# The Kinetic Control of Crystal Growth in Geological Reactions: An Example of Olivine–Ilmenite Assemblage

Anastassia Yu. Borisova, Kirill Lozovoy, Alessandro Pugliara, Teresa Hungria, Claudie Josse, Philippe de Parseval

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Anastassia Yu. Borisova, Kirill Lozovoy, Alessandro Pugliara, Teresa Hungria, Claudie Josse, et al. The Kinetic Control of Crystal Growth in Geological Reactions: An Example of Olivine–Ilmenite Assemblage. Minerals, 2025, 15 (6), pp.569. 10.3390/min15060569. hal-05350362

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Article

# The Kinetic Control of Crystal Growth in Geological Reactions: An Example of Olivine–Ilmenite Assemblage

Anastassia Y. Borisova <fnref n="1" />,* [ID], Kirill Lozovoy <fnref n="2" /> [ID], Alessandro Pugliara <fnref n="3" />, Teresa Hungria <fnref n="3" /> [ID], Claudie Josse <fnref n="3" /> and Philippe de Parseval <fnref n="1" /> [ID]

**Abstract:** The main constituent of the planetary lithosphere is the dominant silicate mineral, olivine α-(Mg,Fe)₂SiO₄, which, along with associated minerals and the olivine-hosted inclusions, records the physical-chemical conditions during the crystal growth and transport to the planetary surface. However, there is a lack of physical-chemical information regarding the kinetic factors that regulate crystal growth during melt-rock, fluid-rock, and magma-rock interactions. Here, we conducted an experimental reaction between hydrated peridotite rock and basaltic melt and coupled this with a structural and elemental analysis of the quenched products by high-resolution transmission electron microscopy. The quenched products revealed crystallographically oriented oxide nanocrystals of ilmenite (Fe,Mg)(Ti,Si)O₃ that grew over the newly formed olivine in the boundary layer melt of the reaction zone. We established that the growth mechanism is epitaxial and is common to both experimental and natural systems. The kinetic model developed for shallow (<1 GPa) crystal growth requires open system conditions and the presence of melt or fluid. It implies that the current geodynamic models that consider natural ilmenite–olivine assemblage as a proxy for deep to ultra-deep (>>1 GPa) conditions should be revised. The resulting kinetic model has a wide range of geological implications—from disequilibrium mineral growth and olivine-hosted inclusion production to mantle metasomatism—and helps to clarify how geological reactions proceed at depth.

**Keywords:** kinetics; melt-rock reaction; nanocrystal growth; epitaxy; olivine; ilmenite

## 1. Introduction

Mantle-derived melts, fluids, and magmas encounter rocks under diverse physical-chemical conditions when they ascend toward the surface of planetary bodies such as the Earth, the Moon, Mars, Venus, and asteroids, among others. Reactions of crystal dissolution and growth take place during these deep processes. Although these open system processes are potentially crucial for regulating rock and magma composition, as well as their physical-chemical properties, the fact remains that the structural and physical-chemical controls on the crystal growth remain unidentified. To study the open system reactions between melts or fluids and rocks occurring in the oceanic mantle, mantle–crust transition zone, and planetary lithosphere, we conducted experiments to

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Academic Editors: Xing Ding, Haibo

Yan, Panlao Zhao and Deshui Yu

Received: 19 March 2025

Revised: 23 May 2025

Accepted: 24 May 2025

Published: 27 May 2025

**Citation:** Borisova, A.Y.; Lozovoy, K.; Pugliara, A.; Hungria, T.; Josse, C.; de Parseval, P. The Kinetic Control of Crystal Growth in Geological Reactions: An Example of Olivine–Ilmenite Assemblage. *Minerals* **2025**, *15*, 569. https://doi.org/10.3390/min15060569

**Copyright:** © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

<footnote n="1">¹ Géosciences Environnement Toulouse—GET, CNRS, IRD, UPS, Université de Toulouse, 14 Avenue E. Belin, 31400 Toulouse, France; philippe.deparseval@get.omp.eu</footnote>

<footnote n="2">² Department of Quantum Electronics and Photonics, Faculty of Radiophysics, National Research Tomsk State University, 36 Lenin Av., Tomsk 634050, Russia; lozovoymailbox@gmail.com</footnote>

<footnote n="3">³ UAR Raimond Castaing, CNRS, INP, UPS, INSA, Université de Toulouse, 3 Rue Caroline Aigle, 31400 Toulouse, France; alessandro.pugliara@toulouse-inp.fr (A.P.); teresa.hungria@ums-castaing.fr (T.H.); claudie.josse@centre-castaing.fr (C.J.)</footnote>

<footnote>* Correspondence: anastassia.borisova@get.omp.eu</footnote>
observe how serpentinite and basaltic melt react under shallow conditions [1,2]. These
reactions have been observed in the modern hydrated oceanic lithosphere at relatively shal-
low depths (≤1 GPa) [1–4] and are likely to have taken place in the early proto-lithosphere,
during the Hadean period of Earth and the Pre-Noachian to Noachian period of Mars [2,4].
However, the primary structural and chemical factors governing crystal growth during
these reactions have not been identified. This study aims to examine the key kinetic factors
that control crystal growth during melt–rock and fluid–rock reactions on nano-atomic to
atomic scales.

Currently, no information is available regarding the primary control of ilmenite crystal
growth under high-temperature conditions during interactions between melts or fluids and
olivine-rich rocks (e.g., mafic–ultramafic cumulates, hydrated peridotites, or serpentinites).
Yet it is common to find ilmenite oxide mineral in combination with other silicate and
oxide mineral phases known as MARID (mica–amphibole–rutile–ilmenite–diopside) in
lithospheric mantle xenoliths [5–7] and kimberlite rocks [8,9]. Ilmenite is frequently found
together with olivine. It has been interpreted as a mineral phase exsolved from olivine
in layered intrusions and serpentinites [10,11]. Ilmenite is frequently present in mare
basalts and ordinary chondrites [12]. As a common accessory mineral in association with
olivine, ilmenite serves as a proxy for the highly debated pressure–temperature conditions
during olivine formation in peridotites and layered intrusions [6,10,11,13,14]. This natural
association was studied to experimentally explore the olivine–ilmenite thermometer [15],
investigate the equilibrium TiO₂ solubility in olivine in synthetic [16] systems as a function
of temperature and pressure, and record the electrical conductivity of the mixed olivine–
ilmenite system [17].

There is considerable disagreement about the physico-chemical conditions for the
formation of this mineral assemblage, which can form under disequilibrium conditions
in nature [18]. The oriented, naturally occurring ilmenite nanocrystals with olivine lattice
orientations of (100, 010, and 001) in garnet peridotite rocks have been proposed to be the
result of topotactic growth or in situ re-crystallization (or solid-state exsolution) of ilmenite
from Ti-enriched olivine. This olivine is thought to be produced at extremely deep mantle
conditions that correspond to the mantle transition zone at a depth of 410 km [13]. Fur-
thermore, the current experimental evidence regarding the equilibrium solubility of Ti in
olivine and the association of ilmenite–olivine is inconsistent. The data point toward a sig-
nificant Ti solubility of more than 1 wt% TiO₂ in olivine at 12 GPa [14], whereas equivalent
concentrations occur in the presence of ilmenite at elevated temperatures (≥1200 °C) and
much lower pressures (1.5–5.5 GPa) [16]. High titanium solubility in olivine was suggested
to be facilitated by the production of Ti-OH-rich olivine or by hydrous titanian clinohumite
with a formula of (Mg,Fe²⁺)₉(Si,TiO₄)₄(F,OH)₂, which possesses a monoclinic-prismatic
structure and belongs to the space group P 2₁/c [17–21]. The presence of clinohumite has
been hypothesized as an explanation for the formation of ilmenite ingrowths in natural
olivine [11,13,16,22,23], based on the interpretation of the surface density of ilmenite in
olivine as a geobarometer. These authors discussed the appropriate setting for the natural
olivine–ilmenite association, considering either the shallow and hydrous subcontinental
mantle (<3.5 GPa) or the ultra-deep mantle (>4 GPa). However, the exact physico-chemical
conditions are not yet known, although open systems are much more common in nature
than closed systems.

**2. Materials and Methods**

2.1. *Experimental Method*

The natural mid-ocean ridge basalt used in the experiments is a typical, moderately
differentiated (8.2 wt.% of MgO) glassy tholeiitic basalt (sample 3786/3) from the Knipovich
Ridge of the Mid-Atlantic Ridge dredged during the 38th expedition of the Research Vessel *Academic Mstislav Keldysh*. The serpentinite used as starting material is a homogeneous rock composed of antigorite with accessory Fe-rich oxides and devoid of relics of primary mantle silicates. It was sampled in Zildat, Ladakh, in the northwest Himalaya. Well-known starting materials [1–3] were used. For the hybrid runs, the serpentinite was prepared as a doubly polished ~1000 µm thick section, which was thereafter cut to 2.7 mm diameter cylinders with a core drill machine. The mid-ocean ridge basaltic (MORB) glass was crushed to powder (<100 µm glass size).

Thermal gradient during serpentinite–basalt interaction exists, and it has been the primary focus in previous papers [1–4]. This experimental study uniquely addresses the high-temperature process happening at the interface of the dehydrated and partially dissolved rock (harzburgite) and the basaltic melt. To experimentally model the melt–rock interaction, we used a similar design to that used in [1–4], with a serpentinite cylinder in the upper part and basaltic glass powder in the lower part of a Au₈₀Pd₂₀ capsule. For starting materials, we used the natural tholeiitic basalt (84.3–86.6 wt%) and natural serpentinite (13.4–15.7 wt%). These experiments were designed to simulate the open system process of basaltic melt infiltration into serpentinite rock, which does not necessarily reach equilibrium. For example, an absence of melt homogeneity in the system indicates conditions out of equilibrium. Two experiments (P15, P18) with different basalt-to-serpentinite ratios (5.4–6.5) were carried out at a pressure of 0.5 GPa and a temperature of 1300 °C using a piston cylinder at the Bavarian Research Institute of Experimental Geochemistry and Geophysics (BGI), Bayreuth, Germany. The elevated melt–rock mass ratio does not affect the out-of-equilibrium reaction happening at the interface due to the olivine growth. The duration of experiments (P15 and P18) was 0.5 and 5 h, respectively. In this work, only P18 was studied at the nanoscale. The experiments were conducted using the “Max Voggenreiter” end-loaded Boyd–England piston–cylinder apparatus at the BGI, Bayreuth, Germany. Talc cells with a ¾-inch diameter and Pyrex sleeves were used. A tapered graphite furnace was inserted in each cell. Alumina (Al₂O₃) spacers were used as a pressure-transmitting medium. A Au₈₀Pd₂₀ capsule loaded with starting materials was set in the central part of the assembly. A 20% pressure correction was applied for the friction between the talc cell and the pressure vessel. A molybdenum disulfide (MoS₂) lubricant was introduced to minimize friction. The temperature in the upper part of the capsules was controlled by a EUROtherM (2404) controller via either a W₃Re₉₇/W₂₅Re₇₅ (type D) or a Pt₆Rh₉₄/Pt₃₀Rh₇₀ (type B) thermocouple, accurate to ±0.5 °C. The sample was compressed to 0.5 GPa over a period of 20 min and then heated up to the run temperature (1300 °C) at a rate of 100 °C/min. The samples were maintained at run conditions for the desired durations. The experiments were quenched by switching off the power supply. We applied decompression during periods from 20 min to 2 h. The rate of quenching to the ambient temperature was ~500 °C/min. After the runs, the Au₈₀Pd₂₀ capsules were mounted in epoxy, cut in half using a diamond micro-saw, and then polished using SiC sand paper and diamond pastes.

## 2.2. *Theoretical Method*

To model the epitaxial growth of ilmenite on olivine, the theoretical model developed for molecular beam epitaxy [24] was applied. The kinetics of the epitaxial formation of nanostructures follows Frank–van der Merwe, Volmer–Weber, and Stranski–Krastanow growth modes. The model allows for an estimation of the growth mechanism for a given epitaxial system, as well as the assessment of some practically valuable parameters such as the thickness of the two-dimensional layer, the surface density, and the average size of nano-islands. There are three modes of epitaxial growth of nanostructures [25,26]:
(a) Frank-van der Merwe mode: layer-by-layer growth of materials in systems in which the lattice constants of the growing material and the substrate are matched; (b) Volmer-Weber mode: island growth of material in systems with a strong mismatch between the lattice constants of the growing material and the substrate; and (c) Stranski-Krastanow mode (intermediate case): at the first stage, layer-by-layer growth of the material on the substrate is realized with the formation of a two-dimensional wetting layer, followed by a transition to the three-dimensional growth of islands.

The proposed kinetic model (Equation (1)) considers the change in Gibbs free energy ΔF during the formation of nano-islands from atoms of the growing material on the surface of the substrate [24–26]. In this case, the competing factors are the change in the energy of atoms due to an increase in surface energy ΔF<sub>surf</sub>, the relaxation of elastic stresses ΔF<sub>elas</sub>, and a decrease in the attraction of atoms to the substrate ΔF<sub>attr</sub>.

$$ \Delta F(h) = \Delta F_{surf} + \Delta F_{elas} + \Delta F_{attr}. \qquad (1) $$

In this process, there is a certain critical value of the thickness of the growing material h at which the formation of ilmenite nano-islands [27] becomes energetically more favorable than the formation of a continuous two-dimensional wetting layer (when ΔF decreases with the thickness h). The value of this critical thickness, h<sub>c</sub> (usually measured in the monolayers of growing material, ML), correlates with the preferred growth mode in the epitaxial system [25,26]. Growth according to the Volmer-Weber mechanism is considered a limiting case of growth according to the Stranski-Krastanow mechanism when the critical thickness of the wetting layer tends toward zero (h<sub>c</sub> < 1 monolayer). Growth according to the Frank-van der Merwe mechanism is a limiting case of epitaxial growth with an unlimited increase in the critical thickness of the transition from two-dimensional to three-dimensional growth (h<sub>c</sub> >> 1 monolayer).

A change in surface energy, ΔF<sub>surf</sub>, does not depend on the thickness of the two-dimensional layer and is always positive [24]. The second term, ΔF<sub>elas</sub>, in (Equation (1)) equals the difference in elastic energies of growing material of volume V (cm<sup>3</sup>) in a 2D layer and in an island, characterizing the relaxation of elastic strain [26]:

$$ \Delta F_{elas} = -(1-Z)\lambda\epsilon_0^2 V, \qquad (2) $$

where λ is the elastic modulus of the growing material (dyn/cm<sup>2</sup>), Z is the coefficient of elastic strain relaxation (unitless) [24], and ε<sub>0</sub> is the lattice mismatch (in %) between the deposited material and the substrate. This term is always negative, being the main factor enhancing island formation.

A change in energy, ΔF<sub>attr</sub>, is defined as the difference in energies of attraction to the substrate of atoms in a 2D layer and in a 3D island, and it depends exponentially on the thickness of the growing material:

$$ \Delta F_{attr} = \frac{\Psi_0}{d_0} \exp\left(-\frac{h}{d_0}\right) V, \qquad (3) $$

where Ψ<sub>0</sub> is the wetting energy density (erg/cm<sup>2</sup>), h is the thickness of the two-dimensional layer (Å), and d<sub>0</sub> is the height of 1 monolayer of growing material (Å) [24–26].

The overall change in free energy, ΔF, decreases with the thickness of the growing material when the sum of the second and third terms in (Equation (1)) becomes negative. Then, the critical thickness h<sub>c</sub> (Å) is defined as the thickness, where ΔF<sub>elas</sub> + ΔF<sub>attr</sub> = 0:

$$ \frac{\Psi_0}{d_0} \exp\left(-\frac{h}{d_0}\right) V - (1-Z)\lambda\epsilon_0^2 V = 0. \qquad (4) $$
From this condition, the critical thickness $h_c$ can be found:

$$h_c = d_0 \ln \left( \frac{\Psi_0}{d_0(1-Z)\lambda\epsilon_0^2} \right) \quad (5)$$

It may be concluded from this expression that the main factor influencing the value of the critical thickness is the lattice mismatch between the growing material and the substrate (Figure S7). It is also known [24,26] that the critical thickness depends slightly on temperature (approximately as ~1/T¹/²). A comparison of critical thicknesses for the transition from 2D layer growth to 3D nano-island growth in the ilmenite–olivine system at 1000 °C and 1300 °C (Figure S7) shows that the temperature dependence of critical thickness is insignificant for high mismatches and leads to no more than a 10% increase in critical thickness for low mismatches.

It is repeatedly stated in the literature [24] that the main factor determining the epitaxial growth mode is the value of the lattice mismatch between the growing material and the substrate. The lattice parameters for olivine (a = 4.762 Å, b = 10.225 Å, c = 5.994 Å) and ilmenite (a = 5.088 Å, c = 14.085 Å) are well known [23,28–32]. Thus, the lattice mismatch in this system may be calculated as the ratio of the largest diagonals of the olivine and ilmenite unit cells and is ~23.8% (Figure S6). Such a large value of mismatch implies that the Volmer–Weber growth mode with the preferential formation of nano-islands should be realized in this case. The kinetic model for epitaxial growth [25,26] was used for a misfit of 23.8% and a growth temperature of 1300 °C. The values of other parameters used in the calculations may be found in Supplementary Dataset S4. The results of the calculation show that the critical thickness of the transition from 2D to 3D growth will be about 0.2 monolayers of growing material, which means that the Volmer–Weber growth mode should be realized. This is confirmed by the results of the TEM microscopy investigations. Indeed, there are only three-dimensional nano-islands of ilmenite on the surface of olivine. The nano-islands have the shape of pyramids. The calculated average surface density of the nano-islands is (3.0 ± 1.3) × 10¹⁰ cm⁻², and the average size of nano-islands is 25 ± 3.7 nm (n = 20), estimated based on the total analyzed surface area of 0.5 µm² for two images (numbers 10 and 16 with ×15k magnification) by using ImageJ (https://imagej.net/ij/, accessed on 18 March 2025) and Digital Micrograph software (version 3.53.4137.0) (Figure S8).

## 2.3. Analytical Methods

A thin micro-section of 20 µm was prepared using the focused ion beam lift-out technique in an FEI NanoLab HELIOS 600i (FEI-Thermo Fisher Scientific, Eindhoven, The Netherlands) focused ion beam (FIB)/scanning electron microscope (SEM) to allow for the preparation of a lamella for transmission electron microscopy (TEM) analysis (high-resolution TEM, HR-TEM; scanning TEM with high-angle annular dark field (STEM-HAADF); and chemical composition mapping (scanning TEM with energy dispersive X-ray spectroscopy, STEM-EDS)). TEM imaging and STEM-EDS studies were performed using a transmission electron microscope JEOL cold-FEG JEM-ARM200 F (JEOL, Tokyo, Japan) operated at 200 kV equipped with a probe Cs corrector (resolution 0.78 Å) and a JEOL CENTURIO SDD EDS detector with a resolution of 129 eV at Mn Kα. Selected area electron diffraction (SAED) patterns were recorded using a JEOL Schottky-FEG JEM 2100 F (JEOL, Tokyo, Japan) transmission electron microscope operated at 200 kV with an aperture of 150 nm. EDS analyses were quantified using the Cliff–Lorimer k-factor method [33]. SAED patterns were simulated with CaRIne Crystallography (version 3.1) [34]. The processing of EDS spectra was carried out with the Analysis Station JEOL Co. software (version 3.8.0.59). In order to identify the zone axis of the diffraction pattern of a structure, the
planar distances (hkl) and angles between each pair of planes were measured and compared to those of simulated diffraction patterns using the CaRIne Crystallography program [34]. Different zone axes of the same structure were also determined using this method. The experimental angles between each pair of zone axes were compared with those determined from simulated diffraction patterns through CaRIne Crystallography to avoid ambiguities in the indexing of the diffraction patterns.

The experimental spectra were acquired at rates between 1000 and 2000 counts per s, with dead time in the range of 25%–10%, and were collected for 150–200 s. The error due to the counting statistics at the 2 s level for Mg and Si is in the range of 0.6%. For Ti, Ni, and Fe, the relative error depends on concentration and varies between 1% and 15%. This leads to a maximum relative error in the determination of Ti concentration of about 12% or, in absolute units, between ±0.07 and ±0.15 wt.%, depending on concentration. The results are presented in Supplementary Figures S1–S8 and Datasets S1–S3.

The major (Mg) and minor (Ti) element compositions of the crystals and glasses were mapped using a CAMECA SX-Five FE (CAMECA) (Figure S9). Elemental X-ray mapping (Ti, Mg) was performed using a CAMECA Five FE (CAMECA). For the mapping of the elemental distributions, operating conditions of 10 kV and 40 nA with a 0.6 s dwell time were applied to obtain images of 256 by 256 pixels with a step size of 0.1 µm and acquisition in beam scan mode.

### 3. Results

#### Ilmenite–Olivine Assemblage Due to Open System Reaction

The basaltic melt–serpentinite reaction performed under controlled conditions of 0.5 GPa and 1300 °C using a piston–cylinder press [1,2] results in the dehydration of serpentinite rock, the liberation of aqueous fluid, the production of harzburgite rock (consisting of olivine, orthopyroxene, and oxides), and the progressive dissolution of the rock into the reacting basaltic melt (L—liquid ~50 wt% SiO₂ and 1.5 wt% TiO₂). It is important to note that the region of interaction between the melt and rock exhibits progressive incongruent dissolution of orthopyroxene and the generation of newly formed euhedral olivine, along with development of the boundary melt (L1), which has higher Si and Ti contents than the initial basaltic melt (L) (Figures S1 and S8). The silica-rich melts are formed due to the incongruent dissolution of orthopyroxene, which is a mineral phase enriched in silica. High Ti contents in the boundary melts L1 are due to Ti incompatibility with olivine (mineral to melt partitioning coefficient below 1). This reaction with the initial basaltic liquid culminates in the creation of an outer olivine-rich zone (Ol + L1) and an inner harzburgite rock zone. In this study, we examine the outer zone rich in olivine, comprising the boundary melt (L1), and newly formed euhedral olivine crystals (Ol) at the nano- and atomic scales.

The scanning transmission electron microscopy–energy dispersive X-ray spectroscopy (STEM-EDS) analysis enables the identification of a mineral nano-phase measuring 25 ± 3.7 nm (n = 20), likely an oxide that is rich in Fe, Ti, and Si. This phase overgrows the olivine crystals (Ol), which are themselves enriched in Ti (0.2–0.8 atom % Ti, equivalent to 0.4 to 1.0 wt% TiO₂) and is associated with Si-Ti-rich boundary melt (L1) (80–85 atom % Si—equal to 83–87 wt% SiO₂ and up to 4.1% Ti or 6.9 wt% TiO₂) (Figure S2, Datasets S1–S3). Further analysis of the structure and chemistry through EDS and identification through selected area electron diffraction (SAED) reveals the presence of olivine and Fe-Mg-Ti-Si oxide with ilmenite structure ((Fe²⁺,Mg)(Ti,Si)O₃). The structural examination indicates the epitaxial overgrowth of the ilmenite nanocrystals on the olivine, α-(Mg,Fe)₂SiO₄ (Figure 1). Using the diffraction pattern analysis, we detected evidence of epitaxial overgrowth of ilmenite nanocrystals on the host olivine for different nanocrystals oriented with the
same zone axis as those shown in this paper. This result, coupled with BF-TEM images of nanocrystals all oriented in the same way, suggests the possibility that all ilmenite nanocrystals are epitaxial overgrowths on the host olivine. All numerical and analytical data are summarized in Datasets S1–S4 and are plotted in Figures 1, 2 and S1–S9.

<figure><img src="image_4.png" /><figcaption>**Figure 1.** Evidence for the 3D epitaxial overgrowth of the experimentally produced ilmenite nanocrystal on the host olivine from selected area electron diffraction (SAED) analysis (Figures S3–S5). A kinetic model of epitaxial growth for the ilmenite crystal overgrowth directly on the olivine crystals is developed in this work.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 2.** A kinetic model of the critical thickness of the transition from 2D layer growth to 3D nano-island growth in the ilmenite–olivine system at 1300 °C as a function of lattice mismatch. The possible growth modes are distinguished by different colors: I—Frank–van der Merwe mode (lattice-matching layer growth); II—Stranski–Krastanow mode (transition from layer to island growth); III—Volmer–Weber mode (island growth). The star symbol corresponds to the experimental mismatch obtained in this work; the circled area denotes the mismatches used [23]. A comparison between the curves for 1300 °C and 1000 °C (see Figure S7) suggests that the 300 °C temperature change is insignificant for high mismatches and leads to no more than 10% critical thickness increase for low mismatches.</figcaption></figure>
# 4. Discussion

## 4.1. Kinetic Experiments

Our experimental runs, conducted under shallow (≤1 GPa) and disequilibrium conditions, indicate that crystallographically oriented ilmenite in olivine-rich samples from mafic-ultramafic terrestrial and planetary crusts and mantle lithosphere does not necessitate equilibrium and high to ultra-high pressure conditions. Such orientations can now be explained by the presence of (Si,Ti)O₂—L1-type silicate boundary layer melt, or fluid, in association with olivine. Building on the established possibility of the Fe-Mg exchange between olivine and ilmenite [15] and Si-Ti substitution within the structure of olivine [16], the open system process (with an absence of equilibrium in the system evidenced by variations in the melt composition) of forming ilmenite nanocrystals can be characterized as follows:

$$ (Fe,Mg)_2(Si,Ti)O_4^{Ol} + (Si,Ti)O_2^{L1} = 2(Fe,Mg)(Ti,Si)O_3^{Ilm}, \quad (6) $$

where Ol is olivine, L1 is boundary melt in association with the growing olivine (Ol, α-(Fe,Mg)₂(Si,Ti)O₄), and Ilm is ilmenite ((Fe,Mg)(Ti,Si)O₃). A trigonal-rhombohedral crystal system, with a space group of R-3 H, characterizes Ilm, whereas an orthorhombic-dipyramidal crystal system, with a space group of Pnma, distinguishes olivine. In our experimental samples, the examined olivine-rich zone, which has a newly formed olivine lattice orientation of (010), is epitaxial, with ilmenite having an orientation of (210). In the second zone, the olivine orientation of (132) is also epitaxial, with ilmenite having an orientation of (601) (Figures 1 and S3-S5). The lattice mismatch between olivine with a lattice orientation of (010) and an ilmenite orientation of (210) in the investigated experimental system is calculated to be 23.8% (Figure S6). In previously investigated natural systems [10,11,13,23], olivine lattice orientations of (100, 010, and 001) are also epitaxial with ilmenite, resulting in ≤3% mismatch with the olivine lattice.

## 4.2. Kinetic Control of Ilmenite Crystal Growth on Olivine

The kinetic model of epitaxial growth developed for nanomaterials [24–26] was used here to simulate the epitaxial growth of ilmenite on olivine (Dataset S4). The constructed kinetic model enables the estimation of the growth mechanism for a specific epitaxial system while also assessing practical parameters such as two-dimensional layer thickness, surface density (number of islands per unit area), and average nano-island size (Figures S7 and S8, Dataset S4). In fact, three forms of epitaxial growth for nanostructures exist [27]. The Frank-van der Merwe mode involves the layer-by-layer growth of materials in a system in which the lattice constants of the growing material and substrate match. The Volmer-Weber mode takes place when there is island growth of the material in systems with a significant lattice constant mismatch between the growing material and the substrate. The Stranski-Krastanow mode [27] occurs when the material grows layer by layer on the substrate, forming a two-dimensional (2D) wetting layer initially before transitioning to the three-dimensional (3D) growth of islands. The experimental epitaxial growth investigated here is limited by 3D Volmer-Weber growth, which is modeled based on the misfit of 23.8% (Dataset S4). In contrast, the similar natural system [10,11,13,23] experiences 2D-3D Stranski-Krastanow mode growth with a misfit of 0.5%–3% (Figures 2 and S7). The critical thickness of transition from 2D layer growth to 3D nano-island growth in the ilmenite-olivine system was calculated with the developed model of epitaxial growth (Equations (1)–(5)). According to (Equation (5)), the critical thickness of the 2D-3D transition significantly decreases with an increase in the lattice mismatch between the growing material and the substrate (Dataset S4).

For the investigated experimental system with a misfit of 23.8%, the critical thickness is notably less than one monolayer, meaning that the epitaxial growth control is limited
by 3D Volmer–Weber growth based on the theoretical model of epitaxial growth [25,26]. Meanwhile, for the natural system with a misfit of 0.5%–3%, the critical thickness calculated with (Equation (5)) is in the order of 3–6 monolayers, which is typical for classical Stranski–Krastanow systems known in nanomaterial science [24–27]. This means that the ilmenite growth in natural samples [10,11,13,23] may occur via 2D–3D Stranski–Krastanow mode growth (Figure 2). It is also known [24,26] that the critical thickness slightly depends on temperature. However, this dependence is insignificant for high mismatches (Figure S7). Our kinetic model for the epitaxial overgrowth of ilmenite directly on the olivine crystals (Dataset S4) suggests that nano- or micro-crystalline ilmenite mineral growth is under epitaxial control during open system reactions in the presence of melt or fluid, instead of closed-system topotaxial re-crystallization in a solid state.

These findings regarding the reaction (Equation (6)) and the epitaxial model of ilmenite
growth developed here influence our comprehension of mineral production across the
upper mantle, mantle–crust transition zone, and planetary lithosphere. We conclude that
the growth mechanism of the ilmenite–olivine association follows an epitaxial law due to a
disequilibrium process described by nanomaterials science. Thus, the epitaxial growth of
ilmenite suggests that the epitaxial mechanisms from nanomaterial physics can be used to
explore and produce the mineral-hosted inclusions, which require melt–rock reactions and
nanoscale investigation rather than microscale studies (Figure S9).

Our research indicates that ilmenite overgrowths form directly on orthorhombic-
dipyramidal olivine crystals in a crystallographic orientation (Figure 3). This occurs due to
the reaction (Equation (6)) with the boundary melt (L1), suggesting melt- or fluid-present
conditions favorable for ilmenite growth during reactions with olivine. To crystallize
Fe-Mg-Ti-Si oxides, such as those found within the MARID mineral association, ilmenite
growth does not require significant Ti enrichment of the initial reacting melt (L), provided
the boundary layer melt (L1) or fluid is formed. Indeed, the reacting mid-ocean basaltic
melt (L) contains 1.5 wt% TiO₂, whereas the boundary layer melt (L1) has Ti contents
up to 6.9 wt% TiO₂ (Figure 3; Dataset S3). This differs from most existing models of Fe-
Ti metasomatism [28], which suggest strong Ti enrichment of the reacting agents, such
as melts or fluids, compared to the regular basaltic melts. Longer experimental runs
provide sufficient timescales for micron-scale growth. The formation of larger crystals
is also described by epitaxial processes. This has been repeatedly observed in epitaxial
experiments with extended durations and larger amounts of deposited material [35].

<figure><img src="image_6.png" /><figcaption>**Figure 3.** A principal schema of the ilmenite–olivine assemblage formation in the mantle–crust transition zone and mantle lithosphere due to an open system reaction between hydrated peridotite</figcaption></figure>
rock and basaltic melt. (A) An initial reaction of the serpentinized rock with a reacting melt or fluid. (B) The final formation of an olivine-rich zone with ilmenite nano-islands associated with boundary layer melt or fluid.

**5. Conclusions**

In conclusion, our experimental laboratory runs performed under shallow conditions resolve the existing disputes over the frequent association of natural ilmenite with olivine. This mineral association reflects the reaction of olivine-rich rocks (such as olivine cumulate, serpentinite, or hydrated peridotite) with silicate melts or fluids in shallow settings (<1 GPa). It is therefore necessary to revise current geological and geodynamic models by considering ilmenite nano- and micro-inclusions in olivine to be a representation of high to ultra-high pressure (>>1 GPa) environments [11,13,22,23]. The presence of ilmenite inclusions in olivine should be considered a strong indicator of open system reactions with melts or fluids. Our kinetic model thus has a wide range of geological implications—from disequilibrium mineral growth and olivine-hosted inclusion production to mantle metasomatism—and helps to elucidate how geological reactions proceed at depth.

**Supplementary Materials:** The following supporting information can be downloaded at https://www.mdpi.com/article/10.3390/min15060569/s1, Dataset S1: Olivine (Ol) composition; Dataset S2: Fe-Ti oxide (Ilm) composition; Dataset S3: The boundary silicate glass (L1) composition; Dataset S4: Parameters of the kinetic model; Figure S1: The FIB SEM section preparation for the HR-TEM (EDS-STEM) in the zone of the experimental olivine growth (Ol + L1) due to basaltic melt-serpentinite rock interaction run; Figure S2: A nanoscale investigation of the experimental olivine-rich zones with the corresponding compositions of mineral and glass phases produced due to hydrated peridotite-basalt reaction by scanning transmission electron microscopy-energy dispersive X-ray spectroscopy (STEM-EDS); Figure S3: Evidence for the epitaxial orientation of the experimental olivine (010) with an ilmenite orientation of (210); Figure S4: Evidence for the epitaxial orientation of the experimental olivine (132) with ilmenite (601); Figure S5: Additional evidence for the epitaxial orientation of the experimental olivine (132) with ilmenite (601) in the same zone as Figure S4; Figure S6: The calculation of the lattice mismatch in the experimental system between olivine, α-(Fe,Mg)₂(Si,Ti)O₄, with an orientation of (010) and ilmenite, (Fe,Mg)(Ti,Si)O₃, with an orientation of (210); Figure S7: The critical thickness (in monolayers of growing material) of the transition from 2D to 3D growth versus the lattice mismatch (in %); Figure S8: The calculation of the experimental nano-island density based on the images. The average size of the nano-islands is 25 ± 3.7 nm (n = 20), estimated based on the total analyzed surface area of 0.5 µm² for two images (number 10 and 16 with ×15k) by using ImageJ and Digital Micrograph software; Figure S9: EPMA-FE maps for the Mg and Ti contents in the olivine-rich zone (L1 + Ol) of the reaction zone in the P18 sample.

**Author Contributions:** Conceptualization: A.Y.B.; Methodology: A.Y.B., K.L., A.P., T.H., C.J. and P.d.P.; Investigation: A.Y.B., K.L., A.P., T.H., C.J. and P.d.P.; Visualization: A.Y.B., K.L., A.P., T.H. and C.J.; Funding acquisition: A.Y.B.; Project administration: A.Y.B.; Supervision: A.Y.B.; Writing—original draft: A.Y.B., K.L., A.P., T.H. and C.J.; Writing—review and editing: A.Y.B., K.L., A.P., T.H., C.J. and P.d.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This article is funded by the European Union (ERC, PLANETAFELSIC, project 101141259) to AYB. The views and opinions expressed are, however, those of the author(s) only and do not necessarily reflect those of the European Union or EUROPEAN RESEARCH COUNCIL EXECUTIVE AGENCY [ERCEA]. Neither the European Union nor the granting authority can be held responsible for them. This work was funded by Institut National de Sciences de Univers (France) PLAGIOGRAN grant to AYB and by Tomsk State University Development Program (Priority 2030) to KL.

**Data Availability Statement:** All data generated or analyzed during this study are included in this published article and its Supplementary Information Files.
**Acknowledgments:** The authors thank A-M Cousin (GET, France) for the help in the figure drawing.

**Conflicts of Interest:** The authors declare no conflicts of interest.

**Abbreviations**

The following abbreviations are used in this manuscript:

EPMA electron probe microanalysis

HR-TEM high-resolution transmission electron microscopy

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