 
Fractals
 CF10: Draw a polymer at several scales, showing fractal behavior. 
 CF20: We wrote Mass ~ Size^{Df} 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
firstgrade children? 
 CS20: Describe how a wavelength shifter (or onepole 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 conundrumyou 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, t_{opt},
for an aqueous sample if we choose Xrays with wavelength 1.54
Angstroms? 
 What is the optimum thickness, t_{opt},
for polyethylene if we choose Xrays with wavelength 1.54 Angstroms? 
 What is the optimum thickness, t_{opt},
for polystyrene if we choose Xrays with wavelength 1.54 Angstroms?

 What is the optimum thickness, t_{opt},
for an aqueous sample if we choose Xrays 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 objectse.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 xrays 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 timedependent
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) ). 
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 10^{14} Hz
and 10^{14} Hz + 10^{3} Hz. So a beat frequency of 10^{3
}Hz is expected. The PMT cannot respond AT ALL to anything like 10^{14}
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:

E_{s} =
Eq. A
In this expression, E_{o} is some essentially meaningless
amplitude, q is the scattering vector magnitude, and R_{j}
is the position of element j with respect to some arbitrary origin.
Realworld detectors do not track the e(iwt)
term. The scattering intensity I_{s} is given by the square (complex
conjugate) of Eq.A:
Eq. B
As we know from our development of the RMS endtoend length of polymer
chains, there are N^{2} 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
Xrays. 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
reexpress 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 SAXSe.g., keeping track of phase shifts as
rays of electromagnetic radiation find their way through a particlework OK
for visible light scattering, but we have to take into consideration the
refractive index, n. Why is n » 1 for Xrays 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 l_{o}/2, where
l_{o} = 488 nm is the wavelength in vacuo of bluegreen
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 aircontaining 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? (n_{H2O}=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 lensaperturepinhole 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 solventi.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
intensityan 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, q^{2}. The plot begins
linearly, but trends upward as q^{2} rises. Give some possible
reasons for this behavior. 
 CDLS60: A DLS practitioner makes a plot of decay rate, Gamma, vs.
squared scattering vector magnitude, q^{2}. The plot begins
linearly, but levels off as q^{2} rises. Give some possible reasons
for this behavior. 
 CDLS60: A mixture of latex spheres is studied by DLS. Luckily, the
distribution is bimodalcharacterized 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 exponentials35%
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 25^{o}C (n=1.33,
h=0.8904 cP) at a scattering angle of
q=45.0^{o} 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
kickout/leanback technique. Will the frequency of the
kickout/leanback 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 oring 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? 
