Yang Fuqian, Schmidt Harald, Hüger Erwin
Materials Program, Department of Chemical and Materials Engineering, University of Kentucky, Lexington, Kentucky 40506, United States.
Clausthal Centre of Material Technology, Clausthal University of Technology, Clausthal-Zellerfeld DE-38678, Germany.
ACS Omega. 2023 Jul 21;8(30):27776-27783. doi: 10.1021/acsomega.3c04029. eCollection 2023 Aug 1.
Understanding mass transport in micro- and nanostructures is of paramount importance in improving the performance and reliability of the micro- and nanostructures. In this work, we solve the diffusion problem in a multilayer structure with periodic conditions under a constant heating rate via a Fourier series. Analytical relation is established between the coefficients of eigenfunctions and the intensity of X-ray or neutron Bragg peak. The logarithm of temporal variation of the intensity of X-ray or neutron Bragg peak is a linear function of the nominal diffusion time, with the nominal diffusion time being dependent on the heating rate. This linear relation is validated by experimental data. The Taylor series expansion of the linear relation to the first order of the diffusion time yields an approximately linear relation between the logarithm of temporal variation of the intensity of X-ray or neutron peak and the diffusion time for small diffusion times, which can be likely used to calculate the activation energy for the diffusion in a multilayer structure. The validation of such an approach is subjected to the fact that the characteristic time for heat conduction is much less than the characteristic time for the ramp heating as well as the characteristic time for diffusion.
理解微纳结构中的质量输运对于提高微纳结构的性能和可靠性至关重要。在这项工作中，我们通过傅里叶级数解决了在恒定加热速率下具有周期性条件的多层结构中的扩散问题。建立了本征函数系数与X射线或中子布拉格峰强度之间的解析关系。X射线或中子布拉格峰强度随时间变化的对数是名义扩散时间的线性函数，其中名义扩散时间取决于加热速率。该线性关系得到了实验数据的验证。将线性关系按扩散时间的一阶进行泰勒级数展开，对于小扩散时间，得到X射线或中子峰强度随时间变化的对数与扩散时间之间近似线性关系，这可能用于计算多层结构中扩散的活化能。这种方法的验证取决于热传导的特征时间远小于斜坡加热的特征时间以及扩散的特征时间这一事实。
