8.1 Transient Diffusion

Transient diffusion describes how the concentration of a species changes with both time and position within a medium. Think of a drop of dye placed in still water: initially concentrated at one point, the dye gradually spreads until the color is uniform everywhere.

Understanding transient diffusion is essential because most real-world mass transfer processes are time-dependent. Steady-state conditions are the exception, not the starting point. This section covers the governing equation (Fick's second law), analytical solution techniques, concentration profile behavior, and the key dimensionless numbers used to characterize diffusion time scales.

Transient Diffusion Fundamentals

Key Concepts and Equations

Fick's second law is the governing equation for transient diffusion. In one dimension it takes the form:

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

where $c$ is the species concentration, $t$ is time, $D$ is the diffusion coefficient (also called diffusivity), and $x$ is the spatial coordinate. The equation states that the local rate of concentration change is proportional to the curvature of the concentration profile at that point.

The diffusion coefficient $D$ quantifies how fast a species spreads through a given medium. Its value depends on temperature, pressure, and the specific species–medium pair. Typical orders of magnitude:

• Gases in air: $D \sim 10^{-5}$ m²/s
• Solutes in liquids: $D \sim 10^{-9}$ m²/s
• Atoms in solids: $D \sim 10^{-12}$ m²/s or smaller

To solve Fick's second law you need two pieces of information beyond the PDE itself:

• Initial condition (IC): the concentration distribution at $t = 0$ throughout the domain (e.g., $c(x, 0) = c_0$, a uniform value).
• Boundary conditions (BCs): constraints at the edges of the domain for all $t > 0$. Common types include a constant surface concentration (Dirichlet), a specified flux (Neumann), or an impermeable wall where $\partial c / \partial x = 0$.

Concentration Profile Evolution Toward Steady State

As diffusion proceeds, the concentration gradient decreases and the profile becomes more uniform. Eventually the system reaches steady state, where $\frac{\partial c}{\partial t} = 0$ and the concentration no longer changes with time.

A useful concept here is the penetration depth $\delta$, the distance over which the concentration has changed appreciably from its initial value. It scales as:

$\delta \propto \sqrt{Dt}$

This square-root dependence means that doubling the penetration depth requires four times as long. It's a direct consequence of the diffusive (parabolic) nature of Fick's second law.

The shape of the evolving profile also depends on system geometry (planar slab, long cylinder, or sphere) and the boundary conditions applied. Each geometry introduces its own coordinate-dependent form of the Laplacian in Fick's second law.

Solving Transient Diffusion Problems

Analytical Methods and Techniques

Several classical techniques exist for obtaining closed-form solutions to Fick's second law. The choice depends on the domain (finite vs. semi-infinite) and the boundary conditions.

Separation of variables is the most common approach for finite domains. The steps are:

1. Assume the solution has the product form $c(x,t) = X(x) \cdot T(t)$.
2. Substitute into Fick's second law and divide through so that one side depends only on $x$ and the other only on $t$. Both sides must equal the same constant (the separation constant).
3. Solve the resulting ODE in $x$ and apply the boundary conditions to find the allowed eigenvalues.
4. Solve the ODE in $t$ (an exponential decay for each eigenvalue).
5. Superpose all eigenfunctions and use the initial condition to determine the coefficients via orthogonality (Fourier series).

The result for a slab of thickness $L$ with $c = 0$ at both surfaces and uniform initial concentration takes the form:

$c(x,t) = \sum_{n=1}^{\infty} A_n \sin\!\left(\frac{n\pi x}{L}\right) \exp\!\left(-\frac{n^2 \pi^2 D t}{L^2}\right)$

Each successive term decays faster (the $n^2$ in the exponent), so at long times only the first term matters.

Laplace transforms offer an alternative: transform the time variable to convert the PDE into an ODE in $x$, solve in the Laplace domain, then invert back. This is especially useful for semi-infinite domains and problems with time-dependent boundary conditions.

Solutions for Different System Geometries

Semi-infinite medium. For a domain extending from $x = 0$ to $x \to \infty$ with a step change in surface concentration to $c_s$ at $t = 0$, the similarity variable $\eta = \frac{x}{2\sqrt{Dt}}$ collapses the PDE into an ODE. The resulting concentration profile is:

$\frac{c(x,t) - c_0}{c_s - c_0} = \text{erfc}\!\left(\frac{x}{2\sqrt{Dt}}\right)$

where $\text{erfc}$ is the complementary error function. This solution is valid whenever the diffusion front has not yet "felt" the far boundary, making it a good short-time approximation even for finite systems.

Finite slab with constant surface concentrations. If the two surfaces are held at $c_1$ and $c_2$, the transient profile evolves from the initial distribution toward the linear steady-state result:

$c(x,\, t \to \infty) = c_1 + (c_2 - c_1)\frac{x}{L}$

The transient part is captured by the Fourier series solution described above, superimposed on this linear profile.

Cylindrical and spherical geometries introduce Bessel functions (cylinders) and spherical harmonics or sinc-type functions (spheres) as the eigenfunctions, because the Laplacian takes a different form in those coordinate systems. The overall solution strategy (separation of variables, eigenvalue problem, series expansion) remains the same.

Concentration Profiles in Transient Diffusion

Factors Affecting Concentration Distribution

Three things control the shape and evolution of a transient concentration profile:

• Initial concentration distribution. A uniform initial condition is the simplest case; non-uniform starting profiles change the Fourier coefficients but not the eigenfunctions.
• Boundary conditions. Constant surface concentration, zero-flux walls, and convective (Robin-type) boundaries each produce qualitatively different profile shapes.
• Diffusion coefficient. A larger $D$ flattens the profile faster. If $D$ varies with direction (anisotropic diffusion), the profile evolves asymmetrically and the scalar $D$ must be replaced by a diffusivity tensor.

Internal sources or sinks (e.g., a chemical reaction consuming the diffusing species) add a generation term to Fick's second law and can sustain non-uniform profiles even at steady state.

Analyzing Concentration Profiles

Plotting $c$ vs. $x$ at several time snapshots is the standard way to visualize transient diffusion. From these curves you can extract two important quantities:

• Local flux. The slope at any point gives the concentration gradient, which drives diffusive flux through Fick's first law:

$J = -D\frac{\partial c}{\partial x}$

A steeper slope means a larger flux at that location.

• Total species content. Integrating the profile over the domain gives the total amount of diffusing species present at time $t$:

$M(t) = \int_0^L c(x,t)\, dx$

Tracking $M(t)$ tells you how much mass has entered or left the system, which is often the quantity of practical interest (e.g., how much solvent has been absorbed by a polymer film).

Comparing profiles across different times, geometries, or boundary conditions reveals how each factor influences the rate and extent of diffusion.

Characteristic Diffusion Time

Definition and Significance

The characteristic diffusion time sets the time scale for a diffusion process:

$\tau = \frac{L^2}{D}$

where $L$ is the characteristic length (typically the half-thickness for a slab, or the radius for a cylinder/sphere). Because $L$ enters as a square, geometry has a powerful effect: doubling the thickness quadruples the time to reach equilibrium.

As a rough benchmark, for a planar slab of half-thickness $L$ with constant surface concentrations, the center reaches about 90% of its steady-state value at roughly $t \approx 0.3\,\tau$.

This time scale is central to the design of diffusion-controlled processes such as case hardening of steel, drying of coatings, and controlled drug release from polymer matrices.

Dimensionless Numbers and Time Scales

Three dimensionless groups appear repeatedly in transient diffusion analysis:

Understanding how $Fo$, $Bi_m$, and $\phi$ interact lets you quickly estimate whether a process is diffusion-limited, reaction-limited, or controlled by external mass transfer, and guides the selection of appropriate simplifications when solving problems.