Generalization of the Unified Analytic Melt-Shear Model to Multi-Phase Materials: Molybdenum as an Example

Abstract

The unified analytic melt-shear model that we introduced a decade ago is generalized to multi-phase materials. A new scheme for calculating the values of the model parameters for both the cold ( T = 0 ) shear modulus ( G ) and the melting temperature at all densities ( ρ ) is developed. The generalized melt-shear model is applied to molybdenum, a multi-phase material with a body-centered cubic (bcc) structure at low ρ which loses its dynamical stability with increasing pressure (P) and is therefore replaced by another (dynamically stable) solid structure at high ρ . One of the candidates for the high- ρ structure of Mo is face-centered cubic (fcc). The model is compared to (i) our ab initio results on the cold shear modulus of both bcc-Mo and fcc-Mo as a function of ρ , and (ii) the available theoretical results on the melting of bcc-Mo and our own quantum molecular dynamics (QMD) simulations of one melting point of fcc-Mo. Our generalized model of G ( ρ , T ) is used to calculate the shear modulus of bcc-Mo along its principal Hugoniot. It predicts that G of bcc-Mo increases with P up to ∼240 GPa and then decreases at higher P. This behavior is intrinsic to bcc-Mo and does not require the introduction of another solid phase such as Phase II suggested by Errandonea et al. Generalized melt-shear models for Ta and W also predict an increase in G followed by a decrease along the principal Hugoniot, hence this behavior may be typical for transition metals with ambient bcc structure that dynamically destabilize at high P. Thus, we concur with the conclusion reached in several recent papers (Nguyen et al., Zhang et al., Wang et al.) that no solid-solid phase transition can be definitively inferred on the basis of sound velocity data from shock experiments on Mo. Finally, our QMD simulations support the validity of the phase diagram of Mo suggested by Zeng et al.

Ab Initio Phase Diagram of Tungsten

“…The Ref. [41], respectively, are similar. In both cases, the ambient bcc solid structure transforms into dhcp on increasing P. Only these two solid phases are confirmed as being present on the two phase diagrams, via ab initio QMD simulations using the Z methodology.…”

“…Hence, most likely, Errandonea's curve is an intermediate DAC transition boundary rather than the true melting curve of Mo. The latter has the initial slope of 34 K/GPa, and both its most recent experimental measurements [17] and theoretical calculations [41,57] agree with each other.…”

“…However, in [40] a transformation to a different solid structure, namely, dhcp, which is thermodynamically more stable than fcc, is predicted to occur, albeit at much lower pressure of 650 GPa. If a bcc-dhcp s-s phase transition does occur in W, just like in Mo [41], the two phase diagrams may look similar. As we demonstrate in what follows, this is indeed the case.…”

Tantalum strength at extreme strain rates from impact-driven Richtmyer-Meshkov instabilities

“…Because the tantalum in these experiments stays solid in the ambient bcc α phase, essentially similar behavior can also be achieved using a Mie-Grüneisen equation of state with a linear U s − U p relation. A temperature-and density-(i.e., pressure) dependent shear modulus treatment was used [55][56][57]. For reference in case such a model is not available, simpler calculations using merely the ambient-pressure temperaturedependent shear modulus caused slight changes to the sound wave speeds, which changed the spike velocities and then final strength estimate by only about 2%.…”

Using density functional theory to construct multiphase equations of state: tin as an example

“…3.13-3.15 are listed inTable 3.2. Note that arguments outlined in Ref [27]. provide an estimate for Γ 1 as case of tin corresponds to a value of 2.375.…”