# Bayesian Data Assimilation of Temperature Dependence of Solid–Liquid Interfacial Properties of Nickel

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Phase-Field Model

_{1}W

_{0}/d

_{0}is the coupling constant with a

_{1}= 5√2/8, W

_{0}is the interface thickness, and d

_{0}is the capillary length represented by d

_{0}= σ

_{0}(T

_{m}c

_{p}/ΔH

^{2}). β

_{0}is the kinetic coefficient. u

_{int}is the dimensionless undercooling at the interface defined as

_{p}is the specific heat. τ (

**n**) and W (

**n**) are relaxation time and interface width, respectively. a

_{c}(

**n**) and a

_{k}(

**n**) are functions based on the interface orientation, respectively, and are expressed as follows.

_{c}and ε

_{k}represent the strength of anisotropy of interfacial energy and the kinetic coefficient. (n

_{x}, n

_{y}) is the normal vector at the interface obtained by

## 3. Data Assimilation Based on Ensemble Kalman Filter

#### 3.1. State Vector and System Model

**x**

_{t}:

**x**

_{t}at time t from the state vector

**x**

_{t}

_{–1}at time t − 1, is defined as

**f**

_{t}is the simulation model at time t and

**v**

_{t}is the system noise representing the imperfectness of simulation model. The observation model, which represents the relationship between observation data

**y**

_{t}and the state vector

**x**

_{t}at time t, is defined as

**w**

_{t}includes measurement error and imperfectness of the simulation model.

#### 3.2. Ensemble Kalman Filter (EnKF)

#### 3.3. Calculation Procedure of Data Assimilation

_{0}, interface energy σ

_{0}, and their anisotropy parameters ε

_{k}and ε

_{c}during the growth of a single crystal under isothermal conditions were estimated by the EnKF procedure. The time evolution equation of the phase-field method (Equation (1)) was used as f

_{t}in the system model (Equation (A1)), which describes the time evolution of state vectors from t − 1 to t. Equation (1) was discretized in a standard finite different scheme with second order accuracy in the space and it was solved in an explicit Euler scheme. The calculation system was divided into 100 × 100 grid points. State variable vector

**x**

_{t}and observation vector

**y**

_{t}are given as:

_{i}

_{,j}at all lattice points (i, j) and four parameters to be estimated (β

_{0}, σ

_{0}, ε

_{k}, ε

_{c}), were used as state variables in state variable vectors. The phase-field variables at each grid point in the observation data were used as the observation variables in the observation vector. Table 1 and Table 2 show the parameters used for the phase-field simulation and for the EnKF data assimilation, respectively. We created 100 phase-field simulations using independent state vectors and optimized the simulations based on the observed data by alternately executing prediction by the system model and filtering by Equation (A6). Equation (A1) was used as a system model, where Equation (1) was employed as a nonlinear operator f

_{t}, and Equation (A2) was used as an observation model. The observation noise

**w**

_{t}was set as a random number vector generated from Gaussian distribution according to covariance matrix R

_{ϕ}which is an identity matrix in the shape of 10,000 × 10,000. Observation matrix

**H**

_{t}in the observation model is given as:

## 4. Molecular Dynamics Simulation for Observation Data

^{2}) of the MD simulation cell was divided into two-dimensional difference grids of 100 × 100. After assigning all atoms in the closest grid, the majority of local atom configurations (i.e., solid or liquid) for assigned atoms were employed as the phase-field variables of each grid point (solid: 1, liquid −1). Since this voxel structure had no interfacial thickness, it was relaxed by solving the phase-field equation without the curvature effect [73] to obtain the phase-field profile with diffuse interface. This conversion procedure was carried out for the time series of atomic configuration of MD simulation with 10 ps interval. Obtained phase-field profiles were employed as the observation data for data assimilation. The observation data between 300 and 600 ps at 10 ps interval were used in the data assimilation in following the sections to avoid the initial relaxation period of solid nucleus in the MD simulation.

## 5. Results and Discussion

_{0}, σ

_{0}, ε

_{k}, and ε

_{c}from the dataset of MD simulation at 1480 K was performed. Figure 3a shows time change of the estimated values of four parameters. Estimated values of three parameters, β

_{0}, ε

_{k}, and σ

_{0}converged to certain values with decreasing the variance. On the other hand, variance of the estimated value of ε

_{c}did not decrease during the estimation although the estimated value itself came close to a certain value. It was expected that accuracy of estimation of ε

_{c}was lower than those of the other parameters.

_{0}, ε

_{k}, and σ

_{0}converged to certain values with decreasing variance, while accuracy of estimation of ε

_{c}was again low compared to the other parameters.

_{0}, ε

_{k}, and σ

_{0}decreased with increasing temperature. The negative temperature dependence of σ

_{0}agreed with our previous estimation of the solid–liquid interfacial energy of bcc-Fe by EnKF [46] and some reports in the literature [42,43,44]. Bayesian inference derived the most probable values of solid–liquid interfacial energy at various temperatures from the results of the MD simulation without any prior knowledge. It is guaranteed that the phase-field model employed in this study reproduces the Gibbs–Thomson effect properly. Therefore, the parameters derived in this study were appropriately within the range where the Gibbs–Thomson effect is valid. One possible reason of discrepancy from some studies of positive temperature dependence may be due to the effect of interface curvature [37,41]. However, it was difficult to find the physical origin of negative temperature dependence directly from our result. The degree of temperature dependence of β

_{0}was smaller than that of σ

_{0}, Incidentally, β

_{0}was nearly independent of the temperature within the examined temperature range in our previous study for bcc-Fe [46]. Moreover, it was difficult to find a clear trend in the temperature dependence of ε

_{c}since the accuracy of estimation of ε

_{c}was lower, as described above. The estimated values of σ

_{0}ranged from 0.27 to 0.38 J/m

^{2}, which basically overlapped experimental and theoretical values of σ

_{0}for Ni at melting point (0.255 [13], 0.284 [75], 0.306 [76], and 0.325 J/m

^{2}[14]). Regarding the kinetic coefficient, β

_{0}took values between 0.0035 and 0.0037 s/m. β

_{0}can be converted into the interfacial mobility μ by the following relation, μ = c

_{p}/β

_{0}ΔH [55]. Using the values of c

_{p}and ΔH in Table 1, β

_{0}= 0.0035 m/s was converted into μ = 0.418 m/sK. This is within the range of reported values, 0.18–0.45 m/sK, which were derived from MD simulations with planar solid–liquid interfaces of Ni [77]. It is convincing that both the solid–liquid interfacial energy and interfacial mobility estimated in this study were consistent with previous reported values from various methodologies. On the other hand, temperature dependence of ε

_{k}took the opposite trend to our previous estimation of bcc-Fe, which was the positive temperature dependence. This difference might come from the difference in the strength of anisotropy. That is, a strong anisotropy appeared in the crystal structure of fcc-Ni in this study, whereas a weak anisotropy appeared in that of bcc-Fe structure [46]. Further study is needed to discuss the anisotropy in interfacial mobility.

_{c}was low compared to the other parameters in the four-parameter estimation. Therefore, ε

_{c}was separately estimated while fixing the other parameters at estimated values of β

_{0}= 0.00275 [s/m], ε

_{k}= 0.338, and σ

_{0}= 0.277 [J/m

^{2}]. For this one parameter estimation, state variable vector and the observation matrix were modified as follows.

**y**

_{t}was the same as Equation (12). The observation matrix is given as:

_{c}starting from different initial distributions at 1505 K. The other conditions were the same as those of the four-parameter estimation. Two estimations did not converge to the same value. That is, it was not successful in obtaining the converged value of the estimation for ε

_{c}, even from the procedure of one parameter estimation.

_{c}= 0.008, 0.010, and 0.012 were employed. The other parameters were the same as listed in Table 1. Figure 5b shows the phase-field profile after 30,000 step simulations. The morphologies of the structures did not change significantly with respect to ε

_{c}. Therefore, it is considered that the effect of anisotropy of solid–liquid interfacial energy on the crystal structure is very small under the condition of large undercooling temperature. In that condition, growth velocity is very fast and the anisotropy in interfacial mobility is dominant. In other words, it is difficult to estimate the parameters of a less influential factor in the framework of the present study. The anisotropy parameter ε

_{c}may be estimated when a near equilibrium structure of the crystal is employed as observation data, which will be investigated in the next step.

## 6. Conclusions

## Supplementary Materials

_{0}, interfacial energy σ

_{0}, and their anisotropy parameters, ε

_{k}and ε

_{c}) using observation data of molecular dynamics (MD) simulation at 1455 K. Figure S2: Estimation of four parameters (kinetic coefficient β

_{0}, interfacial energy σ

_{0}, and their anisotropy parameters, ε

_{k}and ε

_{c}) using observation data of molecular dynamics (MD) simulation at 1505 K. Figure S3: Estimation of four parameters (kinetic coefficient β

_{0}, interfacial energy σ

_{0}, and their anisotropy parameters, ε

_{k}and ε

_{c}) using observation data of molecular dynamics (MD) simulation at 1530 K.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mathematical Expression of Ensemble Kalman Filter

**w**

_{t}is given as Gaussian distribution with zero mean and a covariance matrix of

**R**

_{t}. In the prediction step, time evolution of states vector is calculated as

**y**

_{1:t}

_{−1}represents observational data from time t = 1 to time t − 1. In the filtering step, states vector is updated as [44,52]:

**K**

_{t}is ensemble approximation of Kalman gain, given as

**R**

_{t}are sample covariance matrixes of state vector and observation error, given as:

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**Figure 1.**Schematic image of prediction and filtering processes at time t by ensemble Kalman filter (EnKF).

**Figure 2.**Molecular dynamics simulation of the growth of a single crystal of nickel from an undercooled melt for observation data for data assimilation. (

**a**) Initial configuration of simulation system. (

**b**) Snapshots of atomic configuration during the crystal growth from the undercooled melt of ΔT = 200 K. Green and white atoms represent solid and liquid atoms, respectively, which were identified by polyhedral template matching.

**Figure 3.**Estimation of four parameters (kinetic coefficient β

_{0}, interfacial energy σ

_{0}, and their anisotropy parameters ε

_{k}and ε

_{c}) using observation data of molecular dynamics (MD) simulation at 1480 K. (

**a**) Time changes of the estimated values of four parameters β

_{0}, ε

_{k}, σ

_{0}, and ε

_{c}. (

**b**) Snapshots of observation data from the MD simulation and representative results of the estimated structure.

**Figure 4.**Temperature dependence of the estimated values of four parameters from observed data of 1455, 1480, 1505, and 1530 K. Estimated values on the last filtering step are plotted in the figure.

**Figure 5.**(

**a**) Time changes of the estimated value of the anisotropy parameter ε

_{c}in one parameter estimation starting from two initial distributions. (

**b**) Phase-field profiles from the phase-field simulations with fixed parameters.

Parameter | Symbol | Value |
---|---|---|

Grid size [m] | Δx | 9.0 × 10^{−10} |

Interface thickness [m] | W_{0} | 2.0Δx = 1.8 × 10^{−9} |

Latent heat [J/m^{3}] | ΔH | 2.83966 × 10^{9} [60] |

Constant pressure specific heat [J/(m^{3}K)] | c_{p} | 4.1578 × 10^{6} [60] |

Temperature [K] | T | 1455, 1480, 1505, 1530 |

Time step [s] | Δt | 1.0 × 10^{−14} |

Parameter | Symbol | Value |
---|---|---|

Ensemble number | − | 100 |

Filtering interval [s] | − | 1.0 × 10^{−}^{11} |

Total time [s] | − | 3.0 × 10^{−}^{10} |

System noise of ϕ | ${Q}_{\varphi}$ | 1.0 × 10^{−}^{3} |

System noise of β_{0} | ${Q}_{{\beta}_{0}}$ | 1.0 × 10^{−}^{6} |

System noise of ε_{k} | ${Q}_{{\epsilon}_{k}}$ | 1.0 × 10^{−}^{4} |

System noise of σ_{0} | ${Q}_{{\sigma}_{0}}$ | 1.0 × 10^{−}^{4} |

System noise of ε_{c} | ${Q}_{{\epsilon}_{c}}$ | 1.0 × 10^{−}^{5} |

Observation noise of ϕ | ${R}_{\varphi}$ | 1.0 |

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**MDPI and ACS Style**

Nagatsuma, Y.; Ohno, M.; Takaki, T.; Shibuta, Y.
Bayesian Data Assimilation of Temperature Dependence of Solid–Liquid Interfacial Properties of Nickel. *Nanomaterials* **2021**, *11*, 2308.
https://doi.org/10.3390/nano11092308

**AMA Style**

Nagatsuma Y, Ohno M, Takaki T, Shibuta Y.
Bayesian Data Assimilation of Temperature Dependence of Solid–Liquid Interfacial Properties of Nickel. *Nanomaterials*. 2021; 11(9):2308.
https://doi.org/10.3390/nano11092308

**Chicago/Turabian Style**

Nagatsuma, Yuhi, Munekazu Ohno, Tomohiro Takaki, and Yasushi Shibuta.
2021. "Bayesian Data Assimilation of Temperature Dependence of Solid–Liquid Interfacial Properties of Nickel" *Nanomaterials* 11, no. 9: 2308.
https://doi.org/10.3390/nano11092308