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### Diffusion and Kinetics

### See also Mesoscale simulation tools and RASPA tools

### Scaffolding Simulations in a Rate Processes of Materials Course

This learning resource describes a set of programming assignments that are used in a Rate Processes of Materials course. The assignments are designed around the pedagogical principle of scaffolding, in which students are given initial support structures that are gradually removed. The incremental skill-building helps students become independent learners.

During the term, students complete four programming assignments utilizing MATLAB; two skill modules focused on general programming and simulation methods and two scenario-based memos targeting specific materials science simulations or data processing. These are described below:

- Skill Module 1: The introductory activity provides a review of MATLAB syntax and programming skills. Students complete an online MATLAB tutorial to address any gaps in their prior knowledge, take a brief online quiz based on the tutorial, and analyze a simple data set.
- Skill Module 2: In this module, students are introduced to the Finite Difference Method (FDM), which is a computational method used to estimate the solutions to partial differential equations. Students complete a task requiring them to apply the FDM to solve a basic physics problem. The activity also requires students to perform basic code verification, comparing their simulation result to a known solution.
- Scenario 1: This assignment requires the FDM (from Skill 2) to be used to solve a materials science problem. In this scenario, students take on the role of engineers at a chemical company. They are provided with the concentration of a chemical over time and need to use this information to calculate the reaction order and reaction constant. Finally, students simulate the reaction kinetics assuming a different starting reaction concentration.
- Scenario 2: For the final memo, students are given their choice of assignments.
- Scenario 2A: In this assignment, students extend their knowledge of the FDM to simulate one-dimensional diffusion based on Fick's laws of diffusion. Students must compare their results from data provided in a paper.
- Scenario 2B: This task is based on analyzing a three-dimensional microstructure. Students are provided with the data, but must develop a MATLAB script to analyze the large data set. Similar to Scenario 2A, students must read and summarize the key points of a paper.

Included in the distribution are the student instructions, data files where needed, and corresponding rubrics. Solutions have not been provided as these activities are still used in the course. Contact Susan Gentry (spgentry@ucdavis.edu) if you have any questions.

**2-D Diffusion Game**

The Diffusion Game package introduces students to computational materials modeling by simulating the 2D diffusion of metal atoms. See the Supporting Docs tab for more information, and a paper game board.

**Grain Boundary Diffusion Calculator**

This tool calculates the effective diffusivity in a grain boundary network represented by a three-dimensional Voronoi diagram. Two types of grain boundaries with different diffusivities are randomly distributed in the domain. The effective diffusivity is calculated using the mean squared displacement method, where periodic boundary conditions are applied in all directions. Users are free to choose the fraction of each grain boundary type as well as the activation energy and pre-factor for each grain boundary diffusivity.

**Diffusion Calculator: HCP Dilute Solutes**

The diffusivity of a dilute solute in an HCP lattice is calculated with Ghate's 8-frequency model. Each of the eight frequencies (w_i) are specified by a transition energy barrier (E_i) and transition attempt frequency (v_i). The vacancy formation energy, vacancy-solute binding energy, and the HCP lattice constants are also required. The output temperature range can also be specified in terms of 1000/T.

**Diffusion Calculator: FCC Dilute Solutes**

The diffusivity of a dilute solute in an FCC lattice is calculated using LeClaire and Lidiard's 5-frequency model. Each of the five frequencies (w_i) are specified by a transition energy barrier (E_i) and transition attempt frequency (v_i). The vacancy formation energy and the FCC lattice constant are also required. The output temperature range can also be specified in terms of 1000/T.

**Particle Trajectory Diffusion Analysis**

This tool takes as input particle position data from methods such as molecular dynamics or kinetic Monte Carlo and computes the mean squared displacement for all particles as a function of time. For a system with multiple types of particles, the mean squared displacement is computed for each particle type. The tracer diffusion coefficient is then calculated from the slope of the mean squared displacement vs time curve.

**Scaffolding Simulations in a Rate Processes of Materials Course**

This learning resource describes a set of programming assignments that are used in a Rate Processes of Materials course. The assignments are designed around the pedagogical principle of scaffolding, in which students are given initial support structures that are gradually removed. The incremental skill-building helps students become independent learners.

Skill Module 1: The introductory activity provides a review of MATLAB syntax and programming skills.

Skill Module 2: In this module, students are introduced to the Finite Difference Method (FDM), which is a computational method used to estimate the solutions to partial differential equations.

Scenario 1: This assignment requires the FDM (from Skill 2) to be used to solve a materials science problem. In this scenario, students take on the role of engineers at a chemical company. They are provided with the concentration of a chemical over time and need to use this information to calculate the reaction order and reaction constant. Finally, students simulate the reaction kinetics assuming a different starting reaction concentration.

- Scenario 2A: In this assignment, students extend their knowledge of the FDM to simulate one-dimensional diffusion based on Fick's laws of diffusion. Students must compare their results from data provided in a paper.
- Scenario 2B: This task is based on analyzing a three-dimensional microstructure. Students are provided with the data, but must develop a MATLAB script to analyze the large data set. Similar to Scenario 2A, students must read and summarize the key points of a paper.

**Process Lab: Concentration-Dependent Diffusion**

This simulation tool simulates the dopant diffusion process by solving the partial differential equations. The tool gives users the freedom to adjust critical parameters and conditions in the process, such as the initial doping profile, time, temperature, length, and so on. It also gives users opportunities to choose between the delta or box-shaped dopant source, concentration dependency, as well as the type of dopants among 6 commonly used dopant species.