How do We Know a Black Hole Lives at the Milky Way's Center?

People often wonder how science can get a handle on out-of-this-world things like the properties of the universe many light years away or the small-scale structure of space-time itself.  In 2005, I presented the distinguished faculty address to give my audience a sense of how this is done.  The title of my talk was, "From Black Holes to the Internet:  How We Use The Scientific Method To Understand the Mysteries of Things Unseen."  It was a very enjoyable hour with lots of wonderful audience participation.  Here I summarize one part of my talk on "how we know" that there is a black hole in the center of our galaxy.

Science often proceeds by taking small steps that together build a general understanding. Here I outline how experiments on the earth along with observations of our universe can lead to an amazing understanding of the natural world.

The outline below shows how simple experiments on the earth's surface can be used to detect a massive black hole at the center of our galaxy, the Milky Way.

1.  Establish Newton's Theory of Gravity by measuring forces between hanging masses here on earth.
2. Use Newton's Theory of Gravity to determine the mass of the earth and check if it is consistent with what we know about the earth's composition and size.
3. Predict the orbital shape and period of the moon based on the earth's mass. It is found that the calculated period is consistent with the measured one and the shape of the orbit is accurately predicted (ellipse with the earth at the focus). This evidence suggests that gravity acts on lunar distances.
4. From the orbits of the planets, the mass of the sun is determined. Every planetary orbit gives the same solar mass (with the sun at the focus of every ellipse), so gravity appears to work even on these larger scales. Furthermore, the density of the sun that is determined from its mass is consistent with what we know of the sun's composition from independent spectroscopic measurements.
5. The orbital properties of all bodies in the solar system obey Newton's Theory of Gravity with impeccable precision. This includes comets, asteroids, moons, satellites, and space ships.
6. Mercury's orbit is found to deviate ever so slightly from Newton's predictions. This irks physicists until Einstein formulates the General Theory of Relativity in 1918 which fully accounts for the small deviation. It turns out that Newton's Theory of Gravity is a special case of General Relativity, which predicts the possibility of the existence of black holes.
7. Telescopes that view infrared light are able to penetrate the dust that obscures the center of the Milky Way to visible light to see stars at our galaxy's center.
8. Astronomers measure the orbits of these stars over more than a decade. As predicted by Newton, the orbits of the stars are perfect ellipses. The foci of these ellipses all coincide with an invisible object.
9. Using the orbital data, the calculated mass of the dark object is almost 4 million solar masses.
10. Some of the stars get very close to the dark object, so an upper limit of the object's size is determined from the distance of closest approach.
11. The dark object's mass and density fall in the range predicted by general relativity for a black hole.

The infrared observations of stellar orbits as described above do not prove the existence of a galactic black hole; but provide strong evidence. There are other independent measurements that all point to a black hole (see below). When the pieces of the puzzle are assembled, the picture that emerges is one of a huge black hole at the galactic center.

Incidentally, such massive black holes are found at the centers of other galaxies and even globular clusters. Since evidence of smaller black holes are routinely "observed" in binary star systems, it becomes clear that black holes are out there.

This journey illustrates something fundamental about nature, and about the breadth of physical laws. Richard Feynman said it best, "Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry."

A youtube video shows the motions of the stars around the central black hole that comes from direct telescopic observations.  The data covers over a decade of observations. A more dramatic version can be seen here.

Scientists are slow to accept a new idea or theory unless there are multiple pieces of supporting evidence.

The case for black holes is bolstered by X-ray observations of the hot gas surrounding the galactic black hole at the heart of our Milky Way. The observed X-ray spectrum can be used to determine the gas temperature using the same principles that betray the temperature of glowing orange embers in our fireplaces.

If the gas is to remain stationary, the inward gravitational tug of the black hole must be balanced by the outward pressure of the gas. The calculation is simple enough for a high school physics student, requiring only Newton's Universal Law of Gravity (see below) and the gas laws.

This simple calculation leads to a black hole mass of about 3.4 billion suns, in agreement with the observations of stellar orbits.

Newton's Universal Law of Gravity

Newton's universal law of gravity states that the force of attraction between two objects is proportional to the product of the two masses and inversely proportional to the square of the distance between their centers. The constant of proportionality is G.

G can be determined be measuring the forces between masses that are measured with a torsion balance. This is a common experiment done in most physics departments by students, and even in high schools.

 

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Kepler's Laws (1571 - 1630) Kepler's laws follow from Newton's theory of gravity.   Kepler's Laws state that: Planets travel in elliptical orbits with the sun at one of the foci (shown above). Equal areas are swept in equal times (see diagram below). The time it takes to complete one orbit is proportional to d3/2 All objects in our solar system are observed to obey Kepler's Laws, and therefore confirm Newton's more general theory. Einstein's Theory of General Relativity is the most general theory that makes small corrections to the orbit of Mercury and is required to make the GPS system work. But don't believe the authorities. Anyone can observe the motion of Jupiter's moons to determine the mass of the gas giant. I took the photo below of Jupiter and three of its moons.

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 Cutting Through the Dust

Raleigh found that the degree of scattering is proportional to the inverse of the fourth power of the wavelength of light. Blue light has a shorter wavelength than red light, so is more strongly scattered. Rayleigh scattering explains why the sky is blue.

Yellow fog lights work on the same principle. When white light is filtered to remove the blue light, what remains is green and red, which appears yellow. The longer wavelengths pass further through the fog and the scattered glare from the blue light is eliminated, making it easier to see.

The infrared range of the spectrum is made from light of even longer wavelengths, allowing telescopes to see through the muck. Special detectors are used to image the light. The image below is of the center of the Milky Way, taken by researchers at Max-Planck Intitut fur extraterrestrische Physik.

What I expect of my PhD students

Every graduate student needs to be aware of expectations.  Each adviser is unique and operates on different principles.  A small number view their students as a pair of hands to do the dirty work.  In the old days, some advisers expected their grad students to mow their lawn and do housework. A friend of mine from grad school had an old-school adviser who had him come to his place on the weekend for services.  However, most faculty members by that time had already moved into the modern era and this kind of despicable practice is no longer tolerated.

It is important that each student have an understanding of what they are getting into when they join a research group. Here I will explain what I expect.

My highest principle is thinking of a graduate student as a junior-colleague-to-be.  Even the best students start out incapable of doing real research.  They are clumsy in the lab and need lots of hard work to sharpen their analytical skills.  As such, my first criterion is that they be good students who like to learn.  That doesn't mean that they need to be straight-A students.  In fact, many A students make poor researchers because they lack creativity.

On the topic of classes, I believe that the more the better.  I encourage students who are in the middle of their research phase to take classes.  Ironically, the students themselves resist because they feel it interferes with research.  Even some of my colleagues prefer that their students not take courses that are not of direct help to the research at hand.  I strongly disagree with this premise. The process of learning new things sows new ideas.  I myself enjoy learning because these new nuggets of knowledge invariably get incorporated into my research, which leads to totally new and wonderful directions.

Principle 1.  Always keep learning from classes, reading the literature, and just thinking about crazy ideas.  If you feel yourself to be leaving your comfort zone, you are on the right track.  I expect my students to never stop learning and to be constantly pushing themselves.

An important part of being a PhD scientist is independence.  Grants, which support research activities, expect results.  Many advisers thus give their students a very short leash.  The end result is bad for the student's independence.  I prefer to give the student a specific assignment, and let her and him work on it for a year or so without giving them a detailed map of how to get there.  However, I do give lots of course corrections and teach them things they need to know along the way (or send them to the literature) if their struggles are based on missing information.

I once had a PhD student who complained that I seemed to be giving another student lots of attention while neglecting him.  I treat PhD and masters degrees differently.  The individual with a masters degree needs skills to survive in industry, while a PhD scientist is required to come up with new ideas and find ways to tackle a new problem.  Having said that, some of my best and independent students happened to be masters students who were quite capable of getting a PhD.

Principle 2. Learn to be self reliant early on in your research.  Read the literature and talk to others to gain the skills you need to do your work, but don't wait for someone to tell you what to do with those skills.  Constantly try new things in the lab or with paper and pencil to both sharpen your skills and generate new ideas.  You will make lots of mistakes along the way when not given step-by-step instructions, but making these mistakes and getting through them are the most important part of the experience.

I do not yell at my students, ever.  I may tell them when I am displeased with research performance, but if a student does not perform, (s)he will not get a degree.  Passing the Prelim and being in a research group is no guarantee of success.   I disagree with the idea espoused by the administration that we need to help the students along so that we increase our graduation rate.  Graduating a PhD without the proper skills and talents serves nobody.  A PhD degree is not a ticket to a good job.  It's the skills that the individual has mastered and the ability to think independently that makes him or her valuable to society.  A huge pool of unemployed physicists is not what we want to be generating.

It takes lots of hard work and perseverance to finish a PhD degree.  People often ask me how many hours they should work.  My answer is all the time.  If you are not excited by your work and don't enjoy thinking about physics beyond your area of expertise, then you're in the wrong field.  Academic jobs are tough to get, and real research jobs in industry are rare.  However, PhD physicists have lots of success in engineering jobs, which are more plentiful.  If you like to tinker, then engineering may be an excellent way to earn a good living while having fun.

A PhD degree should not only be a guarantee of skills, but of work effort and perseverance.

Principle 3.  Approach your research with a passion.  The benefit of enjoying your work, aside form the direct rush of endorphins, is that you will put in the time required to do a good job.

One of the most important attributes is perseverance.  Watching Star Trek, or other sci-fi shows/movies gives one the impression that scientists apply skills to very easily solve problems.  This is not the case unless a student is super lucky.  I have a long list of stories on the same theme;  students who would spend months trying to get an experiment to run, only to have to start from scratch to try a different approach.  Aside from being good problem solvers, the PhD degree is an imprimatur of a person that does not give up.

In my PhD work, I had spent quite some time building an experiment on a 5' x 10' table optical table, which was filled with all sorts of laser sources and optics, resembling a Borg city.  Each step in the process often required a step backwards.  After completing the construction of the experiment, it still didn't work.  I then figured out the reason, tore the experiment apart and rebuilt the whole thing on another optical table with another laser.  Luckily, it ended up working.  Only then could I start taking the data that was the topic of my thesis.

Principle 4. Don't give up.  When something doesn't work, don't shy away from the problem.  Work twice as hard.  This will serve you well in all aspects of your lives.  When I was a new faculty member, aware of the importance of getting grants, I would write two new proposals for every one that did not get funded.  After my first two years, I built a healthy portfolio of projects.

Many people do not recognize the creativity behind science.  Creativity is part of choosing a scientific problem to study, helps in problem solving, and leads to interesting new science.

Principle 5.  Be creative.  Always think about neat implications of what you are doing or find new ways of looking at old problems.  This skill is particularity important in the career of young scientists who want to make it to the next level.

Principle 6.  Be meticulously careful in your work. Do not publish sloppy results, which will come back to haunt you, and always apply the highest standards to yourself.  You should be more critical of your own work than I am of you as your  adviser.  On the flip side, do not let this attitude prevent you from finishing a project.  In the end, we can never be sure if we are right, and there will always be a mistake somewhere.  If a paper is perfect, it is not science.  When working in the unknown, there are always huge dark shadows in the areas not exposed by your searchlight.

Principle 7.  Be honest with others, but especially with yourself.  Many very good scientists have fallen into the trap of fooling themselves into believing in something that is false.  Consider cold fusion, N-rays. etc.  Design experiments that are resistant to experimenter bias.  Also, do not try to make data fit your adviser's expectations.  I need to know when an experiment contradicts my viewpoint.  And it goes without saying that you should never fudge data in any way or plagiarize the work of others.

Principle 8. Impress me.  When you graduate, I need to make an honest assessment of your strengths and weaknesses in the letter of recommendation, which will determine your success on the job market.  Follow all of the above principles.  Do not come to my office to ask what you should do next.  Tell me the issues you are having, your line of reasoning, possible explanations, and engage me in debate about the possibilities.  You should act as a junior colleague.  Don't worry about offending me.  I am more interested in getting at the truth than being right.  However, that does not give you the right to be caustic.

What I have posted above mentions nothing of the content of the work, which is of central importance.  A short post cannot cover the nuances.  To zeroth-order approximation, I expect the student to add a new piece of physics to the body of knowledge.  This could be a theory that helps us understand a phenomenon or the discovery of a new phenomena.  Fitting data to a mathematical expression is not enough.  The parameters of the theory must have meaning that is independently testable, be interpretable in terms of fundamental processes, and make predictions well beyond the domain of the original results that generated the theory.  Perhaps I will write more on the topic later.

If you approach everything in life with a passion, it will be a fulfilling one.   When you take a break from physics, make it count.  While I may seem one-dimensional in this post, I do find time for other activities.  Though I am not good at it, I play ice hockey with a passion.  I enjoy playing the piano and writing.  Taking a break from work is, in a sense, work.  During times of alternate activities, things percolate in the brain.  I have had the most profound revelations while driving my car in the middle of nowhere or playing the piano.  So, don't hesitate to take a break with intense activities.

I have to run now.  After I finish packing, I will take a short walk around San Diego, then I have to catch a plane back to Pullman.  Until then, get excited about physics.

Perhaps this time it may be right - taking a big chance

I wrote a while back how Shiva's measurements gave 0.29eV as the binding energies in our polymer/dye material (with an experimental uncertainty of 0.02 eV) which is responsible for forming domains that are at the heart of our theory of self-hearing .  I tried to figure out what interactions between molecules and polymer would give this energy and came up with a possibility.  But because I read the data tables incorrectly, I wrongly thought I had solved the problem.

When preparing my talk for SPIE a couple days ago, I drew the PMMA polymer chain with a molecule drawing program and added a few DO11 tautomer molecules to see where they would fit.  Miraculously, as a plopped the DO11 molecules on the page, I immediately saw that the NH from the DO11 tautomer cozies up to one oxygen in the PMMA polymer chain while the OH group naturally attaches itself to another oxygen in the chain, as shown above.  And he energy?  You got it; the sum of the two hydrogen bound energies is 0.30eV, a match.  The table below shows the energies of four types of hydrogen bonds.




There are always other possibilities that we have not yet considered, but this smells right.  Perhaps we are onto something.  Future experimentalists will allow us to test this hypothesis and zero in on what is going on when a molecule self heals.

This project has been one huge puzzle, were each new experiment presents to us a new piece.  It reminds me of how the discovers of the structure of DNA (Crick, Watson, and Wilson  ) pieced together cardboard cutouts of molecules to guess its molecular structure, and confirmed their results using x-ray scattering data from  Rosalind Franklin.  Incidentally, the story behind Franklin's contributions to the discovery of DNA and not being recognized  at the time makes for interesting reading.  I also recommend readers to check out Schrodinger's guess as the structure of DNA using simple physics principles.  The title of his very thin but fascinating book is

"What Is Life?: with 'Mind and Matter' and 'Autobiographical Sketches'"

I can imagine the thrill of discovery experienced by Crick, Wason, Wilson, and Farklin.  From little cardboard pieces and an "X" on a piece of film from an x-ray scattering experiment (shown above), they revolutionized our understanding of the workings of DNA.  Ironically, the forces that hold together the double helix reside in the hydrogen bond, the very forces that seem to be at work in our molecule/polymer system.

I am preparing my talks this morning, and plan to go on a limb proposing stating that the interaction between a DO11 molecule and a polymer chain  through hydrogen bonding underpins the phenomena of self healing.  I am not a chemist and have a naive view of the intricacies of how molecules interact.  But, I hope that my bold proposal will result in good feedback form my audience that will help us fine tune our models of the mechanisms of self healing.

I have been very excited in recent months by all of the discoveries that we are making.   Even if they end up being wrong, the process of the search for the truth is exhilarating.  Gotta run.  Too much to do.  And again, sorry for the typos!

Even teeny weeny discoveries are great fun

This morning, in the process of editing a paper, I made a small discovery. We have developed a new mathematical framework for determining the properties of quantum graphs in terms of the properties of the pieces. This work provides simple identities on the pieces that will allow us to determine general principles form the ground up rather than having to calculate the properties of the full graph.

I have to run because my wife is calling me to lunch.

Here is my email to my collaborators.

Your introduction to edge states was perfect. I liked the physical approach that leads to the formalism. In fact, its clarity was instrumental in allowing me to make a minor discovery (see below).

With regards to the paper, EDGE STATES ARE WONDERFUL! I am taking a break for lunch now, but FYI, I have been working soley on the appendix because I have done what I think is a really neat calculation which uses the power of the edge state. If you recall, in the past, we used the fact that the sum over all of the edges yields the full sum rule. However, it turns out that there are sum rules on each edge! The edge state formalism has allowed me to do this very easily. The result is given by Equation A20 of the geometry. I have pretty much dropped everything to work on this, but I will need to get back to preparing my talks for SPIE since I still have lots to do.

I suggest the following. I still need to reread the appendix because I was making changes while calculating -- never a good thing in terms of introducing typos. I will work on this after lunch. In the meantime, please check the appendix and let me know if I made any errors. The result is so logical that it seams right. I will then alternate between working on my talks and working on the paper.

Most likely, I will not go in to work today so that I can finish the paper in time to be posted on the archives tonight. Even these small discoveries are great fun.

A correspondence with a colleague on our nonlinear topology/geometry project

We are not resting on our laurels, but rather are continuing to work on new ideas and extending our work to more general cases.  In the process, we call get confused about the physics of the system, and try to find ways to picture what is going on that gives insights into how a system may behave under various conditions.

In working on this project, I realize that I spend lots of time writing emails to colleagues and students about various aspects of the work.  An email may come in at 10:30 at night because my collaborator is hot to figure something out and wants insights.  I too write emails at weird hours asking students for more details on something they observed in an effort to test my newest crazy ideas.  This is not a line of work for those who want to sleep soundly.  Being excited about physics is the best stimulant.

Anyway, today I took a two-hour break to play floor hockey and found a series of emails from a collaborator who is very excited (as am I) on the new direction of our work and the new physics that is implied.   Since we are moving into new territories, many things are not well mapped out, so we have to navigate partially by instinct.  I occasionally like to post excerpts from emails to give readers a sense of the kinds of exchanges that we have in the process of working on a project.  Here is my response to an exchange about a new quantum graph that we are working on, as shown above.

An excerpt from my email response (edited for typos and names removed):

First, I think there is a big difference in taking the limit as the prong length goes to zero and actually having a zero prong length.  In the former, I think that you get a broken vertex, because as the wire gets shorter, the end of the wire is getting closer to the vertex, so indeed the wavefunction on the prong must get vanishing small as the prong length goes to zero.  Then, this will look identically to a break in the loop.  Of course, when there is no prong, you get the traveling wave solutions.  So it is interesting that the limiting case yields a very different result than the case without the prong - again a statement about topology.  As long as the prong is there, no matter how short, the topology is of a loop-star.  Get rid of a prong, and then it is just a loop!


I am a little confused about what you are saying with regards to the ground state energy.  So, let me just make a point that may not be related at all to what you are saying.  If I have a prong with zero wavefunction, the loop cannot have a constant wavefunction, i.e. the wavefunction with n=0.  By continuity, the amplitude of the constant wavefunction will be zero, so there is no particle in the system.  So, I believe that in a loop, you have the zero-energy wavefunction but as long as a prong is there, you will not have a zero-energy wave function.


However, you bring up an interesting possibility of a wavefunction in which there is zero wavefunction in the prong and a standing wave in the loop in which there is a node at the prong.  This seems to be alright in terms of the continuity of the wavefunctions, and probability current is certainly conserved.  So unless there is another constraint that I am not seeing, this looks reasonable.


The problem is that this may be a legitimate wavefunction, but not an energy eigenfunction and is therefore not a stationary state.  Forcing this kind of state would be like having a particle in a box wave function, let's say with multiple nodes.  Having the wavefunction oscillate to he left of the node, then zeroing it to the right of the node obeys all the boundary conditions, but, this is not an energy eigenstate.


So, I think I will stick with my original assessment, though now I have thought about this for an additional 10 minutes for a total of 15.  I am not confident in my view because I have not done anything with paper and pencil, just picturing things in my head - a very dangerous activity!


These kinds of discussions do not come out of the graduate students until the point that they are getting ready to graduate.  Only then does a light bulb turn on, which makes them get it.  Once they become very useful and great fun in terms of being intellectually stimulating, they leave for greener pastures.  Then they can start to challenge and stimulate their new advisers.

Not all triangles are the same

Just the other day I wrote about a revelation I had about the self healing process, a hot topic in our lab these days. As often happens, the first impression is simplistic and not quite right, but eventually, we hopefully converge on the truth. However, my fallacy of yesterday gave me insights today, which I continue to pursue.

On another front, we have completed a new paper that we are submitting to Physical Review Letters, the highest impact physics journal. I always have reservations about sending a manuscript to a journal just because it is prestigious. What counts is the quality of the paper. On the other hand, if our work is as significant as we believe, then appearing in a top journal will give it more visibility.

I am excited by the science, and the possibility that we may have started a new branch of study. At the heart of our work are calculations of the optical nonlinearity of quantum wires. This in itself is totally new (to the best of our knowledge), but we are taking the elevator down a level to the realm of fundamental science. For those more practically minded, our work may also have some useful applications.

Science is often focused on a particular thing. While a researcher may be interested in solving the problem of global warming -- a grand problem, the actual work may involve studying the behavior of a particular kind of electrode dipped in a specific chemical. In fact, many groups around the world may be studying exactly the same thing, trying to work out a detail that could make a battery store 5% more energy. Such a leap would indeed be important.

Rather than focusing on details, our work is painting a big picture. I like ideas that have broad influence; views of the world from unique perspectives; and unexpected results on topics that have not crossed anyone's mind, but that resonate with all scientists as being really neat.

Our new work falls beyond the typical boundaries of what others are doing. We are interested in the abstract concept of how the shape (geometry) and topology of an object determine its optical properties. These ideas go beyond specific molecules or materials. To allow us to focus on the basics, we need to remove other complications. To that end, we study what is often called a toy model -- one that brings out the qualities of interest and suppresses the rest. In our case, we are considering structures made of connected wire segments that carry a sole electron.

Consider a continuous loop of wire in the shape of a triangle. If we deform this triangle into other triangles with differing edge lengths and angles, we find that the nonlinearity changes smoothly and not a hell of a lot. In fact, deform the triangle into a quadrangle and then into a quintangle, and nothing much new happens. Any closed loop, independent of the shape, is of the same topology. Thus, we might conclude that the geometry has little effect on the nonlinear response.

A bent wire that does not form a loop is of a different topology. So, consider the simple experiment of a triangle whose nonlinear-optical response is being measured. Now cut a vertex of the triangle so that two of the edges no longer touch. This is still a triangle but its topology has changed. Interestingly, the nonlinear response is found to be profoundly different with the snip of the wire cutters. Thus a change in topology for fixed geometry leads to a dramatic change of the nonlinear-optical response.

This work has applications in the design of better materials because it suggests that taking a molecule (modeled as a wire) and lopping off just a single bond could yield a dramatic improvement. Or, our work could inform nano-technologists on how to make better quantum wires.

We have only evaluated a small number of shapes, including loops made into triangles, quadrangles, quintangles, bent wires, split triangles, and star graphs. Star graphs, which are lines radiating from a central point, represent a topology that yields the larges hyperpolarizability.

To sample the space of all possible shapes, we let the computer randomly pick triangles, quadrangles, quitangles, star graphs, and whatever other shape we can squeeze in. Then we can see what is possible. With enough random tries -- we usually run our simulations over tens of thousands of configurations -- we can test the influence of any parameter, such as topology.

Below is a plot of the first (left) and second (right) hyperpolarizability, which tells us how strongly two and three photons interact with a molecule. Included are triangles (red), simple quadrilaterals (with no crossing edges - green), and all quadrilaterals (blue). Each point (and there are 10,000 here of each color), represents one configuration. A casual glance at the pattern reveals that geometrical effects do not make a big difference. To see the effects of topology, you'll have to read our paper on The Physics Archives.



I find this work really neat (and I hope the reviewers will agree) because we are sampling a very fundamental property of a molecule in terms of some very simple mathematical concepts that go back hundreds to thousands of years. The ancient Greeks heard the music of the spheres in planetary motion using a the metaphorical geometric ear. In our work, we can literally see the effects with light on our eyes when the system's structure changes so ever subtly. And, we get to enjoy a vision of the underlying process with the minds eye as portrayed in very pretty and colorful plots.

A pendent necklace and a new insight about self-healing molecules

In a recent post on our research on self healing, I discussed our new theory, which is posted in the Physics Archives (see it here). The paper has been accepted for publication in the Journal of Chemical Physics and will appear soon.

We used lots of data as input to construct the model, which took years to complete. Data that seemed to support one model initially would later be contradicted by additional data. Over time, the model evolved into a coherent picture as more hypotheses were eliminated by experiments. Finally, we had a model that fit the data AND had as its cornerstone the formation of domains of molecules that together, would help a damaged molecule heal.

There is no direct evidence for domain formation, though the behavior of all the experiments to date are consistent with this model, and only this model. Remove the domains and the predicitve power of the theory is lost. The burning question pertains to the nature of the domains. What are they? Are they clumps of molecules or molecules that are somehow stuck to the same polymer chain? What is the nature of the force that keeps the domains together, and how is it that a domain of healthy molecules acts to promote healing in a damaged one?

We may be closer to an answer.

The lab is in a wonderful buzz of activity with lots of new measurements -- always an exciting time. There are bold new hypotheses based on initial data that generalize our model, followed by letdowns after new data or a more detailed calculation proves us wrong. The process is highly stimulating. I can just smell it; something new and wonderful is brewing.

In the midst of all this activity, I found myself sitting at my computer writing my conference paper for SPIE, where I will give a couple of papers in August. I completed writing the introduction and then explained our new model. What next? I needed something new that did not detract from the presentations of my students. So, I drew the molecular structures of the polymer and the molecules, and started to play with them, rotating this one this way and that one here, etc.

In less than a few minutes, I realized again that a molecule could stick to a polymer through what is called hydrogen bond -- an attractive force between a hydrogen molecule and in this case, an oxygen, very much like the forces found between water molecules. This thought had crossed my mind in the past, and is indeed a motivation for a subset of projects. However, having all this jumbled data running around my head made me realize that Shiva, my coauthor on the theory paper, had already determined the three parameters of our model, one of which is the force that binds the molecule to a domain. If the molecules are sticking to the polymer chain through a hydrogen bond, the hydrogen bond energy should have the same value as the corresponding parameter in the model.

This is an excellent example of a model that we built to explain the data is now guiding us in figuring out what is going on.

I got on the internet and searched for hydrogen boding and found a table of numbers. The energy between a hydrogen and oxygen was one of the first values listed, at 0.3 eV. Then I nervously clicked through the directory tree on my computer to find its measured value. As I scrolled to the table with the results, my eyes focused on the value of the lambda parameter -- 0.29 eV with an uncertainty of 0.01. The two matched!

It is not often that things work out this easily, so I considered the next question, and that was how self-healing is mediated by molecules attached to a chain. A polymer with molecules connected by hydrogen bonding looks a lot like a necklace (polymer) with pendents (molecules) thrown on the night dresser as shown in the figure below.The hypothesis that I proposed is as follows. (a) When a molecule absorbs a photon, (b) it breaks into two fragments that are charged. There is evidence from earlier work that charged species are involved. One of the fragments is fixed in place by the polymer and (c-e) the other hops from molecule to molecule along the chain (f) until it finds its mate and recombines.

An alternative explanation is that the attached fragment attracts a small fragment from a neighboring molecule. The neighboring molecule then attracts a fragment from its neighbor, and so on, which propogates down chain like a wave of fans at a stadium until the original damaged piece combines with an adjacent fragment. The more molecules in the domain (i.e number of molecules attached to a polymer chain), the bigger the chance that there is a contiguous path for the fragment to find a mate.

This is indeed an exciting time. In addition to this work, there are other very exciting developments that I will post in the near future. Breakthroughs can be addictive. I can't wait for the next one!