Atomistic Modelling of Ti-Nb AlloysChris Ehemann and John Wilkins

The Ohio State University Department of Physics

Ti-Nb alloysTi-Nb alloys in the range of 20-30 at. % Nb can exhibit bone-matching elastic properties and excellent

biocompatibility, and are thus promising materials for biomedical implants. These alloys also form the basis

for a new class of high-strength multi-purpose alloys called gum metals. Martensitic phase transformations,

which are non-diffusive and involve the coordinated motion of many thousands of atoms, are fundamental to

the interesting shape-memory and super-elastic behavior of Ti-Nb and gum metal. We present a molecular

dynamics potential which captures the phase transformations and elastic properties of Ti-Nb alloys. This is

demonstrated through calculations of elastic anisotropy, transformation pathways, and simulations of the

shape memory effect, involving both stress-induced and temperature-induced transformations.

Figure 4: Effect of niobium on the

structure of the α martensite in Ti-Nb

alloys as calculated with the fitted

MEAM potential. (a) shows the

development of an energetic minimum

with respect to a shift of alternate

(0001)𝛼 basal planes along the [1100]𝛼direction beginning around 30 at. % Nb.

(b) shows the cell parameters γ (red) and

c/a (blue) as a function of Nb content. A

symmetric hcp structure is maintained

until 20 at. % Nb, where the symmetry is

reduced to a Bh lattice. Beyond 30 at. %

Nb, the symmetry is further reduced and

the lattice must be described as

orthorhombic. MEAM overestimates the

stability of α and α’ relative to

experiment.

Figure 5: Energy barrier for

the β to ω transition in Ti3Nb

(blue) and Ti (red inset). The

well-known β to ω mechanism

involves a collapse of alternating

pairs of {111}β planes to form the

inter-basal honeycomb ω planes,

corresponding to displacements

from a longitudinal ⅔[111]βphonon. MEAM accurately

describes the barrier in pure

titanium, but underestimates the

stabilization of the bcc phase

against the ω phonon in Ti3Nb.

Figure 6: Energy in the configuration

space of the β to α’’ transition in Ti3Nb

(top) and β to α transition in pure Ti

(bottom). Transformations proceed via

the Burger mechanism for which, as

shown to the right, ‘shuffle’ η

corresponds to a concerted displacement

of atoms in alternating (011)β planes in

the [011] β direction, while ‘shear’ ξ

corresponds to uniaxial compression

along [011] β. MEAM accurately

describes the bcc-hcp transformation in

pure titanium and Ti3Nb.

Static phase transformations

Shape memory

α’’

β

fcc

other

(100)𝛽 || (0 10)𝛼′′

[011]𝛽 (0 11)𝛽shear

(a) (b)

The Embedded-Atom Method (EAM) [1] is an interatomic potential model which parameterizes the energy of

an atom as a functional of local “electron density” of the other atoms based on the Stott-Zaremba corollary to

DFT. The Modified EAM (MEAM) [2] model includes three-body angular terms in the parameterized density.

Functions are described by cubic splines. An optimization scheme combining conjugate gradient and genetic

algorithms is used to fit the spline knot y-values to a database of DFT forces, stresses and energies including

elastic constants, high-temperature snapshots and defect structures [3].

Optimization of MEAM potential

Converged?

Conjugate gradient for each

potential in population

Objective function:Sort population by error and

determine breeding partners

Breed

INPUT:

• Initial potential

• Population size

• Breeding and mutation rates

• Database of forces, stresses

and energies

φTiTi φTiNb

𝑓TiTi 𝑓TiNb 𝑓NbNb ρNb

𝑔TiTiTi𝑔TiTiNb 𝑔TiNbNb

𝑈Ti

𝑔NbTiTi 𝑔NbTiNb 𝑔NbNbNb 𝑈Nb

ρTiφNbNb

YES

NO

ElasticityFigure 2 (below): Anisotropy of Young’s

modulus for given materials. DFT elasticity is

considerably more stiff than the gum metals,

but qualitatively similar. MEAM accurately

predicts 𝐸 111 of 23 at. % Nb gum metal, but

is considerably too soft in the 110 and 100directions. Experimental results are from [4, 5].

Figure 3 (above): BCC (β) elastic constants as a function of niobium

concentration for MEAM (lines) and DFT (points). MEAM results

are for a 2000-atom solid solution, while DFT results are from G1

and B2 supercells for 25 at. % Nb and 50 at. % Nb, respectively.

Figure 1: Schematic representation of the optimization routine, including output error and optimized MEAM functions.

References

1. M.S. Daw and M.I. Baskes, Physical Review B 29(12), 1984.

2. T. Lenosky et al., Simul. Mater. Sci. Eng. 8, 2000.

3. F. Ercolessi and J. B. Adams, Europhysics Letters 28(8), 1994

4. R. J. Talling et al., Scripta Materialia 59, 2008.

5. M. Hara et al., Ti-2007 Sci. and Eng., Niimoni et al., NKG Japan 627, 2007.

Acknowlegements

All computational resources provided by OSC. DFT calculations performed with Vienna Ab-initioSimulation Package with PAW-PBE pseudopotentials. Molecular dynamics simulationsperformed using the LAMMPS package. Structure identification performed using the adaptedcommon-neighbor analysis technique, implemented in Ovito atomistic post-processing suite.

(c)

Figure 7: 256,000-atom simulation of the shape memory effect in Ti75Nb25

using the fitted MEAM potential. (a) shows the cell in the β (blue) phase at

300 K and 0 GPa. The cell is then sheared in the [011]𝛽 (0 11)𝛽 system while

maintaining zero pressure. (b) shows domains of the α’’ (red) martensite

formed by the induced shear stress. (c) shows that a subsequent heating of the

cell to 1000 K destroys the martensite domains, returning to the β phase while

accumulating defects represented by grey atoms. (d) shows the evolution of

atomic fraction (top) and shear stress (bottom). Heating induces a reverse

transformation and relieves the shear stress.

(d)

(a) (b)

𝛽𝐿60 𝛼′′