Will Ultrasonics Help in the Cleanup of Oil Spills?

Since small particle sizes increase the coefficient of diffusion and hence the rate of diffusion one might be able to speed up the dispersion of oil from a spill if one had a means of reducing the particle size of the oil droplets. This is what happens in ultrasonic cleaning. The mechanism is known as sonication in which the mechanical agitation produced by sound waves is used to disrupt the intermolecular forces holding molecules together. Ultasonic cleaning works better if a surfactant is present which helps to alter surface tension. The contaminant ends up as a colloid.

If one wanted to "get rid" of a lot of oil quickly this might prove an effective way of doing so if the object is to protect beaches and ecosystems from being overwhelmed by a coating of oil The trade-off is that a lot of toxins may end up in the water column. Spreading out the toxins over a large volume of water reduces the harm to any particluar location. There may be organisms which can tolerate a small amount of toxins present but as with other pollutants biomagnification can increase the harm done if toxins enter the food chain.

What Can We Learn From a Diffusion Experiment?

A diffusion experiment by itself would just give us an experimental value of the Diffusion Coefficient. If the temperature and particle number density of the solute are known this leaves the molecular mass and cross-sectional radius as unknowns. So we get one curve with two unknowns which is unsolvable. If the molecular mass can be found by some other means such as by using a mass spectrometer then the Diffusion Coefficient will allow us to make an estimate of the molecular cross-sectional radius. This information might help to determine if a molecule was a compact ball or more of a thin rod.

The results of experiments such as these are used to determine the properties of molecules and are useful in identifying the presence of chemical compounds in a mixture. Two analytical tools used in chemistry and biology similar to diffusion are chromatography and electrophoresis.

An Approximate Formula for the Diffusion Coefficient

From the Kinetic Theory of Gases* one can find an approximate formula for the Diffusion Coefficient.

We can use this formula to estimate the rate of diffusion for oil droplets if we substitute the particle mass and cross-sectional radius for the moleculelar mass and cross-sectional radius. The particle mass is also dependent on the cross-sectional radius since it is the product of the density of oil and the droplet's volume. So one would expect the diffusion coefficient to increase rapidly with the decrease in size of the oil droplets. The particle number density would be the number of molecules per unit volume of seawater.

*see Castellan, Physical Chemistry, 3rd ed., p. 756.

Lessons Learned?

The normal distribution applies to many situations where measurements are subject to random errors. At first it may seem strange that some of the same formulas apply to diffusion and statistics but in diffusion we are concerned with concentrations or particle densities and in statistics with populations and frequencies of events. The expansion that Einstein used to derive Fick's Second Law shows the generality of the results. The similar formulas allow us to use tables of confidence intervals to get results for diffusion but we have remember that instead of standard deviations we are dealing with diffusion lengths that change over time.

Certainly errors were made in the Gulf of Mexico and it is likely that there are still lessons to be learned. Both the oil industry and the government need to review what happened and see what can be done to avoid making the same mistakes in the future.

Fick's Second Law of Diffusion and Its Solution

One can find Einstein's papers on Brownian motion in Investigations on the Theory of the Brownian Movement which is still in print. In the first paper from 1905 Einstein derives Fick's Second Law of Diffusion by making a series expansion of the concentration of particles, f (φ in the Wikipedia article), as a function of time and position. Then he gives a solution for this equation which turns out to be a normal distribution with σ equal to Δ (rms), which is also known as the diffusion length. Strictly speaking, this solution is for a point source in one dimension but for a narrow column food coloring in a thin slab of jello one would expect that this would be a close approximation with x replaced by r, the distance from the column and the center of the distribution over time.

edit: For one dimension half of the particles will be within 0.674 σ of the center of the normal distribution. The fraction within 1 σ is 0.683, within 2 σ it is 0.954 and within 3 σ one will find 0.997 of all the particles. The particles slowly spread over time with σ (the diffusion length) acting as a scale factor. In two dimensions the radial distribution giving the density of particles as a function of the radius, r, only is also a normal distribution but with the one dimensional σ replaced by √2 σ.

Dispersion Combining Einstein's Relation and Grahams Law

If we combine Einstein's formula for the dispersion from his Theory of Brownian Motion with Graham's experimental results for diffusion we get the following proportion for the dispersion. The constant of proportionality would have to be determined by experiment and probably is dependent on the absolute temperature, T.

These results suggest that the law should be subject to experimental check. One such experiment might be a vertical cylinder of dye diffusing through a block of jello with measurements of changes in concentration over time.

Problems With Oil Spills and the Use of Dispersants

A problem that was encountered with the response to the Gulf oil spill was the gooey masses which washed up on the beaches. This was likely due to the fact that petroleum (crude oil) has a varied composition. The denser components are known as bitumen and tend to be rather sticky. So what probably happened is that the lighter components were dispersed leaving the gooey mess behind on the surface. One should point out that there are already dispersants in the water since it is a component of the greywater which ends up in the Mississippi.

Another problem is that diffusion is not a very efficient process. The fundamental law of diffusion is known as Fick's law which defines the diffusion coefficient. From Brownian dynamics and Einstein's dispersion relation (1908) we know that the root mean square (rms) dispersion of a number of particles is proportional to the square root of the time, t, and inversely proportional to the square root of the particle mass, M,





So diffusion would be more efficient for smaller particles of oil and molecules with lower molecular mass if γ, a drag coefficient, is relatively constant. A similar dependence of the rate on mass is known as Graham's Law*. One should also note that there are other physical processes such as agitation of the water and shear stresses which tend to disperse the particles.

*edit: Graham studied both effusion and diffusion. He showed that the square of the times of diffusion was proportional to the densities.