Four-body Mistriotis-Flytzanis-Farantos (MFF) model driver v001

Amit K. Singh; Antonis D Mistriotis; Nikos Flytzanis; Stavros C. Farantos; Ellad B. Tadmor; Ronald E. Miller

doi:10.25950/bdc45775

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MFF__MD_514777050453_001

Title A single sentence description.	Four-body Mistriotis-Flytzanis-Farantos (MFF) model driver v001
Description	Four-body Mistriotis-Flytzanis-Farantos (MFF) model driver. This potential is based on a modified Stillinger-Weber form with an additional four-body term. This functional form was originally developed for silicon, where the four-body terms were necessary to obtain the correct melting temperature and the geometry and energies of small clusters. This driver supports up to two species types.
Disclaimer A statement of applicability provided by the contributor, informing users of the intended use of this KIM Item.	None

Contributor	Amit K. Singh
Maintainer	Amit K. Singh
Implementer	Amit K. Singh
Developer	Antonis D Mistriotis Nikos Flytzanis Stavros C. Farantos Ellad B. Tadmor Ronald E. Miller
Published on KIM	2018
How to Cite	This Model Driver originally published in [1-2] is archived in OpenKIM [3-5]. [1] Mistriotis AD, Flytzanis N, Farantos SC. Potential model for silicon clusters. Physical Review B. 1989Jan;39(2):1212–8. doi:10.1103/PhysRevB.39.1212 [2] Tadmor EB, Miller RE. Modeling Materials: Continuum, Atomistic and Multiscale Techniques. Cambridge University Press; 2011. [3] Singh AK, Mistriotis AD, Flytzanis N, Farantos SC, Tadmor EB, Miller RE. Four-body Mistriotis-Flytzanis-Farantos (MFF) model driver v001. OpenKIM; 2018. doi:10.25950/bdc45775 [4] Tadmor EB, Elliott RS, Sethna JP, Miller RE, Becker CA. The potential of atomistic simulations and the Knowledgebase of Interatomic Models. JOM. 2011;63(7):17. doi:10.1007/s11837-011-0102-6 [5] Elliott RS, Tadmor EB. Knowledgebase of Interatomic Models (KIM) Application Programming Interface (API). OpenKIM; 2011. doi:10.25950/ff8f563a Click here to download the above citation in BibTeX format.
Funding	Not available

Short KIM ID The unique KIM identifier code.	MD_514777050453_001
Extended KIM ID The long form of the KIM ID including a human readable prefix (100 characters max), two underscores, and the Short KIM ID. Extended KIM IDs can only contain alpha-numeric characters (letters and digits) and underscores and must begin with a letter.	MFF__MD_514777050453_001
DOI	10.25950/bdc45775 https://doi.org/10.25950/bdc45775 https://commons.datacite.org/doi.org/10.25950/bdc45775
KIM Item Type	Model Driver
KIM API Version	2.0
Programming Language(s) The programming languages used in the code and the percentage of the code written in each one.	100.00% C

Previous Version	MFF__MD_514777050453_000

Models using this Model Driver

MFF_MistriotisFlytzanisFarantos_1989_Si__MO_080526771943_001

Files

CMakeLists.txt
LICENSE.CDDL
MFF.c
README
kimprovenance.edn
kimspec.edn

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This 4-body potential is a modification of the Stillinger-Weber model originally developed for silicon, where a four-body term has been added, and the angle dependence of the 3-body and 4-body terms have been changed. These modifications result in a qualitative improvement when the model was fit to silicon. More specifically, good values for the ground state energies of clusters were obtained for a large range of the parameters of the 3-body and 4-body terms.

The functional form of this potential is:

$$
\begin{align*}
 & \phi(1,2,...,N) =\sum_{1 \leq i \leq N} V_{1}(\textbf{r}_i)+\sum_{1 \leq i \leq j \leq N} V_{2}(\textbf{r}_i,\textbf{r}_j)+\sum_{1 \leq i \leq j \leq k \leq N} V_{3}(\textbf{r}_i,\textbf{r}_j,\textbf{r}_k)+\sum_{1 \leq i \leq j \leq k \leq l \leq N} V_{4}(\textbf{r}_i,\textbf{r}_j,\textbf{r}_k,\textbf{r}_l)&
\end{align*}
$$

where

$$
\begin{align*}
&V_{1}(\textbf{r}_i)=0
\end{align*}
$$

$$
\begin{align*}
&V_{2}(\textbf{r}_i,\textbf{r}_j)=v_{2}(r_{ij})
\end{align*}
$$

$$
\begin{align*}
&v_{2}(r_{ij})=\left\{
\begin{aligned}
&A\left(B/r_{ij}^4-1\right)\exp\left[\alpha(r_{ij}-R)^{-1}\right] , r_{ij}<R \\
&0 , \text{otherwise} \\
\end{aligned}
\right.
\end{align*}
$$

$$
\begin{align*}
&V_{3}(\textbf{r}_i,\textbf{r}_j,\textbf{r}_k)=v_{3}\left(r_{ij},r_{ik},\theta_{jik}\right)+v_{3}\left(r_{ji},r_{jk},\theta_{ijk}\right)+v_{3}\left(r_{ki},r_{kj},\theta_{ikj}\right)
\end{align*}
$$

$$
\begin{align*}
&v_{3}\left(r_{ij},r_{ik},\theta_{jik}\right)=v_{3r}\left(r_{ij},r_{ik}\right)\cdot v_{3\theta}\left(\theta_{jik}\right)
\end{align*}
$$

$$
\begin{align*}
&v_{3r}\left(r_{ij},r_{ik}\right)=\left\{
\begin{aligned}
&\exp\left[\gamma(r_{ij}-R)^{-1}+\gamma(r_{ik}-R)^{-1}\right] ,  r_{ij}<R \quad   r_{ik}<R\\
&0  ,  \text{otherwise}
\end{aligned}
\right.
\end{align*}
$$

$$
\begin{align*}
&v_{3\theta}\left(\theta_{jik}\right)=\lambda_{3}\left\{1-\exp\left[-Q\left(\cos\theta_{jik}+1/3\right)^2\right]\right\}
\end{align*}
$$

$$
\begin{align*}
&V_{4}(\textbf{r}_i,\textbf{r}_j,\textbf{r}_k,\textbf{r}_l)=v_{4}\left(r_{ij},r_{ik},r_{il},\theta_{jik},\theta_{jil},\theta_{kil}\right)+v_{4}\left(r_{ji},r_{jk},r_{jl},\theta_{ijk},\theta_{ijl},\theta_{kjl}\right)+v_{4}\left(r_{ki},r_{kj},r_{kl},\theta_{ikj},\theta_{ikl},\theta_{jkl}\right)+v_{4}\left(r_{li},r_{lj},r_{lk},\theta_{ilj},\theta_{ilk},\theta_{jlk}\right)
\end{align*}
$$

$$
\begin{align*}
&v_{4}\left(r_{ij},r_{ik},r_{il},\theta_{jik},\theta_{jil},\theta_{kil}\right)=v_{4r}\left(r_{ij},r_{ik},r_{il}\right)\cdot v_{4\theta}\left(\theta_{jik},\theta_{jil},\theta_{kil}\right)
\end{align*}
$$

$$
\begin{align*}
&v_{4r}\left(r_{ij},r_{ik},r_{il}\right)=\left\{
\begin{aligned}
&\exp\left[\gamma(r_{ij}-R)^{-1}+\gamma(r_{ik}-R)^{-1}+\gamma(r_{il}-R)^{-1}\right] ,  r_{ij}<R \quad r_{ik}<R\quad r_{il}<R\\
&0  ,  \text{otherwise}
\end{aligned}
\right.
\end{align*}
$$

$$
\begin{align*}
&v_{4\theta}\left(\theta_{jik},\theta_{jil},\theta_{kil}\right)=\lambda_{4}\left\{1-\exp\left[-Q\left(\cos\theta_{jik}+1/3\right)^2-Q\left(\cos\theta_{jil}+1/3\right)^2-Q\left(\cos\theta_{kil}+1/3\right)^2\right]\right\}
\end{align*}
$$

Reference:

A. D. Mistriotis, N. Flytzanis, and S. C. Farantos, “Potential model for silicon clusters”, Phys. Rev. B, vol. 39, 1212-1218, 1989

The Tunable Intrinsic Ductility Potential (TIDP) of Rajan, Warner and Curtin is based on standard Morse potential. The ductility is tuned by altering the tail of $\varphi(r)$ while leaving the energy well unchanged. The functional form is

\[\varphi(r)= \begin{cases} (1-\exp[-\alpha(r-1)])^2-1 & r \le r_1 \\ A_1 r^3 + B_1 r^2 + C_1 r + D_1 & r_1 < r \le r_2 \\ A_2 r^3 + B_2 r^2 + C_2 r + D_2 & r_2 < r \le r_3 \\ 0 & r_3 < r \end{cases}\]

The TIDP model has 12 parameters:

\[\alpha, \quad r_1, \quad r_2, \quad r_3, \quad A_1, \quad B_1, \quad C_1, \quad D_1, \quad A_2, \quad B_2, \quad C_2, \quad D_2.\]