UNESCO/IUPAC Postgraduate Course in Polymer Science

Introduction to X-ray and neutron scattering

Zhigunov Alexander

Institute of Macromolecular Chemistry ASCR, Heyrovsky sq. 2, Prague -162 06http://www.imc.cas.cz/unesco/index.html

unesco.course@imc.cas.cz

Lecture:

Contents

Examples of structural studies

Small-angle x-ray and neutron scattering

Wide-angle X-ray scattering

Common principles

Examples of polymeric structures

Examples of polymer structuresChains and particles

Networks

Semicrystalline and organized structures

Polymer chain (in solution)

LENGTH SCALES

100 - 103Å

Polymer particles

LATEX MICELLE

102 - 103 101 - 102

101 - 103ξξξξ

CRYST.AM.CRYST.

CRYSTAL STRUCTURE: 10-2 - 100 Å

101 - 102

CUBIC, LAMELLAR, HEXAGONAL

LONG PERIOD: 101-102 Å

WavesWave characteristics are: amplitude, frequency, phase, speed

λ - Wavelength of a sinusoidal wave is the spatial period of the wave (the distance over which the wave's shape repeats).

When passing through media:AbsorptionReflectionInterferenceRefractionDiffractionPolarization

Picture by Spigget

Diffraction theoryWhen a wave passes through an opening in a barrier, the wave spreads out, or diffracts. When two waves occupy the same location, they interfere. When this interference results in a larger wave, we call it constructive interference. When the size of the wave is reduced, it is called

destructive interference.

Waves interactionsWhen x-rays are incident on an atom, they make the electronic cloud move as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency (blurred slightly due to a variety of effects); this phenomenon is known as Rayleigh scattering (or elastic scattering). These re-emitted wave fields interfere with each other either constructively or destructively (overlapping waves either add together to produce stronger peaks or subtract from each other to some degree), producing a diffraction pattern on a detector or film. The resulting wave interference pattern is the basis of diffraction analysis.

Picture by Christophe Dang Ngoc Chan

Scattering Experiment

X,N2θθθθ

Sr

0Sr

Scatteringintensity

Scatteringvector

COLL. SAMPLE DETECTORPLANE

)(qIr

)()/2( 0ssqrrr

−−−−==== λλλλππππ

θλπ sin)/4(=q

SOURCE

Wave should be coherent and collimated (parallel waves)

SAXS and WAXS

SAS: Small-Angle Scattering. Material, containing inhomogeneities from 10 to 1000 Ǻscatters radiation into agngles 0-2º

WAS: Wide-Angle Scattering. Single crystal. Material with inhomogeneities with size of inter-atomic distances shows diffraction spots at angles 2-90º

Small and Wide Angle Scattering

Scattering of materials

Homogenous material Material with inhomogeneities Single crystal Polycrystalline material

Primary beam only.No scattering.Vacuum is the only homogenous material.

Inhomogenities 10-1000 AScattering angle ~ 0-2 o

Small Angle Scattering

Interatomic distancesScattering angle ~2-90 o

Wide Angle Scattering

Angle depends on disnatces

Polycrystalline sample

Examples of WAXS patterns

AMORPHOUS SAMPLE

POLYCRYSTALLINE POWDER

SINGLE CRYSTAL

Crystal structure

Simple cubic

Body-centered cubic

Face-centered cubic

Unit cells Miller indices

a1

a2

a3

Crystal structure

5 10 15 20 25 30 35 400

200

400

600

800

1000

1200

1400

Inte

nsity

, a.u

.

2Θ, degree

p-TSA

WAXS ofp-Toluenesulfonic acid

α = β = γ = 90a = b = cCubic

α = β = γ ≠ 90a = b = c--- Rhombohedral division

γ = 120

α = β = 90a = b ≠ c--- Hexagonal division

Hexagonal

α = β = γ = 90a = b ≠ cTetragonal

α = β = γ = 90a ≠ b ≠ cOrthorombic

α = γ = 90

β ≠ 90a ≠ b ≠ cMonoclinic

α ≠ β ≠ γ ≠ 90a ≠ b ≠ cTriclinic

Angels between CrystalAxes (degrees)

Axial Translations(Unit-cell constants)

System

Bragg’s Law2 d sin ΘΘΘΘ = n λλλλ d = interplanar distance

q d = 2ππππ n n = integer

Lattice planes

Geometry of the Bragg reflection analogy:

The waves “reflected” by the two adjacent planes are in phase at scattering angle 2Θgiven by the Bragg equation. For all values of Θ that do not satisfy this equation the diffracted rays are out of phase with each other and no reflection is observed.

The interference is constructive when the phase shift is a multiple of 2π

Examples of polymer structuresScattering from a single atom

Scattering from a group of atoms

Which technics to use?

b = scattering length (s. amplitude)

bX-ray = 0.282 x 10-12 cm x number of electrons

bN = tabulated bN(H) = - 0.374 x 10-12 cm

bN(D) = + 0.667 x 10-12 cm

I = b2

O2θ

B

Sr

0Sr

rr

Incident beam

Scattered))(/2( 0ssq

rrr−−−−==== λλλλππππ

Scattering vector

∑ −==k k

rqik

bqI 2|)exp(|2|AmplitudeTotal|)(rrr

I = I(qr) ⇒⇒⇒⇒ {Short distances ⇔⇔⇔⇔ high q (WAXS)

long distances ⇔⇔⇔⇔ small q (SAXS, SANS)}

WAXS on Polypropylene + 50 wt% Starch

∫

∫∞

∞

=

0

2

0

2

d)(

d)(

qqIq

qqIqx

c

cDEGREE OF CRYSTALLINITY

Θ=

cosβλK

LCRYSTALLITE SIZE

ββββ ≡≡≡≡ breadth of the

reflection

WAXS

WAXS onWAXS onPolymersPolymers

FF

DD BB

CC

EE

Size of crystallites

Degree of crystallinity

Distinguishing between ordered

and disordered structures

Lattice parameters

Identification of crystalline phases

AA

Crystal structure (single crystals, fibres)

Interpretation of SAS data

∫∞

=0

22

2 )sin()(

2)( dqq

qr

qrqI

rrp

π

[ ]∑∑= =

−⋅−=N

j

n

kkj rrqi

NqS

1 1

)(exp1

)(

SCA

TT

ER

ING

PA

TT

ER

N STRU

CTU

REI(q)

q

?

Scattering intensity: I(q) = P(q)S(q)

Distance distribution function:

Structure factor for N beads:

Form factor of sphere:2

33

)(

)cos()sin(3

3

4),(

⋅⋅−⋅∆=qR

qRRqqRRRqP ρπ

m

n

ie

V

Zr∑== 1ρ

Scattering length density:

where Z is the atomic numberre

= 2.81 x 10-13 cm, is the classical radius of the electronV

mis molecular volume

Radius of Gyration

∑ −=ji

jig rrN

R,

22

2 )(2

1

22

5

3RRg =

12

22 L

Rg =

Radius of gyration is the name of several related measures of the size of an object, a surface, or an ensemble of points. It is calculated as the root mean square distance of the object’s parts from its center of gravity.

In polymer physics, the radius of gyration is proportional to the root mean square distance between the monomers:

Sphere Thin rod Thin disc Cylinder

2

22 R

Rg =122

222 LR

Rg +=

Interpretation of SAS dataExperimental SAS curve

I exp(q)

Structure parameters

(e.g., Rg, V, S)A priori information

Structure model

I(experiment) =

I(model) ?

STRUCTURE ( ? )

Other techniques

YES

NO

SAXS vs SANS

??? !!!

Range of scattering vectors: q = 10-3 - 10-1 Å -1

Length scale: D = 101 - 103 Å

Scattering density:ρρρρ = b/V

Scattering contrast: ∆ρ∆ρ∆ρ∆ρ(r) = ρρρρ(r) - ρρρρ02

V)dVrqi)exp(r∆ρ()qI( ∫ −=rrrr

Scattering intensity:

ρρρρ0

ρρρρ

ρρρρ0

ρρρρ1ρρρρ1

ρρρρ2

ρρρρ0 ≠≠≠≠ ρρρρ2 ⇒⇒⇒⇒ I(q) = I 12(q)

ρρρρ0

ρρρρ1

ρρρρ2

ρρρρ0 = ρρρρ2 ⇒⇒⇒⇒ I(q) = I 1(q)

Contrast variation for Multicomponent Particles

Scattering from a polymer chain

q-1

I

I=I(0)~q-2

~q-1

Length of scattering vector q

qD«1 qD≅≅≅≅1 qD≥≥≥≥2

∼∼∼∼ exp(-q2Rg2/3)

qL p≥≥≥≥2

D: Size of chainL p: Persistence

length

0

Guinier Debye

Rod-like

„Magnification” increases

SAXS and SANS on polymers

SAXS and SANSSAXS and SANSon polymerson polymers

BB

EE

CC

DD

AA

Solid polymers:Characterization of

heterogeneities (pores, domains in block copolymers, fractal structures,

....)

SANS: Contrast variation. Studying of multicomponent

particles

Polymer particles:Shape, size

(distribution), mass, surface, internal

structure, degree of swelling

Semicrystallinepolymers: Degree of

crystallinity, long period, size of

crystallites.

Polymer chains: Radius of gyration, mass, persistence

length, cross-sectional parameters.

SANS Example

Thank you!and welcome to our laboratory.

UNESCO/IUPAC Postgraduate Course in Polymer Science

•Institute of Macromolecular Chemistry ASCR, Heyrovsky sq. 2, Prague -162 06•http://www.imc.cas.cz/unesco/index.html•unesco.course@imc.cas.cz