Diffusion Creep of Forsterite and Its Grain Size Effects: New Constraints from High-Precision Gas-Medium Deformation Experiments

1. Introduction

Olivine, the most abundant (50‒60 vol.%) [1,2] and rheologically weakest major mineral in the upper mantle, governs its plastic flow [3,4]. Consequently, extensive experimental studies have investigated the rheology of olivine single-crystal [5,6,7,8], polycrystalline olivine aggregates (eg., [9,10]), and olivine-rich rocks [11,12,13,14] and hot-pressed polycrystalline aggregates [15,16]. Early work focused on natural dunites, while recent studies utilize synthetic or separated natural mineral aggregates. Two primary flow regimes are recognized: grain-size-insensitive dislocation creep and grain-size-sensitive diffusion creep. Both are described by a constitutive flow law:

$$\dot{\mathsf{ϵ}}=\mathrm{A}{\mathrm{d}}^{-\mathrm{p}}{\mathsf{\sigma }}^{\mathrm{n}}\exp \left(-\mathrm{Q}/\mathrm{RT}\right){\mathrm{f}}_{H_2\mathrm{O}}^{\mathrm{r}}{\mathrm{f}}_{O_2}^{\mathrm{m}}$$

(1)

Where $\dot{\mathsf{ϵ}}$ is strain rate (s^-1), A is a pre-exponential constant, d is grain size (μm), p is the grain size exponent (p=0 for dislocation creep), σ is differential stress (MPa), n is the stress exponent, Q is apparent activation energy (J*mol^-1), R is the gas constant (J*mol^-1*K^-1), T is absolute temperature (K), ${\mathrm{f}}_{H_2\mathrm{O}}$ is water fugacity (MPa), r is the water fugacity exponent, ${\mathrm{f}}_{O_2}$ is oxygen fugacity (MPa), and m is the oxygen fugacity exponent.

Based on well-constrained data, especially obtained by gas or molten salt medium deformation apparatus, flow parameters for olivine have been determined. For dry conditions, dislocation creep yields n = 3–5 and Q = 500–540 kJ/mol [11,12,14,17,18,19], while diffusion creep is characterized by n≈1, Q = 240–375 kJ/mol and p=2-3.4 [20,21]. Under wet conditions, the activation energy for dislocation creep decreases significantly (Q= 334–520 kJ/mol) [11,12,13,17], with a water fugacity exponent r = 0.69–1.25 [10,18], while for diffusion creep, the water fugacity exponent r = 0.7–1 [9,17,19].

While experiments on natural olivine-rich rocks and hot-pressed olivine aggregates have constrained flow laws for upper mantle conditions, olivine's rheology is compositionally dependent. As a solid solution between forsterite (Fo) and fayalite (Fa), its properties vary with iron content, with Fe-rich olivine generally exhibiting lower viscosity but higher activation energy. [22,23,24]. Therefore, determining the rheology of its end-members is essential for a complete understanding of upper mantle rheology. This requires high-purity, iron-free forsterite samples, which must be prepared via chemical synthesis (e.g., using MgO and SiO₂ as starting materials), as natural samples inevitably contain Fe and other impurities. For instance, McDonnell et al. [25] reported grain boundary sliding in synthetic wet (0.5 wt.% H₂O) Fo aggregates (n = 1.7 ± 0.4, Q = 302 ± 22 kJ/mol). Nishihara et al. [26], using a D-DIA on sintered nano-powder aggregates, reported parameters for Fo diffusion creep (Q = 506 ± 34 kJ/mol) and dislocation-accommodated grain boundary sliding (n = 3, Q = 519 ± 53 kJ/mol), though stress uncertainties in such experiments necessitate caution. These prior studies, while valuable, exhibit significant scatter in key parameters like activation energy and stress exponent, partly due to experimental challenges (e.g., porosity, stress resolution, water content) and a lack of systematic investigation of grain size effects.

To better constrain forsterite rheology and resolve these discrepancies, we synthesized high-purity (≥98 vol.%), iron-free Fo aggregates via pressureless sintering and deformed them in a high-stress-precision gas-medium Paterson apparatus at ~300 MPa and 1423–1523 K. Our experimental design prioritizes three key improvements: (1) the use of a Paterson apparatus, which offers superior stress resolution compared to D-DIA, (2) systematic variation of initial grain size to directly constrain the grain size exponent, and (3) strict control of dryness to avoid the confounding effects of water. Our results confirm diffusion creep with a well-constrained stress exponent n = 1.0 ± 0.08, an activation energy Q = 365 ± 22.7 kJ/mol, and, most importantly, a precise grain size exponent p = 2.9 ± 0.23, providing strong evidence for Coble creep. As chemically synthesized, iron-free, and high-purity samples lack the Fe and trace impurities that weaken natural olivine, our flow law defines the Mg-end member of the olivine solid solution system and represents the viscosity upper bound for upper mantle peridotites deformed dominantly by diffusion creep.

2. Experimental Methods

2.1. Starting Material and Sample Preparation

Based on the method for synthesizing forsterite aggregates via pressureless sintering reported by Koizumi et al. [27] and Hiraga et al. [28], we appropriately modified the procedure [29] to prepare initial samples for deformation experiments: forsterite cylinders measuring ~7 mm in diameter and ~14 mm in length. First, nano-sized powders of Mg(OH)₂ (~50 nm) and SiO₂ (~30 nm) bought from American Elements Inc. were mixed in an agate mortar on a ball mill at a molar ratio of 1.95:1, and then subjected to a temperature of 1273 K for ~4 hours to synthesize the mineral mixture. The reaction powder was incrementally cold compacted under 100 MPa into a three-piece split mold with a central bore diameter of 13 mm to form a green body. The resulting cylinder was then encased in a rubber sleeve and further consolidated under a cold isostatic pressure of ~170 MPa. After 30 minutes, the compacted cylinder was retrieved, with dimensions of ~10.5 mm in diameter and ~21.0 mm in length. The compacted cylinder was then vacuum-sintered in a tube furnace at approximately 1630 K for either 4 or 8 hours to produce forsterite aggregates of different grain sizes. After sintering, a ~1 mm thick disk was cut from the cylinder for Raman spectroscopy and SEM-BSE analysis. No distinct orthopyroxene peaks were detected in the Raman spectra (Figure 1a), and no second phase with obvious contrast differences was observed in BSE images (Figure 1b), indicating a high purity of the synthesized forsterite, estimated to be ≥98 vol.%. The Raman spectrum of the synthesized forsterite (Figure 1a) is consistent with that reported in our previous study on sample synthesis and characterization [29]. Finally, the sintered cylinder was machined on a lathe to the standard dimensions required for deformation experiments: approximately 7 mm in diameter and 14 mm in length. The density of the forsterite cylinder, calculated from accurate measurements of its mass and volume, is approximately 3.0 g/cm³, which is lower than its theoretical density of 3.2 g/cm³, indicating a porosity of ≤ 7%. To obtain denser samples and eliminate the influence of porosity on rheological properties, the samples were hot-pressed in a Paterson apparatus at ~300 MPa and 1423 – 1523 K for 1–2 hours before each deformation test. Scanning electron microscopy (SEM) analysis shows that the porosity of the samples after hot-pressing was effectively reduced to <2% (Figure 2b).

2.2. Deformation Experiments

Axial compression experiments on forsterite were performed in a Paterson gas medium apparatus at the Guangzhou Institute of Geochemistry, Chinese Academy of Sciences. The sample assembly generally followed the design of Li et al. [30,31], but employed a different sample geometry (~7 mm in diameter × ~14 mm in length) and slightly longer alumina pistons (54 mm) at both ends. Direct contact between the sample and the iron jacket buffered oxygen fugacity close to the Fe–FeO phase boundary during deformation. Temperature was maintained and measured using an R-type thermocouple (Pt–13%Rh/Pt) located adjacent to the upper alumina spacer surface directly above the sample. Pre-run calibrations revealed a stable thermal environment, with temperature variations along the specimen of less than ±2 K. A servo-controlled system applied the differential stress, monitored by an internal load cell, and axial displacement was tracked with a linear variable differential transformer (LVDT) inside the pressure vessel. For more details, please refer to Li et al. [30]. To eliminate potential high porosity arising from pressureless sintering, the sample was held at the target pressure and temperature for ≥1 hour before deformation commenced.

The deformation experiments were carried out under stepwise loading. At each step, a constant load—and therefore an approximately constant differential stress, as axial shortening was limited to <3 % per step—was maintained. Once steady state flow was reached, the corresponding strain rate was recorded. Each loading step can thus be treated as a constant stress creep test.

2.3. Microstructural Analysis

Due to the fine grain size of the synthesized forsterite (~1 μm), which is much smaller than the conventional thin-section thickness of 30 μm, microstructural characterization via traditional optical microscopy was not feasible. We therefore relied on scanning electron microscopy (SEM) to examine grain size, porosity, and related features before and after deformation. The procedure for SEM sample preparation was as follows: a ~1 mm thick disk was cut from one end of either the pressureless-sintered or the deformed sample cylinder using a slow-speed diamond saw. Both surfaces of the disk were polished successively up to 8000-grit sandpaper, followed by a final polish with a ~0.05 μm colloidal silica suspension on a polishing cloth for 3 hours. To enhance the visibility of grain boundaries, the polished surface was etched for 10 seconds using a diluted HF solution (HF: H₂O = 1:100), which preferentially dissolves grain boundary regions and produces topographic contrast under SEM. Both backscattered electron (BSE) and secondary electron (SE) imaging were performed using a TESCAN MIRA3 field-emission SEM operated at an accelerating voltage of 20 kV and a working distance of 15 mm. BSE imaging was primarily used to identify whether any secondary mineral phases (e.g., enstatite) were present in addition to forsterite, whereas SE imaging was employed to examine grain morphology, porosity, and for grain size measurement. The average chord length, obtained by applying the linear intercept method [32] to at least 200 forsterite grains in the acquired images, was determined as the two-dimensional grain size of forsterite.

Given that diffusion creep typically does not induce significant microstructural evolution, additional micro-analytical techniques such as electron backscatter diffraction (EBSD) or transmission electron microscopy (TEM) were not employed in this study.

2.4. Mechanical Data Analysis

During axial compression, the Paterson gas-medium apparatus continuously records parameters such as axial displacement and applied load. After corrections for machine distortion, strength of the iron jacket, and changes in the sample cross-sectional area, the mechanical data can be reduced to strain rate versus differential stress.

Both theoretical models and experimental observations indicate that when a mineral aggregate undergoes plastic deformation, its rheological behavior—whether deformation is governed by dislocation or diffusion creep—can be described by a flow law of the following form:

$$\dot{\mathsf{\epsilon }}=\mathrm{A}{\mathrm{d}}^{-\mathrm{p}}{\mathsf{\sigma }}^{\mathrm{n}}\exp \left(-{\displaystyle \frac{\mathrm{Q}}{\mathrm{R}\mathrm{T}}}\right)$$

(2)

where $\dot{\mathsf{\epsilon }}$is the strain rate (s^-1), A is a material-dependent parameter, d is the grain size (μm), p is the grain size exponent, σ is the differential stress (MPa), n is the stress exponent, Q is the activation energy (kJ/mol), R is the gas constant with a value of 8.314 J/(mol·K), and T is the absolute temperature (K).

During high-temperature deformation, forsterite grains coarsen with time. The initial (before deformation) and final (after deformation) grain sizes can be determined statistically from SEM images. For the real-time grain size, as it is impractical to remove a specimen from the apparatus to measure the grain size after each creep stage, it must be estimated from the initial and final grain sizes according to a grain-growth law [20,21]:

$${\mathrm{d}}_{\mathrm{t}}^2-{\mathrm{d}}_{\mathrm{i}}^2=\mathrm{kt}$$

(3)

Here, ${\mathrm{d}}_{\mathrm{t}}$is the grain size (μm) at time t, ${\mathrm{d}}_{\mathrm{i}}$ is the initial grain size (μm), k is a rate constant (μm²∙s^-1), and t is the time (s). The constant k follows an Arrhenius relation, k =k₀ exp(-Q_gg/RT), with Q_gg = 160 kJ/mol for olivine [33]. For each experiment, we first calculated the average k from the total duration and the measured initial and final grain sizes. Then, using the known Q_gg and the experimental temperature, we determined the pre-exponential factor k₀ from that k value. This k₀ was subsequently applied to compute the real-time grain size d_t for each deformation step at its corresponding elapsed time.

Finally, we fitted the steady-state flow data (strain rate vs. differential stress along with the real-time grain size corresponding to each loading step) to flow law (2) to determine the corresponding rheological parameters.

3. Results

3.1. Microstructures-Grain Size

Scanning electron microscopy (SEM) observations reveal that the initially sintered sample without hot pressing consists of equant polygonal forsterite grains with a grain size of approximately 0.7 or 1 μm (Figure 2a,f). The polished sample surface exhibits ~5% porosity (Figure 2a), which is effectively reduced to <2% after 1–2 hours of hot pressing (Figure 2b). Forsterite grain sizes measured by the linear intercept method are approximately 0.7 μm (Figure 2a; Figure 3a) after 4 hours of sintering and approximately 1 μm (Figure 2f; Figure 3e) after 8 hours of sintering; detailed results are presented in Table 1. Preliminary mechanical data indicate that the dominant deformation mechanism under the experimental conditions is diffusion creep, which typically does not significantly alter the microstructure. Therefore, microstructural analysis of the deformed samples focuses primarily on grain growth relative to the initial grain size. SEM analysis shows that forsterite grains remain equant and polygonal, with grain sizes measured by the linear intercept method ranging from approximately 1.0 to 1.1 μm (Figure 2c‒e; Figure 3b‒d) for 4-hour sintered samples, and 1.2 to 1.3 μm for 8-hour sintered samples (Figure 2g,h; Figure 3f‒h), respectively (Table 1), indicating noticeable grain growth compared to the initial grain sizes. Grain growth rates (Equation (3)) were fitted to the initial and finial grain sizes versus duration time, yield Q_gg ≈ 152 kJ/mol and k₀ ≈ 5 μm²*s^-1. The value of Q_gg is consistent with that suggested by Karato [33] within the error range.

Therefore, the growth constants at different experimental temperatures: k = 1.25×10^-5 μm²·s^-1 at 1423 K, k = 1.94×10^-5 μm²·s^-1 at 1473 K, and k = 2.92×10^-5 μm²·s^-1 at 1523 K. Subsequently, the real-time grain sizes at each stage of data acquisition were calculated, and the results are presented in Table 2.

3.2. Mechanical Results

A total of 6 stress-stepwise deformation experiments (each step can be regarded approximately as a constant-stress creep test) were conducted on 3 samples sintered for 4 h and 3 samples sintered for 8 h (Table 2). Representative stress-time and strain-time curves are shown in Figure 4. The deformation experiments were performed at a confining pressure of ~300 MPa, temperatures of 1423 K, 1473 K, and 1523 K, and applied stresses ranging from 50 to 380 MPa. The measured strain rates ranged from 8 × 10^-7 s^-1 to 1.2 × 10^-5 s^-1. The mechanical data are plotted in a strain rate vs. stress diagram (Figure 5a) and show that the samples sintered for 8 h are overall stronger than those sintered for 4 h, which is attributed to their larger grain sizes. A global fit using the power-law flow law including grain size term (Equation (2)) yields a stress exponent n = 1.0 ± 0.09, an activation energy Q = 365 ± 22.7 kJ/mol, a grain size exponent p = 2.9 ± 0.23, and a material-dependent parameter A = 10^5.4±0.73. The coefficient of determination R²=0.896 indicates a good fit. Accordingly, under the experimental conditions, the sintered forsterite obeys the following flow law:

$$\dot{\mathsf{\epsilon }}=105.4\text{±}0.73{\mathrm{d}}^{-2.9\text{±}0.23}{\mathsf{\sigma }}^{1.0\text{±}0.09}\exp \left(-{\displaystyle \frac{365\text{±}22.7\mathrm{kJ}/\mathrm{mol}}{\mathrm{R}\mathrm{T}}}\right)$$

(4)

To illustrate separately the dependence of strain rate on differential stress, temperature, and grain size, the mechanical data were normalized using the relevant parameters in equation (4). The normalized data are then plotted as log $\dot{\mathsf{\epsilon }}$ vs. log σ, log $\dot{\mathsf{\epsilon }}$ vs. 1000/T, and log $\dot{\mathsf{\epsilon }}$ vs. log d, as shown in the Figure 5b–d. A good linear relationship is observed in each case. The parameters n, Q, and p obtained from these individual fits are in good agreement with those derived from the global fit, with the exception of n value at 1423 K (0.7 ± 0.1), which is slightly lower than the global value. This minor discrepancy may reflect the narrower stress range covered at this temperature.

4. Discussion

4.1. Deformation Mechanism

Scanning electron microscopy (SEM) observations indicate that forsterite remains equant and polyhedral after deformation, with no significant deformation microstructures, such as undulatory extinction or lattice preferred orientation. This is consistent with diffusion creep not significantly altering the material's microstructure. During deformation of forsterite, the strain rate exhibits a near-linear dependence on the applied differential stress (stress exponent n = 1.0 ± 0.09) and a clear grain size dependence (grain size exponent m = 2.9 ± 0.23). The obtained activation energy Q = 365 ± 22.7 kJ/mol, is consistent with values reported for diffusion creep of olivine (Q = 300 kJ/mol [17]; Q = 347 kJ/mol [20]). These features collectively indicate that diffusion creep is the dominant deformation mechanism of forsterite under the experimental conditions.

4.2. Grain Size Effect

Under the experimental conditions, the dominant deformation mechanism of forsterite is diffusion creep, and the strain rate exhibits a strong grain size dependence. To better investigate this dependence, samples with different initial grain sizes were prepared by controlling the high-temperature sintering time. Linear intercept measurements on scanning electron microscopy (SEM) images indicate that the initial grain size of samples sintered for 4 hours is ~0.7 μm, while that of samples sintered for 8 hours is ~1 μm. In the plot of log $\dot{\mathsf{\epsilon }}$ vs. log σ (Figure 5a), under otherwise identical conditions, the strain rate of the 8-hour sintered samples is significantly lower than that of the 4-hour sintered samples, indicating that samples with larger initial grain sizes exhibit higher flow strength. Furthermore, the mechanical data for the 8-hour sintered samples shown considerable scatter: under otherwise identical conditions, later stages of the deformation experiments, which involve larger grain sizes due to grain growth, result in higher flow strength of forsterite. In summary, both the initial grain size and grain growth during deformation significantly influence the rheological behavior of forsterite. Therefore, the real-time grain sizes at each mechanical data acquisition point during deformation were calculated based on the grain growth rates. The power-law flow law was then fitted to analyze the grain size dependence of the strain rate, yielding a high-precision grain size exponent p =2.9±0.23. This value is very close to the theoretical value of 3 for Coble creep, where diffusion occurs along grain boundaries, and clearly distinguishes it from Nabarro‒Herring creep (p=2, lattice diffusion). The grain size dependence of the strain rate can be more directly observed in the plot of log $\dot{\epsilon }$ vs. log d (Figure 5d), after normalizing for deformation temperature and differential stress.

4.3. Compare with Previous Studies and Geological Implication

4.3.1. Comparison with Previous Forsterite Diffusion Creep Studies

Our results provide a refined and robust flow law for diffusion creep in forsterite. A direct comparison with previous studies reveals both agreements and significant advances. McDonnell et al. [25] studied synthetic forsterite aggregates containing ~0.5 wt.% H₂O and reported a stress exponent n = 1.7 ± 0.4 and an activation energy Q = 302 ± 22 kJ/mol. Their higher stress exponent, which deviates from the theoretical value of 1 for diffusion creep, suggests that their deformation mechanism might have been influenced by a component of grain boundary sliding or minor dislocation activity, likely facilitated by the presence of water. In contrast, our experiments were conducted under strictly dry conditions, yielding n = 1.0 ± 0.08, which is a much cleaner signature of pure diffusion creep.

Nishihara et al. [26] used a D-DIA apparatus on sintered forsterite nano-powders and reported a much higher activation energy of Q = 506 ± 34 kJ/mol for what they interpreted as diffusion creep. However, as the authors themselves noted, most of their mechanical data were acquired at a single temperature (1573 K), with only two data points at 1473 K and 1423 K from one run; this limited temperature coverage inevitably leads to substantial uncertainty in the derived activation energy. In addition, the inherent stress uncertainties in D-DIA experiments may also contribute to errors in flow law parameters. In contrast, our study, using a Paterson apparatus known for its superior stress resolution and stability, provides a more reliable activation energy of Q = 365 ± 22.7 kJ/mol.

Theoretical and experimental studies have established that diffusional creep in olivine is rate-controlled by the diffusion of the slowest species, silicon [34,35]. Jaoul. [34] proposed a multicomponent diffusion model in which the activation energy for creep is the sum of that for silicon diffusion and that for the formation of octahedral vacancies (V_Me). For lattice (volume) diffusion of Si in forsterite, the reported activation energy is ~410 kJ/mol [35], whereas Si grain-boundary diffusion exhibits a significantly lower activation energy of 245 ± 10 kJ/mol [36] or 220 ± 30 kJ/mol [37]. Our determined Q = 365 ± 22.7 kJ/mol lies between these two values—higher than that for Si grain-boundary diffusion but lower than that for Si lattice diffusion. This observation suggests that the diffusion creep of our forsterite aggregates is not purely controlled by Si lattice diffusion (which would yield a higher Q approaching ~410 kJ/mol), nor is it exclusively governed by Si grain-boundary diffusion (which would yield a lower Q of ~220–245 kJ/mol). Instead, the intermediate Q value may reflect a combination of grain-boundary and lattice diffusion contributions, or it may indicate that the diffusion of Si is coupled with the formation and migration of other defects (e.g., octahedral vacancies) during creep [34]. This is also consistent with the fact that our experimentally determined grain size exponent (p = 2.9) strongly supports Coble creep (grain-boundary diffusion control), yet the activation energy is somewhat elevated relative to pure grain-boundary diffusion values, possibly due to the contribution from vacancy formation or other defect-related processes. This value is in excellent agreement with the activation energy for grain boundary diffusion in olivine and is consistent with the Coble creep mechanism inferred from our grain size exponent.

To further compare the diffusion creep flow laws of forsterite, we normalized our results and those of Nishihara et al. [26] to common conditions of P = 300 MPa, T = 1573 K, and d = 1 μm, and plotted strain rate against differential stress on a logarithmic diagram (Figure 6). At a given differential stress, the strain rate predicted by the flow law of Nishihara et al. [26] is approximately 3‒4 times higher than that predicted by our flow law, indicating that their forsterite aggregates are significantly weaker. We attribute this discrepancy to the presence of ~10 vol.% enstatite in the starting material of Nishihara et al. [26]; McDonnell et al. [25] reported that increasing enstatite content from 0 to 2.5 vol.% reduces aggregate strength by a factor of 3‒4.

4.3.2. Comparison with Studies on Natural Olivine (Fo₉₀) and Other Compositions

Our forsterite flow law is a critical end-member for understanding the rheology of natural olivine, which typically contains ~10 mol% fayalite (Fo₉₀). Previous studies on diffusion creep in natural olivine aggregates (e.g., [9,20]) have reported similar activation energies (e.g., Q ≈ 347 kJ/mol) and stress exponents (n ≈ 1). However, our study provides a more precise constraint on the grain size exponent for the Mg-rich end-member.

The most significant compositional difference between our pure forsterite and natural olivine is the effect of iron. Zhao et al. [22,23] demonstrated that Fe-rich olivine exhibits lower viscosity but higher activation energy for both diffusion and dislocation creep. Figure 6 confirms that natural San Carlos olivine (~Fo₉₀) is considerably weaker than our pure forsterite, with strain rates approximately two orders of magnitude faster at equivalent conditions. This substantial weakening cannot be solely attributed to the iron effect documented by Zhao et al. [22,23]; other factors are also likely to contribute. First, natural minerals generally contain more trace impurities, which facilitate the generation of lattice defects and thereby enhance diffusivity. Second, natural San Carlos olivine aggregates inevitably contain secondary mineral phases—for instance, spinel [21] that is difficult to eliminate, or enstatite [20] that is purposely added to buffer Si fugacity during sample preparation. Finally, differences in sample fabrication procedures may result in distinct material characteristics, including crystallographic preferred orientation, grain boundary structure, and trace water content. Nevertheless, the iron-induced viscosity reduction observed in olivine solid solutions implies that upper mantle rheology is compositionally heterogeneous; regions with contrasting iron content (e.g., due to melt extraction or metasomatism) will exhibit corresponding viscosity variations.

Critically, because our samples are chemically synthesized, iron-free, and of high purity (≥98 vol.%), they lack not only Fe but also the trace impurities that would otherwise promote defect generation and enhance diffusion in natural olivine. Consequently, our forsterite aggregates exhibit significantly higher strength than natural Fe-bearing olivine. Our pure forsterite flow law therefore represents the highest-viscosity end-member for a given grain size and temperature under dry conditions—defining the viscosity upper bound for olivine-dominated mantle rocks. This end-member is essential for constructing accurate composite rheological models of the upper mantle, particularly for simulating the mechanical behavior of depleted, Mg-rich harzburgites in the lithospheric mantle.

4.3.3. Geological Implications

Our findings carry several profound implications for the dynamics and rheological stratification of the Earth’s upper mantle.

First, our flow law defines a rigorous viscosity upper bound for diffusion creep in olivine-dominated mantle rocks. Extrapolation of our flow law, together with published diffusion-creep laws for olivine and orthopyroxene, to mantle conditions using the 60-Ma oceanic geotherm of Turcotte and Schubert [38] (p. 221) and a reference grain size of 1 mm (Figure 7) shows that our pure forsterite consistently exhibits the highest viscosity at depths >100 km—approximately 2–3 orders of magnitude higher than the San Carlos olivine flow law of Hirth and Kohlstedt [20], and about 5 orders of magnitude higher than that of Karato et al. [21]. The orthopyroxene diffusion flow law [39] yields viscosities comparable to the San Carlos olivine of Hirth and Kohlstedt [20], i.e., ~2 orders of magnitude lower than our pure forsterite. This upper-bound framework provides a baseline for quantitatively evaluating the weakening effects of iron, water, trace impurities, and secondary phases in geodynamic models [40].

Second, the precisely determined grain size exponent (p = 2.9 ± 0.23) provides robust experimental evidence that Coble creep (grain-boundary diffusion) is the rate-controlling mechanism for diffusion creep in forsterite under upper-mantle conditions. This confirms that grain-boundary diffusion, rather than lattice diffusion (Nabarro–Herring creep), dominates in fine-grained mantle regions such as mylonite zones or domains undergoing dynamic recrystallization, resolving a long-standing debate in olivine rheology.

Third, the strong grain-size sensitivity ($\dot{\mathsf{\epsilon }}$ ∝ d^-2.9) implies that modest grain-size reduction can dramatically enhance strain rates—for example, decreasing grain size from 10 mm to 1 mm increases the creep rate by nearly three orders of magnitude. This positive feedback, whereby deformation promotes recrystallization and grain refinement, which in turn facilitates grain-size-sensitive diffusion creep, is a key mechanism for shear-zone localization and lithospheric weakening.

Fourth, the interplay between temperature and grain size governed by our flow law can explain the mechanical transition at the lithosphere–asthenosphere boundary (LAB). In cold, coarse-grained lithospheric mantle, dislocation creep is expected to dominate. However, with increasing temperature and potentially finer grain sizes (due to melt–rock interaction or recrystallization) near the LAB, a switch to Coble creep could occur, creating a low-viscosity zone that facilitates plate decoupling. Using our flow law and a reference grain size of 1 mm, we obtain viscosities on the order of ~10²² Pa·s at LAB depths (Figure 7), about three orders of magnitude higher than the geophysically inferred low-viscosity layer (~10¹⁹ Pa·s) beneath 50-Ma oceanic seafloor [41]. This discrepancy implies that either asthenospheric grain size must be substantially smaller than 1 mm (e.g., tens of micrometers) or that additional weakening mechanisms—such as iron content, water, partial melt, or secondary phases—are required to reconcile our end-member flow law with geophysical observations.

Fifth, our end-member flow law provides a key reference for understanding the compositional dependence of olivine rheology. While a complete description of Fe-induced weakening requires further systematic work on intermediate compositions and careful decoupling of Fe effects from those of impurities and secondary phases, the contrast between our Fo₁₀₀ data and existing results on Fe-bearing olivine [22,23] highlights the substantial role of iron content. This end-member comparison offers a useful starting point for future efforts to parameterize Fe effects in diffusion-creep flow laws, though such parameterization will need to account for the complexity of natural systems. Moreover, integrating our end-member law with grain-growth kinetics and second-phase effects [28,42] will allow geodynamic models to incorporate lateral and vertical variations in mantle composition, moving beyond uniform rheological assumptions.

In summary, our study not only provides a well-constrained diffusion creep law for the Mg-end member of olivine but also establishes a quantitative upper-bound reference for mantle viscosity. This framework is indispensable for evaluating the relative contributions of iron, water, impurities, and grain size to upper-mantle weakening, and for constructing next-generation, compositionally aware rheological models.

5. Conclusions

• We have successfully synthesized high-purity (≥98 vol.%), iron-free, fine-grained (~0.7–1.0 μm) forsterite aggregates and deformed them under dry conditions in a high-precision Paterson gas-medium apparatus. Our experiments confirm that diffusion creep is the dominant deformation mechanism at 300 MPa, 1423–1523 K, and 50–380 MPa.
• robust flow law was determined: $\dot{\mathsf{\epsilon }}=105.4\text{±}0.73{\mathrm{d}}^{-2.9\text{±}0.23}{\mathsf{\sigma }}^{1.0\text{±}0.09}\exp \left(-{\displaystyle \frac{365\text{±}22.7\mathrm{kJ}/\mathrm{mol}}{\mathrm{R}\mathrm{T}}}\right)$. The low stress exponent (n ≈ 1) and activation energy (Q ≈ 365 kJ/mol) are characteristic of diffusion creep.

3.

The precisely determined grain size exponent (p = 2.9 ± 0.23) is in excellent agreement with the theoretical value for Coble creep (grain boundary diffusion), providing the first direct experimental evidence that grain boundary diffusion, not lattice diffusion, is the rate-controlling mechanism for diffusion creep in forsterite under upper mantle conditions.

4.

Our chemically synthesized, iron-free, high-purity forsterite defines the Mg-end member flow law for the olivine solid solution system and represents the viscosity (diffusion creep dominant) upper bound for olivine-dominated mantle rocks under dry conditions. This end-member flow law, combined with the strong grain-size sensitivity of diffusion creep, provides a necessary constitutive framework for modeling shear zone formation, lithospheric weakening, and the rheological stratification of the Earth's mantle, with direct implications for large-scale geodynamic processes.