Dissolution kinetics of strained calcite

Get~,hiraica et Cosm~'himica Acta Vol. 53. pp. 373-382

0016-7037/89/$3.00 + .00

Copyright ~ 1989 Pergamon Press plc. Printed in U.S.A.

Dissolution kinetics of strained calcite JACQUES SCHOTTt, SUSAN BRANTLEY2, DAVID CRERAR3, CHRISTOPHE GUY ~, MARIA BORCSIK3 and CHRISTIAN WILLAIME4 ILaboratoire de ~ochimie, Universit~ Paul Sahatier, 31062 Toulouse Cedex, France 2pennsylvania State University, Department of Geosciences, University Park, PA 16802, U.S.A. 3Princeton University, Department of Geological and Geophysical Sciences, Princeton, NJ 08544, U.S.A. 'Institut de G6ologie, Universit6 de Rennes, 35042 Rennes Cedex, France (Received April 19, 1988; accepted in revised form December 2, 1988) Abstract--Interface-limiteddissolution of minerals occurs non-uniformly with preferential attack at sites of excess surface energy such as dislocations, edges, point defects, microfractures, etc. Strained crystals are predicted to show higher dissolution rates due to the increased internal energy associated with dislocations and due to enhanced nucleation of dissolution pits at dislocation outcrops on the surface. Using calcite strained to different degrees, we have observed a measurable rate enhancement of two to three times relative to unstrained crystals at temperatures from 3 to 80°C. This rate enhancement is large compared to that predicted from the calculated increase in crystal activity due to strain energy, but small compared to the three orders of magnitude difference in dislocation densities for the crystals tested (106-109 cm-2). Measurements over a range ofpH (4.5-8.3) and temperature (3-80°C) showed that the rate enhancement increased with increasing pH and decreasing temperature. Calculations based on the excess free energy of screw dislocations suggest that dissolution rate enhancement should become significant above a critical defect density of roughly 107 cm -2, in apparent agreement with our observations. Crystal dissolution comprises several parallel processes operating in parallel at active sites. The small relative enhancement of dissolution rate with defect density reflects the greater quantity of dissolved material delivered to solution from receding edges and ledges relative to material coming from point defects and dislocations. Our data, coupled with existing information on other minerals, sngg~ that generally applicable kinetic measurements can be made on lowstrain, macroscopic mineral specimens. However, kinetic data on highly strained minerals should include measurement of defect density because of the rate vs. strain correlation. Selective dissolution can be expected to occur in naturallydeformed rocks, where heterogeneity in dislocation distribution could cause solution transfer and deformation. INTRODUCTION

fective surface area of a mineral (HELGESON et aL, 1984). Equation (1) illustrates, in a simple fashion, the problem addressed in this paper. The di~culty lies in the surface area term Se: If dissolution occurs preferentially at active sites such as lattice defects, then this should be incorporated into the estimated surface area. Ostensibly, minerals with greater defect densities should dissolve faster and their effective surface areas should be greater than more perfect specimens of the same compound. This effect is not predictable and, therefore, must be measured in each case. For this reason, HELGESON et ai. (1984 ) recommended including an additional term in Eqn. (1) expressing the relative reactivity of different mineral sites a n d / o r surfaces based on measured rates (see also, HOLDREN and SPEYER, 1987). This raises concerns about the applicability of rate determinations on minerals in general, since natural specimens could cover an enormous range of defect densities (varying by many orders of magnitude). Why should kinetic measurements for a sample of one particular mineral apply to specimens with different growth histories and defects? HOLDREN and SPEYER (1985, 1987) have addressed one important aspect of this problem in their studies of the relationship between measured rates and particle grain size (or exposed surface area) for alkali feldspars and plagioclases. They showed that dissolution rates increase linearly, as expected, with decreasing grain size (and increasing specific surface area) to a critical region where they hypothesize that grain size and distance between adjacent reactive sites become roughly equivalent (,-~50-100 ~tm). For smaller grain sizes (larger specific areas), rates and reactant surface area are not related in any simple fashion. This suggests that rate deter-

THE KINETICSOF MINERALdissolution and precipitation are well explained by transition state theory--see LASAGA(1981) for a review. The framework of this theory and kinetic data for minerals have shown that dissolution occurs preferentially at "active sites" on mineral surfaces. These active sites consist of dislocations, edges, microfraetures, point defects, kinks, grain or twin boundaries, etc. The importance of the concentration or density of such active sites in determining reaction-limited mineral dissolution rates has been stressed repeatedly in the literature (e.g., BERNER, 1978; BERNER el al., 1980; PETROVIC, 1981a,b; AAGARD and HELGESON, 1982; LASAGA, 1983; HELGESON et al., 1984; BRANTLEY et al., 1986; HOLDREN and SPEYER, 1985, 1987). Dissolution at active sites is readily apparent in the development of etch pits and related features on mineral surfaces (HOLDREN and SPEVER, 1987; and references therein). Statement o f the problem The concept of selective dissolution at active sites raises several difficult questions for both theoreticians and experimentalists interested in explaining the kinetic behavior of natural mineral systems. Consider, for example, the net rate, r, for dissolution of a mineral under highly undersaturated conditions: r = k+S, H aT'

(1)

i

where al and ni are the activity and reaction order of the ith species, k+ is the dissolution rate constant and Se is the el373

374

J. Schott et al.

minations can be reported as rates per unit area for the larger grain sizes, where rate and area remain linearly related; however, at finer grain sizes, rates apply exactly only to those specimens for which they were determined. In the present study, we examine a second aspect of this problem: What is the relationship between measured dissolution rates and the density or concentration of active sites? This is accomplished using large, single crystals of calcite for which surface area and rate should be directly related. Varying concentrations of active sites (lattice defects) are imposed by straining individual samples to different degrees. We will show that the dissolution rate increases nonlineafly with increasing dislocation density. We will also derive a simple expression to calculate the critical dislocation density above which such rate enhancements are predicted.

THEORY Following BLUM and LASAGA(1987), we can write the following expression to describe the net rate of dissolution of a perfectly flat crystal dissolving by a reaction-limited mechanism:

r=

'

k I. perfect lufface sites

+

exp[

kT

) /-AG;~11

~

exp|~H

dislocated aufface tltes

\

r~l

|.

(2)

/)J

We have replaced the bulk rate constant and surface area term from Eqn. (1), k+ S,, with a summation over surface sites with different activation energies toward dissolution. In this equation, u is a frequency factor, k is Boltzmann's constant, AG~ is the activation energy for dissolution of perfect crystal surface and AG~ is the activation energy for dissolution of dislocated crystal surface. AG~"will differ from AG~by the magnitude of the strain energy associated with the dislocation. It is apparent from Fxln. (2) that high dislocation densities can increase the dissolution rate in two possible ways: 1) As the number of dislocations increases, the strain energy associated with the defects will increase the potential energy of the solid and increase a ¢ ~ ; 2) As the number of surface sites intersected by dislocations increases, the second summarion term in the brackets becomes more important. The first effect can be thought of as enhancement of dissolution caused by the increased activity of the crystal; the second effect expresses the increased rate caused by a greater number of reacting surface sites. The first effect is likely to be small because, for most minends, the dislocation strain energy is small compared to the crystel's total free energy (HULL, 1975 ). The internal energy of dislocations involves both the elastic strain energy in the lattice around the dislocation and the energy stored in the highly strained region of the core. For a screw dislocation, the energy per unit line, U~i,c, can be expressed as (HIRTH and LOTHE, 1982): Uline

Tb2 In R

----~

'~04" a1"b 2

(J/cm)

(3)

where 7 is the shear modulus, b is the Burgers vector, R is

the radius of the strain field, r0 is the radius of the core, and a is a factor describing the contribution of the core energy. Values of R and r0 used in our calculations are one-half the average distance between dislocations, p-1/2/2 (where p is dislocation density in cm -2) and b, respectively. The total strain energy, Utom,, and activity of the crystal, a, can be calculated from: Ustrain,tot =

Uline,OV

(J/mol)

(4)

and a = exp(Ull.ep V/RT)

(5)

where Vis molar volume. Using appropriate values of the parameters (from HULL, 1975, and MOTOHASHI et all.,1976) we can calculate the strain energy of calcite of known dislocation density (Table I). It is clear that both the strain and core energy contribute insignificantly to the thermodynamic properties of calcite. Even with the extremely high dislocation density of 10 I' c m - : , the free energy of calcite is increased by only 80 J/reel which corresponds to a 25°C activity of 1.04 for calcite (Eqn. [5]). The rate of dissolution of this strained crystal based on its activity should be only 1.04 times faster than that of an unstrained calcite crystal under similar conditions. Calculations of strain energy associated with dislocations in quartz (WINTSCH and DUNNING, 1985), rutile and various silicates (SCHOTT et al., 1988) show a similarly small effect. Note that these figures arc not substantially changed for edge instead of screw dislocations; in this case the strain energy would increase by a factor o f ( l - u) -~ or about 1.5 (where u = Poisson's ratio). In contrast, the second effect could play an important role in bulk dissolution. This is because the core energy computed in Table 1 reduces the activation energy required to form activated complexes at dislocation outcrops. Note that the core energy, which represents 20 to 40% of the strain energy (Table 1), affects an area about 100 times smaller than the strain field. For example, if we could calculate a molar free energy for a core by assuming that this core energy disrupts one calcite formula unit of atoms, it follows that the core

T a b l e I. Calculated excess internal energies per mole calcite as a function o f dislocation density, based on Eqns. (3), (4) and (5) in text. Dislocation density (cm -3)

0 Il. 10o10.101

R = ]/(2~"0)

500

158

50

15.8

5

UIt rain field (J c m - l . l O 11)

3.2

2.7

2.15

1.6

l.l

uco~

5.6

5.6

5.6

5.6

5.6

0.14

1.2

IO.O

70.8 613

(cm.107)

(J c m - L l O 12)

U atrltin,tot (J mo1-1) acalcita 3 "C

I.O0 1.00 I.O04 1.035 1.31

acilclt e 2 5 " C

I.O0 l.O0 1.00 1.03 a 1.28

acalclt e 8 0 " C

I.O0 l.O0 I.O0 1.02 s 1.23

~=0.1 "U = 3.5 x 1010pa r o ffi b ffi 4 x 10"8cm Vmolar, calcite ffi 36.93 cm s

Dissolution kinetics of strained calcite energy for a mole of such formula units would be ,-,40 k J / mol. As a result, below a certain critical solute concentration, etch pits will spontaneously develop at the site of dislocation outcrops on a mineral surface (see, for example, BRANTLEY et al., 1986). We emphasize the word "spontaneously" because below the critical solute concentration there is no nucleation barrier for pit formation. Following BLUM and LASAGA( 1987 ), we can rewrite Eqn. (2) assuming that the net rate of dissolution in undersaturated solutions, r, is the sum of two components:

r

:

I I a'/'(Stotkp + Sdkd)

(6a)

i

What

375

De t e r m in e s

Measured

Dissolution

Rate With parallel

ProCesses?

Fastest process is normally rate-detormminE, unless its contribution to total dissolved concentrl~ion is insignificant •

V

~

Rate

Quantity

Point Defects Dislocations Mlcrofraclures

ID

Kinks Grain or Twin Boundaries

D i

Corners Edges, Ledges v

where k~ : dissolution rate constant per unit area of dislocation-free surface, kd : dissolution rate constant per unit area of dislocated surface, Sd is the area of dislocated surface, and S~otis the total surface area (which is assumed equal to the total perfect surface area Sp, since Sp ~, Sd). We can estimate Sd by noting that Sd = pStot~dwhere Sd is the strained area of one dislocation: r = ]'I aT'S=ot(kp + p~dkd).

(6b)

In Eqn. (6b), the product of activities will normally take the form of the product of crystal activity times reacting solute activity (e.g., activity of H +, O H - , or H20). This equation expresses explicitly the two effects driving dissolution of a strained crystal: the first term expresses the activity effect and the second term expresses the surface site effect. If the first term of Eqn. (6b) dominates, we predict close to a zerothorder dependence on dislocation density; if the second term dominates, then the rate of crystal dissolution will exhibit a linear dependence on p. However, if dislocations are closely spaced, then dislocation strain fields will overlap and Sd will decrease as p increases. In this case, the rate of dissolution will vary nonlinearly with p. The relative contribution of the two terms in Eqns. (6a, b) is determined by the importance of other active surface sites whose dissolution occurs in parallel with dislocation pits. Dissolution processes starting at edges, surface pits, point defects, twin or grain boundaries, microfractures, etc. all contribute to bulk dissolution; the rates and relative quantities dissolved by each of these processes (along with the relative concentrations of these sites) will determine the importance of variation of dislocation density (Fig. 1). Indeed, for a crystal whose surface is not perfectly flat, the general rate in Eqn. (2) could be expressed using several rate constants to describe the energetics of dissolution at the varied sites present on a real surface. In addition, the relative importance of each parallel dissolution process will change as the solution concentration increases during dissolution. For example, at extreme undersaturation ( C < Co,= ~) pits can theoretically be nucleated anywhere on a surface. At slightly higher concentrations (but still at high undersaturation, C,~t t < C < Cc,t2), the driving force for dissolution is large enough to nucleate pits at dislocation outcrops but not at a perfect surface. As the dissolved concentration rises above the C~t2, etch pit nucleation should largely cease, and the relative importance of dislocations to bulk dissolution could decrease (for details, see BRANTLEY

Entire Face Wifh A}l D e f e c t s

F3G. 1. Highly schematic illustration of the parallel processes involved in crystal dissolution. The horizontal length of each arrow indicates the relative rate of each process (actual rates can differ by many orders of magnitude). The vertical thickness of each arrow represents the relative quantity of material dissolved and defivered to aqueous solution by that process. Thus, while point and linear defects react most rapidly, they deliver less dissolved material to solution than slower dissolution of faces and pits occurring at edges, ledges and corners.

et aL, 1986). This may explain the reported enhancement of dissolution rate for deformed polycrystalline calcite which was only significant for the first 30 minutes of dissolution (BIANCHETTI and REEDER, 1985). We note that the relative importance of the two factors discussed above in determining dissolution kinetics will be greatly affected by whether the reaction is surface- or transport-controlled. A surface-controlled reaction is rate-limited by the dissolution step, implying that both the increase in crystal activity and the increase in the number of active sites could be important. A transport-controlled reaction is ratelimited by diffusion of the dissolving species through the fluid boundary layer or a protective surface layer. Such a reaction would only be affected by an increa~ in local equilibrium concentration of a reactive species in the boundary layer; this increase would create a larger concentration gradient across the boundary layer and a correspondingly larger observed rate. EXPERIMENTAL METHODS To quantify the effect of dislocations on the rate of dissolution, single crystals of calcite were strained to high defect densities (up to 10 9 ffl'f1-2) and dissolved in aqueous KC! solutions. Samples consisted of 6.2 mm diameter cylinders drilled in Iceland spar crystals with core axis parallel to the crystal C axis. The samples were maintained at constant temperature by an external furnace and were deformed under atmospheric pressure at constant strain rate (using apparatus located at Centre Armoricain d'Etude Structurale des Socles, Universit6 de Rennes). At the end of the experiment the core sample was quenched and checked microscopically for abnormal surface roughness or cleavages.Three deformed samples were made according to conditions described in Table 2. Dissolution experiments for these samples were performed by rotating the cylinders at 220 rpm, as shown in Fig. 2. The samples were dissolved in KCI solution at constant pH using a pH-stat to add HC1 throughout the run to maintain constant pH. The bases of the calcite cylinder and the external surface close to the bases (where the dislocation density is smaller) were covered in Teflon to prohibit dissolution. A CO2-N2 gas mixture of

J. Sehott et al.

376 Table 2. Strain conditions for calcite samples used in this study. F o r difference between Type 1 and II

samples,referto Fig.2. Sample T(*C) Strainrate MaximumDislocation strain density (sec-I) (cm -s)

Type I 1 2 3

500 500 440

10 -5 1.3.10 -s 1.310 -s

4.75 6.2 6.05

10 ? 5t08 109

T y p e I1 4

855

10 -5

2.2

106-107

known composition was bubbled through the reacting solution continuously during the experiments.Runs were conducted in 0. ! molar KCI solution at 3, 25, and 800C (+0.20C). Dissolution progress was monitored by noting the cumulative addition of HCI by the pH-stat, and dissolution rates were calculated for initial dissolution periods of one to eight hours, depending on pH, temperature and length of the sample. During these periods of time, the observed dissolution rates remained constant. At the end of the dissolution experiments, thin sections of the samples were ion-thinned and carbon-coated for TEM observation. Dislocation densities were evaluated by counting the dislocationsvisibleon areas severalmicrons wide. Rate constants observed are summarized in Table 3. A second sample type (Type 2 sample) was run to investigatethe surface morphology during dissolution. This sample consisted of a rectangular parallelopipeddeformed at 25"C in a triaxial Brace-type rig by Brian Evans at Massachusetts Institute of Technology. This sample was mounted in epoxy with deavage face exposed, and pol. ished (600# carborundum, 6 t~m, 3 ~m, 1 ~m diamond paste). This sample was dissolved in a rotating disc assembly (SJOBERGand RICKARD, 1984;COMPTONand D^LY, 1984;COMPTONeta/., 1987) at 600 rpm under CO2-freeN2 atmosphere in 0.7 molar pH 8.4 KCI solution at 250C under free drift conditions.

for dislocation densities as high as 10 t' cm -2, the variation of calcite activity is insignificant (Table 1 and Fig. 6 ). Thus, we postulate that the observed rate increase at higher defect densities is related to the number of active surface dissolution sites. Corroborating this is the observed effect of temperature and pH on rates. The low-pH regions for which rate constants vary exponentially with pH on Figs. 3 and 4 represent regions of partially diffusion-limited behavior and the most rapid dissolution rates (see SJOBERG and RICKARD, 1984; BUSENBERG and PLUMMER, 1986). At higher pH, the rate becomes surface-reaction limited and for this reason the effect of defect structures on the rate is larger. Similarly, as the temperature decreases, the surface step becomes significantly slower, and the effect of dislocations on dissolution rate increases. These results, manifesting a pH-dependence at low pH and pH-independence at high pH fit nicely with the model of PLUMMER et at (1978). The Plummer-Wigley-Parkhurst model (reviewed and extended in BUSENBERG and PLUMMER, 1986) proposes that calcite dissolution is rate-limited by H + diffusion across the hydrodynamic boundary layer at the surface for pH < 4.5. Above pH 4.5, reaction is partially transport- and partially surface-controlled. Our results show a decreased rate dependence on dislocation density at low pH as expected for a partly transport-limited reaction. At very low pH, we still observe a slight increase in rate with defect density, which implies that defects are influencing a transport-limited rate. PLUMMER el at (1978) suggest that the rate of dissolution (pH < 4.5) can be described as: r = KT(aH+~B)- aH+~o))

RESULTS The rates of dissolution of normal and strained calcite samples are compared in Figs. 3 and 4. Our rates of dissolution of unstrained crystals are nearly identical to published rates of dissolution in pure H20 reported by BUSENBERG and PLUMMER (1986). For unstrained Iceland spar, we found the same rate of dissolution as measured by SJOBERG and RICKARD(1984): r = 5 . 1 0 -I° moles cm -2 s -~ in 0.1 M KCI solutions at 25"C, pH = 4.45, and 220 rpm. Although our experiment was not a true rotating disc experiment, the correspondence of measured rates implies that our experiment was operating in a transport-dominated kinetic regime, at pH < 6 in all cases, as hypothesized by SJOBERG and RICKARD (1983) (see Figs. 3 and 4). For defect densities on the order of 107 to l0 s cm -2 or higher, the effect of dislocations on dissolution rate is significant. For example, at 25"C and pH ffi 7, the rate of dissolution of strained calcite (p = 109 cm-2) is about two times faster than that of normal calcite (Fig. 3). This increase in dissolution rate with strain is most pronounced in the more alkaline solutions (pH > 5.6) and at lower temperatures (Fig. 4; Table 3). An SEM photomicrograph of the surface of a dissolved Type 2 sample is shown in Fig. 5. Overlapping etch pits were found to be numerous on this sample after 6 h of dissolution. DISCUSSION First, it should be emphasized that the increase in rate of calcite dissolution observed in these experiments cannot be explained by an increase in the activity of bulk calcite: Even

where KT is the transport constant, a~+(B) is the H ÷ activity in the bulk fluid and all+(0) is the H ÷ activity at the bottom of the hydrodynamic boundary layer. For a transport-controlled reaction, aH÷<0)will represent the equilibrium activity for solid calcite and the carbonate species in the adsorbed surface layer. Our observation of increased dissolution rates even at low pH imply that aH÷(o)(strained) < aH+(o)(unstrained). This effect would be related to the increased number of reactive surface sites available for adsorption on the strained crystal. Second, our results could support the existence of a critical dislocation density, as suggested by BLUM and LASAGA (1987), above which a crystal shows a rate dependence on r o t o t l n g spindte

a)

of sample d :nc~°;~rr: . . . . . . . . .

Teflon

coverin9

sur~oce) Type ! Sample __

b)

-----

,otatinq ~pindle

t eflon sample hokter

~ombohedron

----

calcite

- -

epoxy ~ o u n t

Type 2 Sample

FIG. 2. The two types of calcite sample mount used in this study.

377

Dissolution kinetics o f strained calcite Table 3.

Measured dissolution rates (k+, moles em -2 sec -1 .101°) on

non-strained and strained calcite at different pHs, temperatures and COs pressures. Type I Samples T - 25"C

pH

4,1

5,

non-strained 7.3 2.s strained

7.8

3.7

8.3

3.5

15.6,1 .6,16.4,16.6"lT.oblT.4'lT.4blT.9" 1.15 l.,9 i.o5 l.o5 1.15 1,9

-

2.04

1.75

1.4

(IO T cm-S) strained (5.10 0 cm-S)

strained

2.1

4.15

2.5

2.2

-

1.8 2,3

1.72 2.3

(10 9 cm-S)

Type I Samples T = YC

Type I Samples T = 80°C

i|Hm

p.

42. I5l.

17.4,

pH

4a I 5" 15.6b16.6blT.4b

non-strained

10.5

4.4

2.6

2.6

non-strained 6.3

1.1

strained (5.10s cm-S)

12.4

5.7

3.5

3.6

strained (5' I 0 s cm -s)

1.35 0.92 0,78 0.82

6.0

0.7

0.45 0.48

pCOs= lO-Satm b: pCOs= lO-Satm

a:

dislocation density. Below this density, the rate of dissolution appears to be independent of dislocation density. In their independent theoretical study, BLUM and LASAGA(1987, p. 285) predi~ed that the critical dislocation density for quartz is on the order of 109 cm-2; this is quite pertinent t o the present study because the parameters required by Eqn. (31 for both calcite (MOTAHASHI e/ el., 1976) and quartz (WINTSCH and DUNNING, 1985 ) are very sjmilCr. The cri'tical dislocation density can also be estimated from Eqns. (6a, b ) and the expression for the core energy in Eqn. ( 31 by a simpler approach, as shown below.

Using Eqn. (6b), we can write an exlm~sion for the critic~d dislocation density, pc,t, at which the dissolution rates of strained and unstrained crystal become equal (equating the two fight-hand terms of Eqn. [6b]): ,o,:nt = k,,/~, kd.

(7)

9.0

-9,0

0 streln.d (107} x strained (10 0 ) strained (5x10 O) normal 2 5 eC

9.5 t

'5 o

~ -9.5

I0.0

n~

--

e •

-I0.0 I 4

I pH

I 6

I

r

I 8

FiG. 3. Rate of dissolution (log rate ~nsUmt ) of calcite as a fon~on of dislo~tion density and pH at 25°C. Note that at lower pH values

where dissolution rate is transport-controlled, the effect of defect density is much less significant.

10.5 4

G

pH

8

I0

FIG. 4. Comparison of the rates of dissolution (log rate constant) of normal (non-grained) calcite and calcite strained to a dislocation density of 0.5. 1 0 9 - 1 . 0 • 109 cm -2 as a function ofpH at 3, 25 and 80°C.

378

J. Schott et al.

FIG. 5. Surface of strained calcite (dislocation density ~ 107 c m - 2 ) after dissolution for six hours at 25"C. Surface was polished to 0.3 ~m before experiment. Scale bar = 10 urn.

dislocation density we predict that dissolution should show a measurable rate increase, which is in reasonable agreement with our experimental data (Fig. 3). Note, however, that the predicted value should be considered only a rough indication because the terms of Eqn. (9b) are not well-defined. For example, if instead of r0 = b, we assume as for quartz, ro = 1.25b (BLUM and LASAGA, 1987), then Pc,t -~ 3 . 1 0 9 c m -2. Using the above approximate approach and the parameters for quartz given by BLUM and LASAGA(1987) (b = 5.2 A, T = 4.84. 10 l° Pa, ro = 1.25b) yields Pcnt "~ 2.2- 109 em -2. This agrees well with the Monte Carlo prediction of the same quantity for quartz by BLUM and LASAGA (1987) (Pcnt 8.109 cm-2). We can also use Eqns. (6a, b) to predict rates of dissolution for strained and unstrained crystals (p ,f pc~t), in order to predict the normalized rate of dissolution of a high dislocation density crystal:

From Eqn. (2), we see that the ratio of/% to kd can be expressed as: /%/kd = e x p ( - A G ~ + A G ' d ) / k T .

(8)

Assuming that the transition-state complex of the dissolution reaction is the same whether it occurs at a dislocation or a perfect surface site, then the argument in Eqn. (8) is exactly equal to the added strain energy, u, associated with one dislocation, and we can insert into Eqn. (7): 1 pcnt= ~ e x p ( - u / k T ) .

(9a)

Approximating u by the core strain energy, we can write: 1 /-a~'b2V / \ P . t = ~d e x p t - - - - - ~ o N / k T )

g

~

.

~ 5A

---.t------4-- . . . . .

------

I

I

I

I

|0 s

107

108

109

DISLOCATION

DENSITY

= a=r[1 + pSd e x p ( u / k T ) ]

runstr

= astrIl -t- P]Pcrit "

(10b)

From Eqns. (9a) and (10b) we can write the following proportionalities: rstr ~ b 2e x p /| T- -V~\ |

(11)

runstr

and Pcrit OC . 2

I [TV\

(12)

"

o exp t--~- )

I 1010

1011

[cm-2J

FIG. 6. Calcite dissolution rate as a function of dislocation density at 25"C for pH 7.4 (+ = experimental results; solid line = calculated rate from Eqns. (6) to (10) due to defects; dashed line = calculated rate from increase in activity of bulk calcite [Eqns. (!), (3), (4) and

(5)1.

r~r

The inverse proportionality between rstr/rnnstr and T, for example, is apparent in the comparison of strained and normal dissolution rates at different temperatures in Fig. 4. Equation (10b) expresses the two effects which we have said are important in enhancing dissolution of high p crystals: the enhancement of activity ( the first term), and the increased number of active sites (second term). With high values of p, the second term will outweigh the first term, as previously discussed. We have calculated the relative i n ~ in calcite dissolution rate for different defect densitiesbased on Eqn. (10b) (see Table 4). These calculated values are compared with our measured rates in Fig. 6. In the absence of experi-

x

~

(lOa)

aunstr kp

where au.,tr = 1 by definition. This equation can be rearranged:

(9b)

where Nis Avogadro's number and V / N r e p r e s e n t s an average molecular volume. Inserting into Eqn. (9b), using ro = b, and the values for calcite parameters summarized here in Table 1 ( from MOTOHASHieta/., 1976), we calculate a critical dislocation density of pent = 2.107 cm -2. At this value of

,~

rst~ = astr(kp + pSdkd) runstr

Table 4. Calculated rate o f calcite dissolution at 25"C and p H 7.4 (normalized to unstrainlgl calcite) as a function of dislocation density, based on Eqn. (10b) in text.

Dislocation density

IO e

IO T

5,10 7

108

l0 g 5.10 s 10 TM

1011

(cm'=) i a) r o = 4A [1.05

1.5

3.5

b) r o = 5A 1.00

1.003

1.015

6

51

26

1,03 1.32 2.61

500 5000 4.2

33

Dissolution kinetics of strained calcite mental results for dislocation densities above 109 cm -2, Fig,. 6 shows a reasonably good agreement between measured and predicted rates (with ro = 5 A). However, it is likely that for p > pent, dependence of rate on dislocation density is not linear, as predicted by Eqn. (10b). In all cases in Fig. 6, rate increases nonlinearly with increasing p. We postulate that r will not increase linearly with p at high densities due to the overlap of dislocation strain fields on the surface (i.e., Sd decreases as p increases). Indeed, dislocations commonly occur in bundles or tangles and are not distributed uniformly throughout individual crystals (e.g., HULL and BACON, 1984, p. 175; BARBER, 1985, p. 167). We note also from Fig. 5 that etch pit area varies drastically from pit to pit, and that overlap of pits (and by implication, dislocation strain fields) is ubiquitous.

Dissolution behavior as etch pits widen It should be emphasized that the above calculations apply only at the beginning of dissolution when the radius of the hollow core still approximately equals that of the original dislocation core. Simple calculations show that it takes less than a few minutes for the radius of the dislocation core to increase, for example, from 5 tO 10 A. This decreases surface strain energy per unit area and should therefore decrease dissolution rate. However, deep hollow cores are produced at dislocations because of the extension of the defect into the crystal (in other words, the rate of dissolution parallel to the surface is much smaller than that normal to the surface); this increases total surface area and should, therefore, increase dissolution rate. After a short transition period, these two effects tend to partially compensate each other. Ultimately, a steady state is reached where the rate of dissolution remains roughly constant. For this steady-state condition, the calculated critical dislocation density, P,=t, at which dissolution rates of strained and unstrained crystals become equal is quite close to that calclJlated for initial dissolution as well as that during the intermediate transient period. Our calculations indicate that during this steady-state period the overall dissolution rate should increase by only a factor of three or four at a maximum, due to an imbalance between decreasing surface strain energy and increasing surface area as dislocation cores dissolve and widen. Higher dissolution rates are possible during the initial transient period, but this period is sufficiently short that such changes are difficult to observe during dissolution experiments. Theoretical calculations demonstrating the above effects are outlined in the accompanying appendix.

Comparison with other minerals There is relatively little information currently available on the dissolution rates of strained materials. BOSWORTH(1981) observed a correlation between net dissolution rates and plastic strain of crystalline halite, but did not report dislocation densities or specific rates. Several studies have been published on the dissolutiOn of shocked materials. In the most recent of these, CASEYeta L (1988) measured the dissolution kinetics of highly shocked rutile in hydrofluoric acid at 25 °C. Sample material varied from unshocked, low dislocation densities (106 crn -2) to shocked, high dislocation densities

379

( > 4 . 1 0 ~ crn-2), while measured dissolution rates for the material differed by onlya factor of two. Applying Eqns. (6) to (9) to futile (with b = 2.7 A, r0 = 3.4 A and y = 7. i0 I° Pa) predicts a critical dislocation density p~t "~ 10 l° cm -2 for rutile, which falls nicely within the range of Casey's observations. MURR and HISKEY (1981) describe leaching experiments with three sized fractions of shocked chalcopyfite in acid-dichromate solution at 50 and 70°C. An unambiguous effect of dislocations was observed at the lowest size fraction at 50°C: the rate constant increased by about a factor of two for dislocation densities increasing from roughly 107 cm -2 to 10 ~ cm -2. BOSLOUGHand CYGAN (1988) report preliminary data for dissolution of shocked feldspar and hornblende. Specific-area normalized rates of dissolution increased up to seven-fold for materials shocked at 22 GPa. No dislocation densities were reported. Data from all of these minerals corroborate the low rate enhancement associated with increased defect density of natural minerals. In all cases, however, rate enhancements are greater than predicted from increased crystal activity alone.

Why are rates and defect densities only weakly related?. It may seem counterintuitive that dissolution rates increase only slightly with orders of magnitude increases in dislocation density. Crystals dissolve preferentially at active sites such as defects; yet the dissolution rates of minerals studied to date are, within a factor often or less, independent of the number of defects in macroscopic crystals. Only at unusually high defect densities is there an observable rate enhancement, and even in these cases the effect is small and nonlinear. This apparent paradox can be rationalized by considering the different physical processes involved in crystal di.~solution. Crystals dissolve at microscopic sites of excess energy ranging from point defects to linear dislocations, kinks, edges and ledges. Dissolution proceeds simultaneously at all such sites. Each of these parallel processes contributes dissolved material to the aqueous solution, but at different rates and in different quantities. This is illustrated schematically in Fig. 1. Dissolution rates are normally measured by analyzing the change in solution concentration with time. Those processes which deliver the greatest quantity of dissolved material will therefore control the measured rate. Referring to Fig. 1, point defects and dislocations might be expected to dissolve most rapidly, but will deliver small quantities of material to solution because they operate at atomic or sub-microscopic scales. However, the continued dissolution of line defects produces etch pits with edges and ledges. The bulk of the crystalline material between and within etch pits is dissolved by the slow retreat of these ledges. Such dissolution processes may be slower, but are in sum rate-limiting or rate-defining, because they deliver much more material to solution. In such cases the rate should depend on total exposed surface area in the conventional manner. The quantity of material delivered by microscopic defects becomes comparable only at unusually high defect densities (>107-10 cm -2 for calcite and >109 cm -2 for quartz). Under such high-strain conditions, dissolution rates will begin to vary with defect density, but will not vary linearly since defects tend to occur in tangles and produce overlapping etch pits.

380

J. Schott et al.

Another possible explanation for the small effect of dislocations on dissolution is that dissolution rate enhancement might be related to increased solid activity around dislocations and not to an increase in the number of activated sites. In fact, in contrast to one's intuitive expectations, the number of active sites per unit surface area could only be weakly related to dislocation density, and could also be roughly the same in undeformed and deformed samples. This is because the dislocation densities in the experimenB reported here or in the literature (# ~ 10 Hem -2) are negligible compared to the total number of active surface sites on minerals, which is typically in the range 1-10.1014 c m -2 (KUMMERT and STUMM, 1980). A rigorous test of this hypothesis must await analysis by surface titration techniques to determine chemical speciation on both strained and unstrained crystals. CONCLUSIONS The rate of dissolution of macroscopic crystals of calcite increases slightly with increasing dislocation density. Similar behavior has been observed with shocked ruffle, c~eopyrite, feldspar, and hornblende. Observed rate enhancements for calcite are on the order of two to three times for dislocation densities varying from 106 to 109 cn1-2 . Rate enhancement increases at higher pH and lower temperature. This suggests that it should be possible to measure meaningful and generally appfieable kinetic properties of most natural, undeformed crystals, resolving the problem raised in the introduction to this paper. On the other hand, kinetic measurements on highly strained crystals must include a determination of defect density. However, it is important to note that for minerals like olivine and Kfeldspar, which exhibit high values of the parameters rb 2 and V in Eqns. ( 3 ), (11) and (12), the calculated critical defect density pc,t is roughly 102 cm -2 (SCHOTT et al., 1988). Therefore, the rate ofdissolution of these particular minerals could very well be controlled by dislocations even when normally crystalli,ed. This might explain the conflicting results for the dissolution rates of olivines and feldspars measured by different authors (e.g., MURPHY, 1985; HOLDREN and SPEYER, 1985). Our results fit, but do not prove, the postulated critical dislocation density for calcite of 2.107 c m -2. We propose that, below this P,~t, the parallel processes of dissolution at edges and ledges, while relatively slow, will produce considerably more dissolved material than rapid dissolution of atomic-scale defects and dislocations. Although observed rate enhancements are small, the observed effect of dislocation density could cause solution transfer of material within a polygranular aggregate of deformed mineral grains. Indeed, such a mechanism has been proposed to explain the replacement of strained quartz grains by carbonates in several Pennsylvania sandstones (SIEVES, 1959); more recently, the same process has been used to explain selective dissolution of calcite crinoid eolumnals in rocks of the Appalachian Plateau (ENGELDER, 1982). Straindriven solution transfer, while distinct from stress-driven solution transfer (pressure solution) could be associated with pressure solution features. For example, within the most

highly strained regions of folds, stylolites and planar regions of dissolution are commonly oriented perpendicular to the original direction of greatest principal stress (susgesting selective dissolution); in the same systems, mineralized veins typically run perpendicular to the least principal stress (e.g., RAMSAYand HUBER, 1983, p. 123). The dislocations in deformed minerals are often not uniformly distributed, but instead tend to occur in bundles or tangles (see discussion above). Such areas would be cente~ for enhanced dissolution in a water-infiltrated rock. Further investigation ofdissolution and precipitationrotesof deformed and undeformed minerals under fluid flow and no-flow conditions would enable us to accurately model solution transfer in deformed zones, as driven by local defect concentrations. The observed enhancement of dissolution rates cannot be explained solely by the increase in chemical activity of the bulk starting material, but rather by the increase in the number of active dissolution sites at high defect densities. We have attributed the nonlinear dependence of calcite dissolution rate on dislocation density to the increasing overlap of dislocation strain fields and the "bundling" of dislocations at higher defect densities. More quantitative treatment of etch pit density, size, and depth correlated with dissolution rates would test this hypothesis. In addition, experiments aimed at measuring dissolution rates under transport-limited conditions could test the relative importance of the two hypothesized effects of dislocation density on dissolution rate: the increased driving force associated with the enhanced chemical activity, and the kinetic effect of an increased number of active surface sites. Under the transport-limited regime, only the first effect could be observable. Similarly, experiments which test near-equih'brium conditions would be of interest in order to more closely simulate natural regimes where siowmoving water infiltrates tight cracks and pore spaces in rocks. Acknowledgements--This reseal~h was supported by a C.N.R.S. grant (ATP C_~echimie) to Schott; NSF grant EAR-8657868, Petroleum Research Fund grant 19455-G2, and a Research Corporation grant to Brantley; and NSF grants EAR-8218726 and EAR-8407651 to Crerar. Brian Evans acknowledges the support of NSF grant EAR8419152. D.A.C. also gratefullyacknowledgessupport from C.N.R.S., France, and the Shell Companies Foundation throughout the course of this work. The manuscript benefitted from reviewsby John Morse, Eurybiades Busenberg and an anonymous reviewer. Editorial handling: E. J. Reardon REFERENCES AAGARDP. and HELGESONH. C. (1982) Thermodynamic and kinetic constraints on reaction among minerals and aqueous solutions: I. Theoretical considerations. Amer. J, Sci. 282, 237-285. BARBER D. J. (1985 ) Dislocations and microstructures. In Preferred Orientation in Deformed Metals and Rocks: An Introduction to Modem Texture Analysis (ed. H.-R. WENK), pp. 149-182. Academic Press. BERNER R. A. (1978) Rate control of mineral dissolution under Earth surface conditions. Amer. J. Sei. 278, 1235-1252. BERNER R. A., SJOBERG E. L., VELBEL M. A. and KROM M. D. ( 1980 ) Dissolution ofpyroxenes and ~rnphiboles during weatherin~ Science 207, 1205-1206. BIANCHETT1 S. F. and I~EDER R. J. (1985) Variable dissolution rates of deformed and undeformcd calcite. Geol. Soc. Amer., Ann. Mtg. Abstracts, 522.

Dissolution kinetics of strained calcite BLUM A. and LASAGAA. (1987 ) Monte Carlo simulations of surface reaction rate laws. In Aquatic Surface Chemistry: Chemical Processes at the Particle- Water Interface ( ed. W. STUMM), pp. 255292, and J. Wiley & Sons. BOSLOUOH M. B. and CYGAN R. T. (1988) Shock-enhanced dissolution of silicate minerals and chemical weathering on planetary surfaces. Proc. Lunar Planet. Sci. Conf. 18th, Cambridge Univ. Press, pp. 443-453. BOSWORTHW. ( 1981) Strain induced preferential dissolution ofhafite. Tectonophysics 78, 509-525. BRANTLEY S. L., CRANE S. R., CRERAR D. A., HELLMANN R. and STALLARD R. (1986) Dissolution at dislocation etch pits in quartz. Geochim. Cosmochim. Acta 50, 2349-2361. BUSENBERG E. and PLUMMER L. N. (1986) A comparative study of the dissolution and crystal growth kinetics of calcite and aragonite. In Studies in Diagenesis (ed. F. A. MUMPTON), pp. 139-168. U.S. Geol. Surv. Bull. 1578. CASEY W., CARR M. J. and GRAHAM R. A. (1988) Crystal defects and the dissolution kinetics of futile. Geochim. Cosmochim. Acta 52, 1545-1556. COMPTON R. G. and DALY P. J. (1984) The dissolution kinetics of Iceland spar single crystals. J. Colloid Interface Sci. 101, 159-166. COMPTON R. B., DALY P. J. and HOUSE W. (1987) The dissolution of Iceland spar crystals: The effect of surface morphology. J. Colloid Interface Sci. 113, 12-20. ENGELDER T. (1982) A natural example of the simultaneous operation of free-face dissolution and pressure solution. Geochim. Cosmochim. Acta 46, 69-74. HELGESON H. C., MURPHY W. M. and AAGARD P. (1984) Thermodynamic and kinetic constraints on reaction rates among minerals and aqueous solutions. Rate constants, effective surface area, and the hydrolysis of feldspar. Geochim. Cosmochim. Acta 48, 2405-2432. HIRTH J. P. and LOTHE J. (1982) Theory of Dislocations. J. Wiley & Sons, New York. HOLDREN G. R. JR. and SPEYER P. M. (1985) Reaction rate-surface area relationships during the early stages of weathering: I. Initial observations. Geochim. Cosmochim. Acta 49, 675-68 I. HOLDREN G. R. JR. and SPEYER P. i . (1987) Reaction rate-surface area relationships during the early stages of weathering: II. Data on eight additional feldspars. Geochim. Cosmochim. Acta 51, 23112318. HULL D. (1975 )Introduction to Dislocations. Pergamon Press, London. HULL D. and BACON D. J. (1984) Introduction to Dislocations, 3rd edn., Pergamon Press, London, 255p. KUMMERT R. and STUMM W. (1980) The surface complexation of organic acid on hydrous ~t-A1203. J. Colloid Interface Sci. 75, 373385. LASAGA A. (1981) Rate laws of chemical reactions. In Kinetics of Geochemical Processes, (eds. A. LASAGAand R. J. KIRKPATRICK), Reviews in Mineralogy, Vol. 8, pp. 1-110. Mineral Soc. Amer. LASAGAA. (1983) Kinetics of minerai dissolution. In Fourth Internl. Sympos. Water-Rock Interactions, pp. 269-274. Misasa, Japan. MOTOHASHI Y., BRAILLON P. and SERUGHETTI J. (1976) Elastic energy, stress field and dislocation parameters in calcite crystals. Phys. Stat. Sol. 37, 263-270. MURR L. E. and HISKEY J. B. (1981) Kinetic effects of particle size and crystal dislocation density on the dichromate leaching ofchaiocopyrite. Metall. Trans. B 1215, 255-267. MURPHY B. ( 1985 ) Thermodynamics and kinetic constraints on reaction rates among minerals and aqueous solutions. Ph.D. dissertation, Univ. Caiifornia, Berkeley, 160p. PETROVIC R. ( 1981a) Kinetics of dissolution of mechanically cornminuted rock-forming oxides and silicates: II. Deformation and dissolution of quartz under laboratory conditions. Geochim. Cosmochim. Acta 45, 1665-1674. PETROVIC R. ( 1981b) Kinetics of dissolution of mechanically cornminuted rock-forming oxides and silicates in the laboratory and at the Earth's surface. Geochim. Cosmochim. Acta 45, 1675-1686. PLUMMER L. N., WIGLEY T. M. L. and PARKHURST D. L. (1978)

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The kinetics of calcite dissolution in CO2-water systems at 5 ° to 60°C and 0.0 to 1.0 arm CO2. Amer. J. Sci. 278, 179-216. RAMSAY J. G. and HUrtER M. I. (1983) The Techniques of Modern Structural Geology. Vol. 1: Strain Analysis. Academic Press, 307p. SCHOTT J., GuY C. and CRERAR, D. A. (1988) Does the Burgers vector control the dissolution of minerals? (abstr.). Goidschmidt Conf., May 1988, Baltimore, 72. SIEVER R. (1959) Petrology and geochemistry of silica cementation in some Pennsyivanian sandstones. SEPAl Spec. Publ. 7 ( ed. H. A. IRELAND), 55-79. SJOBERG L. and 17dCKARDD. T. (1984) Temperature dependence of calcite dissolution kinetics between 1 and 62°C at pH 2.7 to 8.4 in aqueous solutions. Geochim. Cosmochim. Acta 48, 485493. W]NTSCH R. P. and DUNNING J. (1985) The effect of dislocation density on the aqueous solubility of quartz and some geologic implications: A theoretical approach. J. Geophys. Res. 90, 36493657.

APPENDIX Dissolution rates as a function of increasing dislocation core diameter For dissolution occurring at a deep hollow core centered around a dislocation (Fig. A-l), Eqn. (6a) can be rewritten as: rt =/%Sto, + keSd

(A-I)

where rt is the total dissolution rate,/% = dissolution rate constant per unit area of dislocation-free surface, ke = dissolution rate constant per unit area of dislocated surface, S ~ is the total perfect surface area, and Sd is the area of dislocated surface. When considering a cylindrical hollow core (Fig. A-I), one should consider the contributions to dissolution of the base (b) and the inner wall (w). To accomplish this, Eqn. (A-l) can be written as: rt = k p S , o,

+ kd.bSd.b"3t-kd,wSd~w.

(A-2)

Inserting appropriate expressions for kd.b, kd.., S,Lb and Sd.. (see the main text), yields the following expression for the overall dissolution rate: rt = / % S ~ + 2*rp/%Stot

e x p ( U I R T ) r , dr

+ 2,rr2HpSt~/% e x p ( U / R T )

(A-3)

where p is the dislocation density, r0 = 1.25b = 5 A is the radius of the core, r is the radius of the hole (partially dissolved core), H . r is the depth of the hole, and the total strain energy Uis given by Eqns. (3) and (4). Numerical solution of Eqn. (A-3) shows that the value ofthe second term (the integral) depends strongly on the value chosen for the core

mineral surface

FIG. A-1. Sketch illustrating the opening of a hollow core centered about a dislocation.

382

J. Schott et al.

radius, which is frequently disputed. With the value selected in this study (r0 = 5 A), the second term remains constant for • > 8 A, and is equal to 0.7kpS~. With ro = 4.5 A, the cort~ponding values are 8 A and 5kpStot. In any case, this term can be neglected after an initial transient period as the hollow core deepens. Equation (A-3) then reduces to:

Table A - I . Calculated rate of calcite dissolution at 25"C (normalized to unstrained calcite) as a function o f radius r o f the hollow core, based on Eqn. (A-4), for p = 10 TM cm-S. activity at the

SjStot

r

surface of the

(A-4)

(A)

hollow core

Under steady-state conditions, Eqn. (A-4) can be used to predict the critical dislocation density, p~t, at which the dissolution rates of strained and unstrained crystals become equal. Using r = 40 A, H = 200, and the values for calcite parameters summarized in Table 1, we calculate a critical dislocation density pent = 5.109 cm -2 which is the same as that calculated at the beginning of dissolution (before enlargement of the dissolved core). Table A-1 lists calculated rates of calcite dissolution (normalized to unstrained calcite) during the opening of hollow cores based on Eqn. (A-3) for p = 10 to cm-2 and for H = 50 and 200. It can be seen that during the opening of a hollow core from 15 to 60 A (which takes about two hours), the rate of dissolution of strained calcite remains roughly constant and is increased only by a factor of two to four relative to unstrained crystals. The rates calculated here agree well with those given in Table 4 for the beginning of dis-

15

9

0.07

2,3

20

4,6

0.13

2.3

40

1.55

0.5

2.5

60

l.l

1.1

2.9

15

9

0.28

3.5

20

4.6

0.5

3,3

40

1.55

2.0

4,1

60

l.l

4

5

rt = IcpStot + 2*rr2HpStotkp e x p ( U / R T ) .

H

H=

rate of dissolution

- 50

200

solution (before core enlargement). We do not expect these rates to increase notably at higher p because o f the overlap of dislocation strain fields at the surface o f highly strained crystals (p > I0 t i cm-2).