Kinetic of Calcite Growth - De Yoreo

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PII S0016-7037(00)00341-0

Kinetics of calcite growth: Surface processes and relationships to macroscopic rate laws

H. HENRY TENG,1, PATRICIA M. DOVE,1,* and JAMES J. DE YOREO21School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, GA 30332, USA

2Department of Chemistry and Materials Science, Lawrence Livermore National Laboratory, Livermore, CA 94550, USA

(Received June 18, 1999; accepted in revised form January 18, 2000)

AbstractThis study links classical crystal growth theory with observations of microscopic surface processesto quantify the dependence of calcite growth on supersaturation, , and show relationships to the samedependencies often approximated by affinity based expressions. In situ Atomic Force Microscopy was usedto quantify calcite growth rates and observe transitions in growth processes on {1014} faces in characterizedsolutions with variable . When 0.8, growth occurs by step flow at surface defects, including screwdislocations. As exceeds 0.8, two-dimensional surface nucleation becomes increasingly important. Thesingle sourced, single spirals that are produced at lower were examined to measure rates of step flow andthe slopes of growth hillocks. These data were used to obtain the surface-normal growth rate, Rm, by the purespiral mechanism.

The dependence of overall growth rate upon dislocation source structure was analyzed using the funda-mentals of crystal growth theory. The resulting surface process-based rate expressions for spiral growth showthe relationships between Rm and the distribution and structures of dislocation sources. These theoreticalrelations are upheld by the process-based experimental rate data reported in this study. The analysis furthershows that the dependence of growth rate on dislocation source structures is essential for properly representinggrowth. This is because most growth sources exhibit complex structures with multiple dislocations. Theexpressions resulting from this analysis were compared to affinity-based rate equations to show where popularaffinity-based rate laws hold or break down.

Results of this study demonstrate that the widely used second order chemical affinity-based rate laws arephysically meaningful only under special conditions. The exponent in affinity-based expressions is dependentupon the supersaturation range used to fit data. An apparent second order dependence is achieved whensolution supersaturations are very near equilibrium and growth occurs only by simple, single sourceddislocation spirals. These findings indicate the need to apply caution when deducing growth mechanisms andrate laws from temporal changes in bulk solution chemistry. Observations of various types of surface defectsthat give rise to step formation suggest that popular rate laws are sample-dependent composites of rate

contributions from each dislocation structure. Copyright 2000 Elsevier Science Ltd

1. INTRODUCTION

The abundance of calcium carbonate minerals throughoutnatural and engineered earth systems has motivated investiga-tions of calcite crystallization over the last century. It is widelyrecognized that an understanding of the kinetics and mecha-nisms governing growth is of first order importance for pre-dicting mineralization and thus, acquiring the ability to controlit. The significance of harnessing mineralization phenomena isseen in the remarkable ability of organisms to direct the crys-tallization of aqueous ions into biogenic materials that expressa diverse morphologies and biological functions. Renewed

efforts to unravel the physical basis of carbonate biomineral-ization are largely focused on understanding the in vivo pro-cesses that control growth at the interface between a biologicalmatrix and the biominerals that form. Future advances in con-trolling or directing the growth of carbonate minerals alsohinge upon clarifying two uncertainties: (1) The dependence ofgrowth mechanism upon supersaturation; (2) The microscopic

surface processes that control the macroscopic manifestationsof overall growth rates. Answers to these questions establishthe knowledge base for constructing general models that quan-tify growth in complex mixtures of organic and inorganicconstituents. They also establish relationships between micro-scopic processes and macroscopic growth rates determined bywhat are known as bulk methods throughout the scientificliterature. However, a review of the findings published to datesuggests that these issues have not been adequately explored.

Growth studies conducted over a wide range of supersatura-tions have established our collective understanding of growthkinetics and mechanisms for calcite crystallization. Their find-ings can be summarized according to three experimental meth-ods with different length-scales. At a very short scale, directobservations of growth using Atomic Force Microscopy (AFM)confirm that calcite growth occurs by step flow, the advance-ment of individual molecular layers generated at crystal imper-fections (Hillner et al., 1992a,b; Gratz et al., 1993) or bytwo-dimensional surface nucleation (Dove and Hochella,1993). However, these studies report contradictory relation-ships between growth mechanisms and supersaturation. Forexample, Gratz et al. (1993) reports that growth by spiralformation is the only growth mechanism at supersaturations ashigh as 6.9 where the supersaturation index is definedas the ratio of ion activity product (IAP) to solubility product

* Author to whom correspondence should be addressed: P. M. Dove,Department of Geological Sciences, Virginia Polytechnic Institute andState University, Blacksburg, VA 24061. ([email protected]). Present address: Argonne National Laboratory, ER-203, Argonne, IL60439.

Pergamon

Geochimica et Cosmochimica Acta, Vol. 64, No. 13, pp. 22552266, 2000Copyright 2000 Elsevier Science LtdPrinted in the USA. All rights reserved

0016-7037/00 $20.00 .00

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(Ksp). In contrast, Dove and Hochella (1993) report that sur-face nucleation becomes the dominant growth mechanism atconditions much closer to equilibrium ( 12).

Studies investigating growth at larger scales have examinedthe products of step assembly to form polygonized hillockfeatures. By using growth solutions with relatively high super-saturations ( 1024), Paquette and Reeder (1990, 1995),

Staudt et al. (1994), and Reeder (1996) document the formationof growth hillocks on {1014} calcite faces with diameters of upto hundreds of microns. The local supersaturations in thesestudies could not be well characterized due to static hydrody-namic conditions of the growth media. Nonetheless, examina-tions of these mesoscale growth structures after removal fromthe growth media led the investigators to conclude that growthoccurred predominantly by a spiral mechanism.

At the largest scale, macroscopic studies have evaluated therates and mechanisms of calcite crystallization using indirectmethods that monitored changes in solution chemistry over thecourse of growth. These studies establish the dependence ofgrowth kinetics upon chemical and physical parameters such as

supersaturation state, pH, pCO2, ionic strength, and temperature(e.g., Nancollas and Reddy, 1971; Plummer et al., 1978; Busen-berg and Plummer, 1986; Christoffersen and Christoffersen,1990; Shiraki and Brantley, 1995). Growth mechanisms werepostulated in these investigations based upon the rate of solu-tion composition change with time. These bulk kinetic mea-surements could not discern the possibly different contributionsof unique crystal faces to the overall growth rate or relatesolution chemistry to the range of microscopic growth pro-cesses occurring at mineral surfaces.

Three major categories of elaborate kinetic models for crys-tallization have arisen from interpretations of solution compo-sition data obtained by these indirect studies based upon (1)

surface complexation (Nilsson and Sternbeck, 1999; Arakakiand Mucci, 1995); (2) summation of the elementary reactions(Plummer et al., 1978; Busenberg and Plummer, 1986); and (3)chemical affinity (Smallwood, 1977; Reddy, 1977; Reddy andGaillard, 1980; Compton and Daly, 1987; Reddy, 1988). Sur-face complexation-based rate laws take into account the reac-tions involving surface speciation, while elementary reaction-based rate laws describe growth rate as a function of multipleelementary reactions. Affinity-based models, the most widelyused of the three, are developed in terms of free energychanges, G (chemical affinity) or exp(G/RT) (supersatura-tion index, ), of precipitation reactions (e.g., Nancollas andReddy, 1971; Reddy and Nancollas, 1971; Morse, 1978;House, 1981; Nielsen, 1983; Mucci and Morse, 1983; Mucci,1986) to yield two types of rate laws: linear and nonlinear withrespect to G. Linear rate laws have the following generalform:

Rm kexpnG/RT 1 (1a)

where the net rate, Rm, (moles area2 time1), is characterized

by the rate constant of the forward reaction, k

, (moles area2

time1) and the free energy change of the overall reaction. Rand T are the molar gas constant and temperature (K), respec-tively. The parameter n is a constant and has been assumed tocontain information about the growth mechanism. For example,the kinetic behavior described by Eqn. 1a with n 1 has been

attributed to adsorption-controlled growth (Nielsen, 1983). Ex-pressions of this form with n 1/2 and 1 were used todescribe the precipitation kinetics of calcite (Nancollas andReddy, 1971; Kazmierczak et al., 1982).

Nonlinear rate laws are generally expressed as (Lasaga,1981):

Rm kexp G/RT 1n (1b)

Theoretical models have been used to argue that second orderequations of this form (n 2) describe growth at screwdislocations by the spiral mechanism while higher order ones(n 2 3) can be applied to growth at both screw and edgedislocations (Blum and Lasaga, 1987). The kinetics of calciteprecipitation has been fitted to second order expressions atambient (Reddy and Nancollas, 1973; House, 1981) and ele-vated temperatures (Shiraki and Brantley, 1995). Others haveused an expression that combines Eqns. 1a and 1b to describecalcite growth (Inskeep and Bloom, 1985).

Together, these models delineate the currently accepted

quantitative representations of calcite crystallization across arange of chemical compositions and conditions. However,these rate laws stand without substantiative confirmation of theactual growth processes that occur at mineral surfaces. This isa significant deficiency that constrains current models to pro-vide only empirical and semi-quantitative representations ofcalcite precipitation kinetics. A closer analysis raises concernsabout the relevance of these general rate laws to calcitegrowth. For example, Eqn. 1b with n 2 was used in theo-retical (Blum and Lasaga, 1987; Lasaga, 1998) and experimen-tal (Shiraki and Brantley, 1995) studies to describe spiralgrowth without considering the effect of various types of dis-location sources, including the strain factors associated with

these defects, on growth rate. Yet, the classic crystal growththeories predict that the growth rate of a spiral must be stronglycontrolled by the structure of dislocation sources (Burton et al.,1951). Experimental observations have upheld these predic-tions (Rashkovich, 1991; Vekilov and Kuznetsov, 1992; Veki-lov and Rosenberger, 1996; Land et al., 1997; De Yoreo et al.,1997). In relating this fundamental aspect of crystal growththeory to accepted bulk rate expressions, an important issuearises: Chemical affinity-based rate laws are not a function ofdislocation source structures. The only measurable parameter inthese expressions is the quantity exp(G/RT). Hence, rate lawsrepresented by these expressions cannot reflect the controls ofmicroscopic growth parameters (step velocity and slope ofgrowth hillocks) on growth kinetics or reliably indicate theactual growth mechanism. Indeed, the broad range of overallgrowth rates that arise from the complexities of various dislo-cation sources naturally yield a quasi-parabolic behavior thathas little mechanistic significance.

The goal of this study is to explore these concerns bycomparing experimental evidence with the theoretical modelsto show where popular growth expressions are valid or breakdown. By using in situ AFM, this was done by (1) Determiningthe supersaturation range where growth occurs only by spiralformation; (2) Determining the critical supersaturation valuethat marks the activation of growth by a surface nucleationmechanism; (3) Measuring step flow rates and the correspond-ing slopes of growth hillocks in well-characterized solutions;

2256 H. H. Teng, P. M. Dove, and J. J. De Yoreo


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(4) Calculating the overall or normal growth rates of singlespirals using measurements of step speed and hillock slope; and(5) Analyzing the dependence of overall growth rate upondislocation source structure. We show that spiral growth can beapproximated by a second-order chemical affinity-based rate

law only when supersaturation is very low and growth on agiven crystal face is dominated by step flow from simplespirals. We also show that the kinetic data obtained in (4)represent only a minimum overall rate for pure spiral growth incalcite because growth generated by complex dislocationsources usually possesses a higher rate than that for singlesources.

2. EXPERIMENTAL PROCEDURES

2.1. Sample and Solution Preparation

Calcite was grown on {1014} faces of fragments cleaved from acrystal of optical-quality Iceland spar (Wards Scientific, Chihuahua,Mexico). The fragments (approximately 2 2 0.5 mm3) werehandled with tweezers to avoid surface contamination by skin oils andwere cleaned with bursts of N2(g) to remove small particles from thesurfaces. Each experiment began by mounting a freshly cleaved frag-ment onto a glass cover-slip using sealing wax. This cover-slip wassubsequently adhered to a magnetic steel puck using double-sidedadhesive tape.

Supersaturated CaCO3 solutions were prepared by dissolving reagentgrade sodium bicarbonate (NaHCO3, Aldrich) and calcium chloride(CaCl2 H2O, Aldrich) into deionized water. The ionic strength ofeach growth solution was fixed within a narrow range of 0.105 to 0.111M using reagent grade NaCl (Baker). The pH of all solutions wasadjusted to 8.50 using 0.5 N NaOH prior to injection into the Fluid Cellof AFM.

2.2. Solution Speciation and Supersaturation

The chemical speciation of each solution was determined using thenumerical code HYDRAQL (Papelis et al., 1988) assuming that theAFM fluid cell and input reservoir approximated a closed system.Aqueous complexes considered in the calculation were CaCO3,CaHCO3

, CaOH, NaCO3, NaHCO3, CO3

HCO3, and H2CO3. Be-

cause the fluid cell was isolated from the ambient atmosphere in aliquid-full reactor environment, we assumed that the amount ofCO2(g) in the cell was negligible for the aqueous speciation calculation.The activity ratio of Ca2 to CO3

2 was forced to equal 1.04 0.01 byadjusting the amount of NaHCO3 and CaCl2 H2O, and the Daviesequation (Davies, 1962) was used to correct for activity.

The supersaturation, , was calculated by:

kBT ln a

ae ln

aCa2aCO32

Ksp (2)

where is the chemical potential difference between CaCO3 mole-

cules in the aqueous and solid phases, kB is the Boltzmann constant, aiis the activity of the ith species and Ksp is the solubility product ofcalcite at zero ionic strength. The Teng et al. (1998) value of 108.54 forKsp at 25C was used to compute supersaturation. This value is lowerthan the most widely reported values ranging from 108.48 to 108.29

(Mucci, 1983, and the refs. therein; Stumm, 1992; Drever, 1997). Thislow value was used because we observed that steps on {1014} faces inthese experiments still advanced at measurable rates when the ionicactivity product (IAP) of the experimental solution was equal to thelowest reported Ksp. Hence, we determined Ksp from the IAP measure-ment where step velocity equaled zero. The 15% difference between108.54 and the lowest reported value of 108.48 cannot be accountedfor by the experimental errors of 2 to 4% reported in Table 1. Teng etal. (1998) suggests that the value of 108.54 is due to the lower pCO2of the closed experimental system or indicates a value that is specific to{1014} faces. This study presents results for 0.04 to 1.4 ( 1.044.06) using the solution compositions and activities of Ca 2 andCO3

2 given in Table 1.

2.3. Imaging by Fluid Contact AFM

In situ observations of calcite growth were made by Contact Modeusing a Nanoscope IIIa Scanning Probe Microscope (Digital Instru-ments) equipped with a piezoelectric scanner capable of scan areas toa maximum of 120 120 m. Surfaces were imaged using commer-cially available Si3N4 cantilevers that have triangular tips with a lengthof 200 m, and a force constant of approximately 0.38 Newton m1.Radii of lever tips were approximately 30 to 50 nm, which correspondsto a theoretical contact force of 2746 109 N between tip andsample (Eggleston, 1994). To reduce the possibility of artifactualchanges in micro-topography by scanning tip-surface interactions, thecontact force was carefully minimized. Scan rates ranged from 5.8 to29 Hz with 512 lines per scan, which corresponds to a capture time of88 to 18 s per image, respectively. Scanner drift was minimized byallowing the instrument to thermally equilibrate before imaging.

The experiments were conducted by first imaging the mineral sam-ples in air to locate a relatively flat area and to optimize image quality.The reactant solution was then input as a continuous flow through anAFM fluid cell with an internal volume of 50 L using a syringe pump.

Once in solution, the crystallographic orientation of an individualcalcite fragment was established using methods described previously(Teng and Dove, 1997; Stipp et al., 1994). All images were collectedusing flow-through rates greater than 30 ml/hr after preliminary exper-iments showed that step speed on calcite {1014} faces become inde-pendent of flow rate above 30 to 40 ml/hr. In situ measurements oftemperature in this flow-through environment were 25C.

2.4. Step Advancing Rate and Hillock Slope Measurements

Data were collected by first locating a single spiral hillock (e.g., Fig.1A) at each supersaturation. The spiral was usually allowed to grow forat least an hour in a supersaturated, but near-equilibrium, solution toensure its quality and stability. The input solution was then changed toa new one with the desired and maintained for another 10 to 30 min(depending on imaging size and growth rate) whereas the step flow rate

Table 1. Summary of salt concentrations used in each growth experiment and the corresponding solute activities and supersaturations.

Salt concentrations

[CaCl2] (104 M) 1.60 1.70 1.80 1.90 2.00 2.04 2.07 2.10 2.25 2.45 2.75 3.00 3.42

[NaHCO3] (103 M) 5.00 5.25 5.50 5.80 6.15 6.25 6.35 6.40 6.85 7.45 8.25 9.00 10.10

[NaCl] (M) 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10Calculated activities and supersaturationsa

aCa2 (105

) 5.59 5.91 6.23 6.57 6.88 6.99 7.10 7.17 7.68 8.29 9.21 9.95 11.18aCO32 (105) 5.37 5.62 5.95 6.20 6.59 6.70 6.77 6.84 7.32 7.96 8.81 9.58 10.71

0.042 0.140 0.274 0.346 0.450 0.484 0.512 0.534 0.668 0.826 1.030 1.196 1.422SDb (0.02) (0.02) (0.04) (0.04) (0.04) (0.04) (0.04) (0.04) (0.04) (0.04) (0.04) (0.02) (0.02)

a All growth solutions had a fixed pH of 8.5.b Parentheses give standard deviation for each calculated supersaturation.

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and geometry of the spiral adjusted to the new . Subsequently, aminimum of six images were captured in continuous mode for latermeasurements by image analysis.

Measurements of step velocity, vs and vs, hillock slopes, p andp

, were made for the positive directions, [441]

and [481]

, andnegative directions, [441]

and [481]

, respectively. Step velocity was

determined by measuring step displacement over time using one of twomethods: (1) Determining the difference between the distances of twoequivalent step edges referenced to a fixed point (e.g. the dislocationsource in the growth spiral shown in Fig. 1a) in sequential images; (2)

Measuring the advancement of individual steps relative to the y scandirection. This was done by adjusting the scan angle to orient the steptrain parallel to the y axis and then disabling the slow scanning at thecenter of the spiral. This results in an image that records the movementof steps as the evolution of individual points at step edges over theimaging area as shown in Fig. 1b. Using the measured angle, (Fig.1b), vs was then calculated for the step trains on both sides of thehillock by:

vsSr A

N tan (3)

where Sr is the scan rate (line/second), A the scan size, and N thesampling rate (line/scan, 512 in this study).

In a second measurement, the slopes of spirals were calculated as theratios of step height, h, to terrace width, , by

p h

(4)

where h is the height of individual molecular layer (0.31 nm) on{1014} faces.

3. EXPERIMENTAL RESULTS

3.1. Dependence of Growth Mechanism Upon

Supersaturation

Layer growth was observed as the advancement of mono-molecular (3) steps. These steps were generated by one or twomechanisms depending upon the surface structure and thesaturation state of the input solution. At low values ofrangingfrom 0 to 0.8, steps were initiated solely at dislocations, obvi-ous crystal imperfections, and grain boundaries. Thus, crystaldefect-originated growth by step flow from single and complexdislocation sources was the only growth mode. As shown inFigs. 2AD and Figs. 3A, this mechanism resulted in severaltypes of hillocks that became visible within a few minutes after

the introduction of a supersaturated solution and new stepswere formed only at defect sources that evolved into the apexof a hillock. Examination of more than 50 samples revealed thatthe most common types of hillocks (more than 80%) weremultiple spirals generated by individual dislocations with mul-tiple Burgers vectors (single sourced, multiple growth, Figs.2A,B), dislocation groups (multi-sourced, multiple growth, Fig.2D), and those that originated at obvious crystal imperfections(Fig. 2C). Single spirals (less than 20%) were the least commonand often suppressed by multiple ones. In the areas wherepresumably no imperfections intersected the surface, or thehigh energy sources were overgrown by the newly crystallizedlayers, growth occurred only at the existing step edges and nonew steps were generated.

When supersaturation exceeded approximately 0.8, steps be-gan to be also generated by what appeared to be a homoge-neous surface nucleation mechanism (Fig. 3B, C). However,AFM cannot distinguish nucleation on calcite lattice points(homogeneous) from nucleation at randomly distributed impu-rity sites or particles (heterogeneous). In this study, we willrefer to this type of mechanism as homogeneous or two-dimensional surface nucleation.

Growth by surface nucleation occurred at randomly distrib-uted two-dimensional nuclei on the substrate surface. Subse-quent growth by this mechanism did not lead to hillock devel-opment. Nonetheless, growth originated from defects and bytwo-dimensional surface nucleation co-existed over the experi-

Fig. 1. Fluid Cell AFM image of a spiral hillock on a (1014) facewith the slow scanning direction (A) enabled and (B) disabled acrossthe apex of the hillock. The individual steps along each direction aremono-molecular units having a height of 3. The hillock exhibits asymmetry plane parallel to the c-glide as shown by the dashed line in(A). The angles formed by step train and y-axis, designated by in(B), were measured for both sides to estimate the step velocities at[441]

and [441]

. See text for details.



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mental supersaturation of 0.8 as illustrated in Fig. 3B. Whenboth mechanisms were operative, two-dimensional nuclei formedprimarily within flat areas on the surface rather than on terraces ofspiral hillocks. Increasing supersaturation resulted in faster nucle-ation rates and the development of more 2-D nuclei (Fig. 3C),indicating that growth by homogeneous surface nucleation wasincreasingly dominant with increasing supersaturation.

3.2. Growth Rate of {1014} Faces for Single Spirals

The overall growth rate of a crystal face, or, the growth ratenormal to the surface, Rm (length/time), is given by (Burton etal., 1951):

Rm pv s (5)

Measurements of step velocities, vs, and hillock slopes forsingle spirals, p, over the range of supersaturation used in thisstudy were found to be unique to the positive directions, [441]

and [481]

, and the negative directions, [441]

and [481]

(Table 2). For each direction, vs showed a complex dependenceupon the deviation of equilibrium activity, (a ae), where pscaled inversely with supersaturation, . Details of the depen-dence of direction-specific step velocity and slope on saturationstate were discussed elsewhere (Teng et al., 1998; Teng et al.,1999). The normal growth rate of a single spiral, Rm, wasobtained using Eqn. 5 and the measurements of vs and p.

Fig. 2. Growth hillocks on {1014} faces of calcite. (A) Growth generated by a dislocation with a Burgers vector of tworesults in a double spiral. (B) Growth generated by a dislocation with a Burgers vector of three results in a triple spiral. (C)Growth initiated at an obvious surface imperfection to yield a growth hillock with a growth unit of two mono-molecularlayers. (D) A closeup view (1 1 m) of growth initiated by a dislocation with a Burgers vector of two but complicatedby obvious surface imperfections to result in a multi-sourced, multiple hillock.



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Estimates of Rm increased from 1010108 mm/s over the

experimental supersaturation range (Table 2). The dependence

of Rm upon supersaturation for a single spiral is illustrated inFig. 4.

4. DISCUSSION

In situ AFM observations demonstrate that new steps aregenerated only at surface imperfections when supersaturation islow, but form by both crystal defects and two-dimensionalsurface nucleation as supersaturation increases. These observa-tions are consistent with the predictions of classical BCF the-ories (Burton et al., 1951). The evolution in mechanism withsupersaturation has important implications for quantifying theoverall growth rate, Rm, associated with microscopic surfaceprocesses and for interpreting the widely used macroscopic rate

laws that are based upon chemical affinity.

4.1. Growth Rates, Rm

, Controlled by Simple and

Complex Sources

The growth rate of crystal faces and, therefore, the overallgrowth rate of a crystal by a spiral mechanism is described bymicroscopic parameters vs and p through Eqn. 5 (Burton et al.,1951; Rashkovich, 1991). Because experimental observationsshow that growth of calcite at low supersaturations occurs bythe formation of different types of hillocks (Fig. 2), physicallyrepresentative rate expressions must account for the contribu-tions of all of the important types of dislocation source struc-

tures to growth kinetics. This is difficult to achieve usingchemical affinity based rate laws since the only variable inthese expressions is the supersaturation ratio, . This discus-sion examines the types of rate expressions that result fromgrowth by three predominant and increasingly complex modes:single spirals, single-sourced multiple spirals (spirals generatedby one dislocation with a Burgers vector 2), and multi-sourced multiple spirals (spirals generated by multiple disloca-tions).

To begin, first recall that the dependence of step velocityupon solute activity in the absence of an impurity effect isgiven by:

vs a ae (6)

Fig. 3. Fluid Cell AFM images showing the growth mechanism on a(1014) face at different supersaturation states. (A) Observations ofspiral growth collected at 0.4 within minutes after the input ofsolution. Three spirals are observed in the imaging area. Spirals 1 and3 are single ones, and spiral 2 is a convolution of two double ones. Inthe area where dislocations are absent, growth occurred by the advance-ment of existing mono-molecular layers. (B) Co-existence of spiralgrowth (denoted by s) and homogeneous surface nucleation growth

(denoted by n) at 1.0. Spirals occurred in the bottom part of theimage where dislocations were linearly distributed. Two-dimensionalnuclei appear in the upper right portion of the images. Growth byadvancement of existing steps dominated in the rest imaging area. (C)The dominance of growth by two-dimensional surface nucleation wasrecorded at 1.6 within tens of seconds after the input of solution.Two-dimensional surface nuclei at 100s nm scale formed randomly onthe surface with some of them already coalesced to form bigger nuclei.Continuous surface nucleation is also observed at several locations(denoted by n).4



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where is the kinetic coefficient (or rate constant) in length/time, is the specific molecular volume of crystal, and (a ae) is the difference between actual and the equilibrium activ-

ities of the solute (Chernov, 1961; Chernov and Komatsu,1995). From Eqn. 2, the solute activity in a solution can beexpressed as:

a ae expkBT ae expG

RT (7)

Substituting Eqn. 7 into Eqn. 6 gives the dependence of stepvelocity upon chemical affinity:

vs aeexpGRT 1 (8)Next, the terrace width, , of a single spiral can be estimated

from the critical step length, Lc, by (Burton et al., 1951):

4Lc (9)

where relates to the delay of step advancement when steplength is comparable to the magnitude of Lc and is approxi-mately one for calcite (Teng et al., 1998). In addition, Lc can be

expressed as (Burton et al., 1951; Rashkovich, 1991; Teng etal., 1998):

Lc2kBT (10)

where is the step edge free energy (per unit length per unitheight).

From this basis, the properties of the simplest type of spi-ralsa spiral generated by one dislocation with a Burgersvector of one (Fig. 1A)can be quantified. The hillock slope,p, of such spirals can be obtained by combining Eqns. 4, 9, and10 into the expression:

p hkBT

8 G

RT (11)

which indicates that hillock slopes have a positive correlationwith solution supersaturations as illustrated by Fig. 5. Noticethat the definition

G/RT (12)

was used to develop Eqn. 11.It follows from Eqn. 5 that, in the absence of impurities, the

overall growth rate for the simplest type of spiral growth is theproduct of Eqns. 8 and 11:

RmaehkBT

8 expG

RT 1GRT . (13)

This is illustrated in Fig. 6A that shows Rm has a superlineardependence upon supersaturation.A more complex, second type of spiral growth can occur

when the dislocation source has a Burgers vector of m (Fig.2AB). In this case, the growth unit is multiple steps of m.Hence, Eqn. 11 becomes

p mh kBT

8 G

RT (14)

and Rm can be estimated by:

Rmma ehkBT

8 expG

RT 1GRT . (15)

Table 2. Summary of experimental measurements of step velocities, step densities, and calculated overall growth rates.

0.042 0.140 0.274 0.346 0.450 0.484 0.512 0.534 0.668 0.826 1.030 1.196 1.422

vs (nm/s) 0.15 0.35 0.52 1.18 1.63 1.89 2.44 2.52 3.59 5.18 8.10 9.46 13.10SDa (0.04) (0.10) (0.10) (0.16) (0.10) (0.18) (0.20) (0.17) (0.09) (0.14) (0.13) (0.14) (0.10)

vs (nm/s) 0.3 0.83 1.13 1.87 2.15 2.19 2.41 2.54 2.61 3.08 3.69 4.15 4.64SDa (0.04) (0.09) (0.04) (0.12) (0.14) (0.18) (0.20) (0.17) (0.06) (0.06) (0.09) (0.09) (0.09)p

(103) 0.47 0.72 0.71 0.75 0.61 0.58 1.23 1.69 2.23 2.76 3.15 3.15SD (103)a (0.01) (0.01) (0.04) (0.02) (0.01) (0.01) (0.01) (0.01) (0.13) (0.07) (0.25) (0.33)

p

(103) 0.44 0.48 0.58 0.46 0.47 0.54 2.12 2.81 4.27 6.10 6.85 7.09SD (103)a (0.01) (0.04) (0.03) (0.01) (0.03) (0.02) (0.06) (0.10) (0.35) (0.41) (0.33) (0.25)

Rm (109 mm/s) 0.16 0.37 0.84 1.22 1.15 1.41 3.86 6.07 11.55 22.36 29.80 41.27

Rm (109 mm/s) 0.36 0.54 1.08 0.99 1.02 1.30 5.38 7.33 13.05 22.51 28.42 32.90

a Parentheses give standard deviation for each measurement.

Fig. 4. The overall growth rate of a single spiral measured on a(1014) face of calcite in the experimental supersaturation range. Theline is a best fit to the data.



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This is shown in Fig. 6B where the dependence of Rm onsupersaturation is m times greater than for a single spiral.

A third, more complicated type of spiral growth occurs whenthe source is a group of dislocations. Burton et al. (1951)demonstrated that, for a group of screw dislocations, if theseparation between them is smaller than one half of the stepwidth generated by individual dislocations, they co-operate andform a multi-sourced multiple spiral. Consider a group of m

screw dislocations of the same rotation direction lying on a lineof length . Assuming that the separations between thesedislocations are less than /2 generated by individual disloca-tions, a multi-sourced multiple spiral hillock develops (Fig.2D). The magnitude ofp in this case is controlled by m, , andLc through (Rashkovich, 1991, Vekilov et al., 1992):

p mh

4Lc 2. (16)

Combining Eqns. 10 and 12 and then substituting into Eqn. 16yields the dependence of hillock slope upon chemical affinityfor a multi-sourced spiral:

p

mh kBTGRT8 2kBTGRT

. (17)

Assuming step velocity is independent of p, the product ofEqns. 8 and 17 gives an expression for the rate of a multi-sourced multiple spiral growth:

Rm

aehmkBTGRT8 2kBTGRT

expGRT 1 . (18)

Eqn. 17 can also be applied to situations where the dislocationsare not arranged in a straight line. In this case, 2 representsthe perimeter of the area occupied by these dislocations (Rash-kovich, 1991).

It is important to also note that, for this type of spiral growth,rate can have a discontinuous dependence upon supersatura-tion. As pointed out earlier in this section, a multi-sourcedspiral forms from a group of dislocations only when /2

(Burton et al., 1951). Because terrace width, , scales inverselywith supersaturation (Eqns. 9 and 10), this complex spiral isstable at low supersaturations where step width is large. Withincreasing supersaturation and decreasing step width, the spiraldecomposes into a number of single spirals that are generatedby each individual dislocation when /2 becomes smaller than. When this occurs, the hillock slope will, in general, increasediscretely from the value given by Eqn. 17 to that by Eqn. 11.Hence, the overall growth rate, Rm, is no longer a smoothfunction of supersaturation, as illustrated in Fig. 6C and shownby experiment (e.g. Vekilov and Kuznetsov (1992).

4.2. Relationships Between Process-Based and Affinity-

Based Rate Expressions

Because direct observations of calcite growth show that stepflow originates from predominantly complex sources and, to alesser extent, from simple sources, the net rate of spiral growthacross one surfacenatural or synthetic, is necessarily a com-plex composite of contributions from several Rm expressions(e.g., Eqns. 13, 15, and 18), each of which has a uniquedependence upon saturation state. Hence, the question arises:Are the reported macroscopic, chemical affinity-based rate lawsrelated to microscopic processes occurring at the growing sur-face? The obvious conclusion from the preceding analysis isthat the growth rates determined by measuring temporalchanges in bulk solution compositions must also be composites.

Fig. 5. Fluid Cell AFM images of a spiral hillock on a (1014) face taken at 0.3 and 0.8 show the decreasing terracewidth with increasing supersaturation.



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If no, are there conditions where comparisons of the two typesof rate laws are meaningful?

To answer these questions, first consider Eqn. 18, the morecomplicated rate expression for spiral growth. This equationreduces to the simpler Eqn. 15 when m dislocations overlap,i.e., 0, to form a single source with a Burgers vector ofm.Eqn. 15 further reduces to Eqn. 13, the rate expression for thesimplest type of spiral growth, when the Burgers vector of the

dislocation becomes one (m 1). Eqn. 13 can be furthersimplified to:

RmaehkBT

8 expG

RT 1

2

(19)

when supersaturation is very low or (G/RT) 1. The formof this expression is analogous to a second order chemicalaffinity-based rate law (Eqn. 1b with n 2) provided thataehkBT/8 represents the rate constant, k.

The above comparison suggests that spiral growth is approx-imated by a second order chemical affinity based-rate law onlywhen (1) the supersaturation is so low that spiral formation isthe only growth mechanism and (2) the formation of simple

spirals dominates the growth of all crystal faces. Direct obser-vations (Fig. 2) reported in this study show that even in thesupersaturation range where spiral formation is the only growthmode, the second requirement fails because multiple spirals arefar more commonly observed than single ones. This suggeststhat even when growth occurs at low supersaturation and thespiral mechanism is the only valid growth mode, still no singlekinetic expression can describe growth because the dislocation

source structures may be sample specific. Rather, Rm measuredat each supersaturation for pure spiral growth must be within arange that is defined by Eqns. 13 and 18 for the lower and upperboundaries (Fig. 6AB, respectively). The lower boundary forthe growth of calcite {1014} faces in this study is given by theexperimental measurements shown in Fig. 4. Finally, in thepresence of obviously large surface defects such as pits, frac-tures, and grain boundaries (Fig. 2CD), the determination ofRm can be further complicated because the growth is controlledby the specific characteristics of each source. Most signifi-cantly, if the source is at a gross defect where steps mustcircumscribe a structural imperfection of a perimeter , theterm 2 in Eqns. 16 to 18 is replaced by . The result is that

Fig. 6. Schematic view showing the dependence of overall growth rate upon supersaturation index, , in different growthmodes. All plots are referenced to the same scale. (A) Growth controlled by single spirals. (B) Growth controlled by

single-sourced multiple spirals. (C) Growth controlled by multi-sourced multiple spirals. (D) Growth controlled bytwo-dimensional surface nucleation. When spiral growth is the only mechanism, the minimum growth rate may beconsidered as that given for single spirals (e.g., Eqn. 13 and the data in Fig. 4 for calcite). When growth is controlled byboth spiral and surface nucleation mechanism, the overall rate is the sum of the contributions from each growth mode.



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as increases and Lc decreases, the second term in the denom-inator becomes dominant, the slope becomes independent ofsupersaturation and Rm becomes a linear function of.

When supersaturation is high enough that two-dimensionalsurface nucleation mechanism becomes operative (Fig. 6D), thecontributions of two-dimensional nuclei formation to total Rmmust be added. The macroscopic growth rate by a two-dimen-

sional nucleation mechanism has an exponential dependenceupon supersaturation through the expression

Rm 1.137hIv s21/3 (20)

where h is the step height, I the nucleation frequency (area2

time1) (van der Eerden, 1993). Because I increases exponen-tially with increasing supersaturation, contributions of the two-dimensional surface nucleation mechanism are increasinglyimportant with increasing supersaturation and quickly becomethe dominant component in the overall growth rate. In thesupersaturation range where spiral and two-dimensional nucle-ation mechanisms co-exist ( 0.8 or 2.2 for this study),

the overall growth rate will be the sum of all growth modesillustrated in Fig. 6AD.

The clear conclusion from experimental observations andthis analysis is that attempts to relate affinity-based rate laws tomicroscopic growth mechanism are of limited value. Macro-scopic rate laws derived from bulk experiments represent thecomposite average of quite different growth rate expressionsfor the unique reactivities of many different crystal facets andsurface structures. However, affinity-based rate laws comprisemost of our quantitative knowledge regarding calcite growthkinetics and will continue to be employed by the geological andengineering communities. Calcite grains in previous studies aretypically dominated by the {1014} faces or become dominated

by these faces as growth proceeds in the absence of impurities.It is therefore useful to understand the physical basis of theserate expressions by making comparisons to process-based ratelaws that are derived on the {1014} faces. To probe the rela-tionships, we ask the question: What is the dependence of Rmupon ( 1) as a function of the number of dislocations, m,and the separation between these dislocations, ? Using aver-age values of step edge free energy, 1400 erg/cm2 (Tenget al., 1998), and kinetic coefficient of step advancement, 0.4 cm/s (Teng et al., 1999), the dependence of Rm uponsupersaturation ( G/RT) was estimated by Eqn. 18.These calculated values of Rm were fitted by a simple powerlaw ofk( 1)n where k and n are fitting parameters. Resultsof this analysis give insights to the physical meaning of theexponent, n. When dislocation separation is held constant ( 50 nm, Fig. 7A), n is independent of both the net Burgersvector and the supersaturation. Differences in Rm are given bychanges in the rate constant. Yet, a comparison with Fig. 7Bshows that growth by the same dislocation structure yields asmaller value ofn when a broader saturation range is used. Thatis, the value of the exponent is inversely dependent uponsaturation range when growth occurs on complex spirals. Tofurther examine these relations, the net Burgers vector is heldconstant (m 3) and the distance between dislocations isvaried (Fig. 7C). It shows that when the saturation range isnarrow and close to equilibrium, the exponent term exhibits astrong inverse dependence upon dislocation separation. How-

ever, as the saturation range becomes broader, n becomesindependent of dislocation density (Fig. 7D). Notice that onlywhen separation equals zero and the supersaturation range isclose to equilibrium does n approaches 2 (Fig. 7C). Thisprediction is upheld by the model fits of calcite growth data(Shiraki and Brantley, 1995) that obtained a value of n 1.93 0.14 when exp(G/RT) 1.72 or supersaturation 0.54.

These theoretical relationships suggest that the exponent andthe rate constant in chemical affinity-based rate laws are de-pendent upon the details of dislocation distributions and dislo-cation structures as well as the supersaturation range used to fitthe growth rate data. There may not be a unique power lawexpression with the simple form of k( 1)n for spiralgrowth. Rather, each expression may apply only to the specificsupersaturation range over which the model is fitted to theexperimental data. This analysis shows that, in general, Rm canbe described by a chemical affinity-based rate law that assumes1 n 2. The value of n is inversely related to thesupersaturation range over which the fitting is performed. Rm

exhibits a quadratic (n approaches two) dependence upon ( 1) only when the dislocation source is compact ( 0) and thesolutions are near-equilibrium ( 0.4). This analysis indi-cates that experimental results that yield n 2 in a wide rangeof supersaturation that is not sufficiently close to equilibriumare likely indicating a non-equilibrium growth (e.g., growthoccurring on non-equilibrium forms) or a growth combiningspiral and surface nucleation mechanisms.

5. CONCLUDING REMARKS

In situ observations of growth using AFM reveal the mech-anisms of calcite growth across a range of supersaturation. At

lower values of from 0 to 0.8, growth is initiated solely bysurface imperfections including screw dislocations. Thus, crys-tal defect-originated growth, in particular, spiral growth, is theonly manifested mechanism. At the range of 0.8, two-dimensional surface nucleation becomes increasingly importantwith increasing supersaturation. The growth rate of {1014}faces grown by single spirals is estimated from the product ofstep velocities and spiral hillock slopes measured at differentsupersaturations. This estimate corresponds to the minimumrate for pure spiral growth on a calcite (1014) face.

More general rate expressions are considered by taking intoaccount the effect of complex dislocation sources on the slopesof spirals. Such expressions cannot be approximated by thesecond order rate law expressed by chemical affinity unlessgrowth proceeds only by the formation of single spirals at lowsupersaturation range where only the spiral mechanism is op-erative. Even for pure spiral growth, the experimentally mea-sured rate of calcite crystallization cannot be unique becausethe types of dislocation source structures are variable. Thesecond order rate law based upon the chemical affinity of theprecipitation reaction can be considered as a special case ofspiral growth when dislocation sources are simple or compact.Even so, this expression is only valid at low supersaturations(approximately 0.8 or 2.2 for calcite growth in thisstudy). With increasing supersaturation, the magnitude of Rmbecomes the sum of spiral growth and two-dimensional surfacenucleation.



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Results of this study demonstrate the need to apply cautionwhen deducing growth mechanisms and rate laws from tempo-ral changes in bulk solution chemistry. Further, interpretationsof growth mechanism using data collected from these indirectmethods are particularly hazardous without direct evidence forthe growth processes that are occurring at the mineral surfaces.The analysis in this study suggests that popular rate laws areempirical, at best. By combining direct observations with these

methods, improved rate laws with greater predictive capabili-ties may be possible.

AcknowledgmentsThis work was supported by the Geosciences Re-search Program, Basic Energy Sciences, U.S. Department of Energythrough grant number DE-FG05-95-ER14517 and was performed un-der the auspices of Lawrence Livermore National Laboratory undercontract W-7405-Eng-48. The images in Figure 5 were collected by K.Davis of Georgia Tech. We thank Susan Brantley and an anonymousreviewer for thoughtful comments.

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