Theoretical consideration on the application of the Aagaard–Helgeson rate law to the dissolution of silicate minerals and glasses

Chemical Geology 255 (2008) 14–24

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Chemical Geology 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 / c h e m g e o

Theoretical consideration on the application of the Aagaard–Helgeson rate law to the dissolution of silicate minerals and glasses Stéphane Gin ⁎, Christophe Jégou, Pierre Frugier, Yves Minet Commissariat à l'Énergie Atomique (CEA), Marcoule DTCD/SECM, BP 17171, 30207 Bagnols - sur - Cèze Cedex, France

A R T I C L E

I N F O

Article history: Received 18 December 2007 Received in revised form 29 April 2008 Accepted 13 May 2008 Editor: D. Rickard

A B S T R A C T The kinetic laws derived from the work of Aagaard and Helgeson are discussed, notably those applied to aluminosilicate or borosilicate glasses. Aagaard and Helgeson extended the kinetic formalism of an elementary reaction in a homogeneous medium to overall alteration processes in heterogeneous media by assuming they consist of a series of elementary steps. The dissolution rate of a mineral phase can thus be expressed as follows: −nij

Keywords: Silicate Rate law Glass Aagaard-Helgeson Transition state theory

r / ¼ k/ ∏ ai i



 A 1−exp − σRT

ð1Þ

where kϕ is the kinetic constant of hydrolysis of the mineral ϕ, ai the activity of the reactants i in the limiting elementary step j, nij the stoichiometric coefficient for reactant i in the limiting reaction j, A the chemical affinity of the overall dissolution reaction, σ the average stoichiometric number of the overall reaction, R the ideal gas constant and T the temperature. We first illustrate the relation between transition state theory and a kinetic law such as Eq. (1) initially associated with an elementary reaction, using the simple example of hydriodic acid synthesis. We then discuss the extension of Eq. (1) to overall processes, showing that there is no obvious relation between the elementary limiting step and the contents of the affinity function and that this reflects a problem of scale in the Aagaard–Helgeson law between the kinetic constant (based on microscopic theories) and the affinity term (a macroscopic entity derived from classical thermodynamics). The discussion shows the difficulties encountered in attempting to determine the activity of the reactants in the elementary limiting step, particularly in the case of surface groups, and highlights the limited validity of extending a chemical affinity law of the type (1 − exp(−ΔG/RT))—which is theoretically valid for an elementary reaction near equilibrium for an overall process. Our article then reviews Grambow's reasoning and the difficulties he encountered in applying the Aagaard– Helgeson law to the dissolution of nuclear borosilicate glasses. This example shows that the concepts of the Aagaard–Helgeson law are not simple to use, notably for determining the content of the affinity function and calculating the activity of the surface species. With complex (basaltic or nuclear) glasses, formulating the hypothetical series of elementary reactions becomes unrealistic, and the notion of an equilibrium constant remains a difficult problem. From those considerations, one can conclude that the classical first order rate law appears to be a more empirical than theoretical equation. Moreover, even if the affinity function with respect to the silicate network stability is a point to account for the rate drop, other phenomena like slow diffusion of reactive species through the hydrated layer or precipitation of secondary minerals (smectite, zeolite) are likely more important to predict the long-term dissolution rate of natural or nuclear glasses in most of the confined environments. © 2008 Elsevier B.V. All rights reserved.

1. Introduction Some aluminosilicate glasses, such as basalt glasses, are considered to be natural analogs for the glasses that have been developed to confine the fission products and actinides obtained from reprocessing spent nuclear fuel (Allen, 1982; Malow et al., 1984; Byers et al., 1985; ⁎ Corresponding author. Tel.: +33 4 66 79 14 65; fax: + 33 4 66 79 66 20. E-mail address: [email protected] (S. Gin). 0009-2541/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.chemgeo.2008.05.004

Lutze et al., 1985; Ewing and Jercinovic, 1987). Reasoning by analogy is based on the similarity of the short-term behavior of basalt and borosilicate nuclear glasses. Laboratory studies of the alteration kinetics of both types of glass have revealed several characteristic process steps (Byers et al., 1985; Crovisier et al., 1985; Berger et al., 1987; Guy, 1989; Crovisier et al., 1989; Advocat et al., 1991; Vernaz and Dussossoy, 1992; Crovisier et al., 1992; Verney-Carron et al., 2007). The initial dissolution is characterized by ion exchange between protons in solution and glass network modifier cations, resulting in the formation

S. Gin et al. / Chemical Geology 255 (2008) 14–24

of a hydrated layer depleted in alkali metal and alkaline earth cations, through which the aqueous species diffuse (Boksay et al., 1968). At the same time, the glass network—consisting of network forming cations (Si, B, Al, etc)—hydrolyzes, resulting in the release of these elements to solution. As long as the medium remains sufficiently dilute, dissolution occurs at a constant rate, that is, the initial dissolution rate r0. At a more advanced stage of the dissolution reaction (e.g. achievable in a closed reactor system) the glass alteration rate is observed to diminish (Vernaz and Dussossoy, 1992); dissolution then becomes incongruent. Both types of glass produce two types of alteration products: amorphous gels (known as palagonite for basaltic glass) and secondary mineral phases (phyllosilicates, zeolites, calcium silicates, rare earth phosphates, etc.) (Thomassin, 1984; Jercinovic et al., 1990; Advocat et al., 1991; Vernaz and Dussossoy, 1992; Verney-Carron et al., 2007). The kinetic laws used to describe the dissolution of basaltic glass (Guy, 1989; Berger et al., 1994; Daux et al., 1997) and nuclear glass (McGrail et al., 1997; Grambow, 1985; Advocat et al., 1998; Abraitis et al., 1999, Grambow and Muller, 2001) are thus relatively similar and almost always derived from the general law of mineral dissolution proposed by Aagaard and Helgeson in the early 1980s (Aagaard and Helgeson, 1982): −nij

r/ ¼ k/ ∏ ai i





 A 1−exp − σRT

ð1Þ

where rϕ is the dissolution rate of a mineral ϕ, kϕ is the kinetic constant of hydrolysis of the mineral ϕ, ai the chemical activity of species i, nij the stoichiometric coefficient for reactant i in the limiting reaction j, R the ideal gas constant, T the temperature, A the chemical affinity of the overall dissolution reaction, and σ the average stoichiometric coefficient of the overall reaction. Bourcier or Oelkers also proposed a kinetic law based on a chemical affinity function that takes into account an equilibrium between the alteration layer (“gel layer” or “leached layer” depending on the authors) and solution (Bourcier et al., 1990; Oelkers, 2001). Although Eq. (1) is widely used and has become a conventional framework in geochemistry, it is reasonable to reconsider the origin and theoretical signification of each of its terms. We begin by examining the contribution of transition state theory to the development of the Aagaard– Helgeson theory, before reviewing the difficulties encountered in using these theoretical concepts, notably in Grambow's work on nuclear borosilicate glass (Grambow, 1984, 1985). Kinetic laws with the same form as Eq. (1) but with a more general and global scope are also discussed. Our intention is to show that modifying the signification of the terms with respect to the theoretical concepts defined by Aagaard and Helgeson mitigates many of the constraints involved in the use of rate laws. 2. Contribution of transition state theory To discuss the reasoning advanced by Aagaard and Helgeson, consider the simple example of synthesizing hydriodic acid: I2 + H2 ↔ 2HI (Prigogine, 1967). The kinetics of this reversible chemical reaction involving competition between two elementary (forward and reverse) reactions, can easily be expressed by applying the Van't Hoff law to each: 2 rnet ¼ rþ −r− ¼ kþ CH2 CI2 −k− CHI :

ð2Þ

Eq. (2) can also be expressed as follows: i rnet


 Q ¼ kþ CI2 CH2 1− Keq

2 γ 2HI CHI γI2 γH2 CI2 CH2

and Keq ¼ Q when rnet ¼ 0: More generally, the expression for the rate of a reversible elementary reaction i of the type naA + nbB ↔ ncC + ndD takes the form (Prigogine, 1967):

 −Ai i rnet ¼ kþ Cana Cbnb 1−exp RT

ð5Þ

where rinet is the net rate of a reversible elementary reaction i, k+ the kinetic constant of the forward elementary reaction, CA the concentration of reactant A in elementary reaction i, na the reaction order (the stoichiometric coefficient of reactant A in the case of an elementary reaction), Ai the chemical affinity of elementary reaction i, R the ideal gas constant, and T the temperature. Ai is defined as follows:
 Keq Ai ¼ RTln : Q

ð6Þ

The affinity term of the Eq. (5) for an elementary reversible reaction step i has often been presented in the literature (Nagy et al., 1991; Burch et al., 1993) as a direct application of transition state theory (Eyring, 1935). In fact, it may be derived using simple kinetic concepts (notably the Van't Hoff law) irrespective of any hypotheses concerning the reaction mechanisms. The notion of an activated complex associated with an elementary step is theoretically compatible with the kinetic law (5), assuming an equilibrium exists between the activated complex and reactants in the forward and reverse directions (Lasaga, 1981). However, this notion is not required to obtain Eq. (5) and indeed leads to a paradox. This paradox may be illustrated by again considering the example of hydriodic acid, with the notion of an activated complex. The derivation we show follows that given by Lasaga (Lasaga, 1981). The overall mechanism involves two distinct elementary steps in which reactants (I2 and H2 in the forward direction and HI in the reverse direction) cross over two different potential energy barriers with the same “peak” (the activated complex is assumed to be the same in both directions). The kinetic constants determined from transition state theory (Lasaga, 1981) are: kþ ¼


 kb T γ I2 γ H2 ΔG⁎ exp − h kb T γ⁎

ð7Þ

k− ¼


 kb T γ2HI ΔG⁎0 exp − h γ⁎ kb T

ð8Þ

where γ is an activity coefficient (the asterisk refers to the activated complex), kb is the Boltzman coefficient, and h is the Planck constant. These expressions imply the existence of an equilibrium between the reactants and the activated complex in the forward and reverse directions; ΔG⁎ and ΔG⁎′ are the standard free enthalpy values. Hence: ΔG⁎ ¼ μ-⁎−μ˚I2 −μ˚ H2 ΔG⁎0 ¼ μ-⁎−2μ˚HI

ð9Þ ð10Þ

ð3Þ where µ°i is the chemical potential of species i in its standard state. The ratio of the kinetic constants can then easily be obtained:

where Q¼

15

ð4Þ



 γ I2 γ H2 ΔG˚ kþ γ I2 γH2 ΔG⁎−ΔG⁎ 0 ¼ ¼ exp − exp − k− kb T kb T γ2HI γ2HI

ð11Þ

16

S. Gin et al. / Chemical Geology 255 (2008) 14–24

where ΔG° is the standard free enthalpy of the elementary reaction (this is not the reaction affinity, which depends on the reaction progress). ΔG° is expressed as a function of the chemical potentials: ΔG- ¼ μ -I2 þ μ -H2 −2μ˚HI :

ð12Þ

Because reaction I2 + H2 ↔ 2HI is a reversible elementary step, according to the detailed balancing principle (see below), the ratio of the kinetic constants can be expressed as follows: kþ γ I2 γ H2 0 ¼ Keq ¼ Keq k− γ2HI

ð13Þ

where K′eq is the equilibrium constant related to the concentrations of reactants (I2 and H2) and products (HI). From Eq. (11), the ratio of the rates is then:
 ΔG˚ rþi kþ CI2 CH2 γ I2 γH2 CI2 CH2 ¼ ¼ exp − 2 2 r−i kb T k− CHI γ2HI CHI

ð14Þ



 ΔG˚ rþi aI2 aH2 ΔGi ¼ exp − ¼ 2 exp − r−i kb T kb T aHI

ð15Þ

where ΔGi is the free enthalpy of the elementary reaction, depending on the reaction progress ξ. Taking the net rate (rinet = r+ i − r− i) into account yields Eq. (5), Which expresses the rate of elementary step I as a function of the chemical affinity (Ai = −ΔGi):

 ΔGi i rnet ¼ rþi 1−exp RT

 −Ai i : rnet ¼ kþ CI2 CH2 1−exp RT

ð16Þ ð17Þ

The paradox lies in the fact that an equilibrium was assumed between the activated complex and reactants (I2 and H2) in the forward direction, but that a second equilibrium was also assumed between the activated complex and the product (HI) in the reverse direction. This implies an equilibrium between the products and the reactants, so the net rate should be zero. This paradox, of course, does not call into question the expression of the kinetic constants: the forward rate simply offsets the reverse rate. Postulating an equilibrium between the reactants forming the activated complex in both directions and the activated complex therefore implies that Eq. (5) is valid only at equilibrium. At this point it is important to note that transition state theory— which is often cited in the literature to justify Eq. (5) (Nagy et al., 1991; Burch et al., 1993)—can only determine the kinetic constant k+ (or k−) associated with an elementary step. It cannot address the evolution of the energy (affinity) term, which is based on classic thermodynamics. 3. The approach proposed by Aagaard and Helgeson The hypothesis advanced by Aagaard and Helgeson, Eq. (5), which is valid near thermodynamic equilibrium, is also applicable to an overall mechanism comprising a series of elementary steps. If all the elementary steps in the series are at equilibrium except for one (the limiting step j) the overall affinity A is then equal to the chemical affinity Aj of elementary reaction j, and the rate is equal to that of the limiting step j (Aagaard and Helgeson, 1982). A ¼ σAj tot rnet ¼

i rnet : σ

ð18Þ ð19Þ

Apart from the stoichiometric coefficient σ, the inclusion of which in the affinity function is open to debate (Lasaga, 1995), Eq. (1) for thus takes the same form as Eq. (5). Activity product Πa− nij in general Eq. (1) (or the concentration product, if the activity coefficients are included in the intrinsic kinetic constant) refers to the reactants of the limiting elementary step. The nij coefficients thus represent the number of molecules of species i reacting during limiting step j. Another important point in the theoretical approach developed by Aagaard and Helgeson is that the overall activation energy is identified with that of the limiting elementary step: Ea =RT + ΔHj. Eq. (1) takes into account the overall affinity of the reaction, not the affinity of the elementary limiting step which, in the Aagaard– Helgeson hypothesis, is the driving force behind the overall mechanism. Given that the two affinities are hypothetically equivalent, the values of the (1-exp(-A/σRT)) function can be determined even without determining the nature of the limiting elementary step. It is thus apparently not indispensable to calculate the chemical affinity of the limiting step; this is a practical advantage, since it is not always possible to determine the activity of reactants and products of an elementary reaction with heterogeneous kinetics. Conversely, the nature and activity of the reactants in the limiting elementary step must be known to determine the activity product Πa− nij. This review of the terms of Eq. (1) underscores the fact that applying the law within such a theoretical framework raises a problem of scale: it is necessary to determine an affinity term based on the macroscopic notion of solubility and a concentration product for reactants operating at a molecular scale but based on microscopic kinetic theories. This situation raises a number of questions. 3.1. Is the content of the affinity function related to the nature of the limiting activated complex? Aagaard and Helgeson associate the limiting elementary step j with a critical activated complex whose irreversible decomposition limits the overall alteration process: that is, the chemical affinity is thus implicitly related to the nature of the critical activated complex, as expressed in Eq. (18). Some authors (Daux et al., 1997; McGrail et al., 1997) use this implicit relation by expressing the chemical affinity of an overall reaction in terms of Aj. With regard to the dissolution of basaltic glass, Daux considers the activated complex to be stoichiometrically identical with the dealkalinized pristine glass. For McGrail, investigating the dissolution of a nuclear borosilicate glass, the stoichiometry of the activated complex is independent of the pristine glass composition; the stoichiometric coefficients of the activated complex, which appear in the ion activity product of the affinity function, are arbitrarily assumed equal to one. In both approaches, flow-through experiments were used to fit the unknown parameters of the kinetic law (forward rate, equilibrium constant, Temkin coefficient, etc.). Other authors—for example, Oelkers investigating the alteration kinetics of feldspars with a variable Si/Al ratio—consider that the relation between the limiting mechanism and the affinity function is not obvious (Oelkers et al., 1994; Oelkers and Schott, 1995). For albite (NaAlSi3O8), with a Si/Al ratio of 3, preferential hydrolysis of the Si–O– Al bonds produces purely siliceous-limiting surface precursors. Conversely, in the case of anorthite (CaAl2Si2O8), with a Si/Al ratio of 1, successive hydrolysis of the Si–O–Al bonds of the tetrahedrons releases silicon and aluminum directly into solution; the limiting surface precursors in this case are aluminosilicates. Oelkers shows that these differences are reflected directly in the activity products Πa−nij preceding the affinity function in the Aagaard–Helgeson law (Eq. (1)). For albite, this product comprises the activity of protons and Al3+ ions, but the protons in the case of anorthite. Despite the differences in the nature of the limiting mechanisms, Oelkers considered an overall affinity function (A and not Ai) for both minerals taking into account

S. Gin et al. / Chemical Geology 255 (2008) 14–24

the activities of all the constituents (Ca2+, Al3+, SiO2aq for anorthite and Na+, Al3+, SiO2aq for albite). Oelkers was able to show on the basis of flow-through experiments with various dissolved silicon and aluminum concentrations that this reasoning is valid. This manner of perceiving the terms of the Aagaard–Helgeson law converges on the approach adopted by Lasaga (Lasaga, 1981, 1995), who asserts that the affinity function derived from the thermodynamics of irreversible processes is based on the macroscopic notion of solubility, irrespective of any hypothesis concerning the reaction mechanisms (see Section 4. Generalized laws below). These examples taken from published works show that the reasoning proposed by Aagaard and Helgeson is subject to different interpretations. Nevertheless, it appears preferable today to consider that the chemical affinity is unrelated to the nature of the elementary step (Lasaga, 1995) even if this implies that microscopic and macroscopic terms coexist in the rate law (Eq. (1)).

17

(Rimstidt and Barnes, 1980; Ganor et al., 2005). As previously noted, this is not an elementary reaction but rather a series of adsorption and hydrolysis reaction steps affecting the bonds of the silica tetrahedron. A schematic example of the overall silica dissolution mechanism is provided by Lasaga (Lasaga, 1981): K1

H2 O þ ≡Si− ↔ H2 Oads :Si → ≡Si−OH k1

K2

H2 O þ ≡Si−OH ↔ H 2 Oads :Si−OH → ¼ SiðOH Þ2 k2

K3

H2 Oþ ¼ SiðOHÞ2 ↔ H2 Oads :SiðOHÞ2 → −SiðOH Þ3

K4

3.2. How can the activity of the reactants in the limiting elementary step be defined with heterogeneous kinetics?

3.3. To what extent may a chemical affinity law of the type (1 − exp(−A/ σRT)), which is valid for an elementary step near equilibrium, be applied to an overall alteration mechanism? Lasaga contests the validity of such an extrapolation from a mathematical standpoint (Lasaga, 1995), by showing that an error in Boudart's work (Boudart, 1976) makes it impossible to conclude that the chemical affinity function associated with an overall mechanism has the same form—up to and including the Temkin coefficient—as one associated with an elementary reaction. Lasaga thus proposes a rate law in which the analytical form of the chemical affinity function cannot be defined a priori (see Section 4. Generalized laws below). From an experimental standpoint, only the works reported by Rimstidt and Ganor provided a validation of the Aagaard–Helgeson approach for the simple silica hydrolysis reaction mentioned above

k3

H2 O þ −SiðOH Þ3 ↔ H2 Oads :SiðOH Þ3 → H4 SiOads 4 k5

One of the major problems in a strict application of the Aagaard– Helgeson law is to determine the activity of the surface species with heterogeneous kinetics. The problem does not arise in calculating the affinity function, which, according to Aagaard and Helgeson, is an overall function, but it does occur in determining the activity product Πa-nij. As noted above, Oelkers was able to express the product Πa-nij for the feldspar dissolution reaction, but this approach requires assumptions of the structure of the phase involved. This is generally not the case with complex glass formulations. Another difficulty lies in the definition of an elementary step in heterogeneous kinetics. Unlike homogeneous media, for which the Van't Hoff law based on collision theory is applicable, the concepts of order and molecularity are more difficult to define in heterogeneous kinetics (Glasstone et al., 1941; Madé, 1991). The composite and indirect nature (adsorption, hydrolysis, desorption) of heterogeneous reactions makes it difficult to express an elementary reaction. In the case of silicate glasses, silica dissolution (SiO2 + 2H2O ↔ H4SiO4) is not an elementary reaction (Lasaga and Gibbs, 1990), but rather a succession of adsorption and hydrolysis steps affecting the bonds of the silica tetrahedron. It is not possible at the present time to define or measure the activity of a silicon atom partially separated from the surface of the solid. Moreover, successive hydrolysis and recondensation reactions disturb the interface, which is undoubtedly three-dimensional. In as much as the solution composition is more easily measured than that of the mineral surface, the activity product is generally in fact limited to the aqueous species (Daux et al., 1997; Advocat et al., 1998), and only certain activities such as H+ or OH– are explicitly defined. This means that the kinetic constant is not an intrinsic constant for the reaction, because it includes the activities of all the unmeasured reactive species of the limiting step (Nagy et al., 1991).

ð20Þ

aq H4 SiOads 4 → H4 SiO4 :

k4

ð21Þ

ð22Þ

ð23Þ

ð24Þ

The Ki constants are the equilibrium constants for water adsorption on each silicate site; each hydrolysis step is characterized by a kinetic constant ki. The concepts of transition state theory should from a strict point of view be applicable to each elementary step in the hydrolysis of the SiO4 tetrahedron (Lasaga and Gibbs, 1990) and not to the overall SiO2 + 2H2O ↔H4SiO4 reaction (Rimstidt and Barnes, 1980). In other words, the notion of an activated complex cannot be associated with the SiO2 + 2H2O ↔H4SiO4 reaction. However, the validity of “extending” the concept of an elementary reaction—and thus of an affinity function ƒ(A) of the type (1−exp(−A/ σRT))—to the case of silica does in fact follow the detailed balancing principle. This principle stipulates that at thermodynamic equilibrium all the elementary reactions are at equilibrium and thus implies that the same elementary step controls both dissolution and precipitation. The equilibrium constant for the overall reaction comprising a series of s elementary steps is thus: s

K¼ ∏ i¼1

ki k−i

ð25Þ

and the affinity function is characterized by the following properties: • ƒ(0) = 0 • ƒ′(0), the derivative of ƒ(A) with respect to A for A = 0, is defined and has the same value in both directions, that is, for dissolution (A N 0) and for precipitation (A b 0). The existence of ƒ′(0) is consistent with the thermodynamics of linear irreversible processes; this implies linear relations near equilibrium between the forces and flows: the rate varies linearly with the chemical affinity. The detailed balancing principle has not yet been clearly validated experimentally to date even for a simple oxide like silica. Ganor et al. (2005) suggest to apply this principle to the large database of quartz dissolution rates together with appropriate thermodynamic data, for modeling quartz precipitation under natural conditions. They point out that their data are not sufficient to retrieve a function which uniquely describes the precipitation rate dependency on deviation from equilibrium and to propose a mechanism for the precipitation of quartz. Other experimental work (Nagy et al., 1991) on kaolinite did not fully verify this principle near equilibrium and notably was unable to guarantee a continuous slope of the ƒ(A) function near equilibrium. Glass raises a number of inherent difficulties in this respect, as discussed below with regard to Grambow's work.

18

S. Gin et al. / Chemical Geology 255 (2008) 14–24

activity of aqueous species H4SiO4, irrespective of the glass composition (Grambow, 1984, 1985):

4. Generalized laws The reasoning advanced by Aagaard and Helgeson calls for other reservations as well. Their theoretical developments assume that the activation energy Ea can be identified with that of a single elementary step (the limiting step j). For Lasaga (Lasaga, 1981) this is obviously not the case: the experimental activation energy takes into account all the formation enthalpies of the activated complexes involved in the overall mechanism. This point is in contradiction with the work of Aagaard and Helgeson and also casts doubts on the signification of the activity product Πa− nij that is indissociable from the intrinsic kinetic constant of the limiting elementary step j. In order to overcome these difficulties, Lasaga (Lasaga, 1995) proposed a generalized rate law: n

tot rnet ¼ k0 Amin e−Ea =RT aHHþþ gðIÞ ∏ ani i f ðΔGÞ

ð26Þ

i

where k0 is the kinetic constant, Amin the reactive surface area of the mineral, Ea the activation energy of the overall reaction, R the ideal gas constant, T the temperature, aH+ the proton activity in solution, g(I) includes the effects of the ionic strength of the solution on the rate, Πa− ni comprises all the species liable to catalyze or inhibit the overall reaction (these kinetic effects are discriminated from those related to the affinity function), f (ΔG) the affinity function and ΔG, the difference in Gibbs freeenergy between the bulk solid and the dissolved species. The activation energy Ea and affinity function f (ΔG) are defined globally, i.e., for an overall reaction. The affinity function is defined from all the elements of the material. This is a very general function that must be determined empirically. Subsequent work has been done by Lasaga and Lüttge (Lasaga and Lüttge, 2001, 2003, 2004a,b, 2005) and Lüttge (Lüttge, 2006) in order to propose an affinity function in agreement with the dissolution mechanisms modeled at nanoscopic level (Monte-Carlo calculations) and observed on various crystals using atomic-level microscopy (e.g. vertical scanning interferometry of dolomites — Lüttge et al., 2003). An example of nonlinear affinity function derived from such mechanisms and applied at the macroscopic level is as follows:
   B ΔG f ðΔGÞ ¼ 1−e kT tanh g ðΔGÞ ð27Þ g ðΔGÞ ΔGcrit kT ΔG 1−e kT

where B is a constant and g ðΔGÞ ¼ 1− 1−e

. ΔGcrit is a critical value

for ΔG, associated with a transition between local and global dissolution mechanisms. For low ΔG values, the dissolution is governed by etch pits (Lasaga and Lüttge, 2001) and is described by the Cabrera and Levine model (Cabrera and Levine, 1956) similar to the dislocation-driven crystal growth, modeled by the Burton–Cabrera– Frank theory. For ΔG N ΔGcrit, the dissolution is surface-wide and described by dissolution stepwaves. The affinity function proposed in Eq. (27) has been tested by comparison with experimental results on several crystals: albite, gibbsite, labradorite, smectite (Lasaga and Lüttge, 2001), feldspars and kaolinite (Lüttge, 2006). Nevertheless, Lüttge precised that the relatively good agreement isn't as such a validation of the model, but insisted in the necessity to derive the affinity function from “the standard kinetic treatment of the full reaction mechanism” and not only from an elementary reaction assumed to be dominant in order to justify a TST-based rate law which happens to give a good mathematical fit with the observations. 5. Application of the Aagaard–Helgeson law to borosilicate glass dissolution Aagaard and Helgeson (Aagaard and Helgeson, 1982) provided a theoretical framework for the development of a kinetic law of nuclear borosilicate glass dissolution depending only on the pH and the

r ¼ r0 1−

aint H 4 SiO4

!

a⁎H4 SiO4

ð28Þ

where r0 is the initial glass dissolution rate comprising the kinetic constant and the proton catalysis term, aint H4SiO4 is the orthosilicic acid activity at the reaction interface, and a⁎H4SiO4 is the orthosilicic acid activity at saturation. Note that in a more recent paper, Grambow takes into account the diffusion of water within the glassy network as a key mechanism of the nuclear glass dissolution reaction (Grambow and Muller, 2001). Here, we discuss only the theoretical aspects in relation with the application of the Aagaard– Helgeson rate law. This type of law implies that the hydrated glass dissolution can be considered as a pure form of silica and SiO2 + 2H2O ↔ H4SiO4 reaction applied to the glass is reversible. In this section, we discuss the reasoning that led Grambow to postulate this “first-order” law and highlight the difficulties encountered in using the theoretical concepts proposed by Aagaard and Helgeson. In order to apply the Aagaard– Helgeson law, the following must be clearly defined: • the overall mechanism consisting of a series of elementary steps, to determine the affinity term; • the nature of the limiting step, to determine the kinetic constant k and the activity product Πa− nij. The reaction interface of a borosilicate glass can be described schematically according to Grambow as follows:

ðNa2 SiO3 Þmatrix ;ðSiO2 Þmatrix ;↓;Glass þ Aqueous;species↔Matrix–O–SiðOHÞ3 þ H2 O↔ H2 Oads ; Matrix–O–SiðOHÞ3 →H4 SiO4

ð29Þ

The glass is considered as a mixture of simple oxides and silicates. Successive hydrolysis (via a serial mechanism) of the first three bonds of the silica tetrahedron (regardless of the X atom of the Si–O–X bond) leads to the formation of Matrix-O-Si(OH)3 groups. The first three silica tetrahedron hydrolysis reactions are assumed to be very rapid, and hydrolysis of the final bond is supposed to limit the overall mechanism. All the elementary steps are therefore at (dynamic) equilibrium except for the limiting step j. The “generalized” limiting elementary step (that is, for any X) is thus: H2 Oads ;Matrix–O–SiðOHÞ3 →H4 SiO4 þ Matrix–OH

ð30Þ

The activated complex limiting the dissolution is intermediate between the atomic configuration corresponding to the adsorption of a water molecule on the X–O–Si(OH)3 bond and orthosilicic acid in solution. The advantage of considering the final hydrolysis step as the limiting reaction is that the activity of the products from the limiting elementary step is that of a species in solution, and that it can thus be measured. However, the choice of this limiting step has been questioned for pure silica and is still a debate. The hydrolysis of silica by H3O+ is believed to involve several elementary steps including the following: (1) adsorption of H3O+ onto a bridging oxygen atom defined by Xiao and Lasaga (1994) resulted the protonation of the bridging oxygen atom and consequent release of H2O, (2) formation of a pentacoodinate Si intermediate, and (3) cleavage of an Si–O–Si bond (Criscenti et al., 2006). It may be reasonable to consider the hydrolysis of the first bond as the limiting step since the silicon is less acidic and therefore, less reactive (Dove and Crerar, 1990) as confirmed by ab initio calculations (Pelmenschikov et al., 2000). However, Pelmenschikov

S. Gin et al. / Chemical Geology 255 (2008) 14–24

noticed that theoretical activation energy (20 kcal/mol) of the last bond hydrolysis is in better agreement with experiments (Pelmenschikov et al., 2001). These authors consequently proposed a mechanism based on the assumption that the breakage of the first Si–O–Si bond could be followed by the very fast reverse reaction of dehydroxylation of the newly formed Si-OH HO–Si defect. Because of this “self-healing” effect, the probability of both Si–O–Si bonds of the double-linked Si atoms dissociation is very low, which could explains both low activation energy and small rates of dissolution. Within this new mechanism, the measured activation energy should be associated with the hydrolysis of the last Si–O–Si bond of the Si atoms. Nevertheless, recent ab initio calculations of the activation energies of Si–O–Si bonds proved that Pelmenschikov calculations of the activation energies were slightly overestimated and consequently suggest that the breaking of the Q2Si–O–Si bonds best represent the rate limiting step of the overall dissolution process (Criscenti et al., 2006). They also noticed that measured activation energy on various minerals is quite similar whatever the connectivity of the silica units within the mineral. This short review shows that there is still neither real experimental nor theoretical evidence defined that the final hydrolysis step is the limiting step of glass dissolution reaction as suggested by Grambow. Collecting such evidences is not an easy task, all the more if we remind the discussions about the generalized laws: Lüttge (2006) stressed that Lasaga (Lasaga, 1981) and others have explained in detail that the overall activation energy is not necessarily the same as the one of a single elementary reaction. Following a line of reasoning similar to that of Aagaard and Helgeson, the kinetics of the overall alteration mechanism can be expressed as follows: j

rtot ′ f−Si≡ðOHÞ3 g−k′ aH4 SiO4 net ¼ rnet ¼ kþ

aH4 SiO4 : −Si≡ðOH Þ3

ð32Þ

The affinity of the limiting elementary step then becomes:
 Qj Aj ¼ −ΔGj ¼ −RTln Kj

ð33Þ

The activity of the reactants in the limiting elementary step j must be known in order to apply the Aagaard–Helgeson law, in this case the activity of the −Si ≡ (OH)3 (silanol) groups and the overall alteration reaction affinity, that is, the affinity of elementary step j (A = Aj). Grambow (Grambow, 1985) proposed the following expression for the activity of the −Si ≡ (OH)3 groups:
   −ΔGr ðÞ −Si≡ðOHÞ3 ¼ Bexp RT

Based on Eqs. (31), (32), and (34) the glass dissolution rate can be expressed as follows: !   aH4 SiO4 k0 j tot  rnet ¼ rnet ¼ k0þ −Si≡ðOHÞ3 1− −0  ð35Þ kþ −Si≡ðOH Þ3
   Qj j tot ¼ rnet ¼ k0þ −Si≡ðOHÞ3 1− ð36Þ rnet Kj that is, the rate expression for an elementary step in which k'+ is the kinetic constant of the limiting elementary step and {−Si ≡ (OH)3} is the activity product Πa− nij of Eq. (1). At this stage, although it would now appear possible to calculate the affinity of the limiting elementary reaction and of the overall mechanism in Paul's approximation, Grambow advanced the following hypothesis (Grambow, 1985): “In the simplest case, the silicon atom Si-(OH)3 that desorbs is bonded to another silicon atom; the reaction limiting the process is then written: SiO2 + 2H2O → H4SiO4”. The author thus considered that the overall alteration mechanism is identical with that of silica. He therefore abandoned the approach of Paul's model and did not attempt to calculate ΔGi although this calculation appeared to be within his reach. This change in reasoning thus appears to be ambiguous. Considering the overall affinity of the SiO2 + 2H2O → H4SiO4 reaction (which is not an elementary step) implies revising the mechanisms initially proposed to describe the reaction zone. The reaction interface then becomes the following:

ðSiO2 Þmatrix þ Aqueous; species↔Matrix–O–SiðOHÞ3 þ H2 O↔H2 Oads ; Matrix–O–SiðOHÞ3 →H4 SiO4

ð37Þ

ð31Þ

where k+′ =k+/γSi ≡ (OH)3, k′- =k-/γH4SiO4, {−Si ≡ (OH)3} is the activity of the reactants in the limiting elementary step j (a surface species) and aH4SiO4 is the orthosilicic acid activity in solution; the activity of the water is assumed equal to 1. For the limiting step considered here, k'+/k'+ =Kj (the equilibrium constant of the limiting step j). The ion activity product is thus: Qj ¼ 

19

ð34Þ

where B is a constant and ΔGr(ξ) is the free energy of glass dissolution calculated from Paul's model (Paul, 1977), that is, an overall free energy calculated from the sum of the free energies associated with the glass oxide and silicate constituents. It may be noted that at a high degree of glass dissolution reaction progress the activity of the −Si ≡ (OH)3 groups diminishes. It must be emphasized that Eq. (34) takes all the glass elements into account; this expression is discussed in more detail below.

Hence the following general relation: !
 aint −ΔGr ðÞ H SiO tot 1− ⁎ 4 4 : rnet ¼ k0þ Bexp RT aH4 SiO4

ð38Þ

The affinity term refers simply to the affinity function defined for the overall silica dissolution mechanism. Theoretically activities are defined at the glass/water interface. Nevertheless silicic acid activities from the bulk solution can be used when transport processes are neglected. Note that the activity of the −Si(OH)3 groups, which depends on the reaction progress, leads to a kinetic “constant” depending on the degree of reaction progress.
 −ΔGr ðÞ : kðÞ ¼ k0þ Bexp RT

ð39Þ

This is not in contradiction with the work of Aagaard and Helgeson, considering that the chemical affinity of the limiting elementary reaction, which varies with ξ, depends on the activity of the reactants, in this case {−Si ≡ (OH)3}. In the kinetic laws of glass dissolution proposed in recent years (Grambow, 1985; Berger et al., 1994; Daux et al., 1997; McGrail et al., 1997; Advocat et al., 1998; Abraitis et al., 1999; Grambow and Muller, 2001), the dependence of the kinetic “constant” on the reaction progress is not taken into account, and only a single initial dissolution rate r0 is considered. Avoiding the calculation of Eq. (34) (refer to the Discussion below), yields the classic first-order law of Eq. (28). 6. Discussion of relation (34) Calculating surface activities remains a difficult problem, as illustrated by Curti's discussion of the hypotheses postulated in Grambow's model for calculating Eq. (34) (Curti, 1991).

20

S. Gin et al. / Chemical Geology 255 (2008) 14–24

The dissolution of a simple oxide or a silicate ϕ can be symbolized as follows:

at pH 9.9, for example, the first and third hypotheses lead to the following results:

/ þ  1 r1 þ  2 r2 þ … ↔β1 P1 þ β2 P2 þ …

  H3 SiO−4 H3 SiO−4 10−9:9 ¼  þ Y ¼1 ½H4 SiO4  ½H4 SiO4  H

ð49Þ

f≡Si−O− g 10−6:8 f≡Si−O− g ¼  þ Y ¼ 103:1 : f≡Si−OH g f≡Si−OH g H

ð50Þ

ð40Þ

where νi, βi are the stoechiometric coefficients, ri the reactants i and Pi the products of the reaction. The activity product at a state of reaction progress ξ is expressed as follows: ½P1 β1 ½P2 β2 … : Q/ ¼ ½r1 1 ½r2 2 …

ð41Þ

The free energy ΔGr(ξ) of the overall dissolution reaction for a glass, considered to be a mixture of simple oxides, is calculated as the sum of the energy terms for each of the simple oxides and silicates in the glass, according to a semi-empirical approach developed by Paul (Paul, 1977). The free energy is expressed thus:

The two hypotheses are thus clearly incompatible. The first hypothesis was postulated to calculate the overall affinity ΔGr(ξ) of the glass dissolution reaction. The chemical potentials are defined from the interface activities and can thus be recalculated from the activities in solution. Thus, for a simple compound k:



 ΔGr /ðÞ ¼ ∑  i μ i ¼ ∑  i μ 0i þ RTlnfig

ð51Þ

 ΔGr /ðÞ ¼ ∑  i μ 0i þ RTln½i−RTln

ð52Þ

i

ΔGr ðÞ ¼ RT ∑ f/ :ln /

Q/⁎ K/

ð42Þ

i

where fϕ is the molar fraction of solid phase ϕ in the glass, Kϕ is the solubility product, and the asterisk designates the activity at the reaction interface. Grambow considered the following reactions, among others: SiO2 þ 2H2 O→H4 SiO4 1

=6 ð3Al2 O3 : 2SiO2 þ 18Hþ →2H4 SiO4 þ 6Al3þ þ 5H2 OÞ

ð43Þ ð44Þ

1=2ðNa2 SiO3 þ 2Hþ þ H2 O→2Naþ þ H4 SiO4 Þ

ð45Þ

1=2ðB2 O3 þ 3H2 O→2H3 BO3 Þ:

ð46Þ

The following hypotheses were taken into account by Grambow to calculate the interface activities: 1- Each species in solution as a result of glass dissolution has a corresponding surface species: The surface silanol group ≡ Si–OH is associated with H4SiO4. The ≡ Si–O– group is associated with H3SiO–4. 2- The ratio of the interface activities to the activities in solution is assumed constant: fxg fyg ¼ ½x ½y

ð47Þ

where {} represents the surface species and [] the concentrations in solution. 3- The distribution of surface species and species in solution is inversely proportional to the reaction progress: fxg 1 ¼ : ½x 

ð48Þ

As the reaction progress tends toward zero, the activity of the surface sites is maximal, while as ξ tends toward infinity, all the species x are in solution. This rather unrealistic assumption fails to take into account the advancing reaction front and thus the renewal of surface sites during dissolution. 4- The pH is the same at the interface and in solution. Curti showed that the first and third hypotheses are incompatible. Given the following relations: H4 SiO4 ↔H3 SiO–4 þ Hþ with pKa ≈9:9ðSchwartzentruber al:; 1987) 1987Þ (Schwartzentruber et al., ≡Si–OH↔≡Si–O– þ Hþ with pKa ≈6:8ðSchindler (Schindler and Kamber; Kamber, 1968Þ 1968)

i

where νi corresponds to the stoichiometric coefficients of the reactants and products of a “simple” reaction ϕ, and μi is a chemical potential.

Q/ ΔGr /ðÞ ¼ RT ln −∑ i :ln : K/

ð53Þ

The total free energy variation of the water/glass reaction can then be written as a function of the activities of the species in solution, by taking the sum of the ΔGrϕ(ξ) terms:

Q/ ΔGr /ðÞ ¼ RT ∑ f/ ln −∑ i :ln : K/ /

ð54Þ

To justify Eq. (34), Grambow defined the equilibrium constant K of the overall glass dissolution reaction as follows: K¼


 aMatrix−O−SiðOHÞ3 −ΔGr ðÞ ¼ exp RT acompoundð/Þ awater

ΔG-r ¼ −RTlnK:

ð55Þ

ð56Þ

Although Eq. (34) can easily be derived from Eq. (55), the latter is not correct: an equilibrium constant cannot be a function of the reaction progress. Moreover, the overall reaction product is orthosilicic acid and not “Matrix-O–Si(OH)3”. The activity calculation for the {− Si ≡ (OH)3} groups is thus ambiguous for two main reasons: • the contradictory hypotheses concerning the interface chemistry • the unrealistic assumption of a limiting elementary step “generalized” to the entire reaction progress. 7. General discussion of the Aagaard–Helgeson law applied to glass As illustrated in the preceding discussion of Grambow's work, application of the concepts behind the Aagaard–Helgeson law is not simple, notably with regard to the content of the affinity function and the calculation of the activities of the surface species. In addition, in the case of glass, the hypothetical series of elementary reactions is unrealistic (Berger et al., 1994; Emanuel and Knorre, 1974; Berger, 1995, 1998). From the experimental point of view, the first order rate law (Eq. (28)) has been applied to SON68 nuclear glass (inactive glass simulating the French R7T7glass) under conditions excluding as much as possible the potential effects of the alteration products (gel, secondary crystalline phases). The experiments were conducted in an

S. Gin et al. / Chemical Geology 255 (2008) 14–24

open system with high solution renewal rate and variable silica concentrations in order to measure the dependence of the glass dissolution rate to the silica concentration. The tests were carried out at 90 °C and pH 8. Under these conditions, the H4SiO4 activity at saturation was 10− 3.02 (value resulting from static tests at high surface-area-to-solution-volume (SA/V) ratio under the same temperature and pH conditions). Fig. 1 illustrates the discrepancy between measurements and the first order rate law predictions (Eq. (28)). This law fails to fit the experimental data correctly. The explanation usually given attributes this discrepancy to the effect of the gel formed between the pristine glass and the bulk solution (Gin et al., 2001; Frugier et al., in press). This material, when formed in “normal silica saturated conditions”, plays a “protective role” limiting the exchanges of reactive species between the pristine glass and the solution, and controlling the composition of the solution (Frugier et al., in press). In these experiments, the conditions required to form the protective gel are not fully achieved due to the high solution renewal rate. But as we noticed a little consumption of silicon by the glass especially in the case of C0(Si) = 90 and 120 mg L− 1, one can expect to see a low decreasing of the dissolution rate at long time. From this kind of experimental evidence, one can wonder if other elements than silicon have to be considered to account for the glass rate drop. By analogy with work on silicate minerals, some kinetic studies of glass based on the Aagaard–Helgeson law have been undertaken by defining an affinity function including all the glass constituents (Advocat et al., 1998) or some network formers (Daux et al., 1997; McGrail et al., 1997; Abraitis et al., 1999). Once again these approaches can account for a series of experimental results, notably by fitting certain parameter values, but have proved unable to reproduce experiments covering a wide range of variations affecting major parameters (pH, temperature, flow rate, SA/V, leaching solution chemical composition, etc.) (Jégou, 1998). It is important not to lose sight of the limits of this approach when applied to glass. Allowance for an overall affinity function implies the definition of an equilibrium constant K for glass. Unfortunately there are only very few data in the literature on the thermodynamic stability of glasses. Linard and co-workers did experimental measurements to get the Gibbs free energy of formation of sodium borosilicate glass (Linard et al., 2001a) and calculated a log K for the dissolution of this ternary glass at room temperature and 363 K. Comparing thermodynamic equilibria to leaching data obtained in static mode at high SA/ V ratios authors conclude that neither a global affinity nor a first order

21

rate law based on the silica alone (like Eq. (28)) are able to account for the observed rate drop (Linard et al., 2001b). As mentioned above, authors invoked a protective layer playing as a diffusive barrier to explain the difference in chemistry between the reactive interface and the bulk solution. This protective effect has been experimentally confirmed by different ways (Gin et al., 2001; Ledieu et al., 2005; Jollivet et al. submitted for publication). Recently, Zhang and Lüttge have proposed that it could even be a key phenomenon for understanding not only for glass but also mineral dissolution kinetics (Zhang and Lüttge, 2008). Several authors suggest referring to a global affinity not based on the bulk glass but on the alteration layer (Oelkers, 2001; Grambow and Muller, 2001). Although such an assumption is an improvement because hydration and ion-exchange are fast reactions compared to hydrolysis of Si–O–Si bonds, meaning that a hydrated glassy network is subject of dissolution, two difficulties remain: i) thermodynamic stability of this kind of phase has never been directly measured and ii) there is some evidence on borosilicate glass that the composition and the structure of the leached layer evolves with time (Ledieu et al., 2005; Arab et al., 2006; Frugier et al., in press). 8. Other kinetic approaches Applying Eq. (1) to glass implies the definition of an equilibrium constant K, which remains a difficult problem. Another kinetic approach could lead to the definition of a steady-state condition of thermodynamic disequilibrium involving competition among various mechanisms (for example, dissolution of the vitreous phase and precipitation of secondary phases). Kinetic laws that do not imply equilibrium conditions have been proposed, for example, by Chou and Wollast (Chou and Wollast, 1985) and by Sverdrup and Warfvinge (Sverdrup and Warfvinge, 1995): r ¼ ∑ ri;diss − ∑ rj;precip i

ð60Þ

j

where the steady state is defined by: r ¼ 0f ∑ ri;diss − ∑ rj;precip ¼ 0: i

ð61Þ

j

Kinetic laws of this type, which do not include the reverse reaction (precipitation of the initial phase), are difficult to apply to glass today for lack of data on the mechanisms and kinetics of the formation of the alteration products (gel + secondary crystalline phases).

Fig. 1. SON68 glass dissolution rate at 90 °C and pH 8 as a function of the concentration of dissolved silica. Other conditions are: SA/V = 150 m− 1 and flow rate = 0.144 mL min− 1, one month duration. Rates are calculated from the boron concentrations; uncertainties are estimated at ± 25%. The black curve corresponds to the prediction calculated from the first order law (Eq. (28)).

22

S. Gin et al. / Chemical Geology 255 (2008) 14–24

Another approach that ignores the overall solubility of the studied phase has been developed for basaltic glasses (Berger et al., 1994). The solid network-forming oxides (SiO2, Al2O3, etc.) are assumed to react as independent entities. This amounts to considering several parallel overall mechanisms with the fastest mechanism controlling alteration. The Aagaard–Helgeson law can then be applied to each overall reaction. For the complex glasses studied, Berger prefers to postulate the existence of parallel mechanisms and thus to define several affinity functions, each referring to the alteration mechanism of a simple oxide i (Berger et al., 1994).
 Q tot rnet ¼ ∑ kþðiÞ CðiÞ 1− : K ði Þ i

ð62Þ

The following more exhaustive form takes into account the surface species:  
Q  tot þ − þ k þ þ 1− rnet ¼ kþ C þ k C C − SiOH2 SiOH2 SiOH SiOH SiO SiO K ðSiÞ  
Q  þ þ − þ kþ CAlOHþ 1− þ… AlOH CAlOH þ kAlO− CAlO þ kAlOHþ 2 2 K ðAlÞ

ð63Þ

The fastest mechanism thus controls the kinetics, and the content of the affinity function cannot be dissociated from the nature of the alteration-controlling bond (Si–O or Al–O). If hydrolysis of the Si–O bond is considered to be the limiting elementary step, for example, the alteration mechanism may then be identified with the silica hydrolysis mechanism, and only silicon is taken into account in the affinity function. The rate is not necessarily controlled by the step with the lowest energy. The rate of an elementary step is the product of two terms: a kinetic constant and a site density. Even if the elementary step corresponding to the rupture of a type of bond occurs at low energy, the glass alteration will not be controlled by this bond if the site density is insufficient (compared with the site density of another higher energy bond). The site density may depend on the initial material composition and on the reaction progress. An elementary step may control the alteration at a certain degree of reaction progress, but an increase in the release of an element may result in the predominance of another step. A change in the solution composition may also affect the nature of the bond controlling the alteration. Contrary to Aagaard and Helgeson, who took the chemical affinity of the overall dissolution reaction into account (despite the initial hypothesis associated with the notion of a limiting elementary reaction), Berger's approach focuses on the limiting elementary reaction and ignores the overall solubility of the solid. Nevertheless, atomistic modeling of an H6SiOAl cluster (Xiao and Lasaga., 1994) revealed that it was not always necessary to distinguish between the hydrolysis reactions of the Al-O and Si-O bonds, as from an energy standpoint the activated complexes associated with each type of bond are equivalent. This remark underscores the difficulties encountered in writing an elementary reaction in heterogeneous kinetics. Nevertheless, in view of the reservations discussed above with regard to considering a multiple-oxide glass as a mixture of simple oxides, the limits of Berger's approach become apparent in determining the equilibrium constants associated with each oxide, and in the fact that the alteration gel—whose protective role has been established for most silicate glasses (Jegou, 1998; Gin et al., 2001; Jollivet et al., submitted for publication)—is not taken into account in this type of law.

resolved by Grambow's work concerning nuclear glass. The famous and very useful first order rate law (Eq. 28) appears finally to be a more empirical than theoretical equation to predict the rate drop of silicate glasses only in some experimental conditions. The alternatives proposed in the literature and mentioned in this article, such as Berger's law, are in fact variants that do not overcome the fundamental difficulties. In the case of borosilicate and aluminosilicate glasses, the gel (or the hydrated layer for some authors) that forms during alteration can be a protective barrier; its effectiveness depends on the alteration conditions. The formation of this thin, microporous layer by in situ recondensation of silica (Angeli et al., 2001; Ledieu et al., 2005) is no doubt a key mechanism of glass alteration. In the case of silicate minerals, such layers are reported in the recent litterature (Hellmann et al., 2003, 2004; Lee et al., 2007). Future work would be devoted to understand how these interface layers modify the global chemistry of the system. One can suggest for example to study in a comparative way like the dissolution of two silicates with the same chemical composition but different structures such as a well crystallized albite feldspar (NaAlSiO3) and a glassy material of the same composition. By measuring the dissolution kinetics of these two phases and characterizing on various scales of time and space and the evolution of the chemical, structural characteristics and porous textures of the interface layers, it should be possible to understand the coupling of the various processes in better way which control the dissolution kinetics of silicates. One can also say to moderate these conclusions that, even if the affinity function with respect to the silicate network stability is a point to account for the rate drop, other phenomena like slow diffusion of reactive species through the gel layer or precipitation of secondary minerals (smectite, zeolite) are likely more important to predict the long-term dissolution rate of natural or nuclear glasses in most of the confined environments (Gin et al., 2005; Frugier et al., in press). Modeling these mechanisms to simulate the very high reaction progress is no doubt a new challenge for geochemists. Appendix A This Appendix discusses the significance of the Temkin parameter σ and highlights the controversy in the literature concerning the theoretical justification and pertinence of this parameter. It presents and reviews the work of Boudart (Boudart, 1976) in particular, which was later used by Aagaard and Helgeson to substantiate the kinetic law of mineral dissolution and to introduce the σ parameter. Relationship between the Kinetics of an Elementary Reaction and the Kinetics of an Overall Reaction (Boudart, 1976) Consider the complex overall reaction: aA þ bB þ …↔cC þ dD þ eE þ … which can be broken down into a series of s elementary reactions. If the forward rate of each elementary reaction is designated ri and the reverse rate r i (where i = 1,2,…,s), then: ðr1 −r−1 Þr2 r3 … rs þ r−1 ðr2 −r−2 Þr3 … rs þ r−1 r−2 ðr3 −r−3 Þ … rs þ r1 r−2 r−3 … ðrs −r−s Þ ¼ r1 r2 r3 … rs −r−1 r−2 r−3 … r−s

ðA–1Þ

Thus, under steady-state conditions: 9. Conclusion tot ri −r−i ¼ σ i rnet

The discussions proposed in this article highlight the theoretical difficulties involved in establishing kinetic laws for glass alteration. The Aagaard–Helgeson law raises several problems that were not

ði ¼ 1; 2; … ; sÞ

ðA  2Þ

where σi is the stoichiometric coefficient of reaction i and rnet tot the net overall reaction rate.

S. Gin et al. / Chemical Geology 255 (2008) 14–24

The significance of the σi coefficient may be illustrated by the overall reaction A2 + 2B → 2AB (Hollingsworth, 1957), which is the result of two elementary reactions: A 2 ↔ 2A and A + B ↔ AB. The stoichiometric coefficient of the first step is 1, while that of the second is 2. In other words, σi may be defined as the ratio between the reaction progress of reaction i to that of the overall reaction (dξi/dξ). Substituting Eq. (A-2) into Eq. (A-1) yields the expression for the steady-state rate rnet tot of an overall reaction: tot rnet ¼ rþ −r− ¼

r1 r2 r3 … rs −r−1 r−2 r−3 … r−s : ðA  3Þ σ 1 r2 r3 … rs þ r−1 σ 2 r3 … rs þ … þ r−1 r−2 … r−s

If one of the elementary reactions is irreversible in the forward direction (r− 1 = 0), then the overall reaction is irreversible from left to right, and its kinetics are expressed by rnet tot = r+. Similarly, if one of the elementary reactions cannot occur in the reverse direction (ri = 0), then rnet tot = r− . The following relations can then be obtained from Eq. (A-3): rþ ¼

r1 r2 r3 … rs P

ðA  4Þ

r−1 r−2 r−3 … r−s P

ðA  5Þ

23

Discussion of Boudart's work The chemical affinity function associated with an overall reaction mechanism thus appears to have the same form as the function associated with an elementary reaction, except for the Temkin coefficient. This coefficient is often presented in the literature as an adjustable parameter with a value that may be different from 1. In fact, it corresponds to the stoichiometric coefficient of the limiting elementary reaction j (σ = σj) within the scope of the Aagaard– Helgeson hypothesis. The demonstration that led Boudart to Eq. (A-11) and which was subsequently used by Aagaard and Helgeson has been discussed by Lasaga (1995) and more recently by Lichtner (1998). According to Lasaga, the transition from Eqs. (A-4) and (A-5) to Eq. (A-6)—by simplifying the denominator P—is incorrect: the denominator P is different in Eqs. (A-4) and (A-5), invalidating the remainder of the demonstration (Hollingworth, 1957). Conversely, Lichtner (1998) highlights an algebraic error by Lasaga and considers Boudart's work to be valid. This controversy plainly illustrates the difficulties encountered in attempting to express the chemical affinity function of an overall mechanism using concepts that have clearly been demonstrated for an elementary reaction.

and r− ¼

where P is the denominator of Eq. (A-3). Hence: s rþ r ¼ ∏ i : r− i¼1 r−i

ðA  6Þ

Moreover, for an elementary reversible reaction in the case of hydriodic acid, we obtained the following relation between ri and r− i near equilibrium:
 ri A ¼ exp i r−i RT

ðA  7Þ

where Ai is the chemical affinity of elementary reaction i in the forward direction (equal to the Gibbs free energy -ΔGi associated with reaction i). Eq. (A-6) then becomes:
s  rþ A ¼ exp ∑ i : r− i¼1 RT

ðA  8Þ

Let us now define the mean stoichiometric coefficient (Temkin coefficient) of the overall reaction as follows: s

∑ σ i Ai

A σ ¼ i¼1s ¼ s ∑ Ai ∑ Ai i¼1

ðA  9Þ

i¼1

where A is the chemical affinity of the overall reaction: s

A ¼ ∑ σ i Ai ¼ −ΔGr ðÞ:

ðA  10Þ

i¼1

Substituting Eq. (A-9) into Eq. (A-8) yields Eq. (A-11), which is valid for an overall reaction composed of a series of elementary steps:
 rþ A : ðA  11Þ ¼ exp r− σ RT Eq. (A-11) may be rewritten in the more usual form:

 A tot : rnet ¼ rþ 1−exp − σ RT

ðA  12Þ

This relation, which was used by Aagaard and Helgeson, warrants further discussion.

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