We present an equation of state as well as a phase diagram of ammonia at high pressures and ...... Weinert and J. W. Davenport, Physical Review B 45, 13709.
Equation of state and phase diagram of ammonia at high pressures from ab initio simulations Mandy Bethkenhagen, Martin French, and Ronald Redmer Institut f¨ ur Physik, Universit¨ at Rostock, 18051 Rostock, Germany (Dated: 31 May 2013)
We present an equation of state as well as a phase diagram of ammonia at high pressures and high temperatures derived from ab initio molecular dynamics simulations. The predicted phases of ammonia are characterized by analyzing diffusion coefficients and structural properties. Both the phase diagram and the subsequently computed Hugoniot curves are compared to experimental results. Furthermore, we discuss two methods that allow us to take into account nuclear quantum effects, which are of considerable importance in molecular fluids. Our data cover pressures up to 330 GPa and a temperature range from 500 K to 10 000 K. This regime is of great interest for interior models of the giant planets Uranus and Neptune, which contain, besides water and methane, significant amounts of ammonia. Keywords: ammonia, equation of state, phase diagram, Hugoniot curve, diffusion, high pressure I.
INTRODUCTION
Only little is known about the high-pressure phases of ammonia because experimental data are very scarce. So far, five solid phases of ammonia have been discovered.1–6 By compressing the simple cubic phase I, that is present at ambient pressures and below room temperature, the orthorhombic phases IV and V can be obtained. Besides these ordered solids two rotationally disordered phases, the phases II with a hcp and III with a fcc nitrogen lattice exist. Moreover, Pickard and Needs7 suggested an ionic crystal above 90 GPa and at zero temperature which is characterized by alternate layers of NH+ 4 and NH− 2 ions. Additional possible zero temperature structures have been discussed by Griffiths et al.8 A superionic phase of ammonia was first predicted above 1200 K and 60 GPa by Cavazzoni et al.9 This phase is characterized by highly mobile hydrogen ions diffusing through a nitrogen lattice with hcp structure according to Cavazzoni et al.9 Performing x-ray diffraction and Raman scattering experiments on ammonia loaded into a diamond anvil cell (DAC), Ninet et al.10 reported the experimental discovery of the superionic phase above 700 K and 57 GPa. They found the nitrogen lattice to have hcp or fcc structure depending on the probed thermodynamic path. Recently, Ojwang et al.11 also carried out DAC measurements up to 2500 K and 60 GPa, but were not able to identify a superionic phase in that region. Nevertheless, they did not rule out that such a phase might be present above 60 GPa, in agreement with Ninet et al.10 There are still many open questions concerning the superionic phase such as the type of the nitrogen lattice and the location of the phase boundaries, which might have some implications for the structure of the magnetic fields and the interiors of the giant planets Uranus and Neptune. These planets are believed to contain considerable amounts of ammonia, mixed with other compounds like water and methane.12–15 The interior of these planets cannot be probed directly by experiments, so that the only promising way to increase our knowledge
about Uranus and Neptune is theoretical modeling. Such models require accurate phase diagrams and equation of state (EOS) data of the planetary mixture. While there have been efforts to directly examine mixtures of water, isopropanol, and ammonia, referred to as ”synthetic Uranus”,16–18 the systematic understanding of its single compounds needs to be improved as well. There is no EOS for ammonia that covers the wide temperature and pressure range of planetary interior conditions, so far. The literature offers data only up to 0.95 GPa and 723 K.19,20 It is the aim of this paper to examine the high-pressure behavior of ammonia under planetary conditions with focus on the phase diagram and the EOS, covering pressures and temperatures up to 330 GPa and 10 000 K. The quality of our theoretical results is assessed by comparing with the Hugoniot curve which has been probed in several experiments.17,21–23
II.
COMPUTATIONAL METHODS
A. Finite-temperature density functional theory molecular dynamics simulations (FT-DFT-MD)
Our results were obtained by performing FT-DFT-MD simulations with the Vienna Ab Initio Simulation Package (VASP 4.6).24–27 This approach is based on the BornOppenheimer approximation, where the electrons are described via finite-temperature density functional theory (FT-DFT),28–32 while the ions are propagated as classical particles within the framework of molecular dynamics (MD). The ion temperature is controlled with a Nos´e thermostat.33 The interaction between electrons and ions is described by projector augmented wave (PAW) pseudopotentials.34,35 Here we use the standard pseudopotentials for H and N (5 valence electrons, 2 frozen), so that 8 electrons per NH3 molecule are treated self-consistently in the FT-DFT. We employ the PBE exchange correlation functional.36 Our system contained 32 ammonia molecules in periodic
2 boundary conditions. Furthermore, a plane-wave energy cutoff of 1000 eV was used and the Brillouin zone was sampled at the Baldereschi mean value point.37 A simulation run was carried out for each density % and temperature T with time step sizes of 0.5 fs and durations between 5 ps and 25 ps after equilibration. The convergence of all simulation parameters was ensured by performing several additional tests, which are briefly summarized in appendix B. The evaluation of the simulation data leads to the pressure (thermal EOS) p(%, T ) and the internal energy (caloric EOS) u∗ (%, T ). The latter was post-processed in order to include quantum effects of the ions, which will be described in the following subsection. B.
Vibrational correction of the caloric equation of state
In the framework of FT-DFT-MD simulations used here, all ions are treated as classical particles, but quantum effects of the ionic motion are of significant importance, especially for the caloric EOS. The main reason is that the characteristic vibrational temperatures (frequencies) of the ammonia molecule are comparable with the temperature range of our simulations. As shown in a previous work on water, a major quantum contribution can already be included with a simple post-processing scheme using the harmonic oscillator approximation.38 Here we compare two approaches to include the quantum effects of the ion motion in greater detail. Both yield a correction term uvc (%, T ) that is added to the caloric EOS u∗ (%, T ) derived from the FT-DFT-MD simulations via u(%, T ) = u∗ (%, T ) + uvc (%, T ),
(1)
where u(%, T ) is the final result. The vibrational correction term can be expressed via a spectrum-weighted frequency integration that adds the quantum statistical energy of a harmonic oscillator and subtracts the respective classical energy for each frequency interval.39 It is given by Z 3N ∞ vc u (%, T ) = dν S(ν, %, T ) (2) m 0 1 1 × hν + − kB T , 2 exp(hν/kB T ) − 1 P where N is the total number of ions, m = i mi is the total mass of all individual ions mi , ν is the frequency, h and kB are Planck’s and Boltzmann’s constant, respectively. The centerpiece of Eq. (2) is the vibrational frequency spectrum S(ν, %, T ) which we calculated within two different approximations. The first and simplest approximation involves the set of experimentally known normal mode frequencies {νi } for the ammonia molecule40 , Q Nmol X δ(ν − νi ), Sδ (ν) = 3N i=1
(3)
where δ(ν − νi ) is the Delta function, Q is the number of normal modes per molecule and Nmol is the number of molecules. After inserting the above spectrum into Eq. (2), one recovers a formula analogous to that given by French and Redmer38 , which was successfully applied on the EOS of warm dense water.41 However, using constant frequencies is not justified apart from the molecular fluid phase. Dissociation and correlation effects are not taken into account so that its scope of application is rather limited. The respective correction term uvc δ (T ) depends only on the temperature. Alternatively, the spectrum can be calculated using the ab initio vibrational density of states (power spectrum).39 This is obtained from the simulations by Fourier-transforming the velocity autocorrelation functions h~vα (t) · ~vα (0)i of each ion species α = {N, H}. It reads X 4mα Nα Svv (ν, %, T ) = (4) 3N kB T α Z ∞ × dt cos (2πνt) h~vα (t) · ~vα (0)i , 0
where mα is the mass and Nα the number of the ions of species α. The above spectra can be calculated with good accuracy from FT-DFT-MD simulations, and their normalization conditions are typically fulfilled within 1-3%. The power spectrum Svv (ν, %, T ) contains the influence of molecular dissociation or frequency shifts and broadening due to correlations, thus it depends on T as well as on %. Appendix A contains a sample power spectrum for purely molecular ammonia in comparison with experimental frequencies. The density-dependent spectrum Svv (ν, %, T ) also introduces a density dependence in uvc vv (%, T ). This, in turn, requires an additional quantum correction for the pressure pvc vv (%, T ) to be made, so that the fundamental relation between the thermal and caloric EOS, ∂u ∂p 2 −% =T −p (5) ∂% T ∂T % is fulfilled. Such a correction can be made, e.g., by solvvc ing the above equation for pvc vv (%, T ) using uvv (%, T ). In vc our simulations, uvv (%, T ) was only weakly dependent on the density so that the pressure correction pvc vv (%, T ) was small enough to be neglected. However, this should not be understood as a general result since there are known examples where ionic quantum effects play a role also for the thermal EOS, e.g. in cold solids.42 Goldman et al.43 proposed to use a Gr¨ uneisen ansatz for the EOS to obtain a thermodynamically consistent quantum correction based on an ab initio power spectrum. Such an ansatz has limitations though, e.g., the Gr¨ uneisen parameter can change drastically in regions where molecules dissociate.44 Thus, using either of the above spectra introduces both advantages and disadvantages into the quantum correction approach. A detailed quantitative comparison and
3 discussion will be given in section IV. A different way to efficiently include ionic quantum effects into MD simulations, which we did not follow here, is to use a semiclassical thermostat that introduces quantum fluctuations into the ion dynamics.45 Such an approach also leads to thermodynamically consistent EOS data by virtue. An application to methane46 showed that very similar results in comparison to post-processing methods43 as described here are obtained for the respective EOS.
III.
CHARACTERIZATION OF THE PHASE DIAGRAM
In the following, we present the high-pressure phase diagram of ammonia and discuss the calculated diffusion coefficients and pair correlation functions to characterize the phases of ammonia in more detail. Analyses of the mean square displacements of each ion species and the electronic density of states from every simulation run were also made to determine different phases. Furthermore, extensive cooling and heating simulation runs were carried out to locate the phase boundary between the fluid and the superionic phase.
A.
FIG. 1. Phase diagram of ammonia as determined in this work; each symbol represents an equilibrium simulation run. The gray and black curves indicate the isentropes of Neptune and Uranus as given by Redmer et al.14 The background color only serves as guide to the eye since heating and cooling simulations do not yield sharp boundaries.
Phase diagram
The high-pressure phase diagram obtained in this work is shown in Fig. 1. Our phase diagram is divided into four different phases, which are not necessarily separated by first-order phase transitions, and a transient region dominated by diatomic molecules. The fluid is mainly characterized by a molecular phase consisting of NH3 molecules, as it also occurs under ambient conditions, and a dissociated phase which shows a complex chemical behavior. The transient region at low pressures and temperatures between 4000 K and 5000 K (represented by yellow circles in Fig. 1) is dominated by relatively stable diatomic nitrogen and hydrogen molecules which are formed during the dissociation process of the ammonia molecules. Furthermore, a superionic phase and a rotationally disordered solid are identified, both are characterized by a fcc nitrogen lattice (see section III B). Our predicted phase diagram shows a good qualitative agreement compared to earlier work of Cavazzoni et al.9 , albeit their way of classifying the phases differs somewhat from ours. We also obtain deviations concerning the location and properties of the fluid phases. Cavazzoni et al.9 distinguish three fluid phases including a metallic fluid they pedicted due to band gap closure at 300 GPa and at about 5500 K. Our calculations indicate the vanishing of the band gap already at 3000 K due to thermal effects which would lead to an electronically conducting fluid. In all cases where we obtained signs of dissociation from the pair distribution functions, the fluid was referred to as dissociated (see subsection III B) and otherwise as molecular.
FIG. 2. Phase diagram based on measurements as well as simulations performed by Ninet et al.5,10 (colored areas with solid black phase boundaries) in comparison to the present work (colored symbols as in Fig. 1) and experimental data by Ojwang et al.11 (blue and orange dashed lines).
Moreover, we find a fcc nitrogen lattice in the superionic phase instead of the hcp structure predicted by Cavvazzoni et al.9 This is in concordance with the fcc lattice we find in the neighboring rotationally disordered solid, which is also in very good agreement with the experimental and simulation data by Ninet et al.10 as shown in Fig. 2. However, this does not necessarily prove that the fcc nitrogen lattice is the thermodynamically favoured lattice type in the superionic phase. Our simulation technique involves fixed volume and shape of the simulation box, and a fcc structure was chosen as initial condition so that our system is unable to relax into a lattice with hcp symmetry. Ninet et al.10 find the fcc as well as the
4
B.
Pair distribution functions
Several structural characteristics of the different highpressure phases of ammonia can be observed in the radial pair distribution functions (RDF). Especially dissociation processes can be clearly seen in such correlation functions. The set of the three ionic pair distribution functions for 5000 K and several densities located in the dissociated fluid phase is pictured in Fig. 3. The functions gHH and gN N contain characteristic peaks resulting from the formation of H2 and N2 molecules, which is a clear sign that the ammonia molecules are at least partially dissociated. The intramolecular peak in gN H softens in the same way but does not entirely disappear. This effect is particularly strong at the small densities, where the peaks from the diatomic species are very prominent. At higher densities and/or temperatures, the diatomic molecules dissociate as well. To determine the lat-
5
gHH(r)
4
1
r [Å] 3
2 H2
0
4
1
2 N2
(a)
3
4 (b)
16 12
3
8
2 1
4
0
0
gNH(r)
6
0.5 g/cm³ 1.0 g/cm³ 1.5 g/cm³ 2.0 g/cm³ 3.0 g/cm³
(c)
4 2 0
0
1
2
3
gNN(r)
r [Å] 0
r [Å] 0
1
3
2
4
5
6
7
gαβ(r)
9
8
60
(a)
40 20 0 60
gαβ(r)
hcp structure, depending on the thermodynamic path realized in their experiments. In addition, Fig. 2 contains recent results of Ojwang et al.11 , which show strong deviations from our calculations as well as from the data given by Ninet et al.10 While our calculations agree very well with the data by Ninet et al.10 , especially in terms of the phase boundaries, Ojwang et al.11 report very different locations of the phase boundaries. According to our phase diagram Ojwang et al.11 probed conditions that allow the formation of the superionic phase. Furthermore, Fig. 1 contains the isentropes of Neptune and Uranus as calculated by Redmer et al.14 Neither of the isentropes crosses the superionic phase albeit the isentrope of Neptune is located very close to the superionic region at 3000 K. According to our phase diagram, ammonia is very likely to only occur as a fluid under conditions present in the interior of Uranus and Neptune.
(b)
gHH gNH gNN
40 20 0
0
1
2
3
4
5
6
7
r [Å] FIG. 4. Pair distribution functions for averaged ion positions, (a) the rotationally disordered solid (1000 K, 1.5 g/cm3 ) and (b) the superionic phase (2000 K, 3.0 g/cm3 ). The solid curves represent the simulation results, while the dotted black lines indicate the positions of the nitrogen ions on an ideal fcc lattice.
tice type of the superionic and solid phase, special RDFs were calculated using ion positions that were averaged over an entire simulation run. The respective results for the superionic and the rotationally disorded phase are compared to the RDF of an ideal fcc nitrogen lattice in Fig. 4. The mean pair distribution functions of the nitrogen ions agree very well with the ideal fcc lattice in both cases. Furthermore, the behavior of such averaged gN H and gHH allows us to distinguish between the superionic and the rotationally disordered phase. The hydrogen ions show a very uniform distribution in the superionic phase because they are able to diffuse through the nitrogen lattice. In contrast, the hydrogen ions in the rotationally disorded phase behave very differently. In that phase, the correlation functions gN H and gHH have maxima at the nitrogen fcc lattice positions and a pronounced feature at small distances. This feature appears because it is very likely for the averaged hydrogen positions to be located directly on the nitrogen ions and, thus, also on top of each other. It is a characteristic sign that the molecules are intact and that the hydrogen ions of each molecule can rotate around their central nitrogen ion. All simulations referred to as superionic or rotationally disordered show the same fcc nitrogen lattice as shown in Fig. 4, so that the mean RDFs are a reliable criterion to characterize the lattice type of such phases.
4
r [Å] FIG. 3. Pair distribution functions (a) gHH , (b) gNN and (c) gNH at 5000 K for different densities.
C.
Diffusion coefficients
The diffusion coefficients Dα of species α are calculated by integrating over their respective velocity autocorrela-
5 tion functions: -1
dt h~vα (t) · ~vα (0)i .
(6)
0
In general, the diffusion coefficients of both ion species plotted in Fig. 5 show a strong temperature and density dependence. The diffusion coefficients of both hydrogen and nitrogen ions rise with increasing temperature throughout the fluid regime. The dependence on the density is, however, more complicated. The diffusion coefficient of nitrogen always decreases with the density and drops to zero once the superionic or solid phase occurs. At the same time, the diffusion coefficient of hydrogen decreases between 0.5 g/cm3 and 1.5 g/cm3 and stays almost constant at higher densities. At 2000 K and 1.5 g/cm3 , the hydrogen diffusion coefficient has a pronounced minimum coinciding with the phase transition from the molecular fluid to the superionic phase which results from the nitrogen lattice formation. Compared to the molecular fluid, where the hydrogen diffusion solely depends on the mobility of the molecule, the hydrogen ions become slightly less diffusive as the nitrogen lattice forms. They are very likely to stay close to the nitrogen ions that remain on fixed lattice positions in the superionic phase. At 3000 K such a minimum is not observed because the molecules are already partially dissociated when they form the superionic phase. Fig. 5 also shows that the diffusion coefficient of hydrogen vanishes in the rotationally disordered solid phase. The overall behavior of the diffusion coefficients correlates well with the phase diagram; one can clearly identify fluid, superionic, and solid phases this way. In the fluid phase the systematic difference in the diffusion coefficient of each species is a consequence of the dissociation of ammonia molecules. IV.
ρ [g/cm³] 1,5 2,0
2,5 (a)
-2
10
-3
-4
10
-5 -1
10
500 K 1000 K 2000 K
-2
10
10
3000 K 4000 K 5000 K
-3
6000 K 7000 K 8000 K 10000 K
(b)
-4
10
10
-5
0,5
1,0
2,0 1,5 ρ [g/cm³]
2,5
3,0
FIG. 5. Diffusion coefficients of (a) hydrogen and (b) nitrogen for temperatures between 500 K and 10 000 K. The colors of the isotherms are the same in (a) and (b). The dashed lines in (a) indicate first-order phase transitions, while the dotted line serves as guide to the eye, since the diffusion coefficient does not exist in this range.
1.5 g/cm3 slightly changes due to the formation of the superionic and the rotationally disordered phase, respectively, from the molecular fluid. The same effect should occur at the phase transition between the molecular fluid and the rotationally disordered solid at 500 K between 1.0 g/cm3 and 1.3 g/cm3 , but it is probably too small to be seen in the figure. Moreover, the isotherms of the caloric EOS above 3000 K are characterized by changes in their slopes around 1.3 g/cm3 which result from dissociation processes that were discussed in more detail in the previous section. 120
THERMODYNAMIC PROPERTIES
100 80 p/ρ [kJ/g]
Equation of state (EOS)
3,0
10
10
In this section, we present the EOS data and the Hugoniot curve for ammonia calculated from our ab initio simulations. A.
1,0
10 DH [cm²/s]
1 3
Z∞
DN [cm²/s]
Dα =
0,5
60
10000 K 8000 K 7000 K 6000 K 5000 K 4000 K 3000 K 2000 K 1000 K 500 K
40
The high-pressure EOS for ammonia is plotted in Figs. 6 and 7. The isotherms of both quantities show a very smooth behavior throughout most of the considered density and temperature range. However, there is a prominent discontinuity between 1.8 g/cm3 and 2.0 g/cm3 in the 3000 K isotherm which coincides with the phase transition between the superionic and the dissociated phase. This is a strong indication for a firstorder phase transition that we expect since an fcc lattice is formed. The same but behavior can be observed for the 1000 K and 2000 K isotherms, but it is less pronounced. For both isotherms the slope between 1.3 g/cm3 and
20 0 0,5
1,0
2,0 1,5 ρ [g/cm³]
2,5
3,0
FIG. 6. Thermal EOS p(%, T ) divided by the density % for different isotherms between 500 K and 10 000 K. The dashed lines indicate first-order phase transitions. vc The vibrational correction terms uvc δ (T ) and uvv (%, T )
6 -20 -30 -40
4,0 500 K 1000 K 2000 K 3000 K
4000 K 5000 K 6000 K 7000 K
500 K 1000 K 2000 K 3000 K 4000 K
8000 K 10000 K
3,0
5000 K 6000 K 7000 K 8000 K 10000 K
uvc [kJ/g]
u [kJ/g]
-50 -60 -70
2,0
-80
1,0
-90 -100 -110 0,5
1,0
2,0 1,5 ρ [g/cm³]
2,5
FIG. 7. Caloric EOS u(%, T ) for different isotherms from 500 K to 10 000 K. The presented data already include the vibrational correction using the power spectra Svv (ν, %, T ) from the simulations, see Section II B for further details. Again, the dashed lines indicate first-order phase transitions.
are shown in Fig. 8. In both cases the correction terms become smaller as the temperature increases. This is reasonable, because the quantum effects are naturally more important at small temperatures. Both methods used to calculate the correction term yield generally similar results. The most obvious difference is that uvc vv (%, T ) drops to zero more quickly than uvc δ (T ). The reason is that its spectrum is influenced by dissociation effects which result in an effective decrease of high-frequency vibrational modes. The density dependence of uvc vv (%, T ) is very weak and can often not be resolved within the statistical uncertainties. However, since these deviations occur at high temperatures, where the absolute influence of the ionic quantum effects is small anyway, we conclude that either of the correction methods can be applied satisfactorily in the region investigated here.
B.
0,0 0,5
3,0
Hugoniot curve
Thermodynamic states at several GPa and high temperatures can be probed with shock-wave experiments. The states obtained by single-shock techniques are located on the Hugoniot curve, which can be calculated with the following relation: 1 1 1 u1 − u0 = (p1 + p0 ) − . (7) 2 %0 %1 The thermodynamic properties of the initial (0) and compressed states (1) are subscripted accordingly. The Hugoniot curve has been calculated with the unmodified EOS from the FT-DFT-MD simulations as well as with the two distinct quantum-corrected EOS. The resulting Hugoniot curves are shown in Fig. 9. The initial density at 230 K and p0 =1 bar is %0 = 0.6933 g/cm3 , as
1,0
2,0 1,5 ρ [g/cm³]
2,5
3,0
FIG. 8. Vibrational correction for the caloric EOS. The solid lines represent the vibrational correction uvc δ (T ) using the set of experimental frequencies by Tassaing et al.40 , whereas the circles illustrate the vibrational correction uvc vv (%, T ) based on the ab initio spectra from the simulations.
in the experiments.17,22,23 None of our Hugoniot curves intersects with a boundary to the solid or superionic phases. Fig. 9 shows that the choice of the caloric EOS has little effect on the behavior of the p-% relation of the Hugoniot curve. All three curves reproduce the measured data very well. This is a similar behavior as was found earlier in the case of water,38 whereas, it is possible to measure even such small differences in high-precision shock wave experiments.47 In contrast, the p-T relation of the Hugoniot shows a much stronger dependence on the caloric EOS. Although none of the curves is able to reproduce the experimental data points within the symbol-sized error bars, the curves based on the EOS including the vibrational correction are significantly closer to the experimental data. Therefore, we conclude that the vibrational correction leads to a substantial improvement of the caloric EOS, in agreement with earlier studies for other materials.38,43,46 We expect the exchange-correlation functional (PBE36 ), that describes van-der-Waals-like interactions only with limited accuracy, to be the main reason for the residual deviation to the measured Hugoniot temperatures. However, we cannot completely rule out that nuclear quantum effects beyond those from the harmonic approximation are of importance for the EOS of warm dense ammonia, e.g. for the various dissociation processes.
V.
CONCLUSION
In summary, we have characterized the high-pressure properties of ammonia by performing extensive FT-DFTMD simulations. We propose an extended phase diagram in the range of temperatures from 500 K to 10 000 K and
7
0
20
40
60
p [GPa] 80
~
100
120
140
160
2,5
3000
2,0
10 (b)
8
.
1,5
4000
stretching vibrations
1,5
bending vibrations
Mitchell and Nellis (1982) Radousky et al. (1990) Nellis et al. (1997) FT-DFT-MD FT-DFT-MD with exp. vc FT-DFT-MD with spectral vc
1,0
T [10³ K]
1000
(a)
c S(ν) [10-3cm]
ρ [g/cm³]
2,0
0
ν [cm-1] 2000
1,0
6
0,5
4 2 0
0,0 0
20
40
60
80 p [GPa]
100
120
140
160
FIG. 9. Hugoniot curve derived from FT-DFT-MD simulations (solid lines), including different vibrational corrections (vc) for the caloric EOS, compared to experimental data17,22,23 (symbols). The symbols and line colors are the same in (a) and (b).
for pressures up to 330 GPa. It contains fluid phases of different chemical composition, a solid, and a superionic phase. Our phase diagram shows strong similarities compared to the earlier work of Cavazzoni et al.9 The location of the boundary between the superionic phase and the rotationally disordered solid III is in very good agreement with the experiments of Ninet et al.10 Furthermore, we have determined the equation of state for ammonia under conditions as present in Uranus and Neptune. This involved the application and assessment of a quantum correction approach for the caloric EOS. Moreover, our calculated Hugoniot is in very good agreement with data from shock-wave experiments for the p-% relation, while there is still a significant deviation between the respective calculated and experimental temperatures. Superionic ammonia is very unlikely to occur in the interiors of Neptune or Uranus. However, our investigations are an important step toward understanding the influence of ammonia and its chemical dissociation products on the interior structure of these planets.
0
1000
2000
3000 Tvib [K]
4000
5000
6000
FIG. 10. Vibrational spectrum obtained from a simulation at 230 K and 0.6933 g/cm3 (black curve) in comparision with experimental data by Tassaing et al.40 (dashed lines with full circles) over the characteristic vibrational temperature Tvib and the wavenumber ν˜, respectively. The blue colored lines indicate degenerate vibrations, while the orange ones illustrate non-degenerate vibrations.
Appendix A: Vibrational spectra
A sample vibrational spectrum Svv (ν, %, T ) calculated at 230 K and 0.6933 g/cm3 is shown in Fig. 10. The ammonia molecule has six vibrational degrees of freedom, two of those are twofold degenerate. This leads to four characteristic vibrational peaks in the spectrum. This spectrum is in good agreement with the experimental data by Tassaing et al.40 The only noticeable deviation occurs in the the lowest characteristic frequency. Tassaing et al. report of a relatively strong density dependence of this particular vibrational frequency, which shifts to higher wave numbers (characteristic temperatures) as the density increases. Our simulations were made at significantly higher densities compared to the experimental ones, so that such a deviation seems to be reasonable. The good overall agreement with the experimental data is also a sign of well-converged forces in the FT-DFT-MD simulations. All vibrational peaks significantly broaden with the increasing temperature.
Appendix B: Equation of state data ACKNOWLEDGMENTS
We thank W. Lorenzen, A. Becker, S. Hamel and M.P. Desjarlais for helpful discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) within the SFB 652 and the SPP 1488. The calculations were performed at the North-German Supercomputing Alliance (HLRN) and at the IT and media center (ITMZ) of the University of Rostock.
In Tab. I we present our data for the thermal as well as the caloric EOS. The latter one includes the vibrational correction uvc vv (%, T ) based on the power spectra that were computed self-consistently from the simulations. The convergence of the EOS data was extensively checked with respect to the particle number, the plane-wave energy cutoff and the number of ~k points used to sample the Brillouin zone. We also checked that our simulation
8 data are insensitive to a reduction of the size of the time step. Several test calculations with 64 molecules were carried out for up to 5 ps at various temperatures and densities. These calculations yielded pressures and internal energies deviating from the simulations with 32 molecules up to 1.5% and 10 meV/molecule, repectively. Moreover, DFT calculations with fixed ion positions were performed to check the influence of different ~k point sets and the energy cutoff, which was varied between 500 eV and 2000 eV. We found the Baldereschi mean-value point to sufficiently sample the Brillouin zone compared to the Monkhorst Pack48 set 3x3x3. A plane-wave energy cutoff of 1000 eV converges the pressure better than 1% and the internal energy better than 5 meV/molecule compared to a cutoff of 2000 eV. In general, the pressure is converged within an error bar of 2%, except for the data points marked with asterisk, which have an uncertainty of up to 5%. The caloric EOS is of the same quality as the pressure. TABLE I: EOS data of ammonia obtained from FT-DFT-MD simulations. % [g/cm3 ] 0.5 0.75 1.0 1.3 1.5 0.5 0.75 1.0 1.3 1.5 1.8 2.0 0.5 0.75 1.0 1.3 1.5 1.8 2.0 2.5 3.0 0.5 0.75 1.0 1.3 1.5 1.8 2.0 2.5 3.0
T [K] 500 500 500 500 500 700 700 700 700 700 700 700 1000 1000 1000 1000 1000 1000 1000 1000 1000 2000 2000 2000 2000 2000 2000 2000 2000 2000
p [GPa] 0.309 1.575 4.970 12.61 22.13 0.536 1.987 5.595 13.68 23.50 45.07 64.57 0.749 2.514 6.553 16.40 25.07 46.96 66.32 135.1 237.0 1.320 3.696 9.084 20.25 29.40 51.72 71.78 143.1 247.1
u [kJ/g] -105.69 -105.78 -105.33 -104.16 -102.63 -105.12 -104.90 -104.33 -103.33 -101.95 -98.512 -95.843 -103.94 -103.90 -103.24 -101.60 -100.50 -97.240 -94.413 -85.619 -74.050 -100.32 -99.876 -99.037 -96.942 -95.758 -92.251 -89.348 -80.317 -68.708
0.5 0.75 1.0 1.3 1.5 1.8 2.0 2.5 3.0 0.5 0.75 1.0 1.3 1.5 1.8 2.0 2.5 3.0 0.5 0.75 1.0 1.3 1.5 1.8 2.0 2.5 3.0 0.5 0.75 1.0 1.3 1.5 1.8 2.0 2.5 3.0 0.5 0.75 1.0 1.3 1.5 1.8 2.0 2.5 3.0 0.5 0.75 1.0 1.3 1.5 1.8 2.0 2.5 3.0
3000 3000 3000 3000 3000 3000 3000 3000 3000 4000 4000 4000 4000 4000 4000 4000 4000 4000 5000 5000 5000 5000 5000 5000 5000 5000 5000 6000 6000 6000 6000 6000 6000 6000 6000 6000 7000 7000 7000 7000 7000 7000 7000 7000 7000 8000 8000 8000 8000 8000 8000 8000 8000 8000
1.906 5.111 10.78 22.83 34.40 59.40 76.45 150.0 256.0 3.15* 7.64* 14.36* 25.86 38.11 63.67 85.92 162.2 269.2 4.45* 9.00* 16.11 28.48 41.38 67.73 91.34 169.8 279.3 5.499 10.07 17.11 30.92 44.68 72.69 96.70 177.9 289.8 5.942 11.06 18.71 33.88 47.92 77.54 102.7 186.3 300.8 6.261 11.95 20.69 36.77 51.96 82.81 109.0 194.7 311.9
-96.113 -95.462 -94.265 -91.815 -89.106 -84.818 -83.814 -74.607 -62.896 -89.81* -87.98* -85.86* -83.838 -81.254 -77.635 -74.501 -64.754 -52.984 -83.05* -80.90* -78.258 -75.368 -73.493 -69.583 -66.402 -56.982 -45.610 -75.881 -73.339 -70.786 -67.969 -66.017 -62.289 -59.346 -50.496 -39.021 -68.485 -65.817 -63.687 -61.532 -59.690 -56.548 -53.746 -44.838 -33.503 -60.767 -59.071 -57.593 -55.805 -54.356 -51.015 -48.431 -39.822 -28.337
9
0.5 0.75 1.0 1.3 1.5 1.8 2.0 2.5 3.0
1 R.
10000 10000 10000 10000 10000 10000 10000 10000 10000
7.700 14.42 24.75 43.27 60.51 94.24 122.0 212.1 333.2
-46.837 -46.553 -46.675 -45.714 -44.612 -41.540 -38.960 -30.274 -18.953
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