Department of Nuclear Methods, Institute of Physics, Maria Curie-Sklodowska University,

1. Department of Nuclear Methods, Institute of Physics, Maria Curie-Sklodowska University, Lublin, Poland Ps bubble in liquids Bożena Zgardzińska

2. Ps BUBBLE MODEL FOR LIQUIDS To describe the size of free volume in liquids for 54 years thebubble model proposed by Ferrel is in common use. The zero point motion of the particle creates a spherical cavity around this particle. The equilibrium radius corresponds to the minimum of energy : (1) • positronium energy in the bubble; • surface tension; • external pressure. positronium energy energy energy of surface of external tension pressure The bubble represents a potential well for Ps (Ps is selftrapped) . EPs depends on R and well depth U The surface tension decreases with increasing temperature, hence the size of the bubble should increase (with increasing temperature), and the o-Ps lifetime increases too. R. A. Ferrel, Phys. Rev. 108 (1957) 167.

3. Ps BUBBLE MODEL FOR ALKANES The subject of this work is: How Ps behaves in liquid alkanes and their derivatives?

4. EXPERIMENT - ALKANES o-Ps lifetime increases with temperature O-Ps lifetime in alkanes as a function of temperature.

5. O-Ps LIFETIME IN ALKANES The experimental points are arranged along a single curve  C7H16  C9H20  C13H28  C19H40 ≈1 ns Size of free volume in the liquid increases by more than 50% at the change of temperature by 150 K 150 K 0,21 0,33 volume, nm3 O-Ps lifetime in alkanes as a function of the distance from melting point

6. O-Ps LIFETIME IN ALKANES  C7H16  C9H20  C13H28  C19H40 Surface tension as a function of distance from the melting point for some alkanes, of the same lengths of carbon chain as in our experiment (left). O-Ps lifetime in alkanes as a function of the distance from melting point

7. Ps BUBBLE RADIUS IN ALKANES The bubble radius can be found using Tao-Eldrup model.  C7H16  C9H20  C13H28  C19H40 R in alkanes as a function of the distance from melting point. S. J. Tao, J. Chem. Phys. 56, 5499 (1971). M. Eldrup, D. Lightbody, J. N. Sherwood, Chem. Phys. 63 (1982) 51.

8. Ps BUBBLE RADIUS IN ALKANES How to calculate the radius? First, we have to know EPs Inside the bubble electron density is zero; outside – assumed constant. The molecular forces are very shortranged. Rectangular potential well seems to be a good approximation. The radius of electron-less sphere we denote R. For infinitely deep well the energy is: (2) For potential well of finite depthU one can calculate the energy, however, no analytical formula for E(R), needed to differentiate it in Equation (1). There are very few data about the real depth of potential well. It can be estimated for solids from Ps time-of-flight experiments. Morinaka et al. give the values in the range (1-3) eV. R L. I. Shiff, Quantum Mechanics, McGraw Hill, N.Y. (1968). R. Zaleski, dissertation Y. Morinaka, Y. Nagashima, Y. Nagai, T. Hyodo, T. Kurihara, T. Shidara, K. Nakahara, Mat. Sci. Forum 689 (1997) 255-257.

9. BUBBLE MODEL FOR LIQUIDS POSITRONIUM ENERGY infinite potential well potential well of finite depth Vo=1eV R Liquid alkanes Vo=3eV R+Δ Vo=5eV Energy of 1s state in spherical geometry for different depth of potential well

10. BUBBLE MODEL FOR LIQUIDS POSITRONIUM ENERGY infinite potential well potential well of finite depth In the well of depth U the EPs is smaller than in infinite well of the same radius. Vo=1eV R Vo=3eV Liquid alkanes Energy comparison of energy of 1s state in infinite depth of potential well Vo=5eV R+Δ It is interesting, that if we assume, the values like in Tao-Eldrup model (i.e. R+Δ, U=∞) EPs is very close to that for R and U=1 eV. Probably the real U is rather close to 1 eV (see eg. Mogensen’s estimate for liquid benzene, U=0,961 eV) O. E. Mogensen, F. M. Jacobsen, Chem. Phys. 73 (1982) 223.

11. ENERGY OF EXTERNAL PRESSURE If: then: and So for R of several Å: At moderate pressures the last term can be neglected, and the equilibrium radius corresponds to the minimum of energy: (3)

12. Ps BUBBLE MODEL FOR LIQUIDS Let us assume, for convenience, that the depth of potential well is 1 eV and then (instead of real E vs. R dependence), we approximate E vs. R by that for infinitely deep well broadened by Δ. (3) We obtain the equation of the fourth degree, and there are four solutions, but 3 of them are non-physical (complex or negative).

13. Ps BUBBLE MODEL FOR LIQUIDS O-Ps lifetime - experiment and calculations Experimental data C7H16 3 calculations ___ ___ ___ Green curve looks like a good approximation, but adding Δ to bubble radius is artifical (not justified). The range for which the surface tension values have been extrapolated The range for which the surface tension is taken from literature The purple line has the slope exactly like the experimental data.

14. MICRO- AND MACROSCOPIC SURFACE TENSION For bubbles surface tension depends on the radius of curvature with decreasing radius R, the surface tensionσ increases bubble H2O Benzen Cyclohexan Ar N drop alkanes r+ r- r convex concave flat surface W. S. Ahn, M. S. John, H. Pak, S. Chang, Jurnal of Colloid and Interface Science, Vol. 38, No. 3, p.605-608, 1972

15. MICRO- AND MACROSCOPIC SURFACE TENSION For bubbles: We don’t know the value of d * for alkanes ! so micro-surface tension estimation is difficult (impossible) The microscopic surface tension should be greater than the macroscopic one. *d for N2 is about 0,3 nm J. Melrose, Amer.Inst.Chem. Eng.12 (1966) 986. W. S. Ahn, M. S. John, H. Pak, S. Chang, Jurnal of Colloid and Interface Science, Vol. 38, No. 3, p.605-608, 1972

16. Ps BUBBLE MODEL FOR LIQUIDS O-Ps lifetime - experiment and calculations Experimental data C7H16 3 calculations ___ ___ ___ ___ The range for which the surface tension values have been extrapolated The range for which the surface tension is taken from literature

17. Ps BUBBLE MODEL FOR LIQUIDS O-Ps lifetime - experiment and calculations Experimental data C7H16 Macroscopic surface tension 3 calculations ___ ___ ___ ___ The range for which the surface tension values have been extrapolated The range for which the surface tension is taken from literature Microscopic surface tension?

18. Ps BUBBLE MODEL FOR LIQUIDS C7H16 C6H14 C9H20 σ·3,05 σ·2,86 σ·2,9 σ·3,1 σ·3,1

19. O-PS LIFETIME IN ALCOHOLS Analogous experiments as for the alkanes were carried out with alcohols Size of free volume in the liquid increases by more than 16% at temperature increase by 100 K 0,18 0,25 volume, nm3 O-Ps lifetime in alcohols as a function of the distance from melting point

20. Ps BUBBLE RADIUS IN ALCOHOLS R in alcohols as a function of the distance from melting point Surface tension as a function of distance from the melting point for some alcohols, of the same lengths of carbon chain as in our experiment (left).

21. ALKANES AND ALCOHOLS O-Ps lifetime and decay constant

22. ALKANES AND ALCOHOLS Δ0,024 Δ0,015 curves run parallel For given σ the values of λ for alcohol are shifted (upwards). Comparing to respective alkane

23. ALKANES AND ALCOHOLS Δ0,024 Δ0,015 Difference in λ for alkane and alcohol means, that beside surface tension other factors play the role: - Radiation chemical reactions (with the rate chem = Δλ); - Difference of potential well depth U. If U is of the order of (1-1,5) eV, the shift of λ by 0,015 ns-1 corresponds (very rough estimate) to the reduction of U in alcohol by about 0,3 eV.

24. CONCLUSIONS • The positronium lifetimes as a function of temperature above the melting point are identical for all alkanes under study; • Best fit of model to the experiment , we get assuming: • - infinite potential well of radius R+Δ; • - taking into account the surface tension • The difference in the values of decay constants for alcohols and alkanes at the same surface tension is approximately constant. This can be the result of: • - radiation chemical reactions; • - difference of potential well depth U.

25. Thank you for your attention