The Theoretical Foundation of Spin-Echo Small-Angle Neutron

The Theoretical Foundation of Spin‐Echo Small‐Angle Neutron Scattering (SESANS) Applied in Colloidal System

Wei‐Ren Chen, Gregory S. Smith, and Kenneth W. Herwig (NSSD ORNL) Yun Liu (NCNR NIST & Chemistry, University of Delaware) Li (Emily) Liu (Nuclear Engineering, RPI) Xin Li, Roger Pynn (Physics, Indiana University) Chwen‐Yang Shew (Chemistry, CUNY)

UCANS‐II Indiana University July 08th 2011 Bloomington, IN Outline

1. Motivation — why Spin-Echo Small-Angle Neutron Scattering (SESANS)? 2. Basic Theory — what does SESANS measure? 3. Results and Discussions

— what can SESANS do? (1). Straightforward observation of potential (2). Sensitivity to the local structure (3). Sensitivity to the structural heterogeneity 4. Summary Outline

1. Motivation — why Spin-Echo Small-Angle Neutron Scattering (SESANS)? 2. Basic Theory — what does SESANS measure? 3. Results and Discussions

— what can SESANS do? (1). Straightforward observation of potential (2). Sensitivity to the local structure (3). Sensitivity to the structural heterogeneity 4. Summary Neutron Scattering

Structure (Elastic Scatt.) Dynamics (Inelastic Scatt.) Small‐Angle Neutron Quasi‐Elastic Neutron Unpolarized Scattering (SANS), Scattering (QENS), beam Neutron Diffraction, Inelastic Neutron Scattering Neutron Reflectometry (INS) Polarized Spin‐Echo Small‐Angle Neutron Spin‐Echo (NSE) beam Neutron Scattering (SESANS) Neutron Scattering

Comparison to Light Scattering • Extended length scale range • Multiple scattering • Transparent samples

Comparison to Ultra‐Small Angle Neutron Scattering (USANS) • Much higher flux

Comparison to Transmission Electron Microscope (TEM) • Non‐destructive nature

Comparison to Confocal Microscopy • EblEnsemble average iifnforma tion Examples for SESANS

•Highly concentrated systems such as colloidal glass (e.g. PMMA/PS binary glass)

•Large scale structure (polyelectrolyte aggregation) observed in the polyelectrolyte systems (protein, DNA, ionic polymers)

•Optoelectronic soft matters such as polymeric solutions of PLED/OLED

•Precipitate‐strengthening superalloy Outline

1. Motivation — why Spin-Echo Small-Angle Neutron Scattering (SESANS)? 2. Basic Theory — what does SESANS measure? 3. Results and Discussions

— what can SESANS do? (1). Straightforward observation of potential (2). Sensitivity to the local structure (3). Sensitivity to the structural heterogeneity 4. Summary Small Angle Neutron Scattering (SANS) —measure the structure

d



d d area A

 Intensity I polar axis d density N I(Q)  (Q) x d

  Ax  azithliimuthal axis 4     Wave vector Q Q  sin I  A  (Momentum transfer)  2 Scattering cross section Roger Pynn Small Angle Neutron Scattering (http://www.iub.edu/~neutron/) Neutron Spin Echo (NSE) —measure the dynamics

+B ‐B I II I II Polarizer Analyzer Detector

P  SQ,cos  d  SQ,cos d  S(Q, )  II I   cBL3m where the "spin echo time"  2 c  4.63681014T 1m2

Mezei, in Neutron Spin Echo, Ed. Mezei, Springer 1980 Spin‐Echo Small‐Angle Neutron Scattering (SESANS) —measure the structure

I II 2  1

  +B I ‐B II L L Polarizer Analyzer Detector d   d   P  Qcos  d 3Q  QcoszQ d 3Q  Gz  d II I  d z  cBL 2 cot where the "spin echo length" z  2 14 1 2 c  4.636810 T m

Pynn, Lecture 11, Neutron Physics and Scattering Indiana University (http://www.iub.edu/~neutron/) Length scale probed by SESANS Density Profile Debye Correlation Function  1 r γ(r)   ρ(r')ρ(r'  r )d 3 r' V V

  r d 2 Gz  2  r  dr IQ (Q)  4  rJ0 Qrr dr  2 2  z r  z d 0 Fourier z axis Abel

y axis  r Density Profile Debye Correlation Function  1 r γ(r)   ρ(r')ρ(r'  r )d 3 r' V V

  r d 2 Gz  2  r  dr IQ (Q)  4  rJ0 Qrr dr  2 2  z r  z d 0

IQ  nP(Q)S(Q) G(z)  Gauto (z)  nGexcl (z)  nGstruct (z) Interaction in concentrated hard colloidal system

Liu et al. PRL 95 118102 2005 Chen et al. Macromolecules 40 5887 2007 Chen et al. Science 300 619 2003

Likos et al. PRE 58 6229 1998 Huang et al. APL 93 161904 2008 Outline

1. Motivation — why Spin-Echo Small-Angle Neutron Scattering (SESANS)? 2. Basic Theory — what does SESANS measure? 3. Results and Discussions

— what can SESANS do? (1). Straightforward observation of potential (2). Sensitivity to the local structure (3). Sensitivity to the structural heterogeneity 4. Summary Example I: Hard Sphere Potential

 r  D V (r)   0 r  D

V(r)

Li et al., J. Chem. Phys. 132 174509 2010 Example II: Attractive Potential

 r  D  V (r)   u D  r  D(1  ) 0 r  D(1  )

V(r) u



D Li et al., J. Chem. Phys. 132 174509 2010 Example III: Screened Coulomb Repulsion Potential

 r  D  V (r)  exp- Z r  D K 1 r  D  1 r

V(r)

D Li et al., J. Chem. Phys. 132 174509 2010 Outline

1. Motivation — why Spin-Echo Small-Angle Neutron Scattering (SESANS)? 2. Basic Theory — what does SESANS measure? 3. Results and Discussions

— what can SESANS do? (1). Straightforward observation of potential (2). Sensitivity to the local structure (3). Sensitivity to the structural heterogeneity 4. Summary Sensitivity to the local structure

1.6

HS 1.4 Phenomenological Model L 1.2 101.0

100 S(Q) 0.8 10-1 0.6 Phenomenological Model 10-2 I(Q) 0.4 10-3

0.2  10-4

0.0 10-5 0 5 10 15 20 25 30 2QR -0.2 0 5 10 15 20 25 30 L 2QR Huang et al. APL 93 161904 2008 Li et al., J. Chem. Phys. 132 174509 2010 Sensitivity to the local structure

Phenomenological Model



L Huang et al. APL 93 161904 2008 Li et al., J. Chem. Phys. 132 174509 2010 Outline

1. Motivation — why Spin-Echo Small-Angle Neutron Scattering (SESANS)? 2. Basic Theory — what does SESANS measure? 3. Results and Discussions

n   D2O n  n D2O H 2O

Li et al., J. Phys.: Condens Matter (submitted) Outline

1. Motivation — why Spin-Echo Small-Angle Neutron Scattering (SESANS)? 2. Basic Theory — what does SESANS measure? 3. Results and Discussions

— what can SESANS do? (1). Straightforward observation of potential (2). Sensitivity to the local structure (3). Sensitivity to the geometric shape 4. Summary Summary

• SESANS length scale: from tens of nm up to several m

• SESANS correlation function G(z): real space projection

• SESANS advantages: direct observation of the spatial distribution sensitivit y to lllocal stttructure sensitivity to geometric shape high concentrated case… Acknowledgement

LDRD of ORNL 05272 DOE NERI‐C Award No. DE‐FG07‐07ID14889 NRC Award No. NRC‐38‐08‐950

Thank you for your attention! Backscattering —measure the dynamics

How about probing much ħ = Ei ‐ Ef slower dynamics (characteristic time > 1 ns)? Neutron Spin Echo —measure the dynamics

+B ‐B I II I II Polarizer Analyzer Detector

Mezei, in Neutron Spin Echo, Ed. Mezei, Springer 1980 What does SESANS measure?

2  1  +B ‐B PADI II z x

Pynn,Lecture 11, Neutron Physics and Scattering Indiana University (http://www.iub.edu/~neutron/) What does SESANS measure?

2  1  +B ‐B PADI II z  Final Polarization P  cos( II I ) x F. Mezei, Z. Physik, 255 (1972) 145

Pynn,Lecture 11, Neutron Physics and Scattering Indiana University (http://www.iub.edu/~neutron/) What does SESANS measure?

2  1  +B ‐B PADI II z  Final Polarization P  cos( II I ) x F. Mezei, Z. Physik, 255 (1972) 145

cos(II I )  cos(cBL cot )  cos(zQz )

Pynn,Lecture 11, Neutron Physics and Scattering Indiana University (http://www.iub.edu/~neutron/) What does SESANS measure?

2  1  +B ‐B PADI II z  Final Polarization P  cos( II I ) x F. Mezei, Z. Physik, 255 (1972) 145

cos(II I )  cos(cBL cot )  cos(zQz ) spin-echo length cBL 2 cot z   2

Pynn,Lecture 11, Neutron Physics and Scattering Indiana University (http://www.iub.edu/~neutron/) What does SESANS measure?

2  1  +B ‐B PADI II z  Final Polarization P  cos( II I ) x F. Mezei, Z. Physik, 255 (1972) 145

cos(II I )  cos(cBL cot )  cos(zQz ) spin-echo length  2 d  d cBL cot P  (Q)cos(  )d 3 Q  (Q)cos(zQ )d 3 Q z   d II I  d z 2

Pynn,Lecture 11, Neutron Physics and Scattering Indiana University (http://www.iub.edu/~neutron/) What does SESANS measure?

SANS SESANS

d d I (Q)  (Q) G(z)  (Q)cos(zQ )d 3Q d  d z

O. Spalla, in Neutron, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter, edited by P. Linder and Th. Zemb (North-Holland, Amsterdam, 2002), pp. 49–71. What does SESANS measure?

SANS SESANS

d 3 d I(Q)  (Q)   (r)exp(iQ r)d r G(z)  (Q)cos(zQ )d 3Q   z d V d

O. Spalla, in Neutron, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter, edited by P. Linder and Th. Zemb (North-Holland, Amsterdam, 2002), pp. 49–71. What does SESANS measure?

SANS SESANS

d 3 d I(Q)  (Q)   (r)exp(iQ r)d r G(z)  (Q)cos(zQ )d 3Q   z d V d

Debye Correlation Function 1 γ( r )   ρ( r' )ρ(r'  r )d 3 r' V V

O. Spalla, in Neutron, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter, edited by P. Linder and Th. Zemb (North-Holland, Amsterdam, 2002), pp. 49–71. What does SESANS measure?

SANS SESANS

d 3 d I(Q)  (Q)   (r)exp(iQ r)d r G(z)  (Q)cos(zQ )d 3Q   z d V d

Debye Correlation Function 1 Real part of FT in z direction γ( r )   ρ( r' )ρ(r'  r )d 3 r' V V

O. Spalla, in Neutron, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter, edited by P. Linder and Th. Zemb (North-Holland, Amsterdam, 2002), pp. 49–71. What does SESANS measure?

SANS SESANS

d 3 d I(Q)  (Q)   (r)exp(iQ r)d r G(z)  (Q)cos(zQ )d 3Q   z d V d

Debye Correlation Function 1 Real part of FT in z direction γ( r )   ρ( r' )ρ(r'  r )d 3 r' V V     (r)r G(z)  (r)dx  2 dr   2 2  z r  z

O. Spalla, in Neutron, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter, edited by P. Linder and Th. Zemb (North-Holland, Amsterdam, 2002), pp. 49–71. Real Colloidal System One Component Model (OCM)

V(r)

Interaction Potential: Hard Sphere, Attraction, Repulsion, Ultrasoft potential… Calculation of Inter‐molecular Structure

V(r) 2 h23 3 h12 c12 c13

h(r)  c(r)  nc(r)  h(r) Ornstein-Zernike (OZ) equation  Closure equation: g(r)  exp V (r)exph(r)  c(r)  b(r) PY, MSA , RY , HNC , ZH… g(r): the probability to find a particle at a distance of r Density Profile Debye Correlation Function  1 r γ(r)   ρ(r')ρ(r'  r )d 3 r' V V

  r d 2 Gz  2  r  dr IQ (Q)  4  rJ0 Qrr dr  2 2  z r  z d 0 Fourier Abel Density Profile Debye Correlation Function  1 r γ(r)   ρ(r')ρ(r'  r )d 3 r' V V

  r d 2 Gz  2  r  dr IQ (Q)  4  rJ0 Qrr dr  2 2  z r  z d 0 Fourier Abel  (r) auto (r)      (r)   (r) (r)   auto (r)   n excl    n g(r)  struct (r) Density Profile Debye Correlation Function  1 r γ(r)   ρ(r')ρ(r'  r )d 3 r' V V

  r d 2 Gz  2  r  dr IQ (Q)  4  rJ0 Qrr dr  2 2  z r  z d 0 Fourier Abel  (r) auto (r)      (r)   (r) (r)   auto (r)   n excl    n g(r)  struct (r) Fourier Abel

IQ  nP(Q)S(Q) G(z)  Gauto (z)  nGexcl (z)  nGstruct (z) Example IV: Two‐Yukawa Potential

 r  R  V (r)   exp- Z r  2R exp- Z r  2R K 1  K 2 r  R  1 r 2 r 1.0 0.4 HS K = 0 0.3 0.8 2 K2 = 4kBT 0.2 G (z) K2 = 5kBT struct

0.6 0.1  = 0.2 Z = 15 0.0 0.4 2 G(z) K =2= 2k T 1 B -0.1 0123456 Z1 = 5 0.2 z / R

0.0

-0.2

0123456 z / R Li et al., J. Chem. Phys. 132 174509 2010 Sensitivity to radial density profile: spherical core‐shell

homogeneous spherical hard sphere shell

Li et al., J. Chem. Phys. 134 094504 2011 Sensitivity to radial density profile: spherical core‐shell

homogeneous spherical hard sphere shell

Li et al., J. Chem. Phys. 134 094504 2011 Soft colloid

soft colloid homogeneous hard sphere

n*=0.0379 nm-3

Li et al., J. Chem. Phys. 134 094504 2011 Create a contrast: D/H mixture

Li et al., J. Phys.: Condens Matter (submitted) Summary

• SESANS length scale: from tens of nm up to several m

30% Sample in 0.5mm Sample Container • SESANS correlation function G(z): real space projection 1.0 Cubic fit 30% Sample in 0.1mm Sample Container Cubic fit 0.8 30wt% Cubic Fit of 0.5mm PMMA Container Data Cubic Fit of 0.1mm Container Data Percus-Yevick Hard Spheres at • SESANS advantages: 0.6 30% Concentration sensitivity to local structure G(z) 0.4 % (?Y)

sensitivity to structural heterogeneity 0.2

high concentrated case… 0.0

-0.2 0 50 100 150 200 250 300 350 • SESANS disadvantage: no sensitivity to scattering power z (% nm(?X) ) SESANS spectrometers

Delft University Nederland (2002) Asterix at LANSCE LANL US (2009) SESAME at LENS IU US (2010)

OFFSPEC at ISIS UK (2010) MAGIK at NCNR NIST US (2013) Larmor at ISIS UK (2014) Multiple Scattering

Pz() T G '(,) zt d tdQdQn () Q  11yzd 1 n  2n kn0 12  dd dQ dQ() Q dQ dQ () Q T 22y zdd 2 ny nz n n t  T n!

QQ12 Qn  Q Ttexp( ) cos(QQQQoddtermszzznz ) cos(12 )cos( ) cos( ) Multiple Scattering (Continue…)

d tdQdQn ()cos() Q Qz  11yz 1 1 z d  '  d dQ dQ()cos() Q Q z nyzz2n  22 2 1 kn0 12  d  n d Gzt()  dQ dQ( Q )cos( Q z ) T T  ny nzd n1 z n!

n (())Gzt Gzt() Gzt'((), ) n ' T  Te( 1) nn1 n!

ln(P(z)) G(z) 1 , P(z)  et(G(z)1) ln(T ) G()  0, P()  T  et G(0) 1, P(0) 1 Dynamic vs. Structural L     I v  L L      II (v  v)sin(  ) (v  v)(1 cot )   v  v cot  v cot v cot      L  L  L II I v(v  v)(1 cot ) v2 v The ratio: E ~ eV , E ~ meV dynamic v / v E / E    ~ 1o ,   20o structural  cot 2 cot

k T Stokes-Einstein equation: D  B 0 6R