Model Generalized Pseudopotential Theory Research Articles

Efficient wide-range calculation of free energies in solids and liquids using reversible-scaling molecular dynamics

We elaborate a novel and efficient method to obtain multiphase Helmholtz free energies from molecular dynamics (MD) simulations over wide ranges of volume and temperature in materials that can be described by temperature-independent ion forces, with both higher accuracy and order-of-magnitude cost savings compared to direct thermodynamic-integration techniques. Our method leverages and significantly extends the technique of reversible-scaling molecular dynamics (RSMD) proposed by de Koning et al. [Phys. Rev. Lett. 83, 3973 (1999)], which allows a free-energy difference in a given phase at constant volume to be calculated as a function of temperature from a single MD simulation. In mechanically stable solid phases, our approach carefully combines quasiharmonic lattice dynamics at low temperatures with an accurate and fully isolated RSMD simulation of the anharmonic vibrational free energy at high temperatures to produce a seamless free energy from zero temperature to above melt along constant-volume isochores. In the liquid, we combine a unique calculation of the free energy along a high-temperature reference isotherm with isochoric RSMD simulations from that temperature to below melt. In metastable solid phases that are mechanically unstable at low temperature, we use two-phase MD melt simulations together with the liquid free energy to obtain the solid free energy along the solidus melt line and then perform isochoric RSMD simulations to temperatures above and below that point. While our free-energy method is general, we have specifically adapted it here to the case of metals in which the ion forces are well described by model generalized pseudopotential theory (MGPT) multi-ion interatomic potentials, and additive electron-thermal free-energy contributions can be included. Then using refined Ta6.8$x$ MGPT potentials, we have converged total free energies and their components to very high and unprecedented sub-milli-Rydberg (mRy) numerical accuracy in the stable-bcc, liquid, and metastable-fcc phases of tantalum for volumes ranging from up to 26% expansion to nearly twofold compression and for temperatures to 25 000 K. In turn, we have successfully used the free energies so obtained to calculate physically accurate thermodynamic properties and gain new insight into their behavior, including sensitive thermodynamic derivatives, bcc and fcc melt curves, and a multiphase equation of state for tantalum (Ta) over the same temperature range and for pressures as high as 600 GPa. We show that the anharmonic free-energy component in the bcc solid, although only 1--5 mRy in magnitude for Ta, can have a significant (15%--20%) effect on thermal expansivity, the Gr\uneisen parameter, and melt temperatures. We further show that the electron-thermal free-energy component can similarly impact the specific heat and thermal expansivity in both the solid and the liquid, while only minimally affecting (to$\ensuremath{\le}3$%) the bcc and fcc melt curves.

Polymorphism and melt in high-pressure tantalum

Recent small-cell (150 atom) quantum molecular dynamics (QMD) simulations for Ta based on density functional theory (DFT) have predicted a hexagonal \ensuremath{\omega} (hex-\ensuremath{\omega}) phase more stable than the normal bcc phase at high temperature ($T$) and pressure ($P$) above 70 GPa [Burakovsky et al., Phys. Rev. Lett. 104, 255702 (2010)]. Here we examine possible high-$T,P$ polymorphism in Ta with complementary DFT-based model generalized pseudopotential theory (MGPT) multi-ion interatomic potentials, which allow accurate treatment of much larger system sizes (up to \ensuremath{\sim}80 000 atoms). We focus on candidate bcc, A15, fcc, hcp, and hex-\ensuremath{\omega} phases for the high-$T,P$ phase diagram to 420 GPa, studying the mechanical and relative thermodynamic stability of these phases for both small and large computational cells. Our MGPT potentials fully capture the $T=0$ DFT energetics of these phases, while MGPT-MD simulations demonstrate that the higher-energy fcc, hcp, and hex-\ensuremath{\omega} structures are only mechanically stabilized at high temperature by large, size-dependent, anharmonic vibrational effects, with the stability of the hex-\ensuremath{\omega} phase also being found to be a sensitive function of its $c/a$ ratio. Both two-phase and Z-method melting techniques have been used in MGPT-MD simulations to determine relative phase stability and its size dependence. In the large-cell limit, the two-phase method yields accurate equilibrium melt curves for all five phases, with bcc producing the highest melt temperatures at all pressures and hence being the most stable phase of those considered. The two-phase bcc melt curve is also in good agreement with dynamic experimental data as well as with the MGPT melt curve calculated from bcc and liquid free energies. In contrast, we find that the Z method produces only an upper bound to the equilibrium melt curve in the large-cell limit. For the bcc and hex-\ensuremath{\omega} structures, however, this is a close upper bound within 5$%$ of the two-phase results, although for the A15, fcc, and hcp structures, the Z-melt curves are 25$%$--35$%$ higher in temperature than the two-phase results. Nonetheless, the Z method has allowed us to study melt size effects in detail. We find these effects to be either small or modest for the cubic bcc, A15, and fcc structures, but to have a large impact on the hexagonal hcp and hex-\ensuremath{\omega} melt curves, which are dramatically pushed above that of bcc for simulation cells less than 150 atoms. The melt size effects are driven by and closely correlated with similar size effects on the mechanical stability and the vibrational anharmonicity. We further show that for the same simulation cell sizes and choice of $c/a$ ratio, the MGPT-MD bcc and hex-\ensuremath{\omega} melt curves are in good agreement with the QMD results, so the QMD prediction is confirmed in the small-cell limit. But in the large-cell limit, the MGPT-MD hex-\ensuremath{\omega} melt curve is always lowered below that of bcc for any choice of $c/a$, so bcc is the most stable phase. We conclude that for the non-bcc Ta phases studied, one requires simulation cells of at least 250--500 atoms to be free of size effects impacting mechanical and thermodynamic phase stability.

Robust quantum-based interatomic potentials for multiscale modeling in transition metals

First-principles generalized pseudopotential theory (GPT) provides a fundamental basis for transferable multi-ion interatomic potentials in transition metals and alloys within density-functional quantum mechanics. In the central body-centered cubic (bcc) metals, where multi-ion angular forces are important to materials properties, simplified model GPT (MGPT) potentials have been developed based on canonical d bands to allow analytic forms and large-scale atomistic simulations. Robust, advanced-generation MGPT potentials have now been obtained for Ta and Mo and successfully applied to a wide range of structural, thermodynamic, defect, and mechanical properties at both ambient and extreme conditions. Selected applications to multiscale modeling discussed here include dislocation core structure and mobility, atomistically informed dislocation dynamics simulations of plasticity, and thermoelasticity and high-pressure strength modeling. Recent algorithm improvements have provided a more general matrix representation of MGPT beyond canonical bands, allowing improved accuracy and extension to f-electron actinide metals, an order of magnitude increase in computational speed for dynamic simulations, and the development of temperature-dependent potentials.

Kink-pair mechanisms for a/2 〈111〉 screw dislocation motion in bcc tantalum

The structure, energetics, and multiplicity of kinks on an a/2 〈111〉 screw dislocation in bcc Ta have been studied in detail via atomistic computer simulation, using quantum-based multi-ion interatomic potentials derived from model generalized pseudopotential theory (MGPT) together with a robust Green's function simulation technique. The stable core structure of the rigid screw dislocation is predicted to be weakly polarized and spread out on three {110} planes in 〈112〉 directions, with two energetically equivalent configurations. This double degeneracy leads to the possibility of anti-phase defects forming on the dislocation line as well multiple kinks and kink pairs. The zero-stress formation energies of 16 possible kink-pair configurations for bcc Ta have been calculated and are in the range 0.67–1.84 eV. The lowest kink-pair energy of the perfect screw is in good agreement with the best current empirical estimate. Under an applied stress, the corresponding kink–kink interaction energy displays a λ −1 elastic attraction when the separation λ is larger than 7 b=20 Å, while the stress needed to maintain the kink pair varies as λ −1.5.

Atomistic simulation of pressure-dependent screw dislocation properties in bcc tantalum

The pressure-dependent core properties of a/2 〈1 1 1〉 screw dislocations in bcc Ta have been simulated by means of quantum-based, multi-ion interatomic potentials derived from model generalized pseudopotential theory (MGPT). We have considered both ambient-pressure behavior and the change in core properties under high-pressure of 10 Mbar. Robust two- and three-dimensional Green’s function techniques have been used to dynamically relax the boundary forces during the simulations. Our results show that the predicted degenerate core structure is sensitive to pressure, being nearly symmetric and only weakly polarized at ambient pressure, while becoming highly polarized with a distinctive three-fold spread out along 〈1 1 2〉 directions at high-pressure. At the same time, the pure-shear Peierls stress is found to scale linearly with the pressure-dependent 〈1 1 1〉 shear modulus G. The degenerate nature of the core structure leads to multiple possible kinks and kink pairs that can form on the dislocation line at finite temperature. The pressure variation of the lowest kink-pair formation energy reveals a very approximate scaling behavior in terms of G and the Burgers vector.

Accurate atomistic simulation of (a/2) 〈111〉 screw dislocations and other defects in bcc tantalum

Abstract The fundamental atomic-level properties of (a/2)(111) screw dislocations and other defects in bcc Ta have been simulated by means of new quantum-based multi-ion interatomic potentials derived from the model generalized pseudopotential theory (MGPT). The potentials have been validated in detail using a combination of experimental data and ab-initio electronic structure calculations on ideal shear strength, vacancy and self-interstitial formation and migration energies, grain-boundary atomic structure and generalized stacking-fault energy (γ) surfaces. Robust and accurate two- and three-dimensional Green's function (GF) techniques have been used to relax dynamically the boundary forces during the dislocation simulations. The GF techniques have been implemented in combination with a spatial domain decomposition strategy, resulting in a parallel MGPT atomistic simulation code that increases computational performance by two orders of magnitude. Our dislocation simulations predict a degenerate core str...

Atomic structure of the Σ5 (310)/[001] symmetric tilt grain boundary in molybdenum

Atomistic simulations offer an important route towards understanding and modeling materials behavior. Incorporating the essential physics into the models of interatomic interactions is increasingly difficult as materials with more complex electronic structures than f.c.c. transition metals are addressed. For b.c.c. metals, interatomic potentials have been developed that incorporate angularly dependent interactions to accommodate the physics of partially filled d-bands. A good test of these new models is to predict the structure of crystal defects and compare them with experimentally observed defect structures. To that end, the Σ5 (310)/[001] symmetric tilt grain boundary in Mo has been fabricated and characterized by HREM. The experimentally observed structure is found to agree with predictions based on atomistic simulations using angular-force interatomic potentials developed from model generalized pseudopotential theory (MGPT), but disagrees with predictions based on radial-force potentials, such as those obtained from the Finnis–Sinclair method or the embedded atom method (EAM).

High-resolution electron microscopy investigation of the (710) twin in Nb

High-resolution transmission electron microscopy has been used to evaluate theoretical predictions of the atomic structure of a symmetric 16.3° tilt grain boundary with [001] tilt axes which forms a twin about (710) in Nb, a body-centered cubic metal. The boundary has been fabricated by diffusion-bonding single crystals with flat, polished (710) surfaces misoriented by a 180° rotation in ultrahigh vacuum. High-resolution electron microscopy has been performed on the interface along the common [001] direction. Images were recorded at four defocus conditions which gave strong contrast of the crossed {110} fringes in the bulk crystal on either side of the boundary. Models of the grain boundary atomic structure were predicted using interatomic potentials derived using the embedded atom method (EAM) and the model generalized pseudopotential theory (MGPT). The EAM predicts a multiplicity of structures differing by relative translations of the adjacent crystals, while the MGPT predicts only one mirror-symmetric grain-boundary structure. The theoretically predicted structures have been compared with the high-resolution images through image simulation. The identification of focus conditions has been aided by Fourier analysis of the amorphous edge of the specimen and comparison with calculated contrast transfer functions. The boundary was experimentally observed to have mirror symmetry to within relative crystal translations of ±0.02 nm as viewed along the tilt axis, hence most of the structures predicted by the EAM can be ruled out. The angular-dependent interactions modeled in the MGPT thus appear to be important in determining the grain-boundary structure of niobium. But unambiguous structure determination is presently limited by the means available to compare simulated with experimental images, the limited resolution of the microscope, imperfect bicrystal orientation, and the inability to distinguish atom positions in the direction parallel to the electron beam in very thin specimens.

Atomic structure of the (310) twin in niobium: Experimental determination and comparison with theoretical predictions.

The atomic structure of the (310) twin in Nb was predicted using interatomic potentials derived from the embedded atom method (EAM), Finnis-Sinclair theory (FS), and the model generalized pseudopotential theory (MGPT). The EAM and FS predicted structures with crystal translations which break mirror symmetry. The MGPT predicted one stable structure which possessed mirror symmetry. This defect was experimentally determined to have mirror symmetry. These findings emphasize that the angular dependent interactions modeled by the MGPT are important for determining defect structures in bcc transition metals.

Atomic Structure of the (310) Twin in Niobium: Theoretical Predictions and Comparison with Experimental Observation

AbstractHigh - resolution transmission electron microscopy (HREM) has been used to characterize the atomic structure of the symmetric 36.9° tilt grain boundary with [001] tilt axes forming a twin about (310) in Nb. The projected structure was imaged along two different directions in the plane of the boundary and was compared to model structures through high - resolution image simulation. The atomic structure of this Σ5 boundary was predicted with atomistic simulations using interatomic potentials derived from the Embedded Atom Method (EAM), Finnis-Sinclair (FS), and the Model Generalized Pseudopotential Theory (MGPT). The EAM and FS predicted structures with translations of the adjacent crystals which break mirror symmetry. The MGPT predicted one stable structure with mirror symmetry. The atomic structure of the (310) twin in Nb was found by HREM to be mirror symmetric. These findings indicate that the angular dependent interactions modeled in the MGPT are important for determining the grain boundary structures of bcc transition metals.

Hrem Investigation of the Structure of the Σ5(210)/[001] Symmetric Tilt Grain Boundaries in Nb.

AbstractWe have simulated the atomic structures of the Σ 5 (210)/[001 ] symmetric tilt grain boundary using interatomic potentials for Nb developed employing the embedded atom method (EAM) and the model generalized pseudopotential theory (MGPT). These potentials do not predict the same lowest energy structure for the Σ 5 (210)/[001]. Using the ultra high vacuum diffusion bonding process, we have fabricated Σ 5 (210)/[001] bicrystals. The samples have been observed using high resolution electron microscopy and the observed images have been compared with those simulated based on the structures predicted theoretically. The experimental result for the Σ 5 (210)/[001] is in close agreement with the structure predicted using the EAM.

Hrem Investigation of the Structure of the σ5(310)/[001] Symmetric Tilt Grain Boundary In Nb.*

ABSTRACTRecent atomistic simulations using interatomic potentials for Nb developed employing the embedded atom method (EAM) and the model generalized pseudopotential theory (MGPT) have indicated a possible cusp at the Σ5(310) orientation in the energy vs tilt angle curves for<001> symmetric tilt grain boundaries. In addition, the most stable structure predicted using EAM exhibits shifts of one crystal relative to the other along the tilt axis and along the direction perpendicular to the tilt axis lying in the boundary plane. The structure predicted using the MGPT was mirror symmetric across the plane of the grain boundary. This boundary has been prepared for experimental study using the ultra high vacuum diffusion bonding method. A segment of this boundary has been studied using high resolution electron microscopy.