MATE 664 Lecture 09

Atomic Models for Diffusion (II): Gas & Liquid Phases

Author

Dr. Tian Tian

Published

February 2, 2026

Note

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PDF version of course note 👉 Open in pdf
Handwritten notes 👉 Open in pdf

Recap of Lecture 07

Key ideas from last lecture:

Analytical solution to diffusion problems (infinite domain)
- Semi-infinite (half-half) solution
- Line / point source solution
- Superimposition method
Separation of variables method (finite / bounded domain)
Laplace transform (will not be covered in exam)

Learning Outcomes

After this lecture, you will be able to:

Recall the conditions behind the Einstein diffusion equation
Understand why the Einstein relation is universal across states of matter
Connect gas, liquid, and solid diffusion within a unified framework
Derive Arrhenius-type diffusivity from a 1D potential well model
Identify assumptions and missing physics in simple diffusion models

Recap: Linking Atomistic Motion to Diffusion

From last lecture:

Diffusion emerges from random microscopic motion
Mean squared displacement (MSD) is the key observable
Einstein equation links MSD to macroscopic diffusivity

Einstein Diffusion Equation

Define MSD as the normalized second moment of concentration:

\[\begin{align} \langle R^2(t)\rangle &= \frac{\int_0^\infty r^2 c(r,t)\,4\pi r^2 dr} {\int_0^\infty c(r,t)\,4\pi r^2 dr} \end{align}\]

Using solution to point source diffusion, Einstein showed: - 3D: \(\langle R^2\rangle = 6Dt\) - 2D: \(\langle R^2\rangle = 4Dt\) - 1D: \(\langle R^2\rangle = 2Dt\)

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date: "2026-02-02"
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- Slides 👉  [Open presentation🗒️](./slides.html)
- PDF version of course note  👉 [Open in pdf](./L09.pdf)
- Handwritten notes 👉 [Open in pdf](./public/L09_annotated.pdf)
:::

:::


## Recap of Lecture 07 {.center}
Key ideas from last lecture:

- Analytical solution to diffusion problems (infinite domain)
  - Semi-infinite (half-half) solution
  - Line / point source solution
  - Superimposition method
- Separation of variables method (finite / bounded domain)
- Laplace transform (will not be covered in exam)


## Learning Outcomes {.center}

After this lecture, you will be able to:

- Recall the conditions behind the Einstein diffusion equation
- Understand why the Einstein relation is universal across states of matter
- Connect gas, liquid, and solid diffusion within a unified framework
- Derive Arrhenius-type diffusivity from a 1D potential well model
- Identify assumptions and missing physics in simple diffusion models

## Recap: Linking Atomistic Motion to Diffusion {.center}

From last lecture:

- Diffusion emerges from random microscopic motion
- Mean squared displacement (MSD) is the key observable
- Einstein equation links MSD to macroscopic diffusivity

---

## Einstein Diffusion Equation

Define MSD as the normalized second moment of concentration:

```{=tex}
\begin{align}
\langle R^2(t)\rangle
&=
\frac{\int_0^\infty r^2 c(r,t)\,4\pi r^2 dr}
{\int_0^\infty c(r,t)\,4\pi r^2 dr}
\end{align}
```

Using solution to point source diffusion, Einstein showed:
- 3D: $\langle R^2\rangle = 6Dt$
- 2D: $\langle R^2\rangle = 4Dt$
- 1D: $\langle R^2\rangle = 2Dt$