Introduction

Grain size and grain shape parameters have a great influence on the physical properties of minerals as well as on geophysical processes. Rheological properties of minerals are strongly dependent on grain size (Frost and Ashby 1982). With the decrease of grain size the deformation mechanism passes from dislocation-controlled creep to diffusion-controlled creep. Physical properties of earth’s minerals are usually studied at micro- or millimetric scales but not at nanometric scale. Even though nanocrystalline minerals might have an important influence on geological processes (Hochella et al. 2008 for review), they are very poorly studied. Despite their limited occurrences in the crust and the mantle, studies show their importance in geophysics. For instance, mechanical grinding of rocks associated with earthquakes can generate gouge of nanoparticles, which are themselves likely to be important in fault mechanics (Wilson et al. 2005). In the context of subducting slab, experiments show presence of nanoparticles of ringwoodite, which is assumed to play an important role in deep-focus earthquakes (Burnley et al. 1991; Green and Burnley 1989).

This work presents results on compressibility of nanocrystalline forsterite (Mg2SiO4). An equation of state of nanocrystalline forsterite has been established and the difference of compressibility with macro-powder sample has been discussed. This is, to our knowledge, the first study on equation of state of nanocrystalline silicate.

Experimental procedure

Nanocrystalline forsterite (nc-Fo) was synthesized following the sol–gel process described by Saberi et al. (2007). The final calcination was performed at 1,073 K for 6 h under a flow oxygen in order to remove the residual carbon. Macro-powder forsterite (m-Fo) was synthesized from a stoichiometric mixture of SiO2 and MgO, finely grinded and fired at 1,000°C for several hours. Compression data of both nc-Fo and m-Fo were collected using two different experimental set-ups, diamond anvil cell (DAC) and multi-anvil press (MAP), for a good range of pressures and temperatures.

DAC experiments

A Mao-Bell cell with 400 μm anvil culet and 270 μm thick stainless steel gasket pre-indented to 50 μm thickness was used (Mao et al. 1979). The sample (nc-Fo or m-Fo) was loaded in a sample chamber of about 180 μm in diameter. Silicone oil (Aldrich, 5 cSt viscosity) was used as pressure-transmitting medium. Shen et al. (2004) showed that silicone oil with a viscosity of 1 cSt provides a quasi-hydrostatic medium as good as 4:1 alcohol mixture up to 20 GPa. In a more recent study, Klotz et al. (2009) has identified a typical gradient of 0.4 GPa below 12 GPa at ambient temperature. In situ X-ray diffraction experiments were conducted at the X17B3 beamline of the National Synchrotron Light Source (NSLS), Brookhaven National Light Source. In each experiment, one 1 μm chip of ruby was loaded into DAC for pressure measurement using a ruby R1 line fluorescence method (Mao et al. 1986). After collecting a diffraction spectrum at ambient conditions, the pressure was increased stepwise and X-ray diffraction patterns were collected at each step for 30 min. Sample pressure was measured before and after the sample diffraction pattern, and an average of the measured pressures is used in data processing (Table 1). Good quality diffraction data were collected up to a maximum pressure of 7.6 GPa. Above this pressure, diffraction signal from nc-Fo sample became much weaker, and even one-hour exposure time still produced a too low signal-to-background ratio for reasonable cell parameter refinements. The diffraction data were analyzed using Fit2D combined with MAUD for cell refinement (Lutterotti et al. 1999).

Table 1 Experimental conditions and results (see text for uncertainties)
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MAP experiments

The cubic-anvil MAP SAM85 at X17B2 beam line of NSLS was used for in situ high pressure and temperature X-ray diffraction. Amorphous boron mixed with epoxy resin (4:1) was used as pressure medium (Fig. 1). The cell assembly was made of relatively soft materials in order to limit non-hydrostatic stress during the experiment. For direct comparison under the same experimental conditions, a layer of m-Fo and a layer of nc-Fo (mixed with boron nitride powder to avoid grain growth) separated by a layer of NaCl + BN mixture were loaded in a BN sample capsule of 1 mm in diameter and 2 mm in length. The cell pressure was derived from the measured volume of NaCl using X-ray diffraction and Decker scale (Decker 1971), which has an accuracy better than 2% for our experimental P-T ranges (Decker 1965, 1971; Mueller et al. 2003). Temperature was directly measured using a W3%Re–W25%Re thermocouple. The experimental procedure was the following: the press load was increased to 60 ton and the temperature was raised to 1,573 K to anneal the sample stress developed during compression. According to Weidner et al. (1992, 1994) and Wang et al. (1998) the deviatoric stress level becomes undetectable for temperature as low as 723 K. The annealing temperature was maintained for just few minutes to avoid grain growth. After annealing, the temperature was decreased to 1,273 K, and then to ambient temperature by 200 K step. The heating/cooling cycle was repeated at 10-ton load interval from 60 to 0 ton during decompression. Energy dispersive X-ray diffraction spectra of nc-Fo, m-Fo and NaCl were collected at each step of cooling and decompression. The cell parameters of nc-Fo, m-Fo and NaCl were extracted from the diffraction spectra using the software Plot85 (Baldwin 1990).

Fig. 1
figure 1

Schematic of the MAP assembly

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Results and discussion

X-ray diffraction patterns of m-Fo and nc-Fo before experiments (Fig. 2) were collected from 15° to 60° (2θ) using Siemens D5000 diffractometer (Cu Kα radiation). nc-Fo appears to be well crystallized and exhibits peaks broadening resulting from the nanoparticle size of the sample (see subview of Fig. 2 for a typical peak broadening of nc-Fo compare to m-Fo).

Fig. 2
figure 2

X-ray diffraction patterns at ambient conditions of m-Fo (dash line) and nc-Fo (solid line) and forsterite standard peaks position (red sticks). The subview on the upper right corner is a zoom of (131) and (112) peaks

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The grain size of m-Fo was measured to 5 μm from previous experiments in similar conditions using scanning electron microscopy. Using the energy dispersive spectra collected at X17B2 and the Scherrer equation (Gerward et al. 1976), the crystallite size of nc-Fo was determined based on the variation of peak width of (021), (002) (130) (131) and (112) as a function of photon energy. Gerward et al. (1976) show that the crystallite size determined using Scherrer equation is in very good agreement with the size obtained using electron microscope. The technique has been applied for grain size determination for nanocrystalline MgO sample at high pressure using the following scheme (Chen et al. 2005). Line broadening at half the maximum intensity of a Gaussian profile (β) can be caused by elastic strain (βD) or small grain size (βS) (Willets 1965):

$$ \beta^{2} = \beta_{\text{D}}^{2} + \beta_{\text{S}}^{2} $$
(1)

In case of energy dispersive diffraction, βD and βS can be expressed as:

$$ \beta_{\text{D}} = 2eE $$
(2a)
$$ \beta_{\text{S}} = {\raise0.7ex\hbox{${K({\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}hc)}$} \!\mathord{\left/ {\vphantom {{K({\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}hc)} {L\sin \theta_{0} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${L\sin \theta_{0} }$}} $$
(2b)

where e is the approximate upper limit of strain, E is the X-ray photon energy, K is a Scherrer constant that we set at 0.9, h is the Plank’s constant, c is the velocity of light, L is the average crystallite size, 2θ0 is the fixed scattering angle. Considering the instrumental line broadening (βi) (Willets 1965), Eq. 1 can also be written:

$$ \beta^{2} = \beta_{\text{o}}^{2} - \beta {}_{\text{i}}^{2} $$
(3)

where βo is the observed broadening. β i was determined experimentally from the diffraction spectra of NaCl at ambient conditions. Using Eqs. 1 and 3, we obtain:

$$ \beta_{\text{o}}^{2} = \beta_{\text{D}}^{2} + \beta_{\text{S}}^{2} + \beta_{\text{i}}^{2} $$
(4)

The derived average crystallite size (L) of nc-Fo before experiment derived from Eq. 4 is equal to 11 ± 2 nm. No increase of this size was observed during high-pressure high-temperature experiments. The microstrain has been found to be negligible as confirmed by Fig. 3, which presents a typical nc-Fo peak profile (crosses) and its Gaussian fitting using Plot85 (red line). (021) peak was collected under 50 ton (6.5 GPa) at room temperature after annealing. The peak shape is broad but not asymmetric showing no evidence of residual stress. The peak fitting using a Gaussian shape gives a very small residual (green line) showing a good fitting of nc-Fo peaks and an accurate determination of nc-Fo cell parameters.

Fig. 3
figure 3

nc-Fo (021) peak at 6.5 GPa and room temperature. Red crosses data, red line Gaussian fitting, green line residual and black stick measured peak position

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Pressure, temperature and cell parameters a, b and c obtained during experiments are presented Table 1. The error on the pressure is estimated to be ±0.1 GPa for MAP experiments and ±0.5 GPa for DAC experiments. The temperature is accurate within 50 K, taking uncertainty of the thermocouple location and temperature gradient into account. The error on the cell parameters (a, b and c) is estimated to be ±0.001 Å.

We fit the P–V–T data using the well-established third-order Birch-Murnaghan equation of state and the program EosFitV5.2 (Angel 2001). For this fitting, data sets from both experimental techniques were combined as we had too few experimental points using DAC to fit them separately from MAP data. A first fitting was performed with no constrained on the parameters. m-Fo exhibits a bulk modulus at zero pressure (K 0(m-Fo)) of 122.9 (±10.5) GPa while the bulk modulus at zero pressure of nc-Fo (K 0(nc-Fo)) is 114.6 (±11.1) GPa with the pressure derivative of the bulk modulus K′ equal to 5.7 (±2.8) and 6.3 (±3.1), respectively. K 0(nc-Fo) exhibits a difference of −8.3 GPa with respect to K 0(m-Fo). For a correct comparison between K 0(m-Fo) and K 0(nc-Fo), the value of both bulk modulus should be obtained by either allowing K′ to float or constraining K′ to an identical and fixed value. In the present study, both K′ obtained in the first fitting are among the highest values found in the literature (between 3.5 and 5.6; e.g. Knittle 1995; Wentzcovitch and Stixrude 1997; Zha et al. 1996, 1998a, b). Pressure derivatives might not be very well constrained due to the small pressure range explored and the tradeoff between K 0 and K′. A second fitting was then performed assuming K′ is identical for m-Fo and nc-Fo. This assumption might introduce uncertainties on the K 0 values, which are not taking into account thereafter. K′ has been fixed to 3.88. Brodholt et al. (1996) derived this value from ab initio calculation of compressibility of forsterite for pressures up to 70 GPa. Using K′ = 3.88, the authors reproduce very well the experimental compression data of Will et al. (1986), Kudoh and Takeuchi (1985) and Andrault et al. (1995) over the entire experimental pressure range (up to 70 GPa).

The results of the second fitting for m-Fo and nc-Fo with K′ fixed at 3.88 are shown in Table 2. Unit cell volume at zero pressure (V 0), thermal expansion parameters (α0 and α1) and the bulk modulus temperature derivative (dK/dT)P of m-Fo and nc-Fo are very comparable and are similar to the values found in the literature (Anderson and Isaak 1995; Fei 1995; Katsura et al. 2008; Knittle 1995 and herein references), except for α1 and (dK/dT)P of nc-Fo. Notice the negative value calculated for α1(nc-Fo) as well as the low magnitude of (dK/dT)P(nc-Fo). A confirmation of both values is necessary before commenting on the trend and the magnitude as the uncertainties associated with them are great. K 0(m-Fo) is equal to 129.6 (±3.2) GPa, which is consistent with previously determined bulk moduli (Brodholt et al. 1996; Isaak et al. 1989; Knittle 1995; Zha et al. 1996, 1998a, b). However, K 0(nc-Fo) remains 6.4 GPa lower than our K 0(m-Fo) and is equal to 123.3 (±3.4) GPa. Figure 4 presents our experimental data (V/V 0 vs. pressure for the different explored temperatures) of m-Fo and nc-Fo. The solid and dashed lines represent the third-order Birch-Murnaghan equation of state fitting of m-Fo and nc-Fo, respectively.

Table 2 Unit cell volume at zero pressure (V 0), thermal expansion parameters (α0 and α1), bulk modulus at zero pressure (K 0), and bulk modulus temperature derivative (dK/dT) of m-Fo and nc-Fo for K′ fixed at 3.88
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Fig. 4
figure 4

Experimental compression data of m-Fo and nc-Fo for a ambient conditions and b different temperatures as function of pressure. Crosses DAC data. Diamonds and circles MAP data. m- macro-powder forsterite data. nc- nanocrystalline forsterite data. The solid and dashed lines represent the third-order Birch-Murnaghan equation of state that fits best m-Fo and nc-Fo data, respectively

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Experimental studies (Ehm et al. 2009; Erb 1995; Mayo et al. 1990; Wang et al. 2007; Zhou et al. 2003) and theoretical simulations (Latapie and Farkas 2003; Van Swygenhoven and Caro 1998; Zhao et al. 2006) on nanocrystalline metal, alloys, ceramic and oxides have demonstrated the influence of grain size on their elastic properties. In numerous cases, elastic moduli of nano-material are lower than their bulk counterpart and decreases with the decreasing of grain size within a range of 2 nm to a few tens of nanometers, although their plastic behaviors may sometimes show an opposite trend (Chen et al. 2005; Zhao et al. 2007). For comparison in the same grain size range, K 0(nc-Fo) is 4.9% smaller than K 0(m-Fo) while K 0 of nanocrystalline MgO is 8.3% smaller than K 0 of its bulk counterpart (Yeheskel et al. 2004), K 0 of nanocrystalline α-Al2O3 is similar to K0 of its bulk counterpart (Zhao et al. 2001) and K 0 of nanocrystalline γ-Al2O3 is 34.5% smaller than K 0 of its bulk counterpart (Ahuja et al. 2004; Zhao et al. 2001). Therefore, the effect of grain size on the bulk modulus of forsterite is relatively moderate.

One of the characteristics of nano-material is a high surface area to volume ratio. This implies a significant volume fraction of grain boundary and triple junction. Simulations on nanocrystalline α-Fe (Latapie and Farkas 2003) and Ni (Zhao et al. 2006), and experimental studies on nanocrystalline MgO (Yeheskel et al. 2004), Fe, Cu, Ni and (Cu-Ni) (Shen et al. 1995) attribute the softening of their elastic moduli to the elastic weakness of grain boundary and triple junction. Following this hypothesis, we calculate the K 0 for grain boundary and triple junction of forsterite using our data set. This is done by using a simple rule of mixture of crystalline grain body, grain boundary and triple junction based on their volume fractions (Latapie and Farkas 2003; Paul 1960; Shen et al. 1995; Zhao et al. 2006). K 0(nc-Fo) can be expressed as:

$$ K_{{0({\text{nc - Fo}})}} = f_{{({\text{cg}})}} K_{{0({\text{cg}})}} + f_{{({\text{gb}})}} K_{{0({\text{gb}})}} + f_{{({\text{tj}})}} K_{{0({\text{tj}})}} $$
(5)

where K 0(cg), K 0(gb) and K 0(tj) (f (cg), f (gb) and f (tj)) are the bulk moduli (volume fractions) of the crystalline grain body, grain boundary and triple junction, respectively. K 0(cg) is assumed to be equal to K 0(m-Fo) since the contribution of grain boundaries and triple junctions is negligible for a 5 μm grain size sample. We determined the volume fractions of crystalline grain body, grain boundary and triple junction using the modified tetrakaidecahedral grain shape model proposed by Zhou et al. (2007). The grain boundary width was assumed to be 0.5 nm (Ricoult and Kohlstedt 1983). Ricoult and Kohlstedt (1983) have determined the thickness of the olivine grain boundaries using transmission electron microscopy (TEM). Zhou et al. (2003, 2007) demonstrated that f (tj) increases rapidly with the decreasing of grain size for grains less than 30 nm and that the relative contribution of K 0(tj) then becomes significant. In our case, with a grain size of 11 nm and a grain boundary width of 0.5 nm and using Zhou et al. (2007) model, we calculate f (gb) = 0.130 and f (tj) = 0.007; therefore f (tj) ≪ f(gb) and K 0(tj) is assumed to have a negligible contribution to K 0(nc-Fo). f (gb) and f (tj) are then summed and represented by the volume fraction of interface f (in):

$$ f_{(in)} = f_{{({\text{gb}})}} + f_{{({\text{tj}})}} $$
(6)

and Eq. 5 becomes:

$$ K_{{0({\text{nc}} - {\text{Fo}})}} = f_{{({\text{cg}})}} K_{{0({\text{cg}})}} + f_{{({\text{in}})}} K_{{0({\text{gb}})}} $$
(7)

The calculation gives K 0(gb) = 83.5 GPa, which is smaller by 35.5% than K 0(cg). It is important to note that the uncertainty of K 0(cg) is very high (more than 40 GPa) due to its sensitivity to the value of K 0(nc-Fo) and K 0(m-Fo) and their uncertainties. The isothermal bulk modulus (K T = 127.4 (±0.4) GPa) determined by Isaak et al. (1989) from accurate measurements on a large single-crystal can also be used as K 0(cg) in Eq. 7. In doing so, we obtain K 0(gb) = 97.4 (±27.3) GPa, which is still smaller than K 0(cg) by 23.5%. This calculation confirms the tendency highlighted in our study that nc-Fo is more compressible than m-Fo due to an important contribution of grains boundaries and triple junctions with higher compressibility. Molecular dynamics simulation would be a useful method to obtain a more accurate value of K 0(nc-Fo) and therefore of K 0(gb). For comparison with other nano-materials, K 0(gb) of MgO is 46.8% smaller than its K 0(cg) (Yeheskel et al. 2004) and K 0(gb) of Ni is 9.2% smaller than its K 0(cg) (Zhao et al. 2006).

In the context of a subducting slab, experimental studies (Burnley et al. 1991; Rubie 1984; Vaughan and Coe 1981; Yamazaki et al. 2005) and modeling (Riedel and Karato 1997) predict a large grain size reduction due to the phase transformation of the metastable olivine into ringwoodite phase. Fine-grained ringwoodite nucleates by incoherent grain boundary or by intracrystalline mechanism depending on olivine grain size (Kerschhofer et al. 1998) and this grain growth rate is slow according to Yamazaki et al. (2005). In the case of incoherent grain boundary nucleation, Burnley et al. (1991) observed using TEM that the grain size ranges from 0.5 μm to less than 10 nm in γ-Mg2GeO4 after phase transformation. Ten nanometers to tens of nanometer size lamellae of ringwoodite are also observed while intracrystalline mechanism takes place (Kerschhofer et al. 1996, 1998). In cold subducting slab, the rheological properties of the regions, where the olivine to ringwoodite phase transformation occurs (below 300 km) are strongly affected by the grain-size reduction and superpasticity is believed to control the deformation mechanism (Vaughan and Coe 1981). Therefore, weakening of the slab (Karato 1998; Karato et al. 1998; Riedel and Karato 1997) and deep-focus earthquake (Burnley et al. 1991; Green and Burnley 1989) result from the presence of this fine-grained ringwoodite. Grain size can also influence elastic properties of olivine as shown in the present work. If elastic behavior of ringwoodite is influenced by its grain size in a similar way to that of forsterite we observed here, the fine grain region of the subducting slab may show a slower seismic velocity.