Background

Pressure dissolution is an important plasticity mechanism in stressed, fluid-filled rocks (Sorby 1863; Durney 1972; Wintsch & Yi 2002; Passchier & Trouw 2005; Croizé et al. 2013; Gratier et al. 2013) and is particularly important for sediment compaction because it provides a dissolution–precipitation pathway that can reduce porosity more than mechanical processes. As analyzed by Gibbs (1878), stressing a solid reduces its thermodynamic stability and hence an inhomogeneous strain field at the scale of a mineral or a rock provides a thermodynamic driving force for the redistribution of atoms to zones of lower strain. In granular rocks, high stress locations occur at mineral-mineral interfaces that, under shallow crust conditions, are separated, at least between points of direct mineral-mineral contact, by very thin aqueous films (van den Ende et al. 2019). Based on this picture, macroscopic models for pressure solution creep have been developed that consider the rates of dissolution, transport through the intervening films to neighboring pores, and reprecipitation that lowers the pore fluid saturation state relative to the stressed mineral (e.g., Revil 1999; Yasuhara et al. 2003, Croizé et al. 2010; van den Ende et al. 2019). Laboratory studies on granular samples including halite (Hickman & Evans 1995; Lohkämper et al. 2003) and calcite (Zhang et al. 2010) observed trends in strain rate that are in good agreement with conceptual and analytical models. For example, at an axial stress of 30 MPa, a column of dry calcite exhibits negligible compaction but the fully saturated column at the same effective stress exhibits creep at rates that vary according to stress and grain size as expected for a combination of diffusion or precipitation controlled processes. The presence of solution species, Mg⁺ and HPO₃^2-, known to diminish calcite growth rates, is always seen to reduce strain rate (Zhang & Spiers 2005; Neveux et al. 2014).

Although Gibbs (1878) provided an unambiguous definition of thermodynamic equilibrium of heterogeneously stressed mineral–fluid systems, significant outstanding questions remain debated as to the thermodynamic driving force away from equilibrium (Toussaint et al. 2018; Hobbs & Ord 2016). Moreover, it is likely that current models for pressure dissolution do not account for all energy contributions (cf. Section 4.1 Toussaint et al. 2018; Hobbs & Ord 2016). In particular, measurements of single-interface convergence using the surface-forces apparatus (SFA) provide strong evidence that the mineral surface electric potential can drive or catalyze pressure dissolution (Fig. 2), a concept that has been acknowledged (Gratier 2013) but not pursued for model development. There is also debate on the existence of a stress threshold below which pressure dissolution ceases (Revil 1999; van den Ende et al. 2019). Van Noort et al (2008) find a sign error in the leading model (eqn. 22 in Revil 2001) but admit that a competition between pressure dissolution and healing could lead to threshold effects. There is no clear consensus for the most appropriate reaction-rate expression for mechanical and chemical disequilibrium (Croize et al. 2013; Toussaint et al. 2018).

Furthermore, the plastic evolution of hydrated and compressed grain boundaries may compete with pressure dissolution (Green 1984). For example, a coupling between crack processes, surface area generation and enhanced dissolution-mediated creep has been proposed as a mechanism for stylolite generation (Keszthelyi et al. 2016). Single-indenter studies also suggest that under certain, poorly understood conditions micro-cracks can precede dissolution-mediated creep (Zubstov et al. 2005; Croizé et al. 2010).

The factors driving the morphological evolution of compressed hydrated mineral-mineral interfaces are also poorly understood (van den Ende et al. 2019). Interfacial roughness would enhance rates of interfacial transport by contributing channels for solute transport that could be significantly greater than nanometer-thick films (Lehner 1990; de Meer et al. 2002; van Noort et al. 2008). Stress concentration at small bridging asperities (Renard et al. 2012), are likely to be less stable than higher-surface area contacts. Thus, the maintenance of rough mineral contacts may require crack formation (Zubstov et al. 2005) or the development of stress instabilities (Sprunt & Nur 1977; Koehn et al. 2003).

Moreover, there is considerable uncertainty about the controls on the redistribution of elements mobilized from minerals surfaces undergoing pressure dissolution. Morphological studies in single-crystal and column studies, are in some cases consistent with solution transfer and growth (Zhang et al. 2010; Hickmann & Evans 1995). Studies on mineral grains, however, have reported the growth of asperities (Renard et al. 2012), nucleation of calcite in proximity to opposed surfaces under nominally undersaturated fluid conditions (Ben-Itzhak et al. 2015), nucleation of calcite in the interface between compressed surfaces. Studies of compressed rocks can also exhibit major mineralogical alteration such as preferential formation of Mg–rich cements (Neveux et al. 2014).

In summary, the mechanistic uncertainties in pressure dissolution models motivate a renewed effort to clarify the interfacial processes and develop a chemical-mechanical model based on chemical rate laws.

Proposed Work

A range of integrated studies will be undertaken to resolve the pathway(s) and rates of creep at calcite–mineral interfaces under normal stress.

Chemical Kinetics Model for the Pressure Dissolution Strain Rate including Surface Potential

A core effort will be the development and testing of a model for interface strain rate via pressure dissolution that includes surface potential. A focus will be on the effect of overlapping electric double layers (EDLs) that we anticipate will affect both the driving force for pressure dissolution, by adding a positive or negative electrostatic contribution to the energy barrier of dissolution, and the chemical kinetics of surface retreat, by altering the equilibrium speciation of surface sites with different surface species being associated with different rates of underlying mineral dissolution (e.g., for quartz, Nangia & Garrison 2008)

1. Thermodynamics

Prediction of the evolution of a mechanically-stressed mineral in a reactive multicomponent fluid is controlled by the difference between the chemical potential 𝜇_s of mineral atoms in the solid phase and 𝜇_f of the atoms in the fluid, with equilibrium expressed 𝜇_s = 𝜇_f (Gibbs 1906). For the fluid phase, 𝜇_f, will be determined using geochemical modeling in CrunchFlow for the equilibrium solute activities for arbitrary pore fluid composition in equilibrium with the mineral that may include trace element impurities.

For the mineral phase, the thermodynamics of a deforming solid has not been consistently and properly handled over the years with regard to the deviatoric strain energy as highlighted by Lehner & Leroy (2004) and Hobbs & Ord (2016) and detailed by Pride (2020). The total internal energy, U, of the solid mineral phase is

U=TS — PV + V𝜏^D: e^D + 𝜇_s N + 𝛾 A

where S is the total entropy of the solid mineral, V the evolving volume of solid, V_o the volume of the solid in an initial undeformed reference state, e^D the deviatoric strain tensor that quantifies shape changes of the solid phase and that is zero in the initial reference state, N, the total number of mineral molecules in the solid and A, the area of contact between the mineral and the EDL. The intensive state functions are (Pride 2020): T temperature, P pressure, 𝜏^D deviatoric stress tensor, 𝜇_s chemical potential of the mineral and 𝛾 surface energy of the mineral surface that is in contact with the EDL. Taking a derivative of this fundamental function gives the first law for a deforming solid. Defining the Helmholtz free energy, 𝜓 = U - TS.

𝜓 = —PdV + Ve^D:𝜏^D + 𝜇_s N + 𝛾 A

Dividing through by N gives a definition for the chemical potential in a form similar to the common expression (e.g., Croizé et al. 2012) but here with a precise definition of 𝜓.

For the EDL, the corresponding thermodynamics is augmented by the notions of disjoining pressure and electrostatic energy under the assumption that the EDL cannot sustain deviatoric strain, even in the Stern layer (determining whether the Stern layer has a finite rigidity is an important subgoal discussed below and here). we will develop electrostatic models for the free energy embodied in overlapping EDLs. Prior calculations of colloidal interactions (Lyklema 2005; Trefalt et al. 2015) indicate that overlap energies can exceed 1 mJ/m², small but non-negligible compared to the hydrated surface energy of a metal oxide, e.g. ~80 mJ/m² for calcite (Söhnel & Mullin 1982).

Continuum overlapping EDL models will be developed for planar mineral-electrolyte-mineral interfaces and subsequently for surfaces with steps. The equilibrium distribution of charges, including chemisorbed ions in the Stern layer and electrolyte ions in the interlayer, will be calculated assuming chemical equilibrium with respect to the various complexation reactions. Analytical or numerical solutions will be developed for surface and edge-site complexation and net charge, interlayer ion distribution and the electrostatic component of the disjoining pressure. The theoretical models will be quantitatively compared with molecular simulations that can provide the Maxwell electrostatic energy stress tensor. The total overlapping EDL disjoining pressure and free energy will include electrostatic, dispersion and solvation forces (Israelachvili 2011 )

2. Chemical rate law

If the Gibbs condition for equilibrium is not satisfied, non-equilibrium thermodynamics predicts that a material flux will be initiated at a rate proportional to a function f=f(𝜇_s - 𝜇_f). For bulk modeling of coupled processes, f frequently incorporates chemical potential gradients associated with both reaction and transport (Lehner 1990). It is typically assumed that f is given by an Arrhenius form with a reaction activation barrier, E_A. However, the microscopic origin of E_A (i.e., the rate-limiting step of a non-elementary reaction) is typically unknown so either E_A is estimated from temperature-dependent bulk experiments or neglected and f is linearized.

In this proposal, by contrast, we seek to develop an expression for the local flux of atoms from a stressed surface that incorporates the best knowledge and models for the molecular pathways and kinetics for mineral dissolution and growth. Typically, models for the rate of calcite dissolution, R_diss, assume a balance between attachment and detachments proceeding with fixed rate constants k

R_diss = k_detach/{CaCO₃} - k_attach/[Ca²⁺][CO₃^2- ] = k_diss(1-Ω)

where {CaCO₃} is the mineral activity, typically considered to be unity, and Ω is the mineral saturation. However, the linear dependence is not in agreement with the results of Subhas et al. (2017) which indicate that surface fluid chemistry and dissolution mechanism much be considered particularly close to equilibrium, as is expected for sediments, and more detailed models for dissolution must be considered (Dove et al. 2005). Thus, a Cross-Cutting project will develop a new model for Strain Effects on Calcite Dissolution Kinetics

3. Interface structure and transport rates

Any conceptual and quantitative model for grain-contact requires a detailed understanding of the evolution of the morphology, transport properties and mechanical strength of the compressed mineral interface. Rough interfaces will concentrate stress at asperities and can introduce channels for fluid and solute transport (Lehner 1990). Although static and dynamic models have been developed for the morphology of stressed grain contacts, quantitative observation at the relevant length scales is lacking. Prior studies often reported an increase in contact surface area without addressing interfacial roughness. Although we apply and assume only a normal stress, at rough interfaces, shear stresses, micro-cracking and even friction may play a role in the full plastic deformation of the compressed interfaces. In addition, under unsaturated conditions, compressed grains can evolve complex interlocking geometries suggestive of the onset of stylolite formation (Gratier 2005).

Surface-forces Apparatus studies will be performed in collaboration with the Fracture Processes Project to determine the evolution of interface roughness evolution during pressure dissolution, anticipating two regimes. For low stress and chemical equilibrium we anticipate that the roughness will diminishes towards the idealized depiction of crystallographically distinct termination surfaces (with step edges). For high stress, including shear, and undersaturated pore-fluid compositions we anticipate that roughness will trend towards a characteristic self-affine properties (over a certain range of length scales) with surface topological descriptors that are reproducible for stress and chemical conditions.

Pore–Discontinuum Chemical-Mechanical Simulations of Interface Plasticity

The single-interface pressure dissolution strain-rate law will be incorporated into a novel micro-chemical mechanical modeling code. This new simulation capability will be developed by integrating in-house LBNL codes for granular mechanics (the numerical manifold method, NMM; Hu et al. 2017) and aqueous geochemistry (CrunchFlow; Steefel et a. 2015). This code will be applied to the study of pressure dissolution at idealized and rough interfaces using interface- and asperity-scale constitutive laws, including a new strain-rate model for pressure dissolution. Figure 3 shows preliminary results of coupled RMT processes induced by loading on the top surface of a single asperity. We do not see a large gradient in horizontal normal stress, 𝜎_x, within the domain. In contrast, vertical normal stress, 𝜎_y, is concentrated at the contact interface resulting in strong pressure dissolution at this location . Shear stress, 𝜏_xy, is significant at the four corners of the asperity, which contributes to faster rates of pressure-driven dissolution, k_dis, at those locations. At the free surfaces, mineral precipitation occurs.

This simple example demonstrates the promise of the NMM-CrunchFlow for pore- and grain-resolved simulations of pressure dissolution as well as other chemical-mechanical processes studied throughout this proposal. The proposed effort will enable models for mineral dissolution to be developed based on the full local stress tensor, and using new rate laws described above for the dissolution rates at strained calcite surfaces and transport through nanoconfined grain-boundary films.

Single Calcite–Mineral Interface Studies using the Surface Forces Apparatus (SFA)

Collaboration with the group of Dr. Younjin Min will provide surface-forces apparatus measurements of hydrated calcite–mineral forces and stress-driven convergence rates as a function of opposing mineral composition (e.g., one calcite crystal against calcite, quartz and mica surfaces), the geochemical conditions of the bulk aqueous solution (e.g., pH and electrolyte conditions) and temperature.

Although prior studies have developed SFA studies of calcium carbonate (Dziadkowiec et al. 2018), the specimens were fabricated by atomic-layer deposition of metal-organic precursors, generating a microcrystalline and likely highly disordered thin film that could be responsible for surprising observations including the apparent formation of interlayer gel (Dziadkowiec personal communication). To avoid possible artifacts, we will develop a method to mount and study thin single-crystal calcite crystals formed by cleaving and/or polishing. A technical description of the SFA instrument and mineral surface preparation methods is given in the here.

The goals of the SFA studies are:

1. Overlapping EDL forces and energetics

We will test theoretical and simulation predictions for the distance-dependent force for EDL overlap for calcite–mineral interfaces. The force-distance profiles arising between calcite and mineral surfaces such as mica and silica will be obtained using a Surface Forces Apparatus (SFA) at different length scales ranging from µm to molecular contact < 1 nm. The force profiles will be modeled using new extension of EDL described above (Zarzycki, Pride and Steefel) that accounts for arbitrary surface complexation reactions, sometimes referred to as charge regulation.

2. Rheological properties of compressed interfaces

The rheological properties of electrolyte solutions will be investigated as a function of a degree of nanoconfinement (pressures or loads) in conjunction with identifying the breakdown of the continuum behaviors. Two different rheological experimental set-ups (compressive and shear modes) are going to be introduced in this regard as illustrated here. Some of the key expected outcomes under this task is to elucidate how the obtained interaction parameters, in particular, the surface potentials can change as the electric double layers overlap at high pressures (or loads), subsequently influencing rheological responses (e.g. viscosity) and giving rise to promote or suppress the dissolution of calcite-mineral interface in aqueous medium.

3. Mineral–mineral convergence rates

SFA measurements of the rate of single-interface creep as a function of temperature and solution chemistry. The dissolution rate of mineral surfaces and the resulting creep will be probed at the sub-nanometer resolution via multiple-beam interferometry-coupled SFA studies. During a course of the SFA experiments, the changes in electrical conductivity and pH of the solution are also planned to be monitored simultaneously so that the extent of dissolution can be quantified and compared with the changes in absolute thickness at the sub-nanometer level over any period of time from milliseconds to days. In parallel, in-situ optical microscopic observation can be made during SFA experiments to visualize the generation of precipitates in the vicinity of the compressed interfaces.

4. Mineral surface roughness

Optical resolution imaging during the SFA studies and profilometry and Atomic Force Microscopy (AFM) along with elemental analytical technique such as X-ray photoelectron spectroscopy (XPS) will establish roughening, flattening and re-precipitation behaviors. The morphological and texture change of samples will be monitored in situ and ex situ. In situ measurements will rely on the analysis of FECO and optical top view microscopic images which can be simultaneously obtained during SFA experiments. Ex situ analysis will involve the metrological characterization of the mineral samples after dissolution experiments via AFM, profilometry, and XPS. AFM and profilometry will be used to compare rms roughness (height parameter), inter-asperity spacing (lateral parameter) power density function of the surface roughness, fractal dimension of the sample before and after SFA experiments.

Grain-Scale Studies using in situ X-ray Nanotomography

Through collaboration with the Advanced Light Source, this group is constructing a transmission X-ray microscope (TXM) for time-lapse radiography and nanotomography of mineral–fluid samples with resolution of ~50-nm. We have designed and constructed miniaturized sample cells to enable long-term creep studies at elevated temperature and pressure with control of fluid chemistry. The goals of the TXM studies of grain-resolved creep are:

1. Mechanism of single-grain deformation

Time-lapse nanotomography of small grain packs will provide unambiguous observations of grain-scale creep mechanisms, including pressure dissolution, crack growth, and twinning, and insights into potential for coupling between these processes.

3. Material redistribution

The data will be analyzed to identify the location and rates of precipitation of mineral constituents mobilized by pressure dissolution. We will evaluate whether topographic features such as twin planes or interstices preferentially nucleate new material and seek the conditions that lead to cementation.

4. Pseudo-2D studies of limestone asperity creep

In order to provide data sets for developing and testing the grain and pore-resolved Chemical-Mechanical Model, we will perform time-lapse tomographic studies of the evolution of mineral asperities with idealized geometries, controlled normal stress and simple solution chemistry. The mineral asperities will be created using the laser mill facility available at the Advanced Light Source. As needed, smaller features can be created using focused ion beam (FIB) milling available at the Molecular Foundry (Fig. 5).

References