Course #4, 3:30-5:00pm
Introduction to Scattering-Based Polymer Characterization Methods
Paul S. Russo, Department of Chemistry and Macromolecular Studies Group
Louisiana State University Baton Rouge, LA
Determination of macromolecular and supramolecular structure in solutions, gels and solids can be challenging because the materials are often too soft for microscopic visualization at high resolution. Even when microscopy yields images without artifact, it can be tedious to examine a sufficient number of specimens, especially in polydisperse systems. In these cases, scattering of visible or X-ray radiation, or of neutrons, can be used to obtain quantitative information without bias and usually without artifact. Absolute molecular weights, particle sizes, extent of aggregation, stability, and phase behavior are just some of the problems that can be studied. Often scattering methods are coupled to on-line chromatographic separations, so that effectively monodisperse samples can be examined. In this short course, the essentials will be presented. Subjects will include scattering geometries, the scattering vector, angular dependence, particle sizes, correlation length and intensity fluctuations in time. Applications from the literature will illustrate these principles. A brief discussion of available instrumentation will round out the discussion.

From a Graduate Student
1. How to differentiate elastic/inelastic scatterings, as well as coherent/incoherent scatterings, and the relationship between these two groups.
2. the way links crystal structure to wave length is the Bragg's Law (2dsin(theta) = n lambda), where the interpanar d is related to miller indices and lattice parameter.
3. The function difference between pinhole and aperture in DLS or other LS.
From an alumnus who now who HIRES graduate students
1. Applications.
2. Applications.
3. Applications.
3

Morphology of polymer melts, gels, solutions, liquid crystals, foams.
Measure molecular weight, size, interaction, aggregation, polydispersity, stability, degradation.
Works when imaging doesn’t, with fewer anomalies and less user prejudice.
Conversation with an undergrad:
What’s my motivation? 
4

The nature of inverse space
Sounds crazy—why would we do that?
OK…so how would we do that?
When is there not scattering?
Sprinkled throughout:
Applications to bulk polymers: films, fibers, etc.
Applications to molecules and particles.
Applications to polymer dynamics.
Henri Benoit
5

This shows how you see things in a microscope, minus the illumination system. Pretty complex, eh?
Light
Source
Thanks, Wikipedia.
6

To “see” in inverse space, construct a scattering instrument:
1. Eliminate almost everything
2. Make illumination go in one direction.
3. Measure scattering intensity at one or more points.
4. Plot intensity as a function of position, or follow intensity dynamics.
Light
Source
Detector 1
Detector 2
Detector 4
Intensity
7
Angle

So that’s the “straight-line” or “small-angle” kind of scattering instrument. Here is the SAXS at LSU’s synchrotron.
2D
Multipixel
Detector
To X-ray
Source
Sample chamber
SAXS = Small-angle X-ray Scattering
For a lab-based SAXS, see this compact example: http://www.youtube.com/watch?v=5EmK-uW4Ajo
8

http://dx.doi.org/10.1016/j.colsurfa.2010.01.018
, http://www.princeton.edu/cbe/people/faculty/register/group/facilities/sals/
9

What big machines we build to look at tiny things!
Here, a SANS instrument at Oak Ridge National Lab.
http://www.csmb.ornl.gov/biosans/
10

http://www.cell.com/biophysj/retrieve/pii/S000634950076317X
11

Most straight-line scattering instruments are used at low scattering angles.
They handle a very wide variety of samples and produce a wide variety of scattering patterns.
Patterns can be either structured or unstructured, and somehow represent what’s going on with the sample. They can evolve in time, or just fluctuate in time.
12

http://www.cs.unm.edu/~brayer/vision/fourier.html
The website referred to above contains a nice discussion on 2D FTs of various objects.
What’s this one?
View next slide to see!
13

Well, Bob can! (Prof. Robert Pecora of Stanford)
14

The FT can be understood from simple functional forms. FT is the
“recipe” for a pattern or signal. Here, the FT identifies the signal as containing two waves at different frequencies.
SumOfSines.vi by Paul Russo
FT figures out the recipe is 2 sine waves added together.
This is the signal.
15

The SMALL wiggles of the signal correspond to the HIGH-FREQUENCY component in the FT.
The LONG undulations correspond to the LOW-
FREQUENCY component in the FT.
Low Frequency = Large Features
High Frequency = Small Features
16

I ( q )
I ( q,t ).
17

Here’s a real-world example from the Small-Angle Light
Scattering of a Liquid Crystal. Regular lines lead to regular scattering spots. laser LC screen
Quiz: If the banding pattern on top were to anneal into a finer structure, how would the scattering pattern respond?
18

Inverse Space
Direct Space
Classic stuff 
19

Almost 20 years later, better TEM images and
SAXS with rheology.
Ask yourself: can we do
TEM with rheology?
20

S
E
H
It’s as though the electric field and magnetic field leapfrog over each other through space. l
S
The Poynting vector points where the light goes.
21

l
1 k
1

2
 l
1 k
2

2
 l
2
*Each wiggle = 2
 of phase l
2
22

2
1
l
2
l
1 l
1 k
1

2
 l
1
Red: 100000 radians/cm k
2

2
 l
2
Blue: 130000 radians/cm l
2
23

High-resolution light and low-resolution light.
High-resolution ruler and low-resolution ruler.
http://opinion-forum.com/index/2010/07/the-ruler-and-why-it-was/ 24

In scattering, the scale is set by the difference between incident and scattered light vectors. incident k
 s

2
 l s
 n s
 k i

2
 l i
 n i
 q

4
 l i d

2
 q n
 sin(

2
)

(
 n s

 n i
) scattered
The waves for k i and k s really represent light wavelength, and l i
≈ l s
The waves for q are longer, and represent a Fourier component the scattering device will sense.*
We may variously write the scattering vector q with an arrow over the top or a line underneath
*Yeah, I know…this is confusing. Just wait.
25

 
Induced dipole

E
Polarizability Electric field http://www.wyatt.com/theory/rayleighscattering/mass.html
The question is most easily imagined for scattering from a particle, and the web link above leads to a lovely figure provided by Wyatt. As a wave travels (at the speed of light!) past the particle, the electric field oscillates at about 10 14 Hz.
Only the electrons in the particle can respond to this and they do so by flitting from top to bottom of the particle. This creates a transient, rapidly oscillating dipole. We know a lot about those—that’s a radio antenna!
26

Because the electrons move up and down in response to the incident light, the molecule acts like a transmitter: accelerated electrons emit light. http://en.wikipedia.org/wiki/File:DipoleRadiation.gif
27

A vertical antenna or single vertical scattering particle radiates like a closed torus
There is no radiation above the scattering particle. z 
1
Radiation is maximal in the xy plane…and independent of direction (for small particles) x y vertically polarized incident light
Blue light scatters better than red. Purple even better!
For an explanation of why the donut pattern happens: http://macro.lsu.edu/corecourses/msweb4/VirtualBook/Ch14_Scattering/Ch%2014%20RussoLSNotes_UpdatedWithWordDocStuff.pdf
http://www.antenna-theory.com/antennas/norm3D1lam.jpg
http://en.wikipedia.org/wiki/Poynting%27s_vector
The little icon on your cell phone is a 2D approximation to this closed torus pattern.
28

Single particle
Many particles in a dilute gas
Perfect fluid
Actual fluid
Dilute solution of small particles
Dilute solution of large particles
29

30

one particle N -particle gas Pure fluid

N 0
At modest densities, you just multiply the single-particle result by N particles.
I s

N
 2 
N ( Volume )
2 
N ( R
3
)
2 
NR
6
A single object of radius 10 R scatters the same amount as 1,000,000 objects of radius R .
Dust scatters like crazy!
31

Let’s extend that last equation to its more commonly used form.
I s

N
 2 
N ( Volume )
2 
NR
6 
NM
2  cM where c is the concentration as grams per unit volume—e.g., g/mL
Two important consequences:
1. In a polydisperse suspension of particles or polymers, the big ones scatter much more. You actually measure the weight average molecular weight, M w
.
2. If a solution starts to aggregate, c stays the same but M rises because M is the molecular weight of the whole scattering entity in our equation. I s aggregation.
goes up. Scattering is an excellent way to study
32

The linear increase regime does not last forever:
Infinity particles = perfect fluid  no scattering!
That’s right….the scattering from a perfectly uniform fluid is zero.* ,&
But…there are no perfectly uniform fluids; at some wavelength, they all exhibit inhomogeneous behavior—e.g., density fluctuations.
*The scattering from a perfect solid is also zero. There are no perfect solids; by the time you get down to X-ray wavelengths, one angstrom or so, they all exhibit some inhomogeneities.
& Scattering does occur at the edges of a small volume.
33

Before we attempt to understand the absence of scattering in perfect fluids, let us remember wave interference.
In Phase
Constructive Interference
Nothing wave
Out of Phase
Destructive Interference
Bigger wave
34

Instead of asking why things scatter, it is instructive to understand why some don’t! So, let’s return to that hypothetical, perfect fluid.
By hypothesis, let us imagine two regions, A and
B, within the perfect fluid that are positioned such that the total path lengths of two rays going to them, then scattered to the detector, differs by one-half wavelength:

(A1 + A2) – (B1 + B2)

= l
/2
Such a choice of regions will result in complete cancellation of the scattering…if the contents of A are identical to the contents of B. That is, when the fluid is perfectly homogeneous, such cancellation is guaranteed.
For any sub-volume we choose, a companion subvolume that will cancel its scattering can be found in a perfect fluid…as long as we do not probe with a wavelength that is too short (comparable to the distance between the particles) and as long as we don’t look too near the edges (in that case, the necessary volume might lie outside the sample). For light of visible wavelengths and normal sizes, these conditions are met.
35

R.S. Stein
A and B identical  perfect fluid  No Scattering
A
B
A
B
Region A is more dense than Region B, so
B is unable to cancel the scattering from
A, even though it is properly positioned to do so  Some Scattering
A
B
Look carefully! Region A has a different organization than region B
 depolarized light scattering http://macro.lsu.edu/corecourses/msweb4/V irtualBook/Ch14_Scattering/Ch%2014%20Ru ssoLSNotes_UpdatedWithWordDocStuff.pdf
Laser
Vertical
Polarizer
LC
Horizontal
Analyzer screen
36

In a real fluid, density fluctuations lead to scattering. density y

X
This is just a picture of the ocean, but pretend the vertical wave height represents density inside a fluid at some particular depth where the laser goes through. The density varies everywhere to give large and small fluctuations, including undulations too small to see (and others too big to see). The surface can be represented by a Fourier synthesis of various waves (three shown). Scattering picks out one Fourier component—essentially reflecting off that one. Here, the scattering geometry (red arrows) has been set to capture the wave fronts parallel to the y axis, propagating along x.
37

C:\PR\Type\Video\WavesFromWesterdam_WP_20130525_231145Z.mp4
38

Reflection from Fourier components is another way to think of scattering! This is called Bragg’s law, and it results from destructive interference except at

. m l 
2 d
 sin(
 m

0,1,2,3...
/ 2 )
2
2

A great annoyance to everyone:
The scattering and diffraction communities define
 differently. Here, we follow the
Scattering community definition:
 is the scattering angle, not the angle with respect to the wavefront.
39

An electric field continually races past the particle.
The molecules’ electrons create an oscillating dipole.
That sets up a donut-like radiation profile.
In a dilute gas, the donut patterns just add together: I s
~ Number of particles
In a pure fluid, I s
= 0.
In a real fluid, the scattering geometry chooses one Fourier component with:
1) direction in-between incident & scattered light rays
2) characteristic distance: 2

/ q .
40

Now we understand particles in dilute and liquid-like limits. This is a long way from polymers in solution. In a polymer solution, we divide the solution into m subvolumes and add their scattering together. We must follow three things as a function of position: polarizability, concentration, and density. m volume elements, each with volume y
=(
D x ) 3
D x
This development follows Tanford, and it appears in LSU Chem4011 notes: http://macro.lsu.edu/corecourses/msweb4/VirtualBook/Ch14_Scattering/Ch%2014%20RussoLSNotes_UpdatedWithWordDocStuff.pdf
41

Using some assumptions, most of which are actually valid,* we can borrow from the gas result. Just subtract the solvent intensity, use the excess intensity, and look at variations d away from the average polarizability,

.
I s
I o
 m
( d r
2 c
)
2
16
 4 l 4 o
Compare to the equation appearing on slide 25, near the donut image: d c is the fluctuation away from average polarizability due to fluctuations in concentration (not pressure), and r is the distance to the detector.
* Among the assumptions: the polymer solution is not about to separate into distinct phases, the polymer is small compared to the wavelength of the light, density fluctuations and concentration fluctuations are not correlated. The development is found on p. 17 of LSU Chem4010 notes: http://macro.lsu.edu/corecourses/msweb4/VirtualBook/Ch14_Scattering/Ch%2014%20RussoLSNotes_
UpdatedWithWordDocStuff.pdf
42

For convenience, we convert from polarizability
 to refractive index n . This goes through something called the Clausius-Mossotti relation, and involves a Taylor’s expansion in concentration for the refractive index too.
( d
)
2  n o
2 
4
 2
2 d n d c
2
( d c )
2 n o
= solvent refractive index
c = concentration of scatterer in g/mL
The parameter
 is the volume of imaginary elements (N of them) in to which we have divided the solution of detected volume V:

= V/N. We need not worry about it because it will not appear in the final result. See LSU
Chem4011 Notes: http://macro.lsu.edu/corecourses/msweb4/VirtualBook/Ch14_Scattering/Ch%2014%20RussoLSNotes_Update dWithWordDocStuff.pdf
or the text by C. Tanford, Sec. 17; in the notes, symbol m is used for N.
43

1. Local polarizability fluctuations cause the scattering.
2. These can be converted to concentration fluctuations using the parameter, d n /d c , which is how refractive index varies with concentration.
Little wonder then that scattering experts obsess over (dn/dc). It even gets a name: (d n /d c ) is called the “specific refractive index increment” but most people just say dee-en-dee-see (4 syllables instead of 11 – good deal!)
3. The task of evaluating ( d
) 2 has been traded for a theory that yields ( d c ) 2
4. This is an appropriate thing to ask from statistical thermodynamics.
5. Scattering will be large if concentration fluctuations are! (Unless d n /d c = 0)
6. SAXS and SANS people speak not of refractive index but of electron density and neutron scattering cross-section, respectively.
44

Unless d n /d c = 0, scattering is large when concentration fluctuations are. Another technique that follows concentration gradients is membrane osmometry. h

V = n R T n = g/M c = g/V

= c R T
1
(
M

A
2 c

...)
Semipermeable membrane: stops polymers, passes solvent.
45

Light scattering is like osmometry without the semipermeable membrane to enforce concentration fluctuations. Instead, LS relies on small, spontaneous fluctuations. Both methods look at concentration gradients…which are chemical potential gradients.
Osmometer Light Scattering
Semipermeable
Membrane position in sample
One of these Fourier components of the concentration profile
“couples” to the LS experiment, when properly oriented along q. position in sample
46

Light scattering operates on thermodynamics; think of it as an osmometer without the membrane.
Concentration seems uniform, but magnify details by 100,000
  c x

“Uniform” polymer solution q

4
π n l o sin(

/ 2 )
2
 q
I s

1 


 
 c


T , p
 cRT
1
(
M

2 A
2 c

...)
Osmotic modulus: how hard is it to effect a concentration increase?
Virial coefficient
Teaching Light Scattering to Exemplify and Reinforce Basic Principles, D. S. Poche', P. S. Russo, B. Fong, E. Temyanko and H. Ricks, J. Chem. Ed., 1999, 76 (November), 1534-1538.
47

Bonus for LS over osmometry: LS can sense size. Until now, we have imagined only small particles. When polymer size is > ~ l
/50 it becomes possible to measure that size. q > 0 out of phase,
I s goes down q = 0 in phase
I s maximum
Too Small
Maybe
Easy
Big & detailed
I s

1 
 
 c
T , p
( 1
 q
2  2
)

1 cRT (
M
)

( 1
 q
2
R g
2
3

...)

2 A
2 c

...
correlation length
Radius of gyration
Let’s call this the Zimm equation.
48

I s

1 
 
 c
T , p
( 1
 q
2  2
)
 cRT
1
(
M
)

( 1
 q
2
R g
2
3

...)

2 A
2 c

...
A bulk soft polymer view
(gels, concentrated solutions)
Low


  
 c



T , p 
High


  
 c 


T , p

A molecular view
(dilute polymer solutions)
A
2
> 0: good solvent polymers repel, “excluded volume limit ”
A
2
= 0: ideal (theta) solvent
A
2
< 0: poor solvent
R g is the mass-weighted, root mean square distance of all elements from the center of mass, often called
R
R
CM or <s 2 > 1/2
.
A bulk polymer view
(melts, glasses, semicrystalline)
I s

1 
( 1
 q
2  2
)
Osmotic modulus doesn’t really apply to these systems.
R g

L / 12
R g

R

3
5
CM s
1 s
13
49

R g
R h
R
R g

R

5
3
R g
2  s
2  i n

1 

1 n

1 m i i


1 m i s i
2

R
R h

R
CM s
13
Other shapes don’t even have an identifiable radius
R , but they still have an R h
! s
1
R g

L / 12
R g
= <s 2 > 1/2
R h means: the size of a sphere that diffuses as fast as whatever particle you have, no matter what its shape.
50

Sample in cylindrical cell.
Laser

Volume

1/sin(

)
Minimum at
  90 o
 my second-longest love-hate relationship was with this fat lady http://138.238.133.24/sofica.html
51

The volume seen by the detector is not interesting, but its angular dependence must be known anyway.
Laser
I s
I s
(

 sin(


90 o
)
)

1 .
00

0 .
02
Performance of a single-angle rotating instrument should be this good, even in aqueous samples, where reflection problems are troublesome.
Laser
With a multi-angle fixed instrument, we have two reasons for intensity variation with angle:
1) The sample really might be large enough that intensity is going down (or up after going down, in the case of very large samples).
2) The detectors might not be “normalized”; that is, given the same light level, they could report different voltages.
Precision Detectors Expert
Still available? http://mmrc.caltech.edu/PD_Expert/Expert.html
52

Measured signal arises from scattering, stray light and dark signal.
Let G = measured signal (stands for galvanometer)
G
1
= I s,1

N
1
G
2
= I s,2

N
2
+ I stray,1
+ I stray,2
+ I dark,2
+ I dark,2
The corrections are of two types: additive and multiplicative.
G
3
= etc.
Where…
• I s,i
= the true radiant power scattered in the direction of the i th detector (what you want to know).
• N i
= the normalizing (fudge) factor you will determine after measuring dark signal of the detector and ascertaining the stray light is absent (or at least very weak compared to the normalizing standard).
• I dark,i
• I stray,i
= the signal produced by the i th detector when no light hits it.
= the stray light rattling around inside the instrument.
Normalizing the detectors requires us to extinguish any stray light (reflections, refractions, scratches, etc.) . Then insert a Rayleigh normalizer that scatters uniformly in all directions (in the horizontal plane).
The Rayleigh normalizer should be much smaller than l
, and yet it should scatter strongly. It should have the same refractive index as the sample, too.
Typical examples: toluene, 30 kDa monodisperse polystyrene, small protein such as lysozyme, microemulsions and inverse microemulsions, etc.
A more detailed description is available: http://macro.lsu.edu/HowTo/LSBASICS.DOC
53

To see the constraints on normalization, consider how we get excess scattering: Solution Scattering – Solvent Scattering
Subtract the solvent signal from the solution signal:
G
1,solution
= I s,1,solution

N
1,solution
+ I solution,stray,1
+ I solution,dark,2
G
1,solvent
= I s,1,solvent

N
1,solvent
+ I solvent,stray,1
+ I solvent,dark,1
Good assumptions * I solvent,dark
= I solution,dark
N
1,solvent
= N
1,solution
= N
1
Not-so sure assumption! I solvent,stray
= I solution,stray
If both assumptions are true, you can use the normalization:
I s,1
= excess scattered intensity of i th detector =
(I s,1,solution
- I s,1,solution
) = (G
1,solution
- G
1,solvent
)/N
1
*It isn’t easy to imagine a case where the solvent and solution would have different dark signals: maybe magnetic samples or containers?
*It isn’t easy to imagine a case where the detector normalization varies between solvent and solution, but mismatched refractive index or a big difference in the thickness of the container walls could do it.
54

Strategies to Make it So! (eliminate stray light)
Use a flow cell: This ensures steady stray light (neglecting pressure pulses, thermal expansion, etc.). A flow cell also simplifies dust removal. Almost surely, the biggest thing to happen in LS for the last 40 years is the development of good flow cells.
When you must make measurements in batch mode (one sample at a time), leave the cell stationary and add/remove solvents without bumping it.
Insert the cell in a “refractive index matching bath” – same refractive index as glass to eliminate one reflection, or same refractive index as sample to reduce misalignment. Typical bath fluids: toluene, decalin, even water can help. The index matching fluid also reduces stray light due to scratches. You should rotate the cell to minimize this effect.
55

The famous Zimm plot is based on the molecular half of the previous equation.
I s

1 
 
 c
T , p
( 1
 q
2  2
)
 cRT
1
(
M
)

( 1
 q
2
R g
2
3

...)

2 A
2 c

...
First, we introduce the Rayleigh factor,
R
.
R
= r
2
I s
You have my permission to slap anyone who calls it
Rayleigh ratio.
VI o
The purpose of the Rayleigh factor is to “take out” the uninteresting V and r 2 dependences. For a dipole radiator, I s goes down with squared distance to the detector, r 2 , so multiply that back in; I s goes up with detected volume, so divide that out.
56

Using our Rayleigh factor and replacing the proportionality with optical parameters, we get Zimm’s famous equation:
Kc
R

1
RT (
M
)

( 1
 q
2
R g
2
3

...)

2 A
2 c

...
R is the gas constant here.
T is the Kelvin temperature.
K is called the optical constant.
M is the molecular weight.
A
R g
2 is the virial coefficient is the radius of gyration n o
= refractive index
N = Avogadro’s number
I am using this handwritten equation to remind you that a more complete, if less aesthetically pleasing, set of notes exists, here: http://macro.lsu.edu/corecourses/msweb4/VirtualBook/Ch14_Scattering/Ch%2014%20RussoLSNotes_UpdatedWithWordDocStuff.pdf
We maintain a whole web article about Zimm plots, associated with our program, GuiDe.
http://macro.lsu.edu/howto/guide/guide.doc
Polystyrene in toluene, an easy case.
The Zimm part of it is actually in the appendix, because strictly following Zimm’s procedure is no longer beneficial or needed. (Zimm did not have computers or computer graphing!) You can download GuiDe, install, and practice
Zimm plots using real data.
Bruno Zimm
57

Zimm plots combine (by a scaled addition) both independent parameters—the angle and the concentration—on the x-axis, reserving the z-axis for Kc/ R (or just c/ R sometimes).
These points represent an extrapolation to

= 0. The slope gives A
2
.
These points represent an extrapolation to c = 0. The slope gives R g
.
The

= 0 and c = 0 intercepts should be identical. Combined with optical constant, they give
M w
.
This measurement is in 97% H
2
SO
4
…so what’s your toughest solvent? Poly(ether-etherketone),PEEK, in 97% H
2
SO
4
. Not so easy.
In both cases, just by looking at the plot, you can see the polymers are quite large and dispersed in good solvents
Each point in the middle is
Kc/
R for a particular concentration at a particular angle.
Sin 2 (

/2) + kc where k is a scaling constant to give concentration the same
“importance” in the plot as the angle term, Sin 2 (

/2) .
Even though no one follows Zimm’s exact procedure anymore, the plot is more than a polymer science icon: it tells a lot about the sample!
58

Zimm plots of the rich, famous and obscure.
 Doty, Bradbury &
Holtzer, JACS, 1956.
Does your Zimm plot look this good?
↑Random web search: what does it tell us?
59

The Guinier plot is a great tool for getting R g
, independent of shape, but it only applies when qR g
< ~1. Guinier plots are greatly underutilized: you don’t need to know the zero-intercept value because the slope alone gives you R g
! ln( I s
)
 ln( I s , q

0
)

R g
2
3
Too small to measure; you require a shorter l
Hmmm….how good is your system’s angular response?
Easy and simple to get size, but not shape.
q 2
There may be no linear
Guinier regime, but the good news is you will be fitting some functional form to the data, but there is much hope for both size and shape.
In-between the blue and the green is a size where you might be able to fit SOME of the data to a linear form…choose your angles wisely.
60

(a)
1 2 3 4 5 q
2
/10
-4
Å
-2
6 7
A porphyrin assembly by SAXS
20 mM H
4
TPPS 2at pH 4.0
Hollingsworth, et al.
Biomacromolecules 2012, 13 (1), 60–72.
8
Rapid growth of liposomes under action of an antimicrobial polypeptide.
Smith-Wright et al. Rev. Sci. Instrum. 1996, 67, 3645.
61

LS yields weird averages in polydisperse samples.
Molecular weight: you get the weight average,
M w
.
M w


 i
 i


1

1
N i
M
N i
M i
2 i

  i

1 w i
M i
Size: you get the square root of the Z-average of the square of R g
.
R
LS


 i


1

N i

1 i
N
M i
M i
2 i
R
2 g, i
2
Don’t even get me started on what happens if you compare R
LS
M w to to determine shape. Instead, consult Kratochvil’s chapter in the Huglin book.
62

Cases where polydispersity cannot be avoided:
1.
Reacting samples (see Wayne Reed, Tulane)
2.
Aggregating samples (consider DLS)
3.
Gelling samples (DLS again, maybe SALS too)
4.
Samples in hot or aggressive solvents.
63

In the days of yore, we could make monodisperse samples, for research purposes anyway, get a Zimm plot for each, and make a table of R g vs M like the one at left (Berry, 1966) or generate the right-hand side of this plot (data from Einaga & Berry ~1967) at six (6) molecular weights.
Cited ~650 times!
The left side, with hundreds of molecular weights, was produced automatically in about one-half hour, probably while the student was checking
FaceBook. How do we do that?
64

After a hurricane, many trees fall over and bend into a river. Also, a cow and a cat fall into that flooded river. Which one reaches the ocean first, cow or cat? Moo!
Meow
!
65

In GPC, as when throwing a cat through a tree, the big things come out first..
•Solvent flow carries molecules from left to right; big ones come out first while small ones get caught in the pores.
•It is thought that particle volume controls the order of elution.
•But what about shape ?
66

Graduate student
Wayne Huberty explains GPC better than I do.
The big and little enter together. The big ones come out first.
We can hope that someone lifted the grating to rescue the chicks, but sometimes polymers get permanently stuck (permastuck) on the column.
GPC/MALS still works in this case, by the way…the polymers that do elute are properly measured.
67

MALLS
DRI
DRI degas pump injector
68

This waterfall plot shows many “slices” of a chromatogram:
13 angles were recorded ~8 times per second as the sample flowed by the MALS detector. h r ap e G fac S ur
This is 2013: Less noise!
This was ~1995.
3D Plot - PBLG
Scattered intensity
40
30
20
10
L) (m m e lu v o
0.0
) /cm (1 atio h r ig le R ay
-6 0e x1 6.0
-6 0e x1 4.0
-6 0e x1 2.0
8 LS
7 LS
6 LS
5 LS
4 LS
3 LS
2 LS
1 LS
0 LS
LS
LS
0.0
LS
LS
LS
LS
LS
LS
16
15
14
13
12
11
10
9
8
7
6
5
4
69

The scattering “envelope” for a single slice shows how I s decreases with angle.
140000
Intercept  Molecular weight
120000
100000
80000
60000
Slope  Size
40000
20000
0
0.0
c = 0.044 mg/mL
M = 130000 g/mol
0.2
0.4
0.6
sin
2
(

/2)
0.8
1.0
70

How GPC/MALS looked a dozen years ago: persistence
Length of Helical Polypeptides is “Very High”
100
80
60
Temyanko et al.
Macromolecules 2001
Rod a p
= 240 nm a p
= 120 nm a p
= 70 nm
40
20
0
0
What the biggest polymers in our sample would look like at this a p
100000 200000 300000
M
400000 500000
71

Well, Pat Cotts got quieter data on a less-stiff polymer.
Macromolecules, 29, 7323-7328 (1996) a p

15 nm
Much less than PBLG
(note the log-log scale)
72

Today’s sensitive and quiet new MALLS equipment permits much higherM polypeptide samples to be measured:
1,500,000 vs 500,000 previously.
250
Note absence of pesky log scales that hide noise!
200
PBLG
150
100
50
0
0.0
Anhydrous DMF
0.1 M LiBr in DMF
0.1 M LiBr in DMF + 1 % H
2
O
Rod
 But what’s with this leveling out?
Ah….that’s the problem. When you do GPC/MALS, you really have to do the GPC well first.
2.0
0.5
1.0
1.5
Molar mass / 10
6
g-mol
-1
73

With the right chromatography, beautiful signals yield improved stiffness measurements.
300
250
200
PBLG
150
100 a p
= 250 nm a p
= 165 nm a p
= 130 nm
50
0
0.0
0.1 M LiBr in DMF
Rod
0.5
1.0
Molar mass / 10
6
g-mol
-1
1.5
74

GPC/MALS is used to study polymer/particle degradation by ultrasonic energy and shear (right) http://www.wyatt.com/files/literature/Multi_Angle_Light_
Scattering-polydegradebarron.pdf
75

Fractal dimensions can be determined: M dextran

R 3.3
???!!!
This seems unphysical, bu the authors explain that the polymers are packing in mass at larger sizes, due to branching.
76

Field Flow Fractionation, that’s how!
Eeee!
Moo!
77

Here’s a picture of FFF from Wyatt. In FFF, small stuff comes out first. Opposite of GPC.
78

From Wyatt dual-tec brochure, “How FFF works”
79

Handles a wider range of particles, including colloids.
May be easier for some aggressive solvents.
Can switch conditions faster, without concern for damage to stationary phase (there really isn’t one).
80

AF4 can separate large PTFE latex particles…impossible by GPC?
RMS radius vs Time
AF
4
separation of algoflon tm
LS 90 o
180
160
140
120
100
80
60
40
30 35 time (min)
40
1.0
0.8
0.6
0.4
0.2
0.0
81

1.2
1.0
0.8
0.6
0.4
0.2
0.0
60
(c)
70
Time / min
400
300
200
100
80
0
0.1
0.01
1E-3
1E-4
10
1
1E-5
1E-6
1E-7
0 1
63 Minutes: 474 nm diameter
67 Minutes (Peak): 505 nm diameter
76 Minutes: 510 nm diameter
2 3 4 q
2
/10
10 cm
-2
5
Soto-Cantu et al., Langmuir 2012, 28, 5562-5569
6
Here, the scattering profiles were fit using more than just the law of Guinier to get the size; see later in the presentation.
7
82

Cellulose nanocrystals can be fractionated by AF4/MALS:
New liquid crystal phases? Better microrheology probes?
Guan et al., Biomacromolecules 2012, 13, 2671-2679.
83

84

What if your sample, or some internal detail of it, is too small for visible light scattering (smaller than l
/50 = 10 nm = 100 Å) ?????
I just need a size, darn it!
I must have details like internal structure.
DLS: Dynamic
Light Scattering
SAXS or SANS
Your Lab vs
National Lab
85

Normal DLS works by interference among multiple scatterers, detected coherently.
g ( 2 )
( t ) 
2 T
1
I
2
T lim
 
T

 T
I ( t ) I ( t  t ' ) dt '  1  f coh e
 2  t
0  f coh
 1
  q 2 D trans
q set by scattering angle, refractive index, wavelength
Intensity

Normal DLS  Depolarized DLS  Bootstrapping  Applications  Fringe Benefits
86

Waves again: they move. Overall light = constant. Light reflected by one point fluctuates.
C:\PR\Type\Video\WavesFromWesterdam_WP_20130525_231145Z.mp4
87

View your correlation functions in various plots to understand them better.

This is how small, monodisperse particles behave.

Instead, say hydrodynamic radius, or hydrodynamic diameter.
D

 q
2 d
 or dq
2
1-angle multiangle
 kT
6

R

R h
 kT
6

D k is Boltzmann’s constant: 1.38

10 -16 g
 cm 2 s -2 K -1
T is the Kelvin temperature.
 is the viscosity: g
 cm -1
 s -1
Ooops….there was an error here:
R instead of

90

In principle, we should extrapolate to zero scattering angle, but
sometimes we can get by with one angle (q value)—i.e., D trans
=

/q 2 .
6
4 human influenza virus

Campbell, Epand, and RussoBiomacromolecules 2004, 5, 1728-1735
2
0
0 1 silica-homopolypeptide composite particles
2 5 6 3 q
2
/10
10 cm
-2
4
 Very good.
Fong and Russo, Langmuir 1999, 15(13); 4421-4426
Size essentially comes from inverting D .
Size would seem to depend on q for influenza virus: not good.
91

Here, we use one-angle DLS/SLS to follow the popopen capability of human influenza virus.
 Two surface peptides
Neuraminidase
Hemagglutinen
Kansas deaths
1918
300
I had a little bird,
Its name was Enza.
I opened the window,
And in-flu-enza.
--Children’s skip-rope rhyme, 1918 onward
1200
1000
800
250
200
150
600
100
400
More people died of flu in 1918 than in all of World War I. Ironically, the disease spread globally as American soldiers attempted to stop that war.
200
R h
Intensity
0 0
-400 -200 0 200 400 600 800 1000 1200 1400 t /s
Structural Changes and Aggregation of Human Influenza Virus, Jason N. Campbell, Richard Epand, Paul S. Russo, Biomacromolecules 2004, 5, 1728-1735.
http://www.hangtide.com/health/011/image3.jpg
http://www.udel.edu/chem/white/C647/FluVirus.GIF
50
92

You must measure diffusion as a function of concentration before you can convert it to a particle size accurately.
This is also true of other hydrodynamic methods for size determination, such as DOSY NMR.
R. Cong et al., Macromolecules 2003, 36, 204-209
Valid question: why does DLS diffusion rise with concentration, while DOSY diffusion falls?
Sensitivity matters: if you can measure at very low concentrations, then the extrapolation may be less important…a quick check for concentration effects may then suffice.
93

In polydisperse samples, multiple exponential decay rates will be observed. Sometimes, there is a quasi-continuum of them.
These are screen shots of our
Laplace inversion software suite.
We can watch a movie of it later…it’s considerably better than commercial versions.
94

Here’s an interesting study: ~4.5% THF in water is near a phase boundary, so there are significant fluctuations to be studied by DLS and SANS. No polymer necessary. The BHT preservatives matters!
95

Single-angle DLS will soon be a thing of the past.
MADLS = multi-angle simultaneous DLS.
2π/q x
Laser

4-8 angles at once.
Depends on

Typically 1 minute
$20,000
Multiple
Correlator
Single-sample DLS will also be a thing of the past… developments at Tulane ensure it.
96

97
Low-resolution Movie of MADLS instrument in action.

Depolarized DLS
Bootstrapped DLS
Very, very small volumes! (one drop or less)
XPCS = X-ray PCS
98

Depolarized DLS senses rotational motion, in addition to translational motion.


θ


H
Incident polarizer
Always vertical (v)
V
H
Detection analyzer
Vertical (V) or
Horizonal (H)
Now we can measure g (2)
Vv
(t ) or g (2)
Hv
(t) v
• This only happens if the object is
optically anisotropic.
• Many but not all rods qualify.
• Some spheres qualify.
• This rotational motion contributes fluctuations even when q = 0.
Normal DLS  Depolarized DLS  Bootstrapping  Applications  Fringe Benefits
99

Which has stronger depolarized signal, and why?
Depolarized signal strength arises from optical anisotropy, not anisotropy of shape.
TMV: Tobacco Mosaic Virus PTFE Latex
Shape highly anisotropic
Barely depolarizes at all
Shape barely anisotropic
Depolarizes like crazy!
100
Normal DLS  Depolarized DLS  Bootstrapping  Applications  Fringe Benefits

Dynamic Light Scattering Geometries, a Summary
LASER

Vv
= q 2 D trans v
V

Vv Geometry
(Polarized)
LASER v

Hv
= q 2 D trans
+ 6D rot
Requires optical anisotropy
Normal DLS  Depolarized DLS  Bootstrapping  Applications  Fringe Benefits
101
H

Hv Geometry q
(Depolarized)

4
 n sin
 
/ 2
 l o

102

Vv

Hv
D trans q 2
We’ll see real data later.
Normal DLS  Depolarized DLS  Bootstrapping  Applications  Fringe Benefits
D trans
6 D rot q 2

Probe diffusion, a form of microrheology: invert the Stokes-
Einstein relations. Rather than get a size, know the size and solve for the (micro)viscosity.
D trans
 kT
6  R
 

 kT
6  RD trans
D rot
 kT
8  R 3
 


8  R kT
3 D rot
Originators: Turner & Hallett, Phillies & coworkers.
Many others since the 1980’s.
Other ways: fluorescence or phosphorescence depolarization.
Mixed opinions about equality of

 and

.
Normal DLS  Depolarized DLS  Bootstrapping  Applications  Fringe Benefits
103

Bootstrapping : using two transport measurements eliminates viscosity. We get size without knowing viscosity, then we get viscosity anyway. Starting from nothing, you get something!
D
D trans rot

8 kT
6  R kT
 R 3

4
3
R 2  R 
3
4
D
D trans rot
Look, ma!
No viscosity!
Then:



8  R kT
3 D rot
 kT
6  RD trans http://admin.urel.washington.edu/uweek/ archives/issue/images/22_22/large_bike0594.jpg
G. Bantchev et al., Review of Scientific Instruments (2006), 77(4), 043902/1-043902/6
104

Very low volumes (~1 drop) can be measured in multi-angle mode using cost-effective Shigemi
NMR tubes with a flat bottom.
0.5
2000
Latex spheres in Shigemi cell
15 o
to 120 o
R
2
= 0.9999
0.4
R
2
= - 0.09
1500
0.3
1000
0.2
500
0
0 1
Equation
Weight
Residual Sum of
Squares y = a + b*x
No Weighting
647.50108
Pearson's r
Adj. R-Square
0.99994
0.99987
GammaHz
Intercept
Slope
Value
3.16538
Standard Error
3.74812
3.99993E-8 1.46141E-10
5 2 3 q
2
/10
10 cm
-2
4 6
0.1
0.0
0 1
Equation
Weight
Residual Sum of Squares y = a + b*x
No Weighting
1.60777E-18
Pearson's r
Adj. R-Square
0.1419
-0.08874
DFicks
Intercept
Slope
Value Standard Error
4.00582E-8 1.86769E-10
3.13168E-21 7.28221E-21
5 2 3 q
2
/10
10 cm
2
4 6
105

X-ray PCS: when you need to see something move slowly and not very far. http://www.aps.anl.gov/Science/Fut ure/Workshops/Time_Domain_Scie nce_Using_Xray_Techniques/Slides/
Lurio_time_resolved_04.pdf
http://sector7.xray.aps.anl.gov/~dufresne/UofM/xpcs.html
106

Reasons to do SAXS/SANS
Sample has small features.
Sample is not transparent to visible light.
To join up with LS and
SALS data for fractal analysis.
Many SAXS and some SANS resources can be found at the European
Molecular Biology Laboratory, especially by Dmitri Svergun
Basic SAXS: http://www.emblhamburg.de/biosaxs/software.html
http://www.emblhamburg.de/biosaxs/courses/embo20
12ccmb/slides/02-svergun-saxsbasics.pdf
107

q
I s
( q )

S ( q ) P ( q )
Have a little caution using this equation…It’s for
X-rays, and you can go wrong with visible light
Structure Factor: how are the objects that do the scattering arranged in space?
Particle Factor: what is the shape and internal structure of the scattering species?
Structured fluid
1.0
Simple
Fluid
1.0
Too small
“Small”
Crystal q
“Big” q
108

P ( q ) and S ( q )
Think of P ( q ) as the intrinsic ability of individual scattering elements to send a “signal” to a certain scattering angle. This ability may be reduced by destructive intramolecular interference according to size and shape of the scatterers. 0

P ( q )

1.
Think of S ( q ) as a filter set up by the entire system to let the signal out in only certain directions, due to crystallinity or partial structuring of the solution
(e.g., due to near-contact interactions).
109

Here, we can see the structure build in due to crowding of spheres as concentration is raised.
Calculated for hard spheres.
Experiments on soft micelles.
Small Angle Neutron Scattering Studies of the
Structure and Interaction in Micellar and Microemulsion Systems
Annual Review of Physical Chemistry
Vol. 37: 351-399 (Volume publication date October 1986)
DOI: 10.1146/annurev.pc.37.100186.002031
S H Chen
110

P ( q ) collects the phase shifts for all pairs of scatterers within a particle; this happens as we convert from scattered electric field to intensity.
Without rotational averaging
P ( q )
 i
1
N
2
N  i r ij
N  j e
 i q
 r
 ij
If particles rotate freely during the measurement period.
P ( q )

1
N
2 r ij
N  i

 r i
N  j sin( qr ij
) qr ij
 r
 j j
Here, it is assumed that all scattering elements have equal scattering power; otherwise, multiply each term by f i the scattering powers for each. f j
,
111

P ( q ) is known for simple molecular shapes, and can be fitted well to intensity-vs-q scattering data…when the samples are monodisperse. It is often plotted as the inverse, 1/ P ( q ).
Spheres
Rods
Random coils
This plot applies both to LS,
SAXS and SANS but you are more likely to get into the interesting regime
( qR >1) with SAXS or SANS.
Guinier law (low qR g
) Of these 3 basic shapes, spheres give the strongest effect .
q 2 R g
2
112 From the book by Huglin, Ch. 7

Sometimes, the P ( q ) expressions are simple, sometimes they need a numerical integral.
Simple example: uniform spheres of radius R
We do this in Excel
Numerical integral example: thin rods of length L
P ( q )

1 x

2 x
0 sin u du u qL x

2


 sin x x


2
We do this in LabVIEW
Code available on request.
113

Form factor for uniform spheres in Guinier format: the scattering goes to zero at certain angles!
Calculated Form Factor
R = 750 nm, l
=632.8 nm, n=1.33
1
0,1
0,01
0,001
0,0001
0,00001
0,000001
0,0000001
1E-08
0
Very limited linear Guinier region!
Quiz: Should we gather data at lower angles to use Guinier?
2E+10 4E+10 q 2 /cm -2
6E+10 8E+10
Quiz: What does it mean to have such a bumpy plot?
Hint: think inversely!
Quiz: What would polydispersity do to this curve?
Quiz: An experimental curve usually will not decay by 8 orders of magnitude. Give two possible reasons.
114

Here’s a nice representation. The Porod regime is just the inverse space pair of the l
-4 intensity dependence we saw earlier.
http://capsicum.me.utexas.edu/ChE386K/docs/SAS_intro.pdf
115

As you get to very high qR values, you can “fingerprint” structures. Here, SAXS results for core-shell particles.
Guinier regime
Fingerprints
Ballauff, Macromol. Chem. Phys. 2003, 204, 220–234
116

Form factors for two bent rods, same contour length. Quiz: Which is stiffer?
1
1
0 q
0 q
Potschke,D., et al. Macromolecular Theory and
Simulations 2000, 9, 345-353.
Kholodenko,A.; Ballauff,M.; Granados,M.A.
Physica A-Statistical Mechanics and Its
Applications 1998, 260, 267-293
117

Svergun’s ATSAS package can model SAXS data to give probable structures. In this case, rings made of porphyrin rings!
Pair distribution function
Hollingsworth, et al.
118

Hmmm….simple design, commune with the data, use your eyes, throughput matters, beware black boxes of doom, I hate plumbing, use direct space too, avoid polydispersity, many polymers are not pure and do not dissolve, centrifuges are great, you should not need to filter anything more than once, polarized organic solvents are hard, clean water is a blessing, plastic cleans easier than glass, Sigmacoat, Lemo connectors stink, dust in the air is less of a problem than you’d think, dust on a surface is a nightmare, use 0.2
 m or 0.1
 m filters if you can, vertical alignment is harder than other dimensions, mind your error propagation formulae, it’s better to have a powerful source and not need it than to need it and not have it, scatterbrains are wonderful people, build or buy two of everything.
Let’s focus on these 3:
1. Really, really clean samples help! Spend as much effort on a lab to make clean samples as you do on building/buying equipment.
2. Go train somewhere! By all means, read and study the materials here first. Try a little
SLS/DLS/SAXS or SANS…but then stay for awhile and learn it well!
3. Or….hire trained people.
Dr. Cornelia Rosu Melissa Collins Wayne Huberty Dr. Javoris Hollingsworth
119

• It is a 501-3c nonprofit serving polymer science
• Members serve as private individuals, but….
• Most of us work at LSU, Tulane, SELU or USM
• Industrial members wanted!
• No dues
• Academic tools/expertise available to industrial partners without the usual BS
• A really spectacular collection of polymer characterization equipment, especially for scattering!
• One-stop-shopping for your scattering and other polymer needs
• Very flexible payment/donation options
• Very flexible intellectual property policies
• Designed to help members while promoting regional economic development
APTEC = Applied Polymer Technology Extension Consortium http://macro.lsu.edu/aptec/
120

This workshop comes with lifetime support.
Please contact me or Dr. Rafael Cueto for help!
People visit the lab all the time; you will be very welcome!
Paul S. Russo or Rafael Cueto
Department of Chemistry & Macromolecular Studies Group
Louisiana State University
Baton Rouge, LA 70803-1804
225-578-5729 (office)
225-578-3361 (receptionist)
225-588-8104 (mobile) chruss@lsu.edu
http://macro.lsu.edu/russo http://macro.lsu.edu/psrgroup
 Rafael Cueto
121

4.
5.
2.
3.
1.
122

Zimm-Stein-Doty
McIntyre-Gornick
Van Holde
Berry
Wyatt
Burchard
Guinier
Huglin
Roe
Johnson-Gabriel
Berne-Pecora
Chu
Xu
Schmitz
Higgins-Benoit
Tanford
14
15
12
13
16
8
9
10
11
6
7
3
4
5
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The Ultimate Classic!
Collection of classics.
If you only had time to read just one thing…..
Thorough, detailed.
SLS especially
Branching
Classic SAXS
The Bible
A modern text on SAXS and SANS
A nice, inexpensive paperback book
DLS bible of theory
DLS bible of experiment
Various particle methods, in addition to LS
A useful reference, more modern than the early works
SANS
I love it!
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Reference List
Zimm,B.H.; Stein,R.S.; Doty,P. Classical Theory of Light Scattering From Solutions--A Review.
Polym.Bull.(Berlin) 1945, 1, 90-119.
2) McIntyre,F.; Gornick,D.Light Scattering from Dilute Polymer Solutions; Routledge: 1964;
3) Van Holde,K.E.Physical Biochemistry; Prentice-Hall, Inc.: Englewood Cliffs, N. J., 1985;
4) Berry,G.C. Static and Dynamic Light Scattering on Moderately Concentrated Solutions:
Isotropic Solutions of Flexible and Rodlike Chains and Nematic Solutions of Rodlike Chains.
Adv.Polym.Sci. 1994, 114, 234-290.
5) Wyatt,P.J. Light-Scattering and the Absolute Characterization of Macromolecules.
Analytica Chimica Acta 1993, 272, 1-40.
6) Burchard,W. Solution properties of branched macromolecules. Branched Polymers
Ii 1999, 143, 113-194. http://dx.doi.org/10.1021/bm0001291
7) Guinier,A.; Fournet,G.Small-Angle Scattering of X-rays; John Wiley and Sons: New York, 1955;
8) Huglin,M.B. In Light Scattering from Polymer Solutions; Huglin,M.B., Ed.;
Academic Press: New York, 1972; pp 165-203. Ch. 6. Specific Refractive Index Increments
9) Roe,R.-J.Methods of X-ray and Neutron Scattering in Polymer Science; Oxford University Press:
New York, Oxford, 2000;
10) Johnson,C.S., Jr.; Gabriel,D.A.Laser Light Scattering; Dover: 2013;
11) Berne,B.; Pecora,R.Dynamic Light Scattering; Wiley: New York, 1976;
12) Chu,B.Laser Light Scattering; Academic Press: New York, 1991;
13) Xu,R.Particle Characterization: Light Scattering Methods; Kluwer: Dordrectht, Boston, London, 2000;
14) Schmitz,K.S.An Introduction to Dynamic Light Scattering by Macromolecules; Academic:
San Diego, 1990;
15) Higgins,J.S.; Benoit,H.C.Polymers and Neutron Scattering; Clarendon Press: Oxford, 1994;
16) Tanford,C.Physical Chemistry of Macromolecules; John Wiley & Sons: New York, 1961.
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There are many more web resources than these!
CalTech
Multiangle DLS: http://mmrc.caltech.edu/PD_Expert/Expert.html
V. Nice DLS Intro: http://mmrc.caltech.edu/PD_Expert/Intros/dynamiclightscattering.pdf
LSU
General Info on Several Methods: http://macro.lsu.edu/HowTo
Zimm Plots: http://macro.lsu.edu/howto/guide/guide.doc
LS -Osmometer relation: http://dx.doi.org/10.1021/ed076p1534
DLS: http://macro.lsu.edu/howto/DLS_Minicourse/DLS_Minicourse.pdf
GPC: http://macro.lsu.edu/howto/GPC.DOC
GPC with LS power added: http://macro.lsu.edu/howto/GPCLS.DOC
CMU http://www.chem.cmu.edu/groups/Berry/#LINKS
Way too many beautiful LS links to list, and previous ISPAC lectures are among them!
TULANE http://tulane.edu/sse/polyRMC/publications/publications.cfm
Links are organized by topic; note especially SEC links and multi-sample LS links.
NIST http://www.ncnr.nist.gov/programs/sans/tutorials/
SAXIER http://www.saxier.org/forum/
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Theoretical lecture by Professor George Phillies
Worcester Polytechnic Institute http://www.youtube.com/watch?v=txMs3CzuDiA
At YouTube prompt, try typing:
*Wyatt Channel
*Brookhaven Instruments
*Light scattering
*SAXS
*SAXSess http://www.youtube.com/watch?v=5EmK-uW4Ajo
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Wyatt Corporation’s Theory Page http://www.wyatt.com/theory/rayleighscattering/mass.html
LS Instrument’s Technology Page http://www.lsinstruments.ch/technology/ http://www.alvgmbh.de/Support/Formulae/formulae.html
http://www.brookhaveninstruments.com/literature/lit_theory.html
http://www.malvern.com/labeng/products/iwtm/molecular-weight.htm
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http://www.bruker.com/en/products/x-raydiffraction-and-elemental-analysis/x-raydiffraction/nanostar/overview.html
http://www.rigaku.com/products/saxs/biosaxs1000 http://www.jjxray.dk/saxs http://www.saxs-waxs.be/ http://www.xenocs.com/
Additionally, most of the synchrotrons on the planet have one or more SAXS instruments.
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