7. August 2013   1:11 pm
Dr. K. L. Mittal, Dr. Robert H. Lacombe

Dr. K. L. Mittal, Dr. Robert H. Lacombe

CHAPTER 1: DO WE REALLY KNOW WHAT A SOLID IS

As outlined in last month’s issue of the SURFACE SCIENCE CORNER blog the next several issues will be devoted to exploring the invisible properties of surfaces that we are rarely able to detect by simple inspection but nonetheless have rather profound consequences for the way surfaces behave. However, if we are going to uncover the hidden aspects of surfaces we are going to have to look at the atomic and molecular level which is where all the secretive phenomena reside. Before doing this it would be very helpful to look at the behavior of bulk solids at the atomic and molecular level since what goes on at surfaces is basically a subtle variation of what goes on in the bulk.

So what do we really know about the nature of solids? I wrote an encyclopedia article a while back dealing with Basic Concepts in Adhesion Science and as part of that article I tried to introduce some fundamental concepts on the nature of solids at the atomic and molecular level. The editor wrote back that this section could be eliminated since everyone knows what atoms are all about. Really! I wrote back that yes most people have the 19th century view that atoms are some kind of tiny ball bearings and that solid matter is just a colossal agglomeration of these tiny beads held together by some kind of atomic forces. However, how many people are aware that atomic matter is mostly an exceedingly intense electromagnetic field?

Before entering into the realm of quantum electrodynamics it would be a good idea to cycle back to the late 19th century and investigate the far more intuitive concepts of what defines a piece of solid matter. At that point in time the prevailing scientific notion of a solid was anything that had a significant modulus. So what is a modulus? This concept goes back to the 17th century British physicist Robert Hooke who first stated his law of elasticity in 1660 as a Latin anagram, whose solution he published in 1678 as Ut tensio, sic vis; literally translated as: “As the extension, so the force” more commonly “The extension is proportional to the force”. Basically if you pull on a strip of some material the amount of extension achieved will be directly proportional to the level of applied force and the constant of proportionality is what we call the modulus of the material. For our purposes we need to make this concept more quantitative and a simple example will help in this regard.

Say we have a thin strip of a typical engineering thermoplastic such as poly-styrene and apply our stretch test to it. For quantitative purposes we let our strip be 10 centimeters (10cm or 0.1 meter (0.1m)) long by 0.5cm (0.005 m) wide by 0.1cm (0.001 m)thick. Common thermoplastic materials are fairly stiff materials at room temperature so we will need to put it in a tensile test machine which will apply a tensile load to it measured in Newtons. For consistency we need to keep our units within the commonly accepted SI system (The Système international d’unités (SI), or International System of Units, defines seven units of measure as a basic set from which all other SI units are derived) so we use the Newton as the measure of force abbreviated as Nt which in magnitude amounts roughly to the weight of an apple or close to 200 grams. Now our tensile tester applies a force over the entire cross section of our sample which has some finite area and thus applies a stress or force per unit area to the test specimen. Again for consistency with the SI system of units we measure the stress in Newtons per square meter (abbr. Nt/m2) or Pascals the SI derived unit of pressure, stress, modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal (1623-1662).

Upon applying a load to the sample strip we note that it extends by some amount Δ. To be in conformity with common practice we divide the amount of extension Δ by the original length of the sample “l” to get a dimension less quantity ε = Δ/l called the strain. We can now recast Hook’s law as the stress is proportional to the strain where the constant of proportionality is called the modulus E also called the Young’s modulus in honor of Thomas Young an English polymath notable for scientific contributions to the fields of vision, light, solid mechanics, energy, physiology, language, musical harmony and Egyptology (a polymath indeed!). Our formula can now be compactly expressed by the following simple expression:

σ = E x ε or stress is the modulus times the strain.

We use the traditional Greek symbols sigma for the stress and epsilon for the strain. The stress has units Nt/m2and since the strain is dimension less the modulus must also have units Nt/m2.

Now back to our test strip. Say we apply a load of 150 Nt with the tensile tester and note that the sample stretches by 1 millimeter or 0.001 m. since our sample is 0.1 m long we have induced a strain of 0.001/0.1 = 0.01 or by 1 percent of its original length. The cross section of our strip was 0.005 m x 0.001 m = 0.000005 m2 and the applied load was 150 Nt so the applied stress was 150/0.000005 Nt/m2 = 30,000,000 pascals (abbr. Pa). We note that the pascal is a rather small unit since it amounts to roughly 200 grams spread over one square meter. The pascal amounts to roughly one ten millionth of the atmospheric pressure which is squeezing on every one of us living reasonably close to sea level. Rather than deal with such an inconvenient unit and have to carry all those cumbersome zeros around we define a new unit called the mega pascal or 1 million pascals (abbr. MPa). The applied stress is thus 30 MPa. Going back to Hook’s law as formulated above we see that the modulus of our sample has to be the stress divided by the strain or 30 MPa/0.01 = 3000 MPa. Since we abhor carrying around a lot of zeros we define yet a new unit called the giga pascal (abbr. GPa) which is 1000 MPa or 1 billion pascals. The modulus of our sample thus comes to a nice round 3 GPa. This is quite close the modulus of all common polymers from plastic cups and dinner plates to plastic eye glass lenses.

At this point our understanding of the elastic properties of solids is roughly where things stood at the end of the 19th century. All solids can be defined by their elastic modulus which is determined experimentally by stretching a test piece and dividing the applied stress by the induced strain. All was not well, however, since the theory was based on the hypothesis that all solid materials were part of continuum which looked the same at all scales of length. This notion was already under attack by the late 19th century since many of the ideas associated with chemical reactions were best explained by assuming that elements were discreet entities. By the mid 20th century the atomic theory of matter was established beyond any reasonable doubt and not only that but it was further determined that atoms themselves were made up of still smaller units called electrons, protons and neutrons.

So to explore the hidden nature of surfaces or bulk matter we have to look at the atomic level or at spatial separations of
10-10 m or 0.1 nanometers (abbr. nm). This means we must leave the classical mechanics of 19th century continuum theory behind and delve into the nether world of quantum mechanics. Now we certainly do not have the space here to get into the details of quantum theory but we can present the relevant results fairly succinctly. Those interested in a more detailed discussion are invited to go on the MST CONFERENCES web and look up Vol. 2 No. 3 of the conference newsletter under the heading Hamaker Theory or at Atomic Distances the World Follows the Rules of Quantum Mechanics (www.mstconf.com/Vol2No3-2005.pdf)

By the time the 20th century was half over it was well established that the properties of all common materials were fundamentally governed by electromagnetic interactions. Classical electromagnetism was well explained by Maxwell’s Equations and quantum theory explained how the electromagnetic interactions between the fundamental particles such as the proton and the electron served to bind atomic matter together. What was happening within the atomic nucleus was still problematic since an entirely different set of forces were operating but these forces held the typical nucleus together so strongly and within such a small volume that for the purposes of understanding the chemical nature of matter the nucleus could be assumed to be a point mass and a source of positive charge which attracted and bound the electrons. To get some idea of the sizes and distances involved consider that a typical atomic nucleus is on the order of 3.4×10-15m , the electromagnetic radius of the electron comes to about 2.8×10-15m and within a typical atom electrons and protons are separated on average by roughly a Bohr radius which is close to 5.3×10-11m. From these numbers one readily perceives that atomic matter amounts to relatively small point masses separated by huge distances. To put the matter in better perspective simply scale the numbers up by a factor of 1013. The nucleus then would have a radius of 3.4 cm, which is roughly a small orange, and the electron would come to 2.8 cm, or a crab apple, and they would be separated on average by a distance of 530 meters or roughly half a kilometer! Thus the world at the atomic level seems to be mostly empty space. Or is it? What is it that fills all that space and gives the impression that atomic matter is something sturdy and substantive? The answer as stated above is the electromagnetic field. In essence it is the electromagnetic field between the charged proton and electron that provides the substance of the hydrogen atom or any other atom for that matter. It turns out that the square of the electric field is proportional to an energy density typically measured in joules per cubic meter (abbr. J/m3) where the joule is the SI unit of energy named in honor of James Prescott Joule (1818-1889 an English physicist and brewer). Now we know from our elementary physics courses that the joule is formally defined as the force of 1 Nt acting through a distance of 1 m. Thus if you lift an apple off the floor and raise it to a height of 1 m you have expended 1 joule of energy in the process. We see then that the joule has the fundamental units of Newton-meter (abbr. Nt-m). Thus the units of energy density can be written J/m3 = Nt-m/m3 = Nt/m2 or putting it succinctly the units of energy density are the same as the units of stress or modulus. So the energy density associated with the electric field generated by the electron and proton in the hydrogen atom gives rise to a modulus of some sort.

It turns out that the electric field close to an electron or proton is exceedingly intense. The newsletter article mentioned above derives a simple formula for estimating the energy density associated with the electric field close to an electron and the results are summarized in the following table:

TABLE 1: Electromagnetic energy density in a neighborhood of an electron at representative atomic distances estimated from Eq.(10) (Newsletter article www.mstconf.com/Vol2No3-2005.pdf)

DISTANCE FROM ELECTRON (angstrom = 10-10m) ENERGY DENSITY/ ELECTROMAGNETIC MODULUS (GPa) COMMENTS
4.0 2.8 Roughly the modulus of a thermoplastic i.e. polystyrene
1.37 208.0 Modulus of metal such as steel
1.0 732.0 Modulus of refractory material like silicon carbide or diamond

Table 1 clearly shows that the electromagnetic energy density associated with the electron’s electric field can account, at least in a heuristic fashion, for the elastic properties of all forms of common matter.

This seems like a rather remarkable conclusion since it is also the electromagnetic field associated with sunlight that warms us on a sunny day. Even though solar radiation can warm us it certainly does not appear to have any appreciable substance. Nevertheless sunlight also presses down on us with a very week but non zero pressure. In particular if you step out on a sunny day and hold your hand facing the sun it receives a thermal input of about 7 watts which is just enough to detect. What is not detectable but still present is the fact that the solar radiation exerts a pressure on your hand of roughly 4.5×10-6 Pa which is so small that you would need a balance with nano Newton resolution to detect it. So it is essentially a matter of intensity that determines the apparent behavior of electromagnetic fields. The field close to an electron exerts a reactive force so intense that it closely emulates the behavior of solid matter whereas the field associated with common light is so week it seems to have no substance at all.

This basic fact was brought home to me many years ago as a graduate student doing a project on particle physics. At that time long ago we were analyzing bubble chamber tracks from proton collisions generated by the Zero Gradient Synchrotron at Argonne National Labs. We were essentially using highly specialized camera equipment to analyze the tracks of high energy protons as they traversed the chamber. Mostly what you saw on a given exposure were the slightly curved tracks of the protons as they wizzed through the chamber. The tracks had curvature since the chamber was immersed in a strong magnetic field which exerted a force perpendicular to the travel direction of the proton. Every dozen or so frames however, there would be an event whereby one of the incident protons would smash into one of the hydrogen atom protons which filled the chamber as the entire contents of the chamber was liquid hydrogen. A collision event would appear as a proton track suddenly terminating at a point in the chamber and a number of separate tracks would appear going off in all directions. Two protons essentially collided and exploded into a number of other particles indicating that the proton is not an elementary particle but made up of still smaller entities. What caught my eye on occasion, however, was the fact that some of the collision events gave off a phantom particle that would leave no track near the collision point. The only way one would know that such a particle had been emitted is that at some distance from the primary collision point a pair of tracks would appear emanating from a single point and veering off in separate directions leaving a V shaped imprint behind. That phantom particle was in fact a bit of exceedingly high energy electromagnetic radiation called a gamma ray. These gamma rays were so energetic that they were unstable and would spontaneously self annihilate by turning into an electron and a positron. This clearly illustrated the substantive nature of the electromagnetic field by directly turning radiation into two elementary particles.

This basically ends our introductory tutorial on the electromagnetic nature of solid matter. The stage is now set to explore further the consequences of what we have uncovered in relation to the behavior of solids at their surfaces and how they interact with other solids. We commonly deal with plasmas to modify surfaces and indeed the energy density of plasmas is also locked up in intense electromagnetic fields. So we will see that the electromagnetic field has a few more tricks up its sleeve as will be uncovered in future issues of THE SURFACE SCIENCE CORNER.

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