department of nuclear methods, institute of physics, maria curie-sklodowska university, lublin,...

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

Lublin, Poland

Ps bubble in liquids

Bożena Zgardzińska

Ps BUBBLE MODEL FOR LIQUIDS

p

REPs

)(

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) 167.

03/44)( 32 pRRREdRd

Ps

- positronium energy in the bubble;- surface tension;- external pressure.

positronium energy energy energy of surface of external tension pressure

),( UREPsThe 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) . EPs depends on R and well depth U

The subject of this work is:

How Ps behaves in liquid alkanes and their derivatives?

Ps BUBBLE MODEL FOR ALKANES

EXPERIMENT - ALKANES

O-Ps lifetime in alkanes as a function of temperature.

18 0 20 0 22 0 24 0 26 0 28 0 30 0 32 0 34 0 36 0 38 0 40 0T E M P E R A T U R E , K

2 .8

3 .2

3 .6

4 .0

3

ns

m .p.C 7H 16

m .p.C 9H 20

m .p.C 13H 28

m .p.C 19H 40

o-Ps lifetime increases with temperature

O-Ps LIFETIME IN ALKANES

0 40 80 12 0 160T -T m , K

2 .8

3 .2

3 .6

4 .0

4 .4

3

ns

O-Ps lifetime in alkanes as a function of the distance from melting point

C7H16

C9H20

C13H28

C19H40

150 K≈1

ns

volume, nm30,21 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

0 4 0 8 0 12 0 1 60T -T m , K

2 .8

3 .2

3 .6

4 .0

4 .4

3

ns

O-Ps lifetime in alkanes as a function of the distance from melting point

C7H16

C9H20

C13H28

C19H40

0 20 40 60 80 100 120 140 160T -T m , K

16

20

24

28

SUR

FAC

E TE

NSI

ON

, dyn

/cm

alkane m elting point

C 7H 16 -90 ,5oC

C 9H 20 -53oC

C 11H 24 -25oC

C 13H 28 -5oC

C 19H 40 30 ,5oC

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

Ps BUBBLE RADIUS IN ALKANES

R in alkanes as a function of the distance from melting point.

C7H16

C9H20

C13H28

C19H40

0 4 0 8 0 1 2 0 1 6 0T -T m , K

0 .3 6

0 .3 8

0 .4 0

0 .4 2

0 .4 4

0 .4 6R

, nm

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 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 infinitelyinfinitely deep well the energy is:

(2)

For potential well of finite depthfinite 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

massmpositroniummRm

RE

ePs

PsPs

22

, 2

22

L. I. Shiff, Quantum Mechanics, McGraw Hill, N.Y. (1968).R. Zaleski, dissertationY. Morinaka, Y. Nagashima, Y. Nagai, T. Hyodo, T. Kurihara, T. Shidara, K. Nakahara, Mat. Sci. Forum 689 (1997) 255-257.

03/44)( 32 pRRREdRd

Ps

Ps BUBBLE RADIUS IN ALKANES

R

R+Δ

Vo=1eV

Vo=5eV

Vo=3eV

Energy of 1s state in spherical geometry for different depth of potential well

infinite potentia

l well

potential well of finite depth

BUBBLE MODEL FOR LIQUIDSPOSITRONIUM ENERGY

0 0 .2 0 .4 0 .6 0 .8 1R , n m

0

1

2

3

4

5

6

E, e

V

R in a lkanes

0 0 .2 0 .4 0 .6 0 .8 1R , n m

0

1

2

3

4

5

6

E, e

V

R in a lkanes

Liqu

id a

lkan

es

0 .2 0 .3 0 .4 0 .5R , nm

0

0 .2

0 .4

0 .6

0 .8

1

E/E(

R,

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 LIQUIDSPOSITRONIUM ENERGY

0 .2 0 .3 0 .4 0 .5R , n m

0

0 .2

0 .4

0 .6

0 .8

1

E/E(

R,

In the well of depth U the EPs 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=∞) 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.

Liquid alkanes

ENERGY OF EXTERNAL PRESSURE

03/44)( 32 pRRREdRd

Ps

04 2 RRER Ps

oARIf:

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)

2310AeV

3610AeVp catmospheri

33

2

103/4

4

pRR


04 2 RRER Ps

nm

massmpositroniummnminRwheneVRRm

RE ePsPs

Ps

166,0

21879,02 22

22

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

04

22

2

22

R

RmR Ps

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)


0 20 40 60 80 10 0 12 0 14 0T -T m , K

3

4

5

6

7

8

3

ns

0 20 40 60 80 10 0 12 0 14 0T -T m , K

3

4

5

6

7

8

3

ns

0 20 40 60 80 10 0 12 0 14 0T -T m , K

3

4

5

6

7

8

3

ns

The range for which the surface tension is taken from literature

The range for which the surface tension values have been extrapolated

Experimental data3 calculations

___ 04 2 RRER

___ 04 2 RRER

___ 04 2 RRER

C7H16

O-Ps lifetime - experiment and calculations

Green curve looks like a good approximation, but

adding Δ to bubble radius is artifical (not

justified).

0 20 40 60 80 10 0 12 0 14 0T -T m , K

3

4

5

6

7

8

3

ns

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 curvaturewith decreasing radius R, the surface tension σ increases

concave convexr-r+r

W. S. Ahn, M. S. John, H. Pak, S. Chang, Jurnal of Colloid and Interface Science, Vol. 38, No. 3, p.605-608, 1972

-0 .0 6 -0 .0 4 -0 .02 0 0 .0 2 0 .0 4 0 .0 61 /r , 1 /A

0

20

40

60

80

10 0

H2O

BenzenCyclohexan

ArN

drop

bubble

alkanes

flat surface

http://www.pbase.com/helenpb/challenges_gallery&page=4

0 1 2 3r/2 d

0 .0

2 .0

4 .0

6 .0

8 .0

1 0 .0/

For bubbles:

We don’t know the value of d *

for alkanes !so

micro-surface tension

estimation is difficult

(impossible)

rd21

1

*d for N2 is about 0,3 nmJ. 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

drfor 2

drdrfor 2

2

The microscopic surface tension

should be greater than the

macroscopic one.

MICRO- AND MACROSCOPIC SURFACE TENSION


0 20 40 60 80 10 0 12 0 14 0T -T m , K

3

4

5

6

7

8 3

ns

0 20 40 60 80 10 0 12 0 14 0T -T m , K

3

4

5

6

7

8 3

ns

0 20 40 60 80 10 0 12 0 14 0T -T m , K

3

4

5

6

7

8 3

ns

0 20 40 60 80 10 0 12 0 14 0T -T m , K

3

4

5

6

7

8 3

ns

0 20 40 60 80 10 0 12 0 14 0T -T m , K

3

4

5

6

7

8 3

ns



Experimental data

___

3 calculations

___ 04 2 RRER

___ 04 2 RRER

___ 04 2 RRER

086,24 2 RRER

C7H16



0 20 40 60 80 10 0 12 0 14 0T -T m , K

3

4

5

6

7

8 3

ns

0 20 40 60 80 10 0 12 0 14 0T -T m , K

3

4

5

6

7

8 3

ns

0 20 40 60 80 10 0 12 0 14 0T -T m , K

3

4

5

6

7

8 3

ns

0 20 40 60 80 10 0 12 0 14 0T -T m , K

3

4

5

6

7

8 3

ns

0 20 40 60 80 10 0 12 0 14 0T -T m , K

3

4

5

6

7

8 3

ns



Experimental data

___

3 calculations

___ 04 2 RRER

___ 04 2 RRER

___ 04 2 RRER

086,24 2 RRER

C7H16


Macroscopic

surface tension

Microscopic surface

tension?

Ps BUBBLE MODEL FOR LIQUIDSC7H16

0 20 4 0 60 8 0 1 00T -T m , K

3 .0

3 .2

3 .4

3 .6

3 .8

3

ns

C 19H 40

0 2 0 4 0 6 0 80 1 0 0T -T m , K

3 .0

3 .2

3 .4

3 .6

3 .8

3

ns

C 13H 28

σ·2,86

σ·3,1 σ·3,1

C9H20

σ·2,9

0 40 8 0 12 0 16 0T -T m , K

3 .2

3 .6

4 .0

4 .4

3

ns

0 40 8 0 12 0T -T m , K

2 .8

3 .2

3 .6

4 .0

4 .4

3

ns

0 40 80 12 0 1 60T -T m , K

3 .2

3 .6

4 .0

4 .4

3

ns

σ·3,05

C6H14

Alkanes

Correcting coefficient

xC6H14 3,05C7H16 2,86C9H20 2,9C13H28 3,1C19H40 3,1

O-PS 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, nm30,18 0,25

0 2 0 40 6 0 8 0 1 0 0 1 2 0 14 0 16 0 1 80 200T -T m , K

2 .8

3 .0

3 .2

3 .4

3 .6

3 .8

3

ns

C H 3O H m ethanol C 2H 5O H e thanol C 4H 9O H buthano l C 5H 11O H penthano l C 6H 13O H hexanol C 9H 19O H nonanol C 13H 27O H tridecano l

Analogous experiments as for the alkanes were carried out with 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).

Ps BUBBLE RADIUS IN ALCOHOLS

R in alcohols as a function of the distance from melting point

0 20 4 0 6 0 80 10 0 12 0 14 0 16 0 18 0 20 0T -T m , K

1 5

2 0

2 5

3 0

3 5

4 0

SUR

FAC

E TE

NSI

ON

, dyn

/cm

0 20 40 6 0 8 0 1 00 12 0 14 0 16 0 18 0 20 0T -T m , K

0 .34

0 .36

0 .38

0 .40

0 .42

R, n

m

C H 3O H m ethanol C 2H 5O H ethano l C 4H 9O H buthano l C 5H 11O H penthano l C 6H 13O H hexanol C 9H 19O H nonano l C 13H 27O H tridecanol

ALKANES AND ALCOHOLS

0 40 8 0 1 2 0 160T -T m , K

2 .8

3 .2

3 .6

4 .0

4 .4

3

ns

alkanes a lcohols C 6H 1 4 C 6H 13O H C 9H 2 0 C 9H 19O H C 13H 28 C 13H 27O H

0 40 8 0 12 0 1 6 0T -T m , K

0 .2 4

0 .2 8

0 .3 2

0 .3 6

n

s

alkanes a lcohols C 6H 14 C 6H 13O H C 9H 20 C 9H 19O H C 13H 28 C 13H 27O H

O-Ps lifetime and decay constant

Δ0,024 Δ0,015

%5,7 %5

alkanealcohol

For given σ the values of λ for alcohol are shifted (upwards).Comparing to respective alkane

12 1 6 20 24 2 8 3 2, d y n /cm

0 .24

0 .28

0 .32

0 .36

n

s

1 2 16 20 24 28 32, d y n /cm

0 .2 4

0 .2 8

0 .3 2

0 .3 6

n

s

alkane a lcohol C 9H 20 C 9H 19O H


curves run

parallel


Δ0,024 Δ0,015

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.

12 1 6 20 24 2 8 3 2, d y n /cm

0 .24

0 .28

0 .32

0 .36

n

s

1 2 16 20 24 28 32, d y n /cm

0 .2 4

0 .2 8

0 .3 2

0 .3 6

n

s




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.

alkanesforxxR 3,

Thank you for your attention

department of nuclear methods, institute of physics, maria curie-sklodowska university, lublin,...

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