Fick's laws, diffusion mechanisms, and applications in materials processing
Diffusion is the net movement of atoms or molecules from a region of high concentration to a region of low concentration. This phenomenon is fundamental to nearly every aspect of materials science and engineering, from heat treatment of steels to the fabrication of semiconductor devices.
At the atomic level, diffusion occurs because atoms are constantly in motion. Even in a solid crystal, atoms vibrate around their equilibrium positions, and occasionally an atom will have enough energy to jump to an adjacent site. When there is a concentration gradient, more atoms statistically jump from the high-concentration region toward the low-concentration region, resulting in net mass transport.
Understanding diffusion is essential for controlling many materials processes:
The fundamental driving force for diffusion is the reduction of Gibbs free energy. In most cases, this manifests as atoms moving down a concentration gradient. However, diffusion can also be driven by gradients in chemical potential, stress, or electric fields. For this chapter, we will focus primarily on concentration-driven diffusion.
The rate of diffusion depends strongly on temperature. At higher temperatures, atoms have more thermal energy and can more easily overcome the activation energy barrier required to jump between sites. This temperature dependence is described by the Arrhenius equation, which we will explore in detail.
Atoms can move through a solid by several different mechanisms. The dominant mechanism depends on the type of atom (self-diffusion vs. impurity diffusion), the crystal structure, and the temperature.
Vacancy diffusion is the most common mechanism for self-diffusion in metals and for substitutional impurities. An atom moves by jumping into an adjacent vacant lattice site. For this to occur, two conditions must be met: there must be a vacancy next to the atom, and the atom must have sufficient energy to break bonds with its neighbors and squeeze past them into the vacancy.
The diffusion coefficient for vacancy diffusion depends on both the concentration of vacancies (which increases exponentially with temperature) and the jump frequency of atoms. This is why vacancy diffusion has a strong temperature dependence.
Small atoms like carbon, nitrogen, hydrogen, and oxygen typically occupy interstitial sites in metal lattices. These atoms can diffuse by jumping from one interstitial site to another without requiring a vacancy. Because interstitial sites are always available and the atoms are small, interstitial diffusion is generally much faster than vacancy diffusion.
For example, carbon in iron (FCC austenite) diffuses about 10,000 times faster than iron atoms themselves at the same temperature. This difference is exploited in carburizing processes.
Grain boundaries are regions of atomic disorder where atoms are less tightly packed than in the crystal interior. Diffusion along grain boundaries is typically several orders of magnitude faster than through the bulk lattice. However, the total volume of grain boundary material is small, so grain boundary diffusion is most significant at lower temperatures and in fine-grained materials.
Atoms on surfaces have fewer bonds than bulk atoms and can move more freely. Surface diffusion is the fastest type of diffusion and plays an important role in processes like thin film growth and sintering.
| Mechanism | Typical Activation Energy | Relative Speed |
|---|---|---|
| Lattice (vacancy) | High | Slowest |
| Lattice (interstitial) | Medium | Fast |
| Grain boundary | Medium-Low | Faster |
| Surface | Low | Fastest |
Fick's First Law describes steady-state diffusion, where the concentration profile does not change with time. This occurs when atoms enter one side of a material at the same rate they exit the other side.
Fick's First Law:
$$J = -D \frac{dc}{dx}$$Where: $J$ = diffusion flux (atoms/m²·s or kg/m²·s), $D$ = diffusion coefficient (m²/s), $c$ = concentration (atoms/m³ or kg/m³), $x$ = position (m)
The negative sign indicates that diffusion occurs in the direction opposite to the concentration gradient, meaning atoms flow from high to low concentration. The flux $J$ represents the number of atoms (or mass) passing through a unit area per unit time.
The diffusion coefficient $D$ is a measure of how fast atoms can move through a material. It has units of m²/s (or more commonly cm²/s in older literature). Typical values range from about $10^{-20}$ m²/s for slow diffusion (like substitutional impurities at low temperatures) to $10^{-10}$ m²/s for fast diffusion (like hydrogen in metals at high temperatures).
Consider a plate of thickness $L$ with different concentrations maintained at each surface: $c_1$ at $x = 0$ and $c_2$ at $x = L$. At steady state, the concentration varies linearly across the plate:
$$\frac{dc}{dx} = \frac{c_2 - c_1}{L}$$And the flux through the plate is:
$$J = -D \frac{c_2 - c_1}{L} = D \frac{c_1 - c_2}{L}$$A steel plate 2 mm thick separates a high-pressure hydrogen chamber from a vacuum. At 500°C, the hydrogen concentration is 3.0 kg/m³ at the high-pressure surface and essentially zero at the vacuum surface. The diffusion coefficient of hydrogen in steel at 500°C is $1.2 \times 10^{-8}$ m²/s. Calculate the diffusion flux.
Solution:
Using Fick's First Law:
$$J = D \frac{c_1 - c_2}{L} = (1.2 \times 10^{-8}) \frac{3.0 - 0}{0.002}$$ $$J = 1.8 \times 10^{-5} \text{ kg/m}^2\text{·s}$$This may seem small, but over time significant hydrogen can permeate through the membrane. This is why hydrogen containment is challenging in industrial applications.
Most real diffusion situations involve non-steady-state conditions where the concentration at any point changes with time. Fick's Second Law describes how the concentration profile evolves.
Fick's Second Law (1D):
$$\frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2}$$Where: $\partial c / \partial t$ = rate of change of concentration with time, $\partial^2 c / \partial x^2$ = curvature of the concentration profile
This is a partial differential equation (PDE) that relates the time rate of change of concentration to the spatial curvature of the concentration profile. Where the concentration profile is concave up (positive curvature), the concentration increases with time. Where it is concave down (negative curvature), the concentration decreases.
Fick's Second Law can be derived by considering mass conservation. Consider a thin slice of material between $x$ and $x + dx$. The rate of accumulation of atoms in this slice equals the flux in minus the flux out:
$$\frac{\partial c}{\partial t} dx = J(x) - J(x + dx) = -\frac{\partial J}{\partial x} dx$$Substituting Fick's First Law ($J = -D \frac{\partial c}{\partial x}$) and assuming $D$ is constant:
$$\frac{\partial c}{\partial t} = -\frac{\partial}{\partial x}\left(-D \frac{\partial c}{\partial x}\right) = D \frac{\partial^2 c}{\partial x^2}$$Think of the concentration profile as a hill. Where the profile is curved downward (like the top of a hill), material flows away in both directions and the local concentration decreases. Where the profile is curved upward (like a valley), material flows in from both directions and the local concentration increases. Diffusion naturally smooths out concentration variations over time.
For diffusion in three dimensions with constant $D$:
$$\frac{\partial c}{\partial t} = D \nabla^2 c = D \left(\frac{\partial^2 c}{\partial x^2} + \frac{\partial^2 c}{\partial y^2} + \frac{\partial^2 c}{\partial z^2}\right)$$The diffusion coefficient varies strongly with temperature, following an Arrhenius relationship. This is because diffusion requires atoms to overcome an energy barrier to jump between sites, and the probability of having sufficient energy increases exponentially with temperature.
Arrhenius Equation for Diffusion:
$$D = D_0 \exp\left(-\frac{Q_d}{RT}\right)$$Where: $D_0$ = pre-exponential factor (m²/s), $Q_d$ = activation energy for diffusion (J/mol), $R$ = gas constant (8.314 J/mol·K), $T$ = absolute temperature (K)
Taking the natural logarithm of both sides:
$$\ln D = \ln D_0 - \frac{Q_d}{RT}$$This shows that a plot of $\ln D$ versus $1/T$ gives a straight line with slope $-Q_d/R$ and intercept $\ln D_0$. This is how activation energies are determined experimentally.
| Diffusing Species | Host Metal | $D_0$ (m²/s) | $Q_d$ (kJ/mol) |
|---|---|---|---|
| C | FCC Fe | $2.3 \times 10^{-5}$ | 148 |
| C | BCC Fe | $1.1 \times 10^{-6}$ | 87 |
| Fe | FCC Fe | $2.2 \times 10^{-5}$ | 268 |
| Fe | BCC Fe | $2.0 \times 10^{-4}$ | 251 |
| Cu | Cu | $7.8 \times 10^{-5}$ | 211 |
| Ni | Cu | $2.7 \times 10^{-5}$ | 256 |
Notice that interstitial diffusion (C in Fe) has a much lower activation energy than substitutional diffusion (Fe in Fe), which is why it is faster.
Calculate the diffusion coefficient of carbon in FCC iron (austenite) at 1000°C. Given: $D_0 = 2.3 \times 10^{-5}$ m²/s, $Q_d = 148$ kJ/mol.
Solution:
Convert temperature to Kelvin: $T = 1000 + 273 = 1273$ K
Convert activation energy: $Q_d = 148,000$ J/mol
$$D = D_0 \exp\left(-\frac{Q_d}{RT}\right) = (2.3 \times 10^{-5}) \exp\left(-\frac{148000}{8.314 \times 1273}\right)$$ $$D = (2.3 \times 10^{-5}) \exp(-13.99)$$ $$D = (2.3 \times 10^{-5})(8.4 \times 10^{-7})$$ $$D = 1.9 \times 10^{-11} \text{ m}^2/\text{s}$$Solving Fick's Second Law requires specifying initial conditions (the concentration profile at $t = 0$) and boundary conditions (what happens at the surfaces). Several important cases have analytical solutions.
This is the most common case in materials processing. Consider a thick material (semi-infinite) with initial uniform concentration $C_0$ throughout. At $t = 0$, the surface concentration is suddenly changed to $C_s$ and held constant.
Error Function Solution:
$$\frac{C(x,t) - C_0}{C_s - C_0} = 1 - \text{erf}\left(\frac{x}{2\sqrt{Dt}}\right)$$Where: $C(x,t)$ = concentration at depth $x$ and time $t$, $C_0$ = initial concentration, $C_s$ = surface concentration, $\text{erf}$ = error function
The error function is defined as:
$$\text{erf}(z) = \frac{2}{\sqrt{\pi}} \int_0^z e^{-u^2} du$$Key values to remember:
The quantity $\sqrt{Dt}$ has units of length and represents the characteristic distance that atoms diffuse in time $t$. This is extremely useful for estimating diffusion depths and times:
Steel with an initial carbon content of 0.2 wt% is carburized at 950°C. The surface carbon concentration is maintained at 1.0 wt%. How long will it take to achieve a carbon concentration of 0.45 wt% at a depth of 0.5 mm? At 950°C, $D = 1.6 \times 10^{-11}$ m²/s.
Solution:
First, calculate the left side of the error function equation:
$$\frac{C - C_0}{C_s - C_0} = \frac{0.45 - 0.2}{1.0 - 0.2} = \frac{0.25}{0.8} = 0.3125$$So:
$$1 - \text{erf}\left(\frac{x}{2\sqrt{Dt}}\right) = 0.3125$$ $$\text{erf}\left(\frac{x}{2\sqrt{Dt}}\right) = 0.6875$$From error function tables, $\text{erf}(0.71) \approx 0.6875$, so:
$$\frac{x}{2\sqrt{Dt}} = 0.71$$ $$\sqrt{Dt} = \frac{x}{2(0.71)} = \frac{0.0005}{1.42} = 3.52 \times 10^{-4} \text{ m}$$Solving for time:
$$t = \frac{(3.52 \times 10^{-4})^2}{D} = \frac{1.24 \times 10^{-7}}{1.6 \times 10^{-11}} = 7750 \text{ s} \approx 2.15 \text{ hours}$$The idealized treatments of Fick's laws assume a constant diffusion coefficient and simple boundary conditions. Real materials systems often exhibit more complex behavior.
In many systems, the diffusion coefficient depends on composition. This is particularly true for interstitial diffusion at high concentrations and for diffusion in non-ideal solutions. When $D = D(c)$, Fick's Second Law becomes:
$$\frac{\partial c}{\partial t} = \frac{\partial}{\partial x}\left(D(c) \frac{\partial c}{\partial x}\right)$$This equation generally requires numerical solution.
When two metals A and B are joined and heated, atoms of both species diffuse across the interface. If A atoms diffuse faster than B atoms, there is a net flux of atoms in one direction. Since atoms and vacancies exchange places, this creates a net flux of vacancies in the opposite direction.
The vacancies can coalesce to form voids (Kirkendall porosity), and the original interface moves (the Kirkendall effect). This demonstrates that diffusion occurs by vacancy exchange and that different species can have very different diffusivities.
The interdiffusion coefficient is defined as:
$$\tilde{D} = X_B D_A + X_A D_B$$where $X_A$ and $X_B$ are the mole fractions and $D_A$ and $D_B$ are the intrinsic diffusion coefficients of the two species.
When more than two components are present, diffusion becomes much more complex. The flux of each component depends not only on its own concentration gradient but also on the gradients of other components. This is described by a matrix of diffusion coefficients:
$$J_i = -\sum_j D_{ij} \frac{\partial c_j}{\partial x}$$Real materials contain grain boundaries where diffusion is faster than in the bulk lattice. The effective diffusion coefficient in a polycrystalline material depends on grain size and the relative contributions of lattice and grain boundary diffusion:
Diffusion is central to many important materials processing and engineering applications. Understanding diffusion allows engineers to control these processes for desired outcomes.
Case hardening creates a hard, wear-resistant surface on steel components while maintaining a tough, ductile core. The two main processes are carburizing (adding carbon) and nitriding (adding nitrogen).
In carburizing, steel parts are heated to 850-950°C in a carbon-rich atmosphere. Carbon diffuses into the surface, increasing the carbon concentration to 0.8-1.0 wt%. After quenching, this high-carbon surface layer transforms to hard martensite. The depth of the case depends on time, temperature, and the diffusion coefficient of carbon.
The electrical properties of semiconductors are controlled by adding small amounts of dopant atoms. In silicon, phosphorus or arsenic (donors) make n-type material, while boron (acceptor) makes p-type material.
Diffusion doping involves exposing the silicon wafer to a dopant source at high temperature (900-1200°C). The concentration profile follows the error function solution. Modern processes use ion implantation followed by a diffusion anneal to achieve more precise control.
Boron is diffused into silicon at 1100°C for 1 hour to create a p-type region. If the surface concentration is $10^{20}$ atoms/cm³ and the background n-type doping is $10^{16}$ atoms/cm³, estimate the junction depth (where p-type and n-type concentrations are equal). Given: $D$ = $3 \times 10^{-13}$ cm²/s at 1100°C.
Solution:
At the junction, $C = C_0 = 10^{16}$ atoms/cm³:
$$\frac{C - C_0}{C_s - C_0} = \frac{10^{16} - 0}{10^{20} - 0} = 10^{-4}$$So:
$$1 - \text{erf}\left(\frac{x_j}{2\sqrt{Dt}}\right) = 10^{-4}$$ $$\text{erf}\left(\frac{x_j}{2\sqrt{Dt}}\right) = 0.9999$$From tables, this corresponds to approximately $\frac{x_j}{2\sqrt{Dt}} \approx 2.75$
Calculate $\sqrt{Dt}$: $\sqrt{(3 \times 10^{-13})(3600)} = 3.3 \times 10^{-5}$ cm
Junction depth: $x_j = 2(2.75)(3.3 \times 10^{-5}) = 1.8 \times 10^{-4}$ cm $= 1.8$ µm
Sintering is used to consolidate metal or ceramic powders into dense, strong components. The process involves heating the powder compact to a temperature where diffusion is significant (typically 0.5 to 0.8 of the melting temperature).
During sintering, atoms diffuse from regions of high chemical potential (particle surfaces and contact points) to regions of low chemical potential (necks between particles). This causes the necks to grow and the pores to shrink. Surface diffusion, grain boundary diffusion, and lattice diffusion all contribute to densification.
Many oxidation and corrosion processes are controlled by diffusion through the oxide or corrosion product layer. For protective oxide scales (like aluminum oxide on aluminum), slow diffusion through the oxide limits the oxidation rate. The parabolic oxidation law:
$$x^2 = kt$$where $x$ is the oxide thickness, comes directly from diffusion-controlled growth.
Cast metals often have compositional variations due to segregation during solidification. Homogenization involves holding at high temperature to allow diffusion to even out these variations. The time required can be estimated from:
$$t \approx \frac{L^2}{D}$$where $L$ is the characteristic length scale of the segregation (typically the dendrite arm spacing).
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