropriate for many systems and therefore has to be considered cautiously. However it allows for a first order analysis as will be seen in the following. 4.2.2. Case of the diffusion of two isovalent ions Let A and B be two equally charged ions inter-diffusing in a sublattice of a crystal. As expected, Eqs. (25a) and (25b) yield that the fluxes of A and B are opposite JA =− DAA
AcA ; Ax
ð31bÞ
When geometric correlations between the jumps of the ions are neglected (Eq. (27)), except for the correlations that derive from the zero net ion flux and zero net electric current conditions, when the diagonal phenomenological coefficients are related to tracer diffusion coefficients (Eq. (28)) and when the solution is assumed ideal (Eq. (29)), the solution for DAA is (Eq. (30))
4 ðcA + cB ÞD4 A DB 4 cA D4 A + cB DB
;
ð32Þ
which is a well-known result, readily obtained when the Nernst– Planck equation is used for describing the fluxes of A and B (e.g. [4]). It follows that if one of the two ions has a small concentration or diffusivity it will control the interdiffusion, e.g. when cA bb cB, one has DAA = D4 A; as expected, since this is the definition of the tracer diffusion coefficient when correlations in the experimental measurement are ignored. When DA⁎ bb DB⁎, one has DAA =
cA + cB 4 DA : cB
4.2.3. Case of the diffusion of three isovalent ions Let A, B and C be three diffusing ions with identical electric charge. Fluxes are (Eqs. (25a) and (25b)) JA =− DAA
AcA Ac − DAB B ; Ax Ax
ð33aÞ
JB =− DBA
AcA Ac − DBB B ; Ax Ax
ð33bÞ
JC =− JA − JB :
ð33cÞ
When all the approximations are done as in the previous example (Eqs. (27)–(29)), the elements of the chemical diffusion matrix are (Eq. (30)) DAA =
DAB =
DBA =
DBB =
4 4 4 cB D4 A DB + ðcA + cC ÞDA DC 4 4 cA D 4 A + cB DB + cC DC
4 4 4 −cA D4 A DB + cA DA DC
4 4 cA D 4 A + cB DB + cC DC 4 4 4 −cB D4 A DB + cB DB DC
4 4 cA D 4 A + cB DB + cC DC
;
ð34aÞ
;
ð34bÞ
;
ð34cÞ
4 4 4 cA D 4 A DB + ðcB + cC ÞDB DC 4 4 cA D 4 A + cB DB + cC DC
:
ð34dÞ
ð31aÞ
JB =− JA :
DAA =
This result is a consequence of the negative retroaction of both the electric field and the chemical potential gradient of the vacancies on electric charge excess and vacancy accumulation (see Section 3). If one ion diffuses faster than the other one, then the induced electric field and chemical potential gradient of the vacancies will oppose to electric charge excess and vacancy accumulation. The fastest ion will decelerate down to a velocity close to the other one so the electric charge excess and vacancy accumulation is minimized. Thus, the interdiffusion is controlled by the slowest ion.
This result can also be obtained using the Nernst–Planck equation for the fluxes of the three ions and solving for the electric field with the zero net current relation. When the concentration of a species is set to zero, one expects to be left with a binary case. In fact, when cC = 0, the flux of C is obviously zero and the fluxes of A and B becomes opposite, as well as their concentration gradients (Eq. (24)) and one has JA =−ðDAA − DAB Þ
4 ðc + cB ÞD4 AcA A DB AcA =− A ; Ax 4 Ax cA D4 A + cB DB
as in the case of the interdiffusion of two isovalent ions. The same result is obtained when both the diffusivity and the concentration gradient of one ion are much smaller than those of the other two. 5. Diffusion of non-isovalent ions 5.1. General expressions In the case where ions are not all equally charged, the fluxes are described by the general form (Eq. (12)) Ji = −
Acj 1 Xn Au AcV − + z ; L F ij j j = 1 T Ax Ax Ax
where the subscript 1 that refers to the first sublattice is dropped for clarity, here and in the following equations. ∂µV/∂x and ∂φ/∂x are
E. Gardés, B. Poumellec / Solid State Ionics 180 (2009) 1133–1138
determined using the conditions of zero net ion flux (Eq. (17)) and zero net electric current (Eq. (15)) Xn AcV = i;j Ax
= 1
Xn Au = F i;j Ax
Acj βzi − γ Lij ; β2 − αγ Ax
ð35Þ
Xn
L ; i;j = 1 ij
ð36Þ
ð37aÞ
Xn
zL ; i;j = 1 i ij
β=
ð37bÞ
Xn
zzL ; i;j = 1 i j ij
γ=
ð37cÞ
and the Onsager's relations have been used (Eq. (13)). Contrary to the case of isovalent ions, ∂µV/∂x and ∂φ/∂x are independent because the zero net ion flux and zero net electric current relations are two independent constraints when the ions do not have all the same valence. Thus, if the chemical potential gradient of the vacancies is not included in the description, the condition of zero net ion flux leads to a relation between the phenomenological coefficients. In that case, the coefficients have to be chosen properly in order to fulfill that condition and, especially, it is impossible to set the off-diagonal terms to zero. Substituting Eqs. (35) and (36) leads to the fluxes Ji = −
1 Xn j = T
1
It can be remarked that n has to be greater than 2. In fact, the configuration with only two aliovalent ions that ensures the fulfillment of both electroneutrality and site conservation is that in which the two ions have uniform concentrations, i.e. are not diffusing. 5.2. Simplifications
αzi − β Acj ; L = 1 β 2 − αγ ij Ax
where α=
Xn Acj αzk − β Xn βz − γ Xn : Lij + z L − 2k L Lkj k = 1 β 2 − αγ l = 1 l il l = 1 il Ax β − αγ
5.2.1. Approximations The approximations here are identical to those done in Section 4.2.1. That is, the geometric correlations between the jumps of ions are neglected compared to the correlations that derive from the zero net ion flux and zero net electric current conditions (Eq. (27)), the diagonal phenomenological coefficients are related to tracer diffusion coefficients (Eq. (28)) and the mixture of the ions is assumed ideal (Eq. (29)). It follows a considerable simplification of the coefficients of the chemical diffusion matrix
½
Dij = D4 i δij +
Xn − 2 z −z Acn − 1 Aci i n = ; i = 1 z − z Ax n n − 1 Ax
ð39aÞ
Xn − 2 z Acn n − 1 − zi Aci = : i = 1 z − z Ax n n − 1 Ax
ð39bÞ
Using the relation between chemical potential gradients and concentration gradients (Eq. (22)), it is possible to express the first n-2 independent fluxes as a function of the first n-2 independent concentration gradients in the generalized Fick's law form, with the two last fluxes given by the zero net ion flux and zero net electric current relations (Eqs. (15) and (17)) Xn
− 2 j = 1
Jn − 1 = Jn =
Dij
Acj ; Ax
ði = 1; …; n − 2Þ;
Xn − 2 z − z i n Ji ; i = 1 z − z n n−1
Xn
− 2 i = 1
zn − 1 − zi J; zn − zn − 1 i
Xn
ði; j = 1; …; n − 2Þ:
βzj − γ
ci D4 j
2
ð42Þ where α (Eq. (37a)), β (Eq. (37b)), and γ (Eq. (37c)) simplify to Xn i = 1
β=
γ=
Xn i = 1
Xn i = 1
ci D4 i ; R
ð43aÞ
zi ci D 4 i ; R
ð43bÞ
z2i ci D4 i : R
ð43cÞ
It should be recalled that this expression derives from strongly simplifying approximations and therefore has to be considered cautiously (see Section 4.2.1.). 5.2.2. Case of the diffusion of three aliovalent ions Let zA, zB and zC be the three non equal numbers of electric charge of three diffusing ions A, B and C. Because of the two relations of zero net ion flux and zero net electric current, only one flux is independent in this ternary system (Eqs. (40a), (40b), and (40c)) JA =− DAA
AcA ; Ax
ð44aÞ
zA − zC J ; zC − zB A
ð44bÞ
ð40bÞ
JC =
zB − zA J : zC − zB A
ð44cÞ
ð40cÞ
When geometric correlations between the jumps of the ions are neglected (Eq. (27)), except for the couplings that derive from the zero net ion flux and zero net electric current conditions, when the diagonal phenomenological coefficients are related to tracer diffusion coefficients (Eq. (28)) and when the solution is assumed ideal (Eq. (29)), the solution for DAA is (Eqs. (42) and (43a), (43b), and (43b)))
"
f
zi −
JB =
/jk + δjk zj − zn /n − 1k + δn − 1k zn − 1 − zj /nk + δnk + + cj zn − zn − 1 cn − 1 zn − zn − 1 cn Xn αzl − β Xn βzl − γ Xn z L − 2 L Llk ; × Lik + l = 1 β 2 − αγ p = 1 p ip p = 1 ip β − αγ k = 1
2
ð40aÞ
where the coefficients of the chemical diffusion matrix [D] are
Dij = R
αzj − β
α=
From the conservation of sites and electric charge relations (Eqs. (14) and (18)) follows that only n-2 concentrations are independent. When the n-1th and the nth components are chosen such that zn−1 ≠zn, one obtains
R β − αγ β − αγ zj − zn ci D4 αzn − 1 − β βzn − 1 − γ n − 1 zi − + 2 2 z R β − αγ β − αγ n − zn − 1 αzn − β βz − γ zn − 1 − zj ci D4 n + ; ði; j = 1; …; n − 2Þ; zi − 2 n β 2 − αγ β − αγ zn − zn − 1 R
ð38Þ
Ji = −
1137
#
g
ð41Þ
DAA =
2 2 2 T T T cA cB ðzA − zB Þ + cA cc ðzA −zc Þ + cB cc ðzB − zc Þ DA DB Dc cA cB ðzA −zB Þ2 DTA DTB + cA cc ðzA −zc Þ2 DTA DTc + cB cc ðzB −zc Þ2 DTB DTc
: ð45Þ
1138
E. Gardés, B. Poumellec / Solid State Ionics 180 (2009) 1133–1138
This result cannot be obtained when the Nernst–Planck equation is used for describing the fluxes of the three ions. In fact, after solving for the electric field with the help of the zero net electric current relation, the application of the zero net ion flux relation leads to a relation between the tracer diffusivities, which is obviously not acceptable. As in the case of the diffusion of two isovalent ions, a species with a small concentration or diffusivity will control the interdiffusion. One has, when cA bb cB, cC DAA = D4 A; as expected, since this is the definition of the tracer diffusion coefficient when correlations in the experimental measurement are ignored. One has, when DA⁎ bb DB⁎, DC⁎ DAA =
cA cB ðzA −zB Þ2 + cA cc ðzA −zc Þ2 + cB cc ðzB −zc Þ2 4 DA cB cc ðzB −zc Þ2
The interdiffusion is controlled by the slowest species because of the negative retroaction of both ∂φ/∂x and ∂µV/∂x on the electric charge excess and the vacancy accumulation due to the diffusion of the fastest ions (see Section 3 and 4.2.2). 5.2.3. Case of the diffusion of two isovalent ions with one aliovalent ion It should be recalled first that for any non-isovalent system, the ions must be sorted so that the charges of the two last ions are different (Eqs. (39a) and (39b)). Let consider then three diffusing ions A, B and C with electric charges such that zA = zB ≠ zC. It follows from Eqs. (40a), (40b), and (40c) that the flux of C must be zero and that the fluxes of A and B are opposite JA =− DAA
AcA ; Ax
Acknowledgement We would like to thank very much the late Prof. Olivier Jaoul from Université Paul Sabatier in Toulouse for his training in Mineral Physics during our PhDs. This paper is dedicated to his memory.
JB =− JA ; JC = 0: When all the approximations are done as for the previous example (Eqs. (27)–(29)), DAA is (Eqs. (42), (43a) (43b) and (43c)) DAA =
4 ðcA + cB ÞD4 A DB 4 cA D4 A + cB DB
diffusion occurs in one sublattice and is generalized to any number of diffusing ions, carrying any electric charge. One of the main assumptions is that the species of the other sublattices are immobile. Moreover, the flux of vacancies is assumed to be negligible compared to that of ions because of the internal stress induced by their movement and because of their immobility via defect pairing. It follows that the net flux of the ions in the sublattice is zero. Electroneutrality is also assumed to be fulfilled, which implies that the net electric current is zero. It is assumed that these two conditions on the fluxes are both ensured by the electrostatic potential gradient and the chemical potential of the vacancies gradient because they oppose to any departure from these conditions. When the diffusing ions are isovalent, the two conditions of zero net ion flux and zero net electric current are equivalent. Thus, only n-1 fluxes remain independent and the role of the chemical potential of the vacancies is not distinguishable from that of the electrostatic potential. However, the two conditions are not equivalent when the ions are not isovalent. In that case, only n-2 fluxes are independent and the chemical potential gradient of the vacancies is independent from the electric field. It is shown that neglecting the chemical potential gradient of vacancies introduces a relation between the phenomenological coefficients. The consequence is that the off-diagonal terms cannot be generally set to zero. Generalized Fick's laws with expressions of the coefficients of the chemical diffusion matrix are given for both cases of isovalent and nonisovalent diffusing ions. The analysis of simplified cases shows that the Nernst–Planck equation can be used to describe isovalent systems but fails in the description of non-isovalent ones. This is the consequence of setting the chemical potential gradient of vacancies to zero, what is implicit in the Nernst–Planck equation since the electrostatic potential is the only diffusion potential. Our formalism is thus more appropriate to describe non-isovalent systems.
;
which is identical to Eq. (32), derived in the case of the diffusion of two isovalent ion. We are thus left with a binary case. This result is not surprising since the only possibility for having two ions with identical valences and a third ion with a different valence and both electroneutrality (Eq. (14)) and site conservation (Eq. (18)) fulfilled is that the concentration of the third ion is uniform. Thus, it does not participate to diffusion and only the two first isovalent ions inter-diffuse. 6. Conclusion We report a new description of multicomponent diffusion in ionic crystals containing Schottky defects. It is dedicated to the case where
References [1] Y. Adda, J. Philibert, La diffusion dans les solides, Presses universitaires de France, Paris, 1966. [2] J.S. Kirkaldy, D.J. Young, Diffusion in the Condensed State, The institute of metals, London, 1987. [3] P. Shewmon, Diffusion in Solids, The minerals, metals & materials society, Warrendale, 1998. [4] J. Philibert, Atom Movements — Diffusion and Mass Transport in Solids, Les éditions de physique, Les Ulis, 1991. [5] A.R. Allnatt, A.B. Lidiard, Rep. Prog. Phys. 50 (1987) 373. [6] R.E. Howard, A.B. Lidiard, Rep. Prog. Phys. 27 (1964) 161. [7] I.V. Belova, A.R. Allnatt, G.E. Murch, Philos. Mag. (2007) 4169. [8] A.B. Lidiard, Nuovo Cimento D 15 (1993) 229. [9] R. Lindström, J. Phys. C. Solid State Phys. 7 (1974) 3909. [10] D.J. Young, J.S. Kirkaldy, J. Phys. Chem. Solids 45 (1984) 781. [11] D.J. Young, F. Gesmundo, Oxid. Met. 29 (1988) 169. [12] S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics, North Holland, Amsterdam, 1962. [13] V.M. Aguilella, S. Mafé, J. Pellicer, Electrochem. Acta 32 (1987) 483. [14] A.D. MacGillivray, J. Chem. Phys. 48 (1968) 2903. [15] G.B. Stephenson, Acta Metall. 36 (1988) 2663. [16] D.G. Miller, J. Phys. Chem. 71 (1967) 616.
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Solid State Ionics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / s s i
Multicomponent diffusion in a sublattice of an ionic crystal E. Gardés a,⁎, B. Poumellec b a b
Deutsches GeoForschungsZentrum, Section 3.3, Telegrafenberg, 14473 Potsdam, Germany Université Paris Sud 11, ICMMO, UMR CNRS-UPS 8182, Equipe PCES, 91405 Orsay Cedex, France
a r t i c l e
i n f o
Article history: Received 23 March 2009 Accepted 4 June 2009 Keywords: Multicomponent diffusion Chemical diffusion Tracer diffusion Self-diffusion Nernst field Diffusion potential Nernst–Planck equation Ionic crystal Aliovalent ion
a b s t r a c t We report a phenomenological description of multicomponent diffusion in ionic crystals with no restriction on the number and electric charge of species. The diffusion of ions is assumed to occur via a vacancy mechanism in one sublattice while the species of the other sublattices are immobile. Non-equilibrium thermodynamics is used to derive the entropy production and flux expressions during diffusion. Two relations on the fluxes, the zero net ion flux and the zero net electric current, are derived from the conservation of sites and electric charge. It is assumed that the internal electric field and chemical potential gradient of vacancies that rise from the movement of the species are such that they ensure the fulfillment of these two relations. General and simplified generalized Fick's laws are derived in the two cases of isovalent and non-isovalent diffusing ions. It is shown that neglecting the chemical potential gradient of vacancies implies a relation between the phenomenological coefficients when the ions are not isovalent. The consequence is that the off-diagonal phenomenological coefficients cannot be generally set to zero and particularly that the use of Nernst–Planck equation is not appropriate for describing the diffusion of non-isovalent ions in a sublattice of an ionic crystal. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Diffusion in ionic crystals differs from that in metals and alloys since electrostatic forces are important and ions are confined to their own sublattice (see, e.g., [1–4]). In ionic crystals, the diffusion of species gives rise to an internal electric field that exerts a strong force on ions. Furthermore, if species are immobile in at least one of the sublattices then the diffusion of the other species occurs without sublattice drift [5]. Even if much less numerous than for metals, there are several descriptions of the diffusion in ionic solids [5–11]. However, most of them are limited to the diffusion of two or three ions with identical valences. A generalization to the diffusion of any number of ions, and especially when they do not carry the same electric charge, is missing. We report here a phenomenological description of the diffusion of any number of ions carrying any electric charge in the sublattice of an ionic crystal containing Schottky defects. This description is done in the framework of non-equilibrium thermodynamics [12]. The entropy production is established as a function of independent conjugated fluxes and forces. Therefore the Onsager's reciprocal relations apply to the phenomenological coefficients that relate the fluxes linearly to the forces. Two important relations between the fluxes, the zero net ion flux and the zero net electric current, are derived from the conservation of sites and electric charge. Depending on the electric charges of the species, these two relations can be identical or independent so ⁎ Corresponding author. E-mail address: [email protected] (E. Gardés). 0167-2738/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2009.06.004
the treatment of the diffusion of isovalent ions is treated separately from that of non-isovalent ions. In both cases, fluxes are expressed in the form of the generalized Fick's law and the elements of the chemical diffusion matrix [D] are given. Some further (strongly) simplifying approximations are introduced in order to relate the Dij to tracer diffusion coefficients D⁎i and examples for simple cases are finally given and discussed. 2. Source of entropy and fluxes One considers an ionic crystal containing Schottky defects. It is assumed that species flows are non viscous, temperature is uniform, no magnetic field is applied and mechanical equilibrium is reached. Thus, the rate of entropy production per unit volume σ, when diffusion is one-directional, is given by [12] −Tσ =
X n
Xs
+
j = 1
Xm
k = 1
j
i = 1
J k Ak :
Ji; j
Aci; j AcV; j − Fi; j + JV; j − FV; j Ax Ax ð1Þ
The first term on the right hand side is the contribution of diffusion. This is the sum of the products of conjugated fluxes and thermodynamic forces of the ions and vacancies from the s sublattices of the crystal. J is the species flux, µ the molar chemical potential, F the external force exerted on one mole of species and n the number of different ions in the sublattice. The first subscript i or V refers to ions or vacancies and the second subscript j refers to the sublattice. The last term on the right hand side of entropy production is the contribution of the formation of
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Schottky defects. This is the sum of the products of the reaction rate J k and the chemical affinity Ak of the m different Schottky defects. For instance, these defects can be molecular vacancies, that involve vacancies of all of the sublattices in a proportion respective to that of the P crystal stoichiometry Ff sj = 1 mj Vj ;or defects involving vacancies of a smaller number of sublattices, what causes departure from crystal stoichiometry. It should be noted that, even at equilibrium, the chemical potential of the vacancies cannot be set to zero since the equilibrium condition is Ak = 0 but the chemical affinity is the sum of the different chemical potential of the vacancies of the Schottky defect, weighted by the stoichiometric coefficients [5,6]. As the species can be electrically charged, one has to take the electrostatic forces into account ð2Þ
where zi,j is the number of electric charge of ion i in the sublattice j, φ the electrostatic potential and F the Faraday's number. Thus, the thermodynamical forces are the electrochemical potential gradients. The species are described with their formal electric charges so there is no electric charge and therefore no electrostatic force on vacancies. It is assumed that there is no coupling between the species movements of the different sublattices and that the ions and vacancies of only one sublattice, say the first one, are diffusing while the ions and vacancies from the other sublattices are immobile and form a rigid framework. Therefore, Schottky defects creation is impossible since, obviously, it couples several sublattices J k = 0;
Aci; j A ci; j ui; j ; =− Ax At
ðk = 1; …; mÞ:
ð3Þ
As mechanical equilibrium is considered achieved, according to Prigogine's theorem [12], fluxes can be expressed relatively to any arbitrary velocity and thus relatively to that of the framework of sublattices in which species are immobile. If, for simplicity, that framework is assumed immobile in the laboratory frame, fluxes are expressed
the conservation law for vacancies, that is identical to the previous one because there is no vacancy production (Eq. (3)) AcV; j A cV; j uV; j ; =− Ax At
A Xn1 Ji;1 + JV;1 = 0: i = 1 Ax As the fluxes vanish outside the diffusion zone, it follows from the previous relation Xn
J i = 1 i;1 1
Ji; j = ci; j ui; j and JV; j = cV; j uV; j ;
−Tσ =
Ji; j = 0 and JV; j = 0;
ð j = 2; …; sÞ:
ð5Þ
Substituting Eqs. (2), (3) and (5), the rate of entropy production becomes Xn
1
i =
J 1 i;1
Aci;1 AcV;1 Au + JV;1 + zi;1 F : Ax Ax Ax
ð6Þ
It should be remarked that, due to mechanical equilibrium, the forces are only related by Gibbs–Duhem relation X n
Xs j = 1
j
i =
c 1 i; j
Aci; j AcV; j Au + cV; j + zi; j F = 0; Ax Ax Ax
ð7Þ
Xn
1
i =
J 1 i;1
Aci;1 AcV;1 Au − + zi;1 F : Ax Ax Ax
Acj;1 AcV;1 1 Xn1 Au − + zj;1 F ; Lij j = 1 T Ax Ax Ax
Lij = Lji : 3. Further relations between fluxes
The movement of ions with various mobilities can give rise to local electric charge excess. An electric field thus develops, even if the crystal is initially neutral and in the absence of an applied field, and opposes to this electric charge excess. This internal electric field is the Nernst field, at the origin of the drift term in the Nernst–Planck equation commonly used for describing the diffusive fluxes of electrically charged particles (e.g. [2]). Because of the strength of coulombic interactions, a small departure from electroneutrality produces an electric field strong enough for modifying the fluxes [13,14]. Therefore, departure from electroneutrality is much smaller than any concentration and it is thus reasonable to assume that the electric field is such that electroneutrality is fulfilled everywhere and at any time Xs
Xn
Xn
Xn
+ cV;1 = constant in time and space:
ð8Þ
ð12Þ
ð13Þ
or, since the species of the sublattices 2 to s are immobile
1
ð11Þ
and the phenomenological coefficients follow the Onsager's reciprocal relations [12]
thus the forces acting on the species of the first sublattice are independent between themselves. Because the species of the first sublattice are confined to diffuse in their own sublattice and the other species are immobile, the number of sites in the first sublattice remains constant c i = 1 i;1
ð10Þ
We are now left with n1 independent fluxes and n1 independent forces in the first sublattice. Therefore, under the usual assumption that fluxes are linearly dependent on forces, fluxes are expressed
ð4Þ
where c is the molar concentration and u the velocity in the laboratory frame. Thus, the reference frame of the sublattices in which ions are immobile coincides with the laboratory frame. It is follows from that definition that the species fluxes in these sublattices vanish
+ JV;1 = 0:
Thus, there is no drift of the first sublattice relatively to the others (the opposite would have been unacceptable). As the flux of vacancies is opposite to the sum of the fluxes of the ions, it can be eliminated from the entropy source
Ji;1 =−
−Tσ =
ð9Þ
and the definition of the fluxes (Eq. (4)), it is found that
Au ; Ax
Fi; j =− zi; j F
Starting from the time derivative of this expression, using the conservation law for ions
j = 1
c z i = 1 i;j i;j j
c z i = 1 i;1 i;1 1
= 0;
= constant in time and space:
ð14Þ
E. Gardés, B. Poumellec / Solid State Ionics 180 (2009) 1133–1138
Starting from the time derivative of this expression, using the conservation law (Eq. (9)) and the definition of the fluxes (Eq. (4)), it is found that there is zero net electric current in the sublattice Xn
1
i =
ð15Þ
z J = 0: 1 i;1 i;1
The movement of ions with various mobilities can also result in vacancy accumulation. This provokes gradients in the chemical potential of the vacancies, even if they were initially in equilibrium. Analogously with the negative retroaction of the electric field on electric charge excess, we assume ∂µV,1/∂x opposes to vacancy wind. Two phenomena support this assumption. The movement of species with different volumes induces an internal stress gradient (see, e.g., [15]). It is expected to be the highest in the vicinity of vacancy, where the ions repulse each other because of the missing ion with opposite electric charge. Thus, the internal stress gradient due to the movement of vacancies, implicitly included in ∂µV,1/∂x, opposes to vacancy accumulation. The internal stress gradient due to the movement of ions is neglected in comparison. Another phenomenon that will oppose to vacancy flux is the interaction of the vacancies of the first sublattice with those of the other sublattices, then forming immobile complexes (see, e.g., [2]). Thus, we assume that ∂µV,1/∂x ensures that the flux of vacancies is small and negligible compared to the fluxes of ions ð16Þ
JV;1 = 0:
It follows from Eq. (10) that there is zero net ion flux in the sublattice Xn
1
i =
ð17Þ
J = 0: 1 i;1
1135
The fluxes, expressed versus the ∂µj/∂x only, are then
Ji = −
1 Xn j = T
Pn 1
Lij −
Pn k = 1 Lik l = 1 Llj Pn k;l = 1 Lkl
!
Acj ; ði = 1; …; nÞ: Ax
The contributions of ∂φ/∂x and ∂µV/∂x in Eq. (19) cannot be distinguished. The reason is that the fulfillment of the zero net electric current relation also ensures the fulfillment of zero net ion flux in the case of isovalent ions. Thus, the same result would have been obtained in a description where the chemical potential gradient of vacancies is assumed to be zero, or a description without considering vacancies at all (e.g. [8,16]). However, ∂µV/∂x cannot be generally set to zero, especially in the case of the diffusion of non-isovalent ions as will be seen later. In order to find the expression of the chemical diffusion matrix [D], chemical potential gradients can be related to concentration gradients by: Xn Acj Acj Aci = : i = 1 Ac Ax Ax i
ð21Þ
The sum is limited to the n ions of the sublattice since the concentration gradients of the other species are zero. When the chemical potential of the ions has the form 0 cj = cj + RTln γj cj ; where µ0j is a standard value, R the gas constant and γj the activity coefficient, the chemical potential gradients can be written
Thus, there is no lattice distortion. The velocities of the sublattices are all identical and zero. The conservation law (Eq. (9)) implies
Xn Acj /ij + δij Aci = RT ; i = 1 Ax ci Ax
A Xn1 c = 0; i = 1 i;1 At
where δij is the Kronecker delta and ϕij supports non ideality
and, because of Eq. (8), one obtains Xn 1 c = constant in time and space: i = 1 i;1
/ij =
4. Diffusion of isovalent ions
ci Aγj : γ j Aci
Xn − 1 Ac Acn i =− : i = 1 Ax Ax
When all the ions have the same number of electric charge z, the fluxes (Eq. (12)) simplify to
Jn =−
Acj 1 Xn Au AcV − + zF ; L ij j = 1 T Ax Ax Ax
i; j = 1 P n
Lij
Acj Ax
i; j = 1 Lij
:
Xn −
1 j = 1
Dij
Acj ; Ax
ði = 1; …; n − 1Þ;
Xn − 1
J; i = 1 i
ð25aÞ ð25bÞ
where the coefficients of the chemical diffusion matrix [D] are
where the subscript 1 that refers to the first sublattice is dropped for clarity, here and in the following equations. Using the zero net electric current relation (Eq. (15)), or similarly the zero net ion flux relation (Eq. (17)), one obtains a relation between the forces Pn
ð24Þ
It is thus possible to express the first n-1 independent fluxes as a function of the first n-1 independent concentration gradients in the generalized Fick's law form, with the last flux given by Eq. (17) Ji = −
Au AcV − =− Ax Ax
ð23Þ
Because the sum of the concentrations of the n ions is a constant (Eq. (18)), only n-1 concentration gradients are independent
4.1. General expressions
zF
ð22Þ
ð18Þ
The zero net ion flux and zero net electric current relations impose two constraints on the fluxes but they are identical when all the ions have the same valence. Thus, two cases will be developed in the following: the isovalent case, where all the ions have the same valence and n-1 fluxes are independent, and the non-isovalent case, where at least one ion has a different valence from the others and only n-2 fluxes are independent.
Ji = −
ð20Þ
ð19Þ
Dij = R
Xn k = 1
/kj + δkj / + δkn − kn cj cn
! Lik −
! Pn l;p = 1 Lil Lpk P ; ði; j = 1; …; n −1Þ: n l;p = 1 Llp
ð26Þ 4.2. Simplifications 4.2.1. Approximations It may be assumed that the geometric correlations between the jumps of the ions are negligible compared to the correlations that
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E. Gardés, B. Poumellec / Solid State Ionics 180 (2009) 1133–1138
derive from the zero net ion flux and zero net electric current conditions (Eqs. (15) and (17)). Therefore, if there are no supplementary constraints on the movement of the ions, as assumed in the present description, the off-diagonal phenomenological coefficients can be set to zero ði ≠ j with i; j = 1; …; nÞ:
Lij = 0;
ð27Þ
The diagonal phenomenological coefficients are then proportional to the self-diffusion coefficients or to the tracer diffusion coefficients D⁎i if correlation factors are neglected in experimental measurements [3,5] Lii =
ci D4 i ; ði = 1; …; nÞ: R
ð28Þ
A subsequent consistent approximation is to neglect the chemical interactions between the ions, that is to assume the mixture of the ions in the sublattice is ideal /ij = 0;
ði; j = 1; …; nÞ:
ð29Þ
Therefore, the elements of the chemical diffusion matrix (Eq. (26)) simplify considerably to 0
1 4 − D4n c D i j B C Dij = D4 A: i @δij − P n 4 c D k = 1 k k
ð30Þ
It has to be remarked that this expression, because of the above strongly simplifying approximations, can be inappropriate for many systems and therefore has to be considered cautiously. However it allows for a first order analysis as will be seen in the following. 4.2.2. Case of the diffusion of two isovalent ions Let A and B be two equally charged ions inter-diffusing in a sublattice of a crystal. As expected, Eqs. (25a) and (25b) yield that the fluxes of A and B are opposite JA =− DAA
AcA ; Ax
ð31bÞ
When geometric correlations between the jumps of the ions are neglected (Eq. (27)), except for the correlations that derive from the zero net ion flux and zero net electric current conditions, when the diagonal phenomenological coefficients are related to tracer diffusion coefficients (Eq. (28)) and when the solution is assumed ideal (Eq. (29)), the solution for DAA is (Eq. (30))
4 ðcA + cB ÞD4 A DB 4 cA D4 A + cB DB
;
ð32Þ
which is a well-known result, readily obtained when the Nernst– Planck equation is used for describing the fluxes of A and B (e.g. [4]). It follows that if one of the two ions has a small concentration or diffusivity it will control the interdiffusion, e.g. when cA bb cB, one has DAA = D4 A; as expected, since this is the definition of the tracer diffusion coefficient when correlations in the experimental measurement are ignored. When DA⁎ bb DB⁎, one has DAA =
cA + cB 4 DA : cB
4.2.3. Case of the diffusion of three isovalent ions Let A, B and C be three diffusing ions with identical electric charge. Fluxes are (Eqs. (25a) and (25b)) JA =− DAA
AcA Ac − DAB B ; Ax Ax
ð33aÞ
JB =− DBA
AcA Ac − DBB B ; Ax Ax
ð33bÞ
JC =− JA − JB :
ð33cÞ
When all the approximations are done as in the previous example (Eqs. (27)–(29)), the elements of the chemical diffusion matrix are (Eq. (30)) DAA =
DAB =
DBA =
DBB =
4 4 4 cB D4 A DB + ðcA + cC ÞDA DC 4 4 cA D 4 A + cB DB + cC DC
4 4 4 −cA D4 A DB + cA DA DC
4 4 cA D 4 A + cB DB + cC DC 4 4 4 −cB D4 A DB + cB DB DC
4 4 cA D 4 A + cB DB + cC DC
;
ð34aÞ
;
ð34bÞ
;
ð34cÞ
4 4 4 cA D 4 A DB + ðcB + cC ÞDB DC 4 4 cA D 4 A + cB DB + cC DC
:
ð34dÞ
ð31aÞ
JB =− JA :
DAA =
This result is a consequence of the negative retroaction of both the electric field and the chemical potential gradient of the vacancies on electric charge excess and vacancy accumulation (see Section 3). If one ion diffuses faster than the other one, then the induced electric field and chemical potential gradient of the vacancies will oppose to electric charge excess and vacancy accumulation. The fastest ion will decelerate down to a velocity close to the other one so the electric charge excess and vacancy accumulation is minimized. Thus, the interdiffusion is controlled by the slowest ion.
This result can also be obtained using the Nernst–Planck equation for the fluxes of the three ions and solving for the electric field with the zero net current relation. When the concentration of a species is set to zero, one expects to be left with a binary case. In fact, when cC = 0, the flux of C is obviously zero and the fluxes of A and B becomes opposite, as well as their concentration gradients (Eq. (24)) and one has JA =−ðDAA − DAB Þ
4 ðc + cB ÞD4 AcA A DB AcA =− A ; Ax 4 Ax cA D4 A + cB DB
as in the case of the interdiffusion of two isovalent ions. The same result is obtained when both the diffusivity and the concentration gradient of one ion are much smaller than those of the other two. 5. Diffusion of non-isovalent ions 5.1. General expressions In the case where ions are not all equally charged, the fluxes are described by the general form (Eq. (12)) Ji = −
Acj 1 Xn Au AcV − + z ; L F ij j j = 1 T Ax Ax Ax
where the subscript 1 that refers to the first sublattice is dropped for clarity, here and in the following equations. ∂µV/∂x and ∂φ/∂x are
E. Gardés, B. Poumellec / Solid State Ionics 180 (2009) 1133–1138
determined using the conditions of zero net ion flux (Eq. (17)) and zero net electric current (Eq. (15)) Xn AcV = i;j Ax
= 1
Xn Au = F i;j Ax
Acj βzi − γ Lij ; β2 − αγ Ax
ð35Þ
Xn
L ; i;j = 1 ij
ð36Þ
ð37aÞ
Xn
zL ; i;j = 1 i ij
β=
ð37bÞ
Xn
zzL ; i;j = 1 i j ij
γ=
ð37cÞ
and the Onsager's relations have been used (Eq. (13)). Contrary to the case of isovalent ions, ∂µV/∂x and ∂φ/∂x are independent because the zero net ion flux and zero net electric current relations are two independent constraints when the ions do not have all the same valence. Thus, if the chemical potential gradient of the vacancies is not included in the description, the condition of zero net ion flux leads to a relation between the phenomenological coefficients. In that case, the coefficients have to be chosen properly in order to fulfill that condition and, especially, it is impossible to set the off-diagonal terms to zero. Substituting Eqs. (35) and (36) leads to the fluxes Ji = −
1 Xn j = T
1
It can be remarked that n has to be greater than 2. In fact, the configuration with only two aliovalent ions that ensures the fulfillment of both electroneutrality and site conservation is that in which the two ions have uniform concentrations, i.e. are not diffusing. 5.2. Simplifications
αzi − β Acj ; L = 1 β 2 − αγ ij Ax
where α=
Xn Acj αzk − β Xn βz − γ Xn : Lij + z L − 2k L Lkj k = 1 β 2 − αγ l = 1 l il l = 1 il Ax β − αγ
5.2.1. Approximations The approximations here are identical to those done in Section 4.2.1. That is, the geometric correlations between the jumps of ions are neglected compared to the correlations that derive from the zero net ion flux and zero net electric current conditions (Eq. (27)), the diagonal phenomenological coefficients are related to tracer diffusion coefficients (Eq. (28)) and the mixture of the ions is assumed ideal (Eq. (29)). It follows a considerable simplification of the coefficients of the chemical diffusion matrix
½
Dij = D4 i δij +
Xn − 2 z −z Acn − 1 Aci i n = ; i = 1 z − z Ax n n − 1 Ax
ð39aÞ
Xn − 2 z Acn n − 1 − zi Aci = : i = 1 z − z Ax n n − 1 Ax
ð39bÞ
Using the relation between chemical potential gradients and concentration gradients (Eq. (22)), it is possible to express the first n-2 independent fluxes as a function of the first n-2 independent concentration gradients in the generalized Fick's law form, with the two last fluxes given by the zero net ion flux and zero net electric current relations (Eqs. (15) and (17)) Xn
− 2 j = 1
Jn − 1 = Jn =
Dij
Acj ; Ax
ði = 1; …; n − 2Þ;
Xn − 2 z − z i n Ji ; i = 1 z − z n n−1
Xn
− 2 i = 1
zn − 1 − zi J; zn − zn − 1 i
Xn
ði; j = 1; …; n − 2Þ:
βzj − γ
ci D4 j
2
ð42Þ where α (Eq. (37a)), β (Eq. (37b)), and γ (Eq. (37c)) simplify to Xn i = 1
β=
γ=
Xn i = 1
Xn i = 1
ci D4 i ; R
ð43aÞ
zi ci D 4 i ; R
ð43bÞ
z2i ci D4 i : R
ð43cÞ
It should be recalled that this expression derives from strongly simplifying approximations and therefore has to be considered cautiously (see Section 4.2.1.). 5.2.2. Case of the diffusion of three aliovalent ions Let zA, zB and zC be the three non equal numbers of electric charge of three diffusing ions A, B and C. Because of the two relations of zero net ion flux and zero net electric current, only one flux is independent in this ternary system (Eqs. (40a), (40b), and (40c)) JA =− DAA
AcA ; Ax
ð44aÞ
zA − zC J ; zC − zB A
ð44bÞ
ð40bÞ
JC =
zB − zA J : zC − zB A
ð44cÞ
ð40cÞ
When geometric correlations between the jumps of the ions are neglected (Eq. (27)), except for the couplings that derive from the zero net ion flux and zero net electric current conditions, when the diagonal phenomenological coefficients are related to tracer diffusion coefficients (Eq. (28)) and when the solution is assumed ideal (Eq. (29)), the solution for DAA is (Eqs. (42) and (43a), (43b), and (43b)))
"
f
zi −
JB =
/jk + δjk zj − zn /n − 1k + δn − 1k zn − 1 − zj /nk + δnk + + cj zn − zn − 1 cn − 1 zn − zn − 1 cn Xn αzl − β Xn βzl − γ Xn z L − 2 L Llk ; × Lik + l = 1 β 2 − αγ p = 1 p ip p = 1 ip β − αγ k = 1
2
ð40aÞ
where the coefficients of the chemical diffusion matrix [D] are
Dij = R
αzj − β
α=
From the conservation of sites and electric charge relations (Eqs. (14) and (18)) follows that only n-2 concentrations are independent. When the n-1th and the nth components are chosen such that zn−1 ≠zn, one obtains
R β − αγ β − αγ zj − zn ci D4 αzn − 1 − β βzn − 1 − γ n − 1 zi − + 2 2 z R β − αγ β − αγ n − zn − 1 αzn − β βz − γ zn − 1 − zj ci D4 n + ; ði; j = 1; …; n − 2Þ; zi − 2 n β 2 − αγ β − αγ zn − zn − 1 R
ð38Þ
Ji = −
1137
#
g
ð41Þ
DAA =
2 2 2 T T T cA cB ðzA − zB Þ + cA cc ðzA −zc Þ + cB cc ðzB − zc Þ DA DB Dc cA cB ðzA −zB Þ2 DTA DTB + cA cc ðzA −zc Þ2 DTA DTc + cB cc ðzB −zc Þ2 DTB DTc
: ð45Þ
1138
E. Gardés, B. Poumellec / Solid State Ionics 180 (2009) 1133–1138
This result cannot be obtained when the Nernst–Planck equation is used for describing the fluxes of the three ions. In fact, after solving for the electric field with the help of the zero net electric current relation, the application of the zero net ion flux relation leads to a relation between the tracer diffusivities, which is obviously not acceptable. As in the case of the diffusion of two isovalent ions, a species with a small concentration or diffusivity will control the interdiffusion. One has, when cA bb cB, cC DAA = D4 A; as expected, since this is the definition of the tracer diffusion coefficient when correlations in the experimental measurement are ignored. One has, when DA⁎ bb DB⁎, DC⁎ DAA =
cA cB ðzA −zB Þ2 + cA cc ðzA −zc Þ2 + cB cc ðzB −zc Þ2 4 DA cB cc ðzB −zc Þ2
The interdiffusion is controlled by the slowest species because of the negative retroaction of both ∂φ/∂x and ∂µV/∂x on the electric charge excess and the vacancy accumulation due to the diffusion of the fastest ions (see Section 3 and 4.2.2). 5.2.3. Case of the diffusion of two isovalent ions with one aliovalent ion It should be recalled first that for any non-isovalent system, the ions must be sorted so that the charges of the two last ions are different (Eqs. (39a) and (39b)). Let consider then three diffusing ions A, B and C with electric charges such that zA = zB ≠ zC. It follows from Eqs. (40a), (40b), and (40c) that the flux of C must be zero and that the fluxes of A and B are opposite JA =− DAA
AcA ; Ax
Acknowledgement We would like to thank very much the late Prof. Olivier Jaoul from Université Paul Sabatier in Toulouse for his training in Mineral Physics during our PhDs. This paper is dedicated to his memory.
JB =− JA ; JC = 0: When all the approximations are done as for the previous example (Eqs. (27)–(29)), DAA is (Eqs. (42), (43a) (43b) and (43c)) DAA =
4 ðcA + cB ÞD4 A DB 4 cA D4 A + cB DB
diffusion occurs in one sublattice and is generalized to any number of diffusing ions, carrying any electric charge. One of the main assumptions is that the species of the other sublattices are immobile. Moreover, the flux of vacancies is assumed to be negligible compared to that of ions because of the internal stress induced by their movement and because of their immobility via defect pairing. It follows that the net flux of the ions in the sublattice is zero. Electroneutrality is also assumed to be fulfilled, which implies that the net electric current is zero. It is assumed that these two conditions on the fluxes are both ensured by the electrostatic potential gradient and the chemical potential of the vacancies gradient because they oppose to any departure from these conditions. When the diffusing ions are isovalent, the two conditions of zero net ion flux and zero net electric current are equivalent. Thus, only n-1 fluxes remain independent and the role of the chemical potential of the vacancies is not distinguishable from that of the electrostatic potential. However, the two conditions are not equivalent when the ions are not isovalent. In that case, only n-2 fluxes are independent and the chemical potential gradient of the vacancies is independent from the electric field. It is shown that neglecting the chemical potential gradient of vacancies introduces a relation between the phenomenological coefficients. The consequence is that the off-diagonal terms cannot be generally set to zero. Generalized Fick's laws with expressions of the coefficients of the chemical diffusion matrix are given for both cases of isovalent and nonisovalent diffusing ions. The analysis of simplified cases shows that the Nernst–Planck equation can be used to describe isovalent systems but fails in the description of non-isovalent ones. This is the consequence of setting the chemical potential gradient of vacancies to zero, what is implicit in the Nernst–Planck equation since the electrostatic potential is the only diffusion potential. Our formalism is thus more appropriate to describe non-isovalent systems.
;
which is identical to Eq. (32), derived in the case of the diffusion of two isovalent ion. We are thus left with a binary case. This result is not surprising since the only possibility for having two ions with identical valences and a third ion with a different valence and both electroneutrality (Eq. (14)) and site conservation (Eq. (18)) fulfilled is that the concentration of the third ion is uniform. Thus, it does not participate to diffusion and only the two first isovalent ions inter-diffuse. 6. Conclusion We report a new description of multicomponent diffusion in ionic crystals containing Schottky defects. It is dedicated to the case where
References [1] Y. Adda, J. Philibert, La diffusion dans les solides, Presses universitaires de France, Paris, 1966. [2] J.S. Kirkaldy, D.J. Young, Diffusion in the Condensed State, The institute of metals, London, 1987. [3] P. Shewmon, Diffusion in Solids, The minerals, metals & materials society, Warrendale, 1998. [4] J. Philibert, Atom Movements — Diffusion and Mass Transport in Solids, Les éditions de physique, Les Ulis, 1991. [5] A.R. Allnatt, A.B. Lidiard, Rep. Prog. Phys. 50 (1987) 373. [6] R.E. Howard, A.B. Lidiard, Rep. Prog. Phys. 27 (1964) 161. [7] I.V. Belova, A.R. Allnatt, G.E. Murch, Philos. Mag. (2007) 4169. [8] A.B. Lidiard, Nuovo Cimento D 15 (1993) 229. [9] R. Lindström, J. Phys. C. Solid State Phys. 7 (1974) 3909. [10] D.J. Young, J.S. Kirkaldy, J. Phys. Chem. Solids 45 (1984) 781. [11] D.J. Young, F. Gesmundo, Oxid. Met. 29 (1988) 169. [12] S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics, North Holland, Amsterdam, 1962. [13] V.M. Aguilella, S. Mafé, J. Pellicer, Electrochem. Acta 32 (1987) 483. [14] A.D. MacGillivray, J. Chem. Phys. 48 (1968) 2903. [15] G.B. Stephenson, Acta Metall. 36 (1988) 2663. [16] D.G. Miller, J. Phys. Chem. 71 (1967) 616.