Plastic deformation of superionic water ices

Significance

Superionic (SI) ices are high-pressure, high-temperature phases of water in which oxygen ions occupy a rigid crystalline lattice, whereas the protons diffuse in a liquid-like manner. They are believed to be abundant in the universe, in particular in the interiors of Neptune and Uranus, in which they are conjectured to constitute a thick solid mantle. Here, we investigate the mechanical deformation properties of these phases using state-of-the-art computational techniques. The results indicate that SI face-centered cubic ice is very malleable, suggesting that the rheology of the icy internal layers of these planets may be orders of magnitude faster than previously thought, possibly having important implications for the interior dynamics of Neptune and Uranus.

Abstract

Due to their potential role in the peculiar geophysical properties of the ice giants Neptune and Uranus, there has been a growing interest in superionic (SI) phases of water ice. So far, however, little attention has been given to their mechanical properties, even though plastic deformation processes in the interiors of planets are known to affect long-term processes, such as plate tectonics and mantle convection. Here, using density functional theory calculations and machine learning techniques, we assess the mechanical response of high-pressure/temperature solid phases of water in terms of their ideal shear strength (ISS) and dislocation behavior. The ISS results are well described by the renormalized Frenkel model of ideal strength and indicate that the SI ices are expected to be highly ductile. This is further supported by deep neural network molecular dynamics simulations for the behavior of lattice dislocations for the SI face-centered cubic (fcc) phase. Dislocation velocity data indicate effective shear viscosities that are orders of magnitude smaller than that of Earth’s lower mantle, suggesting that the plastic flow of the internal icy layers in Neptune and Uranus may be significantly faster than previously foreseen.

Despite the simplicity of the isolated water molecule, its condensed phases demonstrate an astonishing richness (1–7), with the existence of at least 19 crystalline phases, a number of amorphous solids, and supercooled liquids with different densities. There has been a growing interest in high-pressure, high-temperature superionic (SI) ices (2, 3, 8–21), which are water phases in which the oxygen ions occupy a crystalline lattice, whereas the protons behave in a liquid-like manner. Not only is their comprehension of fundamental importance to a general understanding of water’s phase diagram, it also represents a key element in unraveling the peculiar geophysical properties of Neptune and Uranus (12, 22–26), whose interiors are conjectured to contain a thick solid mantle containing large amounts of SI water ices (11, 21, 24, 27, 28).

The characteristics of this layer, including ionic/thermal conductivities and mantle convection processes, are believed to contribute to the magnetic anomalies displayed by these ice giants. In this light, there has been substantial interest in the transport properties of SI water phases associated with the transfer of energy, charge, and mass (13, 14). For instance, the recent findings of larger than expected electrical conductivity of SI ices (13) may have an appreciable impact on the outcome of dynamo models (26, 29), which are employed to predict the geometry and time evolution of the magnetic fields based on the inner planetary structure. Much less attention, however, has been given to the mechanical behavior of SI ices (17), in particular those that are related to plastic deformation. Even so, such crystal plasticity effects are known to affect long-term geophysical processes, such as plate tectonics and mantle convection on Earth (30, 31), and are also expected to be involved in the evolution of the internal structure of Neptune and Uranus. Indeed, the ease/difficulty of plastic flow under the influence of external stresses should directly affect the dynamics of their solid mantles, playing a potentially important role in such questions as to whether or not these ice masses may contribute to the formation of stratified, nonconvective layers (32), envisioned to be involved in the magnetic anomalies of these ice giants (25, 26).

In this context, the purpose of this work is to investigate the plastic deformation behavior of high-pressure, high-temperature water ices. To this end, we utilize atomistic-level simulation methods, employing a combination of ab initio molecular dynamics (AIMD) calculations and machine learning (ML) techniques. We explore two aspects of mechanical resistance, namely 1) the ideal shear strength (ISS), which is an intrinsic materials property defined as the minimum shear stress needed to cause permanent deformation in an idealized defect-free crystal (33–38), and 2) the behavior of lattice dislocations—the one-dimensional defects that carry plastic deformation in crystalline solids (39, 40). The ISS results are well described by the renormalized Frenkel model (36), which has been shown to provide an approximately universal description of the ISS for different classes of solids characterized by different bonding types. Furthermore, in the Frenkel picture, the face-centered cubic (fcc) SI ice (3) phase is located in the near vicinity of very easily deformable crystals, such as solid ⁴He (41, 42) and the gum metal approximant Ti₃Nb (43), suggesting that it is malleable as well. This is further supported by the observed behavior of lattice dislocations in this phase, whose velocities are found to display a viscous drag regime that gives rise to relatively low effective viscosities. These results indicate that the SI fcc phase can readily flow under external stresses and suggest that the deformation dynamics of internal solid layers in Neptune and Uranus may actually be significantly faster than previously imagined based on its elastic rigidity (3).

Results and Discussion

ISS.

The ISS calculations have been carried out for a number of high-pressure, high-temperature ice phases with body-centered cubic (bcc) and fcc oxygen lattices, employing computational cells containing 144 oxygen ions (Materials and Methods and SI Appendix, Figs. S1 and S2). Typical images from equilibrated structures prior to deformation are depicted in Fig. 1 A–C (SI Appendix, Fig. S4). The bcc structure, at pressures of ~220 GPa, undergoes a transition between insulating (I) and SI proton behavior between 2,000 and 2,500 K (16). Whereas the protons remain localized for the lower temperatures, as in Fig. 1A at 1,000 K, they display liquid-like diffusion at 2,500 K, as shown in Fig. 1B. In contrast, the fcc phase at pressures of ~340 GPa is SI for all considered temperatures, with a typical case for 2,500 K shown in Fig. 1C. The protonic mean-square displacements (MSDs) for the bcc and fcc phases are depicted in Fig. 1 D and E, respectively. For the former, only the case for 2,500 K shows liquid-like Brownian behavior with a diffusion coefficient of $~4.8\times {10}^{-4}$ cm²/s. For the lower temperatures, the MSDs attain plateaus, consistent with insulating behavior. On the other hand, consistent with previous reports (17), the proton subsystem of the fcc phase behaves liquid like for all considered temperatures, with diffusion constants of ~4.7, 7.1, 8.6, and 9.8 × 10^–4 cm²/s for 2,500, 3,000, 3,500, and 4,000 K, respectively.

Fig. 1.

Structures and protonic MSDs for the bcc and fcc phases as visualized using the Ovito package (44). Images in A–C superpose 4,000 MD configurations, with oxygens and protons displayed as red and white dots, respectively. A and B depict the bcc phases at 1,000 and 2,500 K, respectively, showing a transition between an insulating phase and an SI phase. The fcc phase is SI throughout, with a typical image at $T=2,500$ K shown in C. (D) Proton MSDs in bcc phase at $T=2,500$ K and P = 230 GPa. D, Inset shows results for 1,000 K/215 GPa (black), 1,500 K/220 GPa (green), and 2,000 K/225 GPa (blue). (E) Proton MSDs in fcc phase at $T=2,500$ K and P = 334 GPa (black), 3,000 K/340 GPa (blue), 3,500 K/346 GPa (green), and 4,000 K/353 GPa (red).

Next, we assess the ISS values of these ices by subjecting the systems to homogeneous volume-conserving shearing processes, in which one of the periodic repeat vectors of the computational cells is tilted in a direction perpendicular to its original orientation. This gives rise to shear deformation on a specific slip system (36, 40), defined by the combination of a slip plane and a slip direction. For the bcc and fcc phases, we consider the $\left\{11\overline{2}\right\}\langle 111\rangle$ and $\left\{111\right\}\langle 11\overline{2}\rangle$ slip systems, respectively, which are fundamental deformation modes for these lattice types (40, 45). The deformation is imposed at a strain rate of $\dot{\epsilon }=5\times {10}^{-3}$ps^–1, where $\epsilon \equiv \mathrm{\Delta }/h$, with h being the size of the computational cell in the direction perpendicular to the slip plane and Δ being the displacement along the slip direction. As shown in SI Appendix, Fig. S6, doubling the strain rate does not lead to systematic differences in the results. Fig. 2 displays the results for the bcc phases. Fig. 2 A–C depicts typical snapshots of the oxygen ion lattice along the deformation process between the undeformed initial state (Fig. 2A), an intermediate stage characterized by uniform shear strain (Fig. 2B), and immediately after attaining permanent deformation (Fig. 2C), in which one crystal part has slipped past the other by one lattice vector. To quantify this process, we monitor the slip-plane shear-stress component in the deformation direction as a function of the imposed shear strain. These stress–strain relations are shown in Fig. 2 D–G for the four considered temperatures. Qualitatively, the curves are similar, all of them starting with a linear response for small deformations and then, bending off to reach a maximum at the ISS ${\sigma }_{\text{ISS}}$ for a critical strain value ${\epsilon }_{\text{ISS}}$ known as the shearability, followed by a yield drop signaling permanent deformation.

Fig. 2.

Shear deformation of the bcc phases for the $\left\{11\overline{2}\right\}\langle 111\rangle$ slip system. Snapshots in A–C show typical images of oxygen positions of the structure before deformation, an intermediate stage, and immediately after accruing permanent deformation, respectively. Yellow lines serve as guides to the eye to visualize deformation processes and identify the permanent deformation accrued in C. D–G display stress–strain curves obtained for fully atomistic systems at 1,000 K/215 GPa, 1,500 K/220 GPa, 2,000 K/225 GPa, and 2,500 K/230 GPa, respectively. Corresponding results in the absence of explicit protons are shown in H–K.

The initial part of the stress–strain curves corresponds to the linear elastic regime, with its slope giving the shear modulus μ. For the insulating phases, between 1,000 and 2,000 K, thermal softening gives rise to a decrease of $~9$% of μ, as shown in Tables 1 and 2. Upon crossing the insulating to SI transition, however, μ reduces by more than 50% between 2,000 and 2,500 K, showing that the transformation of the localized to diffusive proton configurations strongly reduces the elastic shear resistance of the SI bcc phase. The ISS values ${\sigma }_{\text{ISS}}$ and ${\epsilon }_{\text{ISS}}$, determined by locating the position and value of the maximum stress from running averages of the stress–strain curves over 2,000 molecular dynamics (MD) steps, show similar behavior. Both show relatively small reductions with increasing temperature for the insulating phases, while a much more pronounced drop occurs across the insulating to SI transition, as shown in Tables 1 and 2. In particular, the ISS for the SI phase at 2,500 K is almost three times lower than that for the insulating phase at 2,000 K, reflecting the large influence of the protonic subsystem on the mechanical response.

Table 1.

Values of μ (in units of gigapascal), ${\sigma }_{\text{ISS}}$ (in units of gigapascal), and ${\epsilon }_{\text{ISS}}$ for the $\mathbf{\langle }1\overline{1}2\mathbf{\rangle \{}\overline{1}11\mathbf{\}}$ and $\mathbf{\langle }111\mathbf{\rangle \{}\overline{1}\overline{1}2\mathbf{\}}$ slip systems of the bcc phase

T					bcc
	1,000 (I)		1,500 (I)		2,000 (I)		2,500 (SI)
	All atoms	Oxygen only	All atoms	Oxygen only	All atoms	Oxygen only	All atoms	Oxygen only
μ	183 (3)	133 (4)	178 (3)	132 (5)	167 (2)	130 (5)	93 (4)	127 (6)
${\sigma }_{\text{ISS}}$	25.7 (3)	10 (1)	23.8 (6)	10.3 (8)	19.1 (5)	10 (1)	7.3 (4)	9.6 (9)
${\epsilon }_{\text{ISS}}$	0.230 (5)	0.105 (5)	0.226 (5)	0.104 (5)	0.168 (5)	0.104 (5)	0.107 (5)	0.104 (5)

Numbers in parentheses represent uncertainty in the last digit(s). The error in ${\sigma }_{\text{ISS}}$ was estimated by evaluating the fluctuation in the running average across an interval of 2,500 steps centered around the determined maximum value. Based on this criterion, we estimate the error bar in ${\epsilon }_{\text{ISS}}$ to correspond to the shear strain accrued in 2,500 AIMD steps (i.e., $\mathrm{\Delta }{\epsilon }_{\text{ISS}}=0.005$). For μ, the uncertainty was estimated by the fluctuations of the linear regression results of the stress–strain relations up to 4, 5, and 6%.

Table 2.

Values of μ (in units of gigapascal), ${\sigma }_{\text{ISS}}$ (in units of gigapascal), and ${\epsilon }_{\text{ISS}}$ for the $\mathbf{\langle }1\overline{1}2\mathbf{\rangle \{}\overline{1}11\mathbf{\}}$ and $\mathbf{\langle }111\mathbf{\rangle \{}\overline{1}\overline{1}2\mathbf{\}}$ slip systems of the fcc phases

T					fcc
	2,500 (SI)		3,000 (SI)		3,500 (SI)		4,000 (SI)
	All atoms	Oxygen only	All atoms	Oxygen only	All atoms	Oxygen only	All atoms	Oxygen only
μ	227 (11)	214 (7)	196 (3)	210 (7)	186 (11)	178 (5)	164 (4)	175 (3)
${\sigma }_{\text{ISS}}$	14.2 (7)	14.0 (1)	11.8 (6)	12.1 (7)	12.7 (4)	13 (1)	9.6 (6)	10.1 (3)
${\epsilon }_{\text{ISS}}$	0.079 (5)	0.079 (5)	0.080 (5)	0.080 (5)	0.073 (5)	0.069 (5)	0.056 (5)	0.059 (5)

Numbers in parentheses represent uncertainty in the last digit(s). The error in ${\sigma }_{\text{ISS}}$ was estimated by evaluating the fluctuation in the running average across an interval of 2,500 steps centered around the determined maximum value. Based on this criterion, we estimate the error bar in ${\epsilon }_{\text{ISS}}$ to correspond to the shear strain accrued in 2,500 AIMD steps (i.e., $\mathrm{\Delta }{\epsilon }_{\text{ISS}}=0.005$). For μ, the uncertainty was estimated by the fluctuations of the linear regression results of the stress–strain relations up to 4, 5, and 6%.

To further investigate this point, we carry out a second set of simulations at the same density and temperatures but replace the protons by a uniform neutralizing charge density distribution, retaining only the oxygen ions as atomic degrees of freedom. Comparison of these results with the all-atom simulations provides insight as to whether the structure/dynamics of the proton subsystem affect the mechanical response. The oxygen-only stress–strain curves are shown in Fig. 2 H–K, and in contrast to the all-atom results, they are essentially identical for all four temperatures. This implies that the differences between the all-atom responses in Fig. 2 D–G are due to structural/dynamical changes of the proton subsystem. In particular, as shown in Table 1, for the insulating bcc phases, μ, ${\sigma }_{\text{ISS}}$, and ${\mathrm{ϵ}}_{\text{ISS}}$ in the all-atom simulations are substantially larger compared with the oxygen-only computations. This means that, in addition to enhancing the elastic stiffness, the presence of the localized proton subsystem increases the amount of stress–strain the system can withstand before yielding irreversibly. This is very different for the SI bcc phase. In this case, except for the larger fluctuations due to the presence of SI proton fluid, the all-atom and oxygen-only stress–strain curves (Fig. 2 G and K, respectively) are very similar, with comparable values for ${\sigma }_{\text{ISS}}$ and ${\mathrm{ϵ}}_{\text{ISS}}$. This indicates that the structural and dynamical details of the protonic subsystem in the SI bcc phase play only a minor role and that its influence can be approximated in terms of a uniform charge density field, very much in the spirit of the jellium model for the electron gas.

The same picture is seen for the SI fcc phase, as depicted in Fig. 3. Fig. 3 A–C shows representative snapshots of the oxygen ion lattice along the shear deformation process in the $\left\{111\right\}\langle 11\overline{2}\rangle$ slip system of the fcc lattice, in which permanent deformation occurs by stacking fault nucleation. Fig. 3 D–K shows the fcc stress–strain curves between 2,500 and 4,000 K for the all-atom (Fig. 3 D–G) and oxygen-only (Fig. 3 H–K) simulations. The all-atom simulations show gradual thermal softening, with the shear modulus and the ISS parameters reducing by ~25% from 2,500 to 4,000 K (Table 2). More importantly, when comparing these results with those obtained for the oxygen-only simulations under the same conditions, the stress–strain curves for both approaches, except for the magnitude of the fluctuations, are very similar for all temperatures, with both the shear modulus as well as the ISS parameters in good agreement. This is consistent with the observations for the SI bcc phase at 2,500 K and lends support to the idea that the influence of the protonic diffusivity on the mechanical response of crystalline SI ices is only minor, with the proton fluid acting mostly as a uniform background charge. Indeed, this picture is consistent with the fact that the electronic charge density distributions are similar for both cases, as shown in SI Appendix, Fig. S5, indicating that polarization effects of the oxygen ions in the SI phases are relatively small. Also, the substitution of the proton fluid by a uniform background charge decreases the simulation time by a factor of approximately three, allowing for a significant reduction of the computational cost when investigating the mechanical behavior of SI ices.

Fig. 3.

Shear deformation of the fcc phases for the $\left\{111\right\}\langle 11\overline{2}\rangle$ slip system. Snapshots in A–C show typical images of oxygen positions of the structure before deformation, an intermediate stage, and immediately after accruing permanent deformation, respectively. The red dashed line in C indicates the slip plane across which deformation has occurred. D–G display stress–strain curves obtained for fully atomistic systems at 2,500 K/334 GPa, 3,000 K/340 GPa, 3,500 K/346 GPa, and 4,000 K/353 GPa, respectively. Corresponding results in the absence of explicit protons are shown in H–K.

For a quantitative assessment of these results, it is useful to analyze them in a broader context of the ISS for crystalline materials in general. Based on the work of Ogata et al. (36), the ISS of a large variety of crystal structures and cohesion types, including metallic, covalent, and dispersion-based bonding, displays a correlation involving the degree of electronic valence charge localization as well as the directionality of the bonding. In particular, it was found that the shearability of a crystal increases with the anisotropy of the bonding, such that ${\epsilon }_{\text{ISS}}$ for covalent semiconductors is larger than that of metals, and within the subgroup of metallic crystals, those with some directionality in the bonding are characterized by a larger shearability compared with the noble metals. Moreover, it was found that a renormalized version of the Frenkel model for ideal strength, in which ${\sigma }_{\text{ISS}}$ and ${\epsilon }_{\text{ISS}}$ are related through ${\sigma }_{\text{ISS}}=2\mu \hspace{0.17em}{\epsilon }_{\text{ISS}}/\pi$ (36), shown by the dashed line in Fig. 4 gives an approximately universal description of the data. Of course, the large values of the shearabilities, reaching almost 40% for the case of diamond, are related to the fact that the analysis involves the idealized case of defect-free crystals.

Fig. 4.

${\sigma }_{\text{ISS}}/\mu$ as a function of ${\epsilon }_{\text{ISS}}$ for solids characterized by different crystal structures and bonding types. The line describes the renormalized Frenkel model (in the text). Black squares correspond to data points from refs. 36 and 38 (⁴He) and those from ref. 43 (Ti₃Nb). Blue circles and red triangles correspond to bcc and fcc ices, respectively. Error bars correspond to uncertainties derived from Tables 1 and 2.

The present results show that the renormalized Frenkel model also captures the ISS characteristics of the high-pressure ices, with all data points located close to the dashed line in Fig. 4. Indeed, the ISS results for the high-pressure/temperature ices reveal the same directionality dependence as that seen for common metallic and covalent solids. Specifically, the insulating bcc phases, in which the bonding is manifestly anisotropic due to the localized proton arrangements, are able to withstand substantially larger amounts of shear compared with the SI phases, which are mediated by a much more isotropic proton subsystem. This is also reflected in the predominantly isotropic electron charge density distributions on the oxygen ions, as shown in SI Appendix, Fig. S5.

Although the ISS is a concept associated with a defect-free solid, it is known to correlate with the degree of ductility of real solids. While systems in the upper right part of Fig. 4, such as Si and diamond, are intrinsically brittle in nature, those in the lower left corner, such as the dispersion-bonded ⁴He and the gum metal approximant Ti₃Nb, are known to readily sustain plastic deformation. Solid ⁴He, for instance, is known to display giant plasticity (41, 42), by which dislocations are able to move through the crystal without any measurable lattice resistance, giving rise to extremely malleable behavior. The fact that the ISS parameters of the SI ices, in particular those of the fcc phase, are among the lowest in the entire set of solids in Fig. 4 then suggests that they may be highly ductile as well, readily deforming under the influence of external shear stresses.

Dislocation Glide.

To further investigate this point, we consider the mechanism of dislocation glide, which is one of the principal modes of plastic deformation in realistic crystals (39, 40). It involves the motion of lattice dislocations through a crystal, causing relative slip of adjacent crystal planes, which induces irreversible deformation. The ease/difficulty with which these line defects can move through a lattice under the influence of external stresses then is a fundamental property that is directly linked to the rheology of crystalline solids (31, 46). Since dislocations are extended defects, their simulation involves computational cells that are substantially larger than those employed for the defect-free ISS calculations (47). Accordingly, due to it elevated computational cost, the density functional theory (DFT) approach employed to compute the ISS characteristics is beyond reach for this purpose. Therefore, we resort to the recent advances in the application of ML techniques to the description of interatomic interactions. Specifically, we employ the deep potential (DP) methodology (48–52), which provides a set of tools for developing deep neural network models of interatomic potential energy surfaces, making it possible to carry out first principles–quality molecular simulations at the cost of semiempirical force fields, as has been shown in a number of studies involving the properties of water (53–55). Here, we employ the DP approach to construct a deep neural network model to capture the ISS response obtained from the DFT calculations and subsequently, employ it in MD simulations to determine the mobility of lattice dislocations. We concentrate specifically on the SI fcc phase, which has been predicted to be stable under the pressure/temperature conditions in the interiors of Neptune and Uranus (3, 21).

The DP model is constructed using a training set containing configurations from the first-principles ISS calculations of the SI fcc phase, including the protons. The set holds both undeformed and deformed states at temperatures between 2,500 and 4,000 K and a density of 4.2 g/cm³ (Materials and Methods and SI Appendix). The strained configurations in the set are characterized by different degrees of deformation (SI Appendix), including several states before and after reaching the critical strain value ${\epsilon }_{\text{ISS}}$.

The resulting DP model is then employed to carry out MD simulations to measure dislocation mobility, following the approaches of Rodary et al. (56) and Abu-Odeh et al. (57). The geometry of the employed computational cell is depicted in Fig. 5A, in which the x, y, and z directions are parallel to the $[1\overline{1}0],\hspace{0.17em}[111]$, and $[\overline{1}\hspace{0.17em}\overline{1}\hspace{0.17em}2]$ directions with dimensions of 30.0, 12.5, and 1.5 nm, respectively, containing 80,352 oxygen ions and 160,704 protons. Periodic boundary conditions are applied along the x and z directions, whereas the oxygen ions in the top and bottom y layers, shown in blue, have their y coordinates fixed at $y_{\text{top}}$ and $y_{\text{bottom}}$. Due to the SI nature of the phase, we introduce two reflecting walls at $y=y_{\text{top}}+\mathrm{\Delta }y$ and $y=y_{\text{bottom}}-\mathrm{\Delta }y$ ($\mathrm{\Delta }y$ equals half the interplanar oxygen spacing in the y direction), represented by the red planes in Fig. 5A, to prevent the protons from diffusing out and to maintain their density at the bulk value. The cell contains a single edge dislocation, with Burgers vector $\mathit{b}=\frac{1}{2}[1\overline{1}0]$, dissociated into two Shockley partials (39, 40). The external stress is applied by adding a constant force in the x direction (parallel to b) to all oxygen ions in the top and bottom layers. The magnitude of the force is inversely proportional to the number of oxygen ions in each layer, such that the total added force is the same (but with opposite signs) for both. The corresponding applied shear stress is then given by ${\sigma }_{yx}=N_{\text{layer}}{f}_x/A$, where $N_{\text{layer}}$ is the number of oxygens in a layer, f_x is the magnitude of the added force to the oxygen ions, and $A=L_x\hspace{0.17em}{L}_z$ is the xz area of the computational cell. The simulation cell as specified above corresponds to the largest possible number of particles tractable within our computational limitations, and as analyzed in a recent simulation study including pure fcc Ni (57), its dimensions are sufficiently large to mitigate finite-size effects on dislocation mobilities.

Fig. 5.

(A) Geometry of a computational cell containing dissociated edge dislocation for dislocation mobility MD simulations, consisting of 80,352 oxygen ions and 160,704 protons, carried out at 3,000 K and a pressure of 340 GPa. The x, y, and z directions correspond to the $[1\overline{1}0],\hspace{0.17em}[111]$, and $[\overline{1}\hspace{0.17em}\overline{1}\hspace{0.17em}2]$ crystallographic directions, respectively. Periodic boundary conditions (PBCs) are applied in the x and z directions. The top and bottom layers of oxygen ions, shown in blue, are constrained to move only in the x and z directions. Reflecting walls (red planes) acting on protons are used to prevent them from diffusing out of the cell and to maintain their bulk density. B and C display xy projections of the cell during deformation simulations as obtained using the Ovito visualization package (44). Red dashed lines represent projections of reflecting walls. Green and red oxygens are those classified as being in fcc and hexagonal close-packed (hcp) (stacking fault) surroundings, respectively. Blue oxygens are those in the dislocation cores and in the top/bottom surfaces. Black dots represent protons. White circles indicate average x positions of leading (L; x_L) and trailing (T; x_T) partials determined using the dislocation extraction algorithm (DXA) algorithm (58) implemented in Ovito. (D) Lines describe the average edge dislocation position $x_{\text{edge}}\equiv \frac{1}{2}(x_L+x_T)$ (taking into account rewrappings through PBCs) as a function of time for applied shear stresses of 10 GPa (red), 5 GPa (purple), 2 GPa (green), and 800 MPa (blue). Dashed straight lines indicate the steady-state linear portions from which the dislocation velocities have been determined. (E) Dislocation velocity as a function of applied shear stress as obtained from linear regressions of the data in D. Error bars are smaller than the symbol size. The dashed line represents the linear fit to the dislocation velocity data.

Fig. 5 B and C depicts typical snapshots obtained from an MD simulation at 3,000 K and a pressure of 340 GPa (Materials and Methods). These conditions are close to those believed to exist at a depth of about 8,000 km, which is about 1/3 of Neptune’s and Uranus’ radii, where temperatures and pressure are thought to range between ~3,000 and 6,000 K and between ~200 and 500 GPa, respectively (3). The configuration in Fig. 5B shows the state before an external force is applied, with the leading and trailing Shockley partials separated by a ribbon of stacking fault. The protonic subsystem displays a liquid-like arrangement, consistent with the SI nature of the system. Fig. 5C represents the atomic configuration at an instant after the constant force has been acting during a period of time. Compared with Fig. 5B, the dislocation as a whole has moved to the right (Movie S1 shows an animation of the dislocation glide process).

To quantify this dislocation movement as a function of the applied stress, we monitor the motion of the edge dislocation for a set of MD simulations, in which different levels of shear stress are applied. To this end, we track the time dependence of the mean dislocation x position, defined as $x_{\text{edge}}\equiv \frac{1}{2}(x_L+x_T)$, with x_L and x_T being the x coordinates of the leading and trailing partials averaged along their dislocation lines, respectively. The results are shown in Fig. 5D, in which each line corresponds to the evolution of $x_{\text{edge}}(t)$ along the MD simulations for stress values ranging between 10 GPa and 800 MPa at a temperature of 3,000 K and a pressure of 340 GPa.

In all cases, after reaching a steady state, the movement of the dislocation is uniform in nature, with its velocity v depending on the applied stress σ in a linear manner, as shown in Fig. 5E. This indicates that the dislocation mobility is in the viscous drag regime (39), which is consistent with the fact that the observed dislocation velocities are more than an order of magnitude smaller than the transverse shear wave velocity $C_t=\sqrt{\mu /\rho )}\simeq 6.8\times {10}^3$ m/s (39). In this regime, the dislocation velocity is given by (56)

$$v=(\sigma -{\sigma }_d)\hspace{0.17em}b/B,$$ [1]

where b = 2.1496 Å is the magnitude of the Burgers vector, σ_d is a threshold-stress value below which there is no dislocation motion, and B is the viscous drag coefficient that quantifies the frictional resistance to dislocation glide. From the slope of the linear fit to the data in Fig. 5E, we estimate the damping coefficient to be $B=(1.1\pm 0.3)\times {10}^{-2}$ Pa$\hspace{0.17em}·\hspace{0.17em}$s. This value is high compared with typical fcc metals, such as pure copper and aluminum, for which B is of the order of 10^–5 Pa$\hspace{0.17em}·\hspace{0.17em}$s, albeit at temperatures and pressures that are significantly lower than those of the SI fcc phase (59, 60). The value of σ_d is more difficult to quantify because of the large computational cost required to measure dislocation velocities at low stresses when they move at low speeds. Specifically, at an applied stress of 400 MPa, we did not observe any dislocation motion over a time interval of 100 ps. Accordingly, we estimate the value of σ_d to be between 400 and 800 MPa.

These results now allow us to obtain an estimate for the effective viscosity associated with deformation due to the motion of edge dislocations in the SI fcc phase. To this end, we employ Orowan’s equation (31), which links the strain rate $\dot{\epsilon }$ to the dislocation velocity v through the relation

$$\dot{\epsilon }={\rho }_d\hspace{0.17em}b\hspace{0.17em}v,$$ [2]

where ρ_d is the density of mobile dislocations. Substituting the viscous drag dislocation velocity law of Eq. 1, one obtains

$$\dot{\epsilon }=\frac{{\rho }_d\hspace{0.17em}{b}^2}{B}\tilde{\sigma },$$ [3]

with $\tilde{\sigma }\equiv \sigma -{\sigma }_d$. The corresponding viscosity at 3,000 K and 340 GPa then is (46)

$$\eta \equiv \frac{\tilde{\sigma }}{\dot{\epsilon }}=\frac{B}{{\rho }_d\hspace{0.17em}{b}^2}\simeq \frac{(2.4\pm 0.6)\times {10}^{13}}{{\rho }_d}\text{Pa}·\text{s},$$ [4]

where ρ_d is expressed in units of cm^–2. The value of ρ_d is unknown a priori since it is not an equilibrium property, but rather, depends on the crystallization, thermal, and deformation history of the system. Furthermore, it also varies in time due to dislocation multiplication processes and dislocation interactions (61). However, since such information is unavailable for the interiors of Neptune and Uranus, we are unable to make reliable predictions of this density evolution. Therefore, we assume ρ_d to be fixed and estimate orders of magnitude of the dislocation glide viscosity using Eq. 4 by taking known values for a variety of crystalline solids, ranging from a very low density of 10³ cm^–2 for undeformed ionic and covalent crystals to a density over 10¹² cm^–2 for highly deformed metals (61).

For this range of dislocation densities, the deep potential molecular dynamics (DPMD) simulations predict an effective shear viscosity due to edge dislocation glide of $\eta ~{10}^1$ to 10¹⁰ Pa$\hspace{0.17em}·\hspace{0.17em}$s. Such values are typically encountered for the viscosity of glass-forming liquids at different temperatures (62) and, on a laboratory scale, correspond to a rather sluggish rheology. However, even for the lowest assumed dislocation density value, the corresponding viscosity of 10¹⁰ Pa$\hspace{0.17em}·\hspace{0.17em}$s is comparatively low on planetary timescales, being many orders of magnitude lower than that of, for instance, Earth’s lower mantle estimated at ~10²¹ Pa$\hspace{0.17em}·\hspace{0.17em}$s (63).

Of course, temperature/pressure variations are expected to affect these values, and further simulations are required to analyze this in detail. However, we expect these effects to be minor. Since the MD calculations were carried out at the low-temperature/ pressure ends of the intervals of interest for Neptune and Uranus mentioned earlier, one should consider increases of temperature and pressure up to a factor of approximately two. While an increase in temperature generally facilitates flow, tending to lower the effective viscosity, an increase in pressure typically leads to the opposite effect (31, 64). However, such effects are usually quite small. For instance, for the prototypical fcc metal copper, raising the pressure from 1 bar to 5 kbar results in an increase of the flow stress of only ~1% (65). Therefore, we expect the estimates based on our calculations to be representative for the temperature/pressure intervals of interest for the interiors of Neptune and Uranus.

A further issue to consider is that dislocation-based processes, on which we focus here, are not the only contributors to the effective flow viscosities of crystalline solid. Processes, such as grain boundary sliding, twinning, and point defect diffusion (31), also provide mechanisms for plastic deformation (31). Although we cannot quantify their contributions based on the present data, they do provide additional means for the system to respond to imposed shear stresses and thus, further contribute to the plastic flow.

In this light, the present results contrast earlier indications (3), inspired by the values of elastic stiffness constants, that the dynamics of SI ice flows might resemble the slow evolution of Earth’s solid mantle. Instead, the effective viscosity data suggest that the rheology of SI fcc water ice may be significantly faster, possibly having important implications for the interior dynamics of Neptune and Uranus and its relation to their anomalous nondipolar magnetic fields.

Materials and Methods

DFT Calculations.

All ab initio calculations of the ISS have been executed using Kohn–Sham (KS) DFT as implemented in the Open Source Molecular Dynamics package (CP2K) package (66), employing the Perdew–Burke–Ernzerhof exchange-correlation functional (67), Goedecker–Teter–Hutter (GTH) norm-conserving pseudopotentials (68), and a combination of GTH double-zeta polarization Gaussian functions and a plane-wave basis set with an energy cutoff of 200 Ry for the expansion of the KS orbitals. We use periodic cells containing 144 water molecules at fixed densities of 3.7 and 4.2 g/cm${}^3$ for the bcc and fcc phases, respectively. Further details of these computational cells are described in SI Appendix, Fig. S1. These structures are considered for the temperature intervals from 1,000 to 2,500 K and from 2,500 to 4,000 K, respectively, in which the corresponding oxygen lattices are stable (11, 17) in the pressure range between 200 and 300 GPa. Temperature is controlled using an adaptive Langevin thermostat (69), and the corresponding equations of motion are integrated using a time step of 0.4 fs. Further details and a description of the equilibration procedures adopted prior to deformation are given in SI Appendix.

DP Neural Network Model.

The DP neural network model was generated using the DeepMD kit package (48). The training set was generated using AIMD/CP2K calculations for both undeformed and deformed states of the fcc phase at temperatures of 2,500, 3,000, 3,500, and 4,000 K. For each of the three principal shear directions, $\left\{111\right\}\langle \overline{1}\overline{1}2\rangle ,\hspace{0.17em}\left\{111\right\}\langle \overline{1}10\rangle$, and $\left\{111\right\}\langle \overline{2}11\rangle$, six states were chosen corresponding to different strain values, including at least two with deformations beyond the shearability limit (SI Appendix, Fig. S7). For each deformation state, we carry out 1,000 AIMD with fixed number of particles, volume and temperature (NVT ensemble) steps at a time step of 0.5 fs using the same DFT setup as used for the ISS calculations. For each deformation state, we collect 25 configurations (including atomic positions, energy, virial, and forces to be used in the training set), resulting in a total of 1,900 different configurations. The DP model has architecture sizes of $25\times 50\times 100$ and $240\times 240\times 240$ for the embedding and fitting nets, respectively. We employ the Adam stochastic gradient descent method for the training, with an exponentially decreasing learning rate starting at $1\times {10}^{-3}$ and ending at $1\times {10}^{-8}$, with 5,000 decay steps. The prefactors in the loss function for energy, forces, and virial are taken as $p_e^{\text{start}}=0.02$ and $p_e^{\text{limit}}=1,\hspace{0.17em}{p}_f^{\text{start}}=1000$ and $p_f^{\text{limit}}=1$, and $p_v^{\text{start}}=0.02$ and $p_v^{\text{limit}}=1$, respectively. The descriptor “se_e2_a” was used with a cutoff radius of 5 Å, with maximum numbers of neighbors of 82 and 155 for O and H atoms, respectively. The optimization of the neural network was carried out over 10⁵ epochs and tested on a second set of 1,900 different AIMD configurations, reaching final relative errors of 7, 16, 13, and 12% in the total energy, forces, pressure, and shear stress, respectively, measured with respect to the SD of the test dataset. SI Appendix, Figs. S8 and S9 and Table S2 show comparisons between the results obtained with the optimized DP model and the AIMD calculations. In terms of the computational cost, the DP model achieves an efficiency gain of approximately two orders of magnitude compared with the AIMD simulations.

DPMD Simulations.

The MD simulations for the edge dislocations were carried out using the LAMMPS package (70, 71) interfaced with the DP module (48). The dislocation cell was created using the MD++ package (72). The temperature of all protons and the oxygen ions not in the constrained top and bottom layers is controlled using a Langevin thermostat (73). The corresponding equations of motion were integrated using a time step of 0.4 fs and a Langevin damping constant of 4 fs. The constrained oxygen ions in the top and bottom layers evolve according to Newtonian dynamics without temperature control.

Data, Materials, and Software Availability

CP2K code, ML training/test data, and LAMMPS script data have been deposited in Zenodo (https://zenodo.org/record/6518807) (74).