Lattice Instabilities and Phase Transformations in Fe from Atomistic Simulations

Abstract

The stability of the body- and face-centered cubic lattices corresponding to the α and γ phases of Fe, respectively, as well as the transformation of one phase to the other were investigated by atomistic simulations. Two interatomic potentials were used: the embedded atom method (EAM) potential of Meyer and Entel and the bond order potential (BOP) developed by Müller et al. The suitability of the potentials for investigating structural transformations in Fe was verified using nonequilibrium free energy calculations and molecular dynamics simulations. The results showed that the EAM potential is capable of describing the bcc → fcc and fcc → bcc transformations whereas no transformation was observed for the computationally more expensive BOP potential with the simulation set up used.

Appendix: Free Energy Calculations

Free energy computations in atomistic simulations for solids are usually calculated as the free energy difference between a solid and a reference system with known free energy. In this work, a nonequilibrium method will be used to calculate the free energy of solids described by different interatomic potentials (EAM or BOP). This method is discussed in detail in Ref 27-29 and only an outline will be given here. The starting point for all free energy calculations is the Helmholtz potential (free energy), F, which can be expressed in terms of the canonical partition function:

$$F = - kT\ln Z$$

(1)

In Eq 1 k is the Boltzmann constant, T is temperature and Z is the canonical partition function given by:

$$Z = \frac{1}{{h^{3N} }}\int \cdots \int {\exp \left[ { - \beta H(p,q)} \right]{{d}}p{{d}}q}$$

(2)

Since the determination of the partition function, Z can be done analytically only in a few simple systems, it is more useful to consider the free energy differences rather than absolute values. The free energy difference between two systems with partition functions Z and Z₀ can be written as:

$$\Delta F = F - F_{0} = - kT\ln \frac{Z}{{Z_{0} }}$$

(3)

Note that in Eq 3 the free energy difference depends on the partition function ratio and not on individual partition functions. The two systems can differ in several ways and may be defined by two different Hamiltonians (interatomic potentials). Consider now two systems defined by a Hamiltonian that depends on an external parameter λ. In this case:

$$F\left( {N,V,T,\lambda } \right) = - kT\ln Z\left( {N,V,T,\lambda } \right)$$

(4)

The derivative of F with respect to λ will give:

$$\frac{\partial F}{\partial \lambda } = - \frac{kT}{Z}\frac{\partial Z}{\partial \lambda }$$

(5)

Since the partition function, Z, is given by Eq 2, it can be shown (if the particle masses of the two systems are the same) that:

$$\Delta F = - \int_{{\lambda_{{min} } }}^{{\lambda_{{max} } }} {\left\langle {\frac{\partial H(p,q,\lambda )}{\partial \lambda }} \right\rangle }_{\lambda } {{d}}\lambda$$

(6)

In Eq 6 above <···>_λ indicates an average taken at fixed value of λ. In the most general case, H (p, q, λ) can be defined by the following equation:

$$H\left( {p,q,\lambda } \right) = \left( {1 - \lambda } \right)H\left( {p,q} \right) + \lambda H_{0} \left( {p,q} \right)$$

(7)

The most common choice for the reference system is the (an Einstein crystal). The method outlined above is called Thermodynamic Integration Method. In this method an external parameter, λ, is used to slowly change the system of interest into another with known free energy, typically a harmonic solid where each atom is fixed to its lattice position by a harmonic spring and with no intermolecular interactions. Frenkel and Ladd[30] used the procedure outlined above to calculate the free energy differences between solids using as reference a harmonic solid. This method requires a great number of simulations at several values of the coupling parameter, λ, followed by numerical integration to obtain the free energy of the system of interest. Nonequilibrium methods on the other hand, allow the determination of the free energy in a single simulation.

In nonequilibrium methods the coupling parameter is an explicit function of time λ = λ(t) and the free energy can be computed using Eq 8:

$$\Delta F = \int_{0}^{{t_{s} }} {\left\langle {\frac{{\partial H\left( {p,q,\lambda } \right)}}{\partial \lambda }} \right\rangle_{\lambda \left( t \right)} \frac{{{{d}}\lambda y}}{{{{d}}t}}{{d}}t}$$

(8)

The free energy of the harmonic crystal used in the work is calculated with the following equation:

$$\frac{{F_{0} }}{NKT} = \frac{3}{2}\left( {1 - \frac{1}{N}} \right)\ln \left( {\frac{{\beta \varLambda_{E} \varLambda^{2} }}{\pi }} \right) + \frac{1}{N}\ln \left( {\frac{{N\varLambda^{3} }}{V}} \right) - \frac{3}{2N}\ln N$$

(9)

where Λ_E is the harmonic constant of the Einstein crystal and Λ is the thermal de Broglie wavelength. Λ_E is a critical parameter and cannot be arbitrarily set. In the present work, the value of Λ_E was calculated using the following formula:[31]

$$\varLambda_{E} = \frac{3kT}{{m^{2} }}$$

(10)

where m is the mean-squared displacement at temperature T.

The adiabatic switching method was used to calculate free energy differences between the fcc and bcc phases of Fe. In the nonequilibrium runs each configuration was kept at value of λ = 0 for 0.1 ns. After this period, λ is changed continuously from 0 to 1 according to the function:

$$\lambda (\tau ) = \tau^{5} \left( {70\tau^{4} - 315\tau^{3} + 540\tau^{2} - 420\tau + 126} \right)$$

(11)

The switching is done in this is done in 0.9 ns. Then comes the second equilibration period where the system is kept in the λ = 1 state for 0.1 ns. After that, the system returns to its initial condition in 0.9 ns. The use of integration in both directions is performed to try to eliminate the dissipated heat due to nonequilibrium process and the free energy difference is calculated using an average of the forward and backward directions.