Shapes in real space ––> reciprocal space

Shapes in real space ––> reciprocal space(see Volkov & Svergun, J. Appl. Cryst. (2003) 36, 860-864. Uniqueness of ab initio

shape determination in small-angle scattering)

Can compute scattering patterns for different shape particles for isotropic dilute monodisperse systems

Shapes in real space ––> reciprocal space(see Volkov & Svergun, J. Appl. Cryst. (2003) 36, 860-864. Uniqueness of ab initio

shape determination in small-angle scattering)

Can compute scattering patterns for different shape particles for isotropic dilute monodisperse systems

Approach 1 (small number of parameters)

Represent particle shape by an envelope fcn – spherical harmonics

Shapes in real space ––> reciprocal space(see Volkov & Svergun, J. Appl. Cryst. (2003) 36, 860-864. Uniqueness of ab initio

shape determination in small-angle scattering)

Can compute scattering patterns for different shape particles for isotropic dilute monodisperse systems

Approach 1 (small number of parameters)

Represent particle shape by an envelope fcn – spherical harmonics

Spherical harmonics fcns are angular part of soln to wave eqn

Of the form

Shapes in real space ––> reciprocal spaceApproach 1 (small number of parameters)

Spherical harmonics fcns are angular part of soln to wave eqn

Of the form

Shapes in real space ––> reciprocal space

Approach 2 (large number of parameters)

Represent particle shape by assembly of beads in confinedvolume (sphere)

Beads are either particle (X =1) or 'solvent' (X =0)

To get scattered intensity:

Shapes in real space ––> reciprocal space

bead 'annealing'envelope

Shapes in real space ––> reciprocal space

bead 'annealing'

Shapes in real space ––> reciprocal space

bead 'annealing'envelope

Shapes in real space ––> reciprocal space

bead 'annealing'

Syndiotactic polystyrene(see Barnes, McKenna, Landes, Bubeck, & Bank, Polymer Engineering & Science (1997) 37, 1480. Morphology of syndiotactic polystyrene as examined by small angle scattering)

Semicrystalline PS

Syndiotactic polystyrene(see Barnes, McKenna, Landes, Bubeck, & Bank, Polymer Engineering & Science (1997) 37, 1480. Morphology of syndiotactic polystyrene as examined by small angle scattering)

Semicrystalline PS

Expect peaks in scattering data typical of lamellar structure

Syndiotactic polystyrene(see Barnes, McKenna, Landes, Bubeck, & Bank, Polymer Engineering & Science (1997) 37, 1480. Morphology of syndiotactic polystyrene as examined by small angle scattering)

Semicrystalline PS

Expect peaks in scattering data typical of lamellar structure

non-q–4 slope dueto mushy interface

Syndiotactic polystyreneSemicrystalline PS

Propose absence of peaks due to nearly identical scattering densities of amorphous & crystalline regions

High temperature saxs measurements done

Syndiotactic polystyreneSemicrystalline PS

Propose absence of peaks due to nearly identical scattering length densities of amorphous & crystalline regions

High temperature saxs measurements done

Syndiotactic polystyreneSemicrystalline PS

lamellar thickness = 18 nm

averages of intensity data around azimuth

Syndiotactic polystyreneSemicrystalline PS