Transcript Document

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Small Angle Neutron Scattering
– SANS
Mu-Ping Nieh
CNBC 2009 Summer School (June 16)
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Outline
CNBC
Summer School,
June 16, 2009
• Common Features of SANS
• Scattering Principle
• SANS Instrument & Data Reduction
• Contrast
• Dilute Particulate Systems – Form Factor
Data analysis
Examples
• Concentrated Particulate Systems – Structure Factor
Data analysis
Examples
• Dilute Polymer Solutions
Analysis for Gaussian Chain – Debye Function
More Generalized Analysis – Zimm Plot
• Small Angle Diffraction Analysis
• Model Independent Scaling Method Analysis
• Summary
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CNBC
Summer School,
June 16, 2009
Common Features & Advantages of SANS
1. The q range of SANS : between 0.002 and 0.6 Å-1
– achievable long wavelengths of neutrons and low detecting
angles are important
2. A powerful technique for in-situ study on the global structures
of isotropic samples
3. Easy to play contrast with isotope substitution
4. Data analysis:
(A) Model dependent: well-defined particulate systems
(possible non-unique solutions)
(B) Model independent: general feature of the structure
CNBC
Summer School,
June 16, 2009
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Neutron Scattering Principle - I
General Equation
The number of neutrons scattered per second
into a solid angle dW with the final energy
between E and E+dE
d2s
f
dW
y
dWdE
=
neutron flux of the incident beam • dWdE
For Elastic Scattering
We do not analyze the energy but only count the
number of the scattered neutrons
[time-1]

ds
d2s
=
dE =
dW 0 dWdE
The number of neutrons scattered per second
into a solid angle dW
neutron flux of the incident beam • dW
[time-1area-1]
[area]
Differential cross section
CNBC
Summer School,
June 16, 2009
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Neutron Scattering Principle - II
Scattering vector, q
4p
q
sin
q = ko – ki =
l
2
Wo
ki
ko
The measured intensity at q (or q), Im(q) can be
q
q
expressed as
Beam area
ds
on the
ki
ds
(
) ·A ·t sample
Im(q) = IF · Wo · e · T
dW v
dW
·
Flux
(
Solid
Differential
Detector
angle
cross-section
efficiency Sample
transmission
I
(q) · Astd · tstd · Tstd
ds
ds
)v,sam = ( ) v,std m,sam
dW
dW
Im,std(q) · Asam · tsam · Tsam
Path length
CNBC
Summer School,
June 16, 2009
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-0.3
-0.2
-0.1
Typical SANS Instrument
0.0
4.5
120
0.2
2-D
Detector
100
80
4p
q=
l
4.0
0.1
0.0
60
circularly
averaged
into 1-D
3.5
3.0
sin q
2
2.5
40
Neutron
Guide
-0.1
2.0
20
-0.2
1.5
0
0
20
40
60
80
100
120
4
2
3
10
I(q)
Sample
table
4
2
2
10
3
4
5
6 7 8 9
Velocity
Selector
2
0.1
q
For l = 8 Å and qmin~ 0.25o, qmin ~ 0.003 Å-1.
Max. attainable length scale = 2p/qmin ~ 2000 Å.
Neutron Beam
from Reactor
Current CNBC SANS Setup
CNBC
Summer School,
June 16, 2009
M.-P. Nieh Y. Yamani, N. Kučerka and J. Katsaras
Rev. Sci. Instrum. (2008) 79, 095102
qmin ~ 0.006 Å-1: Max. attainable length = 2p/qmin ~ 1000 Å.
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Contrast – A Key Parameter
CNBC
Summer School,
June 16, 2009
ds
N
2
(
)v = (rp – ro)2· Vp· P( q ) S( q )
dW
V
low contrast
2 ways of increasing contrast
Scattering intensity is proportional to
“the square of the scattering length
density difference between studied
materials and medium” (known as
“contrast factor”).
Ideally, we would like to increase the contrast yet without changing the chemical
properties of the system. This is one of the greatest advantages using neutron
scattering, since the neutron scattering lengths of isotopes can be very different.
7.E-06
3.E-06
2O
D
2O
H
ne
Po
ly
st
yr
e
Pr
-1.E-06
ot
ei
ns
1.E-06
Ph
os
ph
ol
ip
id
s
r, Å-2
5.E-06
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Contrast Variation
J. Pencer, V. N. P. Anghel, N. Kučerka and J. Katsaras,
J Appl Cryst 40 (2007) 513-525
CNBC
Summer School,
June 16, 2009
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ds
(
)=
dW v
SANS Analysis –
Dilute Particulate Systems
(rp – ro) · Vp· P( q ) S( q )
CNBC
Summer School,
June 16, 2009
V
r Vp
orientational average
p
ds = N
-iq· (r - r ) > dr dr
1 2
(
)v V  r(r1)r(r2) <e
dW
vp
1
N
-iq · r > dr
r(
r
)·<e
=
vp
V
N
2
2
= (rp – ro) · Vp· P( q )
V
ro
2
r1
2
O
r2
N particles
r1 and r2 are vectors
of scattering center
from one particle
Particle Form Factor
1 N
2>
<
|A
(q)
|
=
k
S
Amplitude of the Form Factor
V k=1
Ak(q) =  r( r )·<e-iq · r > dr
particle k
The form factor is determined by the structure of the particle.
SANS Analysis –
Dilute Particulate Systems
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CNBC
Summer School,
June 16, 2006
Simulation of a particular system - Spheres
 r( r
2
2
)·e-iq · r dr
Vsphere v
sphere
=
j
r= R q = p j= 2p
1
2
Vsphere
=
Since spheres are isotropic, there is no
need to do orientational average
9

 
q
2
e-iqr·cosq r2 sinq
dj dq dr
r
r= 0 q= 0 j= 0
1
6
[sin(qR) -
qR·cos(qR)]2
(qR)
2
N
2
ds
(
)v = (rsphere – ro) · Vsphere· P( q )
dW
V
2 9
=fsphereVsphereDr
6
(qR)
arbitrary unit
P(q) =
1
10
-1
10
-3
10
-5
10
[sin(qR) - qR·cos(qR)]2
R= 100 Å
Dr = 1x10-6 Å-2
f= 0.1
6 8
0.01
2
4
-1
q, Å
6 8
0.1
2
SANS Analysis –
Dilute Particulate Systems
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CNBC
Summer School,
June 16, 2009
Common form factors of particulate systems
P(q)
Morphologies
9
Spheres
(radius :R)
2
[sin(qR) - qR·cos(qR)]2 =Asph(qR)
6
(qR)
Spherical shells
(outer radius: R1
inner radius: R2)
Triaxial ellipsoids
(semiaxes: a,b,c)
Cylinders
(radius: R
length: L)
Morphologies
[R13·Asph(qR1)– R23·Asph(qR2)]2
(R13 – R23)2
1 1

2
Asph[q a2 cos2(px/2) + b2sin2(px/2)(1-y2)1 + c2y2 ] dx dy
0 0
1 J 2[qR 1-x2 ]
1
4
sin2(qLx/2)
Thin disk
(radius: R)
dx
2
2 ]2
(qLx/2)
[qR
1-x
0
J1(x) is the first kind Bessel function of order 1
2 - J1(2qR)/qR
By setting L = 0
q2R2
Long rod
(length: L)
2
qL
By setting R = 0
qL

0
sin(t)
t
dt -
sin2(qL/2)
(qL/2)2
“Structure Analysis by Small Angle X-Ray and Neutron Scattering” L. A. Feigen and D. I. Svergun
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SANS Analysis –
Dilute Particulate Systems
CNBC
Summer School,
June 16, 2009
Procedure of Data Analysis
Select a model for
possible structure
of the aggregates
A Possible
Structure
Fit the experimental
data using the
selected model
(Fix the values of any
“known” physical
parameters as
many as possible)
Yes
Is it a
good fit
?
No
Change
the model
CNBC
Summer School,
June 16, 2009
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I (arbitrary unit)
Examples of A Dilute Particulate System
Sample: A mixture of dimyristoyl and dihexanoyl phosphatidylcholine (DMPC,
DHPC), and dimyristoyl phosphatidylglycerol (DMPG) in D2O;
total lipid conc. = 0.1 wt.%
In these cases, the known (or constrained)
4
physical parameters are SLDs of lipid and
10
o
D2O and the bilayer thickness. Thus, there
10 C
3
are only 2 parameters to be determined
10
through fitting in each case.
2
o
45 C
10
Demo of SANS data analysis
in three afternoons!
1
10
o
0
10
10 C
Oblate shells: a=180 Å, b=62 Å
Path
-1
10
Spherical shells: R=133 Å, p= 0.15
-2
10
4
6 8
0.01
2
4
6 8
-1
q (Å )
0.1
2
Bilayer disks: R=156 Å, L= 45 Å
CNBC Summer School,
June 16, 2009
SANS Analysis –
Concentrated Particulate Systems
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ds
(
)=
dW v
(rp – ro) · Vp· P( q ) S( q )
V
N
ds
(
)v =
r(r1)r(r2) <e-iq· (r - r ) > dr1dr2

dW
V V
p
1
r Vp
2
p
1 N
2>
<
|A
(q)
|
=
k
S
V k=1
1
+
V
N N
Ak(q)Aj*(q)
S
S
k=1jk
R2
r1 r
2
e-iq· (Rk- Rj)
R1
O
N
1 N
-iq· (R - R )
2> {1+
~
e
<
|A
(q)
|
}
S
S
k
V k=1
jk
k
ro
j
Structure Factor
N
2
2
= (rp – ro) · Vp· P( q ) S( q ) In dilute solution,
V
S( q ) = 1
N particles
CNBC
Summer School,
June 16, 2009
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SANS Analysis –
Concentrated Particulate Systems
u(rjk)
Models for S(q)
+
+++
Coulomb
Repulsion
Q = 20,
csalt=0.01M
Hard
Sphere
2R
1.6
1.4
Structure Factor
+
+++
1.2
1.0
0.8
0.6
R= 50 Å
Conc. = 5%
0.4
-3kT
2.4R
Sqaure
Well
Attraction
0.2
0.00 0.05
0.10
0.15 0.20
-1
q (Å )
0.25
S(q=0) = kT(N/p)
Osmotic compressibility
S(0)>1 more compressible
S(0)<1 less compressible
0.30
CNBC
Summer School,
June 16, 2009
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SANS Analysis –
Concentrated Particulate Systems
Simulation Result using Coulomb Repulsive Model
qmax
2
10
R= 50 Å
Q = 50
Csalt = 0.001M
1
-1
I(cm )
10
Fixed, due
to P(q)
0
10
4

-1
10
-2
10
qmax ~ f
3
qmax
2x10
f
-2
5 67
-3
10
2
0.01
4
6 8
0.01
3
4 5 67
0.1
f
2
0.1
0.05
0.02
0.005
4
6 8
-1 0.1
q (Å )
2
CNBC
Summer School,
June 16, 2009
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Examples of
Concentrated Particulate Systems
“Bicelles” composed of DMPC/DHPC and small amount of Tm3+
10
2
0.25 g/mL
0.05
10
1
0.025
10
0
4
3
2
-1
qmax (Å )
-1
I (cm )
0.0025
10
-1
0.37
0.01
10
-2
6
5
0.001
10
0.01
0.1
Total Lipid Conc. (g/mL)
-3
2
4
6 8
2
4
0.01
6 8
0.1
-1
q (Å )
2
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“Gaussian Chain”
CNBC
Summer School,
June 16, 2009
“Gaussian Chain” Model : The effective bond length, a, of one step
R4
R1
RN
(composed of several segments of the chain) has a Gaussian distribution.
R3
R2
N
a2 = S |Rn - Rn-1|2 N
n=1
The distribution vector between any two “steps” m, n, i.e., (Rn - Rm) is
Gaussian

(Rn - Rm)2
3/2
F(Rn - Rm, n-m) =[
] exp[ ]
2
2
2pa |n-m|
2a |n-m|
This model is adequate for describing long polymer chains in a theta solvent, where
the segment-segment and segment-solvent interaction and the excluded volume
effect are canceled out. Such polymer chains are also known as “phantom” chains.
“The Theory of Polymer Dynamics” M. Doi and S. F. Edwards
CNBC
Summer School,
June 16, 2009
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SANS Analysis –
Dilute “Gaussian Chain” Solutions
Scattering from a Gaussian Chain
ds
2 (e-q2RG2 – 1 + q2R 2)
2
(
)v = Npfp(rp – ro) VM
G
4
4
dW
q RG
Debye Function
-2
I, cm
-1
10
VM: molecular volume of monomer
Np : degree of polymerization
10
10
-3
RG = 80 Å
Radius of Gyration, RG2=
q-2
-4
6 8
0.01
2
4 6 8
0.1
-1
q, Å
2
4 6 8
1
1
S
<(Rn – Rm)2>
2N2 m,n
CNBC
Summer School,
June 16, 2009
Example –
Polymer Solutions
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Debye function is not always adequate to describe the polymer
conformation!
An example is that as polymers are in a good solvent, the intensity
at high q regime will decay with q-5/3 instead of q-2.
10
-1
8
6
Polylactide in CD2Cl2
4
I, cm
-1
2
10
10
-2
8
6
4
4%
2%
2
1%
-3
2
3
4
5 6 7
q, Å
-1
2
0.1
3
fp
Np
1%
111
39
2%
74
32
4%
42
27
RG (Å)
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Zimm Plot
CNBC
Summer School,
June 16, 2009
For dilute polymer solutions, the scattering intensity at low q regime (i.e.,
q·RG < 1) can be expressed as a function of RG, MW (molecular weight),
A2 (second virial coefficient) and fp (volume fraction of the polymer)
2 fp=0 f1 f2 f3 f4
V
f
Dr
2
m p
VmfpDr
1
= (1 + q2RG2/3 + …)(
+ 2fpA2 +
I(q)
I(q) …)
N
q3
p
Vm is the molecular volume of the monomer
q2
q1
Np is the degree of polymerization
1/Np
RG, Np and A2 can be obtained by Zimm plot, where
q=0
VmfpDr2
q2 + cfp
is plotted against (q2 +cfp) and two extrapolation lines
I(q)
for q=0 and fp=0 are also obtained (c is an arbitrary number.).
The slope of the extrapolation line for fp=0 is RG2/3Np.
The slope of the extrapolation line for q=0 is 2A2/c.
The intercept of both extrapolation lines is
1/Np.
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Application of Zimm Plot
Polylactide in CD2Cl2
50x10
2
fpVMDr 
Np = 387
RG = 79 Å
A2= 88.6
-3
fp = 0
40
CNBC
Summer School,
June 16, 2009
Compared to the result
obtained from Debye fitting
30
20
q=0
10
1/N
p 0
0
10
20 2
30
40x10
-3
fp
Np
1%
111
39
2%
74
32
4%
42
27
RG (Å)
q + cfp
It requires at least two concentrations to make up a Zimm plot.
CNBC
Summer School,
June 16, 2009
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SANS Analysis –
Model Independent Scaling Method
Guinier regime – low-q scattering
ds
2
2
N
(
)v =
(rp – ro) · Vp· [1- (qRG)2/3 + …]
dW
V
~ fp(rp – ro)2Vpe-q2RG2/3
ds
ln[(
)v] = ln(Io) – q2RG2/3
dW
ds
ln[ (
)v]
dW
“Polymers and Neutron Scattering” J. S. Higgins and H. C. Benoit
-RG2/3
q2
CNBC
Summer School,
June 16, 2009
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SANS Analysis –
Model Independent Scaling Method
Porod regime – high-q (with respect to the length scale) scattering
The form factor of a sphere (radius of R)
ds
2 9
[sin(qR) - qR·cos(qR)]2
(
)v=fsphereVsphereDr
dW
(qR)6
For qR>>1, i.e, focusing at smooth interface between two bulks 3-dimensionally
2
2
2pDr2ST
N
V
Dr
9
ds
sphere sphere
(
)v ~
4 =
V
Vq4
2(qR)
dW
log I
2
ds
2pDr ST
4
-4
q(
)v =
dW
V
“Polymers and Neutron Scattering” J. S. Higgins and H. C. Benoit
total surface area
log q
CNBC
Summer School,
June 16, 2009
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SANS Analysis –
Model Independent Scaling Method
If Im(q) scales with q-n over a wide range of q. The “possible”
structure can be obtained with some knowledge of the systems.
particles
•Long Rigid rod
•Smooth 2-D Objects
•Linear Gaussian Chain
•Chain with Excluded Volume
•Interfacial Scattering from 3-D Objects
with Smooth Surface (Porod regime)
with fractal Surface
“Polymers and Neutron Scattering” J. S. Higgins and H. C. Benoit
n
1
2
2
5/3
4
3~4
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Small Angle Diffraction
CNBC
Summer School,
June 16, 2009
DOPC
-CD3
-CH3
CNBC
Summer School,
June 16, 2009
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Example: Orientation of
cholesterol in Biomembrane
Harroun, T. A. et al. Biochemistry 45 (2006)
1227-1233.
CNBC
Summer School,
June 16, 2009
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Take Home Messages
• SANS is a powerful tool to study the global structures of
isotropic systems in length scales ranging from 10 to 1000 Å.
• Isotope replacement could enhance/vary the contrast without
changing the chemistry of the systems.
• The scattering function is proportional to the product of contrast
factor, form factor (intraparticle scattering function) and structure
factor (interparticle interaction).
• Structural parameters can be obtained through model dependent
approaches, while model independent scaling law can also reveal
possible morphology of the studied system.