department of nuclear methods, institute of physics, maria curie-sklodowska university, lublin,...
DESCRIPTION
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).TRANSCRIPT
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
Ps BUBBLE MODEL FOR LIQUIDS
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)
Ps BUBBLE MODEL FOR LIQUIDS
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
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
Ps BUBBLE MODEL FOR LIQUIDS
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
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
___ 04 2 RRER
___ 04 2 RRER
___ 04 2 RRER
086,24 2 RRER
C7H16
O-Ps lifetime - experiment and calculations
Ps BUBBLE MODEL FOR LIQUIDS
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
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
___ 04 2 RRER
___ 04 2 RRER
___ 04 2 RRER
086,24 2 RRER
C7H16
O-Ps lifetime - experiment and calculations
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
alkane a lcohol C 13H 28 C 13H 27O H
curves run
parallel
ALKANES AND ALCOHOLS
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
alkane a lcohol C 9H 20 C 9H 19O H
alkane a lcohol C 13H 28 C 13H 27O H
ALKANES AND ALCOHOLS
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