Small-Angle Scattering from Hierarchical Materials Greg Beaucage, University of Cincinnati

(1)

Small-Angle Scattering from Hierarchical Materials Greg Beaucage, University of Cincinnati

Kammler HK, Beaucage G, Kohls DJ, Agashe N. Ilavsky J J Appl. Phys. 97(2005) (Article 054309).

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- Definition of Dilute, Pay Attention to Contrast - Scaling Features in Scattering

- Guinier’s Law and Power-Laws

Derivation of Guinier’s Law from the Correlation Function Limitations of Power-Laws at Low- and High-q

- The Unified Function for Hierarchical Materials - Mass-Fractal Scattering

- Quantification of Branching and Topology - Dilute Unfolded Protein Scattering

- Simulated Structures from Scattering - Hybrid Unified Functions

- Worm-like Micelles - Hybrid Brämer Function - Correlated Lamellae

- Definition of Semi-Dilute Conditions

- Aggregation/Agglomeration (Immiscible System) - Mean Field Behavior in Scattering

- Specific Interactions in Scattering - Summary

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- Definition of Dilute

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For a complex structure in solution there is an inherent concentration associated with the structure since the structure contains some solvent. For a sphere this is **the excluded volume 8V, c* =** r /8

The concentration within a fractal structure:

When the solution concentration matches c

^*

the structures “overlap”

Then an individual structure can not be resolved and the structures entangle The regime above c

^*

is called semi-dilute

and the regime below c

^*

is called dilute

4

�

^∗

= ��

�� = �

�

³

= �

^�^�

�

³

=�

^�^�⁻³

¿ �

�

³

= �

�

^{3 /}^�^�

=�

^{1 − 3/}^�^�

c ^* is smaller

for larger objects

Typically this is 1% or 10 mg/ml

(5)

In concentrated solutions with chain overlap

chain entanglements lead to a higher solution viscosity

J.R. Fried Introduction to Polymer Science

5

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6

Structural Screening At least initially we want to work with dilute

samples You must demonstrate that your sample is dilute The

screening length (mesh size) is a

useful

parameter to obtain

Karsten Vogtt

Structural Screening n is

proportional to the second order virial coefficient, A₂

(7)

≈ ideally diluted

entangled

The “screening effect” and the screening parameter ν

Concentrated micelle system _�

� ( _{� , �} ) ⁼

�

₀

� ( ^{� , �}

0

) ⁺ ^{� υ}

7

= 1 n

(8)

- A Note Concerning Contrast

You need to pay attention to what gives contrast

8

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- Understanding Contrast

9

X-rays image mostly counterions Neutrons and light image hydrocarbons

Similar Worm-Like Micelle Samples

-2 power-law for the shell -4 power-law

for the surface

(10)

Sketch of the micelle cross-section and estimated parameter values involved in the core-shell model. The inset in the upper- right corner is a snapshot of the simulated micelle cross-section with accumulated ions.

Measured, fitted, simulated scattering curves of mixed surfactant sample. The inset in the lower-left corner is a snapshot of the simulated micelle segment.

Understanding the local structure of worm-like micelles SAXS / CRYSOL / Simulation

Jiang, H.; Vogtt, K.; Koenig, P; Jiang, R.; Beaucage, G.; Weaver, M. Coupled simulation and scattering for evaluation of the local structure of worm-like micelles. 2021, in preparation.

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- Definition of Dilute, Pay Attention to Contrast - Scaling Features in Scattering

11

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Scattering Observation of the Persistence Length

A power-law decay of -1 slope has only one structural interpretation.

Scaling Behavior of Dilute Synthetic Polymers

θ

Scaling Regimes over a range of q

At low end: a size for that structural level At the high end a different structural level

12

R

r_i

(13)

Worm-Like Micelles

cross section large scale structure

subunit

R G₁ = 10 cm^-1

R₁ = 20 Å L₁ = 500 Å s_R,1 = 3 Å z = 6

d_min = 1.1 d_f = 2.46 c = 2.2 n_br = 2 C_p = 1.46 R_g,2 = 350 Å G₂ = 60 cm^-1

circular R₁ = 20 Å 7

With a newly developed approach, the local as well as the global structure can be characterized on the basis of I(q). Using a hybrid scattering function, the local structure (cylinders) and the overall structure (tortuous and/or branched network) are determined.

13

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5mm LAT 16mm HAB Typical Branched Aggregate

d_p = 5.7 nm z = 350

c = 1.5, d_min = 1.4, d_f = 2.1 f_br = 0.8

Branched Aggregates

Beaucage G, Determination of branch fraction and minimum dimension of fractal aggregates Phys. Rev. E 70 031401 (2004).

APS UNICAT

Silica Premixed Flames J. Appl. Phys 97 054309 Feb 2005

14

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- Definition of Dilute

- Scaling Features in Scattering - Guinier’s Law and Power-Laws

15

1) Guinier’s law works for all q. That is there are no limits. It is sufficient to describe the structural size for a particular structural level.

2) Power-laws have two limits.

a. Power-laws overestimate the intensity at low-q near the Guinier Regime.

b. Power-laws over estimate the intensity at high-q where the Guinier regime for the next structural level occurs.

3) A structurally-limited power-law is needed to unify (sum) Guinier and power-law scattering. This results in a universal scattering function that can describe structural hierarchy in a general sense.

(16)

- Guinier’s Law and Power-Laws

Derivation of Guinier’s Law from the Correlation Function

16

This will provide a framework for structurally limited power-laws

(17)

Porod’s Characteristic Function

Rotationally averaged

centro-symmetric structures (Debye)

- Scaling Features in Scattering - Guinier’s Law and Power-Laws

Derivation of Guinier’s Law from the Correlation Function

17

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18

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19

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20

For a sphere

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21

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22

At high-q Porod’s Law

I(q) = 9 Nn_e² (qR/(qR)³)² = 9 Nn_e²/(qR)⁴ = 2pNVr_e² (S/V) q^-4

At low-q Guinier’s Law

cos x = 1 – x²/2! + x⁴/4!-…

sin x = x – x³/3! + x⁵/5!-…

exp(-x) = 1-x²/2!+x³/3!-..

I(q) = Nn_e²exp(-q²R_g²/3)

10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

Intensity (cm)-1

0.001 ² ^{3 4 5 6 7}0.01 ² ^{3 4 5 6 7} 0.1 ² ^{3 4 5 6 7} 1

q (Å)^-1 10 nm spheres

Sphere Function

Guinier's Law -4

Porod’s Law

(23)

23

(24)

24

Consider the origin of the vector r ~ 2p/q

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25

-For a wave scattered at an angle q which is in-phase with the incident wave material must be space at a distance r oriented at q/2 -This can occur for points separated by r such that

|r| = 2p/|q| Bragg Condition where

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27

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28

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33

Beaucage, G. "Approximations leading to a unified exponential/power-law approach to small-angle scattering." Journal of Applied Crystallography 28.6 (1995): 717-728.

(34)

34

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Guinier’s Law

Beaucage G J. Appl. Cryst. 28 717-728 (1995).

g_Gaussian

( )

r ^{= exp -3r}

(

² ₂_s2

)

s² =

x_i -m

( )

²

i=1

å

N

N -1 = 2R_g²

I q

( )

^{= I}eNn_e² exp -R_g²q² 3 æ

è ç ö ø ÷

Lead Term is

I (1/r) ~ N r

( )

^{n r}

( )

²

I (0) = Nn

_e²

g₀

( )

r ^{=1- S}

4V r + ...

A particle with no surface

r Þ 0 then d ( g

_Gaussian

( ) r )

dr Þ 0

Consider binary interference at a distance “r” for a particle with arbitrary orientation

Rotate and translate a particle so that two points separated by r lie in the particle for all rotations and average the structures at these different orientations

Binary Autocorrelation Function

Scattered Intensity is the Fourier Transform of The Binary Autocorrelation Function

35

Guinier’s Law describes a particle with no surface or internal structure

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36

Difficult to sketch since it has no

surface

(37)

37

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38

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G = Nn

_e²

Rayleigh, 1914

Scattering Function for Monodisperse Spheres

39

10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

Intensity (cm)-1

0.001 ² ^{3 4 5 6 7} 0.01 ² ^{3 4 5 6 7} 0.1 ² ^{3 4 5 6 7} 1

q (Å)^-1 10 nm spheres

Sphere Function Guinier's Law

-4 Power-law for Porod scattering Solid Surface

(40)

The Debye (1947) Scattering Function for a Polymer Coil

40

-2 Power-law for Gaussian Chain d_f = 2

I(Q) = 2

Q

²

( Q-1+ exp -Q ( ) )

Q = q

²

R

_g²

(41)

- Guinier’s Law and Power-Laws

Derivation of Guinier’s Law from the Correlation Function Limitations of Power-Laws at Low and High q

41

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42

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I(q) ~ N n

_e²

n

_e

Reflects the number of electrons in a particle N is total number of particles

Second Premise of Scattering All Structures Have a Surface.

(44)

The only location for contrast between phases is

at the interface (for every vector r there is a vector r/2 in the bulk) I(q) ~ N n

_e²

n

_e

Reflects the number of electrons in a particle N is total number of particles

Second Premise of Scattering All Structures Have a Surface.

n

_e

= 4 p

3 r

³

(45)

At the other extreme we consider a surface.

We can fill the interface with spheres of size r N = S/(πr

²

)

I(q) ~ N n

_e²

n

_e

Reflects the number of electrons in a particle N is total number of particles

Second Premise of Scattering All Structures Have a Surface.

n

_e

= 4 p

3 r

³

(46)

N = S/(πr

²

)

Porod’s Law can be thought of as the Second Premise of Scattering:

All “Particles” have a surface reflected by S/V.

(d

_p

= (S/V)

^-1

)

n

_e

= 4 p 3 r

³

I q ( ) ^~ ^Nn

e

2

~ S

p r

²

æ

è ç ö

ø ÷ 4 p r

³

3 æ

è ç ö ø ÷

2

~ 16 S p r

⁴

9 Þ

2 p n

_e²

S V æ

è ç ö

ø ÷

Vq

⁴

(47)

47

(48)

At low-q sin(qr)/qr = 1

(sin x = x – x³/3! + x⁵/5!-…)

Integrate g(r)r² => particle volume

Integrals for Gaussian and Sphere match at large r

At high-q only the small r values matter g(r) for a sphere and Porod match at small r Slope at r = 0 is S/4V (0 for Gaussian, particle with no surface; it is the same value for sphere and Porod)

48

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Structurally-Limited Porod’s Law

r x S gives surface volume Consider ½ of surface doesn’t contribute to scattering

Orientation gives a factor of ½

<cos²q> = ½

Model fails at large r (small q) Due to curvature of the surface This leads to an over estimation of the scattering near R_g.

49

10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

Intensity (cm)-1

0.001 ² ^{3 4 5 6 7}0.01 ² ^{3 4 5 6 7}0.1 ² ^{3 4 5 6 7} 1

q (Å)^-1 10 nm spheres

Sphere Function Guinier's Law

-4 Power- law for

Porod scattering

(50)

Particle Translation/Rotation are used to obtain a Gaussian characteristic function and Guinier’s Law

50

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A similar approach can be used to describe the limitation to Porod’s Law at low-q near R_g Consider an “average” Gaussian particle

This sometimes results in a Porod condition (a probability that drops following a Gaussian decay with q or 1/r)

51

Without translation this doesn’t meet the Porod condition

With translation it meets the Porod condition,

effectively

shortening r_b to the same size as r_a. The translation makes low-q mimic high-q

(52)

The translation effectively reduces r for the original figure. Reduced r ~ increased q.

Due to the Gaussian probability we effectively see high-q scattering at low-q

52

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By modification of q rather than the scattering function this becomes a universal function that can work with any object that displays a radius of gyration and follows Guinier’s Law

53

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We consider one orientation of the vector r (1-d) for 3d translation of the structure

Integrate for all displacement distances

54

A Gaussian probability following the Guinier derivation

(55)

This describes the 1-d probability (one r vector) due to finite structure

Cube to get 3-d probability that modifies g(r) in the Debye Function

So qr in the denominator becomes q*r with q* =q/(erf(qR_g/√6)³ As an approximation we also substitute sin(qr) with sin(q*r) Which enables substitution of q* for q in any power-law

55

This integral is an error function

(56)

By modification of q rather than the scattering function this is a universal function that can work with any object that displays a radius of gyration and follows Guinier’s Law

56

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For Porod’s Law we write

57

This can be summed with Guinier’s Law to yield the Unified scattering function.

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When is this an appropriate approximation? (Most of the time)

Approximation is exact at high-q, q > 2p/R_g sin(qr)/qr = sin(qr)/q*r = sin(q*r)/q*r

Approximation is good at very low-q sin(qr)/q*r = sin(q*r)/q*r

(See two thin dashed curves in Fig. 7) Between p/R_g < q < 2p/R_g

(See bottom two solid curves in Fig. 7)

Approximation leads to steeper decay in integral

This is inconsequential for surface scattering For mass-fractal scattering the calculated scattering curve is within 95% of actual. An empirical prefactor for R_g of kMass Fractal = 1.06 improves the match to 99.2%.

58

R_g = 80Å

(59)

High q-Limit to Power laws

Power-laws terminate at the next structural level since the conditions for the power-law, such as mass fractal structure, are no longer descriptive of the material at that size scale. The functional form for this loss of power-law scattering is the same as for Guinier’s Law for the next smaller structural level

This high-q limit to power-laws naturally occurs for the scattering functions for disks and rods

59

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Derivation of Guinier’s Law from the Correlation Function Limitations of Power-Laws at Low and High q

- The Unified Function for Hierarchical Materials - Mass-Fractal Scattering

- Quantification of Branching and Topology

60

(61)

�

₂

( _� ) ₌ ∑

�

( ^�

^�

⁻ ^�^�,�²³^�²

⁺ ^�

^�

⁻ ^�^{�, � −1}² ³ ^�²

⁽ ^�

^�^∗

⁾

⁻^{� �}

)

erf(x): Error function.

Index i=1, 2,… correspond to different structure levels, 1 smallest.

R

_g,i

: Radius of gyration correspond to structural level i P

_i

: Power-law decay slope for level i

k

_i

: Correction factor = 1 for P

_i

> 3 and 1.06 for P

_i

< 3

The Unified Function

61

(62)

- Scaling Features in Scattering - Guinier’s Law and Power-Laws

- The Unified Function for Hierarchical Materials - Mass-Fractal Scattering

62

(63)

Nano-particles form far from equilibrium.

T ~ 2500°K Time ~ 100 ms f

_v

~ 1 x 10

^-6

d

_p

~ 5 to 50 nm

63

(64)

Kammler HK, Beaucage G, Kohls DJ, Agashe N. Ilavsky J

J Appl. Phys. 97(2005) (Article 054309). 64

(65)

X-ray, λ 2θ

d = λ/(2 sin θ) = 2π/q

Branched Aggregates

65

(66)

Guinier’s Law

X-ray, λ 2θ

d = λ/(2 sin θ) = 2π/q

66

÷ ÷ ø ö ç ç

è

= æ -

exp 3 )

(

2 1 , 2

1

R

g

G q q

I

6 2

2

V ~ R

N G = r

_e

6 8 2

R

~ R

R

g

(67)

Guinier and

Porod Scattering

X-ray, λ 2θ

d = λ/(2 sin θ) = 2π/q

Structure of Flame Made Silica Nanoparticles By Ultra-Small-Angle X-ray Scattering

Kammler/Beaucage Langmuir 2004 20 1915-1921 ⁶⁷

)

4

( q = B q

^-

I

_P

S N

B

_P

= 2 p r

_e²

~ R

2

S

3 2

2

I ( q ) dq N R q

Q = ò = r

_e

2 3

2 R

R B

d Q

P

p

= =

p

(68)

Polydispersity

Particle size distributions from small-angle scattering using global scattering functions, Beaucage, Kammler, Pratsinis J. Appl. Cryst.

37 523-535 (2004).

X-ray, λ 2θ

d = λ/(2 sin θ) = 2π/q

68

G R PDI B

^P ^g

62 . 1

4

=

( ) ⁽ ⁾

¹²

12 ln ln

úûù êëé

=

= PDI

s

g

s

2 1

14 2

3

2

5 ú ú û ù ê ê

ë

= é

_s

e

m R

^g

(69)

Linear Aggregates

Beaucage G, Small-angle Scattering from Polymeric Mass Fractals of Arbitrary Mass- Fractal Dimension, J. Appl. Cryst. 29 134-146 (1996).

X-ray, λ 2θ

d = λ/(2 sin θ) = 2π/q

69

÷ ÷ ø ö ç ç

è

= æ -

exp 3 )

(

2 2 , 2

2

R

g

G q q

I

df

R R G

z G ÷÷

ø çç ö

è

= æ

=

1 2 1

2

df

f

q B q

I ( ) =

^-

( ) ²

2 , 2

d f g

f

d

R d B G

f

G

=

(70)

Branched Aggregates

Beaucage G, Determination of branch fraction and minimum dimension of fractal aggregates Phys. Rev. E 70 031401 (2004).

X-ray, λ 2θ

d = λ/(2 sin θ) = 2π/q

70

( ) ²

2 ,

min

2 d f

g

f

d

R d B G

f

G

=

c d

R z

p R

¹

1 2

min

÷÷ = ø çç ö

è

= æ

df

d

Br

R

^-

÷÷ ø çç ö

è - æ

=

min

1

2

f d

min

c = d

^f

(71)

Large Scale (low-q) Agglomerates

71

)

4

( q = B q

^-

I

_P

(72)

Small-scale Crystallographic Structure

72

(73)

5mm LAT 16mm HAB Typical Branched Aggregate

d_p = 5.7 nm z = 350

c = 1.5, d_min = 1.4, d_f = 2.1 f_br = 0.8

Branched Aggregates

Beaucage G, Determination of branch fraction and minimum dimension of fractal aggregates Phys. Rev. E 70 031401 (2004).

APS UNICAT

Silica Premixed Flames J. Appl. Phys 97 054309 Feb 2005

73

(74)

74

(75)

f* same for all flames!

TiCl

₄

Diffusion Flame Nucleation and Growth

75

(76)

Activated growth for z

Titania Diffusion Flame from TiCl₄

Beaucage G, Agashe N, Kohls D, Londono D, Diemer B 76

( )

kT E

d d R

z

p f

D -

÷÷ ø ö çç

è æ

~

ln 2

~ ln

(77)

Silica Diffusion Flame

Axial Particle Growth follows Classic Diffusion Limited Surface Growth, d ~ t

^1/2

77

(78)

- Scaling Features in Scattering - Guinier’s Law and Power-Laws

- Quantification of Branching and Topology - Dilute Unfolded Protein Scattering

78

(79)

p ~ R d æ è ç ö

ø ÷

d_min

s~ R d æ è ç ö

ø ÷

c

Tortuosity Connectivity

How Complex Mass Fractal Structures Can be Decomposed

d _f = d _min c

z ~ R d æ è ç ö

ø ÷

d_f

~ p

^c

~ s

^d^min

z df p dmin s c R/d

27 1.36 12 1.03 22 1.28 11.2

Beaucage, G Determination of branch fraction and minimum dimension of mass-fractal aggregates PRE 70 031401 (2004). 79

(80)

p ~ R d æ è ç ö

ø ÷

d_min

s~ R d æ è ç ö

ø ÷

c

Tortuosity Connectivity

How Complex Mass Fractal Structures Can be Decomposed

d _f = d _min c

z ~ R d æ è ç ö

ø ÷

d_f

~ p

^c

~ s

^d^min

z df p dmin s c R/d

27 1.36 12 1.03 22 1.28 11.2

(81)

Disk Random Coil

d

_f

= 2 d

_min

=1 c = 2

d

_f

= 2 d

_min

= 2 c = 1 Extended β-sheet

(misfolded protein) Unfolded Gaussian chain

81

(82)

Branch Content from Scattering

G Beaucage, Physical Review E (2004)

A S Kulkarni and G Beaucage, Journal of Polymer Science, Part B: Polymer Physics (2006)

R₂

R₁

d_min p

82

df

c

R z R

p ÷÷

ø çç ö

è æ

1

~

2

~

1 2

G

z = G d

_f

= c ´ d

_min ^d ^d^f

br

R

R z

p

z

^-

÷÷ ø çç ö

è - æ - =

=

min

1

2

f

d

f

d c

z p

=

min

1

(83)

Branch Content from Scattering

R₂

R₁

d_min p

83

min

= 1

=

<<

d

d c

z p

f df

c

R z R

p ÷÷

ø çç ö

è æ

1

~

2

~

1 2

G

z = G d

_f

= c ´ d

_min ^d ^d^f

br

R

R z

p

z

^-

÷÷ ø çç ö

è - æ - =

=

min

1

2

f

(84)

Branch Content from Scattering

R₂

R₁

d_min p

c d

_min

84

df

c

R z R

p ÷÷

ø çç ö

è æ

1

~

2

~

1 2

G

z = G d

_f

= c ´ d

_min ^d ^d^f

br

R

R z

p

z

^-

÷÷ ø çç ö

è - æ - =

=

min

1

2

f

d

f

d c

z p

=

min

1

min

= 1

=

<<

d

d c

z p

f

(85)

Benoit integral for polymers of arbitrary dimension modified based on the branched fractal model

Benoit, H, C.R. Hebd. Seances Acad. Sci. 245 2244 (1957).

Can be extrapolated to the high-q power-law regime to obtain an expression for the power-law prefactor If the power-law prefactor, B_f, can be determined in terms of the branched fractal model then

the Unified Function can be used for branched polymers and mass-fractals

(86)

G Beaucage, Physical Review E, 70, 031401 (2004)

AS Kulkarni, G Beaucage, J Polym Sci, Part B: Polym Phy, 44, 1395 (2006)

-2

Topological information can be extracted from this feature arising from combining Local Scattering Laws

d_f = c x d_min

d_min 2 1 2

c 1 2 2

86

( ) ²

2 2 , min

f d g f

d G

R d B

f

= G

(87)

Quantification of Branching

Mole-Fraction LCB Content

•Beaucage, G., Determination of branch fraction and minimum dimension of mass-fractal aggregates. Physical Review E 2004, 70 (3).

•Ramachandran, R.; Beaucage, G.; Kulkarni, A. S.; McFaddin, D.; Merrick-Mack, J.; Galiatsatos, V., Branch Content of Metallocene Polyethylene. Macromolecules 2009, 42 (13), 4746-4750.

Average number of branch points per chain

87

1 1

1 -

^-

- =

=

^c

br

z

z p f z

( )

÷÷

÷ ø ö çç

ç è

æ -

=

-

÷+ ø ç ö

è

æ -

2

1

1 1 32 52

c c

d br

z f

n

z~ p

^c

~ s

^d^min

d

_f

= cd

_min

(88)

Quantification of Branching

Hyperbranch Content (Branch-on-Branch) Average LCB length

•Beaucage, G., Determination of branch fraction and minimum dimension of mass-fractal aggregates. Physical Review E 2004, 70 (3).

•Ramachandran, R.; Beaucage, G.; Kulkarni, A. S.; McFaddin, D.; Merrick-Mack, J.; Galiatsatos, V., Branch Content of Metallocene Polyethylene. Macromolecules 2009, 42 (13), 4746-4750.

•Rai, D.K.; Beaucage, G.; Vogtt K.; Ilavsky J.; Kammler, H.K. In situ study of aggregate topology during growth of pyrolytic

silica J. Aerosol Sci. 118 34-44 (2018). 88

p br br

i

n n

n = -

_,

) (NMR or SANS br

Kuhn br

br

n

M

z ₌ z f

(89)

Branched polystyrene in a good solvent

d

_min

should be 5/3 (1.67) for self-avoiding walk

M_w/M_n ~ 2 Good Solvent Scaling for c = 1 and c > 1

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Mole fraction of Branches

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# Sample R_g (Å) d_f d_min c f_br Theoretical f_br

1 3 Arm Star 6.87 2.00 1.33 1.49 0.39 0.33

2 6 Arm Star 78.27 2.00 1.25 1.59 0.71 0.67

3 18 Arm Star 4.37 2.27 1.61 1.40 0.81 0.88

Horton et al., Macromolecules, 22, 681 (1989) US Jeng, TL Lin, LY et al. App Phys A, 74, S487 (2002).

Multi-Arm Star Polymers

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M_n (g/mol)

Index LCBI

Mole Fraction of Branches

φ

_br

NMR n_br/10⁴

C

PE 0 46,500 0.04 0 0

PE 2 11,500 0.91 0.39±0.005 0.36

PE 3 37,900 2.27 0.63±0.004 0.91

Model Based on Mole Fraction of

Branches

This Approach Can Quantify LCB in Polyolefins.

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- Quantification of Branching and Topology - Dilute Unfolded Protein Scattering

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Unfolded proteins

Is the molten golbule a third phase of proteins? VS Pande & DS Rohksar, PNAS 95 1490-94 (1998)

Simulation Results,

“g” is denaturant

-Unfolded state is analogous to gas state (following Flory random walk model as applied by Pande)

-Molten globule. Dense states not associated with biological activity could be analogous to glassy state.

-Native State. 0 conformational entropy, native state is analogous to crystalline state

-Unfolded state contains some:

Chain persistence associated with fluctuating helices and b-sheets

Crosslink/disulfide/cystine-cystine bonds that act as branching sites

hydrophobic interactions that may appear to be branching sites

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Unfolded proteins

“a Gaussian-like conformation”

Mapping the cytochrome C folding landscape

Julia G. Lyubovitsky

Caltech 2003 Biochemistry

R_g d_f d_min c

125Å 2.05 1.07 1.90

This is almost a regular structure with dimension 2:

A crumpled sheet!!

95

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R_g PDI

94.0Å 1.08

This is and almost sperical domain of

24.3 nm diameter.

Deviation from PDI = 1 can be due to polydispersity or asymmetry.

Cytochrome C Native State

Space Filling Model

96

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R_g PDI

Native 94.0 Å 1.08 MG 99.0 Å 2.90

(Both also

Show Aggregates at Low-q)

Model of cytochrome b562 A) MG B) N

Cytochrome C

Molten Globule State

Alberts B, Bray D, Lewis J, Raff M, Roberts K, Watson JD. Chapter 5: Protein Function. Molecular Biology of the Cell 3rd Ed. Garland Publishing, New York. 1994. pg 213 & 215. 97

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- Simulated Structures from Scattering

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Descriptors

Beaucage, G. Determination of branch fraction and minimum dimension of mass-fractal aggregates. Physical Review E – Statistical, Nonlinear, and Soft Matter Physics 2004, 70, 031401-1–031401-10.

Rai, D., Beaucage, G., Jonah, E. O., Britton, D. T., Sukumaran, S., & Härting, M. Quantitative investigations of aggregate systems. J. Chem. Phys. 2012, 137, 044311–

1–044311-6.

Ramachandran, R., Beaucage, G., Kulkarni, A. S., McFaddin, D., Merrick-Mack, J., & Galiatsatos, V. Branch content of metallocene polyethylene.

Macromolecules 2009, 42, 4746–4750. 99

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Unified Fit :

Parameters from Unified Fit used to determine topological parameters:

Scattering

Beaucage, G. Approximations Leading to a Unified Exponential/Power-Law Approach to Small-Angle Scattering J. Appl. Cryst. 1995, 28 (6), 717–728.

Herrmann, H. J., & Stanley, H. E. The fractal dimension of the minimum path in two- and three-dimensional percolation. Journal of Physics A:

Mathematical and General 1988, 21, L829–L833.

Meakin, P., Majid, I., Havlin, S., & Stanley, H. E. Topological properties of diffusion limited aggregation and cluster-cluster aggregation. Journal of Physics A: Mathematical and General 1984, 17, L975–L981.

Witten, T. A., & Sander, L. M. Diffusion-limited aggregation. Physical Review B 1983, 27, 5686–5697.

Sorensen, C. M. Light scattering by fractal aggregates: A review. Aerosol Sci. Tech. 2001, 35, 648–687.

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Algorithm:

Input z and a sticking probability Randomly grow aggregates

Compute the scattering parameters, p, R, n

_Br

etc.

Iterate by varying sticking probability until computed matches experimental

A. Mulderig, G. Beaucage, K. Vogtt, H. Jiang and V. Kuppa, Quantification of branching in fumed silica, J. Aerosol Sci.

2017, 109, 28–37.

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2017, 109, 28–37.

Experimental

Simulation

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2017, 109, 28–37. 103

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2017, 109, 28–37. 104

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2017, 109, 28–37. 105

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- Simulated Structures from Scattering - Hybrid Unified Functions

- Worm-like Micelles - Hybrid Brämer Function - Correlated Lamellae

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We often have an ideal nanoscale structure that aggregates, agglomerates, becomes convoluted, becomes branched. For example, protein aggregation and

agglomeration.

-Form factor at nanoscale is available

-Unified can describe larger-scale power-law scaling regimes composed of perfect structures

Three examples:

-Worm-Like Micelles: Polydisperse rods form tortuous and branched chains.

-Polymer Crystals: Stacked lamellae (form factor and structure factor from Brämer Function) form fibers and spherulites.

-Agglomerated Mass-Fractal Spheres

Hybrid Unified Functions:

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Fit model:

• Unified model for large scale structure, form factor and G₁ = I₀ for cylindrical subunits

• Assuming that micelles can be structurally approximated by solid, homogenous

cylinder or a cylindrical shell

• Self avoiding walk in good solvent ® fixed d_min = 1.67

• radius R₁ polydisperse (Log-Normal distribution with scale parameter s_R1)

• No end-caps taken into account

A Hybrid Unified Function: The Cylindrical Subunits

108

I '(q) = G

₂

e

^-

R_g,2² q²

3

+ B

₂

e

^-

R_g,1² q²

3

q*

^-d^f

+G

₁

ò N(R

₁

)P

_cyl

(q, R

₁

, L

₁

) dR

₁

r

B₂ = d_min,2G₂C_p R_g,2^d^{f ,2} G(d_f,2

2 ) æ

èçç ö

ø÷÷ ; q* = q (erf(q 1.06 R_g,2

6 ))³ ; G₂

G₁ +1= z

(109)

fitting

parameter meaning L₁ length of cylinders

R₁ radius of cylinders

s_R1 standard deviation of radii distribution

n screening parameter (virial coefficient)

G₂ zero-angle scattering of global structure

R_g,2 radius of gyration of global structure

d_f,2 fractal dimension of global structure

input

parameters meaning

f volume fraction

Dr

scattering length density contrast; from mass den- sities of micelles or bead density from MD-

simulations (shell model)

d_min,2 (fractal) dimension of chain

without branches (= 1.67) C_p polydispersity factor for size

of large scale structure (M_z/M_w)

Input and fitting parameters

Based on the model a function describing the scattering patters can be formulated. The model distuingishes between the local, cylindrical structure (index 1) and the large scale assembly of the cylinders (index 2).

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Structural hierarchies of worm-like micelles: impact on rheological properties

Dreiss (2007), Soft Matter, 3:956

Contour and subunit lengths of WLMs have a large impact on the macroscopic viscoelastic properties. Interestingly, branching reduces the zero-shear viscosity.

⟶ WLM-systems are vastly employed in liquid soaps, drag reducers and fracking fluids

⟶ structure-function relationship

k

110

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Structural hierarchies of worm-like micelles (WLMs): An assembly of rods

- The WLMs are approximated as a chain of cylinders with length L₁ and radius R₁ - With symbol z as number of subunits the contour length L equals z⨉L1

- The contributions of the cylinders and the overall chain to the total scattered intensity I(q) can be separated ⟶ “hybrid function” for fitting

i struct. element 1

cylindrical sub-

unit ⟶ P(q) of a cylinder

2

Overall chain (branching, tortuosity) ⟶ Unified Function

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The scattered intensity I(q) and the structural domains

cross section large scale structure

subunit

R G₁ = 10 cm^-1

R₁ = 20 Å L₁ = 500 Å s_R,1 = 3 Å z = 6

d_min = 1.1 d_f = 2.46 c = 2.2 n_br = 2 C_p = 1.46 R_g,2 = 350 Å G₂ = 60 cm^-1

circular R₁ = 20 Å 7

With a newly developed approach, the local as well as the global structure can be characterized on the basis of I(q). Using a hybdrid scattering function, the local structure (cylinders) and the overall structure (tortuous and/or branched network) are determined.

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Structural hierarchies of WLMs: Fitting I(q)

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Structural hierarchies of WLMs: Addition of salt

⟶ upon addition of NaCl basically just the large scale structure changes

⟶ number of subunits increases

⟶ possibly onset of branching at 5.0% NaCl

0.232% mixed surf.

Vogtt et al., Langmuir 31:8228-8234 ¹¹⁴

Small-Angle Scattering from Hierarchical Materials Greg Beaucage, University of Cincinnati