An analytical bond-order potential for the copper–hydrogen binary system

Abstract

Despite extensive studies in the past, deterioration of mechanical properties due to hydrogen environment exposure remains a serious problem for structural materials. More effective improvement of a material’s resilience requires advanced computational methods to elucidate the fundamental mechanisms of the hydrogen effects. To enable accurate molecular dynamics (MD) studies of the hydrogen effects on metals, we have developed a high-fidelity analytical bond-order potential (BOP) for the copper–hydrogen binary system as a representative case. This potential is available through the publically available MD code LAMMPS. The potential parameters are optimized using an iterative process. First, the potential is fitted to static and reactive properties of a variety of elemental and binary configurations including small clusters and bulk lattices (with coordination varying from 1 to 12). Then the potential is put through a series of rigorous MD simulation tests (e.g., vapor deposition and solidification) that involve chaotic initial configurations. It is demonstrated that this Cu–H BOP not only gives structural and property trends close to those seen in experiments and quantum mechanical calculations, but also predicts the correct phase transformations and chemical reactions in direct MD simulations. The correct structural evolution from chaotic initial states strongly verifies the transferability of the potential. A highly transferable potential is the reason that a well-parameterized analytical BOP can enable MD simulations of metal-hydrogen interactions to reach a fidelity level not achieved in the past.

Appendices

Appendix 1: Detailed mathematics of BOP

Equation (1) is expressed in terms of U _ij (r _ij ), V _ij (r _ij ), V _2,ij (r _ij ), Θ _ij and Θ _2,ij . U _ij (r _ij ), V _ij (r _ij ), and V _2,ij (r _ij ) are expressed in a general form as

$$ U_{ij} \left( {r_{ij} } \right) = U_{0,ij} \cdot f_{ij} \left( {r_{ij} } \right)^{{m_{ij} }} \cdot f_{{{\text{c}},ij}} \left( {r_{ij} } \right), $$

(2)

$$ V_{ij} \left( {r_{ij} } \right) = V_{0,ij} \cdot f_{ij} \left( {r_{ij} } \right)^{{n_{ij} }} \cdot f_{{{\text{c}},ij}} \left( {r_{ij} } \right), $$

(3)

$$ V_{2,ij} \left( {r_{ij} } \right) = V_{2,0,ij} \cdot f_{ij} \left( {r_{ij} } \right)^{{n_{ij} }} \cdot f_{{{\text{c}},ij}} \left( {r_{ij} } \right), $$

(4)

where U _0,ij , V _0,ij , V _2,0,ij , m _ij , and n _ij are pairwise parameters, f _ij (r _ij ) is a pair function [56], and f _c,ij (r _ij ) is a cutoff function. f _ij (r _ij ) is written as

$$ f_{ij} \left( {r_{ij} } \right) = \frac{{r_{0,ij} }}{{r_{ij} }}\exp \left[ {\left( {\frac{{r_{0,ij} }}{{r_{{{\text{c}},ij}} }}} \right)^{{n_{{{\text{c}},ij}} }} - \left( {\frac{{r_{ij} }}{{r_{{{\text{c}},ij}} }}} \right)^{{n_{{{\text{c}},ij}} }} } \right] $$

(5)

with r _0,ij , r _c,ij , and n _c,ij being pairwise parameters. The cutoff function is expressed as follows:

$$ f_{{{\text{c}},ij}} \left( {r_{ij} } \right) = \left\{ {\begin{array}{*{20}l} {\frac{{\exp \left( { - \alpha_{ij} \cdot r_{ij}^{{\gamma_{ij} }} } \right) - \exp \left( { - \alpha_{ij} \cdot r_{{{\text{cut}},ij}}^{{\gamma_{ij} }} } \right)}}{{\exp \left( { - \alpha_{ij} \cdot r_{1,ij}^{{\gamma_{ij} }} } \right) - \exp \left( { - \alpha_{ij} \cdot r_{{{\text{cut}},ij}}^{{\gamma_{ij} }} } \right)}},} \hfill & {r_{ij} < r_{{{\text{cut}},ij}} } \hfill \\ {0,} \hfill & {r_{ij} \ge r_{{{\text{cut}},ij}} } \hfill \\ \end{array} } \right., $$

(6)

where r _1,ij , r _cut,ij are independent pairwise parameters, and α _ij and γ _ij are dependent pairwise parameters that can be calculated as $\gamma_{ij} = \frac{{\ln \left[ {\ln \left( {0.99} \right)/\ln \left( {0.01} \right)} \right]}}{{\ln \left( {r_{1,ij} /r_{{{\text{cut}},ij}} } \right)}}$ and $\alpha_{ij} = - \frac{{\ln \left( {0.99} \right)}}{{\left( {r_{1,ij} } \right)^{{\gamma_{ij} }} }}$.

The local variable Θ _ij is calculated as

$$ \varTheta_{ij} = \varTheta_{{{\text{f}},ij}} \left( {\varTheta_{ij}^{{\left( {1/2} \right)}} } \right) \cdot \left[ {1 - \left( {f_{ij} - \frac{1}{2}} \right) \cdot k_{ij} \cdot \frac{{V_{ij}^{2} \left( {r_{ij} } \right) \cdot R_{ij} }}{{V_{ij}^{2} \left( {r_{ij} } \right) + \frac{{V_{ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi_{{}}^{i} + V_{ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi^{j} }}{2} + \varsigma_{2} }}} \right], $$

(7)

where $\varTheta_{ij}^{{\left( {1/2} \right)}}$, $\varPhi^{i}$, $\varPhi^{j}$, and $R_{ij}$ are also local variables, as defined below, f _ij is valence filling parameter (0 ≤ f _ij  ≤ 1), k _ij is another pairwise parameter, $\varsigma_{2}$ (and $\varsigma_{1}$, $\varsigma_{3}$, $\varsigma_{4}$ below) are small numbers designed to avoid singularities of the functions, and $\varTheta_{{{\text{f}},ij}}$ is the valence shell filling function. $\varTheta_{{{\text{f}},ij}}$ (as a function of $\varTheta_{ij}^{{\left( {1/2} \right)}}$) is defined as follows:

$$ \varTheta_{{{\text{f}},ij}} \left( {\varTheta_{ij}^{{\left( {1/2} \right)}} } \right) = \frac{{\varTheta_{0} + \varTheta_{1} + S \cdot \varTheta_{ij}^{{\left( {1/2} \right)}} - \sqrt {\left( {\varTheta_{0} + \varTheta_{1} + S \cdot \varTheta_{ij}^{{\left( {1/2} \right)}} } \right)^{2} - 4\left( { - \varepsilon \sqrt {1 + S^{2} } + \varTheta_{0} \cdot \varTheta_{1} + S \cdot \varTheta_{1} \cdot \varTheta_{ij}^{{\left( {1/2} \right)}} } \right)} }}{2}, $$

(8)

where

$$ \left\{ {\begin{array}{*{20}l} {\varepsilon = 10^{ - 10} } \hfill \\ {\varTheta_{0} = 15.737980 \cdot \left( {\frac{1}{2} - \left| {f_{ij} - \frac{1}{2}} \right|} \right)^{1.137622} \cdot \left| {f_{ij} - \frac{1}{2}} \right|^{2.087779} } \hfill \\ {S = 1.033201 \cdot \left\{ {1 - \exp \left[ { - 22.180680 \cdot \left( {\frac{1}{2} - \left| {f_{ij} - \frac{1}{2}} \right|} \right)^{2.689731} } \right]} \right\}} \hfill \\ {\varTheta_{1} = 2 \cdot \left( {\frac{1}{2} - \left| {f_{ij} - \frac{1}{2}} \right|} \right)} \hfill \\ \end{array} .} \right. $$

(9)

When Θ _2,ij is set to zero, the valence shell filling function $\varTheta_{f,ij} \left( {\varTheta_{ij}^{{\left( {1/2} \right)}} } \right)$ allows the potential to be adjusted for different types of elements. To examine this, $\varTheta_{{{\text{f}},ij}} \left( {\varTheta_{ij}^{{\left( {1/2} \right)}} } \right)$ is plotted as a function of $\varTheta_{ij}^{{\left( {1/2} \right)}}$ at different valence shell filling parameters f _ij in Fig. 11. It can be seen that when the filling parameter f _ij  = 0.5, $\varTheta_{{{\text{f}},ij}} \left( {\varTheta_{ij}^{{\left( {1/2} \right)}} } \right)$ reduces to $\varTheta_{ij}^{{\left( {1/2} \right)}}$. This means that the potential is equivalent to the original model derived for half-full covalent systems. When f _ij  = 0.0, $\varTheta_{{{\text{f}},ij}} \left( {\varTheta_{ij}^{{\left( {1/2} \right)}} } \right)$ = 0. This means that Θ _1,ij  = 0 according to Eq. (7), which in turn means that the BOP reduces to a repulsive interaction suitable for inert elements. When f _ij is near 0.1, $\varTheta_{{{\text{f}},ij}} \left( {\varTheta_{ij}^{{\left( {1/2} \right)}} } \right)$ is near constant. This means that the BOP reduces to a pair potential that tends to predict the lowest energy for closely packed structures. For other f _ij values, $\varTheta_{{{\text{f}},ij}} \left( {\varTheta_{ij}^{{\left( {1/2} \right)}} } \right)$ accounts for local environment effects including the angular dependence, which is similar to ideas used within MEAM type potentials.

Fig. 11

$\varTheta_{f,ij} \left( {\varTheta_{ij}^{{\left( {1/2} \right)}} } \right)$ as a function of $\varTheta_{ij}^{{\left( {1/2} \right)}}$ at a different valence shell filling parameter f _ij

The local variable $\varTheta_{ij}^{{\left( {1/2} \right)}}$ is calculated as

$$ \varTheta_{ij}^{{\left( {1/2} \right)}} = \frac{{V_{ij} \left( {r_{ij} } \right)}}{{\sqrt {V_{ij}^{2} \left( {r_{ij} } \right) + c_{ij} \cdot \left[ {V_{ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi^{i} + V_{ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi^{j} } \right] + \varsigma_{1} } }}, $$

(10)

where c _ij is a pairwise parameter. The $\varPhi^{i}$ and $\varPhi^{j}$ terms used in Eqs. (7) and (10) have the same formulation except that they are evaluated at the center of atom i and atom j, respectively. In addition, Eqs. (7) and (10) only require $V_{ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi^{i}$ and $V_{ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi^{j}$. Correspondingly, only $V_{ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi^{i}$ is given as

$$ V_{ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi^{i} = \sum\limits_{\begin{subarray}{l} k = i_{1} \\ k \ne j \end{subarray} }^{{i_{N} }} {g_{jik}^{2} \left( {\theta_{jik} } \right)} \cdot V_{ik}^{2} \left( {r_{ik} } \right), $$

(11)

where θ _jik is the bond angle at atom i spanning atoms j and k, and the three-body angular function g _jik (θ _jik ) is written as

$$ g_{jik} \left( {\theta_{jik} } \right) = \sum\limits_{n = 0}^{7} {g_{n,jik} \cdot \left( {\cos \theta_{jik} } \right)^{n} } , $$

(12)

where g _n,jik , (n = 0, 1, 2, …, 7) are 8 three-body-dependent parameters.

To perform calculations using Eq. (7), the product $V_{ij}^{2} \left( {r_{ij} } \right) \cdot R_{ij}$ is required. This is expressed as

$$ V_{ij}^{2} \left( {r_{ij} } \right) \cdot R_{ij} = \sum\limits_{\begin{subarray}{l} k = i_{1} \\ k,j = n \end{subarray} }^{{i_{N} }} {g_{jik} \left( {\theta_{jik} } \right) \cdot g_{ijk} \left( {\theta_{ijk} } \right) \cdot g_{ikj} \left( {\theta_{ikj} } \right) \cdot } V_{ik} \left( {r_{ik} } \right) \cdot V_{jk} \left( {r_{jk} } \right), $$

(13)

where k, j = n in the summation indicates that k and j are neighbors.

The local variable $\varTheta_{2,ij}$ is evaluated as

$$ \varTheta_{2,ij} = \frac{{a_{2,ij} \cdot V_{2,ij} \left( {r_{ij} } \right)}}{{\sqrt {V_{2,ij}^{2} \left( {r_{ij} } \right) + c_{2,ij} \cdot \left( {\frac{{V_{2,ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi_{2,ij}^{i} + V_{2,ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi_{2,ij}^{j} }}{2} + \sqrt {V_{2,ij}^{4} \left( {r_{ij} } \right) \cdot \varPhi_{4,ij} + \varsigma_{3} } } \right) + \varsigma_{4} } }} + \frac{{a_{2,ij} \cdot V_{2,ij} \left( {r_{ij} } \right)}}{{\sqrt {V_{2,ij}^{2} \left( {r_{ij} } \right) + c_{2,ij} \cdot \left( {\frac{{V_{2,ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi_{2,ij}^{i} + V_{2,ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi_{2,ij}^{j} }}{2} - \sqrt {V_{2,ij}^{4} \left( {r_{ij} } \right) \cdot \varPhi_{4,ij} + \varsigma_{3} } + \sqrt {\varsigma_{3} } } \right) + \varsigma_{4} } }}, $$

(14)

where $c_{2,ij}$ and $a_{2,ij}$ are pairwise parameters, and $\varPhi_{2,ij}^{i}$, $\varPhi_{2,ij}^{j}$, and $\varPhi_{4,ij}$ are additional local variables. $\varPhi_{2,ij}^{i}$ and $\varPhi_{2,ij}^{j}$ have the same expression except that they are evaluated at different atoms. Equation (14) can be calculated if expressions of $V_{2,ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi_{2,ij}^{i}$ and $V_{2,ij}^{4} \left( {r_{ij} } \right) \cdot \varPhi_{4,ij}$ are known. $V_{2,ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi_{2,ij}^{i}$ can be calculated as follows:

$$ V_{2,ij}^{2} \left( {r_{ij} } \right) \cdot \varPhi_{2,ij}^{i} = \sum\limits_{\begin{subarray}{l} k = i_{1} \\ k \ne j \end{subarray} }^{{i_{N} }} {\left[ {p_{i} \cdot V_{ik}^{2} \left( {r_{ik} } \right) \cdot \sin^{2} \theta_{jik} + \left( {1 + \cos^{2} \theta_{jik} } \right) \cdot V_{2,ik}^{2} \left( {r_{ik} } \right)} \right]} , $$

(15)

where $p_{i}$ is a species-dependent parameter. $V_{2,ij}^{4} \left( {r_{ij} } \right) \cdot \varPhi_{4,ij}$ is expressed as

$$ \begin{aligned} V_{2,ij}^{4} \left( {r_{ij} } \right) \cdot \varPhi_{4,ij} & = \frac{1}{4}\sum\limits_{\begin{subarray}{l} k = i_{1} \\ k \ne j \end{subarray} }^{{i_{N} }} {\sin^{4} \theta_{jik} \cdot \hat{V}_{ik}^{4} \left( {r_{ik} } \right)} + \frac{1}{4}\sum\limits_{\begin{subarray}{l} k = j_{1} \\ k \ne i \end{subarray} }^{{j_{N} }} {\sin^{4} \theta_{ijk} \cdot \hat{V}_{jk}^{4} \left( {r_{jk} } \right)} \\ & \quad + \frac{1}{2}\sum\limits_{\begin{subarray}{l} k = i_{1} \\ k \ne j \end{subarray} }^{{i_{N} }} {\sum\limits_{\begin{subarray}{l} k' = k + 1 \\ k' \ne j \end{subarray} }^{{i_{N} }} {\sin^{2} \theta_{jik} \cdot \sin^{2} \theta_{jik'} \cdot \hat{V}_{ik}^{2} \left( {r_{ik} } \right) \cdot \hat{V}_{ik'}^{2} \left( {r_{ik'} } \right) \cdot \cos \left( {\Delta \psi_{kk'} } \right)} } \\ & \quad + \frac{1}{2}\sum\limits_{\begin{subarray}{l} k = j_{1} \\ k \ne i \end{subarray} }^{{j_{N} }} {\sum\limits_{\begin{subarray}{l} k' = k + 1 \\ k' \ne i \end{subarray} }^{{j_{N} }} {\sin^{2} \theta_{ijk} \cdot \sin^{2} \theta_{ijk'} \cdot \hat{V}_{jk}^{2} \left( {r_{jk} } \right) \cdot \hat{V}_{jk'}^{2} \left( {r_{jk'} } \right) \cdot \cos \left( {\Delta \psi_{kk'} } \right)} } \\ & \quad + \frac{1}{2}\sum\limits_{\begin{subarray}{l} k' = i_{1} \\ k' \ne j \end{subarray} }^{{i_{N} }} {\sum\limits_{\begin{subarray}{l} k = j_{1} \\ k \ne i \end{subarray} }^{{j_{N} }} {\sin^{2} \theta_{jik'} \cdot \sin^{2} \theta_{ijk} \cdot \hat{V}_{ik'}^{2} \left( {r_{ik'} } \right) \cdot \hat{V}_{jk}^{2} \left( {r_{jk} } \right) \cdot \cos \left( {\Delta \psi_{kk'} } \right)} } , \\ \end{aligned} $$

(16)

where

$$ \hat{V}_{ik}^{2} \left( {r_{ik} } \right) = p_{i} \cdot V_{ik}^{2} \left( {r_{ik} } \right) - V_{2,ik}^{2} \left( {r_{ik} } \right) $$

(17)

and $\Delta \psi_{kk'}$ defines a dihedral angle by the four atoms i, j, k, k′, which can be calculated as

$$ \cos \left( {\Delta \psi_{kk'} } \right) = \frac{{2\left( {\cos \theta_{kik'} - \cos \theta_{jik'} \cdot \cos \theta_{jik} } \right)^{2} }}{{\sin^{2} \theta_{jik} \cdot \sin^{2} \theta_{jik'} }} - 1,\quad {\text{or}}\quad \frac{{2\left( {\frac{{\overrightarrow {ik'} \cdot \overrightarrow {jk} }}{{\left| {\overrightarrow {ik'} } \right| \cdot \left| {\overrightarrow {jk} } \right|}} + \cos \theta_{ijk} \cdot \cos \theta_{jik'} } \right)^{2} }}{{\sin^{2} \theta_{ijk} \cdot \sin^{2} \theta_{jik'} }} - 1. $$

(18)

Equations (1)–(18) fully define the BOP.

Appendix 2: Parameter bounds

The parameters are bounded within physical ranges during parameterizations and these constraints are listed in Table 5 in three groups representing parameterizations of Cu, H, and Cu–H, respectively.

Table 5 Bounds on BOP parameters

Appendix 3: Numerical values of cohesive energies and atomic volumes of various Cu–H structures

See Table 6 in appendix.

Table 6 Cohesive energies E _c (eV/atom) and atomic volumes Ω (ų/atom) of various Cu–H structures