MATE 664 Lecture 10

Atomic Models for Diffusion (III): Ideal Crystals

Author

Dr. Tian Tian

Published

February 4, 2026

Note

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

Recap: atomic model for diffusion

Random walk picture
Einstein relation links jump statistics to diffusivity
Correlation factor $f$ corrects for non-independent motion

\[\begin{align} D = \frac{\Gamma \langle r^2 \rangle}{6}\, f \end{align}\]

Recap: diffusion in gases and liquids

Gas diffusion from kinetic theory ($D = f(T, p)$)
Liquid diffusion from mobility and viscous drag ($D = f(\mu, T)$)
- Often $\eta \propto \exp(B/T)$

\[\begin{align} D_{\mathrm{gas}} &= \frac{1}{3}\langle u \rangle \lambda \propto T^{3/2}/p \\ D_{\mathrm{liquid}} &= M k_B T \approx \frac{k_B T}{6\pi \eta R} \end{align}\]

Learning outcomes

After this lecture, you will be able to:

Link thermodynamic properties to the diffusivity in ideal solids
Recall the potential well model for solid diffusivity
Analysis of the Arrhenius equation / plot for solids

Diffusion models in solids

Multiple mechanisms exist (what is a machanism?)
Mechanism depends on lattice, bonding, charge, and size mismatch
Two broad classes of diffusing species:
- substitutional: A occupies the lattice site of B
  - interstitial: A is inserted between neighbouring B-sites

Two basic mechanisms

Substitutional diffusion - atom replaces a lattice site - usually vacancy-mediated

Interstitial diffusion - atom moves between host sites - solute squeezes through interstitial positions

Barrier picture for diffusion

Both mechanisms encounter an energy barrier
Goal remains to estimate jump frequency and jump length

\[\begin{align} D = \frac{\Gamma \langle r^2 \rangle}{6} \end{align}\]

Generalized well potential

Particle resides in a potential well
It must cross an activation barrier $E_a$
Activated population follows Boltzmann statistics

\[\begin{align} \frac{P_{\mathrm{activated}}}{P_{\mathrm{well}}} = \exp\!\left(-\frac{E_a}{k_B T}\right) \end{align}\]

Step 1: crossing time

Ask how long one activated particle needs to cross the barrier region
Estimate by distance over average speed

\[\begin{align} \tau_{\mathrm{cross}} = \frac{L_A}{\langle v \rangle} \approx \frac{L_A}{\sqrt{k_B T/(2\pi m)}} = L_A \sqrt{\frac{2\pi m}{k_B T}} \end{align}\]

Does it make sense? A few implications: - Larger $L_A$ gives longer crossing time - Higher $T$ gives shorter crossing time - Heavier particles cross more slowly

Step 2: how many particles can cross?

Crossing rate is number of activated particles divided by crossing time

\[\begin{align} R_{\mathrm{cross}} = \frac{N^\#}{\tau_{\mathrm{cross}}} \end{align}\]

where the total number of crossing $N^\#$ follows

\[\begin{align} N^\# \approx N \frac{\tau_A}{\tau_W} \end{align}\]

$\tau_A$: time spent in activated region
$\tau_W$: time spent in well

Step 3: crossing rate and jump frequency

Combine activated fraction with crossing time
Jump frequency behaves like a first-order rate constant

\[\begin{align} R_{\mathrm{cross}} &= N \left(\frac{\tau_A/\tau_W}{\tau_{\mathrm{cross}}}\right) \\ \Gamma' &= \frac{\tau_A}{\tau_W}\frac{1}{\tau_{\mathrm{cross}}} \end{align}\]

Step 4: probability ratio from the well shape

Ratio of residence times scales with probability of occupying each region For a simple well model, we have

\[\begin{align} \frac{\tau_A}{\tau_W} = \frac{L_A}{L_W}\exp\!\left(-\frac{E_a}{k_B T}\right) \end{align}\]

$L_W$: well width
$L_A$: activated-region width

Arrhenius jump frequency

Substituting the crossing time removes explicit dependence on $L_A$, so that the jump frequency $\Gamma'$ follows a Boltzmann distribution!

\[\begin{align} \Gamma' &= \sqrt{\frac{k_B T}{2\pi m}}\frac{1}{L_W} \exp\!\left(-\frac{E_a}{k_B T}\right) \end{align}\]

If we define the attempt frequency $\nu$ (frequency dependent on thermal kinetic energy and well geometry)

\[\begin{align} \nu = \sqrt{\frac{k_B T}{2\pi m}}\frac{1}{L_W} \end{align}\]

Final form for the jump frequency:

\[\begin{align} \Gamma' = \nu \exp\!\left(-\frac{E_a}{k_B T}\right) \end{align}\]

From jump frequency to diffusivity

In 1D, the random-walk expression becomes

\[\begin{align} D = \frac{\Gamma' \langle r^2 \rangle}{2} \end{align}\]

What is $\langle r^2 \rangle$? For this well, $r \sim L_W + L_A$:

\[\begin{align} D = \sqrt{\frac{k_B T}{8\pi m}} \frac{(L_W+L_A)^2}{L_W} \exp\!\left(-\frac{E_a}{k_B T}\right) \end{align}\]

More realistic potential landscape

Real barriers are not rectangular
Near a minimum, a parabolic approximation is often better 👉 parabolic well model

\[\begin{align} E(x) = E_{\min} + \frac{\beta}{2}(x-x_{\min})^2 \end{align}\]

Barrier height (activation energy) $E_a$ approximated by:

\[\begin{align} \frac{\beta}{2}\left(\frac{L_W}{2}\right)^2 = E_a \end{align}\]

Modified attempt frequency for a parabolic well

The jump frequency in a parabolic well follows:

\[\begin{align} \Gamma' = \frac{1}{2\pi}\sqrt{\frac{\beta}{m}} \exp\!\left(-\frac{E_a}{k_B T}\right) \end{align}\]

Can be rewritten with $\nu = \dfrac{1}{2\pi}\sqrt{\beta/m}$
Still follows the Boltzmann distribution!

What is missing from the simple picture?

One-particle picture may be too simple
Energy landscape can be many-body
Landscape may change with local environment
Many configurations can share the same barrier
Real lattices add symmetry factors
Jumps may be correlated

General activated form

More generally, write the jump frequency using activation free energy

\[\begin{align} \Gamma' &= \nu \exp\!\left(-\frac{G^m}{k_B T}\right) \\ &= \nu \exp\!\left(\frac{S^m}{k_B}\right) \exp\!\left(-\frac{H^m}{k_B T}\right) \end{align}\]

$G^m = H^m - T S^m$: superscript $m$ denotes migration

Entropy contribution in migration

Activation entropy $S^m$ modifies the prefactor
It usually varies weakly with temperature, because it’s usually estimated from all vibrational modes with angular momenta $\omega$

\[\begin{align} S^m = k_B \left[ 2\ln\!\left(\frac{\omega_J^\ddagger}{\omega_J}\right) + \sum_{i=1}^{3N-7}\ln\!\left(\frac{\omega_i}{\omega_i^\ddagger}\right) \right] \end{align}\]

Analog: angular momenta of springs near the stationary point.

Arrhenius equation for diffusion

Using our previous analysis, it can be shown that plugging the jump frequency into the random-walk diffusivity equation is basically the famous Arrhenius form of diffusivity

\[\begin{align} D = D_0 \exp\!\left(-\frac{H^m}{k_B T}\right) \end{align}\]

$D_0$ contains geometric and entropic terms
$H^m$ is the activation enthalpy
In an Arrhenius plot ($\ln\!D$ vs $1/T$), the slope is only related with $H^m$! (see Lecture 01)

Application to real materials

For real materials, the Einstein relation / random-walk analog still holds, while a few more factors need to be considered:

\[\begin{align} D = \frac{\Gamma \langle r^2 \rangle}{6}\,f = \frac{z\,\Gamma'\langle r^2 \rangle}{6}\,f \end{align}\]

$z$ counts equivalent jump paths (e.g. equivalent neighbouring sites in f.c.c. lattice)
$f$ is the correlation factor (usually around 0.7)

Case 1: interstitial diffusion

Interstitial jumps are often uncorrelated - After one interstitial jump, the next jump is not strongly biased - No vacancy left behind to pull the atom back - Successive jumps are approximately independent

\[\begin{align} D_I = \frac{z\,\nu \langle r^2 \rangle}{6} \exp\!\left(\frac{S_I^m}{k_B}\right) \exp\!\left(-\frac{H_I^m}{k_B T}\right) \end{align}\]

Case 2: vacancy self-diffusion

Atom-vacancy exchange makes the vacancy appear to move
Diffusion still uses the migration barrier (for vacancy)
Usually uncorrelated ($f \approx 1$)

\[\begin{align} D_V = \frac{z\,\nu \langle r^2 \rangle}{6} \exp\!\left(\frac{S_V^m}{k_B}\right) \exp\!\left(-\frac{H_V^m}{k_B T}\right) \end{align}\]

Case 3: vacancy-assisted solute diffusion

Solute motion requires a nearby vacancy
Combine vacancy availability with vacancy migration
Correlated motion!

\[\begin{align} D_A &= X_V \, D_V f \end{align}\]

Vacancy concentration is related with the vacancy formation free energy $G_V^f$

\[\begin{align} X_V = \exp\!\left(-\frac{G_V^f}{k_B T}\right) = \exp\!\left(\frac{S_V^f}{k_B}\right) \exp\!\left(-\frac{H_V^f}{k_B T}\right) \end{align}\]

Vacancy-assisted diffusion: final form

After combining vacancy formation and migration, $D_A$ becomes dependent on both vacancy formation enthalpy and vacancy migration barrier!

\[\begin{align} D_A = \frac{z\,\nu \langle r^2 \rangle}{6} \exp\!\left(\frac{S_V^f + S_m}{k_B}\right) \exp\!\left(-\frac{H_V^f + H_m}{k_B T}\right)\,f \end{align}\]

Why can $f < 1$ in vacancy-mediated diffusion?

Vacancy jumps create “memory”
A just-moved atom is likely to jump back into the nearby vacancy
- This reduces net transport efficiency
Approximate estimate from jump-back bias:

\[\begin{align} f \simeq \frac{1+\langle \cos\theta \rangle}{1-\langle \cos\theta \rangle} \end{align}\]

Correlation factor in practice

Simple estimate with $z$

\[\begin{align} f \simeq \frac{1-\frac{1}{z}}{1+\frac{1}{z}} \end{align}\]

For substitutional diffusion, $f$ is commonly below 1
For fcc, a typical value is about $0.78$
In many cases, using $f \sim 0.7$–$0.8$ is reasonable

Other correlated mechanisms

Ring mechanisms
Interstitialcy mechanisms
Cooperative multi-atom motion (e.g. push-out mechanism)

Summary

After today’s lecture, you should be able to - recognize diffusion in solids as an process associated with activation energy - link diffusivity with potential well picture - analyze the enthalpy dependency in the Arrhenius-type diffusivity equations

Next topics

In next lecture, we will discuss diffusion processes with varied energy barriers, in particular:

Diffusion in ionic solids
Diffusion in imperfect solids
Short-circuit diffusion

--- title: "MATE 664 Lecture 10" subtitle: "Atomic Models for Diffusion (III): Ideal Crystals" author: "Dr. Tian Tian" date: "2026-02-04" format: html: {} revealjs: output-file: slides.html pdf: output-file: L10.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L10.pdf) - Handwritten notes 👉 [Open in pdf](./public/L10_annotated.pdf) ::: ::: ## Recap: atomic model for diffusion - Random walk picture - Einstein relation links jump statistics to diffusivity - Correlation factor $f$ corrects for non-independent motion ```{=tex} \begin{align} D = \frac{\Gamma \langle r^2 \rangle}{6}\, f \end{align} ``` ## Recap: diffusion in gases and liquids - Gas diffusion from kinetic theory ($D = f(T, p)$) - Liquid diffusion from mobility and viscous drag ($D = f(\mu, T)$) - Often $\eta \propto \exp(B/T)$ ```{=tex} \begin{align} D_{\mathrm{gas}} &= \frac{1}{3}\langle u \rangle \lambda \propto T^{3/2}/p \\ D_{\mathrm{liquid}} &= M k_B T \approx \frac{k_B T}{6\pi \eta R} \end{align} ``` ## Learning outcomes {.center} After this lecture, you will be able to: - Link thermodynamic properties to the diffusivity in ideal solids - Recall the potential well model for solid diffusivity - Analysis of the Arrhenius equation / plot for solids ## Diffusion models in solids - Multiple mechanisms exist (_what is a machanism_?) - Mechanism depends on lattice, bonding, charge, and size mismatch - Two broad classes of diffusing species: - substitutional: A occupies the lattice site of B - interstitial: A is inserted between neighbouring B-sites ## Two basic mechanisms :::{.columns} :::{.column width="50"} Substitutional diffusion - atom replaces a lattice site - usually vacancy-mediated ::: :::{.column width="50"} Interstitial diffusion - atom moves between host sites - solute squeezes through interstitial positions ::: ## Barrier picture for diffusion - Both mechanisms encounter an energy barrier - Goal remains to estimate jump frequency and jump length ```{=tex} \begin{align} D = \frac{\Gamma \langle r^2 \rangle}{6} \end{align} ``` ## Generalized well potential - Particle resides in a potential well - It must cross an activation barrier $E_a$ - Activated population follows Boltzmann statistics ```{=tex} \begin{align} \frac{P_{\mathrm{activated}}}{P_{\mathrm{well}}} = \exp\!\left(-\frac{E_a}{k_B T}\right) \end{align} ``` ## Step 1: crossing time - Ask how long one activated particle needs to cross the barrier region - Estimate by distance over average speed ```{=tex} \begin{align} \tau_{\mathrm{cross}} = \frac{L_A}{\langle v \rangle} \approx \frac{L_A}{\sqrt{k_B T/(2\pi m)}} = L_A \sqrt{\frac{2\pi m}{k_B T}} \end{align} ``` Does it make sense? A few implications: - Larger $L_A$ gives longer crossing time - Higher $T$ gives shorter crossing time - Heavier particles cross more slowly ## Step 2: how many particles can cross? Crossing rate is number of activated particles divided by crossing time ```{=tex} \begin{align} R_{\mathrm{cross}} = \frac{N^\#}{\tau_{\mathrm{cross}}} \end{align} ``` where the total number of crossing $N^\#$ follows ```{=tex} \begin{align} N^\# \approx N \frac{\tau_A}{\tau_W} \end{align} ``` - $\tau_A$: time spent in activated region - $\tau_W$: time spent in well ## Step 3: crossing rate and jump frequency - Combine activated fraction with crossing time - Jump frequency behaves like a first-order rate constant ```{=tex} \begin{align} R_{\mathrm{cross}} &= N \left(\frac{\tau_A/\tau_W}{\tau_{\mathrm{cross}}}\right) \\ \Gamma' &= \frac{\tau_A}{\tau_W}\frac{1}{\tau_{\mathrm{cross}}} \end{align} ``` ## Step 4: probability ratio from the well shape Ratio of residence times scales with probability of occupying each region For a simple well model, we have ```{=tex} \begin{align} \frac{\tau_A}{\tau_W} = \frac{L_A}{L_W}\exp\!\left(-\frac{E_a}{k_B T}\right) \end{align} ``` - $L_W$: well width - $L_A$: activated-region width ## Arrhenius jump frequency Substituting the crossing time removes explicit dependence on $L_A$, so that the jump frequency $\Gamma'$ follows a Boltzmann distribution! ```{=tex} \begin{align} \Gamma' &= \sqrt{\frac{k_B T}{2\pi m}}\frac{1}{L_W} \exp\!\left(-\frac{E_a}{k_B T}\right) \end{align} ``` If we define the attempt frequency $\nu$ (frequency dependent on thermal kinetic energy and well geometry) ```{=tex} \begin{align} \nu = \sqrt{\frac{k_B T}{2\pi m}}\frac{1}{L_W} \end{align} ``` Final form for the jump frequency: ```{=tex} \begin{align} \Gamma' = \nu \exp\!\left(-\frac{E_a}{k_B T}\right) \end{align} ``` ## From jump frequency to diffusivity In 1D, the random-walk expression becomes ```{=tex} \begin{align} D = \frac{\Gamma' \langle r^2 \rangle}{2} \end{align} ``` What is $\langle r^2 \rangle$? For this well, $r \sim L_W + L_A$: ```{=tex} \begin{align} D = \sqrt{\frac{k_B T}{8\pi m}} \frac{(L_W+L_A)^2}{L_W} \exp\!\left(-\frac{E_a}{k_B T}\right) \end{align} ``` ## More realistic potential landscape - Real barriers are not rectangular - Near a minimum, a parabolic approximation is often better 👉 parabolic well model ```{=tex} \begin{align} E(x) = E_{\min} + \frac{\beta}{2}(x-x_{\min})^2 \end{align} ``` - Barrier height (activation energy) $E_a$ approximated by: ```{=tex} \begin{align} \frac{\beta}{2}\left(\frac{L_W}{2}\right)^2 = E_a \end{align} ``` ## Modified attempt frequency for a parabolic well The jump frequency in a parabolic well follows: ```{=tex} \begin{align} \Gamma' = \frac{1}{2\pi}\sqrt{\frac{\beta}{m}} \exp\!\left(-\frac{E_a}{k_B T}\right) \end{align} ``` - Can be rewritten with $\nu = \dfrac{1}{2\pi}\sqrt{\beta/m}$ - Still follows the Boltzmann distribution! ## What is missing from the simple picture? - One-particle picture may be too simple - Energy landscape can be many-body - Landscape may change with local environment - Many configurations can share the same barrier - Real lattices add symmetry factors - Jumps may be correlated ## General activated form - More generally, write the jump frequency using activation free energy ```{=tex} \begin{align} \Gamma' &= \nu \exp\!\left(-\frac{G^m}{k_B T}\right) \\ &= \nu \exp\!\left(\frac{S^m}{k_B}\right) \exp\!\left(-\frac{H^m}{k_B T}\right) \end{align} ``` - $G^m = H^m - T S^m$: superscript $m$ denotes _migration_ ## Entropy contribution in migration - Activation entropy $S^m$ modifies the prefactor - It usually varies weakly with temperature, because it's usually estimated from all vibrational modes with angular momenta $\omega$ ```{=tex} \begin{align} S^m = k_B \left[ 2\ln\!\left(\frac{\omega_J^\ddagger}{\omega_J}\right) + \sum_{i=1}^{3N-7}\ln\!\left(\frac{\omega_i}{\omega_i^\ddagger}\right) \right] \end{align} ``` Analog: angular momenta of springs near the stationary point. ## Arrhenius equation for diffusion Using our previous analysis, it can be shown that plugging the jump frequency into the random-walk diffusivity equation is basically the famous Arrhenius form of diffusivity \begin{align} D = D_0 \exp\!\left(-\frac{H^m}{k_B T}\right) \end{align} - $D_0$ contains geometric and entropic terms - $H^m$ is the activation **enthalpy** - In an Arrhenius plot ($\ln\!D$ vs $1/T$), the slope is only related with $H^m$! (see [Lecture 01](../L01)) ## Application to real materials For real materials, the Einstein relation / random-walk analog still holds, while a few more factors need to be considered: ```{=tex} \begin{align} D = \frac{\Gamma \langle r^2 \rangle}{6}\,f = \frac{z\,\Gamma'\langle r^2 \rangle}{6}\,f \end{align} ``` - $z$ counts equivalent jump paths (e.g. equivalent neighbouring sites in f.c.c. lattice) - $f$ is the correlation factor (usually around 0.7) ## Case 1: interstitial diffusion Interstitial jumps are often uncorrelated - After one interstitial jump, the next jump is not strongly biased - No vacancy left behind to pull the atom back - Successive jumps are approximately independent ```{=tex} \begin{align} D_I = \frac{z\,\nu \langle r^2 \rangle}{6} \exp\!\left(\frac{S_I^m}{k_B}\right) \exp\!\left(-\frac{H_I^m}{k_B T}\right) \end{align} ``` ## Case 2: vacancy self-diffusion - Atom-vacancy exchange makes the vacancy appear to move - Diffusion still uses the migration barrier (_for vacancy_) - Usually uncorrelated ($f \approx 1$) ```{=tex} \begin{align} D_V = \frac{z\,\nu \langle r^2 \rangle}{6} \exp\!\left(\frac{S_V^m}{k_B}\right) \exp\!\left(-\frac{H_V^m}{k_B T}\right) \end{align} ``` ## Case 3: vacancy-assisted solute diffusion - Solute motion requires a nearby vacancy - Combine vacancy availability with vacancy migration - Correlated motion! ```{=tex} \begin{align} D_A &= X_V \, D_V f \end{align} ``` - Vacancy concentration is related with the vacancy formation free energy $G_V^f$ ```{=tex} \begin{align} X_V = \exp\!\left(-\frac{G_V^f}{k_B T}\right) = \exp\!\left(\frac{S_V^f}{k_B}\right) \exp\!\left(-\frac{H_V^f}{k_B T}\right) \end{align} ``` ## Vacancy-assisted diffusion: final form After combining vacancy formation and migration, $D_A$ becomes dependent on both vacancy formation enthalpy and vacancy migration barrier! ```{=tex} \begin{align} D_A = \frac{z\,\nu \langle r^2 \rangle}{6} \exp\!\left(\frac{S_V^f + S_m}{k_B}\right) \exp\!\left(-\frac{H_V^f + H_m}{k_B T}\right)\,f \end{align} ``` ## Why can $f < 1$ in vacancy-mediated diffusion? - Vacancy jumps create "memory" - A just-moved atom is likely to **jump back** into the nearby vacancy - This reduces net transport efficiency - Approximate estimate from jump-back bias: ```{=tex} \begin{align} f \simeq \frac{1+\langle \cos\theta \rangle}{1-\langle \cos\theta \rangle} \end{align} ``` ## Correlation factor in practice Simple estimate with $z$ ```{=tex} \begin{align} f \simeq \frac{1-\frac{1}{z}}{1+\frac{1}{z}} \end{align} ``` - For substitutional diffusion, $f$ is commonly below 1 - For fcc, a typical value is about $0.78$ - In many cases, using $f \sim 0.7$–$0.8$ is reasonable ## Other correlated mechanisms - Ring mechanisms - Interstitialcy mechanisms - Cooperative multi-atom motion (e.g. push-out mechanism) ## Summary After today's lecture, you should be able to - recognize diffusion in solids as an process associated with activation energy - link diffusivity with potential well picture - analyze the enthalpy dependency in the Arrhenius-type diffusivity equations ## Next topics In [next lecture](../L11), we will discuss diffusion processes with varied energy barriers, in particular: - Diffusion in ionic solids - Diffusion in imperfect solids - Short-circuit diffusion