A general formula for expressing the dissolution or precipitation of any feldspar is:

if the activities of the solid and water are taken to equal unity. If considering only pure albite, y and z equal zero in Eqs. (1) and (2). At chemical equilibrium, the ion activity product is equivalent to the equilibrium constant K. The saturation state of a fluid is often expressed in terms of the ratio (Q/K); if by common convention the dissolving mineral appears on the left side of the reaction, values of (Q/K) < 1 indicate undersaturation of the fluid with respect to the mineral, and conversely, (Q/K) > 1 is representative of supersaturation. In this study, however, we quantify the degree of undersaturation in terms of a thermodynamic parameter, the Gibbs free energy of reaction, ΔG_r. The Gibbs free energy is defined as:

The following general equation (adapted from (Lasaga, 1995); see also (Aagaard and Helgeson, 1982)) is often used to integrate rate-influencing parameters into an overall dissolution rate law for a given temperature:

In the above rate law, the overall, surface area-normalized rate R (units of mol m⁻² s⁻¹) is a function of the forward rate constant k₊ (units of mol m⁻² s⁻¹), the activity a of either H⁺ or OH⁻ raised to the corresponding exponential power n (note that if the reaction occurs in ‘pure’ water, the activity term refers to water and thus can be considered to equal unity), the ionic strength g(I), the catalytic or inhibiting effect of certain aqueous species Πa, and the Gibbs free energy function f(ΔG_r). Very few studies have examined the effect of all of the above terms on mineral dissolution rates; in fact, the majority of mineral dissolution studies have determined rates only in terms of their pH and temperature dependence. Moreover, most experimental studies have been conducted at conditions far from equilibrium, where the f(ΔG_r) term is generally neglected under the assumption that f(ΔG_r) ≈ 1, which implies that the reaction rate is independent of the saturation state of the fluid. Because so few studies have addressed the influence of the f(ΔG_r) term, the driving force of our two studies was measuring the R–ΔG_r relation for albite feldspar, a common rock-forming mineral, and then determining if different conditions of dissolution change the nature of the dependence of dissolution rate on free energy.

The results we report here are based on two separate studies of albite dissolution kinetics as a function of ΔG_r, the first at basic pH, the second at acid pH. Both of these studies are still in progress: study 1 is now in its 10th year, study 2 is in its 4th year. When dissolution reactions occur at conditions near equilibrium in flow through reactors, fluid residence times in the reactor can be very long (e.g. on the order of months), and mineral reaction rates are significantly slower than at far-from-equilibrium conditions. Because of these two reasons, we have paid particular attention to ensuring that each measured rate and corresponding ΔG_r is obtained at conditions representative of both hydrodynamic equilibrium (i.e., passage of at least 2–3 reactor volumes before measurement) and dynamic chemical equilibrium (i.e., steady-state concentrations in reactor over time intervals of several weeks at near equilibrium conditions). This reasoning explains the unusually long time periods associated with these experimental studies.

The experiments are run using flow through reactors based on a CSTR design (continuous stirred tank reactor; for reactor details, see ref. (Hellmann, 1999; Hellmann et al., 1997)). In all experiments, mm-sized albite grains are used that have been ultrasonically cleaned in alcohol. The specific surface area is measured by BET. The quantity of albite used for each experiment is such that only a monolayer of grains covers the floor of the reactor, thereby ensuring fluid circulation between the grains. The kinetics determined in each experiment always represent steady-state rates measured over time spans generally ranging from 1–2 weeks at far-from-equilibrium conditions, to month-long (and greater) periods closer to equilibrium. In all cases, rates are measured after a minimum of three reactor volumes have passed through the reactor. Dissolution rates are continuously determined as a function of time and are based on the aqueous constituent elements collected at the output of the reactor, as well as mass loss; each rate reported from the first study represents the unweighted mean of three rates: R_{mass loss}, R_Si and R_Al; in the second study, only R_Si is used. The free energy of the dissolution reaction can be varied either by changing the solid to solution ratio, or by changing the residence time of the fluid in contact with the minerals. This is effected by changing the flow rate of the pump (decreasing the flow rate results in ΔG_r values closer to equilibrium).

In the first study, the dissolution of albite is measured at 150 °C, pH 9.2 (pH 10.0 25 °C), and 1.5 MPa fluid pressure. The input fluid is injected into the reactor (2 reactors used, 300 or 50 mL volume) with a HPLC pump. The input solution is a boric acid pH buffer (0.05 M H₃BO₃) that is adjusted to pH 10.0 (25 °C) using 1N NaOH. Because the Na concentration is fixed by the buffer ([Na] ≈ 0.039–0.043 m), only the concentrations of Al and Si vary with time and with free energy, and thus only these two elements are used to calculate the dissolution rate. However, the concentrations of all three elements (Na, Al, Si) are used to calculate free energies that correspond to discreet steady-state dissolution rates.

The second study is based on dissolution at acid pH conditions (i.e. high pCO₂) at 100 °C. These conditions were chosen since they are relevant for geological storage of CO₂ (Bachu, 2000). The experimental setup is based on two reactors connected in series. The first reactor is a 2 L autoclave that contains two separate phases: a denser phase of CO₂-saturated H₂O, and a less dense, supercritical CO₂ phase. During each experiment, pure H₂O is continuously pumped into the saturation autoclave where it contacts the supercritical CO₂. The contact (residence) time is sufficiently long to ensure the uptake and equilibration of CO₂ with the injected H₂O (residence times of 10⁴–10⁶ min, depending on flow rate). The CO₂-saturated water prepared at 25 °C is continuously injected into the main reactor (300 mL volume) where dissolution takes place at 100 °C. The pressure of the entire flow system, which includes the back pressure regulator, the main reactor, and the CO₂-saturation autoclave, is 9.0 MPa. This pressure corresponds to the partial pressure of CO₂ in the CO₂–H₂O solution. Using empirical correlation diagrams from Spycher et al. (Spycher et al., 2003) to determine the fugacity coefficient, the corresponding aqueous fugacity (f CO₂) and pH are 7.2 MPa and 3.25, respectively, as calculated by the geochemical code EQ3NR (Wolery, 1992). Because of assumed precipitation of Al as a discrete secondary phase during the experiment, only Si is used to calculate the rates of dissolution. However, the concentrations of all three elements (Na, Al, Si) are used to calculate free energies.

The results we report on here are derived from two ongoing albite feldspar dissolution investigations. A large part of the data from the dissolution study at basic pH and 150 °C has been reported in a recent publication (Hellmann and Tisserand, 2006). Here we extend these results by comparing them with unpublished data from the second study that is based on dissolution at acid pH and 100 °C. This is the first publication that we know of that explicitly compares the R–ΔG_r relation for the same mineral at significantly different conditions of dissolution. As discussed in detail further on, even though the dissolution rates are different, both studies show dissolution rate-free energy relations (R–ΔG_r) that are very similar.

The data from the first study describe a sigmoidal relation between the rate R and the free energy ΔG_r, as shown in Fig. 1. The data are characterized by three distinct R–ΔG_r regions: a: far-from-equilibrium; b: transition; c: near-equilibrium. The data at far-from-equilibrium conditions, ΔG_r = −149 to −70 kJ mol⁻¹, have a relatively high degree of dispersion, in large part attributable to analytical uncertainty associated with measurements at very low Si and Al concentration levels (10⁻⁷ m, ppb range). Because dissolution is most rapid at these conditions, each rate datum represents a separate experiment (note that only ΔG_r > −80 kJ mol⁻¹ data are represented in Fig. 1). This experimental protocol was used to avoid significant changes in surface area over time (i.e. pristine grains were used for each experiment run at a specific flow rate). A constant rate of approximately 1.0 × 10⁻⁸ mol m⁻² s⁻¹ describes the kinetic data in this free energy range. The constancy of the rates in this ΔG_r region represents an important kinetic phenomenon because the rates are independent of ΔG_r, and by definition, also independent of the concentrations of the aqueous dissolution products (in particular Si and Al). In this region, characterized by a ‘rate plateau’, rate laws can be written that do not include a free energy term (i.e. because f(ΔG_r) ≈ 1).

The second set of data lies nearer to equilibrium and extends from ∼−70 to ∼−25 kJ mol⁻¹ (Fig. 1). In this free energy range, the dissolution rates are no longer constant, but rather abruptly decrease in a roughly linear manner with increasing ΔG_r; for this reason, this part of R–ΔG_r relation is called the ‘transition’ region. In this ΔG_r interval, the recorded rate closest to the upper ΔG_r bound is 3.1 × 10⁻⁹ mol m⁻² s⁻¹, which was measured at a ΔG_r = −29.6 kJ mol⁻¹. Thus, in the transition region the slowest recorded rate is only one third of the plateau rate. As was the case at far-from-equilibrium conditions, each datum point represents a separate experiment; however, the data are significantly less dispersed. Overall, dissolution is stoichiometric in the transition region, as well as at far from equilibrium. The sudden decrease in the dissolution rate in the transition region is not due to pH, as the pH remained constant in the reactor based on EQ3NR calculations. Moreover, the effect of secondary precipitates on the rates can also be excluded.

The third set of data represents near-equilibrium conditions, and is delimited by free energies that extend from ∼−25 to 0 kJ mol⁻¹, where the latter value represents chemical equilibrium. At the lower bound of this region (∼−25 kJ mol⁻¹), the slope of the R–ΔG_r relation becomes much shallower, and is characterized by a much weaker, inverse dependence of the rate on free energy. The recorded rates that were measured are clustered with respect to ΔG_r between −24.6 and −15.6 kJ mol⁻¹. The minimum recorded rate in this region, 6.2 × 10⁻¹¹ mol m⁻² s⁻¹, is slightly more than two orders of magnitude lower than the plateau rate, 1.0 × 10⁻⁸ mol m⁻² s⁻¹. In the near-equilibrium region, five separate experiments were used to generate the rates that are plotted in Fig. 1; however, because these rates were by far slower than those at conditions further from equilibrium (i.e., transition, plateau rate regions), each separate experiment yielded up to four different and separate R–ΔG_r data points. This was achieved by varying the flow rate. This experimental protocol required that each rate measurement be acquired at steady-state conditions. Because dissolution at near-equilibrium conditions occurs much more slowly, this necessitated single experiments that were run over periods of up to several hundred days in order to ensure steady-state rate conditions at each flow rate (e.g., the longest single experiment ran for 641 d). In order to measure rates even closer to equilibrium, current experiments are being run at ΔG_r > −15 kJ mol⁻¹.

Results from the second investigation are based on dissolution at acid pH and 100 °C. At present, we can only report on five R–ΔG_r data that were obtained from one single experiment that has been in continuous operation for almost 600 d. Nonetheless, despite the still limited nature of this data set, the overall shape of the data points to a non-linear, sigmoidal R–ΔG_r relation. The three rates representative of conditions furthest from equilibrium, obtained at ΔG_r between −68 and −39 kJ mol⁻¹, represent a dissolution rate plateau, where the average rate is 5.1 × 10⁻¹⁰ mol m⁻² s⁻¹. The two other rates (4.1 × 10⁻¹⁰, 2.8 × 10⁻¹⁰ mol m⁻² s⁻¹), obtained, respectively, at ΔG_r = −37 and −34 kJ mol⁻¹, define the transition region since they mark a sharp decrease in the dissolution rate compared to the plateau rate. These preliminary results are plotted in Fig. 1 (note the use of different rate scales for both studies).

The results at acid pH can be examined with respect to several points. The plateau rate of 5.1 × 10⁻¹¹ mol m⁻² s⁻¹ (log R = −9.3) can be compared to an extrapolated rate of 1.2 × 10⁻⁹ mol m⁻² s⁻¹ (log R = −8.9) at pH 3.25, 100 °C, based on the dissolution of albite using HCl/H₂O solutions (see results in (Hellmann, 1994; Hellmann, 1995)). Given the scatter in the rates in both studies, the two rates are very comparable. However, the uncertainty in the rates makes it difficult, if not impossible, to determine whether an intrinsic kinetic effect can be attributed to the presence of aqueous CO₂ at high concentrations. A second point of importance concerns the abrupt decrease in rates (Fig. 1) that occurs over a range in free energies between −39 and −37 kJ mol⁻¹. This notable decrease in rates is real and cannot be attributed to an increase in pH. Albeit the weak buffering capacity of the CO₂–H₂O solution, the pH of the solution in the reactor did not increase by more than 0.2 pH units, based on calculations using EQ3NR. At ΔG_r > −40 kJ mol⁻¹, the stoichiometry of the dissolution process became increasingly incongruent, based on the measured elemental release rates (R_Si ≈ R_Na ≫ R_Al). The precipitation of a secondary Al-bearing phase(s) (e.g. Al -[oxi]hydroxide) is the most likely reason, which is supported by EQ3NR-derived mineral saturation indices calculated from the aqueous chemistries of samples. In addition, past experience with albite dissolution at elevated temperatures and at acid pH conditions reveals that secondary Al-bearing phases, such as boehmite, predominate (Hellmann, 1999; Hellmann et al., 1989), but because of their high porosity, they do not appear to have an effect on the rate of dissolution.

The continuous and highly non-linear, sigmoidal relation between the dissolution rate R and the free energy ΔG_r described by the data from the first study, and partially from the second study, can be fit with any number of mathematical relations. We have chosen the approach of Burch et al. (Burch et al., 1993) and fit the data (first study only) to a rate law that is based on the sum of two parallel rate processes:

The parallel rate law developed by Burch and co-workers (Burch et al., 1993) that was applied to their experimental data for albite dissolution at basic pH and 80 °C has both experimental and theoretical underpinnings. According to these authors, the use of a parallel rate law implies that, “dissolution proceeds simultaneously at each ‘site’ on the surface and at a rate governed by the energetics of the local surface environment and the bulk solution undersaturation, ΔG_r”. These authors made the observation that extensive etch pit formation only occurred at high undersaturations, whereas at near-equilibrium conditions, most of the grains appeared unaltered, with a very low density of etch pits that were small and shallow. This observation, in conjunction with an abrupt change in rate with decreasing free energy at approximately −30 kJ mol⁻¹, led them to suggest that a major change in dissolution mechanism occurs between the near-equilibrium and far-from-equilibrium regions. They quantified this by proposing that when the free energy of the dissolution reaction is below a certain critical value, extensive etch pit development occurs, i.e., ΔGr<ΔGrcrit (equivalent to high degrees of undersaturation), and conversely, when ΔGr>ΔGrcrit, etch formation is not favored, and dissolution occurs at other sites, such as cleavage steps and grain edges. It is interesting to note that the existence of a critical free energy (‘critical supersaturation’) and an associated change in mechanism was originally postulated in a theoretical study of crystal growth by Burton, Cabrera and Frank (Burton et al., 1951); for additional details on how BCF theory has been applied to dissolution, see refs. (Lasaga and Blum, 1986; Lasaga and Lüttge, 2001; Lasaga and Lüttge, 2004).

Referring back to Fig. 2, the two parallel rate processes that comprise the overall rate law (Eq. (5)) are denoted by curves B and C. At near-equilibrium conditions, rate process C predominates, whereas at conditions far from equilibrium process B predominates. The intersection of the two curves, at −15 kJ mol⁻¹, represents the free energy where there is a switch in the predominant mechanism of dissolution, and hence is equivalent to ΔGrcrit. However, the scatter in the experimental data set (Fig. 2) suggests that the value of ΔGrcrit cannot be exactly determined, but rather should be assigned to a range in free energies extending from −15 to −25 kJ mol⁻¹. It is interesting to note that the microtopography of the grains reflects the influence of ΔG_r, as evidenced by well-developed, geometric etch pits and angular edges formed at ΔGr<ΔGrcrit, and irregular pitting and rounded edges at conditions ΔGr>ΔGrcrit. Fig. 3a,b amply show the difference in grain morphology, at the millimeter scale, between two samples that underwent dissolution at ΔG_r = −17 and −25 kJ mol⁻¹, respectively (see also Figs. 6 and 7 in ref. (Hellmann and Tisserand, 2006)). This example is important in that it readily illustrates how a small difference in free energy (8 kJ mol⁻¹) can shift surface reactions from one predominant mechanism to another when dissolution occurs in the vicinity of ΔGrcrit.

At a fundamental, mechanistic level, one of the most important results from our two studies is the similar nature of the dissolution rate dependence on free energy; in both cases the relation is a highly non-linear, sigmoidal function, despite differences in the conditions of dissolution (i.e., basic vs. acid pH, 150 vs. 100 °C, respectively). The rate-retarding effect of aqueous Al is often cited by Oelkers and co-workers (Oelkers et al., 1994; Oelkers, 2001) as being one of the principal kinetic factors causing the decrease in the overall rate with increasing solution saturation. It is therefore interesting to note that the evolution of bulk Al concentrations as a function of increasing solution saturation is quite different in our two studies. At basic pH conditions [Al] increases continuously with increasing solution saturation, whereas at acid pH, [Al] concentrations decrease significantly over the same free energy range (ΔG_r > −40 kJ mol⁻¹) where the rates begin to decrease sharply. Thus, despite the differences in the behavior of [Al] with increasing ΔG_r, the measured R–ΔG_r relations are very similar. This observation is at odds with the role of Al proposed by Oelkers et al. (Oelkers et al., 1994; Oelkers, 2001). Moreover, in the aforementioned study by Burch et al. (Burch et al., 1993), the authors show by way of isokinetic diagrams that the decrease in rates with approach to chemical equilibrium cannot be attributed to individual mechanisms of rate inhibition by either aqueous Si or Al.

The non-linear, sigmoidal behavior of the R–ΔG_r relations measured in our two studies, coupled to the observation that the sharp decreases in rates occur within a free energy interval of ∼10 kJ mol⁻¹ of each other, leads to the question as to whether this represents some kind of universal behavior (i.e. pH and temperature independent) behavior for albite, and perhaps other feldspars, as well. The aforementioned study by Burch et al. (Burch et al., 1993) also shows a R–ΔG_r relation for albite that is non-linear and sigmoidal, with a ΔGrcrit of ∼−30 kJ mol⁻¹, which compares favorably with our estimate of −15 to −25 kJ mol⁻¹ at basic pH conditions. Moreover, published studies in the literature of other feldspar phases, such as K-feldspar, sanidine, and labradorite, carried out at different pH and temperature conditions, are in accord with a non-linear, sigmoidal dependence of dissolution rate on free energy (Alekseyev et al., 1997; Beig and Lüttge, 2006; Berger et al., 2002; Taylor et al., 2000). This perhaps lends support to the idea of a certain universal behavior describing the dependence of the dissolution rate on the free energy with respect to all of the feldspars. Nonetheless, some studies show R–ΔG_r data for feldspars that do not show this non-linear, sigmoidal behavior (Gautier et al., 1994; Oelkers, 2001). Nonetheless, an in-depth analysis of the discrepancies between the various feldspar studies is beyond the scope of this publication.

If a non-linear, sigmoidal relation between dissolution rate and free energy is, arguably, the most accurate general description for the kinetic behavior of feldspars as a function of solution saturation state, many important questions remain. One of these concerns the nature of the transition region, and in particular, the influence of both the aqueous environment (pH, temperature, ionic strength, presence of inhibitors or catalysts) and the solid phase on the value of ΔGrcrit. This has obvious consequences with respect to the mechanism of dissolution as a function of solution saturation state. Another point of importance is the continuity of the R–ΔG_r relation. In our first experimental study (and partially with respect to the second study), the experimental data show good continuity, extending from conditions at far from equilibrium, traversing the transition region, and continuing at near-equilibrium conditions. In nearly all studies (ours and most other published studies), the experimental protocol is such that to achieve measurements at near-equilibrium conditions, fluids in contact with the solids evolve towards higher solution saturations (i.e., increasing ΔG_r), and thus the measured dissolution rates (in transition and near-equilibrium regions) may be influenced by etch pits and other surface topography features that form during the initial stages of dissolution at conditions far from equilibrium. Lüttge and co-workers have argued that the R–ΔG_r relation is dependent on the reaction history of the solid phase (Beig and Lüttge, 2006; Lüttge, 2006), such that different rates (for the same mineral) are theoretically possible at an identical ΔG_r, depending on whether the particular ΔG_r value is approached from lower or higher free energies. In fact, Lüttge (Lüttge, 2006) makes the case that any rate data falling in the steep transition region represent non-steady state phenomena, and can be attributed to experimental protocol (i.e., initiation of experiments at far-from-equilibrium conditions). To summarize, Lüttge (Lüttge, 2006) proposes that mineral systems dissolve at near-equilibrium conditions with one mechanism (curve C in Fig. 2), and at ΔGrcrit, the dissolution rate jumps to the plateau rate in a discontinuous manner (i.e., in this case, the transition region is not sigmoidal, but rather a step function jump), and remains at the plateau rate for all ΔGr≤ΔGrcrit. Thus, for dissolution conditions where ΔGr≤ΔGrcrit holds, the rate mechanism is the sum of the near-equilibrium rate and the plateau rate. This controversy points out that more experimental work needs to be done in order to verify or disprove this idea.

One last, but very important point needs to be made concerning the mathematical form of the free energy parameter, f(ΔG_r). In our two studies, the data and the fitted rate curve are non-linear and sigmoidal. However, by far the vast majority of geochemical codes used to model non-equilibrium water–rock interactions use kinetic rate laws that incorporate a f(ΔG_r) parameter that is loosely based on ‘transition state theory’ (TST). So-called TST-based free energy terms have the following general form (Aagaard and Helgeson, 1982; Lasaga, 1981):