Combined Experimental and Theoretical Approach to the Kinetics of

Feb 22, 2017 - (3, 4) In turn and just to name a few, synthetic processes of crystallization range from single-crystal breeding(5) to materials produc...
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Letter

A Combined Experimental and Theoretical Approach of the Kinetics of Magnetite Crystal Growth from Primary Particles Marc Widdrat, Emanuel Schneck, Victoria Eleonore Reichel, Jens Baumgartner, Luca Bertinetti, Wouter J.E.M. Habraken, Klaas Bente, Peter Fratzl, and Damien Faivre J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b02977 • Publication Date (Web): 22 Feb 2017 Downloaded from http://pubs.acs.org on February 22, 2017

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A Combined Experimental and Theoretical Approach of the Kinetics

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of Magnetite Crystal Growth from Primary Particles

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Marc Widdrat, Emanuel Schneck, Victoria Reichel, Jens Baumgartner, Luca Bertinetti, Wouter

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Habraken, Klaas Bente, Peter Fratzl and Damien Faivre*

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Department of Biomaterials, Max Planck Institute of Colloids and Interfaces, Science Park Golm,

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14424 Potsdam, Germany.

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Email: [email protected]

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Abstract

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It is now recognized that nucleation and growth of crystals can occur not only by the addition of

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solvated ions but also by accretion of nanoparticles, in a process called non-classical crystallization.

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The theoretical framework of such processes has only started to be described, partly due to the lack

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of kinetic or thermodynamic data. Here, we study the growth of magnetite nanoparticles from

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primary particles – nm-sized amorphous iron-rich precursors – in aqueous solution at different

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temperatures. We propose a theoretical framework to describe the growth of the nanoparticles and

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model both a diffusion-limited and a reaction-limited pathway to determine which of these best

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describe the rate-limiting step of the process. We show that based on the measured iron

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concentration and the related calculated concentration of primary particles at the steady state,

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magnetite growth is likely a reaction-limited process, and within the framework of our model, we

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propose a phase diagram to summarize the observations.

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Table of content image

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Crystallization is a fundamental process, which plays a key role in both natural and artificial

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phenomena. Natural examples encompass e.g. mineral formation in geological settings1, snowflakes2

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or biomineral formation by organisms3, 4. In turn and just to name a few, synthetic processes of

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crystallization range from single crystal breeding5 to materials production for electronics6 or thin

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films7. Processes behind crystal nucleation and growth have been investigated experimentally 8, 9 as

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well as theoretically10, 11, 12. However, the exact mechanisms associated with crystallization have

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often remained unclear, specifically in the case of aqueous processes. The classical nucleation

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theory13 considers crystals being formed from single ions or molecules and was successfully utilized

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to describe crystallization for many years. Recent studies showed, however, strong evidences for

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alternative, so called non-classical nucleation routes14. These include the aggregation of ion-

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association complexes (calcium phosphate15), clusters (calcium carbonate16) or primary particles (PP)

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(iron oxides17). The theoretical framework in which such processes take place has started to

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emerge18, 19. For example, the classical nucleation theory has been amended to take into account the

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presence of this nanoparticulate matter in the pre-nucleation stage of magnetite formation17.

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Magnetite is an ubiquitous naturally occurring mineral with unique magnetic properties. It was

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shown by cryogenic transmission electron microscopy imaging (cryo-TEM) that it forms from PP,

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which aggregate and subsequently nucleate17. It was also suggested that the crystal growth is PP-

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mediated and that despite the involvement of PP, it follows the predictions of the classical theories 2 ACS Paragon Plus Environment

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and in particular a reaction-limited mechanism was proposed17. However, experimental data are

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lacking for a quantitative description of the pathway20. Here, we thus present a study of magnetite

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nanoparticle growth at different temperatures to determine the growth rates and the rate-limiting

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step.

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In our assay, magnetite was formed by the addition of an iron solution in a reactor kept at constant

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pH by the concomitant addition of a NaOH solution (see experimental section for details)21, 22. The

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reactor was kept at constant temperature. Magnetite crystal growth was investigated at five

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different temperatures (5, 10, 15, 20 and 25 °C) and each synthesis was reproduced in

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quadruplicates. X-ray diffraction patterns show typical magnetite peaks at every time point (Figure

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1a) and a closer view of the 311-peak reveals its continuous narrowing (Figure 1b). The mean particle

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diameter was calculated from the Scherrer equation23 and particle growth was observed over time as

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shown in Figure 1c. Transmission electron microscopy shows typical, highly aggregated magnetite

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nanoparticles (Figure 1d).

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Figure 1: Summary of experimental results representative for all samples (a) The XRD diagram

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shows a typical magnetite pattern at every time point (black solid lines). NaCl (black dotted lines) is

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sometimes observed besides magnetite. The chronological evolution is indicated by a color scale

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from yellow (early state) to red (late state), with the magnetite peaks indexed. (b) The insight into

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the 311-peak reveals the decrease of the full width at half maximum. (c) Radius of the

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nanoparticles (r) with standard error as a function of time determined by averaging data for all

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syntheses at 15 °C and (d) TEM image of one representative sample (scale bar: 200nm) showing

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highly aggregated nanoparticles of magnetite (inset: SAED pattern with magnetite rings).

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Particle growth over time is observed for all temperatures (Figure 2, see Supplementary Information

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(SI) for the full data set). After eight hours, larger particles are obtained at higher temperatures

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(compare purple points at 25 °C with red points obtained at 5 °C). In the experimental scenario

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stemming from our earlier cryo-EM observations17, the concentration of the PP ([PP]) is time

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dependent during the first minutes of the reaction prior to magnetite nucleation. Then [PP] reaches a

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steady-state where growth of the initially nucleated particles of a size rinit takes over nucleation of

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new particles. Here, we first measure the particle size after one hour and therefore already are in the

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steady state regime.

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Figure 2: Evolution of the particle radius as obtained experimentally for various temperatures

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(symbols with standard errors) and as predicted by two different models for particle growth with

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adjustable model parameters (solid lines). (a) Diffusion-limited growth. (b) Reaction-limited

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growth.

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The evolution of the particle size was modeled using the different scenarios qualitatively discussed

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above in order to extract more quantitative insights. The model assumes that the PPs (radius

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r0 = 1 nm17, constant bulk concentration c 0 ) are homogeneously distributed within the medium,

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which is justified as the solutions were kept under continuous stirring, and bind to the growing

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magnetite particles with frequency c0 k . Here, k is a rate as specified below that may depend on 5 ACS Paragon Plus Environment

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temperature and the radius r of the growing particles. Each binding event increases the volume V of

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a growing particle by an increment V0 = 4π r03 3 , corresponding to the volume of a PP, so that

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dV dt = V0 c 0 k . For an isotropic growth of the forming particles, as expected for a mineral

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crystallizing in a cubic system, we have dV = 4πr dr and obtain a non-linear differential equation

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for the evolution of the particle radius:

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dr V0 c 0 k (r , T ) , = dt 4πr 2

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For the rate k , we consider two mechanisms, diffusion-limited ( k = k diff ) and reaction-limited (

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k = k reac ) growth. We further assume that the magnetite particles grow from an initial particle with

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a radius rinit. For the diffusion-limited case the Smoluchowsky description is used24:

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(Equation 1)

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k diff = 4π (r + r0 )(D + D 0 ) .

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Note that in this treatment neither the extension of the PPs nor the diffusion of the initially formed

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magnetite nanoparticles is neglected. The diffusion coefficients are represented with the common

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Stokes-Einstein result involving the temperature-dependent solution viscosity ƞ:

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D=

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For ƞ , we used the tabulated values of water at 5°C, 10°C, 15°C, 20°C, and 25°C25. For the reaction-

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limited case, we set the reactive area equal to the surface of a particle with the effective radius for

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the two-particle interaction (r + r0), in analogy with Eq. 2, and we employ a simple Arrhenius factor26

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in order to account for an activation energy barrier ∆U:

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k reac = 4π ( r + r0 ) 2 k 0 e − ∆ U

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Here, k0 is a temperature-independent pre-exponential factor reflecting both a finite reaction

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attempt-frequency f0 and the effect of an entropic contribution to the reaction free energy barrier

(Equation 2)

k BT k T and D0 = B . 6πηr 6πηr0

(Equation 3)

( k BT )

(Equation 4)

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∆S, k 0 = f 0 e ∆ S / k B 27. It should be noted that in Eq. 1 the growth of the magnetite particles is

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assumed to be proportional to the concentration c0 of the PP, i.e., it is assumed that all PP adsorb

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and react independently. Eq. 1 thus does not explicitly account for crowding effects relevant in the

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reaction-limited case for high c0. In practice, however, when using Eq. 1 in combination with Eq. 4,

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such crowding effects would merely manifest in a reduced numerical value of k0.

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The experimental data on the particle radii obtained at the different temperatures were modeled

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simultaneously based on common sets of model parameters. The free parameter rinit was common to

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all temperatures, since it did not exhibit any significant temperature-dependence (see supporting

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information). In the diffusion-limited scenario, the adjustable parameters were c0 and rinit. In the

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reaction-limited scenario ∆U, a prefactor A = c0k0, and rinit were adjustable. Equation 1 was solved

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numerically and the parameters were adjusted in a least-square fit to achieve the best agreement,

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i.e. with minimal χ2-deviation between experimental and modeled values of r(t) for the 5 different

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temperatures. The best-matching model parameters and the corresponding reduced χ2 values are

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summarized in Table 1.

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In the diffusion-limited case (Figure 2a), the splitting of the theoretical curves for different

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temperatures is due to the temperature-dependence of the diffusion coefficients (Equation 3)

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involving the temperature-dependent solution viscosity. In the activation-limited case (Figure 2b),

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the splitting is linked to the height of the activation barrier ∆U (Equation 4). It is seen that our

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experimental data are reproduced reasonably well by both the diffusion limited model and the

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reaction limited model, although the diffusion-limited model under-represents the temperature-

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dependence. Even if e.g. the 5 °C curve presented a poorer match, we restrained our analysis here to

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the global data set and to the two main processes typically envisaged for such process to get first

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hints towards a mechanistic picture. The obtained best-matching initial radii (Table 1: 7.2 nm for

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diffusion-limited, 9.4 nm for a reaction-limited process) are slightly larger than but comparable to

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those of the smallest magnetite particles found in earlier experiments (5 nm17). The reduced χ27 ACS Paragon Plus Environment

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2 2 ≈ 2.9) and reaction-limited scenarios ( χ red ≈ 3.2) deviation is similar for diffusion-limited ( χ red

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(Table 1). Therefore, without measurement of an additional experimental parameter, the modelling

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is also not able to decipher the origin of the process.

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One such crucial parameter is the concentration of the PPs: It indeed appears as fitting parameter in

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the diffusion-limited model and is concomitantly experimentally accessible. We thus determined the

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PP concentration via the iron amount as measured by ICP-OES. To this end, we considered as in the

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model only three iron-based components: the ions, the PPs and the magnetite nanoparticles. Due to

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the very low solubility of iron in water, there is less than 1 % iron left in the solution in the form of

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ions already from pH 420. Therefore, virtually all iron is contained in PPs or larger magnetite

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nanoparticles. We magnetically separated the magnetite nanoparticles from the solution using a

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100 mT magnet. The gradient produced by such a magnet attracts the magnetite nanoparticles

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within few minutes, whereas days would be necessary for the PPs to be removed (See SI for full

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calculations). The measured iron concentration therefore originates from the PPs only. Assuming the

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PPs are “proto-magnetite” following the results obtained on calcium carbonate system28, we

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obtained: c0 ≈ 2.3 x 1020 m-3 (see SI for full calculations). If we however choose a ferrihydrite model

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based on earlier considerations17, and using a dedicated model for a hypothetic ferrihydrite spherical

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particles20 with crystallographic data from Michel et al.29, a similar order of magnitude is obtained (c0

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≈ 15 x 1020 m-3). As an outcome of the modelled diffusion-limited scenario, an at least 4 orders of

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magnitude lower value of c0 is obtained (Table 1). Therefore, the growth of magnetite is almost

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certainly not diffusion-limited but lies instead in the reaction-limited regime. Therefore, a free-

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energy barrier to PP adsorption is required, which we found as low as 24 kJ × mol-1 (table 1). For

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activation-limited crystal growth, an activation energy between 40 and 80 kJ mol-1 is expected in the

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case of a process based on the addition of ions5. Alternatively, in a system involving poorly soluble

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species, oriented attachment of ZnS via nanoparticles leads to activation energies in the range of 125

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kJ × mol-1 30. In this case, the activation energy for this surface activation-limited process becomes

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energetically more demanding because of the large number of atoms being integrated into the

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crystal surface simultaneously.

Model

2 χ red

c0

∆U

k0

rinit

diffusion-limited

2.9

1.3 x 1016 m-3 a

-

-

7.2 nm

reaction-limited

3.2

-

24 kJ x mol-1

0.002 s-1 b

9.4 nm

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Table 1: Parameters of diffusion-limited and reaction-limited models of the evolution of the

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particle radius. a The experimental value is c0 = 2.3 x 1020 m-3 assuming PPs with magnetite

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stoichiometry. b Using c0 = 2.3 x 1020 m-3.

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Using the most conservative experimental estimate of c0 (c0 = 2.3 x 1020 m-3 for PP with magnetite

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stoichiometry), the reaction-limited model yields an estimate of the pre-exponential factor,

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k0 ≈ 0.002 s-1 and, together with ∆U and rinit, a comprehensive description of the particle growth

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process and its temperature dependence.

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As pointed out above, the described growth of the magnetite particles by adsorption of PPs takes

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place in the reaction-limited regime, i.e., kreac