How long will a steel pillar support a bridge before rust eats deep enough to let it snap? How long will a container of corrosive acid hold before it springs a leak? A group of scientists at the University of Rochester has found a mathematical basis for predicting when a single point on a surface will erode to a critical depth. The findings are being published in the Sept. 24 issue of Physical Review Letters.
Predicting when the lowest point in a surface will erode to a critical depth or when the highest point will build up to a critical height is challenging because there is no way to gain statistics on these extreme points. It would be like asking what the average height of the tallest person in a room is-there is only one tallest person, and so your sample is always limited to one.
To produce information on these extreme points, Yonathan Shapir, professor of physics and chemical engineering at the University, and his team have combined scaling math, also known as fractal math, with a recent branch of mathematics called "extreme-value statistics." Evolving in just the last few decades, extreme-value statistics provides a way to determine the probability of extreme events, such as to forecast severe weather conditions or predict floods.
"This is the first time we've had a way to predict how these extreme points grow based on nothing more than the roughness of the surface they're on," says Shapir. An extreme point could be the deepest point of rust in a steel girder or the highest point of metal accumulation inside a battery that leads to a short. Understanding them can lead to better material designs and more reliable devices, as well as cutting the time needed to test such designs.
The process is like trying to find how tall the tallest person in the world is by measuring the height a roomful of people. Obviously, the tallest person in the world probably isn't in the room, but extreme-value statistics offers a way to estimate how tall that tallest person is likely to be.
The key to applying this approach to surfaces is the idea of scaling, or fractals, across both space and time. A pattern that can be scaled is one that has the same shape no matter how far you may "zoom in" on it. The basic pattern repeats itself on any scale at which you view the pattern. Fractals are visual patterns that display infinite scaling. Likewise, when a surface grows or is eroded, its overall pattern repeats itself with time. So a magnified piece of the surface will look like the whole surface after a long period of erosion or accumulation.
The three physicists and one mathematician visualized how the roughness of a surface changed as it wore away-by an acid, for instance-or accumulated, such as inside a battery. They concluded that the extreme point of a section changed in a non-linear relation to the roughness. It would be as if the tallest person grew faster than everyone else in the room. Shapir and his team could then scale the section to the whole of the surface being affected by the corrosion or accumulation. In essence, they could extrapolate the height of the tallest person in the world by scaling up the relationship between average and tallest to account for the world's population.
The next step in the research was to model the growth or erosion process backward in time to determine what a small section of the surface originally looked like, by studying the way the whole surface wound up. In collaboration with Michael Cranston, mathematics professor at the University, they were able to generalize the way the extreme points wound up by solving a mathematical model that should work similarly for any other fractal surface.
Graduate student Subhadip Raychaudhuri and undergraduate physics major Corry Przybyla then ran computer simulations of different growth and erosion processes thousands of times to gather enough statistics for the extreme points. Their results confirmed the fractal-based predictions.
"These findings show that we can determine when an extreme point will hit a critical height with much more accuracy than just trial and error," says Shapir. The research was funded by the National Science Foundation and the Office of Naval Research.