FEM Simulations

Numerical simulations were performed using the transport of diluted species module of the commercial finite element method (FEM) modeling package Comsol Multiphysics 5.2 (Comsol AB, Sweden). Three-dimensional models were formulated using inputs from experimental data to estimate the interfacial concentration of species during the dissolution process and determine the dissolution rate constant of the first-order heterogeneous reaction in terms of interfacial proton concentration.1, 22,

23

The models simulate the mass transport of species during the dissolution of a calcite crystal in a quiescent solution at pH 3.1. At this pH, the proton-promoted dissolution process at the crystal surface can be described by (eq. 5.1 and 5.2):1, 5

CaCO3 + H+ Ca2+ + HCO3- (5.1)

HCO3- + H+ →_← H2CO3 →CO2 (aq) + H2O (5.2)

The models also take account of chemical equilibria in the solution that are relevant under the conditions of our experiment35, 36 _{(Table 5.1). These reactions are expressed}

in the simulations by activity corrected mass action rate equations23, 37 _{and the rate}

constants are defined to be fast enough to maintain the solution processes at equilibrium at a particular point in space. The mass transport of species is governed exclusively by diffusion and assumed to be effectively at a steady-state, for which the following is solved (eq. 5.3):

(5.3) where Ji is the flux, Diis the diffusion coefficient, ciis the concentration of species i,

and Ri is the reaction rate expression for the species i. The use of a steady-state model

is appropriate because the characteristic diffusional time for the mass transport of protons from bulk solution to the microscopic crystal surface, estimated using a semi- infinite diffusion model according to eq. 5.4, is about 0.14 s, which is 4 orders of magnitude faster than the duration of a typical crystal dissolution experiment (30 min for the complete dissolution of a crystal). tdiff,i denotes the steady-state diffusion time

and d is crystal largest dimension size.

(5.4) The diffusion coefficients of the individual species at infinite dilution were obtained from the literature36 _{and corrected for the ionic concentration in bulk solution.}

Diffusion coefficients were assumed to be constant over the entire domain (Table 5.2). → ← ∇J_i=∇ ⋅(D_i∇c_i)+R_i=0 tdiff,i≈ d2 Di

Table 5.1. Equilibrium Reactions for the Calcite-Water System open to the Atmosphere.

Reaction pKeq* 36

H2O H+ _{+ OH}-

14

CO2 (aq) + H2O H2CO3 1.446

CO32- + H+ HCO3- -10.33

HCO3- _{+ H}+ _H2CO3° _-6.35

Ca2+ + CO32- CaCO3 (aq) -3.20

Ca2+ _{+ HCO}

3- CaHCO3+ -1.00

* Value before activity correction

H2CO3° signifies the total concentration of dissolved carbonate in the form CO2 and H2CO3

Table 5.2. Diffusion Coefficients of the Species Considered in the FEM Model.

Species Ca2+ CO32- HCO3- H2CO3 CaCO3 CaHCO3+ H+ OH-

D (×10-9 m2 s-1) 0.760 0.886 1.137 1.137 0.818 1.039 8.939 5.062

The three-dimensional domain used in the simulations is shown in Figure 5.1 and was parameterized with the average crystal dimensions and geometry of ten independent crystals determined by optical microscopy and scanning ion conductance microscopy. The models simulated the dissolution process described in eq. 5.1-5.2 and were developed by applying a flux of protons into the calcite surface that causes a release of calcium and bicarbonate ions, representing the flux of species leaving the dissolving crystal faces. The model, denoted here as Model 1, used as the flux value for each individual crystal face, the experimental dissolution rate of the {104} face,

Jobs, obtained from the time-lapse optical images. The model designated Model 2,

defined the flux of species to and from the crystal surface as a result of a first-order process in near-surface (surf) proton concentration,1, 22, 23 applying eq. 5.5, where ksurf

is the intrinsic dissolution rate constant.

Ji=ksurf[H +_] surf (5.5) → ← → ← → ← → ← → ← → ←

Using appropriate boundary conditions (Table 5.3), the concentrations of species near the calcite-solution interface and in the solution around the dissolving crystal were simulated and the results obtained from each model were compared. The solutions of the partial differential equations for both models were acquired using the direct solver MUMPS in the Comsol environment, with a relative error tolerance of 10-6. Simulations were carried out with optimized tetrahedral mesh elements.

Figure 5.1. Three-dimensional domain used for FEM simulations (not to scale). The numbers correspond to the boundaries described in Table 5.3.

Table 5.3. Boundary Conditions used in the FEM Simulations

Boundary Characteristics Conditions Model 1 Conditions Model 2

1-5 Calcite {104}

face

6 Petri dish

surface

7-11 Bulk solution

The boundary numbers are in accordance with the boundaries defined in Figure 5.1

n denotes the vector normal to the surface

ci,bulkis the bulk concentration of the species i governed by the equilibrium reactions in Table

5.1 n⋅D_H+∇c_H+ =Jobs −n⋅D_Ca2+∇c_Ca2+ =Jobs −n⋅D_HCO 3 −∇c_HCO 3 − =J_obs n⋅D_H+∇c_H+ =ksurf[H +_] surf −n⋅D_Ca2+∇c_Ca2+ =ksurf[H +_] surf −n⋅D_HCO 3−∇cHCO3− =ksurf[H + ]surf n⋅Di∇ci=0 n⋅Di∇ci =0 ci =ci,bulk ci=ci,bulk