# New Analyzing Approaches for In Situ Interdiffusion Experiments to Determine Concentration-Dependent Diffusion Coefficients in Liquid Al–Au

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Simulation of Experiment-like Data Sets

#### 2.2. Analyzing Models

#### 2.2.1. Common Erf-Function Model

#### 2.2.2. Left/Right Erf-Function Model

#### 2.2.3. Local Erf-Function Approach

#### 2.2.4. Numerical Fick’s Law Approach

#### 2.3. In Situ Shear-Cell Experiment

## 3. Results

#### 3.1. Noiseless Simulated In Situ Data

**1st**$\mathit{D}\left(\mathit{c}\right)$ is a diffusion coefficient exemplary for a steadily increasing/decreasing concentration dependence, which is for example observed in liquid Al–Ag [38]. The

**2nd $\mathit{D}\left(\mathit{c}\right)$**has a local maximum at ≈60 at%, which was found for example in simulations of Al–Ni melts [11]. The

**2nd $\mathit{D}\left(\mathit{c}\right)$**is exemplary for a diffusion coefficient with a local minimum/maximum at a certain concentration. The

**3rd $\mathit{D}\left(\mathit{c}\right)$**one is a random distribution, that we use to show the sensitivity of the analyzing approaches and will only be presented in the evaluation of the noiseless data set.

#### 3.2. Robustness against Noise

#### 3.2.1. Noise on Local Erf-Fit

#### 3.2.2. Noise on Numerical Fick’s Law Approach

#### 3.3. Performance on Experimental Data Sets of Al–Au System

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Three concentration dependent diffusion coefficients $D\left(c\right)$ used as input for the simulation of the concentration profiles: 1st steadily increasing, 2nd with a local maximum, and 3rd completely random distribution. The order of magnitude was chosen to match other binary melts. (

**b**–

**d**) Comparison of results for the analysis of the noise-free simulated data-sets. Local Erf-fit and Numerical Fick’s Law Approach are run on $\Delta x=5$ px and $\Delta t=5$ s.

**Figure 2.**Interdiffusion simulation of a concentration profile starting from a step function of maximum concentration difference (0 and 100 at% of specimen B in A) propagating with time. The diffusion coefficient $D\left(c\right)$ ranges from 6 to 10 [$\times {10}^{-9}$ m${}^{2}$/s] (referred to 1st $D\left(c\right)$ of Figure 1).

**Figure 3.**Effect of noise on the Local Erf-fit analyzing approach. Error bars exceed the chosen scale of the graph. They have not been plotted in favor of a clearer layout. The inaccuracy due to the fluctuations in the results of the algorithm is around 5–10%, increasing with noise. However, the systematic error of the evaluation method is greater at least for the 2nd $D\left(c\right)$. The algorithm coefficient is chosen to be $\Delta x=25$ px.

**Figure 4.**Effect of noise on the Numerical Fick’s Law Approach. Error bars exceed the chosen scale of the graph. They have not been plotted in favor of a clearer layout. The inaccuracy due to the fluctuations in the results of the algorithm is around 5–40%, increasing with noise. The error is also larger for center concentrations around 50 at%. Algorithm coefficients are chosen to be $\Delta x=25$ px and $\Delta t=25$ s.

**Figure 5.**Al–Au concentration profile propagating with time measured using X-ray radiography at 1000 ${}^{\xb0}$C. The statistic noise of the data set is ∼2.0%.

**Figure 6.**Diffusion coefficients from different analyzing approaches performing on a 0/9 at% Au-content in Al experimental data set at 1000 ${}^{\xb0}$C. The error bars of the Numerical Fick’s Law Approach are of the same size, but not all were plotted in favor of a clearer layout. Reference measurements were done for smaller concentration gradients 0/3-3/6-6/9 at% Au-content in Al. The black dotted line represents an approximated concentration dependence for the diffusion coefficient in Al–Au from all the results of the analyzing algorithms.

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**MDPI and ACS Style**

Schiller, T.; Sondermann, E.; Meyer, A.
New Analyzing Approaches for In Situ Interdiffusion Experiments to Determine Concentration-Dependent Diffusion Coefficients in Liquid Al–Au. *Metals* **2021**, *11*, 1772.
https://doi.org/10.3390/met11111772

**AMA Style**

Schiller T, Sondermann E, Meyer A.
New Analyzing Approaches for In Situ Interdiffusion Experiments to Determine Concentration-Dependent Diffusion Coefficients in Liquid Al–Au. *Metals*. 2021; 11(11):1772.
https://doi.org/10.3390/met11111772

**Chicago/Turabian Style**

Schiller, Toni, Elke Sondermann, and Andreas Meyer.
2021. "New Analyzing Approaches for In Situ Interdiffusion Experiments to Determine Concentration-Dependent Diffusion Coefficients in Liquid Al–Au" *Metals* 11, no. 11: 1772.
https://doi.org/10.3390/met11111772