Subscriber access provided by University of Newcastle, Australia
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
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.
The Journal of Physical Chemistry Letters 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 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
The Journal of Physical Chemistry Letters
1
A Combined Experimental and Theoretical Approach of the Kinetics
2
of Magnetite Crystal Growth from Primary Particles
3
Marc Widdrat, Emanuel Schneck, Victoria Reichel, Jens Baumgartner, Luca Bertinetti, Wouter
4
Habraken, Klaas Bente, Peter Fratzl and Damien Faivre*
5 6
Department of Biomaterials, Max Planck Institute of Colloids and Interfaces, Science Park Golm,
7
14424 Potsdam, Germany.
8
Email:
[email protected] 9
Abstract
10
It is now recognized that nucleation and growth of crystals can occur not only by the addition of
11
solvated ions but also by accretion of nanoparticles, in a process called non-classical crystallization.
12
The theoretical framework of such processes has only started to be described, partly due to the lack
13
of kinetic or thermodynamic data. Here, we study the growth of magnetite nanoparticles from
14
primary particles – nm-sized amorphous iron-rich precursors – in aqueous solution at different
15
temperatures. We propose a theoretical framework to describe the growth of the nanoparticles and
16
model both a diffusion-limited and a reaction-limited pathway to determine which of these best
17
describe the rate-limiting step of the process. We show that based on the measured iron
18
concentration and the related calculated concentration of primary particles at the steady state,
19
magnetite growth is likely a reaction-limited process, and within the framework of our model, we
20
propose a phase diagram to summarize the observations.
1 ACS Paragon Plus Environment
The Journal of Physical Chemistry Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1 2
Table of content image
3 4
Crystallization is a fundamental process, which plays a key role in both natural and artificial
5
phenomena. Natural examples encompass e.g. mineral formation in geological settings1, snowflakes2
6
or biomineral formation by organisms3, 4. In turn and just to name a few, synthetic processes of
7
crystallization range from single crystal breeding5 to materials production for electronics6 or thin
8
films7. Processes behind crystal nucleation and growth have been investigated experimentally 8, 9 as
9
well as theoretically10, 11, 12. However, the exact mechanisms associated with crystallization have
10
often remained unclear, specifically in the case of aqueous processes. The classical nucleation
11
theory13 considers crystals being formed from single ions or molecules and was successfully utilized
12
to describe crystallization for many years. Recent studies showed, however, strong evidences for
13
alternative, so called non-classical nucleation routes14. These include the aggregation of ion-
14
association complexes (calcium phosphate15), clusters (calcium carbonate16) or primary particles (PP)
15
(iron oxides17). The theoretical framework in which such processes take place has started to
16
emerge18, 19. For example, the classical nucleation theory has been amended to take into account the
17
presence of this nanoparticulate matter in the pre-nucleation stage of magnetite formation17.
18
Magnetite is an ubiquitous naturally occurring mineral with unique magnetic properties. It was
19
shown by cryogenic transmission electron microscopy imaging (cryo-TEM) that it forms from PP,
20
which aggregate and subsequently nucleate17. It was also suggested that the crystal growth is PP-
21
mediated and that despite the involvement of PP, it follows the predictions of the classical theories 2 ACS Paragon Plus Environment
Page 2 of 13
Page 3 of 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
The Journal of Physical Chemistry Letters
1
and in particular a reaction-limited mechanism was proposed17. However, experimental data are
2
lacking for a quantitative description of the pathway20. Here, we thus present a study of magnetite
3
nanoparticle growth at different temperatures to determine the growth rates and the rate-limiting
4
step.
5
In our assay, magnetite was formed by the addition of an iron solution in a reactor kept at constant
6
pH by the concomitant addition of a NaOH solution (see experimental section for details)21, 22. The
7
reactor was kept at constant temperature. Magnetite crystal growth was investigated at five
8
different temperatures (5, 10, 15, 20 and 25 °C) and each synthesis was reproduced in
9
quadruplicates. X-ray diffraction patterns show typical magnetite peaks at every time point (Figure
10
1a) and a closer view of the 311-peak reveals its continuous narrowing (Figure 1b). The mean particle
11
diameter was calculated from the Scherrer equation23 and particle growth was observed over time as
12
shown in Figure 1c. Transmission electron microscopy shows typical, highly aggregated magnetite
13
nanoparticles (Figure 1d).
14
3 ACS Paragon Plus Environment
The Journal of Physical Chemistry Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1 2
Figure 1: Summary of experimental results representative for all samples (a) The XRD diagram
3
shows a typical magnetite pattern at every time point (black solid lines). NaCl (black dotted lines) is
4
sometimes observed besides magnetite. The chronological evolution is indicated by a color scale
5
from yellow (early state) to red (late state), with the magnetite peaks indexed. (b) The insight into
6
the 311-peak reveals the decrease of the full width at half maximum. (c) Radius of the
7
nanoparticles (r) with standard error as a function of time determined by averaging data for all
8
syntheses at 15 °C and (d) TEM image of one representative sample (scale bar: 200nm) showing
9
highly aggregated nanoparticles of magnetite (inset: SAED pattern with magnetite rings).
10
4 ACS Paragon Plus Environment
Page 4 of 13
Page 5 of 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
The Journal of Physical Chemistry Letters
1
Particle growth over time is observed for all temperatures (Figure 2, see Supplementary Information
2
(SI) for the full data set). After eight hours, larger particles are obtained at higher temperatures
3
(compare purple points at 25 °C with red points obtained at 5 °C). In the experimental scenario
4
stemming from our earlier cryo-EM observations17, the concentration of the PP ([PP]) is time
5
dependent during the first minutes of the reaction prior to magnetite nucleation. Then [PP] reaches a
6
steady-state where growth of the initially nucleated particles of a size rinit takes over nucleation of
7
new particles. Here, we first measure the particle size after one hour and therefore already are in the
8
steady state regime.
9
10 11
Figure 2: Evolution of the particle radius as obtained experimentally for various temperatures
12
(symbols with standard errors) and as predicted by two different models for particle growth with
13
adjustable model parameters (solid lines). (a) Diffusion-limited growth. (b) Reaction-limited
14
growth.
15 16
The evolution of the particle size was modeled using the different scenarios qualitatively discussed
17
above in order to extract more quantitative insights. The model assumes that the PPs (radius
18
r0 = 1 nm17, constant bulk concentration c 0 ) are homogeneously distributed within the medium,
19
which is justified as the solutions were kept under continuous stirring, and bind to the growing
20
magnetite particles with frequency c0 k . Here, k is a rate as specified below that may depend on 5 ACS Paragon Plus Environment
The Journal of Physical Chemistry Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 6 of 13
1
temperature and the radius r of the growing particles. Each binding event increases the volume V of
2
a growing particle by an increment V0 = 4π r03 3 , corresponding to the volume of a PP, so that
3
dV dt = V0 c 0 k . For an isotropic growth of the forming particles, as expected for a mineral
4
crystallizing in a cubic system, we have dV = 4πr dr and obtain a non-linear differential equation
5
for the evolution of the particle radius:
6
dr V0 c 0 k (r , T ) , = dt 4πr 2
7
For the rate k , we consider two mechanisms, diffusion-limited ( k = k diff ) and reaction-limited (
8
k = k reac ) growth. We further assume that the magnetite particles grow from an initial particle with
9
a radius rinit. For the diffusion-limited case the Smoluchowsky description is used24:
2
(Equation 1)
10
k diff = 4π (r + r0 )(D + D 0 ) .
11
Note that in this treatment neither the extension of the PPs nor the diffusion of the initially formed
12
magnetite nanoparticles is neglected. The diffusion coefficients are represented with the common
13
Stokes-Einstein result involving the temperature-dependent solution viscosity ƞ:
14
D=
15
For ƞ , we used the tabulated values of water at 5°C, 10°C, 15°C, 20°C, and 25°C25. For the reaction-
16
limited case, we set the reactive area equal to the surface of a particle with the effective radius for
17
the two-particle interaction (r + r0), in analogy with Eq. 2, and we employ a simple Arrhenius factor26
18
in order to account for an activation energy barrier ∆U:
19
k reac = 4π ( r + r0 ) 2 k 0 e − ∆ U
20
Here, k0 is a temperature-independent pre-exponential factor reflecting both a finite reaction
21
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)
6 ACS Paragon Plus Environment
Page 7 of 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
The Journal of Physical Chemistry Letters
1
∆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
2
assumed to be proportional to the concentration c0 of the PP, i.e., it is assumed that all PP adsorb
3
and react independently. Eq. 1 thus does not explicitly account for crowding effects relevant in the
4
reaction-limited case for high c0. In practice, however, when using Eq. 1 in combination with Eq. 4,
5
such crowding effects would merely manifest in a reduced numerical value of k0.
6
The experimental data on the particle radii obtained at the different temperatures were modeled
7
simultaneously based on common sets of model parameters. The free parameter rinit was common to
8
all temperatures, since it did not exhibit any significant temperature-dependence (see supporting
9
information). In the diffusion-limited scenario, the adjustable parameters were c0 and rinit. In the
10
reaction-limited scenario ∆U, a prefactor A = c0k0, and rinit were adjustable. Equation 1 was solved
11
numerically and the parameters were adjusted in a least-square fit to achieve the best agreement,
12
i.e. with minimal χ2-deviation between experimental and modeled values of r(t) for the 5 different
13
temperatures. The best-matching model parameters and the corresponding reduced χ2 values are
14
summarized in Table 1.
15
In the diffusion-limited case (Figure 2a), the splitting of the theoretical curves for different
16
temperatures is due to the temperature-dependence of the diffusion coefficients (Equation 3)
17
involving the temperature-dependent solution viscosity. In the activation-limited case (Figure 2b),
18
the splitting is linked to the height of the activation barrier ∆U (Equation 4). It is seen that our
19
experimental data are reproduced reasonably well by both the diffusion limited model and the
20
reaction limited model, although the diffusion-limited model under-represents the temperature-
21
dependence. Even if e.g. the 5 °C curve presented a poorer match, we restrained our analysis here to
22
the global data set and to the two main processes typically envisaged for such process to get first
23
hints towards a mechanistic picture. The obtained best-matching initial radii (Table 1: 7.2 nm for
24
diffusion-limited, 9.4 nm for a reaction-limited process) are slightly larger than but comparable to
25
those of the smallest magnetite particles found in earlier experiments (5 nm17). The reduced χ27 ACS Paragon Plus Environment
The Journal of Physical Chemistry Letters
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1
2 2 ≈ 2.9) and reaction-limited scenarios ( χ red ≈ 3.2) deviation is similar for diffusion-limited ( χ red
2
(Table 1). Therefore, without measurement of an additional experimental parameter, the modelling
3
is also not able to decipher the origin of the process.
4
One such crucial parameter is the concentration of the PPs: It indeed appears as fitting parameter in
5
the diffusion-limited model and is concomitantly experimentally accessible. We thus determined the
6
PP concentration via the iron amount as measured by ICP-OES. To this end, we considered as in the
7
model only three iron-based components: the ions, the PPs and the magnetite nanoparticles. Due to
8
the very low solubility of iron in water, there is less than 1 % iron left in the solution in the form of
9
ions already from pH 420. Therefore, virtually all iron is contained in PPs or larger magnetite
10
nanoparticles. We magnetically separated the magnetite nanoparticles from the solution using a
11
100 mT magnet. The gradient produced by such a magnet attracts the magnetite nanoparticles
12
within few minutes, whereas days would be necessary for the PPs to be removed (See SI for full
13
calculations). The measured iron concentration therefore originates from the PPs only. Assuming the
14
PPs are “proto-magnetite” following the results obtained on calcium carbonate system28, we
15
obtained: c0 ≈ 2.3 x 1020 m-3 (see SI for full calculations). If we however choose a ferrihydrite model
16
based on earlier considerations17, and using a dedicated model for a hypothetic ferrihydrite spherical
17
particles20 with crystallographic data from Michel et al.29, a similar order of magnitude is obtained (c0
18
≈ 15 x 1020 m-3). As an outcome of the modelled diffusion-limited scenario, an at least 4 orders of
19
magnitude lower value of c0 is obtained (Table 1). Therefore, the growth of magnetite is almost
20
certainly not diffusion-limited but lies instead in the reaction-limited regime. Therefore, a free-
21
energy barrier to PP adsorption is required, which we found as low as 24 kJ × mol-1 (table 1). For
22
activation-limited crystal growth, an activation energy between 40 and 80 kJ mol-1 is expected in the
23
case of a process based on the addition of ions5. Alternatively, in a system involving poorly soluble
24
species, oriented attachment of ZnS via nanoparticles leads to activation energies in the range of 125
25
kJ × mol-1 30. In this case, the activation energy for this surface activation-limited process becomes
8 ACS Paragon Plus Environment
Page 8 of 13
Page 9 of 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
The Journal of Physical Chemistry Letters
1
energetically more demanding because of the large number of atoms being integrated into the
2
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
3
Table 1: Parameters of diffusion-limited and reaction-limited models of the evolution of the
4
particle radius. a The experimental value is c0 = 2.3 x 1020 m-3 assuming PPs with magnetite
5
stoichiometry. b Using c0 = 2.3 x 1020 m-3.
6 7
Using the most conservative experimental estimate of c0 (c0 = 2.3 x 1020 m-3 for PP with magnetite
8
stoichiometry), the reaction-limited model yields an estimate of the pre-exponential factor,
9
k0 ≈ 0.002 s-1 and, together with ∆U and rinit, a comprehensive description of the particle growth
10
process and its temperature dependence.
11
As pointed out above, the described growth of the magnetite particles by adsorption of PPs takes
12
place in the reaction-limited regime, i.e., kreac
