# What Is Diffusion?

Diffusion is the time-dependent movement of particles, leading to mass transport by atomic motion. In solid materials, atoms can migrate by different mechanisms and are motivated by the reduction of Gibbs free energy, which controls rates of many phase transformations.

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# Atomic Mechanisms of Diffusion

## **Vacancy Diffusion**:
- Atoms jump into vacant lattice sites; requires activation energy and is governed by vacancy concentration which rises exponentially with temperature.
<img src=https://www.doitpoms.ac.uk/tlplib/diffusion/images/vacancy.png>
## **Interstitial Diffusion**:
- Small atoms (like C or H) move from one interstitial site to another; generally faster than vacancy diffusion due to the smaller size of diffusing atoms.
<img src=https://cms-media.bartleby.com/wp-content/uploads/sites/2/2021/10/20043020/Interstitial-diffusion-1024x575.jpg>
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# Types of Diffusion: Self-Diffusion and Interdiffusion

## **Self-Diffusion**:
**Movement of host atoms in pure metals**, often experimentally measured using isotopic tracers.
## **Interdiffusion**:
Atoms of **one material diffuse into another**, leading to concentration changes (e.g., Cu-Ni alloy system); occurs when there is a concentration gradient.
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# Fick’s Laws of Diffusion

## **Fick’s First Law (Steady-State)**:
The diffusion flux $J$ is proportional to the negative concentration gradient:

$$
J = -D \frac{dC}{dx}
$$

where $D$ is the diffusion coefficient and $\frac{dC}{dx}$ is the concentration gradient.
## **Fick’s Second Law (Non-Steady-State)**:
scribes how concentration changes with both time and position:

$$
\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}
$$

Solutions involve error functions and are used for practical problems like case harning or doping of semiconductors.

## Factors Affecting Diffusion Rate
### **Temperature**:
The diffusion coefficient increases exponentially with temperature, modeled as:

$$
D = D_0 e^{-Q_d/(RT)}
$$

where $Q_d$ is the activation energy, $R$ is the gas constant, and $T$ is the absolute temperature.
### **Bonding and Structure**:
Stronger bonding and close-packed structures restrict atomic motion, making diffusion slower in solids than in liquids or gases.

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### Concentration Profiles and Migration

1. Diffusion leads to migration of atoms from regions of high concentration to low.
2. Concentration profiles (before and after diffusion) show the redistribution of atoms.
3. Self-diffusion and interdiffusion can be visualized by tracing labeled atoms or using radioactive tracers.

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### Practical Applications

- **Case Hardening**: Carbon atoms diffuse into surface layers of iron/steel, improving hardness and fatigue resistance.
- **Doping**: Impurities (e.g., phosphorus in silicon) are diffused into semiconductors, critical for device fabrication.
- **Sintering**: Diffusion drives neck growth and densification in ceramic materials.

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### Special Effects and Experimental Observations

- **Kirkendall Effect**: The displacement of a marker in diffusion couples (e.g., Cu-Zn alloys) confirms that different atomic species move at different rates due to diffusion, not uniform exchange.
- **Brownian Motion**: Atoms and molecules perform random walks, providing a microscopic basis for diffusion.

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### Summary Table of Core Concepts

| Concept | Explanation | Key Equations/Relations | Examples/Applications |
| :-- | :-- | :-- | :-- |
| Diffusion Mechanisms | Vacancy, Interstitial | $J, D, Q_d$, exponential temp dependence | Case hardening, doping, sintering |
| Fick’s Laws | 1st: Steady-state; 2nd: Non-steady-state | $J = -D \frac{dC}{dx}$, $\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}$ | Alloy formation, protective barriers |
| Kirkendall Effect | Different rates for different species | Marker displacement | Cu-Zn diffusion experiments |
| Temperature Effect | Diffusion coefficient increases with T | $D = D_0 e^{-Q_d/(RT)}$ | Hardening, annealing treatments |
| Applications | Materials design/processing | Engineering problem examples | Electronics, chemicals, metals |