X-Ray Scattering Studies ofThin Polymer Films

Scattering and Diffraction

Sunil Sinha
University of California San Diego
Cheiron School 2009
Spring-8

People want pretty pictures(in Space and Time!)

X-Ray Scattering• Indirect• Global Statistical Information• Non-destructive• Buried Interfaces• Magnetic and Structural• Dynamics• Can use with high pressures, magnetic
fields,etc.

S.R. based research can help
us to understand:• How the constituent molecules selfassemble to form nanoparticles.
• How these self-organize intoassemblies• How structure and dynamics lead tofunction• How emergent or collective properties
ariseIntrinsic Cross Section:X-Rays i ( k⋅r −ω t )Ein = E0 e

Erad ( R, t ) =ex(t − R / c )24πε 0 c R

eα (ω ) Ein e iωR / c cosψm
Erad ( R, t )e i kRcosψ
= − r0 α (ω )Ein
Rx(t − R / c ) = −Thomson Scattering Lengthof the Electron(classical electron radius) :e2−152.
8210m
r0 ==×2
4πε 0 mc

EradIntrinsic Cross

Erad ( R, t )EinSection: X-Rays dσ ∝ω0=202rα (ω ) 2 P (ψ ) =Rf (Ω )

2R21
 dσ =(1 + cos 2 ψ ) r02 α (ω ) 2
 dΩ  0 2

 dΩ  0RayleighScattering2ω2α (ω ) = 2
ωr − ω 2 − iηωResonance
Scattering

Thomson Scatteringω ≈ ωr4

ω >> ωr ⇒123

 dσ 2r=

0 P
 dΩ  0ωωr4

Adding up phases at the detector of the
wavelets scattered from all the scattering
centers in the sample:

Wave vector transfer is defined asq =kf - ki

X-raysdσ = r02[1 + Cos2(2θ)] S(q)dΩ2S(q) = 〈Σij exp[-iq.(r i-r j )]〉{r i} == electron positions.

Now, Σi exp[-iq.R i] = ρel(q) Fourier Transform of electron density

Proof:
ρel(r ) = Σi δ( r - R i)
ρel(q) = ∫ ρel(r ) exp[-iq.r ] dr = ∫ Σi δ( r - R i) exp[-iq.r ] dr= Σi exp[-iq.R i]

So, for x-rays, S(q) = 〈 ρel(q) ρel*(q) 〉∑el− iq . Rl=(2π )V03∑ δ (q − G)GqzqyqxX-Ray Scattering from Surfaces and Interfaces

Applications of Surface/Interface
Scattering•
••••
•••
•
•

study the morphology of surface and interface roughness
wetting films
film growth exponents
capillary waves on liquid surfaces (polymers, microemulsions,liquid metals, etc.)
islands on block copolymer films
pitting corrosion
magnetic roughness
study the morphology of magnetic domains in magnetic films.
Nanodot arrays
Tribology, Adhesion, Electrodeposition

Scattering Geometry & Notation
qxqReflectivity:qx= qy = 0
qz= (4π/λ)sinαiqz

kfki
Wave-Vector: q = kf – ki

Perfect & Imperfect „Mirrors“Basic Equation: X-Rays

Helmholtz-Equation & Boundary Conditions

Refractive Index: X-Rays & Neutrons
magnetic+
partmagnetic+
part

Minus!!

Dispersion

AbsorptionRefractive Index: X-Rays & Neutrons
magnetic+
partmagnetic+
part

Minus!!

Dispersion

AbsorptionRefractive Index: X-Rays

Electron Density
Profile !
E = 8 keV

λ = 1.54 ÅSingle Interface: Vacuum/Matter
FresnelFormulae

Reflected
AmplitudeTransmitted
Amplitude

WaveVectors

X-Ray Reflectivity: Principle
Visible Light
Reflectivity:n2 > 1
X-Ray
Reflectivity:n2 < 1n1

n1<n2n2n1>n2n1n2

Fresnel Reflectivity: RF(αi)

Total External
Reflection
Regime

The „Master Formula“
Reformulation for Interfaces∫
Fresnel-Reflectivityof the Substrate

Electron Density Profile

R(qz) = RFexp(-qz2σ2)

Roughness Damps Reflectivity

σj = 10 Å
λ = 1.54 Å

X-Ray Reflectivity:
Water Surface

Difference
ExperimentTheory:
Roughness !!
Braslau et al.
PRL 54, 114 (1985)

Fresnel Reflectivity

MeasurementExample: PS Film on Si/SiO2
Density Profile

X-Ray Reflectivity (NSLS)
λ = 1.19Å d = 109ÅData & Fit

Calculation of Reflectivity
Slicing of Density Profileε ~ 1Å

Slicing&Parratt-Iteration
Reflectivity
fromArbitrary
Profiles !
• Drawback:Numerical Effort !

Grazing-Incidence-Diffraction

X-Ray Reflectometers
Laboratory
Setup

Synchrotron
Setup
HASYLAB: CEMO

Reflectivity from Liquids I
Synchrotron
Setup (APS)Coherence of Light

Coherence Lengths

ξl = λ2 / ∆λ= λ(∆λ /λ)-1ξt = λ R / s(ξhor., ξvert.)

Photon Correlation
Spectroscopy
Brownian Motion of 100 particles

QuickTime™ and aVideo decompressor
are needed to see this picture.

Diffraction Pattern

SpecklesQuickTime™ and aVideo decompressor
are needed to see this picture.

Intensity-intensity auto correlation
1.15

g 2 ( q, τ ) =I ( q, t ) I ( q, t + τ )1.1

2

2I ( q, t )gQuickTime™ and aVideo decompressor
are needed to see this picture.

1.05

1010

1210

10
delay

310

Photon Correlation Spectroscopy
sampleX-ray speckle pattern from a static silica aerogel

detector1002

1.10

5000g

Intensity (cps)coherent
beam

1020time (sec)g 2 ( q, t ) =I ( q, t ′ ) I ( q, t ′ + t )2I ( q, t ′ )30Siegert Reln.
g2(q,t) = g1 (q,t) 2

1.05

1.00
0.1

1.0

10.0

100.0

time delay (sec)g2 (t ) = 1 + β exp( −2 Γt )= 1 + β exp( −2t / τ )β:speckle contrast

T. Thur n-Albrecht et al., PRL 1996

Statistical Description of Surfacesz( r|| ) = z + δz( r|| )C ( R, τ ) = δ z(0)δ z( R, τ )• Surface profile• Height-height correlation function for a homogeneous,isotropic and ergodic surface• [C(R)] ( actually the Fourier Transform of a function relatedto C(R) ) is measured in a surface scattering experiment.
S.K. Sinha et al., Phys. Rev. B 38, 2297 (1988)

Propagating Capillary Waves
qx =Small contrast !

Experiment Setup at Sector 8ID-I, APS

1.10-1

Intensity
Autocorrelationq||= 3.85E-3 nm
τ = 124.0 sec

1.05

1.00
1.10-1q||= 5.87E-3 nm
τ = 36.1 sec

g 2 ( q,τ ) =I ( q, t ) I ( q, t + τ )I ( q, t )g 2 ( q,τ ) = 1 + β e − 2 t / τ2g2

1.05

1.00
1.10
-1q||= 8.92E-3 nm
τ = 7.8 sec

1.05

1.00
1.10-1q||= 1.23E-2 nm
τ = 1.5 sec

1.05

1.00
0.1

1.0

10.0

100.0

time delay (sec)Hyunjung Kim, et.al., PRL 90, 68302 (2003)

Over-damped Relaxation Time Constants••

Identical thickness for allfilms (h=160 nm)
When Mw>30k g/mol,surface dynamics are– dependent on Mw at T>>Tg;– independent on Mw at nearTg.

••When Mw≤30k g/mol,surface dynamics are
dependent on Mw, andsingle exponential for all
T>Tg.
Capillary wave relaxation:2η J. Jäckle, J. Phys: Condense. Matter 10, 7121 (1998)
 H. Kim, et al., Phys. Rev. Lett. 90, 68302 (2003)
 Z. Jiang, et al., Phys. Rev. E 74, 11603 (2006)[cos(q|| h ) 2 + (q|| h ) 2 ]τ (q|| ) ≅γq|| [sinh(q|| h )cosh(q|| h ) − q|| h ]“Apparent” Shear Viscosity•••

Consistent with bulk values from rheological measurements at T>>Tg.
Orders of magnitude less than bulk values and independent of Mw near Tg.
Intersect with reptation theory (η∝Mw3.4) at Mw*– Mw* =Mc (Mc=31.2k g/mol: critical molecular weight for entanglement).

•Same Mw* found at 106 ºC, where surface dynamics are 7 times slower.

G ′′(ω )η = limω →0ω

Z.Jiang et al., PRL 2008F. C. Frank, Proc. R. Soc. London A 215, 43 (1952).
P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Phys.
Rev. B 28, 784 (1983)

P.Wochner et al.
PNAS 2009

Why go lensless?
• A technique for 3D imaging of 0.5 – 20 µm isolated objects•

Too thick for EM (0.5 µm is practical upper limit)•

Too thick for tomographic X-ray microscopy (depth of focus < 1 µm at 10 nm
resolution for soft X-rays even if lenses become available)•Goals<10 nm resolution (3D) in 1 - 10µm size biological specimens(small frozen hydrated cell, organelle; see macromolecular aggregates)

Limitation: radiation damage!

•2 nm resolution in less sensitive nanostructures(Inclusions, porosity, clusters, composite nanostructures, aerosols…)
eg: molecular sieves, catalysts, crack propagation

Image reconstruction from thediffraction pattern•Lenses do it, mirrors do it– but they use the full complex amplitude!
•Recording the diffraction intensity leads to the“phase problem”!
•Holographers do it – but they mix in a reference
wave, need very high resolution detector or
similar precision apparatus•Crystallographers do it – but they use MAD,
isomorphous replacement, or other tricks(plus the amplification of many repeats)Miao, Charalambous, Kirz, Sayre, Nature 400, 342 (1999).

λ=1.8 nmsoft x-ray
diffraction
pattern

Low angle data
From optical
micrograph

Scanning
electronmicrographof object

Image reconstructedfrom diffraction
pattern (θmaxcorresponds to 80nm). Assumed
positivity

Basic principles• Single object, plane wave incident,
scattered amplitude is Fourier transform of (complex)
electron density f(r)F(k) = ∫ f(r) e-2πi k ∙ r dr

• Assume: Born Approximation• Assume coherent illumination:for object size a, resolution d,– spatial coherenceδθ < λ/4a– temporal coherenceδλ/λ < d/4a

Reconstruction
Equations can still not be solved analytically
Fienup iterative algorithm

Reciprocal space•Positivity ofelectron
density helps!
11/3/2009

Imposediffraction
magnitudes

Miao thesis

Real space

Imposefinitesupport

42

DIFFRACTION IMAGING BY J. MIAO ET AL• From Miao, Ishikawa, Johnson,
Anderson, Lai, Hodgson PRL
Aug 2002• Diffraction pattern taken at
2 Å wavelength at SPring 8• Both levels show because
the depth of focus issufficient• SEM image of a 3-D Ni
microfabricated object with twolevels 1 µm apart• Only top level shows to usefulextent
11/3/2009• 2-D reconstruction withFienup-type algorithm• Resolution = 8 nm (newrecord)from Howells

43

MIAO ET AL 3-D RECONSTRUCTIONS• Miao et al 3-D
reconstruction of thesame object pair
• a and b are sections
through the image• c is 3-D density• Resolution = 55 nm

JCHS 7

Successful reconstruction of image from soft X-ray speckle alone.
SEM Image

X-ray reconstruction

50 nm diameter Gold Balls on transparent SiN membrane.

No “secondary image” was used
Approximate object boundary obtained from autocorrelation fn.
*How to make an isolated object ? Use AFM to remove unwanted balls.

.

He, Howells, Weierrstall, Spence Chapman, Marchesini et al. Phys Rev B In press. 03, Acta A.59, 143 (2003)

Imaging of individual nanoparticles at the APS
Ross Harder, University of Illinois, Champaign

Coherent diffraction patternfrom 170 nm Ag particle

170 nm silver cubes

inversion ofdiffraction pattern‘lensless imaging’I.K. Robinson, et al., Science 298 2177 (2003)

The Future?
• XPCS withcompletely coherent
sources, eg. XFEL’sor ERL’s• Large q• Down tomicroseconds tonanosecs

••

••

Atomic resolution
structures known
for few mammalian
membraneproteins!
Collect many single
molecule diffraction
patterns from fast
x-ray pulses, andreconstruct?
Lysozyme explodes
in ~50 fsec
R. Neutze et al.,Nature 406, 752(2000)

Single molecule
imaging?