1 / 22

Model structures

Model structures. (Read Roe, Chap. 5) Remember:  ( r ) ––> I ( q ) possible I ( q ) ––>  ( r ) not possible. Model structures. (Read Roe, Chap. 5) Remember:  ( r ) ––> I ( q ) possible I ( q ) ––>  ( r ) not possible I ( q ) ––>    ( r ) only

nissim-wall
Download Presentation

Model structures

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Model structures (Read Roe, Chap. 5) Remember: (r) ––> I (q) possible I (q) ––> (r) not possible

  2. Model structures (Read Roe, Chap. 5) Remember: (r) ––> I (q) possible I (q) ––> (r) not possible I (q) ––> (r) only Thus, use a reasonable model to fitI (q) or (r)

  3. Model structures (Read Roe, Chap. 5) Remember: (r) ––> I (q) possible I (q) ––> (r) not possible I (q) ––> (r) only Thus, use a reasonable model to fitI (q) or (r): dilute particulate system non-particulate 2-phase system soluble blend system periodic system

  4. Model structures • Use reasonable model to fitI (q) or (r): • Dilute particulate system • polymers, colloids • dilute - particulates not correlated • if particulate shape known, can calc I (q) • if particulate shape not known, can calc • radius of gyration

  5. Model structures • Use reasonable model to fitI (q) or (r): • Non-particulate 2- phase system • 2 matls irregularly mixed - no host or matrix - • crystalline & amorphous polymer phases • also, particulate system w/ no dilute species • get state of dispersion, domain size, info on • interphase boundary

  6. Model structures • Use reasonable model to fitI (q) or (r): • soluble blend system • single phase, homogeneous but disordered • two mutually soluble polymers, solvent + solute • get solution properties

  7. Model structures • Use reasonable model to fitI (q) or (r): • periodic system • crystalline matls, semi-crystalline polymers, block • copolymers, biomaterials • crystallinity poor • use same techniques as in high angle diffraction, • except structural imperfection prominent

  8. Model structures • Dilute particulate system

  9. Model structures • Dilute particulate system

  10. Model structures • Dilute particulate system

  11. Model structures • Dilute particulate system • dilute - particulates not correlated • matrix presents only uniform bkgrd • Itotal = Iindividual particle • isotropic • Radius of gyration, Rg: • Rg2= ∫r2 (r) dr/∫(r) dr • (r) = scattering length density distribution • in particle

  12. Model structures • Dilute particulate system • Radius of gyration, Rg: • Rg2= ∫r2 (r) dr/∫(r) dr • (r) = scattering length density distribution • in particle • If scattering length density is constant thru-out particle: • Rg2= (1/v)∫r2 (r) dr • (r) = shape fcn of particle; v = particle volume

  13. Model structures • Dilute particulate system • Simple shapes • For a single particle • A(q)= ∫(r) exp (-iqr) dr • I (q) = A2(q) • Average over all orientations of the particle v

  14. Model structures • Dilute particulate system • For a single particle • A(q)= ∫(r) exp (-iqr) dr • I (q) = A2(q) • Sphere: • (r) = for r ≤ R & 0 elsewhere • Then • A(q)= ∫(r) 4πr2 (sin (qr))/qr dr • A(q)=  ∫(r) 4πr sin (qr) dr • A(q)= (3v/(qR)3)(sin (qR) - qR cos (qR)) v ∞ o R o

  15. Model structures • Dilute particulate system • For a single particle • A(q)= ∫(r) exp (-iqr) dr • I (q) = A2(q) • Sphere: v

  16. Model structures • Dilute particulate system • Thin rod, length L, ∠ betwn q& rod axis =  • I(q) = (v)2(2/qL cos )2 sin2 ((qL/2)cos )

  17. Model structures • Dilute particulate system • Thin rod, length L, ∠ betwn q& rod axis =  • I(q) = (v)2(2/qL cos )2 sin2 ((qL/2)cos ) • Averaging over all orientations • I(q) = (v)2 2/qL (∫ (sin u)/u du - (1- cos qL)/ qL) qL o

  18. Model structures • Dilute particulate system • Thin disk, radius R • I(q) = (v)2 2/(qR)2 (1- (J1(2qR))/qR) 1st order Bessel fcn

  19. Model structures • Dilute particulate system • Polymer chain w/ N + 1 independent scattering "beads" • Gaussian - w/ one end of polymer chain at origin, • probability of other end at drobeys Gaussian distribution • bead volume is vu ; chain volume is v = (N + 1) vu

  20. Model structures • Dilute particulate system • Polymer chain w/ N + 1 independent scattering "beads" • Gaussian - w/ one end of polymer chain at origin, • probability of other end at drobeys Gaussian distribution • bead volume is vu ; chain volume is v = (N + 1) vu • scattering length of each bead is vu • A(q) = vu-iqrj) N+1 j=0

  21. Model structures • Dilute particulate system • Polymer chain w/ N + 1 independent scattering "beads" • Gaussian - w/ one end of polymer chain at origin, • probability of other end at drobeys Gaussian distribution • bead volume is vu ; chain volume is v = (N + 1) vu • scattering length of each bead is vu • A(q) = vu-iqrj) • I(q) = (vu)2 ∫P(r)-iqr)dr • P(r) is # bead pairs r apart N+1 j=0

  22. Model structures • Dilute particulate system • Polymer chain w/ N + 1 independent scattering "beads" • Gaussian - w/ one end of polymer chain at origin, • probability of other end at drobeys Gaussian distribution • bead volume is vu ; chain volume is v = (N + 1) vu • scattering length of each bead is vu • A(q) = vu-iqrj) • I(q) = (vu)2 ∫P(r)-iqr)dr • P(r) is # bead pairs r apart • Averaging over all such chains: • I(q) = (vu)2 ∫P(r)-iqr)dr N+1 j=0

More Related