Introduction

Knowledge about the structure and properties of silicate liquids under relevant pressure and temperature conditions of the deep interior is critical to our understanding of the chemical and thermal evolution of our planet. Geodynamical models used to describe the cooling and crystallization of earth’s early magma ocean rely on physical properties of component liquids (e.g., Rigden et al. 1984; Solomatov 2007; McNamara et al. 2010). Seismic low-velocity zone at 410 km depth (Revenaugh and Sipkin 1994; Song et al. 2004) and ultra low-velocity zone at 2,890 km depth (Williams and Garnero 1996; Rost et al. 2005) are generally associated with the presence of partial melts at the corresponding depths. It is thus important to improve our understanding of the origin and stability of such deep melts. For instance, the density determines whether the melt will rise or sink and the diffusivity/viscosity controls the rate of cooling and transport of magma.

During the last several years, we have studied silicate liquids of different compositions using density-functional theory-based molecular dynamics simulations (e.g., Karki 2010). Here, we choose to simulate CaO and CaSiO3 liquids, which are yet to be studied from first principles. Alkaline earth oxide CaO is considered to be one of the major components of the mantle (Jeanloz et al. 1979). CaSiO3 is the calcium end member of pyroxene, which is a major component of basalt, and its high-pressure phase is considered to be a major component of the lower mantle. Although their crystalline phases have widely been studied, our knowledge about the molten CaO and CaSiO3 is limited. Available experimental data are usually limited to the ambient or low pressures (e.g., Lange and Carmichael 1987, 1990; Rivers and Carmichael 1987; Lange 1997; Funamori et al. 2004; Benmore et al. 2010; Skinner et al. 2012). Zhang et al. (2010) reported the classical molecular dynamical simulations of MgO–CaO–SiO2 ternary system at pressures up to 25 GPa. For being a modifier cation of relatively large size, it is important to understand the effects of Ca on the properties and crystal-melt density contrasts of silicate melt system. Here, we report the first-principles calculations of the structural, thermodynamic and transport properties of CaO and CaSiO3 liquids as a function of pressure and temperature, and make appropriate comparisons with their crystalline properties.

Computational details

We use first-principles molecular dynamics (FPMD) approach within the local density approximation and projector augmented wave method as implemented in Vienna ab initio simulation package (VASP; Kresse and Furthmüller 1996). Our simulations were performed using the canonical NVT ensemble with Nosé thermostat and cubic supercells consisting of 64 atoms for CaO liquid and 160 atoms for CaSiO3 liquid (see Supplementary Fig. S1). We also simulated CaO liquid using 216 atoms to test the finite size effect, but the differences between 64- and 216-atom results are within statistical uncertainty (see Supplementary Table S5). Volume ranges of V/V X = 1.3 to 0.5 (where the reference volume, V X is 24.35 cm3/mol for CaO and 44.23 cm3/mol for CaSiO3) considered correspond to pressure ranges of around 0 to over 150 GPa. At each volume, the initial structure was first melted at 8,000 K and then quenched down to lower temperatures of 6,000–2,500 K. We also simulated crystalline CaO and CaSiO3 polymorphs. A time step of 1 fs was used with simulation durations ranging from 15 ps at 8,000 K to 50 ps or longer at 3,000 K. The total pressure was calculated as a sum of the pressure directly obtained from FPMD simulation, the ideal gas contribution, and a couple of correction terms. The value of Pulay stress correction for the plane wave cutoff of 400 eV and Γ point used increases from 2 to 5 GPa for CaO, and 3.5 to 6.5 GPa for CaSiO3 for the volume ranges considered. In addition, an empirical correction of 1.1 (1.9) GPa was added to CaSiO3 (CaO) to account for the usual over-binding tendency of the local density approximation used (e.g., Stixrude and Karki 2005; Karki and Stixrude 2010a, b). The partial radial distribution functions (Fig. 1, Supplementary Fig. S2) and mean-square displacements (Supplementary Fig. S3) were carefully examined to ensure the liquid state of the simulated systems. Dynamical properties were found to converge more slowly with respect to time than the equilibrium properties. For instance, at two state points: (3,000 K, 0.55 V X, 69 GPa) and (4,000 K, 0.5 V X, 119 GPa), the calculated MSD remains below 10 Å2 even with long durations of over 50 ps (Supplementary Fig. S3), implying very slow dynamics of the melts at high pressure. We follow the similar methodology as described in several previously published works (e.g., Karki et al. 2006, 2010, Karki 2010 and also see Supplementary Description D1).

Fig. 1
figure 1

All partial radial distribution functions (RDFs) of CaSiO3 (solid lines) and CaO (dashed lines) liquids at 3,000 K and V X (~2 GPa). Also shown are CaSiO3 results at 3,000 K and 0.5V X (112 GPa) and CaO results at 3,000 K and 0.47V X (74 GPa) by magenta solid and dashed lines, respectively

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Results and discussion

Structural properties

For both the CaO and CaSiO3 liquids simulated here, all partial radial distribution functions (RDFs) show an initial peak followed by rapidly decreasing fluctuations and finally approaching unity at larger distances (Fig. 1). These features clearly illustrate the anticipated short-range order and long-range disorder characteristics of the liquid state. In particular, the cation–anion functions are well structured with g SiO(r) having much sharper peak and deeper minimum than g CaO(r)’s (Fig. 1). The strong charge order thus controls the melt structure. The other RDF’s have relatively broader and shorter peaks, and also the peaks are located at larger distances (Fig. 1). On isochoric heating, the peaks become shorter, wider, and more asymmetric, and the RDF value at the minimum increases (see Supplementary Fig S2 and Table T1), all implying an increased structural disorder. As the liquid is compressed, generally the peaks shift to shorter distance as shown in Fig. 1 for CaSiO3 liquid. In g SiO(r), the peak height gradually decreases, and a new peak appears at shorter distance than the original second peak. This intermediate peak is due to branching out of some of the second nearest neighbor oxygen atoms to shorter distances than the original second peak. These oxygen atoms may eventually get bonded to Si’s (i.e., fall under the first peak) with a slight increase in pressure. In g CaO(r), the peak height decreases initially and then increases on compression for both liquids. After small initial changes, the shape of the O–O RDF changes qualitatively at high compression with a shoulder appearing before the second peak. This feature in g OO(r) also predicted for silica and other silicate liquids, suggesting significant rearrangement of O atoms at high pressure.

With increasing pressure along 3,000 K, the mean Ca–O bond distance monotonically decreases from 2.44 (2.51) Å at zero pressure to 2.35 (2.37) Å at high pressure for CaO (CaSiO3) liquid (see Supplementary Fig. S4). However, the mean Si–O bond distance shows a weak anomalous behavior (discussed later in the context of coordination species): The distance increases initially from a value of 1.68–1.73Å at ~25 GPa, then gradually decreases to 1.69 Å at 112 GPa (Supplementary Fig. S4). The average bond lengths are systematically larger than the corresponding RDF first peak distances because of the asymmetric peak shapes and nonzero values at the minima (see Supplementary Tables T1 and T2). Not only Ca–O bonds are relatively long, they are also relatively weak and are hence broken at much higher rates than Si–O bonds (Supplementary Table T3). These differences mean that the two liquids show significant differences in their structural and dynamical properties as calculated here.

Our results suggest that the liquid structure is highly sensitive to change in volume. On isochoric heating, the mean coordination numbers of all types remain essentially unchanged. However, they considerably and linearly increase with increasing compression (Fig. 2). The Ca–O coordination numbers of CaO and CaSiO3 liquids increase from about five and seven, respectively, at zero pressure to about eight and ten, respectively, at high pressure (Fig. 2). Such large coordination difference between the two liquids, like the large Mg–O coordination difference between MgO and MgSiO3 liquids (e.g., Karki et al. 2006, 2010) can be associated with closely packed structure of oxides as opposed to silicates. The mean Si–O coordination number is close to four at the ambient pressure, which is consistent with tetrahedrally coordinated crystalline silicate phases stable at low pressures. The number gradually increases to six with the coordination change occurring over a wide pressure interval. The increased structural packing is also reflected in the increased O–Ca and O–Si coordination on compression (Fig. 2). The mean O–Si coordination of other liquids is lower than pure silica liquid indicating that Mg/Ca disrupts the inter-polyhedral linkages. The Si–Si coordination in these liquids is also lower compared to the pure silica melt (Karki et al. 2007). The predicted coordination-pressure trends are consistent with those for other silicate liquids including MgSiO3.

Fig. 2
figure 2

Computed mean coordination numbers along different isotherms: 3,000 K (circles), 4,000 K (squares) and 6,000 K (triangles). Filled symbols represent the CaO liquid numbers. Black lines show the temperature averaged trends, compared with the calculated trends of other silicate liquids: silica (red lines, Karki et al. 2007), diopside (blue lines, Sun et al. 2011), anorthite (green lines, de Koker 2010) and enstatite (purple lines, Karki et al. 2010). Colored dashed lines represent Si–O and O–Si coordination, whereas dotted lines represent Ca–O and O–Ca coordination

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The Ca/Si and O coordination environments consist of various species (states) with their abundances being sensitive to temperature and pressure (Fig. 3). With increasing temperature, coordination distribution becomes wider with the appearance of new species (both low- and high-coordination side) but without significantly affecting the mean coordination numbers on isochoric heating. On the other hand, pressure tends to systematically shift coordination distribution to high value with appearance of high-coordination species at the cost of low-coordination species (Fig. 3).

Fig. 3
figure 3

Calculated Ca–O (top, middle) and O–Ca (bottom) coordination distributions in CaSiO3 and CaO liquids at 3,000 K (thick solid lines), 4,000 K (thin solid lines), and 6,000 K (thin dashed lines). The numbers represent different coordination species. Note that the O–Ca and Ca–O coordination distributions are essentially the same in CaO liquid

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In CaO liquid, the fivefold Ca–O coordination species (42 % at 0.4 GPa, 3,000 K) dominates at low pressure, whereas the eightfold coordination species (67 % at 40 GPa, 3,000 K) dominates at high pressure (Fig. 3, top). The abundances of sixfold and sevenfold states successively become the maximum at mid pressures. The coordination abundances of CaSiO3 liquid (Fig. 3, middle) are systematically shifted to higher values for the coordination species compared to CaO liquid values: sevenfold dominating (35 % at 0 GPa, 3,000 K), whereas ninefold dominating (39 % at 112 GPa, 3,000 K). A significant fraction of O atoms is either singly coordinated or not bonded (26 %) with Ca, while the rest are doubly (44 %) and triply (27 %) coordinated O atoms at 0 GPa and 3,000 K (Fig. 3, bottom). Increased coordination between Ca and O on compression arises from an increased sharing of O atoms; almost all O atoms (96 %) are coordinated with two or more Ca atoms at high pressure (112 GPa).

For the Si–O coordination along 3,000 K, the fourfold species dominates at 0 GPa (with 92 % tetrahedral), whereas sixfold species dominates at high compression (80 % octahedra at 112 GPa). This four-to-six transition occurs gradually through preponderance of fivefold (pentahedral) species with four and sixfold species present in considerable abundances around 40 GPa (Fig. 4, top). The predicted anomalous behavior of Si–O bond length (d) can be related to relative abundances of Si–O coordination species. Our analysis shows that there is always pressure-induced decrease in the average bond length within any identical (tetrahedra, pentahedra or octahedra) polyhedra. However, we see a clear trend of d 4SiO < d 5SiO < d 6SiO for the entire pressure range considered (where d nSiO corresponds to average bond lengths for the n-fold Si–O’s). At initial compression, more fivefold Si–O’s are formed at the cost of fourfold Si–O’s. As can be seen from Fig. 4 (upper panel), the abundance of these pentahedral states increases with increasing pressure up to ~40 GPa where sixfold Si–O’s also starts to shoot up and soon dominates the coordination. Until around 40 GPa the effect of “increase in the average bond lengths due to the increased number of higher coordination polyhedral units” overcomes the normal compression effect of “decrease in the bond length in identical polyhedra.” Once the sixfold domination is reached, the former effect mostly vanishes as there is not so much activity in terms of polyhedral transformation and the system displays normal pressure-induced decrease in the average bond lengths. The predicted existence of some sevenfold species at higher compression is consistent with the experimental suggestions based on silicate glasses (Murakami and Bass 2011). Also at high compressions the abundance of the sevenfold coordinated Si seems to be directly correlated with the temperature, i.e., higher the temperature, higher is the percentage of sevenfold Si’s. The O–Si coordination distribution at the ambient pressure is dominated by non-bridging oxygen (NBO); about 2/3rd of the oxygen are NBO and the remaining are bridging (which is equivalent to Wollastonite structure). Also, free oxygen (FO, not bonded to any Si) is present (5–10 %) at higher temperatures (Fig. 4, bottom). With increasing compression, FO and NBO are rapidly suppressed with the appearance of more BO and even oxygen tri-cluster (O bonded to 3 Si).

Fig. 4
figure 4

Calculated Si–O (top) and O–Si (bottom) coordination distribution in CaSiO3 liquid at 3,000 K (thick solid lines), 4,000 K (thin solid lines), and 6,000 K (thin dashed lines). The numbers represent different coordination states

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The high degree of structural packing and change in its nature is reflected by relatively high mean values of other coordination as well. The Ca–Ca coordination number along 3,000 K increases monotonically from 6.8 to 8.2 between 0 and 50 GPa and decreases to 7.9 at 112 GPa (see Supplementary Table T4). Similarly, the mean O–O coordination number increases (8.2–10.9) up to 50 GPa and then gradually decreases with pressure reaching 9.6 at 112 GPa (Supplementary Table T4). Such non-monotonic changes can be attributed to the pressure-induced pushing of the first peak and minimum in RDFs to smaller distances and also to the eventual appearance of a new second peak.

Our calculated structural parameters compare favorably with the available experimental data (Supplementary Table T2) for CaSiO3 liquid (Waseda and Toguri 1977) and CaSiO3 glass (Taniguchi et al. 1997). Also, the calculated X-ray static structure factor S(q), where q is the scattering vector, obtained by using the Fourier transformation of total radial distribution function agrees excellently with the experiment results (Fig. 5). The first diffraction peak (FSDP) in S(q) is due to cation–cation correlation, the height of which systematically decreases with increasing temperature. Shoulder after the first peak is due to cation–anion (mainly from Ca–O) correlation, whereas the second broad peak is due to both cation–cation (mainly Si–Si) and cation–anion (mainly Si–O) correlations. Structural differences between CaO and CaSiO3 melts are also reflected in computed X-ray structure factors (Fig. 5, inset). The FSDP in CaO is solely due to Ca–Ca while second peak is due to both cation–cation and cation–anion contributions.

Fig. 5
figure 5

X-ray structure factor of CaSiO3 liquid from our simulations at 0 GPa: 6,000 K (purple), 4,000 K (blue), 3,000 K (black), and 2,500 K (green). Gray line shows the X-ray diffraction result at 1,900 K (Benmore et al. 2010). Inset compares the calculated structure factors between CaO and CaSiO3 liquids at 3,000 and 6,000 K. Curves are shifted upward in y direction for clarity

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Thermodynamics

Pressure–volume–temperature (P–V–T) results obtained from FPMD simulations (Fig. 6) are successfully represented with the Mie–Grüneisen equation of state:

$$P\left( {V,T} \right) \, = P\left( {V,T_{0} } \right) \, + P_{\text{TH}} \left( {V,T} \right)$$
(1)
Fig. 6
figure 6

Equation of state results of CaO and CaSiO3 liquids with FPMD results at 3,000 K (circles), 4,000 K (squares), 6,000 K (triangles), and 8,000 K (diamonds). Lines represent the fit to the third-order Birch–Murnaghan equation. Inset shows the results for solid phases at 1,000, 2,000 and 3,000 K in comparisons with the previous calculations at 2,000 K (open circles) both CaO (Karki and Wentzcovitch 2003) and CaSiO3 (Kawai and Tsuchiya 2014). Also shown are the experimental data (open triangles, top inset) for solid CaSiO3 at 2,000 K (Ricolleau et al. 2009)

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Here, P(V,T 0) is the pressure along the reference isotherm (T 0 taken to be 3,000 K) described with the third-order Birch–Murnaghan equation (Table 1). The thermal pressure P TH(V,T) is linear in temperature described with volume dependent coefficient, B(V). Comparisons of the liquid and solid results (Fig. 6, inset) suggest that the liquid density can exceed the solid density within the stability pressure field of the crystalline CaO B1 and B2 phases and CaSiO3 cubic phase. To further constrain the predicted liquid–solid density crossovers, we need to fully explore the solid phase diagram and also partitioning of incompatible elements between the liquid and solid.

Table 1 EOS fit parameters for CaO and CaSiO3 melts at 3,000 K along with their co-existing solid phases at 1,000 K
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With the calculated pressure P(V,T) and internal energy E(V,T) both varying linearly with temperature along isochore, we can derive three key thermodynamic quantities from FPMD simulations, which are isochoric heat capacity (C V ), thermal Grüneisen parameter (γ) and thermal expansion coefficient (α):

$$C_{V} = \left( {\frac{\partial E}{\partial T}} \right)_{V}$$
(2)
$$\gamma = \frac{V}{{C_{V} }}\left( {\frac{\partial P}{\partial T}} \right)_{V}$$
(3)
$$\alpha = \frac{1}{V}\left( {\frac{\partial V}{\partial T}} \right)_{P}$$
(4)

Note that the thermal pressure coefficient can also be defined as \(B\left( V \right) = \left( {\frac{\partial P}{\partial T}} \right)_{V} = \frac{{\gamma C_{V} }}{V}\), which increases almost linearly with compression for both liquids.

The calculated isochoric heat capacity (C V ) increases and decreases upon compression, respectively, for CaO and CaSiO3 liquids (Fig. 7, right). On the other hand, C V for crystalline phases tends to remain nearly insensitive to volume and also closely obeys the Dulong–Petit limit, but liquid C V can exceed this limit. Similar trends were previously predicted for MgO and MgSiO3 liquids. The relatively large heat capacity of the liquid state can be attributed to the structural changes due to temperature (discussed earlier). Whereas the mean coordination number is insensitive to temperature at constant volume, the range of coordination environments increases with increasing temperature (Figs. 3, 4).

Fig. 7
figure 7

Liquid coefficient of thermal expansivity (left), isochoric heat capacity (right-top) and thermal Grüneisen parameter (right-bottom) of CaO and CaSiO3 melts, compared with the corresponding results for solid phases. Experimental data from Stebbins et al. (1984) for CaSiO3 liquid (green triangles) and Wang et al. (1996) for cubic CaSiO3 (blue triangles) are also shown. Previous first principle results for cubic CaSiO3 (gray lines) are from Kawai and Tsuchiya (2014)

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The calculated Grüneisen parameter (γ) of CaO liquid increases from 1.0 to 1.4, whereas that of CaSiO3 liquid increases from 0.7 to 1.6 over the volume ranges considered here (Fig. 7, right). Comparisons with previous MgO (1–1.4) and MgSiO3 (0.4–1.2) liquids suggest that the silicates show larger increase than the oxides do (Stixrude and Karki 2005; Karki et al. 2006). Increasing γ with compression also predicted for several other liquids is in contrast to crystalline phases for which γ always decreases with compression as also found here for CaO and CaSiO3 polymorphs (Fig. 7, right). We attribute this behavior to the pressure-induced changes in the structure of the liquid (Fig. 2): Higher coordination is expected to result in larger values of the Grüneisen parameter.

The values of thermal expansion coefficient calculated at 3,000 K are large at the ambient pressure: 12 × 10−5 K−1 for CaO liquid and 9 × 10−5 K−1 for CaSiO3 liquid. As pressure increases, α of each liquid initially decreases rapidly and then decreases more gradually (Fig. 7, left). The values of two liquids remain close to each other and approach 2 × 10−5 K−1 in the high-pressure regime. They are also comparable to the thermal expansivities of MgO and MgSiO3 liquids. The solid phases have much smaller values (3 to 9 × 10−5 K−1) at zero pressure which become <1 × 10−5 K−1 at high pressure (Fig. 7, left). The overall agreement of our results with previous experiments (e.g., Lange and Carmichael 1987, 1990; Rivers and Carmichael 1987; Wang et al. 1996, Lange 1997) and theoretical calculation (e.g., Matsui 1996; Karki and Wentzcovitch 2003; Kawai and Tsuchiya 2014) is good (See Figs. 6, 7; Table 1). In particular, the equation of state and thermodynamic parameters of cubic CaSiO3 compare well with the recent first-principles study (Kawai and Tsuchiya 2014).

Diffusivity

Self-diffusion coefficient for each atomic species (α = Ca, Si or O) can be readily calculated from FPMD position-time series using the Einstein relation:

$$D_{ \propto } = \mathop {\lim }\limits_{n \to \infty } \frac{{{\text{MSD}} (t)}}{6t} = \mathop {\lim }\limits_{n \to \infty } \frac{1}{6t}\left\langle {\frac{1}{{N_{ \propto } }}\mathop \sum \limits_{i = 1}^{{N_{ \propto } }} \left| {\overrightarrow {{r_{ \propto ,i} }} \left( {t + t_{0} } \right) - \overrightarrow {{r_{ \propto ,i} }} (t_{0} )} \right|^{2} } \right\rangle$$
(5)

Here, the mean-square displacement (MSD) at time t corresponds to the differences in positions at the time interval of t, and it is averaged over different time origins (t 0’s). Our long simulation durations assure that the liquid system reach diffusive regime where the log–log plot of MSD and time has the slope of the unity (Supplementary Fig. S3).

Self-diffusion coefficients for both CaO and CaSiO3 liquids were calculated as a function of temperature between 2,500 and 8,000 K at zero pressure (Fig. 8). For CaO liquid, the O diffusion was predicted to be systematically faster than the Ca diffusion at all temperatures. This trend satisfying the mass order (m O < m Ca) arises because of symmetric coordination environments between Ca and O, which perhaps imply essentially the same mechanism (bond events) for both species. However, the situation with CaSiO3 liquid is different and more complex: At the lowest temperature of 2,500 K studied, Ca is much more mobile than Si and O, with Si being the slowest species and the D Ca/D Si ratio being 3.1. Strong Si–O bonding and asymmetric cation–anion coordination environments do not only result in weak Si diffusion but also slow down O movement. With increasing temperature over the range considered here, the Ca diffusivity increases much less (by a factor of 25) compared to Si diffusivity, which increases by a factor of 85. The O diffusivity shows an intermediate level of variation of 60-fold with temperature. Three diffusivities of CaSiO3 liquid become increasingly similar and then undergo crossover eventually satisfying the mass order at the highest temperature of 8,000 K (i.e., D O > D Si > D Ca). This behavior suggests a flow-like motion of particles facilitated by high rates of bond breaking events at high temperatures (Supplementary Table T3). Our results thus show that with decreasing temperature, the movements of different species in CaSiO3 liquid decouple more and more from each other and diffusion mechanisms are essentially controlled by local structural environments which differ between different atomic species. The Ca–O bonds are broken at much higher rates than the Si–O bonds (Supplementary Table T3). The relative importance of strong Si–O and weak Ca–O bonding is enhanced as temperature decreases. Similarly, Mg was previously predicted to be the fastest species in MgSiO3 liquid at low temperatures (Karki et al. 2010).

Fig. 8
figure 8

Calculated temperature variations of different self-diffusivities of CaSiO3 and CaO liquids at zero pressure (±1 GPa). Symbols represented the calculated data points at the temperatures 2,500–8,000 K and lines represent the Arrhenius fits (see Table 2)

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Despite large differences in the extents of temperature variations, the calculated self-diffusion coefficients of all species closely follow the Arrhenian relation: \(D_{\alpha } \left( T \right) = D_{\alpha 0} \exp \left[ { - E_{\alpha } /(RT)} \right]\), where α represents atomic species. The activation energies fall in the range 97–152 kJ/mol (Table 2) in the E Ca ≤ E O order for CaO liquid and the E Ca < E O ≤ E Si order for CaSiO3 liquid. The calculated results are generally smaller than the values derived from low-temperature (<2,500 K) experimental data for silicate liquids, which vary from ~180 to ∼270 kJ/mol (Dunn 1982; Shimizu and Kushiro 1991; LaTourrette et al. 1996; Lesher et al. 1996; Reid et al. 2001; Tinker et al. 2003). Our Ca activation energy is comparable to the value inferred for Na-rich calcium-aluminosilicate melt (Watson 1979). Large differences between computation and experiment have previously been reported in other silicate melts as well (e.g., Karki et al. 2010, 2011; Ghosh and Karki 2011; Verma and Karki 2012).

Table 2 Arrhenius fit parameters for the temperature dependent self-diffusion coefficients of CaSiO3 and CaO liquid at zero pressure (0 ± 1 GPa)
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The calculated self-diffusivities of different species for both liquids considerably vary with pressure being sensitive to temperature (Fig. 9). A normal trend is one in which the diffusivity decreases linearly (in log scale) with pressure and the slope decreasing with increasing temperature. However, at high temperatures, the calculated diffusivity-pressure trends show upward bending, whereas at low temperatures like 3,000 K or 4,000 K they bend downward, indicating deviation from typically considered linear behavior. In addition, the Si and O diffusivities of CaSiO3 liquid along 3,000 K slowly decrease in low-pressure regime. Such changes in slope of the diffusivity isotherms have been predicted in other silicates, particularly, in the case of highly polymerized liquids including silica and anorthite (Karki and Stixrude 2010b, 2011). This means that the calculated pressure–temperature variations of self-diffusivities (Fig. 9) require the extended form of Arrhenius relation as in previous studies:

$$D_{\alpha } \left( {P,T} \right) = D_{0\alpha } \left[ { - \frac{{E_{\alpha } + PV_{\alpha } }}{RT}} \right]$$
(6)

where V α  = (V 0 + V 1 T) + P(V 2 + V 3 T + V 4/T). These five fit parameters, which represent the pressure, temperature and cross-derivatives of the activation volumes are shown in Table 3. Note that we adopt the values of the pre-exponential and activation energies from the zero pressure Arrhenius fit (Table 2). As pressure increases, all activation volumes increase at 3,000 K, remain nearly constant at 4,000 K, and decrease at higher temperatures (6,000–8,000 K). Although the opposite role of pressure and temperature on activation volume is intuitive, their exact relation is unknown. This P–T dependence of activation volumes best describe the diffusivity data over wide P–T ranges studied here. A slightly upward bending trend of the higher temperature fits suggests that the pressure-induced suppression of diffusivity becomes sluggish, compared to lower temperatures. The effective activation volumes are positive for all species and follow the V Ca > V O > V Si order.

Fig. 9
figure 9

Calculated pressure variations of self-diffusion coefficients of different species in CaSiO3 and CaO melts along several isotherms (black circles, 3,000 K; blue squares, 4,000 K; purple triangles, 6,000 K; red diamonds, 8,000 K). Lines represent the modified Arrhenius fits to the computed values (symbols). Previous liquid MgSiO3 FPMD results (Karki et al. 2010) by gray solid lines and CaSiO3 MD results (Zhang et al. 2010) by gray dotted lines are shown for comparisons. Also shown are previous FPMD results for anorthite (light green, Karki et al. 2011) and silica (orange, Karki and Stixrude 2010b) liquids at 3,000 K for comparison

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Table 3 Activation volumes and their pressure and temperature derivatives from the modified Arrhenius equations of the calculated self-diffusion coefficients
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The calculated diffusivity trends of silicate and oxide liquids over the wide range of pressure and temperature suggest a variety of mechanisms that are relevant for melt dynamics. At low pressures and high temperatures, the diffusivity is similar among all species. The diffusion appears to occur as flow-like motion of atoms facilitated by frequent bond-breakings caused by high temperature. At lower temperatures, the diffusion is controlled by local structure, and different species see different local environments. Si atom is highly confined with tetrahedral cages and cannot move easily. In contrast, non-framework cation Ca is relatively weakly bonded to O atoms and thus can move easily in open space available. Note that the Ca–O bond breaking events are much frequent than Si–O bond breaking events (Supplementary Table T3). At high pressures, the increased coordination of Ca atoms means that Ca atoms also become more and more confined in the cages formed by surrounding O atoms so DCa decreases more rapidly at higher pressure. Along 3,000 K, the D Ca/D O and D Si/D O ratios are 1.37 and 0.68, respectively, at zero pressure, and 0.50 and 0.44, respectively, at high pressure (69 GPa). In CaO liquid, D Ca/D O ratio at 3,000 K increases from 0.70 at zero pressure to 0.94 at 39 GPa. Thus, O becomes the fastest species at high pressure for both liquids.

Comparison between CaO and CaSiO3 self-diffusion coefficients helps assess the effects of SiO2 component on the transport properties. The SiO2 component considerably lowers the Ca and O diffusivities at low pressures and temperatures below 8,000 K (Fig. 9). At higher pressures, it becomes closely packed like CaO, and the open network of stable Si–O polyhedra is no longer applicable. The diffusivities of two liquids first become comparable as pressure increases, and eventually cross each other, i.e., CaO liquid eventually becomes slower than CaSiO3 liquid at high pressure (see Supplementary Fig. S5). We further compare our results for CaSiO3 liquid with previous calculations on other silicate liquids. The values of both D Si and D O for CaSiO3 liquid are higher than those for highly polymerized liquids including anorthite and silica. No anomalous behavior was predicted in CaSiO3 liquid. The differences with MgSiO3 liquid (Karki et al. 2010) are rather small (Fig. 9) as expected because of the similar extents of polymerization, for instance, NBOs are 65.2 % (64.4 %) for CaSiO3 (MgSiO3) liquids at zero pressure and 3,000 K (Fig. 10). Our analysis thus suggests that Ca and Mg as the network modifier cations have similar effects on the transport properties of the silicate melt.

Fig. 10
figure 10

Pressure-induced total self-diffusion coefficient (right X-axis) and O–Si coordination distribution (left X-axis) of CaSiO3 melt at 3,000 K. Dashed lines show similar data for MgSiO3 melt (Stixrude and Karki 2005)

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Conclusions

We have simulated CaO and CaSiO3 liquids using first-principles molecular dynamics method to study their structural, thermodynamic and transport properties over wide ranges of pressure and temperature. Good agreement between our calculations and available experimental data demonstrates the accuracy of density-functional theory in the study of Ca-bearing phases. The simulated liquid structure is highly sensitive to compression. The mean Ca-O coordination number increases from 5 (7) near the ambient pressure to 8 (10) at high pressure for CaO (CaSiO3) liquid, whereas the mean Si–O coordination number increases from 4 to 6 over the same pressure regime. The calculations show that both liquids are much more compressible than their solid counterparts, and their Grüneisen parameter and thermal pressure coefficient increase with increasing pressure. The calculated self-diffusion coefficients are strongly dependent on temperature and pressure so they require non-Arrhenian representation with variable activation volume. The diffusivity differences between the two liquids are large in low-temperature and low-pressure regime. An improved knowledge about compositional effects as shown here for Ca-bearing liquids is expected to enhance our understanding of deep magma ocean from which the present deep earth had been originated.