24. Diffusion

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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 (60Co), cobalt-59 (59Co), lead (Pb), aluminum (Al), gold (Au), silver (Ag), copper (Cu), iron (Fe), graphite, carbon (C), calcia (CaO), zirconia (ZrO2), hydrogen (H), manganese (Mn), fused silica (SiO2), borosilicate glass (SiO2+B2O3), soda-lime glass (SiO2+Na2O+CaO), lead borate (PbO+B2O3), borate (B2O3) phosphate (P2O5), platinum (Pt)
Applications doping of semiconductors, oxygen sensor for catalytic converters, outgassing, drying


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.


Archived Lecture Notes #9 (PDF)

Book Chapters Topics
Buy at Amazon [JS] 5.2, "Thermal Production of Point Defects." Activation energy of vacancies vs. interstitials; Arrhenius plot; thermal expansion
Buy at Amazon [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
Buy at Amazon [JS] 5.4, "Steady-State Diffusion." Linear solution to diffusion at constant concentration
Buy at Amazon [JS] 5.5, "Alternate Diffusion Paths." Bulk, surface, and grain boundary diffusion

Lecture Video

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This resource may not render correctly in a screen reader.Lecture Slides (PDF - 3.0MB)

Transcript (PDF)

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.


Problems (PDF)

Solutions (PDF)

For Further Study

Supplemental Readings

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

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


Adolf Fick

Peter Debye


Wagner, Richard. "Ride of the Valkyries." Die Walk├╝re, WWV 86B.

Buy at Amazon 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.


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