MATE 664 Lecture 11

Diffusion in Defect and Material Imperfections

Author

Dr. Tian Tian

Published

February 9, 2026

Note

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Recap of Lecture 10

Key ideas from last lecture:

Diffusion equations for solids
Link Einstein’s equation to Arrhenius equation
Emergence of activation enthalpy / entropy
Vacancy formation free energy and diffusion

Learning outcomes

After this lecture, you will be able to:

Apply the free energy-dependent diffusion equations to systems with defects and material imperfections
Understand how intrinsic and extrinsic defect in ionic materials are formed
Analyze intrinsic / extrinsic diffusion regimes in ionic materials
Understand material imperfections as the diffusion shortcuts

What does diffusion in real materials look like?

Real solids are not perfect crystals
Defects control transport properties
Ionic diffusion shows rich temperature dependence
Same Einstein framework, new physics inside

Base line: vacancy-mediated diffusion in metals

Diffusion = random walk of atoms via vacancies
Diffusivity related to both vacancy density (controlled by $G_v^f$) and vacancy jumping (controlled by $G_v^m$)

\[\begin{align} D_A &= \frac{z \langle r^2 \rangle \nu}{6} \exp\!\left(\frac{S_v^{f} + S_v^{m}}{k_B}\right) \exp\!\left(-\frac{H_v^{f} + H_v^{m}}{k_B T}\right) \, f \end{align}\]

Defect-mediated diffusion in ionic crystals

Defects in ionic crystals are more complex than in metal
Charge neutrality has to be conserved
More than one single defect species are involed
In other words, defects always come in pairs in ionic solids

Schottky and Frenkel defects in ionic crystals

Schottky defects: missing both positive and negative species
Frenkel defects: one charged species moved to other site

Comparison between Schottky and Frenkel defects

Ionic defects: Kröger-Vink (KV) notation

Formula $X^Z_Y$
X = what is at the site (Element or Vacancy)
Y = what site is defective (Element or i)
Z = effective charge at the site (• = +; ’= –)

KV notation for Schottky defects

Example: Schottky defects in MgO (anion + cation vacancies)

Formation reaction

\[ \text{null} \rightarrow V_{\mathrm{Mg}}^{''} + V_{\mathrm{O}}^{\bullet\bullet} \]
Equilibrium condition
- $G_S^f$: formation free energy for Schottky defects (per reaction)
\[ K_S = [V_{\mathrm{Mg}}^{''}][V_{\mathrm{O}}^{\bullet\bullet}] = \exp\!\left(-\dfrac{G_S^{f}}{kT}\right) \]
Charge neutrality

\[ [V_{\mathrm{Mg}}^{''}] = [V_{\mathrm{O}}^{\bullet\bullet}] \]

KV notation for Frenkel defects

Example: Frenkel pairs in LiF (Lithium escaping to interstitial sites)

Formation reaction

\[ \mathrm{Li}_{\mathrm{Li}}^{\times} \rightarrow V_{\mathrm{Li}}^{'} + \mathrm{Li}_i^{\bullet} \]
Equilibrium condition
- $G_F^f$: formation free energy for Schottky defects (per reaction)
\[ K_F = [V_{\mathrm{Li}}^{'}][\mathrm{Li}_i^{\bullet}] = \exp\!\left(-\dfrac{G_F^{f}}{kT}\right) \]
Charge neutrality
$[V_{\mathrm{Li}}^{'}] = [\mathrm{Li}_i^{\bullet}]$

Intrinsic vs extrinsic defects

Intrinsic defects are those controlled by thermodynamics $G_S^f$, $G_F^f$ or $G_V^f$
Extrinsic defects are those defects added to the ionic crystal via doping

Example of Extrinsic defects: CdCl$_2$ in NaCl

Dopant incorporation reaction
\[ \mathrm{CdCl}_2 \rightarrow \mathrm{Cd}_{\mathrm{Na}}^{\bullet} + 2\,\mathrm{Cl}_{\mathrm{Cl}}^{\times} + V_{\mathrm{Na}}^{'} \]
Extrinsic defect concentration
\[ [V_{\mathrm{Na}}^{'}]_{\text{ext}} = [\mathrm{Cd}_{\mathrm{Na}}^{\bullet}] = [\mathrm{CdCl}_2] \]
Total vacancy concentration
\[ [V_{\mathrm{Na}}^{'}] \approx [V_{\mathrm{Na}}^{'}]_s + [V_{\mathrm{Na}}^{'}]_{\text{ext}} = \frac{\exp\!(-\dfrac{G_S^{f}}{k_B T})}{ [\mathrm{CdCl}_2]} + [\mathrm{CdCl}_2] \]

Diffusivity of ionic species

We can still use the vacancy exchange mechanism for the diffusivity of ionic species
For f.c.c. lattice the neighbor sites of same charge, the multiplicity $z=6$.
E.g. Na$^+$ exchanges with its own vacancy $V_{\mathrm{Na}}^{'}$
$[V_{\mathrm{Na}}^{'}]$ depends on the $T$-regime!

\[\begin{align} D_{\mathrm{Na}} &= [V_{\mathrm{Na}}^{'}]\, f\,\lambda^2\,\nu\, \exp\!\left(-\frac{G_{\mathrm{Na}}^{m}}{kT}\right) \end{align}\]

Intrinsic vs extrinsic regimes

Extrinsic: vacancy dominated by doped materials
- Low-$T$ regime (high $1/T$)

\[\begin{align} D_{\mathrm{Na,ext}} &= [\mathrm{CdCl}_2]\, f\,\lambda^2\,\nu\, \exp\!\left(\frac{S_{\mathrm{Na}}^{m}}{k}\right) \exp\!\left(-\frac{H_{\mathrm{Na}}^{m}}{kT}\right) \end{align}\]

Intrinsic: vacancy dominated by thermal dissociation
- High-$T$ regime (low $1/T$)

\[\begin{align} D_{\mathrm{Na,int}} &= f\,\lambda^2\,\nu\, \exp\!\left(\frac{S_{S}^{f}}{2k}\right) \exp\!\left(\frac{S_{\mathrm{Na}}^{m}}{k}\right) \exp\!\left(-\frac{H_{S}^{f}}{2kT}\right) \exp\!\left(-\frac{H_{\mathrm{Na}}^{m}}{kT}\right) \end{align}\]

Two-regimes in the Arrhenius plot

More regimes in Arrhenius plot

Example: cation diffusion in FeO during oxidation
Equilibrium depends on both $G^f$ and $p(O_2)$
Multiple regimes!

Multiple-regime diffusion in polycrystalline materials

Arrhenius plot for diffusion on imperfections

Diffusion paths at crystal imperfections

Diffusion can occur along non-bulk pathways
Typical crystal imperfections
- Grain boundary and interface diffusion: 2D
- Free surface diffusion: 2D
- Dislocation (pipe) diffusion: 1D
- Vacancy / defect: 0D
Imperfections are associated with lower migration / activation energy!
Think as “shortcuts” during diffusion

Imperfection 1: grain boundaries

Grain boundary diffusion

Harrison’s ABC model for GB diffusion

A foreign material is coated on the top of a polycrystalline metal. The degree of penetration can be studied by comparing the timescale $t$, interatomic distance $\lambda$ and grain size $s$

All regime
- $D_{XL} t > s^2$
- $D_B t > s^2$
Boundary regime
- $D_{XL} t \approx \lambda^2$
- $D_B t > \lambda^2$
- coupled short-circuit and bulk diffusion
Core regime
- $D_{XL} t < \lambda^2$
- $D_B t > \lambda^2$
- diffusion confined to imperfections

Dislocation imperfections (line defect)

Example of diffusion along imperfection: deposition on graphene

See Vagli and Tian et al. Nat Commun 2025, 16, 7726.
Diffusivity change on free graphene surface can be probed by deposition geometry!

Deposition on Graphene - theoretical simulations

Kinetic Monte Carlo (kMC) assuming different diffusion barrier on imperfections
Faster diffusion direction –> lower density

Deposition on Graphene - theoretical simulations

Kinetic Monte Carlo (kMC) assuming different diffusion barrier on imperfections
Faster diffusion direction –> lower density

Deposition on Graphene - KMC vs experiments

Summary

In this lecture, we reviewed a few sample cases where diffusion is dominated by crystal defects and imperfections

Diffusion by ionic defects – multiple Arrhenius regimes
Diffusion by imperfections – shortcut diffusion compared with bulk diffusion
Example of interface mediated diffusion – 2D material

--- title: "MATE 664 Lecture 11" subtitle: "Diffusion in Defect and Material Imperfections" author: "Dr. Tian Tian" date: "2026-02-09" format: html: {} revealjs: output-file: slides.html pdf: output-file: L11.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L11.pdf) - Handwritten notes 👉 [Open in pdf](./public/L11_annotated.pdf) ::: ::: ## Recap of Lecture 10 {.center} Key ideas from last lecture: - Diffusion equations for solids - Link Einstein's equation to Arrhenius equation - Emergence of activation enthalpy / entropy - Vacancy formation free energy and diffusion ## Learning outcomes {.center} After this lecture, you will be able to: - Apply the free energy-dependent diffusion equations to systems with defects and material imperfections - Understand how intrinsic and extrinsic defect in ionic materials are formed - Analyze intrinsic / extrinsic diffusion regimes in ionic materials - Understand material imperfections as the diffusion shortcuts ## What does diffusion in real materials look like? - Real solids are not perfect crystals - Defects control transport properties - Ionic diffusion shows rich temperature dependence - Same Einstein framework, new physics inside ## Base line: vacancy-mediated diffusion in metals - Diffusion = random walk of atoms via vacancies - Diffusivity related to both vacancy density (controlled by $G_v^f$) and vacancy jumping (controlled by $G_v^m$) ```{=tex} \begin{align} D_A &= \frac{z \langle r^2 \rangle \nu}{6} \exp\!\left(\frac{S_v^{f} + S_v^{m}}{k_B}\right) \exp\!\left(-\frac{H_v^{f} + H_v^{m}}{k_B T}\right) \, f \end{align} ``` ## Defect-mediated diffusion in ionic crystals - Defects in ionic crystals are more complex than in metal - Charge neutrality has to be conserved - More than one single defect species are involed - In other words, defects always come in **pairs** in ionic solids ## Schottky and Frenkel defects in ionic crystals - Schottky defects: missing both positive and negative species - Frenkel defects: one charged species moved to other site ![Comparison between Schottky and Frenkel defects](./public/compare-defect.png){width="65%"} ## Ionic defects: Kröger-Vink (KV) notation - Formula $X^Z_Y$ - X = what is at the site (Element or Vacancy) - Y = what site is defective (Element or i) - Z = effective charge at the site (• = +; '= –) ![Example of KV notation](./public/kv-notation.png){width="60%"} ## KV notation for Schottky defects Example: Schottky defects in MgO (anion + cation vacancies) - Formation reaction $$ \text{null} \rightarrow V_{\mathrm{Mg}}^{''} + V_{\mathrm{O}}^{\bullet\bullet} $$ - Equilibrium condition - $G_S^f$: formation free energy for Schottky defects (per reaction) $$ K_S = [V_{\mathrm{Mg}}^{''}][V_{\mathrm{O}}^{\bullet\bullet}] = \exp\!\left(-\dfrac{G_S^{f}}{kT}\right) $$ - Charge neutrality $$ [V_{\mathrm{Mg}}^{''}] = [V_{\mathrm{O}}^{\bullet\bullet}] $$ ## KV notation for Frenkel defects Example: Frenkel pairs in LiF (Lithium escaping to interstitial sites) - Formation reaction $$ \mathrm{Li}_{\mathrm{Li}}^{\times} \rightarrow V_{\mathrm{Li}}^{'} + \mathrm{Li}_i^{\bullet} $$ - Equilibrium condition - $G_F^f$: formation free energy for Schottky defects (per reaction) $$ K_F = [V_{\mathrm{Li}}^{'}][\mathrm{Li}_i^{\bullet}] = \exp\!\left(-\dfrac{G_F^{f}}{kT}\right) $$ - Charge neutrality $[V_{\mathrm{Li}}^{'}] = [\mathrm{Li}_i^{\bullet}]$ ## Intrinsic vs extrinsic defects - **Intrinsic** defects are those controlled by thermodynamics $G_S^f$, $G_F^f$ or $G_V^f$ - **Extrinsic** defects are those defects added to the ionic crystal via doping Example of Extrinsic defects: CdCl$_2$ in NaCl - Dopant incorporation reaction $$ \mathrm{CdCl}_2 \rightarrow \mathrm{Cd}_{\mathrm{Na}}^{\bullet} + 2\,\mathrm{Cl}_{\mathrm{Cl}}^{\times} + V_{\mathrm{Na}}^{'} $$ - Extrinsic defect concentration $$ [V_{\mathrm{Na}}^{'}]_{\text{ext}} = [\mathrm{Cd}_{\mathrm{Na}}^{\bullet}] = [\mathrm{CdCl}_2] $$ - Total vacancy concentration $$ [V_{\mathrm{Na}}^{'}] \approx [V_{\mathrm{Na}}^{'}]_s + [V_{\mathrm{Na}}^{'}]_{\text{ext}} = \frac{\exp\!(-\dfrac{G_S^{f}}{k_B T})}{ [\mathrm{CdCl}_2]} + [\mathrm{CdCl}_2] $$ ## Diffusivity of ionic species - We can still use the vacancy exchange mechanism for the diffusivity of ionic species - For f.c.c. lattice the neighbor sites of same charge, the multiplicity $z=6$. - E.g. Na$^+$ exchanges with its own vacancy $V_{\mathrm{Na}}^{'}$ - $[V_{\mathrm{Na}}^{'}]$ depends on the $T$-regime! ```{=tex} \begin{align} D_{\mathrm{Na}} &= [V_{\mathrm{Na}}^{'}]\, f\,\lambda^2\,\nu\, \exp\!\left(-\frac{G_{\mathrm{Na}}^{m}}{kT}\right) \end{align} ``` ## Intrinsic vs extrinsic regimes - Extrinsic: vacancy dominated by doped materials - Low-$T$ regime (high $1/T$) ```{=tex} \begin{align} D_{\mathrm{Na,ext}} &= [\mathrm{CdCl}_2]\, f\,\lambda^2\,\nu\, \exp\!\left(\frac{S_{\mathrm{Na}}^{m}}{k}\right) \exp\!\left(-\frac{H_{\mathrm{Na}}^{m}}{kT}\right) \end{align} ``` - Intrinsic: vacancy dominated by thermal dissociation - High-$T$ regime (low $1/T$) ```{=tex} \begin{align} D_{\mathrm{Na,int}} &= f\,\lambda^2\,\nu\, \exp\!\left(\frac{S_{S}^{f}}{2k}\right) \exp\!\left(\frac{S_{\mathrm{Na}}^{m}}{k}\right) \exp\!\left(-\frac{H_{S}^{f}}{2kT}\right) \exp\!\left(-\frac{H_{\mathrm{Na}}^{m}}{kT}\right) \end{align} ``` ## Two-regimes in the Arrhenius plot ![](./public/two-regimes-plot.png) ## More regimes in Arrhenius plot - Example: cation diffusion in FeO during oxidation - Equilibrium depends on both $G^f$ and $p(O_2)$ - Multiple regimes! ![](./public/kom-feo-reaction.png) ## Multiple-regime diffusion in polycrystalline materials ![Arrhenius plot for diffusion on imperfections](./public/multi-regime-polycrystalline.png) ## Diffusion paths at crystal imperfections - Diffusion can occur along non-bulk pathways - Typical crystal imperfections - Grain boundary and interface diffusion: 2D - Free surface diffusion: 2D - Dislocation (pipe) diffusion: 1D - Vacancy / defect: 0D - Imperfections are associated with lower migration / activation energy! - Think as "shortcuts" during diffusion ## Imperfection 1: grain boundaries ![](./public/gb.png) ## Grain boundary diffusion ![](./public/gb-diff.png) ## Harrison's ABC model for GB diffusion A foreign material is coated on the top of a polycrystalline metal. The degree of penetration can be studied by comparing the timescale $t$, interatomic distance $\lambda$ and grain size $s$ :::{.columns} :::{.column width="50%"} - **A**ll regime - $D_{XL} t > s^2$ - $D_B t > s^2$ - **B**oundary regime - $D_{XL} t \approx \lambda^2$ - $D_B t > \lambda^2$ - coupled short-circuit and bulk diffusion - **C**ore regime - $D_{XL} t < \lambda^2$ - $D_B t > \lambda^2$ - diffusion confined to imperfections ::: :::{.column width="50%"} ![](./public/harrison-abc.png) ::: ::: ## Dislocation imperfections (line defect) :::{.columns} :::{.column width="50%"} ![Edge and screw dislocations](./public/dislocation.png) ::: :::{.column width="50%"} ![Partial dislocations](./public/partial-dislocation.png) ::: ::: ## Example of diffusion along imperfection: deposition on graphene - See Vagli and Tian et al. _Nat Commun_ 2025, 16, 7726. - Diffusivity change on free graphene surface can be probed by deposition geometry! ![](./public/ncomm-sem.png) ## Deposition on Graphene - theoretical simulations - Kinetic Monte Carlo (kMC) assuming different diffusion barrier on imperfections - Faster diffusion direction --> lower density ![](./public/ncomm-kmc1.png) ## Deposition on Graphene - theoretical simulations - Kinetic Monte Carlo (kMC) assuming different diffusion barrier on imperfections - Faster diffusion direction --> lower density ![](./public/ncomm-kmc2.png) ## Deposition on Graphene - KMC vs experiments ![](./public/ncomm-kmc-exp.png) ## Summary In this lecture, we reviewed a few sample cases where diffusion is dominated by crystal defects and imperfections - Diffusion by ionic defects -- multiple Arrhenius regimes - Diffusion by imperfections -- shortcut diffusion compared with bulk diffusion - Example of interface mediated diffusion -- 2D material