# A computational investigation of the effect of alloying elements on the thermodynamic and diffusion properties of fcc Ni alloys, with application to the creep rate of dilute Ni-X alloys.

Open Access

- Graduate Program:
- Materials Science and Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 12, 2012
- Committee Members:
- Zi Kui Liu, Dissertation Advisor
- Zi Kui Liu, Committee Chair
- Long Qing Chen, Committee Member
- Paul Raymond Howell, Committee Member
- Jorge Osvaldo Sofo, Committee Member

- Keywords:
- Ni-base superalloys
- thermodynamic modeling
- self-diffusion
- impurity diffusion
- non-dilute impurity diffusion
- first-principles

- Abstract:
- Ni-base superalloys have become ubiquitous in the materials science community in the past half-century because of their superior ability to resist chemical and mechanical degradation at temperature upwards of 70 % of their melting temperature. Future generations of Ni-base superalloys with increased service lifetimes and higher efficiencies will require the development of more complex, multi-component alloys carefully engineered to meet specific materials properties specifications, such as high temperature creep resistance. This can only come from an intimate knowledge of the exact effects of each individual alloying element on the thermodynamic and kinetic properties of the Ni-base superalloys. In this dissertation, two computational techniques have been employed to understand the alloying effects of various transition elements in Ni and its alloys. Thermodynamics and phase stability has been investigated through use of the CALculation of PHAse Diagram (CALPHAD) modeling technique, supplemented by first-principles calculations based on density functional theory (DFT). To better understand the kinetics involved in materials transport, the self-diffusion in ferromagnetic fcc Ni is calculated by first-principles, followed by a systematic investigation of the effects of 26 alloying elements on dilute Ni-rich binary alloys. Mechanisms causing specific diffusion behavior are explored, and the usefullness of such a database of knowledge is demonstrated by applying the generated data to a secondary creep rate model to show how each alloying element affects the creep behavior of the dilute Ni alloy systems. Finally, the next frontier of diffusion coefficient calculations by first-principles is explored by extending the calculation to non-dilute impurity concentrations, employing the fourteen-frequency model applied to the Ni-Al system. To aid in the process of narrowing down the large composition space for the design of future Ni-base superalloys, a thermodynamic model using the CALPHAD approach is developed, where Gibbs energy functions of individual phases are parameterized based on fittings to experimentally measured phase equilibria or thermochemical data and computationally predicted thermochemical data. Multi-component Ni-base superalloys can be accurately described within the CALPHAD approach through the extrapolation of the Gibbs energy functions of the simpler sub-systems which are modeled where experimental and computational data is usually more abundant. The Re-Y and Re-Ti systems, integral binary alloy systems in the Ni-base superalloy database, are modeled in the present work. Since little thermochemical data was available for either system, first-principles calculations were used to improve the thermodynamic models of the solid solution phases and the compounds. Both phase diagrams show excellent agreement with available experimental phase equilibria data. To further demonstrate the utility of first-principles calculations, an investigation on the phase stability of the ReTi compound is performed, because it was reported to have one crystal structure through experimental measurements and different crystal structure through first-principles calculations. To demonstrate the ability of the CALPHAD method for successful extrapolation to higher order systems, the Ni-Re-Y is modeled by combining the Re-Y system with the previously modeled constituent binaries. Good agreement is found with an experimental isothermal prediction of a two- and three- phase boundary at 1000 K. In addition to studying thermodynamic and phase stability properties of Ni-base superalloys, this thesis also highlights the importance of the kinetic properties of these materials through their diffusion coefficients. Vacancy mediated self-diffusion coefficients are calculated on ferromagnetic and non-magnetic fcc Ni as a function of temperature. Within Eyring's reaction rate theory, minimum energy pathways for the diffusing atom is calculated using the Nudged Elastic Band method. It is observed that ferromagnetism is necessary for reproducing both Arrhenius and thermodynamic diffusion parameters, while the corresponding non-magnetic calculations show significantly poorer agreement. It is also observed that the use of the Debye-Grunseisen model for calculating the finite temperature entropic contributions to the diffusion coefficient reproduces the experimental self-diffusion more accurately than phonon calculations based on the supercell approach. Reasons for this surprisingly result are discussed in detail. Based on the success of the first-principles calculation of self-diffusion coefficient in pure Ni, the approach using ferromagnetic spin polarization and the Debye-Grunseisen model to calculate the entropic contributions to the diffusion coefficient are employed for 26 alloying elements including: Al, Co, Cr, Cu, Fe, Hf, Ir, Mn, Mo, Nb, Os, Pd, Pt, Re, Rh, Ru, Sc, Si, Ta, Tc, Ti, V, W, Y, Zn, and Zr. The five-frequency model is implemented to accommodate the various jump frequencies associated with impurity diffusion coefficients in fcc Ni. The present work demonstrates that the mid-row 5d transition row element impurities have the highest activation barriers for impurity diffusion, and subsequently are the slowest diffusers in Ni. The fastest diffusers in Ni coupled with the lowest activation barriers for impurity diffusion are demonstrated to be at the far left of the 3d and 4d transition element rows on the periodic table. The present work also demonstrates that the primary mechanism driving the variation in the impurity diffusion coefficient from element to element is the migration barrier for impurity diffusion. In addition, the correlation of the impurity diffusion coefficients is not found to be to the size of the element as previously predicted in the literature, but rather, that the impurity diffusion coefficients have a much stronger correlation to the compressibility of the associated Ni-X dilute alloy. A charge density analysis on the transition state of six of the twenty-six systems shows how the impurity affects surrounding Ni atoms. Assessments of the validity of the five-frequency model and the relaxation techniques for the treatment of the maximum energy point along the diffusion pathway are discussed using the Ni-Al system as a model case. First, an analysis of the assumptions made for the relaxation scheme of the three saddle configurations are made. It is shown that by using a new relaxation scheme, the calculated impurity diffusion coefficient can be improved with respect to experimental data. Additionally, an alternate calculation of the correlation factor for the impurity diffusion coefficient calculation is performed that assumes interactions of the solute and vacancy go beyond the first nearest neighbor shell. The alternate method includes the jump frequency associated with the migration of the host atom in the presence of an impurity at a second nearest neighbor position, as opposed to the original method which assumes this type of jump is analogous to self-diffusion in the host system. The result shows an increased agreement with the experimental data in the case of the Ni-Al system. In order to provide quantitative measures for the improvement of future generations of Ni-base superalloys, a model from the literature for the secondary creep rate typically applied to Ni-base superalloys is selected. The impurity diffusion results as a function of temperature are combined with previous first-principles calculations of elastic and stacking fault energy properties to predict a creep rate for each of the 26 systems normalized to the creep rate of pure Ni. Results from the application of the creep model reveal similar results to the impurity diffusion coefficients in fcc Ni. Mid-row 4d and 5d transition row element impurities decrease the creep rate compared when to pure Ni. In particular, Mo, Tc, Ru, and Re impurities cause the dilute alloys to retain more creep resistance at higher temperatures. Four elements that show significantly faster impurity diffusion coefficients relative to self-diffusion in pure Ni are Hf, Ti, Nb, and Ta. The creep rates of these four elements, however, are equal to or slower than the creep rate of pure Ni, indicating greater influence from stacking fault energy and elastic properties on these dilute alloys. Finally, the first-principles methodology for calculating dilute impurity diffusion coefficients in Ni-X alloy systems is extended further to impurity diffusion in more concentrated Ni-X alloys systems. The fourteen-frequency model is a natural extension of the five-frequency model to take solute-solute interactions into account. Using the Ni-Al system as a benchmark, all of the atomic jump frequencies as a function of temperature associated with the proposed fourteen-frequency model for diffusion in non-dilute systems are calculated using the ferromagnetic spin polarization and the Debye-Grunseisen model approaches. Solute and solvent enhancement factors to show the effect of impurity concentration on the diffusion coefficient are calculated. The impurity diffusion coefficient is fit to an empirical equation to show its dependence on composition in terms of the solute and solvent enhancement factors. Trends in the migration barriers and atomic jump frequencies for all fourteen jump frequencies are examined. The goal of the present thesis is to provide a better understanding of the thermodynamic and kinetic parameters of Ni-base alloys to aid in the future development of more advanced Ni-base superalloy systems. The methodology, results, and analysis presented in this thesis provide a better understanding of the effects of alloying elements on the diffusion properties of dilute and non-dilute Ni alloys, and establish a benchmark for effects and trends of impurity diffusion in other magnetic alloy systems, such as bcc Fe as the host matrix.