Multicomponent diffusion in a basaltic melt: Temperature dependence
Eighteen successful diffusion couple experiments in 8-component SiO2–TiO2–Al2O3–FeO–MgO–CaO–Na2O–K2O basaltic melts were conducted at 1260°C and 0.5 GPa and at 1500°C and 1.0 GPa. These experiments are combined with previous data at 1350°C and 1.0 GPa (Guo and Zhang, 2018) to study the temperature dependence of multicomponent diffusion in basaltic melts. Effective binary diffusion coefficients of components with monotonic diffusion profiles were extracted and show a strong dependence on their counter-diffusing component even though the average (or interface) compositions are the same. The diffusion matrix at 1260°C was obtained by simultaneously fitting diffusion profiles of all diffusion couple experiments as well as appropriate data from the literature. All features of concentration profiles in both diffusion couples and mineral dissolution are well reproduced by this new diffusion matrix. At 1500°C, only diffusion couple experiments are used to obtain the diffusion matrix. Eigenvectors of the diffusion matrix are used to discuss the diffusion (exchange) mechanism, and eigenvalues characterize the diffusion rate. Diffusion mechanisms at both 1260 and 1500°C are inferred from eigenvectors of diffusion matrices and compared with those at 1350°C reported in Guo and Zhang (2018). There is indication that diffusion eigenvectors in basaltic melts do not depend much on temperature, but more »
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Publication Date:
NSF-PAR ID:
10161017
Journal Name:
Chemical geology
Volume:
549
Issue:
119700
Page Range or eLocation-ID:
1-22
ISSN:
0009-2541
3. Abstract Covariance matrices are fundamental to the analysis and forecast of economic, physical and biological systems. Although the eigenvalues $\{\lambda _i\}$ and eigenvectors $\{\boldsymbol{u}_i\}$ of a covariance matrix are central to such endeavours, in practice one must inevitably approximate the covariance matrix based on data with finite sample size $n$ to obtain empirical eigenvalues $\{\tilde{\lambda }_i\}$ and eigenvectors $\{\tilde{\boldsymbol{u}}_i\}$, and therefore understanding the error so introduced is of central importance. We analyse eigenvector error $\|\boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2$ while leveraging the assumption that the true covariance matrix having size $p$ is drawn from a matrix ensemble with known spectral properties—particularly, we assume the distribution of population eigenvalues weakly converges as $p\to \infty$ to a spectral density $\rho (\lambda )$ and that the spacing between population eigenvalues is similar to that for the Gaussian orthogonal ensemble. Our approach complements previous analyses of eigenvector error that require the full set of eigenvalues to be known, which can be computationally infeasible when $p$ is large. To provide a scalable approach for uncertainty quantification of eigenvector error, we consider a fixed eigenvalue $\lambda$ and approximate the distribution of the expected square error $r= \mathbb{E}\left [\| \boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2\right ]$ across themore »