As a particle travels, the molecule is jostled by collisions with other molecules which prevent it from following a straight line. If the path is examined in close detail, it will be the approximation of a random walk. Mathematically, a random walk is a series of steps, one after another, where each step is taken in a completely random direction from the one before. This kind of path was famously analyzed by Albert Einstein in a study of Brownian motion and he showed that the mean square of the distance traveled by particle following a random walk is proportional to the time elapsed. In two dimensions this relationship can be written as:
For a list of common mathematical concepts used to model diffusion see: Diffusion Mathematics
Types of Diffusion
- Anisotropic diffusion, also known as the Perona-Malik equation, enhances high gradients
- Anomalous diffusion in porous medium
- Atomic diffusion, in solids
- Eddy diffusion, in coarse-grained description of turbulent flow
- Effusion of a gas through small holes
- Electronics|Electronic diffusion, resulting in an current (electricity)|electric current called the diffusion current
- Facilitated diffusion, present in some organisms
- Gaseous diffusion, used for isotope separation
- Heat equation, diffusion of thermal energy
- Itō diffusion, mathematisation of Brownian motion, continuous stochastic process.
- Knudsen diffusion of gas in long pores with frequent wall collisions
- Momentum diffusion ex. the diffusion of the hydrodynamic velocity field
- Photon diffusion
- Plasma diffusion
- Random walk
- Reverse diffusion, against the concentration gradient, in phase separation
- Rotational diffusion, random reorientations of molecules
- Surface diffusion, diffusion of adparticles on a surface
- Turbulent diffusion, transport of mass, heat, or momentum within a turbulent fluid
Diffusion of reagents on the surface of a catalyst may play an important role in heterogeneous catalysis. The model of diffusion in the ideal monolayer is based on the jumps of the reagents on the nearest free places. This model was used for CO on Pt oxidation under low gas pressure.
The system includes several reagents (A₁ , A₂ , ... A⩋) on the surface. Their surface concentrations are (c₁ , c₂ , ... c⩋). The surface is a lattice of the adsorption places. Each reagent molecule fills a place on the surface. Some of the places are free. The concentration of the free paces is z=c₀. The sum of all cᵢ (including free places) is constant, the density of adsorption places b.
The jump model gives for the diffusion flux of Aᵢ (i=1,...,n):
The corresponding diffusion equation is:
Due to the conservation law, and we have the system of m diffusion equations. For one component we get Fick's law and linear equations because . For two and more components the equations are nonlinear.
If all particles can exchange their positions with their closest neighbours then a simple generalization gives
Various versions of these jump models are also suitable for simple diffusion mechanisms in solids.
Brownian Motion describes the stochastic movement of particles as they travel through space. This type of random movement is often referred to as a random walk, which is typical of particles that diffuse about a 1D 2D or 3D space filled with other particles or barriers. To understand Brownian motion, lets start by characterizing this phenomenon in 2D space.
The mathematical description of this process often includes these terms:
Δ): diffusion rate coefficient
D = L2 / 2d⋅t
L: step length
D (in units: µm²/s)is the mean diffusion rate per unit time, often in µm²/s for biological motion on a molecular level. This refers to how fast, on average, the particle moves along its trajectories. This value is often of ultimate interest, particularly for simulating Brownian motion; however under most circumstances, the diffusion rate cannot be observed directly in empirical experiments - this would require the ability to visualize all the microscopic particles and collisions that dictate the particle's movement, and would have to be done on a nanosecond timescale. In the animation on the right, the diffusion rate can actually be quantified directly; but what is often seen when observing particle diffusion through a microscope would more closely resemble this: