S=k*ln(W)

On the Energy Barrier

I’m of the opinion that the first physical system that every engineer and scientist should understand is the energy barrier.  Energy barriers are one of those things that tend to appear in nature with a certain regularity as just about anything which is thermally activated tends to depend on them.

Are you interested in how the viscosity or diffusivity of a molecule changes with temperature?  Well, there’s an energy barrier associated with molecular flow.  Once a molecule has enough energy to move past its nearest neighbors, it is capable of jumping to another point in space.  Perhaps you’re more interested in the ability of an electron to jump out of it’s orbital and into space?  Well, there’s an energy barrier associated with that jump which has to do with the coulombic attraction of electrons to the nucleus of an atom.  What about the speed of a chemical reaction?  Well, the rate of reaction is temperature dependant because as molecules run into each other, there is generally an energy maximum associated with their approach which they must overcome in order to get into a stable minimum energy configuration associated with bonding.  It’s this energy barrier which keeps reactions from being spontaneous at absolute temperatures equal to 0K (or - more generally - thermal energies less than the height of the barrier).

So, how can we model the energy barrier?  Well, let’s say we have some distribution of particles sitting at a point with low energy (E1).  Right next to this low energy point is a point with higher energy (E2) such that the difference between the two is E2-E1.  There’s our barrier.  These particles have a distribution of different energies because they are almost certainly bouncing around this barrier and off of one another with different velocities.  If you wonder why it might be that these differing velocities lead to different particle energies, you can draw a parallel between velocity and kinetic energy by remembering that for a general hypothetical particle, kinetic energy is equal to half the mass times the squared velocity.  The only particles which can surmount the energy barrier are those particles in the distribution with energies greater than E2.

Now, the distribution of energy that these particles possess is based on the type of particle we’re looking at.  If these particles are atoms, they follow a Boltzmann Distribution.  If they’re electrons, Pauli Exclusion applies and we have to follow the Fermi-Dirac Distribution.  If these particles are photons and are trapped somehow, they cannot “feel” one another and the Bose-Einstein Distribution applies.  All of these distributions act in such a manner that the energy difference is divided by the product of the temperature and Boltzmann’s Constant (“k” - a handy number which allows us to convert between temperature and thermal energy) and is set inside an inverse exponential.  These distributions express the percentage of particles residing with energies above the barrier.  So, if we want to know how many particles surmount the barrier and are “free” we just use the distribution itself multiplied by the number of particles (N).  In the case of a Boltzmann Distribution, we get P=N/exp((E2-E1)/(kT)).

So, how do we, smart engineers and scientists that we are, manipulate energy barriers?  The easiest way to do it is by varying a temperature.  Raise the temperature and you will have more particles with energies greater than the barrier which will be “free”, so your processes tend to move faster.  In solid state chemistry, the act of raising the temperature so that a reaction occurs at a faster rate is called “annealing”.  Of course, there are some situations where you can’t raise the temperature enough to get the reaction to move fast enough.  Impatient being that you are, you decide to change the only other variable in the equation that isn’t a physical constant, the energy of the barrier.  Now how do you go about doing this? The barrier is defined by the matter that you’re working with, so you choose a different physical scenario.

In chemistry, if you have a reaction which has a really big energy barrier, you can sometimes break this reaction into smaller chain reactions which are a little less efficient but have a smaller energy requirement.  As an alternative, you can introduce a catalyst which will reduce the energy barrier.  When we grow carbon nanotubes, we grow them on a metal catalyst like gold or nickel.  By doing this, we can reduce the reaction temperature required to form nanotubes from somewhere in the vicinity of 1000 degrees Centigrade to a paltry 400C to 600C.

In electrical engineering, the diode is one of those basic units that we use to create much more complex digital systems - including transistors.  It exists simply as an energy barrier to electrons.  Because temperature is generally one of those fixed constants in electrical systems (unless you want to light your motherboard on fire), we have to vary the size of the energy barrier.  This is done by applying a larger voltage to the “low energy” side.  Applying a larger voltage shrinks the barrier and allows more electrons through, increasing current.  Diodes will act as an element in a circuit which will generally only pass current when the voltage applied is either a large negative value or a large positive value.  This energy barrier acts as a fundamental “switch”.

So, my message to all scientists and engineers is to learn how energy barriers work and impress your boss/professor.  You see these barriers everywhere and if you understand how they function, it’s much easier to understand how physical systems behave.