R. M. R. Boundry Value Problems
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BOUNDARY VALUE PROBLEMS - By Rene M. Rogers

      The partial differential equations (PDEs) of physics are the most general and inclusive statements of physical laws that we have been able to formulate.  Specific solutions in specific cases must satisfy not only the underlying PDE, but the specific boundary conditions as well.  The art of finding specific solutions is the main thrust of a senior level course in Boundary Value Problems (BVP) which used to be required for any advanced engineering degree when I went to college.  This is where we learn that analytic solutions are often possible for physics problems involving rectangular, cylindrical, or spherical symmetry.  The solutions tend to involve the EXP function in all cases in some combination with SIN and COS functions in the case of rectangular coordinates, Bessel Functions in the case of cylindrical coordinates, and LeGendre Polynomials in the case of spherical coordinates.  Modern desktop computers have made it possible to solve the PDEs with almost arbitrary boundary conditions, but there is nothing like the old analytic methods to activate our intuition.  In fact, computer solutions to thermal, mechanical, and electrical stress problems tend to be so lacking in intuitive value that an approximate analytical solution is usually required to verify that the computer solution is at all plausible.  Once we are reasonably confident that the computer solution is in the right ballpark, however, we tend to find that it is more accurate than any analytic solution based upon simplifying assumptions.

 

      When I studied BVP in college our professor began the course by saying that the best way to solve these problems was to know the answer.  This is not as facetious as it may sound.  I would modify the idea slightly by saying that knowing the form of the answer was most helpful.  Suffice it to say that many brilliant mathematical minds have covered the subject very thoroughly and a great body of literature has been compiled over the last century or so. CONDUCTION OF HEAT IN SOLIDS, by Carslaw & Jeager is an ideal reference book for the serious student.  Since most of us have a lot more intuition regarding the flow of heat, I will begin with a BVP in heat transfer at first and relate the results to the diffusion of Atomic Particles (Aps)in solids later.

 

      Perhaps the simplest BVP of interest here is the case of one dimensional flow in a slab of uniform thickness.  The slab is thin compared to its extent in the two dimensions at right angles to an axis in the direction between the two faces.  Of course, heat and APs diffuse in all directions, but if the slab is great in the lateral dimensions there will be negligible net flow in those directions.  We can thus devote our attention to flow between the faces in the direction of the thin axis.  Consider the specific case of a metal disk 3 inches in diameter and 1/4 inch thick.  Suppose the disk is at room temperature when it is suddenly immersed in boiling water.  The temperature at the surface will rise almost instantly to the temperature of the water and heat will flow inward at all points on the surface.  The temperature midway between the surfaces will remain constant for a short while except at the outer edge, but it will ultimately come to the temperature at the surfaces.  We can for all practical purposes neglect the effect of radial heat flow except near the outer diameter.  If the water is boiling in turbulent fashion, we can probably neglect the cooling effect on the water due to the introduction of the cooler plate.  If, however, the water is stagnant and the temperature is only slightly different from the temperature of the plate when it is immersed, we may have to consider the thermal conductivity of the water in order to find a completely satisfactory solution. 

 

      At the instant of immersion, the thermal gradient at the surface is infinite if we take the temperature on the surface to be the temperature of the boiling water and the temperature just inside the metal to be the initial temperature of the plate.  This is a mathematical fiction which goes away when we consider the atomic scale and realize that the dependent variable in the heat equation, the temperature T(x,y,z,t), is really the expected value of a random variable proportional to the random thermal energy at each point.  In any case, the infinite value of the temperature gradient does not cause us any mathematical grief since the thermal gradient is quite manageable after the 1st tick of the clock.  After the 2nd tick of the clock, the 2nd atomic layer under the surface begins to get the news as a thermal front starts to propagate inward from all surfaces.

 

      It turns out, in this case, that the temperature near the surface can be expressed by a simple analytic function as:

 

T(x,t) = T0 + {Tmax - T0} * ERF(x/(4Dt))

 

where Tmax is the temperature of the boiling water, T0 is the temperature of the plate before immersion, and ERF is the error function.  ERF is not a common function on pocket calculators, but it is a simple matter to set up the function on a computer spread sheet or make a table using a simple basic program.  The first derivative of the error function is simply the normal distribution Gaussian function, from which it follows that the error function is the area under the normal bell curve with the appropriate scaling factor.  Since the bell curve is rather flat on top, the error function increases almost linearly while the argument is small, but it curves downward rapidly and converges to a maximum value of 1 as the argument gets larger than 2. 

 

      The following table is intended as an aid to the intuition:

 

             u           ERF(u)             ERFC(u) = 1 - ERF(u)

               0.0             0.0              1.0

               0.2             0.2227                 0.7773

               0.4             0.4284                0.5716

               0.6             0.6039           0.3961

               0.8             0.7421          0.2579

               1.0             0.8427                 0.1573

               1.5             0.9661                 0.0339

               2.0             0.9953                 0.0047

     

      The argument of the error function, x/(4Dt), suggests a convenient parameter of scale, namely (4Dt) Cm, sometimes referred to as the diffusion skin depth or the thermal skin depth.[1]  If we know both D, Cm^2/Sec, and the time t, Sec, we can readily estimate the extent below the surface where most of the action takes place, either heat flow or the diffusion of APs.  If the thickness of the work is small compared to this diffusion skin depth, the solution to the boundary value problem takes a different form than the one suggested here.  In the case of the 3 inch diameter plate 1/4 inch thick, the solution along the axis in the thin direction takes the form of a series of exponentially decaying COS functions.  This series can be shown, after some heavy mathematical lifting, to converge to the ERF solution at the surfaces when the skin depth is small compared to the thickness of the plate. 

 

      The diffusion of trace elements in a flaked stone tool is quite analogous to the flow of heat suggested above.  In the two arrowheads I have which show distinct diffusion patterns in their broken cross sections, it seems reasonable to assume that the trace elements responsible for the color variations we see were more or less uniformly distributed throughout the blank when the tool was being made.  Once finished, however, those elements with binding energies low enough to allow for a significant number of random walk events continued to walk on as before.  Those at or near the surface were apt to leave the artifact sooner rather than later and reveal the age of the surface to the skilled and thoughtful observer.  The dependent variable in the diffusion equation is the concentration C(x,t), APs/Cm^3, being distinct and independent[2] for each such element.  As a boundary condition before the surface is created, we assume C(x,t) = C0, the initial concentration, for all t <= 0.  When the binding energy at the surface is significantly less than the heat of diffusion, we can assume that the surface concentration will be negligible, thus C(0,t) = 0 for all t > 0.  In this case the solution is: 

 

C(x,t) = C0 * ERF(x/(4Dt))  Atomic Particles/Cm^3

 

as long as the diffusion skin depth, (4Dt), Cm, is small compared to the thickness of the artifact. 

 

      At some point we must address a number of practical problems when it comes to applying this theory to measuring the age of artifacts.  How do we measure the diffusion skin depth?  How do we know the heat of diffusion for a specific trace element in a specific rock sample?  How do we know the surface binding energy for each trace element for a specific host?  When confronted with these questions with regard to sulfur diffusion in stainless steel, we did not become concerned with the concentration profile except to confirm that the diffusion skin depth was thin compared to the thickness of the sample taken for analysis.  We were able to calculate the total amount of sulfur lost in terms of the concentration gradient at the surface as a function of time in the furnace and compare this to the measured results.  From this data we were able to calculate the diffusion parameter, D = Do * EXP(-Qd/kT) for several values of the temperature T.  A plot of LOG(D) vs 1000/T turned out to be a straight line which gave us both Qd and Do.  From these values, we were able to calculate the diffusion skin depth and verify that the sample in each case was thick enough to include all of the sulfur lost.  We assumed at the outset that C(0,t) = 0, i.e. that the statistics of migration to the surface were far more tortuous than the statistics of leaving the surface so that each sulfur atom reaching the surface was quickly carried away in one form or another by the hydrogen atmosphere in the furnace.  The linearity of the plot of LOG(D) vs 1000/T suggests that this assumption is reasonable.  We note that this is not always the case.  For example, Hydrogen atoms, H1, diffuse rapidly in iron but are never observed to leave the surface unless paired with another atom to form a molecule, H2. 

 

      There was very little data in the literature regarding diffusion parameters in 1958, but after 1970 there appeared an entire section on the subject in the CRC Handbook.  This data was derived from studies of the migration of radioactive isotopes of metals in metals, but it is not clear precisely how the specimens were prepared and the data taken.  In any case, I later had occasion to electroplate a small aluminum wire with gold on one end and copper on the other end and, using the scanning electron microscope, measure the diffusion profile of these elements into the aluminum after a specific time and temperature exposure.  The intent was to construct an integrating thermometer to measure the temperature in a vacuum tube collector.  We never got around to the intended use, but we were able to measure the diffusion profiles at both ends of the wire and roughly confirm the data given in the CRC handbook.

 

      The scanning electron microscope is the ideal device to measure concentration profiles of certain elements, particularly the metals.  The idea is that a sharply focused electron beam impinging on a carefully prepared specimen will excite the atoms on the surface to emit x-rays which can be analyzed according to wavelength and thus generate a precise picture of the distribution of specific atoms.  Once we can observe changes in the concentration profile of a trace element as a function of time and temperature, we have all we need to determine the diffusion parameters Do and Qd as well as the binding energy at the surface.  It might also be possible to measure the concentration profile by taking very thin samples from a surface and analyzing them using flame photometry or old fashioned wet chemistry.  Suffice it to say that there are a variety of methods available to the determined investigator with adequate resources such as we would expect to find in any materials science laboratory.

 

      In August, 1997, I had occasion to revisit some of the sites where I had found flaked tools in the 1930's and 40's and found most of them covered with housing or heavily wooded, but I did manage to find one point along a footpath and a number of chert flakes as well as a few blank nodules.  I broke one of the flakes last evening and found, as no surprise, a beautiful diffusion pattern related to the surfaces.  This piece has no artistic value, but it is large enough to provide a number of samples for analysis.  Given the resources, we could easily determine the structure of the rock as a whole, the types and relative abundance of the trace elements, and their distributions in relation to the various surfaces.  We could then bake the samples at various times and temperatures and note the changes in the distributions of the trace elements.  Given enough such data we could then calculate the diffusion parameters of each trace element in this specific host structure and, using computer modeling of the odd boundary conditions, make as many estimates of the elapsed time and temperature since the surfaces were formed as we can find trace elements.  If there is only a single trace element whose diffusion parameters are known, the concentration profile tells us only a combination of time and temperature.  We can assume a temperature and calculate a time, but if there is more than one tracer element both the time and the average temperature is determined.  If there are a number of tracer elements, the time, temperature, and the statistical uncertainties of the entire process become evident.

 

      Such a study, often repeated, will lead to the formation of a large database of the diffusion parameters of a wide variety of trace elements in a wide variety of host structures.  There will, almost certainly, be many surprises along with a few seemingly impossible puzzles, if past experience is any guide.  The highlights of any scientific career are those occasions where the data tempts us to believe in the supernatural until the demands of intellectual integrity forces us to take yet another look until the truth leaps out on the stage with dazzling clarity. 

 

      There are other experiments in diffusion which deserve mention.  I once had occasion to observe the evolution of gas from the surface of a microscope slide in a vacuum bell jar.  I was using a well focused electron beam to evaporate the glass with the intent of forming an array of tiny pits in the surface.  I never did any detailed analysis, but it was apparent that the amount of gas evolved depended on how deep the pit was and how long the slide had been exposed to the atmosphere since being baked at 400 DgC.  I went to the literature and read about the permeability of glass vacuum tube envelopes to the noble gases in the atmosphere, notably Argon, Neon, and Helium.  The data in those articles was roughly consistent with what I was observing.  We have already mentioned Obsidian Hydration whereby this volcanic glass, a favorite material for making flaked tools wherever it is available, takes up water from the atmosphere.  Whereas the amount of water taken up depends on the relative humidity in the environment and other factors, the concentration of the noble gases in the atmosphere is relatively constant, or so we can suppose over the past few million years. 

 

      Most of us have some experience with how sharp the edges of broken glass are.  I have heard the claim that fresh obsidian flakes hold the sharpest edges known and I have no reason to doubt it.  I also know that nature abhors a sharp edge.  Chemical activity and abrasion will certainly dull any blade in time.  I believe that we are very likely to be able to observe the rate at which fresh obsidian, or chert etc., flakes lose their edge in vacuum by constructing an experiment whereby the force required to sever a standard fiber can be precisely measured as a function of time after the flake is formed.  This data can tell us how frequently the atoms on the surface of the flake move about as a function of temperature and thus the binding energy which holds the material together.  I have had any number of people come to show me trophies of their vacations having bought arrowheads at the trading post.  My sense is always that the edges are too sharp and the surface strains too evident for the artifacts to be other than recent creations made especially for the tourist trade.  I doubt that a study of surface migration will lead to a reliable age dating technique, but knowledge of the self binding energies must be generally useful in any study of diffusion. 

 

      Finally, question of forgeries in art works, sculpture as well as painting, come up in the media from time to time, with large sums of money being at stake.  I must believe, in the case of painting, that studies of the diffusion of pigments in contact with one another must give an unequivocal answer to the question of when the painting was made.  In the case of sculpture, the question as to when the surface was created must yield to the study of diffusion profiles of trace elements within the material as well as the absorption of noble gases from the atmosphere.  In the case of brass castings, it is well known that the initial thermal stresses relieve themselves over time in all types of castings.  The amount or residual stress energy must be a dead giveaway of the time elapsed since the casting was made.  This is a matter of self diffusion according to the binding energy which holds the basic structure together at the atomic level.



    [1]The thermal skin depth on the surface of the earth due to seasonal variations in the air temperature and solar output is a familiar concept to pipeline engineers and others.  Water pipes in the northern and southern climates must be buried below the frost line where the temperature remains relatively constant the year around.

    [2]I have been told by a reliable source that the diffusion parameters of some trace elements in nickel cathodes are detectably affected by the presence of other trace elements.  I suspect that we will have come a long way indeed when we are able to make such distinctions in the project at hand.

 

 
 

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