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.

Dispersants, Emulsions, Diffusion & the Oil Spill

Dispersants are used in an oil spill to break up the oil and help in its reduction through natural processes. The dispersant is basically a surfacant which makes the oil more soluble in water by creating an emulsion consisting of tiny droplets of oil in water. Dispersants rated by the hydrophilic-lipophilic balance. In nature one often finds proteins whose function are to mix substances which tend to be immiscible. For example milk is an emulsion in which oils and proteins are held in suspension as miscelles which are microscopic and can be smaller than a cell. The white color of milk is due to the fact that its particles scatter all frequencies of light equally. The particles are what makes milk "milky" and allow it to be thinned by adding water. This is what those trying mitigate the oil spill do with the oil. As a result it ends up in the water column.

The dispersant initially used by BP was Corexit 9527 which is slightly toxic and the EPA has asked BP to switch to the less toxic Corexit 9500. Another dispersant used in oil spills is Dispersit. Before BP can do anything in the Gulf it has to file a plan with the government which needs to be approved. BP has posted its plans for the use of dispersants as part of its response to the oil spill on its website. These plans included the subsea injections of dispersants during the initial capping operation and topkill. The intent was to let nature take its course. They also note the need to monitor the water column.

Oil can also be broken up by mechanical means as was discovered by Joseph Plateau about 140 years ago. He altered the physical properties of oil in water and rotated the oil breaking it up into smaller particles. Physical forces act on the particles of oil and allow it to disperse through diffusion which is driven by the random motions of the particles like that in Brownian motion. One would expect the oil in suspension in the water column to spread both vertically and horizontally through diffusion and also by convection and currents.

There are still concerns about the persistence of the oil in the water column and its subsequent effect on the environment.

Doing Bisections with Google SketchUp

(click to enlarge)

The image above shows how one can do both line and angle bisections with Google SketchUp. A circle is used to mark off equal lengths along the sides of an angle and then the protractor tool is used to create intersecting guide line with the same included angle. One can use a pair of isosceles triangles to bisect a line.

Doing Constructions with Google SketchUp

This afternoon I discovered Google SketchUp and tested its ability to do geometric constructions. The most difficult problem I had was doing line and angle bisections but these tasks can be accomplished by using the protractor tool to draw guide lines at fixed angles to find points of intersection similar to those found with a compass. The figure did not come out as precise as I would have liked but tinkering with the settings would probably help.

This construction of the pentagon is my first effort using SketchUp. It might help if there was a compass tool for marking off fixed distances along a line and drawing arcs a fixed distance from a point.

The Pentagon and the Golden Section

The cosine of 144° in magnitude equals 1/2 the golden ratio, φ, which was used in the construction of the pentagon. Euclid in his Elements, circa 300 B.C., also uses the golden ratio to construct the pentagon. See Euclid, Elements, Book IV, Propositions 10-11. But Eudoxus, an astronomer and a contemporary of Plato, is given credit for discovery of the golden section which defines the golden ratio. Much of the content of Euclid's Elements was known to the Pythagoreans who shared their knowledge with the Athenian philosophers.

How to Construct a Pentagon

If one wants to make a pentagonal frame it may help to construct a pentagon on paper using a compass and straightedge. For a pentagon the vertices are points on a unit circle separated by θ = 360°/5 = 72° but this angle is not easily found without a protractor. One can use complex numbers to find the projection of this angle, cos θ, onto the horizontal axis. Squaring the complex number, which is equivalent to doubling the angle, produces a length which is easier to construct.

One can find one half the square root of 5 since it is the hypotenuse of a right triangle with sides 1 and 1/2. This length is added to one half the radius of the circle to find cos 2θ. Drawing a perpendicular at half this distance from the center gives two of the points of the pentagon at angles 2θ and 3θ. Bisecting this the angles between these points and the horizontal axis gives the points at angles θ and 4θ. The final point is at angle θ = 0°.

The image above shows the actual construction with additional information to make it easier to follow the construction.

A Short Reading List

Surface tension

Plateau's problem

Plateau's laws

J. Plateau - Wikipedia

J. Plateau - U. Ghent

Liquide glycérique

The Plateau Experiment

Emulsion

"Minimal Surfaces" a Misnomer?

Here is a side view of the soap films for the pentagonal frame. It looks like all the film intersections are curved lines as can be seen in the two nearer vertical intersections. The soap films do not appear to be minimal surfaces as claimed but may more correctly be called equilibrium surfaces. The stresses and forces in the films have to balance out everywhere. The films themselves are subject to gravitational forces and the intersections dropping from the vertices of the pentagon look similar to catenaries. Pressure changes are negligible except for the spherical bubble in the background. Internal stresses in the films are probably not uniform due to the weight of the films as one finds for the tension in a hanging string.

Pentagon Soap Films

I made a wire frame pentagon using #14 copper wire, some styrofoam packing material as a base and some hot glue to hold everything together. Using the styrofoam packing material allows more freedom in the position of the parallel wires. Ordinary dish washing detergent in a large pot was used to create the soap films. One has to dip the frame upside down in order to form the Steiner tree. Right side up produces only five parallel sheets along the perimeter.

The two lower side sheets which connect with the lower base sheet have a curved intersection which drops away from the upper level of the pentagon. If one looks closely one can see where these sheets intersect with the styrofoam at the bottom. The tree produced is very close to what was computed. The surface tension may have caused the soap films to adapt to balance the forces on them.

One has to be quick to capture these films with a weak soap solution. I had to hold the camera in one hand and dip the wire frame with the other. It took several attempts and a little coordination to successfully capture the trees before they disappeared.

A Complicating Factor

The rule about there being just three angles doesn't apply if more than one link ends on one of the points. In the exceptional case below the Steiner tree is a combination of two three point trees. One has to make assumptions about the nature of the tree for a particular set of points. The assumptions affect the values of the objective function and there is no guarantee that the type of tree chosen will yield the global minimum. The reason for the exception in this case is that the point near the center lies on one of the links of the tree of the other four points. The central point would have to be far enough away from the central link for there to form a 120° angle in order for there to be another intermediate point. The result is that there may be some play in the structure of the trees.

A Solution of the Steiner Tree Problem

The solution of a Steiner Tree Problem can be rather difficult. The reason is that unknown magnitudes of the links are functions of an unknown angle. For an example with the five points, the Ps, in the diagram below one can assume three intermediate points, the Xs, and label the seven links a through g. The tree has only three directions since there is a fixed angular separation of 120° between them and these are labeled α, β = α + π/3, and γ = α - π/3.

From the paths between the points, a-b, a-c-d, etc. we get a set of equations involving the unknown lengths and the directions (an e subscripted by one of the angles). From the way in which the paths were chosen the first three equations each introduce a new pair of unknown lengths. One can solve these equations for one pair of unknowns at a time since we can use the previously found functions for the lengths.

To solve for a pair of unknown lengths we can take the dot product of the equation with two independent vectors, in this case the unit vectors with the angles α and β, and it is useful to introduce the notation for a pair of dot products. (U V) is essentially a matrix and its product with a vector is a vector.

The resulting sets of equations for the unknown lengths are,

And by multiplying both sides by the inverse of the matrix on the left we get the linear functions for the unknown lengths.






Once the functions have been found we can then solve the last equation for the unknown angle, α.





If the points are the vertices of a pentagon, we find that α = 90° and we get a plot that looks more like a honey comb structure.

Steiner Trees and Minimal Surfaces

There is a physical analog to the Steiner Tree Problem that allows one to find the solutions without doing a single calculation. The soap films of a wire frame model will form a minimal surface. If one uses the positions of a set of parallel wires in the frame to represent the points and eliminates the unwanted surfaces of the bubbles produced by dipping the frame in a soap solution, a set of minimal surfaces will be formed which is analogous to the Steiner tree. The analogy is even better if one looks just at a plane perpendicular to the parallel wires. See Soap Bubbles by C.V. Boys for more information about soap bubbles on wire frames.

Bubble Kits

Bubble Science Kit - Amazon

Bubble Science Kit - Target

Bubble Builder - Scientifics

Steiner Tree Problem

The generalization of the minimal links problem is known as the Steiner Tree Problem which tries to find the network of minimum length which will connect a given number of points and allowing the selection of arbitrary intermediate points. Below are two Steiner trees for the same set of four points. They correspond to two local minima with the first being the global minimum. The two functions, f1 and f2, are the objective functions and give the sum of the length of the lines in the network. The two intermediate points are based on different choices for circles associated with opposing pairs of points. Pairing points that lie closer together yields the global minimum. Note the honey comb like structure of the trees.