# Local-Basis-Function Equation of State for Ice VII–X to 450 GPa at 300 K

^{*}

## Abstract

**:**

## 1. Introduction

#### Phase Behavior of Ice Beyond 2.2 GPa

_{1}at 930 GPa and P2

_{1}at 1.3 TPa. Eventually a metallic transition to C2/m is predicted at 4.8 TPa. At higher temperatures, ice X transforms into super-ionic ice XVIII above 2000 K [14]. At pressures below 100 GPa, anomalies are reported associated with changes in derivatives or discontinuities in optical spectra, bulk moduli, proton dynamics, and electrical conductivity. The most prominent features include a suggested softening of ice VII volume around 40 GPa [15,16,17] and the onset of the ice X transition by proton symmetrization above about 65 GPa. The latter is supported by changes in the IR reflectivity trend of ν

_{3}’ and translation modes ν

_{Τ}[18], optical reflectivity [19] and from H-NMR experiments [20].

## 2. Materials and Methods

#### 2.1. Helmholtz Energy-Based Equations of State

_{i}contain appropriate derivatives of F evaluated at η = 0. The first term on the right establishes the energy datum. Pressure follows as

_{2}, F

_{3}, F

_{4}or K

_{o}, K

_{o}’, K

_{o}”, plus the 1 bar volume) adequately fits many data sets. The weakness lies in the requirement that the underlying energy potential be represented by a physics-blind series expansion arbitrarily truncated to a small order. Only for a sufficiently small interval will a low order polynomial provide a complete (able to replicate all data within uncertainties) representation. Furthermore, high-pressure behavior may be governed by physics that is not represented in the near ambient pressure potential. In the following, a framework is developed that introduces the use of physics-based constraints and allows construction of equations of state using basis functions having greater flexibility.

#### 2.2. Local-Basis-Function Representation of Helmholtz Energy

- B-spline basis function values are available in all computer environments as a call to a function/subroutine. Analogous to the use of exponential or trigonometric functions, no custom (user) programing is necessary for use of b-spline basis functions. The evaluation of equation of state properties then uses universal calling functions that are not material specific.
- The calculation of values and derivatives of a b-spline model are based on linear programing. Interpolation using b-splines is essentially a weighted average of neighboring model parameters with the basis functions providing the normalized weights. This enables efficient computer algorithms for both construction and evaluation of spline models. Arbitrary precision is possible in representing any functional behavior.
- B-spline basis functions are localized. Unlike global polynomial fits of data, spline model parameters pertain to the behavior of the underlying function in a separate restricted regime of the independent variable.
- Details of how intervals are defined allow flexibility in the behavior of function derivatives at interval boundaries. It is possible to allow discontinuities of the function or specified derivatives of the function at a location to meet the needs of a particular equation of state that might involve higher-order transitions.

#### 2.3. Determination of Helmholtz Energy by Collocation

## 3. Results

#### 3.1. Equations of State for NaCl

#### 3.1.1. Data and Representations

#### 3.1.2. Discussion

#### 3.2. Equations of State for High Pressure Ice (Ice VII–X System)

#### 3.2.1. Data and Representations

#### 3.2.2. Discussion

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. B-Spline and Inverse Method Details Related to Equation of State Representations

#### Appendix A.1. B-Spline Basis Functions

^{3}is both a third degree and third order polynomial. However, using accepted convention, a kth-order spline is associated with polynomial of degree k-1. To avoid confusion in the following discussion, the spline order is parenthetically followed by explicit articulation of the underlying degree of an associated polynomial.

_{1}, t

_{2}, …, t

_{p+k}), where p is the number of coefficients (the model parameters) and k is the order of the b-spline (degree k-1), a recursion relationship (Equation (A1)) defines the spline basis function, B

_{j,k}(x), where x lies between the first and last knot and the first order basis function B

_{j},

_{1}is equal to 1 for x within the interval between t

_{j}and t

_{j+k}and 0 otherwise.

_{jk}evaluated at specified locations, are numerically obtained and represented as arrays with a column count equal to the number of spline coefficients (model parameters) and a row count equal to the number of locations of the function evaluation. The underlying algorithms are robust and available as functions/subroutines in all numerical environments. The use of b-spline basis functions requires no more concern on the part of the user than the use of any standard numerical functions (i.e., trigonometric or logarithmic functions). However, in order to develop new equations of state, the user needs to know how to create an acceptable knot series for a specific application. The knots (t

_{i}) used to construct the spline basis functions must meet several requirements. (1) The total number of knots must equal k + p. (2) Each repetition of a knot reduces by one the number of continuous derivatives at the interval boundary. (3) The first and last knot are repeated k-fold times in order for arbitrary data to be fit. Further discussion of how knots are chosen is given below.

#### Appendix A.2. Evaluation of B-Spline Representations

_{l}and y

_{l}), vector $\overline{m}$ contains p spline coefficients (the model parameters), and B

_{j,k}(x

_{l}) values are determined using Equation (A1). Using vector and matrix notation Equation (A2) is simply a linear equation of the form:

_{j,k}(x

_{l}):

#### Appendix A.3. Details of B-Spline Knots and Control Points

#### Appendix A.4. Local-Basis-Function Equations of State Representations

**Figure A1.**Equation of state example using a cubic b-spline representation on three intervals. Panel (

**a**) (left side): Helmholtz energy (F) as a function of volume (V) is shown as a thick line. Panel (

**b**) (right side): Pressure, as the negative of the first derivative of Helmholtz energy, is shown. Vertical dotted lines show the internal spline boundaries between three intervals of volume. B-spline basis functions are shown at the bottom of each panel with labels B

_{i}and dB

_{i}. The six model parameters (labeled M

_{i}) required to represent the function are shown in panel (a), plotted at their control point locations. Note that the control point locations are implicit in the distribution of the intervals and are not separately tabulated in the model description. The evaluation of the function at a specified volume (vertical dashed line) is illustrated in each panel where the function is given as a sum of the product of basis function values and model parameters. The evaluation of the functions for any other value of volume is undertaken by the same linear analysis. The model parameters, in units of energy, can be determined through a fit of pressures and volumes based on the linear relationships as shown on the right side.

#### Appendix A.5. Defining Equation of State Properties with Derivatives of Helmholtz Energy

#### Appendix A.6. Inverse Techniques to Find Model Parameters

_{o}) shown on the right side of Equation (4) (main text). For local-basis-function representations, the b-spline basis functions evaluated for each independent variable location, are placed in $\stackrel{=}{B}$.

^{t}denotes the transpose operation and the negative one power implies determination of a matrix inverse. Equation (A9) can usually be solved if more data than model parameters are available (not rank-deficit). However, even with an adequate amount of data, solutions may be poorly-conditioned when data do not adequately span the parameter space of the model. The typical example of a poorly-conditioned solution is the exercise of trying to fit a linear distribution of data using a higher-order polynomial. Problems can also arise in constraining model parameters for a local-basis-function representation if data are not adequately distributed in regimes containing measurements or in regions of extrapolation.

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**Figure 1.**Equation of state properties for NaCl as a function of pressure. (

**a**) Density. (

**b**) Isothermal bulk modulus. (

**c**) Pressure derivative of the isothermal bulk modulus. Solid circles in all panels are the tabulated values given in [44]. Lines are predictions based on differing parameterizations as listed in Table 2. Solid lines: local basis functions using different strain metrics (essentially indistinguishable). Dotted line: global third order finite strain. Dashed line: global fourth-order finite strain. Dash-dot line: global ninth-order finite strain.

**Figure 2.**Data and equations of states of ice VII and X at 300K. In all panels, predictions based on the “low structure” local-basis-function (LBF) are shown with thick solid lines, the dotted and dot-dashed lines are the “agnostic” LBFs with lighter or heavier damping, the dashed lines are “transition informed”, the gray lines are the global fourth-order fit. (

**a**) Specific volume as a function of pressure. All measurements are represented with solid circles. Prior fits are identified in the legend. (

**b**) Residuals relative to the preferred LBF representation as a function of the logarithm of pressure. (

**c**,

**d**) Bulk moduli as a function of pressure. All symbols are identified in the legend. (

**e**,

**f**) Pressure derivative of the bulk moduli for the four current representations. The 5/3 limit for high-pressure behavior of K’ in represented as a horizontal dotted line. The behavior of K’ based on a quadratic fit to Shimizu et al. [60] is shown for pressures between 2 and 8 GPa a thick line. Vertical bars indicate identified pressure ranges (Table 1) for higher order transitions.

**Figure 3.**Tradeoff between root-mean-square (rms) data misfit and model smoothness as measured by the root-mean-square average of the fourth derivative of Helmholtz energy. The line defines the locus of solutions obtained by varying the damping factor for regularization. Circles mark points with rms misfits of 1.7 GPa, 2 GPa, and 3 GPa that are associated with fits listed in Table 3.

Pressure of Transition | Suggested Transition | Type of Measurements | References |
---|---|---|---|

5 GPa | |||

5 GPa | Tetragonal distortion | Powder X-Ray diffraction (XRD) | Grande et al. [21] |

10–15 GPa | |||

11 GPa | Lattice distortion | XRD peak splitting | Hirai et al. [22] |

14 GPa | Strain in cubic lattice | Powder X-Ray diffraction | Somayazulu et al. [23] |

11 GPa | - | Changes in Raman line width trends for the ${\nu}_{1}\left({A}_{1g}\right)$ band | Hirai et al. [22], Pruzan et al. [24] |

13–15 GPa | - | Raman line pressure trends | Zha et al. [25] |

13 GPa | - | Neutron diffraction (220/110 ratio) | Guthrie et al. [26] |

10–14 GPa | Lattice distortion | c/a ratio changes in ice VIII | Yoshimura et al. [27] |

10–14 GPa | - | Maximum in electrical conductivity | Okada et al. [28] |

10–15 GPa | - | Maximum in proton diffusion | Noguchi et al. [29] |

20–25 GPa—Possible transition to Ice VII’ with proton dynamic disorder (tunneling and thermal hoping) | |||

23–25 GPa | - | Bump in the 220/110 ratio from Neutron diffraction | Guthrie et al. [26] |

25 GPa | Proton tunneling: ice VII’ | IR reflectivity trend of ${\nu}_{3}$ and ${\nu}_{3}\prime $ trend | Goncharov et al. [18] |

20–25 GPa | Proton tunneling: ice VII’ | H-NMR | Meier et al. [20] |

27 GPa | - | Raman line pressure trends | Zha et al. [25] |

40 GPa | |||

40 GPa | Softening | Drop in volume reported based on XRD | Hemley et al. [15], Loubeyre et al. [16], Sugimura et al. [17] |

44 GPa | - | Raman line pressure trends | Zha et al. [25] |

44 GPa | - | Discontinuity in the pressure dependence of Brillouin esound speeds | Noguchi et al. [30] |

40 Gpa | - | Changes in trend of reflective index | Zha et al. [19] |

40 GPa | - | Drop in Brillouin transverse wave speeds over a narrow P range (<2 GPa) in compression and decompression | Asahara et al. [31] |

>60 GPa transition to ice X | |||

60 GPa | Proton symmetrization | IR reflectivity trend of ${\nu}_{3}\prime $ and translation modes ${\nu}_{T}$ | Goncharov et al. [18] |

62 GPa | - | Raman line pressure trends | Zha et al. [25] |

60 Gpa | - | Changes in trend of reflective index | Zha et al. [19] |

59 GPa | - | Drop in Brillouin transverse wave speeds in compression | Asahara et al. [31] |

70 GPa | Proton symmetrization | H-NMR | Meier et al. [20] |

90 GPa | Proton symmetrization | Emergence of the p20 Raman mode | Zha et al. [25] |

**Table 2.**Equation of state parameters and root-mean-square (rms) misfits of pressure (P

_{rms}) and the bulk modulus (K

_{rms}) for representations of the NaCl 0 K isotherm. A 1 bar density of 2.226 Mg/m

^{3}is used in all representations. Units for all parameters are shown.

Type of Function | Parameters | (rms) | |||
---|---|---|---|---|---|

Global-Basis-Function | |||||

Third order/degree Eulerian Finite Strain | K_{o} = 28.0 GPa | K_{o}’ = 4.5 | P_{rms} = 0.15GPa K _{rms} = 2.3 GPa | ||

Fourth order/degree Eulerian Finite Strain | K_{o} = 27.4 GPa | K_{o}’ = 5.4 | K_{o}” = −0.44 GPa^{−1} | P_{rms} = 0.02 GPaK _{rms} = 0.3 GPa | |

Ninth order/degree Eulerian Finite Strain | K_{o} = 27.8 GPa | K_{o}’ = 5.4 | K_{o}” = −0.67 GPa^{−1} | P_{rms} = 0.01 GPaK _{rms} = 0.2 GPa | |

(Five more parameters for the ninth order fit are not reported here. See Supplementary Materials) | |||||

Local-Basis-Function: | |||||

Knots (strain units): | Coefficients (GPa m^{3}/Mg) | ||||

Eulerian Finite Strain Order: 6 (degree 5) | [−0.08, −0.035, 0.24, 0.67] (first and last knots are repeated six times) | [0.308, 0.242, −0.155, −0.292, 4.10, 10.9, 19.8, 26.1] | P_{rms} = 0.01 GPaK _{rms} = 0.2 GPa | ||

log Strain Order: 5 (degree 4) | [−0.09, −0.036, 0.2, 0.42] (first and last knots are repeated five times) | [0.326, 0.241, −0.141, −0.326 3.92 16.4 26.0] | P_{rms} = 0.01 GPaK _{rms} = 0.2 GPa |

**Table 3.**Equation of state parameters and root-mean-square (rms) pressure misfits (P

_{rms}) for representations of high-pressure ice VII and X 300 K isotherm. Units for all parameters are shown. V

_{o}for ice VII is taken from Klotz (2017): 12.7218 cm

^{3}/mol or 42.25 Å

^{3}at 300 K.

Type of function | Parameters | (rms) | |
---|---|---|---|

Global-Basis-Function | |||

Fourth order/degree Eulerian Finite Strain | K_{o} = 19.2 GPa, K_{o}’ = 3.8, K_{o}” = −0.09 GPa^{−1} | P_{rms} = 3.0 GPa | |

Local-Basis-Function: | |||

knots (dimensionless strain): | Coefficients (GPa cm^{3}/mole) | ||

“Agnostic” log Strain low damping Order: 6 (degree 5) | [−0.01, 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.15, 0.17, 0.20, 0.24, 0.27, 0.31, 0.36, 0.42] (first and last knots are repeated six times) | [−0.10, −0.07, 0.05, 0.40, 1.37, 3.75, 7.13, 12.5, 20.0, 30.7, 46.7, 72.5, 103, 157, 254, 379, 520, 647, 723] | P_{rms} = 1.7 GPa |

“Agnostic” log Strain higher damping Order: 6 (degree 5) | [−0.01, 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.15, 0.17, 0.20, 0.24, 0.27, 0.31, 0.36, 0.42] (first and last knots are repeated six times) | [0.02, −0.03, −0.03, 0.23, 1.16, 3.55, 6.96, 12.3, 19.7, 30.4, 46.9, 71.5, 103, 157, 253, 375, 516, 645, 724] | P_{rms} = 2.0 GPa |

“low structure” log Strain (seven intervals) Order: 7 (degree 6) | [−0.01, 0.08, 0.12, 0.16, 0.16, 0.24, 0.3, 0.42] (first and last knots are repeated seven times) | [−0.09, −0.04, 0.43, 2.21, 6.50, 17.5, 46.3, 93.1, 153, 258, 433, 595, 723] | P_{rms} = 2.0 GPa |

“transition informed” log Strain (seven intervals) Order: 7 (degree 6) | [−0.01, 0.12, 0.16, 0.16, 0.21, 0.23, 0.26, 0.42] (first and last knots are repeated seven times) | [0.10, −0.27, 0.26, 2.86, 10.6, 25.6, 57.3, 104, 153, 263, 396, 556, 722] | P_{rms} = 1.7 GPa |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Brown, J.M.; Journaux, B.
Local-Basis-Function Equation of State for Ice VII–X to 450 GPa at 300 K. *Minerals* **2020**, *10*, 92.
https://doi.org/10.3390/min10020092

**AMA Style**

Brown JM, Journaux B.
Local-Basis-Function Equation of State for Ice VII–X to 450 GPa at 300 K. *Minerals*. 2020; 10(2):92.
https://doi.org/10.3390/min10020092

**Chicago/Turabian Style**

Brown, J. Michael, and Baptiste Journaux.
2020. "Local-Basis-Function Equation of State for Ice VII–X to 450 GPa at 300 K" *Minerals* 10, no. 2: 92.
https://doi.org/10.3390/min10020092