## Session Overview

Modules |
Reactions and Kinetics |

Concepts |
diffusion: Fick's first law and steady-state diffusion, dependence of the diffusion coefficient on temperature and atomic arrangement, Fick's second law and transient-state diffusion, error function solutions to Fick's second law |

Keywords |
error function, diffusion, mass transport, mass flow rate, flux, Fick's first law, concentration gradient, diffusivity, concentration profile, rate of ingress, jump frequency, Debye frequency, activation energy, barrier energy, vacancy formation, atom migration, melting point, substitutional atom, interstitial atom, self-diffusion, random walk, equilibrium, diffusion coefficient, surface diffusion, grain boundary, bulk diffusion, effusion, Fick's second law, permeability, ideal gas law, normal distribution, void fraction, steady-state, transient, heat transfer |

Chemical Substances |
cobalt-60 (^{60}Co), cobalt-59 (^{59}Co), lead (Pb), aluminum (Al), gold (Au), silver (Ag), copper (Cu), iron (Fe), graphite, carbon (C), calcia (CaO), zirconia (ZrO_{2}), hydrogen (H), manganese (Mn), fused silica (SiO_{2}), borosilicate glass (SiO_{2}+B_{2}O_{3}), soda-lime glass (SiO_{2}+Na_{2}O+CaO), lead borate (PbO+B_{2}O_{3}), borate (B_{2}O_{3}) phosphate (P_{2}O_{5}), platinum (Pt) |

Applications |
doping of semiconductors, oxygen sensor for catalytic converters, outgassing, drying |

### Prerequisites

Before starting this session, you should be familiar with:

- Crystal lattice structures and point defects (Session 15 through Session 20)
- Thermal excitation and the Maxwell-Boltzmann distribution (Session 14)
- Activation energy, Fick's first law (Session 23)
- Basic differential equations and calculus

### Learning Objectives

After completing this session, you should be able to:

- Sketch the
**concentration profile**as a function of time for simple**diffusion**situations. - Describe how diffusion occurs at the
**atomic level**, and identify factors which affect the rate. - Use
**Fick's first and second laws**to solve common diffusion problems. - For a given system, identify some method(s) to increase or decrease the
**diffusion rate**, without adversely affecting other material properties of interest. - Name 3
**industrial applications**of diffusion.

## Reading

Archived Lecture Notes #9 (PDF)

Book Chapters | Topics |
---|---|

[JS] 5.2, "Thermal Production of Point Defects." | Activation energy of vacancies vs. interstitials; Arrhenius plot; thermal expansion |

[JS] 5.3, "Point Defects and Solid-State Diffusion." | Diffusion and vacancy migration; Fick's first and second laws; the error function; concentration profiles for common geometries |

[JS] 5.4, "Steady-State Diffusion." | Linear solution to diffusion at constant concentration |

[JS] 5.5, "Alternate Diffusion Paths." | Bulk, surface, and grain boundary diffusion |

## Lecture Video

> Download from iTunes U (MP4 - 213MB)

> Download from Internet Archive (MP4 - 213MB)

### Resources

### Lecture Summary

Last lecture, Prof. Sadoway introduced the concept of **diffusion **to describe **mass transport** in solid materials. **Thermal vibrations **cause atoms to jump randomly through the lattice, so a **concentration gradient** results in a **net flux **towards areas of low concentration; at **equilibrium**, the random motion in one direction equals the motion in the opposite direction, so no net flux occurs. The **energy **required for this motion depends on specific details of the atomic-level structure, such as: **substitutional **vs. **interstitial **travel; number/strength of bonds to break; amount of free volume in **close-packed** bulk vs. **grain boundaries** vs. **glass **with different levels of network formers.

**Fick's first law** describes the flux when the concentration gradient is constant (steady-state), while **Fick's second law **describes the concentration profile when the gradient changes over time. Prof. Sadoway sketches the **steady-state** and **transient **concentration profiles for simple systems, and introduces the **error function** to describe random walk processes, which follow the **normal distribution**. Continuing last lecture's exploration of **catalytic converters**, he explains how oxygen sensors use diffusion into doped zirconia to monitor the exhaust, giving feedback about the air/fuel ratio to optimize the catalysis.

## Homework

## For Further Study

### Supplemental Readings

Fick, Adolf. "Ueber Diffusion." *Annalen der Physik* 170 (1855): 59-86. (Note: this article is in German.)

Carslaw, Horatio S., and John C. Jaeger. *Conduction of Heat in Solids*. Oxford, England: Clarendon Press, 2004. ISBN: 9780198533689.

### People

### Culture

Wagner, Richard. "Ride of the Valkyries." *Die Walküre*, WWV 86B.

Hornsby, Bruce, and John Hornsby. "The Way It Is." *The Way It Is*. Performed by Bruce Hornsby and the Range. RCA, 1986.

### Other OCW and OER Content

Content | Provider | Level | Notes |
---|---|---|---|

Diffusion | DoITPoMS | Undergraduate | |

Diffusion | Connexions | Undergraduate | |

1.061/1.61 Transport Processes in the Environment | MIT OpenCourseWare | Undergraduate (elective) / Graduate | Explore lecture notes, animations, and worked examples focusing on environmental systems. |