Concentration gradient encyclopedia topics

The diffusion equation

To verify any microscopic model we may think up, we need to calculate its consequences and compare these to observation. Another way of arriving at a microscopic model is to write down a general equation and solve it mathematically (i.e. start from what you already know). This general equation, not referring to any microscopic model, is the diffusion equation

$partial_t c (mathbf{r},t) = Dnabla^2 c(mathbf{r},t)$ .

This equation is composed out of two true statements. One of these is the continuity equation

$partial_t c(mathbf{r} , t) = - mathbf{nabla} cdot mathbf{J}(mathbf{r} , t)$ .

And the other Fick's law

$mathbf{J} (mathbf{r} , t) = - D mathbf{nabla} c (mathbf{r}, t)$ ,

where $mathbf{J} (mathbf{r} , t)$ is the flux, $D$ is the diffusion constant, and $c (mathbf{r}, t)$ is the concentration of diffusing material.

The continuity equation is the mathematical equivalent to a piggybank. Your savings increase by the amount that you put in, they decrease by the amount you take out, no more and no less. Fick's law, on the other hand, was born as an empirical law which means that it describes observations and is not derived from any argument.

One general solution to the diffusion equation is a Gaussian one. This suggests an uncorrelated random walk as a microscopic model, completely in line with Robert Brown's observations.

Einstein relation

Einstein showed that Fick's law (empirical) can be derived by noting that the flux due to diffusion only can depend on the chemical potential, and taking this potential to be that of an ideal gas. This last step is justified because the final stage of a spreading concentration may be described as an ideal gas. The result is

$mathbf{J} (mathbf{r} , t) = - frac{kT}{gamma}mathbf{nabla} c (mathbf{r}, t)$ ,

where $gamma$ is the drag coefficient (the inverse of the mobility). The Einstein relation follows directly to be

$D = frac{kT}{gamma}$ ,

which is the most general expression for the diffusion coefficient, not referring to any microscopic model.

Entropy and diffusion

Diffusion increases the entropy of a system. This is nothing else than saying that diffusion is a spontaneous and irreversible process. Something can spread out by diffusing, but it won't spontaneously 'suck back in'.

In biology

In cell biology, diffusion is a main form of transport for necessary materials such as amino acids through cell membranes.

No equilibrium system

Because diffusion is a transport process of particles, the system in which it takes place is no equilibrium system (i.e. it is not at rest yet). For this reason thermodynamics and statistical mechanics are of little to no use in describing diffusion. However, there might occur so-called quasi-steady states where the diffusion process does not change in time. As the name suggests, this process is a fake equilibrium since the system is still evolving.

Types of diffusion

The spreading of any quantity that can be described by the diffusion equation or a random walk model (e.g. concentration, heat, momentum, ideas, price) can be called diffusion. Some of the most important examples are listed below.

Atomic diffusion
Brownian motion, for example of a single particle in a solvent
Collective diffusion, the diffusion of a large number of (possibly interacting) particles
Ellusion of a gas through small holes.
Electronic diffusion, resulting in electric current
Facilitated diffusion, present in some organisms.
Gaseous diffusion, used for isotope separation
Heat flow
Itō diffusion
Knudsen diffusion
Momentum diffusion, ex. the diffusion of the hydrodynamic velocity field
Osmosis is the diffusion of water through a cell membrane.
Photon diffusion
Reverse diffusion
Rotational diffusion
Self-diffusion
Surface diffusion
Active transport, pumping material across a cell membrane
Pinocytosis, "cell drinking" intake of small droplets of lipids
Phagocytosis, carrier proteins are used to transport glucose

Metabolism and respiration rely in part upon diffusion in addition to bulk or active processes. For example, in the alveoli of mammalian lungs, due to differences in partial pressures across the alveolar-capillary membrane, oxygen diffuses into the blood and carbon dioxide diffuses out. Lungs contain a large surface area to facilitate this gas exchange process.

An experiment to demonstrate diffusion

Diffusion is easy to observe, but care must be taken to avoid a mixture of diffusion and other transport phenomena.

It can be demonstrated with a wide glass tubed paper, two corks, some cotton wool soaked in ammonia solution and some red litmus paper. By corking the two ends of the wide glass tube and plugging the wet cotton wool with one of the corks, and litmus paper can be hung with a thread within the tube. It will be observed that the red litmus papers turn blue.

This is because the ammonia molecules travel by diffusion from the higher concentration in the cotton wool to the lower concentration in the rest of the glass tube. As the ammonia solution is alkaline, the red litmus papers turn blue. By changing the concentration of ammonia, the rate of color change of the litmus papers can be changed.