Interdiffusion in the Ni-Re System: Evaluation of Uncertainties

Abstract

Diffusion couple experiments between Ni and Re at 1200 and 1350 °C were performed. These experiments established the limits of the two-phase FCC + HCP region. No intermediate phase was observed at these temperatures. Composition-dependent interdiffusion coefficients and associated uncertainties were estimated by three methods. The first employed fitting of the penetration curves in conjunction with the Sauer-Freise (SF) method. The second method employed a numerical solution of the Boltzmann-Matano ordinary differential equation for composition-dependent interdiffusion coefficient functions whose parameters were optimized by a least squares fitting to the data. Discrepancies between the results of these methods indicate typical uncertainties in experimental determination of diffusion coefficients. To further assess such discrepancies, a third method was employed to perform an uncertainty quantification of the diffusion coefficients via a statistical analysis based on the SF method.

Appendices

Appendix 1: The ODE Method

Given an estimate for the diffusion coefficient $\tilde{D}\left( {x_{\text{Re}} } \right)$, the ordinary differential equation (ODE) (4) is solved[18,19] by a shooting method. Since the diffusivity is nonlinear and can vary by over an order of magnitude, the ODE is stiff and care must be exercised in the numerical procedure.

Starting from provisional values for $C(\xi_{1} )$ and its derivative $C'(\xi_{1} )$ at a given point $\xi_{1}$ in the FCC phase, the ODE is integrated to the left hand endpoint $\xi = \xi_{L}$ and the resulting concentration $C_{L} = C(\xi_{L} )$ is recorded. The ODE is also integrated rightward from $\xi_{1}$ until the phase boundary at $\xi = \xi_{\text{phase}}$ is detected, at which point the jump conditions are applied and the integration is resumed until the right hand endpoint is reached at $\xi = \xi_{R}$. The resulting values of $C_{L}$ and $C_{R}$ are then compared to the desired end conditions $c_{\text{Re}}^{ \pm }$, and the process is repeated by iterating on the values of $C(\xi_{1} )$ and $C'(\xi_{1} )$ until convergence is obtained. A good starting guess is required for $C(\xi_{1} )$ and $C'(\xi_{1} )$, and the steps taken by the root solver must be small enough that the ODE solver does not fail via solution blow-up from a bad guess. Suitable starting guesses can be obtained by continuation from a previous solution or an analytical solution for constant $\tilde{D}$.

The resulting composition profiles are compared with the measured profile at the experimental data points, and the least square error between the profiles is used to iterate on $\tilde{D}$ until convergence is obtained. Again it is necessary to control the size of the correction to $\tilde{D}$ in order to avoid blow-up of the ODE solution. In practice, the initial guesses for $\tilde{D}$ can be obtained from a previous solution or an analytical solution, and small trial-and-error changes to $\tilde{D}$ suffice to obtain an approximate solution. An optimization routine[18,19] can then be employed to determine a final solution, again being careful to avoid large steps by the optimizer that can lead to solution blow-up.

The solution to the ODE is smooth in each phase, and the fit to the experimental data generates a least square error that is comparable to that obtained by the purely functional fits used in the Sauer-Freise and UQ methods as shown by Fig. 4 and 5. The error is dominated by the scatter in the experimental data.

The compositions at the phase boundaries are prescribed during this procedure and are not determined self-consistently by the iteration procedure (although the location of the phase boundary itself is an output of the procedure). The reason for prescribing the compositions in advance is that the least square error in the fit is not sensitive to these values once they are given to the precision of the scatter in the data. Instead, provisional values of the phase boundary compositions are determined by visual inspection of the raw data, and the resulting fit to the data is examined in the locality of the phase boundary. Small changes to the prescribed phase boundary compositions that improve the local fit are then applied, and the overall procedure is repeated. The possible ranges of the phase boundary compositions that do not change the least square error can be taken as a measure of their uncertainty.

Finally, there is an additional complication that is particular to the two-phase problem. If the current numerical solution computes an interface location that is one or more mesh points distant from the location indicated by the experimental data, there is an associated local error in the fit that is of order unity. On the other hand, if the computed and experimental phase boundaries occur at the same mesh point, this component of the error can be significantly smaller. The computed interface position is most sensitive to the value of the Matano position $z_{0}$, but also depends on the diffusion parameters $A_{j}$ and $B_{j}$. As an optimizer scans parameter space in search of an accurate fit, the computed error in the fit can appear to change discontinuously as the numerical solution goes into and out of alignment with the experimental position of the phase boundary, which can significantly interfere with the optimizer’s smoothness assumptions. It is primarily this feature of the two-phase couple that prevents the success of an automated global optimization approach for the ODE method for this two phase diffusion couple.

Appendix 2: The UQ Method

In the uncertainty quantification and machine learning communities, Gaussian-processes (GPs) are well known tools for modeling noise and uncertainty in data. As there are many references that provide comprehensive reviews of the subject, we only introduce the main ideas and highlight issues that are relevant for constructing synthetic datasets.[17,20,21]

To begin, let ${\mathbf{z}} = \left( {z_{1} , \ldots ,z_{N} } \right)$ be a vector of the N positions at which we know the residuals ${\mathbf{r}}\left( {\mathbf{z}} \right) = \left( {r\left( {z_1} \right), \ldots ,r\left( {z_N} \right)} \right)$; see Fig. 4 and 5. The starting point of our analysis is to assume that the ${\mathbf{r}}\left( {\mathbf{z}} \right)$ are sampled from a Gaussian stochastic process, i.e. a random function whose joint probabilities between finitely many values are multivariate Gaussian random variables. Mathematically, this means that the probability density of the residuals taking values ${\mathbf{r}}$ at positions ${\mathbf{z}}$ is given by

$$ P\left[ {{\mathbf{r}}\left( {\mathbf{z}} \right) } \right] = \frac{1}{{\left( {2\pi } \right)^{N/2} \left| {K\left( {{\mathbf{z}},{\mathbf{z}}} \right)} \right|}}{ \exp }\left\{ {\frac{ - 1}{2}{\mathbf{r}}^{\text{T}} K\left( {{\mathbf{z}},{\mathbf{z}}} \right)^{ - 1} {\mathbf{r}}} \right\}, $$

(A1)

where $K\left( {{\mathbf{z}},{\mathbf{z}}} \right)$ is defined as

$$ K\left( {{\mathbf{z}},{\mathbf{z}}} \right) = \left[ {\begin{array}{*{20}c} {K\left( {z_{1} ,z_{1} } \right)} & \cdots & {K\left( {z_{1} ,z_{N} } \right)} \\ \vdots & \ddots & \vdots \\ {K\left( {z_{N} ,z_{1} } \right)} & \cdots & {K\left( {z_{N} ,z_{N} } \right)} \\ \end{array} } \right]. $$

(A2)

$K\left( {z_{i} ,z_{j} } \right)$ is a covariance matrix element indicating the extent to which the residual at $z_{i}$ is correlated with the residual at $z_{j}$, and $\left| {K\left( {{\mathbf{z}},{\mathbf{z}}} \right)} \right|$ is the determinant of $K\left( {{\mathbf{z}},{\mathbf{z}}} \right)$. The goal of our noise modeling is to use the experimental residuals to determine the function $K$.

Now, it is generally impossible for data to effect a regression by itself; we must invoke some additional or outside information. In the case of GP modeling, one postulates a functional form of the covariance $K$, parameterized by some unknown set of constants $\varphi$. The constants are often referred to as hyperparameters. For all of the regression in this work, we use the Matérn covariance function

$$ K\left( {z_{i} ,z_{j} ;\sigma ,\nu ,\rho } \right) = \sigma^{2} \frac{{2^{1 - \nu } }}{\varGamma \left( \nu \right)}\left( {\sqrt {2\nu } \frac{{\left| {z_{i} - z_{j} } \right|}}{\rho }} \right)^{\nu } \kappa_{\nu } \left( {\sqrt {2\nu } \frac{{\left| {z_{i} - z_{j} } \right|}}{\rho }} \right), $$

(A3)

where $\kappa_{\nu }$ is the modified Bessel function of the second kind of order $\nu$, $\varGamma$ is the gamma function, $\sigma$ is the amplitude of the noise, and $\rho$ is a length specifying the characteristic distance over which residuals are correlated. Here we identify the free parameters as $\varphi = \left( {\sigma ,\nu ,\rho } \right)$. Notably, this covariance is sometimes recommended for spatial data that may not be very smooth,[22] as is the case with our data, especially on the Re-rich HCP side of the diffusion couple.

To generate synthetic data, we first determine the set $\varphi$ via a maximum likelihood analysis. Specifically, we insert $K\left( {{\mathbf{z}},{\mathbf{z}}} \right)$ in Eq A1 and maximize the probability our residuals as a function of these three parameters; that is, we determine the set of $\varphi$ that is most likely to reproduce our experimental residuals. This computation can be done using canned routines in, e.g. Matlab. Given these parameters, we then compute the Cholesky decomposition (i.e. matrix square root) L such that $K\left( {{\mathbf{z}},{\mathbf{z}}} \right) = \varvec{LL}^{\text{T}}$. Using a random number generator, one requests N independent, identically distributed Gaussian random variables with unit variance. Denoting the jth realization as a vector ${\mathbf{H}}^{{\mathbf{j}}}$, we then compute the jth realization of synthetic residuals as the matrix product $\varvec{\eta}^{j} =$ L ${\mathbf{H}}^{{\mathbf{j}}}$ for $\varvec{\eta}^{j} = \left( {\eta^{j} \left( {z_{1} ,\varphi } \right),\eta^{j} \left( {z_{2} ,\varphi } \right), \ldots } \right)$ appearing in Eq 15. Repeating this process with new random numbers generates different synthetic residuals.[17]

Given that the FCC and HCP phases generally exhibit different residuals, we use the functions f(z) and h(z) (cf. Eq 13) to construct separate noise models associated with each phase. Then, we generate a full synthetic dataset by combining one realization from both models. This full dataset is subsequently analyzed using the procedure described above in the section, Assessment of $\tilde{D}$ and uncertainty using a statistical method.