MATE 664 Lecture 04

Introduction to Diffusion

Dr. Tian Tian

2026-01-14

Recap of Lecture 03

Key ideas from last lecture:

Entropy flux and production
Rewritting entropy production in flux and driving force terms
Direct and coupling coefficients
Analysis of several cross-coupling effects in material science

Learning Outcomes

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

Identify driving forces and fluxes for diffusive mass transport
Derive Fick’s first law from irreversible thermodynamics using chemical potential
Estimate magnitudes of diffusivity in gases, liquids, and solids
Explain temperature dependence of diffusivity using Arrhenius-type relations
explain the thermodynamic origin of self-diffusion

Recap: Driving Forces in Irreversible Thermodynamics

entropy balance: \[ \frac{ds}{dt} = -\nabla \cdot \vec{J}_s + \dot{\sigma} \]
entropy flux: \[ \vec{J}_s = \sum_i \frac{\psi_i}{T}\vec{J}_i \]
entropy production: \[ T\dot{\sigma} = -\sum_i \vec{J}_i \cdot \nabla \psi_i \ge 0 \]

When Do \(\dot{\sigma}\) Diminish? Orthogonality of Flux and Driving Force

We will show one example that has non-trivial solution to \(\dot{\sigma} = 0\)

entropy production vanishes if: \[ \vec{J}_i \cdot \nabla \psi_i = 0 \]

Example: Hall effect
- Current flows while electric potential gradient is orthogonal
- Generalized for thermomagnetic and galvanomagnetic effects (Callen Phys. Rev. 1948, 73, 1349)
- Magnetic field induced symmetry breaking
- \(L_{ij}(H) = L_{ji}(-H)\)

Chemical Potential as Driving Force

Consider one chemical species with chemical potential \(\mu\)
Definition: \[ \mu = \left(\frac{\partial U}{\partial N}\right)_{S,V} \]
\(\mu\) represents energy cost of adding more molecules
Diffusion driven by gradients in \(\mu\)

See analog to a water tank in handwritten notes

Entropy Production and Mass Flux

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

From Mobility to Fick’s Law (Determine \(L_MM\))

See analog in handwritten notes

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

Chemical Potential in Mixtures

See analog in handwritten notes

For constant \(T,P\): \[ \mu_i = \left(\frac{\partial G}{\partial N_i}\right)_{T,P} \]
Chemical potential in a mixture solution: \[ \mu_i = \mu_i^0 + k_B T \ln \gamma_i x_i \]
activity coefficient \(\gamma_i = 1\) for ideal solution (Raoult’s law)

Fick’s First Law

Substitute \(\mu_i\) into flux. For species \(i\) \[ \vec{J} = -D \nabla c \]
Assumptions:
- ideal solution
- constant \(T\)
- isotropic medium (\(D_{\alpha \beta}=D=\text{Const}\))
concentration gradient is a special case of \(\nabla \mu\)

What Does Diffusivity Depend On?

temperature
concentration
spatial position (??)
general diffusion driven by \(\nabla \mu\)
Fick’s law valid under restricted conditions

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

One-Dimensional Diffusion in Isotropic Homogeneous Medium

1D equation: \[ \frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2} \]
Steady state: \[ \frac{\partial^2 c}{\partial x^2} = 0 \]
- Linear concentration profile!

Physical Meaning of Laplace Operator

\(\nabla^2 c\) measures curvature (sort of…)
Steady state implies zero curvature
Transient diffusion requires nonzero curvature
Curvature and Second Derivative
- concave profile: \[ \frac{d^2 c}{dx^2} < 0 \]
- convex profile: \[ \frac{d^2 c}{dx^2} > 0 \]
- curvature determines smoothing direction

Typical Magnitudes of Diffusivity

gases: \[ D \sim 10^{-5}\ \mathrm{m^2/s} \]
liquids: \[ D < 10^{-9}\ \mathrm{m^2/s} \]
solids: \[ D < 10^{-13}\ \mathrm{m^2/s} \]

Typical \(D\) Range

A Common Misconception

Search for any youtube video with “diffusion experiment dye”

Example Video

Can we verify whether the scenario seen is Ficknian diffusion?

Diffusion vs Convection Length Scale

Length \(L\) traveled in time \(t\):

Diffusion: \(L = 6\sqrt{D_{AB}\,t}\) (Einstein, ~1905)
Convection: \(L = v_m\,t\)

What is typical \(D_{AB}\) in a liquid?
- Often \(D_{AB}\sim 10^{-9}\) to \(10^{-10}\,\mathrm{m^2/s}\)

Assuming \(D_{AB} = 10^{-10} \mathrm{m^2/s}\) \(v_{m} = 10^{-3} \mathrm{m/s}\)

Temperature Dependence of Diffusivity

Arrhenius form: \[ D = D_0 \exp\!\left(-\frac{\Delta H^a}{k_B T}\right) \]
\(\Delta H^a\): activation enthalpy
Why do we measure \(\Delta H_a\), not the \(\Delta G^a\)?
How can you read the plot?

Physical Interpretation of Activation

atoms hop between sites
energy barrier must be overcome
jump frequency: \[ \Gamma = \nu \exp\!\left(-\frac{\Delta G^\ddagger}{k_B T}\right) \]
diffusion proportional to hop rate and distance

Multiple Diffusion Mechanisms

Intrinsic and extrinsic mechanism in doped halides

Different diffusion paths
Different activation energies
Dominant mechanism controls slope of \(\ln D\) vs \(1/T\)

Diffusion in Gases

Kinetic theory description
Chapman–Enskog result: \[ D \propto \frac{T^{3/2}}{P} \]
Collisions limit transport

Diffusion in Liquids

Stokes–Einstein equation: \[ D = \frac{k_B T}{6\pi \eta r} \]
\(\eta\): viscosity
\(r\): particle radius
Is it accurate enough in polymer solutions?

Diffusion in Solids

Diffusion in solids is much more complex!
Discrete lattice sites
Mechanisms (non-exhaustive)
- Vacancy
- Ring mechanism
- Push-out mechanism
- Interstitial
Strong temperature dependence

Vacancy Diffusion Mechanism

Atoms exchange with vacancies
Jump only if vacancy is adjacent
Vacancy sources:
- Surfaces
- Grain boundaries
- Dislocations

Types of Diffusivity

As the previous example of diffusion + convection video shows, the measurement of diffusivity really depends on which reference frame we use.

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)

Measuring Self-Diffusion

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: \[ D^{*} = k_B T\left(\frac{L_{ii}}{c_i} + \frac{L_{ii^*}}{c_{i^*}}\right) \]

Why Self-Diffusion Occurs

No macroscopic concentration gradient!
Chemical potential varies locally
Random walk lowers free energy
Entropy maximization drives motion

Summary

Diffusion is driven by chemical potential gradients
Fick’s laws follow from irreversible thermodynamics
Laplace operator reflects curvature and smoothing
Diffusivity varies strongly with phase and temperature
Self-diffusion exists even in homogeneous systems