Fractals

CF10:  Draw a polymer at several scales, showing fractal behavior.
CF20:  We wrote Mass ~ SizeDf where Df is the fractal dimension.
 What are some practical uses of this equation in polymer science? Write a relationship for density of a fractal object: Density ~ Size???
CF30: Suppose you measure mass and size (from light scattering) for five different polystyrene samples in the same solvent and at the same temperature. All are atactic and all monodisperse. How do you get from this information to the fractal dimension of polystyrene?
CF40: Which has a higher fractal dimension, a polymer in a good solvent or a polymer in an ideal solvent?
CF50: You are reviewing a GPC/MALS paper for a journal. GPC/MALS can report mass vs. radius as the sample elutes, essentially monodisperse, from the column. The author claims a fractal dimension of 8 (wow). Is it possible?

SAXS Intro

CS10: How would you describe how a synchrotron produces light to first-grade children?
CS20: Describe how a wavelength shifter (or one-pole wiggler) enhances the output of a synchrotron.
CS30: Xrays are heavily absorbed, which reduces the scattered intensity (the light on the way to the center of the container is reduced, and so the scattered light on the way out of the container). You would like to keep the sample thin to prevent signal loss (it is important to keep the container walls thin too). But if you make the sample too thin, there won't be any scattering. It's a conundrum--you must determine the optimum sample thickness. The intensity expected for a given thickness (t) can be written as I(t) = I(0)·t·exp(-mt/r) where I(0) is the incident intensity and the quotient m/r is called the mass absorption coefficient.
 What is the optimum thickness, topt, for an aqueous sample if we choose X-rays with wavelength 1.54 Angstroms? What is the optimum thickness, topt, for polyethylene if we choose X-rays with wavelength 1.54 Angstroms? What is the optimum thickness, topt, for polystyrene if we choose X-rays with wavelength 1.54 Angstroms? What is the optimum thickness, topt, for an aqueous sample if we choose X-rays with wavelength 1.0 Angstroms?

You can consult the mass absorption tables here: http://physics.nist.gov/PhysRefData/XrayMassCoef/tab4.html

CS40: The figure below was generated at CAMD by Dr. Challa Kumar and coworkers. A decaying signal eventually turns around and gives a "bump". What distance do we associate with that bump? (Hint: the work is published).

 CS50: It's rather amazing, but SAXS can measure pretty large objects--e.g., colloidal particles. One problem, though: the particles may sink and fall out of the beam! Sketch a design of a SAXS sample cell that maintains the particles in suspension. Keep in mind that the cell walls must be very thin in order to avoid absorption. CS60: Describe how a CCD detector works.

Waves

 CW10: The index of refraction of x-rays in some materials can be (slightly) less than one, meaning that light can move faster in the solid than it does in vacuum. Explain why this is not a failure of the law which says nothing moves faster than the speed of light (if indeed that is what the law says). CW20: How does the speed of sound in a gas depend on density? CW30: What sound frequency corresponds to a wavelength that matches the mean free path between gas atoms in air? CW40: Write the wave equation, Schrodinger equation for time-dependent systems, and Fick's second law. Compare and contrast. CW50: A typical graduate student can hear frequencies as high as 18kHz (you did better when you were younger). It is no problem for good headphones and good tweeters to reach this frequency and even higher. Suppose you listen to a test tone at 25,000 Hz mixed with another test tone of equal amplitude at 25,002 Hz. What will the pressure wavefront look like, and what will you hear? CW60: In an orchestra, the first chair violinist is responsible for tuning. Using a tuning fork or electronic resonator, she has used the beat frequency method to tune her instrument to 440 Hz on the A string and 660 Hz on the E. After intermission, she bows both A and E simultaneously and, to her surprise, hears a beat note at 2 Hz. She turns the tensioning knob on the violin and notes the beat frequency rises. Explain. CW70: If you listen to a beat note derived from two equally intense sources at 440 Hz and 442 Hz, your ear still says 440Hz (actually, 441 Hz) but you can definitely tell the modulation frequency (2 Hz). Some people say the direct sine term at 441 Hz is modulated by the cosine envelope at 2 Hz. That makes sense when you look at the equation and figure below from Wikipedia (http://en.wikipedia.org/wiki/Beat_(acoustics) ).

${ \sin(2\pi f_1t)+\sin(2\pi f_2t) } = { 2\cos\left(2\pi\frac{f_1-f_2}{2}t\right)\sin\left(2\pi\frac{f_1+f_2}{2}t\right) }$

You could not hear a 2 Hz direct sine wave at all (maybe pigeons or elephants or whales could) so it is fortuitous these beat patterns extend our detection limits, even if it is only the cosine envelope we hear. Now let's shift to detection of photocurrent off a photomultiplier tube. Imagine we have two slightly different colors of light due to Doppler broadening in DLS (only two frequencies does not normally happen; we might try to arrange for just two by placing very slowly diffusing particles in uniform electrophoretic motion and mixing the scattered light with a smidgen of the direct light from the laser; or you could do something like laser velocimetry). Let our two light frequencies be precisely 1014 Hz and 1014 Hz + 103 Hz. So a beat frequency of 103 Hz is expected. The PMT cannot respond AT ALL to anything like 1014 Hz. Unlike sound at 441 Hz, the "sine term" is undetectable. How then can you detect the cosine envelope? Won't the exceedingly rapid oscillations of the sine term just go flying by the detector, leaving it effectively zeroed out? Even if little wiggles occur, why don't they average to zero within the envelope provided by the cosine term?

SAXS Form Factor

 CSF10:  The electric field due to scattering N discrete elements in a polymer chain is given by by a sum of plane waves, one for each element:

Es =        Eq. A

In this expression, Eo is some essentially meaningless amplitude, q is the scattering vector magnitude, and Rj is the position of element j with respect to some arbitrary origin. Real-world detectors do not track the e(iwt) term. The scattering intensity Is is given by the square (complex conjugate) of Eq.A:

Eq. B

As we know from our development of the RMS end-to-end length of polymer chains, there are N2 terms in a double sum like this. For the N terms where i = j, what is the value and physical significance?

 CSF20: We sloppily slogged through a development of Eq. B above to yield, for systems that are able to assume any orientation they wish with respect to the scattering vector, this equation for the particle form factor (everything:

Eq. C

Go back and recopy your notes to develop this equation carefully.

 CSF30: This challenge will look at the two previous problems from the standpoint of solid particles, not constructed from discrete bits. The general form of the electric field will be to sum up the contributions of each subvolume of the particle. The picture shows one such subvolume located at x, y, z with volume dx·dy·dz.

The symbol  r(x,y,z) represents the density of electrons, which do the scattering if we are talking about X-rays. In this case, the electric field will be a triple (volume) integral representing the scattering out of each subvolume, weighted for scattering power by r(x,y,z), and adjusted for phase by its position relative to the origin.

It is understood that vectors q and r can be broken into x,y and z components, although in practice we would welcome a little symmetry to bail us out of that mess. Clearly, the particle is not infinitely big, so we expect  r(x,y,z) = 0 everywhere except inside of the particle. Finally, the question! Take this equation, assume a spherically symmetric dependence [i.e., r(x,y,z) = r(r)] and re-express the integral in terms of sines to Drrive at:

 CSF40: We already have assumed spherical symmetry. That means our particle cannot look like a grapefruit that swallowed a celery stick. It can look like a grapefruit that swallowed an orange, though...these are both round. Let's just assume it's a grapefruit, period. Not only spherically symmetric, but also uniformly dense throughout: r(r) = r = constant. That lets us factor out the density. Solve the integral using integration by parts. The result is real, so you can just square it up to get I(q) for a sphere of uniform density. Write the expression for I(q) an plot it from qR  » 0 to qR = 15.

Instead of solving the integral analytically, use Mathematica or Wolfram Alpha to do it.

Make a program in LabView to plot I(q) vs q from ~0 to qR = 15.

LS Form Factor/Aggregation

 CLS10: The basic ideas of SAXS--e.g., keeping track of phase shifts as rays of electromagnetic radiation find their way through a particle--work OK for visible light scattering, but we have to take into consideration the refractive index, n. Why is n » 1 for X-rays but not visible light? Hint: put a magnetic stirbar into a beaker with some viscous liquid and slowly increase the rate of rotation. What happens? CLS20: What is the RGD limit? CLS30: Imagine latex particles that have been functionalized in specific locations to make "bonds". Suppose these particles are 25 nm in radius (about lo/2, where lo = 488 nm is the wavelength in vacuo of blue-green light). Use whatever software you wish to generate form factor curves in the range 5 degrees of angle to 175 degrees of angle for single spheres, dimers, trimers in a triangle, trimers in a straight line, tetramers in a planar array, and tetramers as tetrahedra. Assume n=1.33 for water. CLS40: So, I am giving a talk about air-containing bubbles and a questioner from MIT suggests perhaps it isn't air in the bubble, but some other liquid. Are there any liquids whose refractive index is lower than that of water? (nH2O=1.33) CLS50: Polyethylene has a refractive index of about 1.5 (depends on wavelength). PTFE has a much lower refractive index of about 1.35. Why is the refractive index so low in PTFE (almost as low as water!). What property of liquids seems to confer very low refractive index?

DLS Introduction

 CDLS10: In a typical lens-aperture-pinhole DLS setup, what will reducing the size of the aperture do to the coherence area at the detector? What does it do to the intensity? CDLS20: A student new to DLS is measuring a very large latex sample. The correlation function immediately suggests the presence of two components, one of which seems very slow (large). Filtering the sample to remove dust makes the problem worse. What's wrong, and what can we do about it? CDLS30: A DLS user measures a filtered 1% solution of some polymer thought to have M=1,000,000 but finds it scatters only weakly compared to the solvent--i.e., only about 10% more than solvent. What are some possible explanations and solutions to the problem? CDLS40: Detectors used in photon correlation have limited count rate abilities. For example, if the light striking the detector doubles in intensity--an increase of 100%--the detector may record only a 90% increase. If the light triples, the detector may show an increase of only 2.5X. What effect doest his have on the "coherence parameter", f, in the equation g(2)=B(1+f x [g(1)]2)? CDLS50: A DLS practitioner makes a plot of decay rate, Gamma, vs. squared scattering vector magnitude, q2. The plot begins linearly, but trends upward as q2 rises. Give some possible reasons for this behavior. CDLS60: A DLS practitioner makes a plot of decay rate, Gamma, vs. squared scattering vector magnitude, q2. The plot begins linearly, but levels off as q2 rises. Give some possible reasons for this behavior. CDLS60: A mixture of latex spheres is studied by DLS. Luckily, the distribution is bimodal--characterized by two discrete sizes. Also, the sizes are distinctly different. Yay! The measured heterodyne (electric field) correlation function g(1) fits to a sum of two exponentials--35% signal has decay rate 425 Hz and 65% of the signal has decay rate 1520 Hz. The two latex particles are made from the same monomer (styrene). If the measurements are performed in water at 25oC (n=1.33, h=0.8904 cP) at a scattering angle of q=45.0o and laser wavelength of 532 nm, what are the two sizes? What are the two molecular weights? Estimate the number of small particles to large. For example, in some hypothetical sample containing 100,000 particles, how many would be small and how many large? CDLS70: Would GPC/MALS be able to solve the above problem? How about AF4/MALS? If so, what are the relative advantages compared to DLS? Disadvantages compared to DLS?

Harmonic Oscillators

 HARM10: In lecture, we showed a video (still available in L20 Notes) where a mom is pushing her little girl on a swing. Her nudges restore the energy lost to friction, and they come at the natural frequency of the swing. A few questions about this:  When the family returns to the park again, the child is older and heavier. Does the frequency change? Up or down? Later the child will learn to propel itself on the swing using a kick-out/lean-back technique. Will the frequency of the kick-out/lean-back motion be higher or lower than mom's pushes? Imagine a row of swings large enough to accommodate 7 children of different sizes and weights. We get 7 burly offensive linemen off the LSU football team and tell each one to draw back one child to approximately the same height. On command, all 7 children are released and they begin swinging. Compare this situation to modern FT NMR. HARM20: Write some computer code to simulate the motion of a simple harmonic oscillator, following the method given by Feynman (Chapters 9 and 21). HARM30: How can I resist a Challenge question about the Challenger disaster? After the Challenger space shuttle blew up, a panel of experts was called in to study the failure. From the very beginning, engineers "knew" (to the degree anything is known) the culprit was a little o-ring that failed to seat properly due to the cold weather. I don't think Feynman, father of quantum electrodynamics and Feynman integrals, was a rubber expert, but there he was sitting on the panel. Based on what you know about him from TFLP, was Feynman a good choice for the panel?