Dynamics of a dissolution front for solids under stress

Authors

Chaouqi Misbah,

Groupe de Recherche sur les Phénomènes Hors de l'Equilibre, Laboratoire de Statistique et Probabilités, Université Joseph Fourier (CNRS), Grenoble, France

[1] A kinetic study including dissolution process and diffusion of the dissolved molecules for a stressed solid in contact with its solution is analyzed. We estimate a condition fixing the prevailing dissipation mechanism and analyze it with a linear stability criterion. This criterion depends on which process is limiting the rate of dissipation: dissolution at the solid-liquid interface or diffusion in the fluid. For definiteness we focus on recent experiments on various salts, which have shown that grooves, oriented perpendicular to the main compressive stress, develop on the free surfaces of crystals. We provide the characteristic length scale and the time scale for the development of this stress-induced roughening of solid surfaces, which are consistent with the experiments. Finally, we estimate these parameters for relevant geological conditions.

[2] The morphological instability [Asaro and Tiller, 1972; Grinfeld, 1986; Srolovitz, 1989] of a free surface of a solid, which is nonhydrostatically stressed, is a very general problem, which is encountered in various circumstances in physics and geophysics. There are several situations (see below) where the instability is limited both by the growth/dissolution process, and by mass diffusion in the liquid bulk phase. This is encountered in many geomaterials where the present instability has been the focus of recent studies with the aim of elucidating the stress-driven morphological instability [Gal et al., 1998; den Brok and Morel, 2001].

[3] In experiments on various salt crystals, it has been shown that the free surfaces of crystals under uniaxial stress can develop morphological instabilities [den Brok and Morel, 2001; den Brok et al., 2002; Koehn et al., 2004]. The surface of a crystal in contact with its solution does not remain flat when the crystal is pressed and tends to roughen and develop parallel grooves with time. The instability is driven by a competition between mechanical forces on the surface and capillary effects. Extrapolated to crustal conditions, the experiments show that elastic strain energy does play an important role as such energy can modify the minimization of the free energy of a fluid-rock system and drive dissolution-precipitation processes to dissipate stress heterogeneities at the grain scale. Moreover, the grooves may evolve into cracks [Yang and Srolovitz, 1993; Kassner and Misbah, 1994] and modify strongly the mechanical properties of the rocks.

[4] Here we present a linear stability analysis and apply it to natural examples with conditions corresponding to rocks in the Earth's crust. We provide the characteristic wavelength and the characteristic growth time of the instability. We find that both dissolution and diffusion in the liquid play an essential role.

2. Thermodynamic and Kinetic Analysis

[5] Let us focus on the situation where a semi-infinite solid phase is uniaxially stressed [den Brok and Morel, 2001] in contact with its saturated solution (Figure 1). The driving force for the dissolution is the stress, and we first specify how the chemical potential evolves with the stress. We imagine that a mass Δm of the solute is transformed into the solid phase (or vice versa), and compute the Gibbs free energy, ΔG, involved in that transformation. The chemical potential is defined per unit volume, Δμ = ΔG/ΔV.

where V is the volume and p_{f} the solute pressure. ΔF is the Helmholz free energy change. If only mechanical work is considered, ΔF is obtained upon integration of the infinitesimal work involved in this transformation.

where σ_{ij} is the stress tensor and u_{ij} the deformation tensor. Thus

For plain strain, [Cantat et al., 1998] show, upon using Hooke's law and integration over a finite transformation ΔV, that ΔG = ΔμΔV, where

is the chemical potential (μ_{s} − μ_{l}), the difference between the chemical potential of a molecule in the solid and that in the solution, referred to unit volume. ν is the Poisson ratio, E [Pa] is the Young modulus, γ [Pa · m] is the surface tension, κ [m^{−1}] is the surface curvature, σ_{nn} = n_{i}σ_{ij}n_{j}, σ_{tt} = t_{i}σ_{ij}t_{j}, where n_{i} and t_{i} are the ith components of the normal and the tangent unit vectors to the solid surface.

[6] Let Q denote the ionic concentration product of the dissolution reaction into liquid, and K_{eq} is the thermodynamic equilibrium constant in the absence of stress [Alkattan et al., 1997]. Considering the solute to be an ideal solution, and expanding the chemical potential about the stress-free equilibrium concentration we can write Δμ = −(Q − K_{eq})RT/(K_{eq}), where R is the universal gas constant, and [m^{3} · mole^{−1}] is the molar volume. The minus sign tells us that if the solid chemical potential is increased (Δμ > 0) then a solid dissolution is implied. Using equation (4) together with the above result, we can write

It is sometimes customary to write Δμ = −(Q − K_{eq}*)RT/(K_{eq}), where

The dissolution speed, v_{n} in m.s^{−1} is proportional to the actual difference in chemical potential. We set

where v_{n} is the normal front velocity, and k is a dissolution rate constant [mole · m^{−2} · s^{−1}]. From equations (6) and (7) we obtain the dissolution speed as a function of the stress and the actual solute concentration

The relationship between the ionic product Q and the concentration depends on the order of the dissolution-precipitation reaction. Usually Q ∼ c^{n} where c (in mole.m^{−3}) is the concentration in the fluid and n is the order of the reaction; n = 1 for quartz, whereas n = 2 for sodium chlorate and sodium chloride, and n = 4 for K-alum. We focus on the case n = 1; situations with n ≠ 1 can be dealt with along the same line. Thus we set Q = c and K_{eq} = c_{eq}.

[7] Now the stress configuration in the solid phase for a given surface profile must be determined. When the front is deformed (a dissolution-recrystallization wave) the solute concentration is affected; it depends on the actual front profile. Therefore, the concentration field must also be solved for in a consistent manner.

[8] Let us first address the question of the stress. For a uniaxial stress applied along x we expect the front morphology to be invariant along y (Figure 1). For 2D elasticity one can use the Airy function χ (x, z) and this quantity is known to obey a bi-harmonic equation ∇^{4}χ = 0. For an arbitrary solid profile, the problem can be solved numerically. However, if one is interested in the early stage of the instability, a linear stability analysis is sufficient [Kassner et al., 2001]. In the linear regime the front profile h(x, t) can be written as

where ε is the amplitude of deformation, assumed to be small enough for a linear theory to make a sense, q (m^{−1}) is the wavenumber of the deformation and ω (s^{−1}) is the growth (or attenuation) rate that we wish to determine. An instability is signaled by a positive ω.

where we have used the fact that for small deformations, the curvature has the form κ = −∂^{2}h/∂x^{2} (the minus sign ensures that for a concave solid the chemical potential is increased). σ_{0} ≡ σ_{xx}^{0} − σ_{zz}^{0} is the difference between the horizontal and the vertical stresses in the initial planar configuration. This is the source of the planar front instability.

[10] Now we address the question of the concentration field. The concentration obeys the diffusion equation in the fluid

Far ahead of the front the concentration is that corresponding to the planar front under stress. The concentration field perturbation due to the interface deformation decays with z sufficiently away from the interface, while along x it follows the interface deformation. That is, we must have c = f(z)cos (qx) e^{ωt}. Plugging this into equation (12) one obtains f(z) = Ae^{−βz}, β = , and A is an integration factor to be determined below. Note that the solution that increases exponentially with z has been removed. The concentration field takes the form

Reporting this along with the contributions of stress (equation (10)) and surface tension (equation (11)) into equation (8), one obtains a relationship between A and ε

Because c is a small perturbation we have evaluated the above equation at z = 0 and not at z = h (this is sufficient if one is only interested in the leading contribution which is linear in the deformation ε). Note also that for small deformation, v_{n} ≃ ∂h/∂t = ωε cos(qx)e^{ωt}. Finally the closure condition follows from mass conservation, stating that the dissolution mass current across the interface is proportional to the normal surface velocity, namely

where c_{s} is the concentration in the solid phase. If the concentration is counted as a number per unit volume, we simply have c_{s} = 1/Ω where Ω is the molar volume of the solid. Since the volume occupied by a solute molecule is much larger than that in the solid phase, we have c_{s} ≫ c. In the linear regime we are interested in, the mass conservation equation reads

where use has been made of relation (13). This provides us with another relation between A and ε. Compatibility with equation (14) yields the sought after dispersion relation

Since c_{s} is the concentration of a molar solid mass, we have c_{s} = 1. Note that we have set β = ≃ q. This means that ω (the inverse of the time scale for the instability development) is small in comparison to the diffusion of molecules in the solute. More precisely, the diffusion is fast in comparison to the interface evolution time scale. This is the quasi-steady approximation. Depending on which process (diffusion or dissolution) is the slowest one, two limiting forms of the dispersion relation are obtained from equation (17); see Figure 2.

[11] Let us now discuss the implication of our results for the case of salt crystals under stress [den Brok and Morel, 2001; Koehn et al., 2004], and other geological systems. The dispersion relationship (17) tells us that ω > 0 for

This is nothing but the thermodynamically optimal wave-number (or wavelength λ_{c}) [Asaro and Tiller, 1972; Grinfeld, 1986; Srolovitz, 1989]. Equation (17) informs us, in addition, on the kinetics of the instability as well as on the relative effects of the two competing dissipations, (i) the dissolution process characterized by the kinetic rate constant k, and (ii) the diffusion process signaled by the presence of the diffusion coefficient D. The slowest factor limits the instability development. The instability is limited by diffusion if

In the opposite limit the process would be limited by dissolution. We need to evaluate the wave-number in order to test the above inequality. At the initial stage of the instability it is reasonable to expect this to be given by the fastest growing mode. Let q* refer to that wave-number. Let us suppose that the process is limited by dissolution. In that case one obtains from equation (17) that q* = q_{c}/2. Using the data given by den Brok and Morel [2001] for K-alum, we find λ* = 2π/q* = 51 μm, which is close to the experimental wavelength of 60 μm. Using Table 1 and the above result we find, on the basis of condition (19), that the dynamics is limited by diffusion. For sodium chlorate (Table 2), we find for large stresses that the dynamics is limited by dissolution. Upon lowering the stress diffusion becomes competing (stress 8 MPa and 4 MPa in Table 2). At later time coarsening is expected [Kassner and Misbah, 1994], so that relation (19) is reinforced due to the decrease of q (coarsening). Thus one expects the diffusion process to override dissolution as the limiting mechanism. For the case of quartz (Table 1), and for the same typical length-scales, one finds that q*Dc_{eq}/k ≫ 1, so that here dynamics is, beyond any doubt, limited by the dissolution process.

Table 1. Parameters Used to Calculate the Characteristic Length Scale and Growth Time of the Instability

Table 2. Comparison Between the Experimental and Theoretical Characteristic Length and Growth Time of the Instability on Different Crystals^{a}

Mineral

σ_{0} (MPa)

T (C)

λ_{diss}* (μm)

T_{diss}*

T_{diff}*

a

The largest time (in bold) between T_{diss}* and T_{diff}* characterizes the slowest step that controls the formation of the instability. Application to quartz and NaCl under geological conditions.

[12] The time scales for the instability evolution are obtained from the dispersion relation. Consider the case where the dynamics is limited by dissolution. The time scale T* = 2π/ω* (where ω* = ω(q*); see Figure 2) for the birth of the instability is given by

Using the theoretical values of q* and the other parameters (Table 1), one obtains that T* falls in the typical experimental range within a factor 2 (Table 2). With a similar length-scale, for quartz at 200°C and 25 MPa, (Table 1) one finds that T* = 7800 years. For smaller length-scales (higher stresses) the time scale is lower. For lower temperature, at 100°C, the time scale is found to be of about 23 Myrs.

[13] If the instability is limited by diffusion, from equation (17) one obtains that the fastest growing mode ω* is obtained for q* = 2q_{c}/3. The time scale for this mode is given by

This situation should occur in [den Brok and Morel, 2001] experiments where the characteristic time for diffusion (4.9 h) is greater than for dissolution (0.5 h). If the solution was not stirred, or that the hydrodynamic boundary layer is large in comparison to the ripple wavelength, then diffusion would control ripple formation.

3. Conclusions

[14] The rationale of this study is to use thermodynamic and kinetics derivation to study morphological instabilities on the surface of stressed crystals in contact with a reactive aqueous fluid. The linear stability analysis provides the wavelength and the characteristic growth time of the instability. The characteristic length scales are compatible with recent experimental results on salt crystals [den Brok and Morel, 2001; den Brok et al., 2002; Koehn et al., 2004]. An interesting feature is that, depending on systems we encounter, both dissolution and diffusion can act as limiting factors. For sodium chlorate we expect a cross-over from dissolution-limited to diffusion-limited as coarsening proceeds. We find a consistent picture with available experimental data regarding length and time scales, aside from a factor 2 for time.

[15] Under geological conditions, depending on the material, the length-scale of the instabilities is in the range 0.1–1 mm, and the characteristic time scales varies between several hours (sodium chloride), up to several million years (quartz at shallow depth); see Table 2. Applied to various crustal conditions, these results show that this instability should be considered over geological time scales as it modifies the free-energy of a stressed fluid-rock system.

Acknowledgments

[16] The project has been supported by the Centre National de la Recherche Scientifique (ATI to F. Renard). We would like to thank C. Pequegnat and D. Tisserand for their technical help. We acknowledge J. Schmittbuhl, D. Koehn, D. Dysthe, and D. Clamond for very fruitful discussions, and B. den Brok for the constructive review.