The Phase Diagram of a Deep Potential Water Model
aa r X i v : . [ phy s i c s . c h e m - ph ] F e b The Phase Diagram of a Deep Potential Water Model
Linfeng Zhang
Program in Applied and Computational Mathematics,Princeton University, Princeton, NJ 08544, USA
Han Wang ∗ Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics,Fenghao East Road 2, Beijing 100094, P.R. China
Roberto Car † Department of Chemistry, Department of Physics,Program in Applied and Computational Mathematics,Princeton Institute for the Science and Technology of Materials,Princeton University, Princeton, New Jersey 08544, USA
Weinan E
Department of Mathematics and Program in Applied and Computational Mathematics,Princeton University, Princeton, NJ 08544, USA andBeijing Institute of Big Data Research, Beijing, 100871, P.R. China
Using the Deep Potential methodology, we construct a model that reproduces accurately thepotential energy surface of the SCAN approximation of density functional theory for water, fromlow temperature and pressure to about 2400 K and 50 GPa, excluding the vapor stability region.The computational efficiency of the model makes it possible to predict its phase diagram usingmolecular dynamics. Satisfactory overall agreement with experimental results is obtained. The fluidphases, molecular and ionic, and all the stable ice polymorphs, ordered and disordered, are predictedcorrectly, with the exception of ice III and XV that are stable in experiments, but metastable inthe model. The evolution of the atomic dynamics upon heating, as ice VII transforms first intoice VII ′′ and then into an ionic fluid, reveals that molecular dissociation and breaking of the icerules coexist with strong covalent fluctuations, explaining why only partial ionization was inferredin experiments. The phase diagram of water is extremely rich. In thetemperature and pressure domain with T /
400 K and P /
50 GPa, there are ten stable phases, nine solid (iceIh, II, III, V, VI, VII, VIII, XI and XV) and one liquid,in addition to five metastable phases (ice IV, IX, XII,XIII and XIV) [1–3]. This large variety of structures aremade possible by hydrogen bonded arrangements of themolecules. In ice, the oxygen sublattice is crystalline, butthe hydrogen sublattice can be either ordered or disor-dered, due to the vast number of nearly degenerate hydro-gen (proton) configurations allowed by the ice rules. Thecorresponding configurational or residual entropy stabi-lizes disordered polymorphs at high temperature. Thus,near melting, all the stable phases are disordered (ice Ih,III, V, VI, and VII). In ice Ih, VI, and VII, disorder iscomplete and the residual entropy is well approximatedby k B ln 1 . ≈ . k B /H O [4]. In ice III and V, dis-order is partial [5, 6] and the entropy is less than Paul-ing’s estimate but still significant [7]. Upon cooling, iceIh, VI, and VII become less stable than their orderedcounterparts, ice XI [8], XV [3], and VIII [9], respec-tively. Ordered polymorphs are ferroelectric (XI) or anti-ferroelectric (II, XV, and VIII). Interestingly, ice II doesnot have a disordered counterpart. See, e.g., Ref. [10] fora review of ice polymorphism. At high pressure, the stability of the solid phasesextends to higher temperatures, the hydrogen bondsweaken, and molecular dissociation into ions is promotedby the increasing thermal fluctuation. Molecular to ionictransformation is continuous in the fluid. In the solid,for T '
850 K and pressures above ≈
14 GPa, ice VIItransforms into ice VII ′′ , a superionic phase in which theBCC oxygen sublattice of ice VII coexists with mobileprotons. Upon further heating, ice VII ′′ melts into anionic fluid [11–15].Molecular dynamics (MD) simulations give micro-scopic insight into the water phases and complement ex-periments with atomistic details [16–22]. The key in-gredient of MD is the potential energy surface (PES),which can be constructed either by fitting a physicallymotivated force field to experiment, or, non-empirically,from quantum theory ( ab initio MD (AIMD)). Compar-ing the phase diagram predicted by MD to experiment isthe ultimate accuracy test of a model PES. Due to thehigh computational cost of AIMD, extensive studies ofthe water phase diagram have only been possible so farwith empirical force fields, which, however, face severedifficulties with the ionic phases. By contrast, in AIMD,the PES is constructed on-the-fly from density functionaltheory (DFT) and can describe molecular dissociationprocesses. Indeed, this approach has been particularlyuseful in modeling proton transfer in the liquid at am-bient conditions [21, 23], or the superionic ice phases athigh pressure and temperature [14].Advances in machine learning (ML) are making possi-ble MD simulations of ab initio quality at a cost of empir-ical force fields. Applications to water studied the phasebehavior at ambient [24, 25] and deeply undercooled [26]conditions, isotopic effects [25, 27, 28], infrared and Ra-man spectra [29–32], etc. A recent calculation reportedthe phase diagram in the (
T, P ) range from 150 K to300 K and from 0 .
01 GPa to 1 GPa, at the hybrid DFTlevel, including nuclear quantum effects [33]. However, tothe best of our knowledge, no attempt has been made todescribe water in a wide thermodynamic range includingordered and disordered ice, superionic ice, molecular andionic fluid phases.Here this goal is achieved with Deep Potential Molec-ular Dynamics (DPMD) [34, 35], using an iterative con-current learning scheme, Deep Potential (DP) Genera-tor [36, 37], to construct the PES with SCAN-DFT asthe reference. SCAN [38] is a non-empirical functionalthat describes well several properties of water [39]. Wefind that a unique DP model can reproduce closely DFTin a vast thermodynamic range, extending from ambientpressure to ≈
50 GPa and from ≈
50 K to ≈ T, P ) region the simulations reveal keyfeatures of the temperature induced transitions from iceVII to ice VII ′′ and from the latter to an ionic fluid.To construct the model PES, a trial DP is built fromconfigurations of the liquid, at ambient conditions, andof all the experimentally known stable and metastableice polymorphs for P /
50 GPa (Ih, Ic, II, III, IV, V, VI,VII, VIII, IX, XI, XII, XIII, XIV, XV). The model is usedby DP Generator to explore a wide region of the phasespace with isothermal-isobaric (
N P T ) DPMD trajecto-ries. The protocol is iterated to refine the model withnew DFT data until satisfactory accuracy is achieved.The visited states can be roughly classified into threegroups: the low pressure (A), the high pressure (B), andthe superionic group (C). Group (A) includes states inthe range 50 ≤ T ≤
600 K and 10 − ≤ P ≤ ≤ T ≤
600 K and 0 . ≤ P ≤
50 GPa,starting from configurations of ice VII and VIII. Group(C) includes states in the range 200 ≤ T ≤ ≤ P ≤
50 GPa, starting from ice VII and the fluid.DPMD samples almost uniformly the thermodynamic do-mains of the three groups. The deviation in the predictedforces within a set of representative DP models is used to label configurations for which new DFT calculations ofthe energy, forces, and virial are necessary. The new dataare added to the training dataset and serve to refine therepresentative DP models entering the next iteration.After 36 concurrent learning iterations the error in theforce is satisfactorily reduced and the procedure ends.The accumulated number of snapshots in the trainingdataset is 31058, a tiny fraction ( ∼ . ab initio simulation package (VASP) ver-sion 5.4.4 [40, 41] is used for the DFT calculations.DeePMD-kit [42] is used for DP training and for runningDPMD, interfaced with LAMMPS [43]. DP-GEN [37] isused for the concurrent learning process. See details inSM. Accuracy of the DP model.
The error relative to DFT isquantified with an independent testing dataset including5141 configurations along 67 isothermal-isobaric DPMDtrajectories spanning the relevant thermodynamic do-main (SM Fig. S2). In most cases, the root mean squareerror (RMSE) of energy and force is ∼ O and ∼ ∼
10% orless.
Phase diagram.
Thermodynamic integration is used tocompute the absolute Gibbs free energy of a single statepoint of each phase [44, 45]. The algorithm of Ref. [46]is used to generate the fully disordered structures of iceIh, IV, VI, and VII, and Pauling’s residual entropy con-tribution (0 . k B /H O) is added to their free energy.For the partially disordered structures of ice III and Vand the corresponding entropies we follow Ref. [7]. Tominimize finite size effects we use cells with at least 128molecules. Taking into account finite size, entropy ap-proximation, DP error, and statistical uncertainty we es-timate that the free energy error should be approximately1 meV/H O. Then, using thermodynamic integrationwith the composite Simpson rule we trace a family ofcurves representing the variation with pressure along anisotherm, or with temperature along an isobar, of thefree energy of each phase. The intersections betweenpairs of curves define phase coexistence points. Finally,the phase boundary lines stemming from the coexistencepoints are traced by integrating the Gibbs-Duhem equa-tion [47] with a second order Runge-Kutta method. SeeSM, Sec. SIII A.The numerical accuracy of the predicted phase bound-aries can be gauged from the consistency of the predictedtriple points (TPs). Each TP can be inferred in threeindependent ways from the intersection of two boundary -3 -2 -1
0 100 200 300 400T[K]P[GPa]
IhII VVI
III
VIII VIIF XV (a )XI I h → X I X I → I h -2 -1
0 200 400
VIII Plastic
VIIVIII Ih F V III (b)
TIP4P/2005
500 1000 1500 2000
VII"(a ) F DP (this work)ExperimentTP Ih-II-XI TP II-VI-XVTP VI-VIII-XVTP VII-VII"-F
FIG. 1: Phase diagram of water. (a ) DP model (red solidlines) and experiment (gray solid lines) for T <
420 K. Blackletters indicate phases that are stable in experiment andmodel. Ice III and XV (stable in experiment but metastablein model) are gray. Experimental coexistence lines are fromRef. [48] (melting curves), and from Refs. [9, 49, 50] (solid-solid curves). The gray triangle indicates the postulated Ih-II-XI TP [50]. The two dashed lines indicate the experimen-tally observed transitions Ih → XI and XI → Ih [8]. The graysolid circle and square denote the VI-VIII-XV and the II-VI-XV TPs, respectively [3]. (a ) Phase diagram at high T and P . The experimental melting lines are from Ref. [51] andRef. [13]. The VII-VII ′′ -F TP is from Ref. [13]. (b) Phasediagram of TIP4P/2005 water [52]. lines between the three coexisting phases. The average ofthese estimates defines a TP, and the standard deviationgives the estimated error. From the TPs in SM Tab. SV,we infer that the numerical uncertainty of the calculatedphase boundaries is less than 5 K in temperature and lessthan 0.02 GPa in pressure.Overall, the DP phase diagram in Fig. 1 (a) agrees wellwith experiment. All the stable ice phases are predictedcorrectly, with two exceptions, ice III and XV, which aremetastable in the DP model. The Ih-F coexistence lineis displaced by ≈
40 K to higher temperature than ex-periment, while the Ih-II line is displaced by ≈ / O should be considered degener-ate. This happens to IV and VI in part of the stability domain of the latter (SM Sec.SIII B). The coexistencelines for P ' ′′ boundary, and the VII-VII ′′ -F TP are also ingood qualitative agreement with experiment. At pres-sures higher than reported in Fig. 1, ice VII transformsinto ice X [53]. This regime is beyond the domain ofvalidity of the present DP model and is not investigated.It is instructive to compare the DP phase diagram withthe one derived from one of the most accurate empiricalwater models, TIP4P/2005 [17], which assumes rigidmolecules and is parameterized with experimental obser-vations, such as, e.g., the temperature of maximal liq-uid density at ambient pressure, the densities of ice II,III, and V at different thermodynamic conditions, etc.As shown in Fig. 1 (b), TIP4P/2005 works well at lowand intermediate pressures. At higher pressures, how-ever, significant deviations from experiment affect theboundary lines between ice VIII, VII, and VI. Moreover,the rigid molecule approximation does not allow ionizedwater configurations. At high pressure and temperatureTIP4P/2005 predicts a first-order transition from ice VIIto a plastic phase, in which the BCC oxygen sublatticecoexists with freely rotating molecules [52, 54]. No ex-perimental evidence has been found so far for this phase,nor was such behavior observed in our DP simulations. Ionic phases.
According to the DP model, at low T , ice VII is a molecular crystal with full proton dis-order and insignificant atomic diffusion. Upon heating,however, H diffusion grows exponentially with T , whileO diffusion remains insignificant. This behavior is illus-trated in Fig. 2(a) for the isobar at 30 GPa. Eventually Hdiffusion saturates and remains approximately constantover a finite T interval. At even higher T the diffusivitiesof H and O jump to distinct macroscopic values signalingtransformation to a fluid. Ice VII has been referred to asice VII ′ and as ice VII ′′ in the thermodynamic domainsof exponential growth and saturation of the H diffusiv-ity [14]. The enthalpy evolution along the 30 GPa isobaris depicted in Fig. 2(b). It shows a smooth reversible vari-ation in ice VII ′ followed by a more rapid change whenice VII ′ turns into ice VII ′′ . This affects enthalpy andvolume, and occurs spontaneously without apparent hys-teresis over timescales of 100 ps in simulations with 438molecules. Simulations with up to 3456 molecules show asharpening of the rapid change (SM Fig. S5), as expectedof a weakly first-order phase transition. Since we cannotassociate a separate thermodynamic phase to ice VII ′ , weretain for it the name of ice VII hereafter and in the phasediagram. The VII-VII ′′ transition shows the typical be-havior of a type II superionic transition [55, 56]. A jumpin O diffusivity signals melting of ice VII ′′ , a transitionundetectable by monitoring the enthalpy on timescalesof 100 ps in a heat-until-melt simulation. Thus, to deter-mine the melting temperature we used a two-phase sim-ulation of 1728 molecules at 50 GPa, and extrapolatedthe melting temperature to lower T with Gibbs-Duhem -12.8-12.6-12.4-12.2-12.0-11.8 900 1000 1100 1200 1300 1400 1500 1600 1700 E n t ha l p y [ e V / H O ] T [K] (b) from VIIfrom VII"from Fluid10 -2 -1 D [ A / p s ] Onset of H diffusion Onset of O diffusion (a) D O D H FIG. 2: Two-step melting of ice VII along the P = 30 GPaisobar, showing a solid-solid (VII-VII ′′ ) followed by a solidfluid (VII ′′ -F) transition. (a) H (blue open square) and O(red solid square) diffusion coefficients as a function of tem-perature. (b) Enthalpy as a function of temperature. Theestimated temperatures of solid-solid and solid-fluid transi-tions are indicated by the black arrows. integration. The predicted VII-VII ′′ -F TP is located at(774 K 10.6 GPa), in relatively good agreement with themost recent experimental result (850 K 14.6 GPa) [13].Importantly, the same experiment confirmed the first-order nature of the VII-VII ′′ transition, signaled by adiscontinuous change of the lattice parameter in X-raydiffraction (XRD).A magnified view of the DP phase diagram in the VII-VII ′′ -F domain is shown in Fig. 3, together with exper-imental and AIMD results. Overall there is good qual-itative agreement: experiments confirm the presence oftwo first-order transitions, a solid-solid and a solid-fluidone. Possibly, the significant scatter in the experimentaldata reflects the difficulty of detecting weakly first-orderphase transitions at challenging thermodynamic condi-tions. A two-step melting process for ice VII, with asuperionic intermediate, was first proposed in Ref. [11]based on AIMD simulations. The corresponding solid-solid phase transition was confirmed experimentally inRef. [57], without structural details on the new solidphase. These were provided recently by XRD experi-ments that verified the BCC lattice structure of VII ′′ [13].The DP results are in semi-quantitative agreement withAIMD simulations for the VII-VII ′′ and VII ′′ -F bound-aries [14, 15, 56]. The differences between DPMD andthese earlier studies should be attributed mainly to theadopted exchange-correlation functionals and to the rela-tively small size and time scales of the AIMD simulations.DP simulations give insight on the atomistic processesthat underlie the two-step transition from ice VII to ionicfluid. The O-O, O-H, and H-H pair correlation functionsalong the 30 GPa isobar shown in SM Fig. S4 illustratethe progressive loss of long-range order as the systemprogresses through ice VII, VII ′′ and ionic fluid. Inter-estingly, in spite of the large diffusivity of H in ice VII ′′ P [ G P a ] T [K]
VII VII"Fluid
This work SFThis work SSExp. SF [57]Exp. SS [57]Exp. SF [13]Exp. SS [13]AIMD SF [56]AIMD SF [14]AIMD SS [14,15]
FIG. 3: Phase diagram in the superionic region. SF andSS indicate VII ′′ -F solid-fluid and VII-VII ′′ solid-solid phasetransitions, respectively. Solid and dashed lines indicate theSF and SS coexistence lines according to this work (red) andan earlier AIMD simulation (green). The solid green line isan upper bound for the melting T . and of H and O in the fluid, the running O-H coordinationnumber retains a well defined shoulder at a value equalto 2, indicating that strong covalent fluctuations favor-ing neutral water molecules remain effective in presenceof ionization and breaking of the ice rules. The O sublat-tice is BCC in ice VII and VII ′′ . Thus, in ice VII, beforethe onset of H diffusion each O has 8 O nearest neighborsalong the half-diagonals of a cube, 4 of which are occu-pied by an H atom satisfying the ice rules and 4 of whichare empty. Upon heating, ice rule breaking fluctuationsoccur, in which the H atoms oscillate along a bond cre-ating OH − -OH +3 defect pairs that either rapidly recom-bine or dissociate as the defects move further apart viaGrotthuss-like mechanisms [58]. A rapid increase of theproton mobility with T follows defect pairs dissociation.This process is accompanied by partial occupation of theempty O-O bonds due to molecular rotations, which oc-cur along specific directions and are far from the freerotations hypothesized for the plastic phase. As a con-sequence, the H population of the empty bond networkincreases, that of the occupied bond network decreases,and the overall H diffusion increases. The occupationof interstitial sites outside the bonds remains negligiblethroughout. This trend continues until all the O-O bondsare equally occupied and ice VII transforms to ice VII ′′ ,a process marked by a saturation of the H diffusivity anda concomitant volume expansion due to diminished hy-drogen bonding forces. Proton diffusion is associated torapid hops along the bonds with Grotthuss like mecha-nisms not only in ice VII ′′ but also in the ionic fluid. Theaverage population of ionic defects at 30 GPa is approx-imately 7.0 percent at 1250 K in ice VII ′′ , and becomes10.8 percent at 1450 K in the fluid. Thus, full ionizationis never achieved at these pressures, in agreement withexperiment [59].In conclusion, we have shown that DP has made itpossible to predict the phase diagram of water from abinitio quantum theory, over a vast range of temperaturesand pressures. With further training the potential con-structed here could be extended to other thermodynamicconditions, including the vapor and phases at higher tem-peratures and pressures. Extensions to model solutionsand interfacial water [60–62] are also possible. Compet-ing stable and metastable phases may have free ener-gies within 1 meV/H O or less, posing a severe challengeboth to the accuracy required from the reference quan-tum model, and to the faithfulness of its neural networkrepresentation. Here we adopted the SCAN approxima-tion of DFT in view of its good balance of efficiency andaccuracy, but more accurate functional approximationsand/or higher level quantum chemical methods would bepossible, in principle. Finally, the present study was en-tirely based on classical MD simulations, but it is knownthat nuclear quantum effects are responsible for the ob-served isotopic shifts in the thermodynamic propertiesof water. These shifts are typically smaller than thedeviations from experiment of the present classical for-mulation. In future studies one can include these ef-fects using path integral MD methods, as done, e.g., inRef [25, 28, 33].
Acknowledgement
The work of H.W. is supported by the National ScienceFoundation of China under Grant No.11871110 and Bei-jing Academy of Artificial Intelligence(BAAI). We thankthe Center Chemistry in Solution and at Interfaces (CSI)funded by the DOE Award DE-SC0019394 (L.Z., R.C.and W.E), as well as a gift from iFlytek to PrincetonUniversity and the ONR grant N00014-13-1-0338 (L.Z.and W.E). ∗ Electronic address: wang [email protected] † Electronic address: [email protected][1] Percy Williams Bridgman. Water, in the liquid and fivesolid forms, under pressure.
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