# An introduction to the process of diffusion

Additional Information In lieu of an abstract, here is a brief excerpt of the content: Rabiega and Roland G. Extensive manufacturing development in the southeastern United States is generally accepted to have taken place in this century. Indeed, study of the relocation of the textile industry and its subsequent expansion in the Appalachian Piedmont has almost become hackneyed through its wide use to illustrate 1 the effects of manufacturing introduced into a relatively underdeveloped and agrarian region and 2 the role of labor costs in attracting specific types of manufacturing.

Diffusion, a Mass Transfer Phenomenon Diffusion is a mass transfer phenomenon that causes the distribution of a chemical species to become more uniform in space as time passes. In this case, species is a chemical dissolved in a solvent or a component in a gas mixture, such as the oxygen in air.

The mass transfer of a species is the evolution of its concentration in space and time. If the concentration of a species is initially not uniform the concentration might be greater in one region of a vessel than another, for example then, over time, diffusion causes mass transfer in favor of a more uniform concentration.

The driving force for diffusion is the thermal motion of molecules. At temperatures above absolute zero, molecules are never at rest. Their kinetic energy means that they are always in motion, and when molecules collide with each other frequently, the direction of the motion becomes randomized.

In most cases, these collisions are common; even in air at atmospheric pressure, which hardly seems a "dense" fluid, each molecule collides with a neighbor every few nanoseconds. A Simple Example When molecules are moving but also constantly changing direction, diffusion occurs because of the statistics of this movement.

## Stochastic Systems

The image below shows a volume of a solution in which there is a nonuniform concentration. The red color indicates a high concentration of solute, whereas the blue color indicates nearly pure solvent.

After some time, diffusion causes the concentration in the vessel to become uniform: In the figure below, the initial condition is shown with arrows whose size and direction show the number of molecules moving in a particular direction at one time — remember that their motion is random, so they'll be moving equally in all directions from any point: The size of the arrows indicates the number of molecules randomly moving in a given direction from a given point.

At most points in the system, the uniform concentration means that the number of molecules moving in opposite directions is the same. As you can see, however, close to the boundary between the region of high and low concentration, there will be many more molecules moving to the right than moving to the left: Number of molecules moving right across the central boundary and left across the central boundary when the concentration profile is nonuniform.

This is not because the molecules "prefer" to move in one direction, but just because there are more of them on one side of the boundary than the other. As a consequence, there is a net flux of material from left to right. In this case, mass moves from left to right so that the concentration becomes globally more uniform.

Because diffusion drives a net flux of material from regions of high concentration to low concentration, we often speak of diffusion as acting "down a concentration gradient". Once the concentration has become uniform, the molecules are all still in motion in different, random directions.

However, there are now the same number of molecules moving across the boundary in either direction: Number of molecules moving right across the central boundary and left across the central boundary when the concentration profile is uniform.

Even though the molecules are in random motion, there is no statistical driving force for them to start accumulating anywhere if their distribution is uniform.

There is no net flux or change in concentration. Laws of Diffusion Although diffusion occurs because of statistical effects, when modeling diffusion, we normally use continuous partial differential equations PDEs to describe this statistical process.

The relation of the above statistical process to the observed macroscopic phenomenon of "diffusion down a concentration gradient" was elucidated by Albert Einstein in one of his annus mirabilis papers of 3. There, he considered the related phenomenon of Brownian motion, i. The PDEs used to model diffusion problems might include Fick's laws, the convection-diffusion equationor more complex methods for concentrated mixtures, like Maxwell-Stefan diffusion.

How Fast Is Diffusion? Fick's laws contain only one parameter: This is the measure of the rate of the diffusion process. Steady-State Diffusion In a finite vessel with no sources or sinks of mass, the diffusion layer where the concentration is nonuniform eventually reaches the walls, after which a uniform, steady concentration will be attained.

But in an infinite space or in the presence of a constant supply of material, a uniform concentration may not be attained.

Compare the two animations below:There is also another, purely "probabilistic", approach to diffusion processes, based on the representation of the process as the solution of the Itô stochastic differential equation where is the standard process of Brownian motion.

MSE Introduction to Materials Science Chapter 5, Diffusion 3 Interdiffusion (or impurity diffusion)occurs in response to a concentration gradient. Self-diffusion is diffusion in one-component material, when all atoms that exchange positions are of the same.

2 Introduction to the Diffusion Process Random Motion, Concentration Gradients and Diffusion So far, with the aid of the kinetic theory of gases, the reader should be acquainted.

## Mathematics and statistics online

The diffusion process describes the manner in which different members of the target market often accept and purchase a product. It spans the time from product introduction until market saturation: 1.

Introduction. We consider the so-called backward equation of diffusion theory d2 d The most general diffusion process associated with semi-groups (cf. [2]) is now obtained by starting from an elastic barrier process and superimposing on it the elementary return process in .

The exposition is based on the theory of stochastic analysis. The diffusion processes discussed are interpreted as solutions of Itô's stochastic integral equations.

The book is designed as a self-contained introduction, requiring no background in the theory of probability or even in measure theory.

Chapter 11 The Product and Developing New Products