Light Atom Curriculum (LAC)

• Light Atom Curriculum (LAC) is an integrative program focused on non-steady state diffusion using Fick’s second law, spreadsheet analysis, and OpenFOAM® simulations.
• The curriculum methodically transitions from theoretical lectures to hands-on computational exercises and open-source simulation, ensuring robust model validation with experimental data.
• LAC bridges analytical, numerical, and experimental approaches, equipping students with practical skills applicable to challenges in materials science and engineering.

The Light Atom Curriculum (LAC) is an integrative educational program designed to facilitate the learning of non-steady state diffusion phenomena involving lightweight atoms within heavy atom matrices. Developed around the principles of Fick’s Second Law and computational simulation, LAC combines theoretical lectures, hands-on spreadsheet calculations, and open-source simulation experiments to cultivate both conceptual and applied understanding. The curriculum leverages contemporary computational resources and real experimental datasets to bridge theoretical models, numerical solutions, and empirical outcomes in the context of fluid mechanics and materials engineering.

1. Physical Model of Non-Steady State Diffusion

Central to LAC is the quantitative treatment of diffusion of a light atomic species (A, e.g., hydrogen) migrating into a heavy atom substrate (B, e.g., vanadium). The concentration of A is maintained at its boundary, and diffusion proceeds as a function of concentration gradient, governed by Fick’s Second Law. The general form for the evolution of concentration $c(x, t)$ is:

$\frac{\partial c}{\partial t} - \nabla^2(Dc) = 0$

For the one-dimensional case with constant $D$:

$$\frac{\partial c}{\partial t} - D \frac{\partial^2 c}{\partial x^2} = 0$$

The diffusion coefficient $D$ is expressed as an Arrhenius function of temperature:

$D = D_0 \exp\left(-\frac{Q}{RT}\right)$

Analytical solutions for the evolving concentration profile use the error function:

$$\frac{c_s - c_x}{c_s - c_0} = \mathrm{erf}\left(\frac{x}{2\sqrt{Dt}}\right)$$

and equivalently, the complementary error function:

$$c(x, t) = c_0 \operatorname{erfc}\left(\frac{x}{2\sqrt{Dt}}\right)$$

These expressions embody the predictive basis for light atom penetration over time into the heavy matrix.

2. Sequential Educational Framework

LAC employs a tripartite, session-based structure that progresses from theory through computation to simulation, providing both foundational knowledge and practical engagement.

• Session 1: Comprised of a theoretical lecture (Session 1-A) introducing diffusion physics, Fick’s law, and the error function solution, followed by an in-depth analysis of a real-world example (hydrogen diffusion in vanadium) which grounds abstract principles.
• Session 2: Focuses on spreadsheet-based computation. Students utilize empirical hydrogen concentration data and apply the analytical error function to validate theoretical predictions, conduct error analysis, and explore sensitivity to input parameters.
• Session 3: Incorporates computer simulation on open-source platforms, where participants use OpenFOAM® to numerically solve the Laplacian equation for diffusion (laplacianFoam), establish boundary conditions, and postprocess results with ParaView (ParaFOAM), culminating in direct comparison to theoretical and experimental data.

This framework ensures rigorous exposure to both abstract models and practical implementation.

3. OpenFOAM®-Based Simulation Protocol

LAC integrates OpenFOAM® as its computational suite for simulating the spatiotemporal evolution of light atom diffusion. Simulation configuration entails:

• Construction of a mesh discretizing the sample geometry (e.g., a $250 \times 200 \times 5$ cell grid).
• Implementation of laplacianFoam solver to resolve Fick’s second law with specified initial and boundary conditions.
• Modulation of mesh density, simulation duration, and boundary profiles for hypothesis testing and exploration.

Visualization, using ParaView (ParaFOAM), includes:

• Snapshots of evolving concentration fields.
• Spatial profiles (concentration versus position) and contour mapping of diffusion trajectories.
• Quantitative exportation of computed results to facilitate comparative analysis.

The simulation tool enables virtual experimentation and high-fidelity modeling without the logistical limitations of physical laboratory setups.

4. Correlation with Experimental and Analytical Data

The curriculum explicitly compares numerical simulations, analytical error function solutions, and experimental data. For instance, hydrogen diffusion experiments conducted at 423 K over 465 seconds (Uppsala University) yielded concentration profiles that were juxtaposed with analytical and simulated results. The observed agreement exhibited an average error below 0.90%, substantiating both the theoretical derivations and computational methods. This triangulation validates the efficacy and reliability of simulation tools in educational environments, and ensures that students appreciate the nuances of model-experiment agreement.

5. Student Engagement and Active Learning

LAC emphasizes student agency and collaborative problem-solving via:

• Cooperative group work at shared computer resources.
• Flexibility to adjust critical parameters (diffusion coefficient, concentrations, sample dimensions) through spreadsheet computations and simulation input.
• Open-ended virtual experimentation in OpenFOAM®, where students construct hypotheses, test variations, and scrutinize results with published benchmarks.
• Guided questioning and stepwise exercises that promote mastery of both the physical concepts and simulation methodology.

This approach fosters both deep learning and skill acquisition applicable to computational physics and engineering.

6. Implications for Materials Science and Engineering

The penetration of lightweight atoms into heavy matrices has acute relevance for various domains:

• Predictive modelling of mechanical and chemical properties in materials subject to atomic diffusion, such as hydrogen-induced embrittlement in metals.
• Optimization of hydrogen storage systems, leveraging understanding of atomic transport through metallic layers.
• Application to real-world challenges in nuclear fusion, corrosion resistance, and permeability control in advanced materials.

The multi-modal methodology—combining experimental data, analytical models, and numerical simulation—prepares practitioners to address complex material processes where diffusion underpins functional performance.

7. Synthesis and Outcome

The Light Atom Curriculum synthesizes theoretical physics, analytical methodology, spreadsheet computational analysis, and open-source simulation within a stepwise educational format. Participants emerge with a comprehensive understanding of Fick’s second law and its application to non-steady state diffusion of light atoms, reinforced through rigorous comparison to experimental data and validated simulation. This integrative approach cultivates not only theoretical cognition, but also computational proficiency and data analysis acumen—essential competencies for advanced paper and professional practice in fluid mechanics and materials engineering.