Stochastic Simulation of Isotopic Exchange ... - ACS Publications

Subscriber access provided by FLORIDA ATLANTIC UNIV

Article

Stochastic Simulation of Isotopic Exchange Mechanisms for Fe(II)-Catalyzed Recrystallization of Goethite Piotr Zarzycki, and Kevin M. Rosso Environ. Sci. Technol., Just Accepted Manuscript • Publication Date (Web): 12 Jun 2017 Downloaded from http://pubs.acs.org on June 12, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Environmental Science & Technology is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 29

Environmental Science & Technology

1

Stochastic Simulation of Isotopic Exchange

2

Mechanisms for Fe(II)-Catalyzed Recrystallization

3

of Goethite

4

Piotr Zarzycki, 1,2,* Kevin M. Rosso3,*

5 6 7

1

Energy Geoscience Division, Lawrence Berkeley National Laboratory, Berkeley, CA, U.S.A. 2

Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland 3

Pacific Northwest National Laboratory, Richland, Washington, U.S.A.

8

KEYWORDS isotopic exchange, iron oxide, aggregation, reductive dissolution, redox, electron

9

transfer, goethite, iron

10

ABSTRACT. Understanding Fe(II)-catalyzed transformations of Fe(III)-(oxyhydr)oxides is

11

critical for correctly interpreting stable isotopic distributions and for predicting the fate of metal

12

ions in the environment. Recent Fe isotopic tracer experiments have shown that goethite

13

undergoes rapid recrystallization without phase change when exposed to aqueous Fe(II). The

14

proposed explanation is oxidation of sorbed Fe(II) and reductive Fe(II) release coupled 1:1 by

15

electron conduction through crystallites. Given the availability of two tracer exchange data sets

16

that explore pH and particle size effects (e.g., Handler et al. Environ. Sci. Technol. 2014, 48,

17

11302–11311; Joshi and Gorski Environ. Sci. Technol. 2016, 50, 7315–7324), we developed a

ACS Paragon Plus Environment

1


Page 2 of 29

18

stochastic simulation that exactly mimics these experiments, while imposing the 1:1 constraint.

19

We find that all data can be represented by this model, and unifying mechanistic information

20

emerges. At pH 7.5 a rapid initial exchange is followed by slower exchange, consistent with

21

mixed surface- and diffusion-limited kinetics arising from prominent particle aggregation. At pH

22

5.0 where aggregation and net Fe(II) sorption are minimal, that exchange is quantitatively

23

proportional to available particle surface area and the density of sorbed Fe(II) is more readily

24

evident. Our analysis reveals a fundamental atom exchange rate of ~10-5 Fe nm-2 s-1,

25

commensurate with some of the reported reductive dissolution rates of goethite, suggesting

26

Fe(II) release is the rate-limiting step in the conduction mechanism during recrystallization.

27

INTRODUCTION

28

Iron (III) oxides and oxyhydroxides are some of the most reactive and abundant minerals in

29

soils and sediments, comprising a dominant sorbent and catalytic substrate for contaminant metal

30

uptake and transformations. Their chemical behavior is tied closely to their mineralogy, which

31

in turn depends on redox potential, pH, water activity, temperature, and particle size.1 Redox

32

transformations from one phase to another are particularly facile, and understanding mechanisms

33

of these transformations is key for reconstructing diagenetic history or predicting the mobility of

34

an environmental contaminant. For example, natural reductive processes such as the anaerobic

35

respiration of dissimilatory metal reducing bacteria produce and juxtapose aqueous Fe(II) with

36

their parent Fe(III) phase. This condition triggers transformation from less stable forms (e.g.,

37

ferrihydrite) to more stable forms (e.g., lepidocrocite, goethite, hematite, magnetite), a process

38

that directly impacts metal binding and selectivity.2-6

39

Despite the importance of these processes, mechanisms of Fe(II)-catalyzed phase

40

transformations remain poorly understood. The difficulty arises in part because pathways for iron


2

Page 3 of 29


41

atom exchange between Fe(II) in solution and oxide Fe(III) are obscured by the prospect of

42

Fe(II)/Fe(III) electron exchange at the interface. Until recently, the interaction of Fe(II) with

43

stable Fe(III)-oxides, such as during sorption/desorption, has largely been deemed inert because

44

of apparent reversibility and lack of phase change.1 However, recent 57Fe isotope tracer studies,

45

such as those reported by Handler et al.7-9 for goethite and Frierdich et al.10-12 for hematite and

46

goethite, demonstrate that these stable Fe(III) minerals undergo recrystallization when exposed

47

to Fe(II).13 They show 56Fe-57Fe isotopic equilibration between the solid and solution phases up

48

to 100% on the time scale of days, with no secondary mineral formation or significant change in

49

particle size or shape.7-14 Although constituting an important advance because of its implications

50

for metal uptake (e.g., by incorporation into the structure),10-14 these macroscopic tracer studies

51

alone do not reveal the interfacial processes controlling the isotope exchange dynamics.

52

Mechanistic insights are needed.

53

The concept put forward to qualitatively explain isotopic equilibration without phase change is

54

the conduction mechanism,15,16 where any given crystallite oxidative adsorption of Fe(II) by

55

surface Fe(III) is coupled by bulk15,17 or surface18 conduction of electrons to remote Fe(III) that

56

is reductively released as Fe(II). This nominally 1-to-1 coupled redox growth/dissolution process

57

could thereby facilitate progressive exchange between goethite and solution iron isotopes

58

without altering solid mass or physical characteristics necessarily. Implicit elementary steps

59

including Fe(II) adsorption and interfacial electron transfer to oxide Fe(III), as well as electron

60

transport through the lattice to distinct Fe(III) sites, have all been indirectly validated through

61

Mössbauer measurements19,20 and molecular modeling.18,21-24

62 63

A recent

55

Fe tracer and microscopy study by Joshi and Gorski27 of Fe(II) interaction with

goethite qualitatively corroborated the applicability of the conduction mechanism for exchange,


3


Page 4 of 29

64

but also showed that additional processes such as particle coarsening could also be operative. In

65

particular, through detailed microscopy work that validated molecular modeling predictions18

66

that oxidative Fe(II) adsorption occurs preferentially on nanorod edge faces while reductive

67

release occurs preferentially on nanorod tips, consistent with the conduction mechanism of

68

exchange.

69

In the present study we quantitatively test how well the simulation based on the conduction

70

mechanism fits measured equilibration data across its breadth of conditions, from two

71

comprehensive studies on Fe(II)/goethite.

72

have direct implications for understanding metal uptake during Fe(II)-catalyzed Fe(III)-oxide

73

recrystallization. For example, resolving the extent to which the conduction mechanism plays a

74

role in the redox behavior of Fe(III)-oxides could improve mobility prediction for toxic metals

75

that strongly associate with these minerals.

8,27

The new insights derived from our simulations

76 77

COMPUTATIONAL METHODS

78

In order to provide a mechanistic insight into Fe-exchange between aqueous and solid phase,

79

we developed a stochastic simulation that computes the shape requirement for the isotopic

80

equilibration curves evolving simultaneously based on the 1:1 conceptual model for exchange.

81

We consider two pools of exchangeable iron and we neglect the interface for the computational

82

simplicity. The latter approximation is justified because all experimental and theoretical reports so far are

83

consistent with the moving dissolving-precipitating interfaces,7-19,24-27 which continually exposes a fresh

84

layer of oxide to be exchanged in aqueous solution. If on the other hand, only the surface atoms are

85

exchangeable, isotopic exchange is minimal in stark contrast to the experimental data at pH 7.5.7,8,27


4

Page 5 of 29


86

For simplicity, the simulation reduces the steps of the conduction mechanism into a single 1:1

87

exchange event: one iron isotope from solution (i.e., Fe(II)) is randomly swapped with an iron

88

isotope from the solid (i.e., Fe(III)), preserving the mass of iron in the solid and that in solution.

89

While this simplification is consistent with the conduction mechanism (oxidative Fe(II)

90

adsorption coupled to reductive Fe(III) release) it is not necessarily exclusive to it (e.g., in the

91

real system acidic Fe(III) release and subsequent encounter/electron exchange with Fe(II) in

92

solution could yield the same 1:1 isotopic exchange signature, though such pathways are unlikely

93

to be significant given the low aqueous solubility of Fe(III)).

94

Note also that the observed equilibration rate is not the rate of iron atom exchange nor

95

interfacial electron transfer, per se, because the experiments are blind to the concomitant like

96

isotopic exchanges (e.g.,

97

encompasses all possible exchanges, including the ones to which the experiments are blind, and

98

treats them on an equal basis. The total isotopic compositions of the solid and solution phase Fe

99

are evolved stochastically through unitless numeric time steps, yielding the shapes of the coupled

100

isotope exchange curves. The time step, as a sole fitting parameter, is then scaled to fit the

101

measured equilibration data. Thus, the fitted time step has the prospect of carrying information

102

on rates of processes underlying observed tracer exchange rates.

103

Simulation input and assumptions

56

Fe-56Fe,

57

Fe-57Fe, etc.) that also undoubtedly occur. Our simulation

104

The stochastic modeling algorithm begins with experimental mass ratios as inputs and consists

105

of the following iterative sequence of steps: (i) random selection of one iron atom in the aqueous

106

phase and another in the solid phase; (ii) assignment of their isotopic identity (57Fe or 56Fe) with

107

probabilities proportional to current isotopic compositions of these phases; (iii) carry out an

108

exchange step (i.e., atoms are swapped) and (iv) update the isotopic composition of the two


5


Page 6 of 29

109

phases (as well as any other macroscopic descriptors such as fractionation factors). In addition to

110

this basic algorithm, adaptation has been implemented to explore the effect of evolving

111

aggregation state on isotopic equilibration behavior, as described later.

112

The algorithm assumes a constant, mass-independent exchange rate that swaps Fe isotopes

113

between the two Fe compartments (solid, solution) initially populated at the δ57/56Fe levels used

114

in the reported experiments. Note, that the solid and aqueous phases should also differentiate in

115

terms of mass-dependent Fe-fractionation after reaching mixing equilibrium; however, it will

116

result in small fractionation that becomes evident over a longer period than reported in the

117

experiments.7,8,27

118

In practice, at every simulation step we select two Fe atoms in both iron compartments with

119

their isotopic identity assigned probabilistically based on the current δ57/56Fe ratio in each

120

compartment. The isotopic compositions are updated every successful isotope swap with

121

exchange dynamics expressed in solid recrystallization cycles (1 cycle is defined to have

122

occurred when the number of successful swaps equals the number of oxide Fe atoms in the

123

system).

124

The isotopic compositions are expressed in the δ notation (reported in ‰): [ఱళ ୊ୣ/ఱల ୊ୣ]

125

ߜ ହ଻/ହ଺ Fe = 10ଷ ቀ [ఱళ ୊ୣ/ఱల ୊ୣ]೛೚೚೗ − 1ቁ

126

where [ହ଻ Fe/ହ଺ Fe]௣௢௢௟ is isotopic ratio of a given iron compartment (i.e., aqueous Fe(II) or solid

127

Fe(III)), [ହ଻ Fe/ହ଺ Fe]௦௧ௗ is the reference isotopic ratio (i.e., average for terrestrial igneous rocks,

128

[ହ଻ Fe/ହ଺ Fe]௦௧ௗ =0.023).

ೞ೟೏

(1)


6

Page 7 of 29

129


Here, we discuss primarily just the effect of pH on Fe(II) sorption and exchange-active

130

surface site density as the most plausible explanation of the incomplete exchange at pH 5 and

131

complete exchange at pH 7 without change in the mass of the goethite (discussed below). The

132

incomplete exchange at pH 5 could also be explained by an excluded exchange volume due to

133

aggregation or mass transfer (e.g., via precipitation), but these processes are not observed

134

experimentally8 and they are unlikely to be significant at pH 5.31 The observed incomplete Fe-

135

exchange is most likely due to limited interaction of aqueous Fe(II) with the net positively

136

charged goethite surface,8,27 as implemented in our simulation approach.

137 138

RESULTS AND DISCUSSION

139

Stochastic simulation of isotopic exchange

140

In Figure 1a we illustrate the generic Fe isotopic exchange system, as modeled in our

141

simulation. Each exchange event has been postulated to proceed via a sequence of elementary

142

steps at the solid/solution interface, yielding 1:1 atom exchange. 7-18 For the sake of simplicity,

143

our model considers only the net outcome of pairwise Fe atom swapping between the solid and

144

solution phases, maintaining a constant ratio of Fe in the solid phase to Fe in the solution phase.

145

As shown in Figure 1b, a large number of such swaps ultimately yields isotopic the same

146

isotopic composition in both phases that depends on the mole ratio of iron in the solution relative

147

to the solid and their respective initial isotopic compositions (δ57/56Fe).

148

The time step in the simulations is unitless, and is scaled to establish the actual time step that

149

shows best possible correspondence with the real-time equilibration data. To portray this scaling

150

in terms of an additional physically useful quantity, we express time in terms of recrystallization


7


Page 8 of 29

151

cycles: one cycle corresponds to a total number of swap events that equals the number of Fe

152

atoms in the solid phase (Fig. 1b). At the end of one cycle every Fe atom in the solid, in effect,

153

will have resided in the solution phase at least once. The scaling of the time constant can be

154

considered a simple stretching operation along the cycles/time axis that does not modify the

155

functional form of the simulated equilibration curves. The solid and solution equilibration curves

156

are fit simultaneously with this single adjustable parameter.

157

Fitting pH 7.5 data reported by Handler et al.8

158

In Handler et al.,8 despite a relatively high initial

57

Fe(II) isotopic enrichment in the solution

159

phase (840‰), the low total amount of Fe in the solution phase relative to that in the solid phase

160

(Fe(II):Fe(III) = aqueous:solid = 1:21.3) means that the overall change in δ57/56 for the solid

161

phase is small. Modeling the evolution of δ57/56 in the solid phase alone is rather insensitive to

162

simulation details. Thus we focus attention on the evolution of the solution phase δ57/56 for the

163

highest resolving power in the experiments; the corresponding δ57/56 evolution for the solid phase

164

is always well fit within measurement error limits.

165

We first examine the pH 7.5 equilibration data for both the goethite microrods and nanorods,

166

both of which are based on 1 mM Fe(II) and a constant 2 g/L solids loading.8 At this pH, within

167

approximately 10 minutes almost a half (49% nanorods, 44% microrods) of the aqueous Fe(II) is

168

sorbed to the goethite solids, the remaining aqueous Fe(II) concentration remained relatively

169

constant for the rest of experiment; the first isotopic exchange measurement was collected at 10

170

minutes.8 In Figure 2 we show best fits of simulated 1:1 exchange curves to the experimental

171

results for goethite microrods (reported BET surface area = 40 m2/g) (Fig. 2a) and nanorods (110

172

m2/g) (Fig. 2b). At this pH, for both microrods and nanorods, an equal tracer enrichment in solid


8

Page 9 of 29


173

and solution phase iron is observed within 30 days, with no significant change in the physical

174

characteristics of the goethite detected at the end of the reaction.8

175

For pH 7.5, using a single time constant the 1:1 model is unable to provide a comprehensive fit

176

to the whole time-domain of the experimentally reported δ57/56 values. The deviation is resolved

177

most prominently in δ57/56 for the aqueous phase in the intermediate time domain of 1-6 days for

178

the microrods (Fig. 2a) and 1-5 days for the nanorods (Fig. 2b). Prior to this domain, initial rapid

179

iron isotope exchange within the first day is observed for both the microrods and nanorods (87 %

180

equilibrated by day 1), and this early stage for both particle sizes are well fit with an equal

181

exchange rate of 6.5 days/cycle. This initial rapid exchange is followed in both cases by the

182

requirement of a slower exchange rate, that was furthermore found to be particle-size dependent.

183

In this intermediate time domain, the equilibration rates of the microrods and nanorods remain

184

similar, albeit with an intriguing slower time constant for the smaller nanorods (12.5 days/cycle)

185

compared to that for the microrods (8 days/cycle), which we address below. Because the fits

186

become increasingly insensitive to the time constant in the late stages of equilibration,

187

determining whether the decrease in exchange rate is temporary or persists to experiment

188

completion is not possible. In any event the decrease in exchange rate and the emergence of

189

particle-size dependence in the intermediate domain is clear from the data, and does not preclude

190

near complete equilibration of both microrods and nanorods within 30 days. Assuming a two-

191

stage process, the 1:1 model comprehensively describes this system.

192

Fitting pH 7.5 data reported by Joshi et al.27

193

The recent data set by Joshi and Gorski26 on a nearly identical system (nanorods, pH 7.5,

194

with key differences discussed below) allows a further test of the 1:1 model. This system also

195

entails 2 g/L solids loading, contacted with 0.8 mM Fe(II) which shows substantial rapid Fe(II)


9


Page 10 of 29

196

sorption (60% within the first day), and isotopic equilibration within 30 days. Following the

197

same fitting procedure except applied to this data set (see Supporting Information) also yields

198

two clear time-domains: A fast initial equilibration stage is followed by a substantially slower

199

one that leads to equality of Fe-fractionation in solid and aqueous phases. The intermediate time

200

domain is most evident between 1-15 days. This is striking overall similarity to the data in

201

Handler et al.8 However, the time constants obtained from fitting the Joshi and Gorski27 data are

202

significantly larger than those obtained by fitting the Handler et al.8 data (see Supporting

203

Information). For example, comparing the initial rapid equilibration domain that is evident to 1

204

day duration, the nanorods in the Joshi and Gorski27 system equilibrate (76% equilibrated) about

205

3 times more slowly (19.5 days/cycle) than the nanorods in the Handler system (6.5 days/cycle).

206

These time constants can be directly compared because they intrinsically account for the

207

differences in the exact initial Fe isotopic compositions of the solutions and solids and their

208

evolution over time, between the two studies. They do not yet, however, account for the slightly

209

lower Fe(II) concentration used in Joshi and Gorski,27 which entails a 28:1 solid:solution Fe ratio

210

in contrast to a 21.3:1 ratio in Handler et al.8 If we additionally account for this difference (see

211

Supporting Information) the time constant distinction in the initial equilibration stage between

212

the studies is reduced from a factor of 3 to a factor of 2.5. Note that, the exchange rates depend

213

on the initial composition. Thus, they are normalized with respect to the surface Fe-sites density

214

and initial Fe(II) concentration. The intrinsic differences in the oxide particles (e.g.,

215

defects/impurities density) may also contribute to the exchange rate discrepancies.

216

Aggregation affects exchange

217

Conceptually, it is logical to attempt to explain the equilibration rates similarities (nanorods to

218

microrods in Handler et al8) and differences (Handler et al8 nanorods and Joshi and Gorski27


10

Page 11 of 29


219

nanorods) in terms of available reactive surface area (based on the reported Fe-site density8, and

220

Fe sorption31-32). As deduced in Handler et al,8 rates of exchange should be controlled by the

221

probability of Fe(II) interaction with the goethite surface. However, particle aggregation is a

222

documented but poorly quantified major system observable in all pH 7.5 experiments discussed

223

so far between the two studies.8,27

224

interactions between particles, to the extent that attractive van der Waals interactions start to

225

dominate

226

Aggregation is observed for pH near the point of zero charge.29-33 At pH 7.5 goethite is well

227

within its expected point of zero charge range, typically pH = 6.7-9.5.33 While this net neutral

228

surface charge facilitates Fe(II) interaction with the goethite surface31,32 (hence the strong

229

observed Fe(II) sorption throughout all experiments), prior to this interaction goethite particles

230

will have condensed into aggregates with poorly predictable partially occluded surface area.

231

Compiling the observations about aggregation reported in both Handler et al8 and Joshi and

232

Gorski,27 at pH 7.5 nanorods and microrods formed aggregates of similar sizes (Handler et al8),

233

and nanorod “bundles” comprised ~50% of nanorod suspensions, a value that remained

234

unchanged upon exposure to Fe(II) and throughout the 30 days of isotopic equilibration Joshi

235

and Gorski.27 Fe(II) interaction with goethite aggregates, as opposed to individual particles, is an

236

important enduring characteristic in all pH 7.5 experiments during the observed Fe isotope

237

exchange timescales.

(see

It originates from a decrease of repulsive electrostatic

Derjaguin-Landau-Verwey-Overbeek

theory

of

suspension

stability28).

238

The obscuring effect of aggregation on actual reactive surface makes full reconciliation

239

of the pH 7.5 isotopic equilibration rates difficult. On the one hand, the fact that equilibration

240

goes to completion suggests that occluded surface area is only partially occluded and remains

241

accessible to Fe(II). On the other hand, the extent to which it is accessible between the systems


11


Page 12 of 29

242

can only be qualitatively assessed. It is quite likely that the found two-stages of equilibration

243

arise directly from the effect of aggregation; the initial rapid stage derives largely from surface-

244

controlled interaction between Fe(II) and well exposed goethite particles along the periphery of

245

aggregates, whereas the trailing slower stage derives more from diffusion-limited interaction in

246

interior domains. Other explanations must invoke some unexpectedly unique properties of the

247

nanorod crystallites themselves, relative to the microrods, such as a lower defect density or the

248

prospect of electrostatic coupling of interfacial processes across the ~ 10 nm rod widths that slow

249

the rate of exchange.17,22,24 The same notion of smaller particles yielding a higher proportion of

250

partially occluded interior domains can also explain the slowest equilibration rates found, for

251

Joshi and Gorski27 nanorods. In that study nanorods were both geometrically smaller (254%

252

lower geometric surface area per particle on average by TEM) and exposed less BET surface

253

area (by 18%) than Handler et al8 nanorods. Aggregates of these smallest particle sizes yet

254

considered would be expected to possess the highest proportion of partially occluded interior

255

domains, and conceptually this is consistent with the slowest rates found overall in Joshi and

256

Gorski’s study.27 However, this does not necessarily explain the 2.5x slower initial stage for

257

their nanorods, if it indeed primarily reflects exchange via the most accessible surface area. The

258

extent to which aggregation could partly encumber even this initial rapid stage of equilibration is

259

indeterminate.

260

Fitting pH 5.0 data reported by Handler et al.8In contrast to the experiments at pH 7.5, the pH

261

5 experiments of Handler et al8 are much more useful for mechanistic interpretation because the

262

aggregation is not observed in the reported experimental studies.8 At pH 5, isotopic exchange is

263

observed to be incomplete, converging to a substantially lower amount of exchange over 30

264

days.8 At pH 5 the oxide surface is positively charged due to H+ binding to surface hydroxyl


12

Page 13 of 29


265

groups (e.g.,29-31 ≡FeOH-1/2 + H+ ↔ ≡FeOH2+1/2). Inverse to the situation discussed above for pH

266

7.5, this not only reduces the tendency to aggregate but also limits Fe(II) sorption to the surface.

267

In the pH 5 experiments, no net Fe(II) sorption was detected8. However, other relevant Fe(II)

268

sorption experimental studies and their surface complexation modelling31,32 suggest net Fe(II)

269

sorption on goethite in the vicinity of 7% of the aqueous Fe(II) at this pH, for similar Fe(II)

270

loading and particle sizes. At pH 5 the experiments are thus on the cusp of the Fe(II) sorption

271

edge, and it is reasonable to assume that net Fe(II) interaction with the goethite surface is

272

minimal but not zero.

273

Despite these small amounts of net Fe(II) sorption, the aqueous and goethite phases still

274

exchange Fe and some isotopic exchange occurs (i.e., δ57/56Fe(II) drops from 840‰ to 334‰ for

275

the nanorods, and to 588‰ for the microrods – see ref.,8 Supporting Information and Fig. 1).

276

Here the observed particle-size dependence is free of the counterintuitive behavior found for pH

277

7.5. At pH 5 higher surface area nanorods exchange faster and to a greater extent than the lower

278

surface area microrods, on an equal mass basis. In the absence of the complicating effects of

279

aggregation, the rate and extent differences should therefore be more clearly relatable to

280

available particle surface area, which is about 2.8 times higher for the nanorods than the

281

microrods. If the mechanism of exchange is indeed controlled by interfacial processes, one can

282

anticipate that this ratio might manifest in either the relative rate of approach to equal isotopic

283

fractionation in solid and aqueous phase, the extent of equilibration, or both. In order to fit

284

simulated 1:1 exchange to the pH 5 experimental data8 the model must account for both the

285

observed rate of equilibration and the extent of exchange (Fig. 3). To do so without changing

286

goethite mass requires that only a fraction of the total goethite mass is exchange-active. The

287

equilibration rates for the nanorods and microrods are well fit each with a single time constant in


13


Page 14 of 29

288

our model, with recrystallization cycle times of 205 days for the nanorods, and 615 days for the

289

microrods. Although the available data are more sparse than those available for pH 7.5, the time

290

constants are well constrained by the intermediate time point. The rate of nanorod equilibration

291

is thus about 3 times faster than that for the microrods, consistent with the surface area ratio

292

between them.

293

Fe(II) sorption dictates exchange

294

Furthermore, from the observed extents of isotopic exchange, the fraction of exchange-active

295

sites over 30 days can be estimated at 7% for the nanorods, and 2% for the microrods (Fig. 3).

296

These values are tightly constrained by the measured isotopic compositions at the final time

297

point, and should be considered to be determined primarily from the data itself, as opposed to a

298

fitting outcome. They are calculated from the difference in the aqueous and solid isotopic

299

compositions (i.e., ∆= ∆δ= δ(57/56Fe(II)) - δ(57/56Fe(III))) after 30 days. In the case of nanorods,

300

this difference (∆=309.5‰) corresponds to 7% Fe(III) being exchangeable (~35% of Fe is

301

exposed at nanorod surfaces), whereas in case of microrods (∆=576.1‰) only about 2% of oxide

302

Fe(III) is exchangeable (28% of Fe is exposed at the surface) (see Supporting Information). Here

303

again the surface area ratio of 2.8 appears to manifest directly, this time in the relative amounts

304

of exchange-active sites (7% / 2%), based on exchange extent.Given the clear role of interfacial

305

area in these experiments, and the proposed conduction mechanism, it is logical to hypothesize

306

that the observed limited exchange-active mass is directly proportional to the sorbed Fe(II)

307

density at goethite surfaces. Further, if this density can be independently estimated, because our

308

numerical simulation provides the real-time frequency of the process underlying tracer mixing

309

and because the surface area is known (particularly at pH 5), we can estimate rates of

310

microscopic processes per sorbed Fe(II) that nominally give rise to the observed atom exchange


14

Page 15 of 29


311

behavior.

The reader is here reminded that because our simulation encompasses all atom

312

exchange events, including the exchanges to which the experiments are blind, the computed

313

exchange frequency is that associated with the underlying process step sustaining the

314

macroscopically observed tracer mixing rate.

315

Exchange-active surface site density

316

According to Handler et al.8, the nominal crystallographic density of surface-exposed Fe is

317

about 12.2 Fe/nm2. Surface complexation models31,32 used to fit pH-dependent Fe(II) sorption at

318

two types of goethite particles assume lower surface sites densities of 2 sites/nm2 or 3 sites/nm2,

319

respectively. In those studies, the experimental Fe(II) sorption data32 was obtained for particle

320

sizes (i.e., specific surface area, SSA of 54 and 78 m2/g) very similar to those in Handler et al.,8

321

(40 and 110 m2/g), and the resulting sorption isotherms provide a systematic basis predicting

322

both pH- and particle-size dependent sorbed Fe(II) densities. We therefore emphasize this

323

approach, and use the nominal crystallographic density as the bounding upper limit.

324

instance, at pH 7.5 predicted sorbed Fe(II) densities are 1.54 Fe/nm2 (particles SSA= 78m2/g)31

325

and 1.91 Fe/nm2 (SSA=54m2/g)32. This equates to 12.7% and 15.6% of surface exposed Fe(III)

326

atoms in contact with sorbed Fe(II), for nano- and microgoethite, respectively. At pH 5, sorbed

327

Fe(II) densities are much smaller: 0.2 Fe/nm2 (SSA=78 m2/g)31 and 0.24 Fe/nm2 (54 m2/g)32, and

328

corresponding surface Fe(III) contact values are 1.64% and 2%, respectively. By using these

329

Fe(II) surface densities as the those of exchange-active surface sites, we can estimate the atom

330

exchange rate per sorbed Fe(II) species, for all explored experimental conditions.

331

Fundamental rate of exchange

For

332

Table 1 shows calculated Fe atom exchange rate constants per exchange-active site, i.e., a

333

sorbed Fe(II) site, computed from our simulation results for exchange experiments in Handler et


15


Page 16 of 29

334

al.8 Computed rates encompassed the breath of assumed sorbed Fe(II) densities discussed above,

335

to establish sensitivity to such assumptions. For the pH 7.5 experiments, we approach the

336

complication of aggregation by assuming that because of its larger particle size the microrod data

337

is least prone to kinetics that are not surface controlled. The fitted results for the microrods show

338

an exchange rate per Fe(II) site that ranges from order 10-5 s-1 for the upper limit of 12.2

339

sites/nm2 to 10-4 s-1 for the likely more realistic SCM-fitting based estimate of 1.91 sites/nm2,

340

irrespective of the fast or slower domains. In an attempt to extract meaningful information from

341

pH 7.5 nanorod data, if we assume the exchange rate constant for the fast domain of the

342

microrod data also applies to the nanorod data in the fast domain we can back calculate the

343

effective active surface area in the highly aggregated nanorod experiments (down to 49.61 m2/g

344

from 110 m2/g). Using this effective surface area in the slow domain of the nanorod experiments

345

yields an exchange rate per site on the order of 10-5 s-1. Turning now to the more straightforward

346

pH 5 data, despite their markedly slower observed tracer exchange kinetics and lower sorbed

347

Fe(II) densities, our numerical simulations predict that the underlying exchange rate constants

348

are also in the vicinity of 10-5 s-1, for both nanorods and microrods. Thus, although the observed

349

tracer mixing rates and extents are substantially different across the range of experiments in

350

Handler et al.,8 our simulations reveal that the rate of the underlying process necessary to

351

describe all observed mixing rates is rather constant, ranging between 10-4 s-1 to 10-5 s-1. Taking

352

the pH 5 data as the most reliable, the average calculated exchange per Fe(II) site is

353

2.3±1 × 10-5 s-1.

354 355


16

Page 17 of 29

356 357 358


Table 1. Exchange rates estimated by fitting the stochastic simulation to experimental data from Handler et al.8 For pH 7.5 we distinguish two kinetic regimes nominally arising from aggregation (see text); no such effects pertain at pH 5. Specific results discussed in the text are highlighted. Recrystallization cycle time (days)

Specific Surface Area (SSA) [m2/g]

Exchange-Active Site Density (NFe) [Fe/nm2]

Exchange rate per site [s-1]

pH 7.5 nanorods 6.5 (fast)

12.5 (slow)

110

a

49.61e 49.61h

12.2b 3c 1.54d 1.54d 1.54d

8.993E-6 3.657E-5 7.124E-5 1.580E-4f 8.214E-5g

pH 7.5 microrods a

6.5 (fast)

40

8 (slow)

40a

12.2b 3c 1.91d 12.2b 3c 1.91d

2.473E-5 1.006E-4 1.580E-4 2.009E-5 8.171E-5 1.283E-4

pH 5.0 nanorods 205

110

a

12.2b 3c 0.2d

2.851E-7 1.160E-6 1.739E-5

pH 5.0 microrods 615 359 360 361 362 363 364 365

40

a

12.2b 3c 0.24d

2.614E-7 1.063E-6 1.329E-5

a

surface area determined (BET method) as reported by Handler et al.8,bexchange site density (Fe/nm2) estimated from crystallographic data by Handler et al.8,cFe(II) sorption density used in surface complexation modeling of Fe(II) sorption on goethite, dactive surface site density determined from the SCM fitting31,32 of the Fe(II) sorption edge based on most relevant particle sizes and at specific respective pH values,,eeffective surface area for (aggregated) nanorod suspension in the fast domain determined by fassuming the microrod fast domain exchange rate constant at the same pH, gexchange rate approximately corrected for surface area occlusion hthe effective surface area estimated for the (aggregated) nanorod suspension in the fast domain.

366


17


367

Page 18 of 29

Identifying the rate-limiting step

368

In the 1:1 conceptual model, because each atom exchange event is facilitated by a set of

369

reactions nominally linked in series (Fe(II) sorption, interfacial ET to goethite Fe(III), solid-state

370

electron transport, Fe(II) release), this estimated rate would thus correspond to the rate-limiting

371

step in the series. Using molecular dynamics simulations, Zarzycki et al.18 showed that Fe(II)

372

can form strong inner-sphere surface complexes to low-index goethite rod prismatic faces (e.g.,

373

101 and 100, space group Pnma), and that formation of these complexes from more distal outer-

374

sphere complex locations is largely free of energy barriers. This suggest Fe(II) adsorption into

375

surface-associated configurations compatible with interfacial electron transfer is not kinetically

376

inhibited. These inner-sphere Fe(II) surface complexes were found to be capable of exchanging

377

electrons with underlying goethite Fe(III) at rates ranging from ~ 1-100 s-1, at least five orders of

378

magnitude faster than the exchange rate that appears to govern tracer mixing based on the

379

present study. Likewise, molecular dynamics simulations revealed relatively fast subsurface

380

electron hopping pathways on these goethite faces as well (up to 106 s-1).18 By elimination, this

381

implicates the reductive Fe(II) release step of the conduction mechanism as the most likely rate

382

limiting process, the one that gives rise to the discovered ~ 2 × 10-5 s-1 fundamental exchange

383

rate underlying atom exchange.

384

Exchange and reductive dissolution rates

385

The rate of iron oxides reductive dissolution strongly depends on the type of reductant,1 however

386

in the case of dissolution by ascorbic acid, the process is believed to be controlled by Fe-release.

387

Therefore, we used the reported rates of goethite dissolution by ascorbate as proxies for Fe-

388

release/Fe-exchange rate. Handler et al.8 measured the reductive dissolution rate of their

389

nanorods and microrods in 10 mM ascorbic acid at pH 3 and reported very similar surface-area


18

Page 19 of 29


390

normalized rate coefficients for both particles, ~ 3 × 10-4 g m-2 d-1, which corresponds to an Fe

391

atom release rate of 3.7 × 10-5 nm-2 s-1. The reductive release rate of Fe(II) from goethite by

392

ascorbic acid is known to be relatively insensitive to pH changes in the acidic domain.34 Re-

393

expressing the fundamental exchange rate in similar terms for pH 5 data (microrods, slow

394

domain) yields 5.5 × 10-5 Fe nm-2 s-1.

395

dissolution rates of goethite8 are consistent with the rate of the underlying process revealed by

396

our stochastic simulations. This suggest that reductive Fe-release is in control of macroscopically

397

observed tracer exchange behavior.

398

Mechanistic insights

The correspondence is striking– these

reductive

399

Stochastic simulation analysis of two comprehensive experimental data sets for Fe isotopic

400

tracer mixing in the Fe(II)/goethite system reveal behavior consistent with the conduction

401

mechanism of atom exchange. Application of a 1:1 exchange constraint between solid and

402

solution phase Fe yields equilibration curves that can largely reproduce the macroscopically

403

observed shapes. Under pH 7.5 conditions aggregation appears to play a significant role in the

404

mixing kinetics, consistent with speculation in the original experimental studies.8,27 Two kinetic

405

regimes are present at pH 7.5, initial rapid Fe-exchange is followed by much slower, size-

406

dependent exchange that nonetheless enables complete equilibration of the isotopic compositions

407

within 30 days. At this pH, the mixing kinetics for smaller particle sizes is likely more strongly

408

overprinted by aggregation effects, suggesting care needs to be taken in their mechanistic

409

interpretation.

410

simulations reveal an underlying fundamental exchange rate consistent with the known release

411

rate of Fe(II) during reductive dissolution by ascorbic acid, one that is otherwise incompatible

412

with the remaining steps implicit in the conduction mechanism. The collective findings thus

At lower pH values where aggregation is not significant, our stochastic


19


Page 20 of 29

413

validate to some extent the applicability of the conduction mechanism of atom exchange in this

414

system, with the reductive Fe(II) release step as rate-limiting.

415

Environmental implications

416

The findings support a conceptual model for trace, heavy and/or radioactive ion fate and

417

transport in the subsurface with kinetics potentially dominated by the kinetics of incorporation

418

into and release from Fe(III)-(oxyhydr)oxide minerals. This geochemical pathway would be

419

driven by changes in redox conditions in iron rich soils and sediments and depend upon oxide

420

phase, particle size, relative proportions of Fe(II) and Fe(III), and solution pH.

421 422

ASSOCIATED CONTENT

423

Supporting Information. The results of the extraction and accumulation procedures applied to

424

the experimental data reported by Handler et al.8 Details of computations and fitting the

425

simulation to data reported by Handler et al.8 and Joshi et al.26).

426 427

AUTHOR INFORMATION

428

Corresponding Author

429

*e-mail: [email protected], tel. +48 730 628 237, fax. +48 22 343 0000;

430

[email protected]

431

Author Contributions

432

The manuscript was written through contributions of all authors. All authors have given approval

433

to the final version of the manuscript.


20

Page 21 of 29

434


ACKNOWLEDGMENT

435

This work is based on research sponsored by the U.S. Department of Energy (DOE), Office of

436

Science, Office of Basic Energy Sciences (BES), Division of Chemical Sciences, Geosciences

437

and Biosciences, through its Geosciences program at Pacific Northwest National Laboratory

438

(PNNL). The authors acknowledge helpful discussions with Clark Johnson, Michelle Scherer,

439

and Andrew Frierdich during conceptualization of this study. A portion of this research was

440

performed using EMSL, a national scientific user facility sponsored by the DOE's Office of

441

Biological and Environmental Research and located at PNNL. PNNL is a multiprogram national

442

laboratory operated for DOE by Battelle.

443 444

REFERENCES

445

[1] Cornell, R. M.; Schwertmann, U. The Iron Oxides 2nd edition, Wiley-VCH, Weinheim,

446

2003.

447

[2] Tamaura, Y.; Ito, K.; Katsura, T. Transformation of γ-FeO(OH) to Fe3O4 by Adsorption of

448

Iron(II) Ion on γ-FeO(OH). J. Chem. Soc., Dalton Trans. 1983, 189–6.

449

[3] Yang, L.; Steefel, C. I.; Marcus, M. A.; Bargar, J. R. Kinetics of Fe(II)-Catalyzed

450

Transformation of 6-line Ferrihydrite under Anaerobic Flow Conditions. Environ. Sci. Technol.

451

2010, 44, 5469–5475.

452

[4] Hansel, C. M.; Benner, S. G.; Fendorf, S. Competing Fe(II)-Induced Mineralization Pathways

453

of Ferrihydrite. Environ. Sci. Technol. 2005, 39, 7147–7153.


21


Page 22 of 29

454

[5] Boland, D. D.; Collins, R. N.; Miller, C. J.; Glover, C. J.; Waite, T. D. Effect of Solution and

455

Solid-Phase Conditions on the Fe(II)-Accelerated Transformation of Ferrihydrite to

456

Lepidocrocite and Goethite. Environ. Sci. Technol. 2014, 48, 5477–5485.

457

[6] Pedersen, H. D.; Postma, D.; Jakobsen, R.; Larsen, O. Fast Transformation of Iron

458

Oxyhydroxides by the Catalytic Action of Aqueous Fe(II). Geochim. Cosmochim. Acta 2005, 69,

459

3967–3977.

460

[7] Handler, R. M.; Beard, B. L.; Johnson, C. M.; Scherer, M. M. Atom Exchange between

461

Aqueous Fe(II) and Goethite: An Fe Isotope Tracer Study. Environ. Sci. Technol. 2009, 43,

462

1102–1107.

463

[8] Handler, R. M.; Frierdich, A. J.; Johnson, C. M.; Rosso, K. M.; Beard, B. L.; Wang, C.;

464

Latta, D. E.; Neumann, A.; Pasakarnis, T.; Premaratne, W. A. P. J.; et al. Fe(II)-Catalyzed

465

Recrystallization of Goethite Revisited. Environ. Sci. Technol. 2014, 48, 11302–11311.

466

[9] Beard, B. L.; Handler, R. M.; Scherer, M. M.; Wu, L.; Czaja, A. D.; Heimann, A.; Johnson,

467

C. M. Iron Isotope Fractionation between Aqueous Ferrous Iron and Goethite. Earth Planet. Sci.

468

Lett. 2010, 295, 241–250.

469

[10] Frierdich, A. J.; Scherer, M. M.; Bachman, J. E.; Engelhard, M. H.; Rapponotti, B. W.;

470

Catalano, J. G. Inhibition of Trace Element Release During Fe(II)-Activated Recrystallization of

471

Al-, Cr-, and Sn-Substituted Goethite and Hematite. Environ. Sci. Technol. 2012, 46,

472

10031−10039.

473

[11] Frierdich, A. J.; Beard, B. L.; Reddy, T. R.; Scherer, M. M.; Johnson, C. M. Iron Isotope

474

Fractionation between Aqueous Fe(II) and Goethite Revisited: New Insights based on a Multi-


22

Page 23 of 29


475

direction Approach to Equilibrium and Isotopic Exchange Rate Modification. Geochim.

476

Cosmochim. Acta 2014, 139, 383–398.

477

[12] Frierdich, A. J.; Helgeson, M.; Liu, C.; Wang, C.; Rosso, K. M.; Scherer, M. M. Iron Atom

478

Exchange between Hematite and Aqueous Fe (II). Environ. Sci. Technol. 2015, 49, 8479–8486.

479

[13] Gorski, C. A.; Fantle, M. S. Stable Mineral Recrystallization in Low Temperature Aqueous

480

Systems: A Critical Review. Geochim. Cosmochim. Acta 2016, 198, 439–465.

481

[14] Catalano, J. G.; Luo, Y.; Otemuyiwa, B. Effect of Aqueous Fe(II) on Arsenate Sorption on

482

Goethite and Hematite. Environ. Sci. Technol. 2011, 45, 8826–8833.

483

[15] Yanina, S. V.; Rosso, K. M. Linked Reactivity at Mineral-Water Interfaces through Bulk

484

Crystal Conduction. Science 2008, 320, 218–222.

485

[16] Rosso, K. M.; Yanina, S. V.; Gorski, C. A.; Larese-Casanova, P.; Scherer, M. M.

486

Connecting Observations of Hematite (α-Fe2O3) Growth Catalyzed by Fe(II). Environ. Sci.

487

Technol. 2010, 44, 61–67.

488

[17] Gorski, C. A.; Scherer, M. M. Fe2+ Sorption at the Fe Oxide-Water Interface: A Revised

489

Conceptual Framework. In Aquatic Redox Chemistry; ACS Symposium Series; American

490

Chemical Society: Washington, DC, 2011; Vol. 1071, pp. 315–343.

491

[18] Zarzycki, P.; Kerisit, S.; Rosso, K. M. Molecular Dynamics Study of Fe(II) Adsorption,

492

Electron Exchange, and Mobility at Goethite (α-FeOOH) Surfaces. J. Phys. Chem. C 2015, 119,

493

3111–3123.

494

[19] Williams, A. G. B.; Scherer, M. M. Spectroscopic Evidence for Fe(II)−Fe(III) Electron

495

Transfer at the Iron Oxide−Water Interface. Environ. Sci. Technol. 2004, 38, 4782–4790.


23


Page 24 of 29

496

[20] Larese-Casanova, P.; Scherer, M. M. Fe(II) Sorption on Hematite: New Insights Based on

497

Spectroscopic Measurements. Environ. Sci. Technol. 2007, 41, 471–477.

498

[21] Iordanova, N.; Dupuis, M.; Rosso, K. M. Charge Transport in Metal Oxides: A Theoretical

499

Study of Hematite α-Fe2O3. J. Chem. Phys. 2005, 122, 144305–144311.

500

[22] Kerisit, S.; Zarzycki, P.; Rosso, K. M. Computational Molecular Simulation of the

501

Oxidative Adsorption of Ferrous Iron at the Hematite (001)–Water Interface. J. Phys. Chem. C

502

2015, 119, 9242–9252.

503

[23] Alexandrov, V.; Rosso, K. M. Ab Initio Modeling of Fe(II) Adsorption and Interfacial

504

Electron Transfer at Goethite (α-FeOOH) Surfaces. Phys. Chem. Chem. Phys. 2015, 17, 14518–

505

14531

506

[24] Zarzycki, P.; Smith, D. M.; Rosso, K. M. Proton Dynamics on Goethite Nanoparticles and

507

Coupling to Electron Transport. J. Chem. Theory Comput. 2015, 11, 1715–1724.

508

[25] Chatman, S.; Zarzycki, P.; Rosso, K. M. Surface Potentials of (001), (012), (113) Hematite

509

(α-Fe2O3) Crystal Faces in Aqueous Solution. Phys. Chem. Chem. Phys. 2013, 15, 13911–13921.

510

[26] Chatman, S.; Zarzycki, P.; Rosso, K. M. Spontaneous Water Oxidation at Hematite (α-

511

Fe2O3) Crystal Faces. ACS Appl. Mater. Interfaces 2015, 7, 1550–1559.

512

[27] Joshi, P.; Gorski, C. A. Anisotropic Morphological Changes in Goethite during Fe2+-

513

Catalyzed Recrystallization. Environ. Sci. Technol. 2016, 50, 7315–7324.

514

[28] Israelachvili, J. N. Intermolecular and Surface Forces, 3rd edition, Academic Press,

515

Burlington, 2011.


24

Page 25 of 29


516

[28] Venema, P.; Hiemstra, T.; Weidler, P. G.; van Riemsdijk, W. H. Intrinsic Proton Affinity of

517

Reactive Surface Groups of Metal (Hydr)oxides: Application to Iron (Hydr)oxides. J. Colloid

518

Interface Sci. 1998, 198, 282–295.

519

[29] Hiemstra, T.; Venema, P.; Riemsdijk, W. H. V. Intrinsic Proton Affinity of Reactive Surface

520

Groups of Metal (Hydr)oxides: The Bond Valence Principle. J. Colloid Interface Sci. 1996, 184,

521

680–692.

522

[30] Hiemstra, T.; van Riemsdijk, W. H. Adsorption and Surface Oxidation of Fe(II) on Metal

523

(Hydr)Oxides. Geochim. Cosmochim. Acta 2007, 71, 5913–5933.

524

[31] Dixit, S.; Hering, J. G. Sorption of Fe(II) and As(III) on Goethite in Single- and Dual-

525

Sorbate Systems. Chem. Geol. 2006, 228, 6–15.

526

[32] Kosmulski, M. Surface Charging and Points of Zero Charge, CRC Press, Boca Raton, 2009.

527

[33] Larsen, O.; Postma, D. Kinetics of reductive bulk dissolution of lepidocrocite, ferrihydrite,

528

and goethite. Geochim. Cosmochim. Acta 2001, 65, 1367–1379.

529


25


Page 26 of 29

530 531

Figure 1. Stochastic model of Fe isotopic exchange between solid (δ57/56Fe(III)) and aqueous

532

(δ57/56Fe(II)) pools of iron (a). The simulated isotopic exchange dynamics for systems with a

533

varying ratio of solid to liquid iron pools (Fe(II) : Fe(III)) and initial isotopic compositions equal

534

to δ57/56Fe(II)=840‰ (aqueous) and δ57/56Fe(III)=-0.12 (iron oxide), respectively.

535 536


26

Page 27 of 29


537 538

Figure 2. Fitting of stochastic modeling simulations to the experimental data reported by Hander

539

et al. for goethite microrods (a) and nanorods (b) at pH=7.5. Fast initial exchange is shown in

540

black curves, the slower second stage of exchange is shown in red.

541


27


Page 28 of 29

542 543

Figure 3. Fitting of stochastic exchange simulations to experimental data reported by Hander et

544

al.8 for goethite nanorods (a) and microrods (b) at pH=5.0. That exchange is incomplete after 30

545

days is well constrained by the data, indicating substantially less than 100% exchangeable Fe

546

(%exc.). A recrystallization cycle time of 205 days with 7% exchangeable Fe fits the nanorod

547

data, whereas 615 days and 2% exchangeable Fe fits the microrod data. Nanorods (110 m2/g)8

548

should have about 3 times more Fe exchanged than microgoethite (40 nm2/g)8, which is

549

consistent with the kinetics and differences in δ57/56Fe(III)solid (7% nanorods and 2 %

550

microrods) found here. For comparison, system behavior expected for 100% exchange at the

551

same respective recrystallization cycle times is shown in red curves.

552 553 554


28

Page 29 of 29


555 556 557

TOC GRAPHIC

558


29

Stochastic Simulation of Isotopic Exchange ... - ACS Publications

Recommend Documents