Department of Nuclear Methods, Institute of Physics, Maria Curie-Sklodowska University, Lublin, Poland P s bubble in liquids Bo ż ena Zgardzi ń ska

P s BUBBLE MODEL FOR LIQUIDS To describe the size of free volume in liquids for 54 years the bubble 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) R. A. Ferrel, Phys. Rev. 108 (1957) positronium energy in the bubble; -surface tension; -external pressure. positronium energy energy energy of surface of external tension pressure 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. The bubble represents a potential well for Ps (Ps is selftrapped). E Ps depends on R and well depth U

P s BUBBLE MODEL FOR LIQUIDS H2OH2O water The Ps behaviour is more complex than it follows from Eq. 1 (given above) Stepanov et al.: additional processes (radiation chemistry), should be considered. V. M. Byakov, S. V. Stepanov, Rad. Phys. Chem. 58 (2000) 687. S. V. Stepanov, V. M. Byakov, B. N. Ganguly, D. Gangopadhyay, T. Mukherjee, B. Dutta-Roy, Phys. B 322, 68 (2002).

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

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

O -P s LIFETIME IN ALKANES O-Ps lifetime in alkanes as a function of the distance from melting point  C 7 H 16  C 9 H 20  C 13 H 28  C 19 H K ≈1 ns volume, nm 3 0,21 0,33 Size of free volume in the liquid increases by more than 50% at the change of temperature by 150 K The experimental points are arranged along a single curve

O-Ps lifetime in alkanes as a function of the distance from melting point  C 7 H 16  C 9 H 20  C 13 H 28  C 19 H 40 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 -P s LIFETIME IN ALKANES

P s BUBBLE RADIUS IN ALKANES R in alkanes as a function of the distance from melting point.  C 7 H 16  C 9 H 20  C 13 H 28  C 19 H 40 The bubble radius can be found using Tao-Eldrup model. S. J. Tao, J. Chem. Phys. 56, 5499 (1971). M. Eldrup, D. Lightbody, J. N. Sherwood, Chem. Phys. 63 (1982) 51.

How to calculate the radius? First, we have to know E Ps 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. infinitely For infinitely deep well the energy is: (2) finite depth For potential well of finite depth U 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) P s BUBBLE RADIUS IN ALKANES

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

Energy comparison of energy of 1s state in infinite depth of potential well R R+ Δ Vo=1eV Vo=5eV Vo=3eV infinite potential well potential well of finite depth BUBBLE MODEL FOR LIQUIDS POSITRONIUM ENERGY In the well of depth U the E Ps is smaller than in infinite well of the same radius. It is interesting, that if we assume, the values like in Tao-Eldrup model (i.e. R+ Δ, U=∞) E Ps 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. Liquid alkanes

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)

P s BUBBLE MODEL FOR LIQUIDS We obtain the equation of the fourth degree, and there are four solutions, but 3 of them are non-physical (complex or negative). 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)

P s BUBBLE MODEL FOR LIQUIDS

The range for which the surface tension is taken from literature The range for which the surface tension values have been extrapolated  Experimental data  3 calculations ___ C 7 H 16 O -P s lifetime - experiment and calculations Green curve looks like a good approximation, but adding Δ to bubble radius is artifical (not justified). The purple line has the slope exactly like the experimental data.

MICRO- AND MACROSCOPIC SURFACE TENSION For bubbles surface tension depends on the radius of curvature with decreasing radius R, the surface tension σ increases concave convex r-r- r+r+ rr W. S. Ahn, M. S. John, H. Pak, S. Chang, Jurnal of Colloid and Interface Science, Vol. 38, No. 3, p , 1972 H 2 O Benzen Cyclohexan Ar N drop bubble alkanes flat surface

For bubbles: We don’t know the value of d * for alkanes ! so micro-surface tension estimation is difficult (impossible) *d for N 2 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 , 1972 The microscopic surface tension should be greater than the macroscopic one. MICRO- AND MACROSCOPIC SURFACE TENSION

P s BUBBLE MODEL FOR LIQUIDS The range for which the surface tension is taken from literature The range for which the surface tension values have been extrapolated  Experimental data ___  3 calculations ___ C 7 H 16 O -P s lifetime - experiment and calculations

P s BUBBLE MODEL FOR LIQUIDS The range for which the surface tension is taken from literature The range for which the surface tension values have been extrapolated  Experimental data ___  3 calculations ___ C 7 H 16 O -P s lifetime - experiment and calculations Macroscopic surface tension Microscopic surface tension ?

P s BUBBLE MODEL FOR LIQUIDS C 7 H 16 σ·2,86 σ·3,1 C 9 H 20 σ·2,9 σ·3,05 C 6 H 14 Alkanes Correcting coefficient x C 6 H 14 3,05 C 7 H 16 2,86 C 9 H 20 2,9 C 13 H 28 3,1 C 19 H 40 3,1

O -P S LIFETIME IN ALCOHOLS O-Ps lifetime in alcohols as a function of the distance from melting point Size of free volume in the liquid increases by more than 16% at temperature increase by 100 K volume, nm 3 0,18 0,25 Analogous experiments as for the alkanes were carried out with alcohols

O -P s lifetime in alcohols and alkanes O-Ps lifetime as a function of the distance from melting point alkanes alcohols

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). P s BUBBLE RADIUS IN ALCOHOLS R in alcohols as a function of the distance from melting point

ALKANES AND ALCOHOLS O -P s lifetime and decay constant

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

Alkanes and ALcohols The dependence of λ on σ was approximated by straight line fitted to experimental data. The Δλ is drawn here as the difference between fitted curves. In this case the difference of λ is so large that explanation by potential well modification is not sufficient.

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.

Thank you for your attention

n-alkany S. J. Tao, J. Chem. Phys. 56, 5499, (1971). D. M. Schrader, Y. C. Jean, Positron and Positronium Chemistry, Nowy York, Elsevier, str. 174 (1988).

C 11 H 24 C 19 H 40 J. Doucet et al., J. Chem. Phys. 80 (1984) M. Maroncelli et al., J. Am. Chem. Soc. 104 (1982) Koji Nozaki et al., J. Chem. Phys. 103 (1995) T. Goworek et al., Chem. Phys. Letters 394 (2004) 90. liquid ALKANES in liquid PHASE TEMPERATURE, K CHAIN NUMBER (n)

ALKANES in liquid PHASE Rigid phaseRotator phase Liquid phase M. Marconelli, S. P. Qi, H. L. Strauss, J. Am. Cem. Soc. 104, , (1982). T. Goworek, R. Zaleski, J. Wawryszczuk, Chem. Phys. 295, (2003). B. Zgardzińska, R. Zaleski, J. Wawryszczuk, T. Goworek, Phys. Letter A. 333, (2004). H. Z. Li, T. Yamamoto, J. Chem. Phys. 114, (2001). Computer symulation temperature

MEASURING EQUIPMENT Standard „fast – slow” coincydence spectrometer The source activity 0,5 MBq Schematic representation of the construction of the measuring chamber. Cu LN 2 heater thermocouple AC HEATER THERMOREGULATOR POWER LED POMP KOMPUTER Rs-232 LED diod source and sample ”cold finger” dewar with nitrogen

SURFACE TENSION, dyn/cm J.J. Jasper, The Surface Tension of Pure Liquid Compounds, WSU, Detroit, Michigan, 48202, J. Phys. Chem. Ref. Data, Vol. 1, No. 4, temperature Linear dependence of surface tension on temperature for the alkanes with a specified number of carbons in the molecule σ in temp. 0 0 C TEMPERATURE, O C Surface tension Macroscopic surface tension

BUBBLE MODEL FOR LIQUIDS There are four solutions, but 3 of them are non-physical.

convex concave d for N is about 0,3 nm W. S. Ahn, M. S. John, H. Pak, S. Chang, Jurnal of Colloid and Interface Science, Vol. 38, No. 3, , 1972 r, nm 14,81,05 21,49 11,70 boubble drop Melrose J (1966) Amer.Inst.Chem. Eng.12, 986 Micro- and macroscopic surface tension

Surface tension as a function of distance from the melting point for the alkanes with a specified number of carbons in the molecule 12_Nap(T-Tm)_ALLalkany.grf Surface tension

As the change of  ∞ to  R alters ?  C 7 H 16  C 13 H 28 S. J. Tao, J. Chem. Phys. 56, 5499, (1971). D. M. Schrader, Y. C. Jean, Positron and Positronium Chemistry, Nowy York, Elsevier (1988). C 13 H 28  0,0800,050  0,42 C 7 H 16  0,0720,046  0,44 Pick-off annihilation rate vs. surface tension

About and σ S. J. Tao, J. Chem. Phys. 56, 5499, (1971). D. M. Schrader, Y. C. Jean, Positron and Positronium Chemistry, Nowy York, Elsevier (1988). number of C atoms  10,0930,36 20,1000,33 40,1660,21 50,1550,23 60,1200,30 90,0980,37 130,0870,40

About and σ S. J. Tao, J. Chem. Phys. 56, 5499, (1971). D. M. Schrader, Y. C. Jean, Positron and Positronium Chemistry, Nowy York, Elsevier (1988). alkanes alcohols