MATE 664 Lecture 12

Recitation 1: Kinetics Fundamentals & Diffusion Theory

Author

Dr. Tian Tian

Published

February 11, 2026

Note

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

Learning outcomes

This is a recitation class. We will review the main topics covered in the course so far.

After this lecture, you will be able to:

Recall the major topics covered in the kinetics course
Describe links between irreversible thermodynamics and kinetics
Identify analogies between diffusion and other kinetic problems
Interpret important charts used in diffusion analysis
Apply course ideas to modern materials research examples

Outlines

Kinetics $\approx$ non-equilibrium –> Equilibrium L01
How to describe non-equilibrium process: Force–flux relation L02 & L03
Diffusion laws from driving force L04 & L05
Solving Fick’s equations L06 & L07
Numerical solution to Fick’s equations L08
Atomic models for diffusion – Einstein relations L08 & L09
Diffusion in ideal crystals L10 & defects/short-circuit L11

Conceptual difference: kinetics vs thermodynamics

Kinetics vs thermodynamics on a phase diagram

Entropic view of irreversible thermodynamics

Any non-equilibrium system going back to equilibrium 👉 entropy generation
For local system, entropy generation is always non-zero
Entropy can flow between local systems 👉 root cause of diffusion

\[ \dot{\sigma} = \frac{\partial s}{\partial t} + \nabla\cdot\vec{J}_{s} \geq 0 \]

Entropy flux and generation

Entropy in the system is nothing but some descriptor of how energy and quantities flow
Flow of entropy ⇔ Flow of quantities
Entropy generation ⇔ Magnitude of quantity flow

\[\begin{align} \vec{J}_s &= \sum_i \frac{\partial s}{\partial \xi_i} \vec{J}_{\xi_i} \\ \dot{\sigma} &= \sum_i \vec{J}_{\xi_i} \cdot \nabla \left(\frac{\partial s}{\partial \xi_i}\right) \end{align}\]

Quantity – flux – potential relation

Link in thermodynamics \[ ds = \frac{1}{T}du - \frac{p}{T}dv - \sum_i \frac{\psi_i}{T} d\xi_i \]

$\xi_i$: extensive variables
- volume $v$
- charge $q$
- concentration $c$
- surface area $A$
- dipole moment $\mathbf{p}$
- magnetic moment $\mathbf{b}$

$\psi_i$: conjugate intensive variables
- pressure $p$
- electric potential $\phi$
- chemical potential $\mu$
- surface energy $\gamma$
- external electric field $\mathbf{E}$
- magnetic field $\mathbf{H}$

Each $(\psi_i,\xi_i)$ pair contributes to entropy change.

Flux – potential relation: driving force

Extensive quantities: $\xi_i$
Conjugate driving forces: \[ \vec{F}_i \equiv - \nabla \psi_i \]
Associated fluxes: $\vec{J}_i$

Matrix Form

\[ \vec{J} = \mathbf{L}\,\vec{F} \]

equivalent: \[ \vec{J}_i = \sum_j L_{ij}\,\vec{F}_j \]

Kinetics vs thermodynamics: behaviour

Probability follow Boltzmann distribution $p \propto \exp\left(-\frac{E + C N}{T}\right)$
Scaled potential / entropy $C = T \mu = T \partial S/\partial N$

Demo link Copyright: Vilas Winstein (UC Berkeley)

Mass transfer from driving force

Chemical potential $\mu = \left(\frac{\partial U}{\partial N}\right)_{S,V} = \left(\frac{\partial G}{\partial N}\right)_{P,T}$
$\mu$ is driving force for mass transfer / diffusion
Entropy production due to diffusion: \[ T\dot{\sigma} = -\vec{J}_m \cdot \nabla \mu \]
Linear law: \[ \vec{J}_m = -L_{MM}\nabla \mu \]
$L_{MM}$: phenomenological mobility coefficient

Macroscopic mass transfer: diffusivity – mobility

Force balance and drift velocity ($M$: mobility): \[ v = M \nabla \mu \]
Mass flux: \[ \vec{J} = c v = -M c \nabla \mu \]
Diffusion coefficient: \[ D = M k_B T \]

Solving mass transfer: Fick’s first law

Substitute $\mu$ with $c$
Works for ideal mixture / dilute system
Isotropic medium ($D_{\alpha \beta}=D=\text{Const}$)
Concentration gradient is a special case of $\nabla \mu$

For species $i$ \[ \vec{J} = -D \nabla c \]

Solving mass transfer: Fick’s second law

Mass conservation (no source term) \[ \frac{\partial c}{\partial t} = -\nabla \cdot \vec{J} \]
Substitution: \[ \frac{\partial c}{\partial t} = \nabla \cdot (D \nabla c) \]
If $D_i$ is constant: \[ \frac{\partial c}{\partial t} = D_i \nabla^2 c \]
- $\nabla^2$: Laplace operator

Overview of solutions to mass transfer

Mass transfer equation as engine for kinetic problems.

Using diffusivity $D$: different scenarios

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

Comparison between C- and V-frame

Implications of V-frame interdiffusion

Kirkendall effect
- Movement of interface
- Creation of vacancy voids
Darken’s equation for interdiffusion
- Flux form: \[ J_i^{V} = -\tilde{D} \frac{\partial C_i}{\partial x} \]
- Interdiffusivity: \[ \tilde{D} = D_1 X_2 + D_2 X_1 \]

Solving Fick’s equation

Steady-state solution
- Geometry (planar / spherical / cylindrical)?
- Integration over non-isotropic
Time-dependent (unsteady-state) solution
- Analytical: semi-infinite / point source
- Superimposition of BC and solutions
- Separation of variables method – Fourier transform
- Laplace transform – analysis of temporal decay

Finite difference for Fick’s equation

Diffusion length scale $L_D \approx \sqrt{4Dt}$

derivative	finite-difference approximation	scheme
$\displaystyle \frac{\partial c}{\partial t}$	$\displaystyle \frac{c(i,\,j+1) - c(i,\,j)}{\Delta t}$	forward (time)
$\displaystyle \frac{\partial c}{\partial x}$	$\displaystyle \frac{c(i+1,\,j) - c(i-1,\,j)}{2\Delta x}$	central (space)
$\displaystyle \frac{\partial^2 c}{\partial x^2}$	$\displaystyle \frac{c(i+1,\,j) - 2c(i,\,j) + c(i-1,\,j)}{(\Delta x)^2}$	central (space)

MOL algorithm (explicit time marching)

Define spatial grid with $x_i$ ($N$ points), choose $\Delta x$
Apply boundary conditions at $i=0$ and $i=N$
Discretize spatial derivatives using central difference
Obtain ODE system for $c(i,t)$
Choose time step $\Delta t$
Advance solution in time using forward Euler:

\[\begin{align} c(i,j+1) &= c(i,j) + \Delta t \, \frac{d c(i,t)}{d t} \\ &= c(i,j) + \Delta t \cdot D \frac{c(i+1,t) - 2c(i,t) + c(i-1,t)}{(\Delta x)^2} \end{align}\]

Macroscopic $D$ ⇔ atomic movements

Einstein relation: from continuum solution <–> diffusivity
Random walk model (moluchowski): diffusivity from probability model
Same conclusion

\[ D = \frac{\Gamma \langle r^2\rangle}{2d} \]

3D: $d=3$
2D: $d=2$
1D: $d=1$

Diffusivity in different environments / states

All follow Einstein relation $D = \frac{\Gamma \langle r^2\rangle}{6}$

Gas: kinetic theory ($T, P$)
Liquid: Stokes-Einstein equation ($\mu, M, R$)
Solid: activation energy / potential well ($S^m, H^m$)

Diffusion in solid: general equation

Vacancy mechanism

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

Diffusion in ionic crystals

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 $1/T$)

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

Diffusion in imperfections

Generally $D^{\text{imp}} \gg D^{XL}$
Shortcuts in diffusion pathways
Lengthscale comparison

Seminal Topics In Diffusion

Up-hill diffusion

Can diffusion happen against concentration gradient?

Determination of diffusion in varying $D$ system

Boltzmann-Matano analysis of interdiffusion

Diffusion models in machine learning

AI image generation follows a diffusion process!

What to learn next

In the second half of the course, we will cover the following questions

How do materials evolve when chemical potentials are not at equilibrium?
- In-depth study of phase diagrams
- Continuous phase transformation: spinodal decomposition
- Phase transformation with barrier: nucleation
How do material interface evolve in non-equilibrium process?
- Heat / mass-transfer at interfaces: solidification
- Surface-energy-mediated transformation: sintering
- Interplay of diffusion & reaction: aggregation phenomena
How do we simulate / predict material kinetics?
- Macroscopic pattern formation: phase-field method
- Kinetic Monte-Carlo (KMC) simulations
- Molecular dynamics (MD) simulations
- Obtaining thermodynamic parameters from first principles calculations and machine learning

Summary

This recitation reviewed the major kinetics topics covered in the first half of the course
Irreversible thermodynamics provides a common framework for diffusion and related kinetic problems
Diffusion concepts, solution methods, and atomistic pictures connect across many materials systems

--- title: "MATE 664 Lecture 12" subtitle: "Recitation 1: Kinetics Fundamentals & Diffusion Theory" author: "Dr. Tian Tian" date: "2026-02-11" format: html: {} revealjs: output-file: slides.html pdf: output-file: L12.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L12.pdf) - Handwritten notes 👉 [Open in pdf](./public/L12_annotated.pdf) ::: ::: ## Learning outcomes {.center} This is a recitation class. We will review the main topics covered in the course so far. After this lecture, you will be able to: - **Recall** the major topics covered in the kinetics course - **Describe** links between irreversible thermodynamics and kinetics - **Identify** analogies between diffusion and other kinetic problems - **Interpret** important charts used in diffusion analysis - **Apply** course ideas to modern materials research examples ## Outlines - **Kinetics** $\approx$ **non-equilibrium** --> Equilibrium [L01](../L01) - How to describe **non-equilibrium process**: Force–flux relation [L02](../L02) & [L03](../L03) - **Diffusion laws** from driving force [L04](../L04) & [L05](../L05) - Solving **Fick's equations** [L06](../L06) & [L07](../L07) - **Numerical** solution to Fick's equations [L08](../L08) - Atomic models for diffusion -- **Einstein relations** [L08](../L08) & [L09](../L09) - Diffusion in **ideal crystals** [L10](../L10) & **defects/short-circuit** [L11](../L11) ## Conceptual difference: kinetics vs thermodynamics ![Kinetics vs thermodynamics on a phase diagram](./public/phase-diag-compare.svg) ## Entropic view of irreversible thermodynamics - Any non-equilibrium system going back to equilibrium 👉 entropy generation - For local system, entropy generation is always non-zero - Entropy can flow between local systems 👉 root cause of diffusion $$ \dot{\sigma} = \frac{\partial s}{\partial t} + \nabla\cdot\vec{J}_{s} \geq 0 $$ ## Entropy flux and generation - Entropy in the system is nothing but some **descriptor** of how energy and quantities **flow** - **Flow of entropy** ⇔ **Flow of quantities** - **Entropy generation** ⇔ **Magnitude of quantity flow** ```{=tex} \begin{align} \vec{J}_s &= \sum_i \frac{\partial s}{\partial \xi_i} \vec{J}_{\xi_i} \\ \dot{\sigma} &= \sum_i \vec{J}_{\xi_i} \cdot \nabla \left(\frac{\partial s}{\partial \xi_i}\right) \end{align} ``` ## Quantity -- flux -- potential relation - Link in thermodynamics $$ ds = \frac{1}{T}du - \frac{p}{T}dv - \sum_i \frac{\psi_i}{T} d\xi_i $$ ::: columns ::: {.column width="50%"} - $\xi_i$: **extensive variables** - volume $v$ - charge $q$ - concentration $c$ - surface area $A$ - dipole moment $\mathbf{p}$ - magnetic moment $\mathbf{b}$ ::: ::: {.column width="50%"} - $\psi_i$: **conjugate intensive variables** - pressure $p$ - electric potential $\phi$ - chemical potential $\mu$ - surface energy $\gamma$ - external electric field $\mathbf{E}$ - magnetic field $\mathbf{H}$ ::: ::: Each $(\psi_i,\xi_i)$ pair contributes to entropy change. ## Flux -- potential relation: driving force - Extensive quantities: $\xi_i$ - Conjugate driving forces: $$ \vec{F}_i \equiv - \nabla \psi_i $$ - Associated fluxes: $\vec{J}_i$ Matrix Form $$ \vec{J} = \mathbf{L}\,\vec{F} $$ equivalent: $$ \vec{J}_i = \sum_j L_{ij}\,\vec{F}_j $$ ## Kinetics vs thermodynamics: behaviour ::: {.content-visible when-format="html" unless-format="revealjs"} ```{=html} <iframe width="100%" height="500" src="https://vilas.us/simulations/liquidvapor/" title="Webpage example"></iframe> ``` ::: - Probability follow Boltzmann distribution $p \propto \exp\left(-\frac{E + C N}{T}\right)$ - Scaled potential / entropy $C = T \mu = T \partial S/\partial N$ [Demo link](https://vilas.us/simulations/liquidvapor/) _Copyright: Vilas Winstein (UC Berkeley)_ ## Mass transfer from driving force - Chemical potential $\mu = \left(\frac{\partial U}{\partial N}\right)_{S,V} = \left(\frac{\partial G}{\partial N}\right)_{P,T}$ - $\mu$ is driving force for mass transfer / diffusion - Entropy production due to diffusion: $$ T\dot{\sigma} = -\vec{J}_m \cdot \nabla \mu $$ - Linear law: $$ \vec{J}_m = -L_{MM}\nabla \mu $$ - $L_{MM}$: phenomenological mobility coefficient ## Macroscopic mass transfer: diffusivity -- mobility - Force balance and drift velocity ($M$: mobility): $$ v = M \nabla \mu $$ - Mass flux: $$ \vec{J} = c v = -M c \nabla \mu $$ - Diffusion coefficient: $$ D = M k_B T $$ ## Solving mass transfer: Fick’s first law - Substitute $\mu$ with $c$ - Works for ideal mixture / dilute system - Isotropic medium ($D_{\alpha \beta}=D=\text{Const}$) - Concentration gradient is a special case of $\nabla \mu$ For species $i$ $$ \vec{J} = -D \nabla c $$ ## Solving mass transfer: Fick’s second law - Mass conservation (no source term) $$ \frac{\partial c}{\partial t} = -\nabla \cdot \vec{J} $$ - Substitution: $$ \frac{\partial c}{\partial t} = \nabla \cdot (D \nabla c) $$ - If $D_i$ is constant: $$ \frac{\partial c}{\partial t} = D_i \nabla^2 c $$ - $\nabla^2$: Laplace operator ## Overview of solutions to mass transfer ![Mass transfer equation as engine for kinetic problems.](./public/uss-engine.png) ## Using diffusivity $D$: different scenarios | Diffusivity | Frame | Meaning | |---|---|---| | $D_i^{*}$ | lattice | tracer / self-diffusion | | $D_i$ | C-frame | intrinsic diffusivity | | $\tilde{D}$ | V-frame | interdiffusivity | ## Comparison between C- and V-frame ![Reference frames analog: stream](./public/stream-analog.png) ## Implications of V-frame interdiffusion - Kirkendall effect - Movement of interface - Creation of vacancy voids - Darken's equation for interdiffusion - Flux form: $$ J_i^{V} = -\tilde{D} \frac{\partial C_i}{\partial x} $$ - Interdiffusivity: $$ \tilde{D} = D_1 X_2 + D_2 X_1 $$ ## Solving Fick's equation - Steady-state solution - Geometry (planar / spherical / cylindrical)? - Integration over non-isotropic - Time-dependent (unsteady-state) solution - Analytical: semi-infinite / point source - Superimposition of BC and solutions - Separation of variables method -- Fourier transform - Laplace transform -- analysis of temporal decay ## Finite difference for Fick's equation - Diffusion length scale $L_D \approx \sqrt{4Dt}$ | derivative | finite-difference approximation | scheme | |-----------|----------------------------------|--------| | $\displaystyle \frac{\partial c}{\partial t}$ | $\displaystyle \frac{c(i,\,j+1) - c(i,\,j)}{\Delta t}$ | forward (time) | | $\displaystyle \frac{\partial c}{\partial x}$ | $\displaystyle \frac{c(i+1,\,j) - c(i-1,\,j)}{2\Delta x}$ | central (space) | | $\displaystyle \frac{\partial^2 c}{\partial x^2}$ | $\displaystyle \frac{c(i+1,\,j) - 2c(i,\,j) + c(i-1,\,j)}{(\Delta x)^2}$ | central (space) | ## MOL algorithm (explicit time marching) 1. Define spatial grid with $x_i$ ($N$ points), choose $\Delta x$ 2. Apply boundary conditions at $i=0$ and $i=N$ 3. Discretize spatial derivatives using central difference 4. Obtain ODE system for $c(i,t)$ 5. Choose time step $\Delta t$ 6. Advance solution in time using forward Euler: ```{=tex} \begin{align} c(i,j+1) &= c(i,j) + \Delta t \, \frac{d c(i,t)}{d t} \\ &= c(i,j) + \Delta t \cdot D \frac{c(i+1,t) - 2c(i,t) + c(i-1,t)}{(\Delta x)^2} \end{align} ``` ## Macroscopic $D$ ⇔ atomic movements - Einstein relation: from continuum solution <--> diffusivity - Random walk model (moluchowski): diffusivity from probability model - Same conclusion $$ D = \frac{\Gamma \langle r^2\rangle}{2d} $$ - 3D: $d=3$ - 2D: $d=2$ - 1D: $d=1$ ## Diffusivity in different environments / states All follow Einstein relation $D = \frac{\Gamma \langle r^2\rangle}{6}$ - Gas: kinetic theory ($T, P$) - Liquid: Stokes-Einstein equation ($\mu, M, R$) - Solid: activation energy / potential well ($S^m, H^m$) ## Diffusion in solid: general equation - Vacancy mechanism ```{=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} ``` ## Diffusion in ionic crystals - 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} ``` ## Diffusion in imperfections - Generally $D^{\text{imp}} \gg D^{XL}$ - Shortcuts in diffusion pathways - Lengthscale comparison # Seminal Topics In Diffusion ## Up-hill diffusion - Can diffusion happen against **concentration gradient**? ![KOM Q3.3](./public/uphill.png) ## Determination of diffusion in varying $D$ system - Boltzmann-Matano analysis of interdiffusion ![KOM Q4.1](./public/boltzmann-matano.png) ## Diffusion models in machine learning - AI image generation follows a diffusion process! ![Copyright U Tokyo](./public/diffusion-model.png) ## What to learn next In the second half of the course, we will cover the following questions - How do materials evolve when chemical potentials are not at equilibrium? - In-depth study of **phase diagrams** - Continuous phase transformation: **spinodal decomposition** - Phase transformation with barrier: **nucleation** - How do material interface evolve in non-equilibrium process? - Heat / mass-transfer at interfaces: **solidification** - Surface-energy-mediated transformation: **sintering** - Interplay of diffusion & reaction: **aggregation phenomena** - How do we simulate / predict material kinetics? - Macroscopic pattern formation: **phase-field method** - **Kinetic Monte-Carlo** (KMC) simulations - **Molecular dynamics** (MD) simulations - Obtaining thermodynamic parameters from **first principles calculations** and **machine learning** ## Summary - This recitation reviewed the major kinetics topics covered in the first half of the course - Irreversible thermodynamics provides a common framework for diffusion and related kinetic problems - Diffusion concepts, solution methods, and atomistic pictures connect across many materials systems