MATE 664 Lecture 05

Diffusion (II)

Author

Dr. Tian Tian

Published

January 19, 2026

Note

Slides 👉 Open presentation🗒️
PDF version of course note 👉 Open in pdf
Handwritten notes 👉 Open in pdf

Recap of Lecture 04

Key ideas from last lecture:

Driving forces for diffusion
Fick’s 1st and 2nd laws of diffusion
Time-dependent behavior of Fick’s 2nd equation
Temperature-dependency of diffusivity and activation mechanism
Introduction to diffusivity measurements

Learning Outcomes

After today’s lecture, you will be able to:

Recall different types of diffusivities in solids
Describe basic mechanism in interdiffusion systems
Analysis the driving force for various diffusion mechanisms
Analysis of change of reference system in diffusion experiments.

Recap: Fick’s Laws of Diffusion

1st law: steady-state diffusion

\[ \vec{J}_i = -D_i \nabla c_i \]

2nd law: time-dependent diffusion

\[ \frac{\partial c}{\partial t} = D_i \nabla^2 c_i \]

Recap: Different definitions of diffusivities

Self-diffusivity $D^*$
Intrinsic diffusivity $D_i$ (lattice frame / C-frame; $\mathtt{C}$ 👉crystal)
Inter-diffusivity $\tilde{D}$ (laboratory frame / V-frame; $\mathtt{V}$ 👉volume-fixed)

We will see a few examples today for more clarification.

Self-Diffusion: Chemically Homogeneous Material

Isotope tracer experiments
Lattice constraint: \[ c_i + c_i^* + c_v = \text{const} \]
- General “network-constrained” problem
Vacancy concentration often at equilibrium
- Vacancy balance with the source (surface / grain boundary / dislocation)

Flux Relations in Network-Constrained Systems

Flux driven by chemical potential differences (1D)
- Non-radioactive species: \[ J_i = -L_{ii}\frac{\partial (\mu_i - \mu_v)}{\partial x} -L_{i{i}^*}\frac{\partial (\mu_{i^*} - \mu_v)}{\partial x} \]
- Radioactive species: \[ J_{i^*} = -L_{{i^*}{i}}\frac{\partial (\mu_{i} - \mu_v)}{\partial x} -L_{{i^*}{i^*}}\frac{\partial (\mu_{i^*} - \mu_v)}{\partial x} \]
- Vacancy (zero-flux, why?): \[ J_v = 0 \]

Vacancy Equilibrium Assumption

$\mu_v = \text{const}$
$J_v = 0$
total flux balance: \[ J_i + J_i^* + J_v = 0 \]

Resulting Self-Diffusion Flux

Chemical potential gradient: \[ \frac{\partial \mu_{i^*}}{\partial x} = k_B T \frac{1}{c_i}\frac{\partial c_i}{\partial x} \]
Self-diffusion coefficient: \[ D^{*} = k_B T\left(\frac{L_{ii}}{c_i} - \frac{L_{ii^*}}{c_{i^*}}\right) \]

Self-diffusion summary

No macroscopic concentration gradient!
Chemical potential varies locally for $i$ and $i^*$
Follows the Fick equation!

Self-diffusion in a homogeneous binary solution

Binary alloy / solution between 1 and 2
Isotope tracer for 1
Chemically homogeneous, no lattice change during diffusion

Setup for isotope diffusion for binary mixture

Self-diffusion in binary solution: diffusivity

We have self-diffusion for 4 species (1, 1$^*$, 2 and V)
General flux balance still holds
Vacancy concentration is uniform
Still follows Fick’s law.

\[\begin{align} J_{1^*} &= -k_B T \left[ \frac{L_{11}}{c_1} - \frac{L_{11^*}}{c_{1^*}} \right] \frac{\partial c_{1^*}}{\partial x} \\ &= - D_1^* \frac{\partial c_{1^*}}{\partial x} \end{align}\]

Diffusion in Inhomogeneous Materials

Real materials are rarely homogeneous
Concentration gradients exist by construction
Diffusion fluxes differ locally in space
Leads to imbalance of material transport

Historical Observation: Boyle (17th Century)

Robert Boyle observed solid-state diffusion
Zn penetration into Cu coin → brass formation
Clear macroscopic evidence of diffusion in solids
Mechanism not understood until 20th century

Imbalance of Diffusion Fluxes

Consider a binary diffusion couple A–B
If $D_A \neq D_B$
Then intrinsic fluxes differ: \[ J_A^{C} \neq - J_B^{C} \]
Mass transport is locally unbalanced

Vacancy Mechanism and Imbalance

Substitutional diffusion proceeds via vacancies
Site conservation: \[ J_A^{C} + J_B^{C} + J_v^{C} = 0 \]
Unequal atomic fluxes force a vacancy flux
Vacancy flux is opposite to faster species

Vacancy Accumulation Issue

Vacancy flux implies transport of empty sites
But vacancy concentration remains near equilibrium
Fast sources/sinks (dislocations, GBs, surfaces)
Therefore: no long-term vacancy buildup

Kirkendall Effect

Unequal intrinsic diffusivities
Net vacancy flux in crystal frame
Lattice planes shift to accommodate vacancy flow
Inert markers move relative to sample ends

C-frame View: Inhomogeneous Materials

Crystal (C) frame attached to lattice sites
Atomic fluxes are Fick-like: \[ J_i^{C} = -D_i \frac{\partial C_i}{\partial x} \]
Vacancy flux is kinematically constrained
Lattice assumed locally intact

Intrinsic Diffusivity

Defined in the crystal frame
Measures diffusion relative to lattice sites
Denoted $D_i$
Depends on:
- jump frequency
- vacancy availability
- thermodynamic factor

\[\begin{align*} J_1^C &= -k_BT [\frac{L_{11}}{c_1} - \frac{L_{12}}{c_2}]\Phi \frac{\partial c}{\partial x} \\ &= -D_1 \frac{\partial c}{\partial x} \end{align*}\]

Relation to Self-Diffusivity

Self-diffusion: tracer in homogeneous system
No net chemical potential gradient
Intrinsic diffusivity reduces to self-diffusivity: \[ D_i = D_i^{*} \, \Phi \]
$\Phi$: thermodynamic factor

Thought Experiment: Vacancy Diffusion (C-frame)

Assume $C_v \approx C_{v,\mathrm{eq}}$
Yet $J_v^{C} \neq 0$
Continuity cannot be satisfied by accumulation
Requires an additional velocity

Origin of Lattice Velocity

Transform flux to lab frame: \[ J_v^{\text{lab}} = J_v^{C} + C_v v \]
Impose no net vacancy transport in lab frame: \[ J_v^{\text{lab}} = 0 \]
Lattice velocity: \[ v = -\frac{J_v^{C}}{C_v} \]

Vacancy (V) Frame Basics

V-frame moves with vacancy flux
By definition: \[ J_v^{V} = 0 \]
Atomic fluxes include convective term: \[ J_i^{V} = J_i^{C} + C_i v \]

Governing Equation in V-frame

Impose volume conservation: \[ J_1^{V}\Omega_1 + J_2^{V}\Omega_2 = 0 \]
Leads to single interdiffusion coefficient
Darken equation emerges naturally

Interdiffusivity

Defined in volume-fixed (V) frame
Flux form: \[ J_i^{V} = -\tilde{D} \frac{\partial C_i}{\partial x} \]
Interdiffusivity: \[ \tilde{D} = D_1 X_2 + D_2 X_1 \]

Special Case: Interstitial Diffusion

Interstitial atoms do not occupy lattice sites
No site conservation constraint
No lattice drift required
Interdiffusivity reduces to: \[ \tilde{D} \approx D_{\text{interstitial}} \]

Summary: Types of Diffusivities

Diffusivity	Frame	Meaning
$D_i^{*}$	lattice	tracer / self-diffusion
$D_i$	C-frame	intrinsic diffusivity
$\tilde{D}$	V-frame	interdiffusivity

Summary

Inhomogeneity leads to flux imbalance
Vacancy mechanism enforces site conservation
Lattice velocity resolves vacancy continuity
Choice of reference frame is essential
Kirkendall effect is a frame-dependent phenomenon

--- title: "MATE 664 Lecture 05" subtitle: "Diffusion (II)" author: "Dr. Tian Tian" date: "2026-01-19" format: html: {} revealjs: output-file: slides.html pdf: output-file: L05.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L05.pdf) - Handwritten notes 👉 [Open in pdf](./public/L05_annotated.pdf) ::: ::: ## Recap of Lecture 04 {.center} Key ideas from last lecture: - Driving forces for diffusion - Fick's 1st and 2nd laws of diffusion - Time-dependent behavior of Fick's 2nd equation - Temperature-dependency of diffusivity and activation mechanism - Introduction to diffusivity measurements --- ## Learning Outcomes {.center} After today's lecture, you will be able to: - Recall different types of diffusivities in solids - Describe basic mechanism in interdiffusion systems - Analysis the driving force for various diffusion mechanisms - Analysis of change of reference system in diffusion experiments. --- ## Recap: Fick's Laws of Diffusion - 1st law: steady-state diffusion $$ \vec{J}_i = -D_i \nabla c_i $$ - 2nd law: time-dependent diffusion $$ \frac{\partial c}{\partial t} = D_i \nabla^2 c_i $$ ## Recap: Different definitions of diffusivities ![Lattice (C) vs Lab (V) frame](./public/c-l-frame-nthu.png){width="55%"} - Self-diffusivity $D^*$ - Intrinsic diffusivity $D_i$ (lattice frame / C-frame; $\mathtt{C}$ 👉crystal) - Inter-diffusivity $\tilde{D}$ (laboratory frame / V-frame; $\mathtt{V}$ 👉volume-fixed) We will see a few examples today for more clarification. ## Self-Diffusion: Chemically Homogeneous Material :::{.columns} :::{.column width="50%"} - Isotope tracer experiments - Lattice constraint: $$ c_i + c_i^* + c_v = \text{const} $$ - General "network-constrained" problem - Vacancy concentration often at equilibrium - Vacancy balance with the source (surface / grain boundary / dislocation) ::: :::{.column width="50%"} ![Setup for isotope diffusion](./public/isotope-diff.svg){width="100%"} ::: ::: --- ## Flux Relations in Network-Constrained Systems - Flux driven by chemical potential differences (1D) - Non-radioactive species: $$ J_i = -L_{ii}\frac{\partial (\mu_i - \mu_v)}{\partial x} -L_{i{i}^*}\frac{\partial (\mu_{i^*} - \mu_v)}{\partial x} $$ - Radioactive species: $$ J_{i^*} = -L_{{i^*}{i}}\frac{\partial (\mu_{i} - \mu_v)}{\partial x} -L_{{i^*}{i^*}}\frac{\partial (\mu_{i^*} - \mu_v)}{\partial x} $$ - Vacancy (zero-flux, why?): $$ J_v = 0 $$ ## Vacancy Equilibrium Assumption - $\mu_v = \text{const}$ - $J_v = 0$ - total flux balance: $$ J_i + J_i^* + J_v = 0 $$ --- ## Resulting Self-Diffusion Flux - Chemical potential gradient: $$ \frac{\partial \mu_{i^*}}{\partial x} = k_B T \frac{1}{c_i}\frac{\partial c_i}{\partial x} $$ - Self-diffusion coefficient: $$ D^{*} = k_B T\left(\frac{L_{ii}}{c_i} - \frac{L_{ii^*}}{c_{i^*}}\right) $$ ## Self-diffusion summary - **No** macroscopic concentration gradient! - Chemical potential varies locally for $i$ and $i^*$ - Follows the Fick equation! ## Self-diffusion in a homogeneous binary solution :::{.columns} :::{.column width="50%"} - Binary alloy / solution between 1 and 2 - Isotope tracer for 1 - Chemically homogeneous, no lattice change during diffusion ::: :::{.column width="50%"} ![Setup for isotope diffusion for binary mixture](./public/isotope-diff-binary.svg){width="100%"} ::: ::: ## Self-diffusion in binary solution: diffusivity - We have self-diffusion for 4 species (1, 1$^*$, 2 and V) - General flux balance still holds - Vacancy concentration is uniform - Still follows Fick's law. ```{=tex} \begin{align} J_{1^*} &= -k_B T \left[ \frac{L_{11}}{c_1} - \frac{L_{11^*}}{c_{1^*}} \right] \frac{\partial c_{1^*}}{\partial x} \\ &= - D_1^* \frac{\partial c_{1^*}}{\partial x} \end{align} ```              ## Diffusion in Inhomogeneous Materials - Real materials are rarely homogeneous - Concentration gradients exist by construction - Diffusion fluxes differ locally in space - Leads to imbalance of material transport --- ## Historical Observation: Boyle (17th Century) - Robert Boyle observed solid-state diffusion - Zn penetration into Cu coin → brass formation - Clear macroscopic evidence of diffusion in solids - Mechanism not understood until 20th century --- ## Imbalance of Diffusion Fluxes - Consider a binary diffusion couple A–B - If $D_A \neq D_B$ - Then intrinsic fluxes differ: $$ J_A^{C} \neq - J_B^{C} $$ - Mass transport is locally unbalanced --- ## Vacancy Mechanism and Imbalance - Substitutional diffusion proceeds via vacancies - Site conservation: $$ J_A^{C} + J_B^{C} + J_v^{C} = 0 $$ - Unequal atomic fluxes force a vacancy flux - Vacancy flux is opposite to faster species --- ## Vacancy Accumulation Issue - Vacancy flux implies transport of empty sites - But vacancy concentration remains near equilibrium - Fast sources/sinks (dislocations, GBs, surfaces) - Therefore: no long-term vacancy buildup --- ## Kirkendall Effect - Unequal intrinsic diffusivities - Net vacancy flux in crystal frame - Lattice planes shift to accommodate vacancy flow - Inert markers move relative to sample ends ![](./public/kirkendall.png) --- ## C-frame View: Inhomogeneous Materials - Crystal (C) frame attached to lattice sites - Atomic fluxes are Fick-like: $$ J_i^{C} = -D_i \frac{\partial C_i}{\partial x} $$ - Vacancy flux is kinematically constrained - Lattice assumed locally intact --- ## Intrinsic Diffusivity - Defined in the crystal frame - Measures diffusion relative to lattice sites - Denoted $D_i$ - Depends on: - jump frequency - vacancy availability - thermodynamic factor ```{=tex} \begin{align*} J_1^C &= -k_BT [\frac{L_{11}}{c_1} - \frac{L_{12}}{c_2}]\Phi \frac{\partial c}{\partial x} \\ &= -D_1 \frac{\partial c}{\partial x} \end{align*} ``` --- ## Relation to Self-Diffusivity - Self-diffusion: tracer in homogeneous system - No net chemical potential gradient - Intrinsic diffusivity reduces to self-diffusivity: $$ D_i = D_i^{*} \, \Phi $$ - $\Phi$: thermodynamic factor --- ## Thought Experiment: Vacancy Diffusion (C-frame) - Assume $C_v \approx C_{v,\mathrm{eq}}$ - Yet $J_v^{C} \neq 0$ - Continuity cannot be satisfied by accumulation - Requires an additional velocity --- ## Origin of Lattice Velocity - Transform flux to lab frame: $$ J_v^{\text{lab}} = J_v^{C} + C_v v $$ - Impose no net vacancy transport in lab frame: $$ J_v^{\text{lab}} = 0 $$ - Lattice velocity: $$ v = -\frac{J_v^{C}}{C_v} $$ --- ## Vacancy (V) Frame Basics - V-frame moves with vacancy flux - By definition: $$ J_v^{V} = 0 $$ - Atomic fluxes include convective term: $$ J_i^{V} = J_i^{C} + C_i v $$ --- ## Governing Equation in V-frame - Impose volume conservation: $$ J_1^{V}\Omega_1 + J_2^{V}\Omega_2 = 0 $$ - Leads to single interdiffusion coefficient - Darken equation emerges naturally --- ## Interdiffusivity - Defined in volume-fixed (V) frame - Flux form: $$ J_i^{V} = -\tilde{D} \frac{\partial C_i}{\partial x} $$ - Interdiffusivity: $$ \tilde{D} = D_1 X_2 + D_2 X_1 $$ --- ## Special Case: Interstitial Diffusion - Interstitial atoms do not occupy lattice sites - No site conservation constraint - No lattice drift required - Interdiffusivity reduces to: $$ \tilde{D} \approx D_{\text{interstitial}} $$ --- ## Summary: Types of Diffusivities | Diffusivity | Frame | Meaning | |---|---|---| | $D_i^{*}$ | lattice | tracer / self-diffusion | | $D_i$ | C-frame | intrinsic diffusivity | | $\tilde{D}$ | V-frame | interdiffusivity | --- ## Summary - Inhomogeneity leads to flux imbalance - Vacancy mechanism enforces site conservation - Lattice velocity resolves vacancy continuity - Choice of reference frame is essential - Kirkendall effect is a frame-dependent phenomenon