Shapes of the free volume holes in amorphous polymers as estimated by PALS

1. Shapes of the free volume holes in amorphous polymers as estimated by PALS G. Consolati Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, Via La Masa, 34 – 20156 Milano – Italy 3rd Jagiellonian Symposium on fundamental and applied subatomic physics, Krakow, 23 – 28 June 2019

2. Outlook 1. Introduction 2. The methodology 3. Some results 4. Conclusions

3. PALS: effective tool to detect intermolecular spaces. Probe: positronium (Ps), formed by a positron and an electron of the medium In many materials e+ and e- may form Ps in spaces with reduced electron density (cavities, pores, holes)

4. In condensed matter ortho-Ps decays also by ‘pickoff’ o-Ps lifetime may greatly reduce, from 142 ns up to about a few ns Annihilation by pickoff occurs in two gammas 511 keV 511 keV

5. An example of timing spectrum: PET

6. The Tao-Eldrup equation • Hole: spherical void • ΔR describes the penetration of the Ps wavefunction into the bulk • Electron density: 0 for r < R; constant for r > R. The equation: Relation lifetime-radius: τ3= 1/λ3 = 1/ (λp + λi)

7. Other relationships for non-spherical holes • Prism with square cross section (Jasinska, B., Koziol, A.E., Goworek, T., 1996. J. Radioanal. Nucl. Chem. 210, 617 – 623). a : side of the square cross section; s = height k = s/a: aspect ratio

8. Finite cylinder with radius R and height u: (Olson, B. G.; Prodpran, T.; Jamieson, A. M.; Nazarenko, S. 2002.Polymer 43: 6775 – 6784).

9. Use of non-spherical geometries - Prisms and layered cavities were used in studies of clays • Cubic structures were used in discussing pore size distributions in low k dielectric thin films. • Ellipsoidal holes were introduced to explain the free volume in semicrystalline polymers subjected to tensile deformation. In polymers spherical holes are the standard, although their shapes are irregular. Are all the geometries equivalent in terms of free volume?

10. Estimation of the free volume fraction • V = N’vh + Vocc • f = N’vh/ (N’vh + Vocc) N’ andVoccfromdilatometry PALS f h Dilatometry Simha-Somcynski EOS

11. The Simha-Somcynsky eos: T*,V*: scaling parameters, obtained by fitting SS eos to dilatometric data. y: volume fraction of occupied sites, h=1-y: hole fraction.

12. A second equation is obtained minimizing the Helmoltz free energy (polymer: at equilbrium) Fitting the two equations with V – T data: T* , V*

13. Example: o-Ps lifetime in two rubbers Fluoro rubber cis 1,4-polyisoprene

14. Specific volume from dilatometry Fluoro rubber cis 1,4 polyisoprene

15. Fitting the S-S e.o.s. • Fluoro rubber: V* = 0.5163 ± 0.0003 cm3 g-1 T* = 9161 ± 26 K • Cis 1,4 polyisoprene: V* = 1.0535 ± 0.0004 cm3 g-1 T* = 9547 ± 29 K.

16. Vsp vs. vh (spherical shapes) Slope: number density of holes N’ • V = N’vh + Vocc Intercept: occupied volume

17. f = N’vh / (N’vh + Vocc) • Building the free volume fraction f: • and comparison with the theoretical fraction h:

18. Fit with cylindrical holes • aspect ratio (heigth/radius): ξ = h/r (free parameter) 1) For a given ξ , obtain the hole volume from τ3 2) Plot Vsp vs. vh to get N’ and Vocc 3) Build f and compare with h 4) Repeat by changing ξ to find a best fit • ξ < 1: flattened holes; ξ > 1: elongated holes.

19. Fit with cylindrical elongated holes • aspect ratio (heigth/radius): • 2.4 (fluoro rubber) • 4 (isoprene)

20. Example: polyether-polyester polyurethane

21. Perfluoropolyethers: puzzling results HO-CH2CF2-O-(CF2CF2-O-)a-(CF2-O)b-CF2CH2-OH

22. Perfluoropolyethers:

23. o-Ps lifetime in perfluoropolyethers

24. Specific volume vs. vh

26. A way to tackle the problem Isotropic growth:

27. Alternative growths: Cylinder with fixed height a0 : Cylinder with fixed radius s0:

28. Anisotropic growth: Model of anisotropic cylinder: a s k = 1: isotropic cylinder initial aspect ratio r = a0/s0

29. Empty circles: anisotropic holes. Full squares: isotropic holes.

30. Exponent p, hole density N’, experimental (Vocc) and theoretical (vo,th) occupied volumes in perfluoropolyethers

31. Free volume fractions in polypropylene glycols Full squares: isotropic holes. Empty circles: anisotropic holes.

32. Poly(ester-adipate)urethane Full squares: spherical holes. Empty circles: anisotropic holes (p = 2.3) .

33. Conclusions • N’ and hole volume depend on the geometry • The amount of free volume depends on the geometry • The same o-Ps lifetime can produce different free volume fractions, according to the adopted model • Comparison between f and h allows us: • to gain insight on hole (approximate) shape • to check the guess of isotropic growth • Elongated holes are in agreement with molecular simulations of polymers • Anisotropic growth: crude model, but compatible with dynamics of macromolecules

34. Thank you for your attention!

35. Back up slides

36. The beta spectrum of 22Na Radioactive source: 22Na decay Source: prepared from a droplet of 22NaCl aqueous solution, (1-20 μCi) deposited between two Kapton foils, 7 μm each, glued together. Inserted in the sample in a ‘sandwich’ configuration. Thickness: generally between 1 to 2 mm.

37. The timing spectrometer

38. A discrete component: τ: lifetime; f: intensity A distribution of lifetimes: τ: centroid; σ: second moment; f: intensity

39. The experimental spectrum S(t) (only discrete components): • R(t): resolution function • B: background • Ii: intensity of the i-th component • τi: lifetime of the i-th component τpara≈ 0.15 ns;τfree≈ 0.4 ns;τortho≈ 1 – 10 ns

40. The dilatometer: - a bulb-capillary tube system - small pieces of the sample are immersed in mercury - a thermostatic bath grants a constant temperature - level variations in the capillary tube are viewed by means of a telescopic system • Variations of volume are tranformed into absolute values by knowing the specific volume at room temperature (buoyancy method)