department of nuclear methods, institute of physics, maria curie-sklodowska university,
DESCRIPTION
Department of Nuclear Methods, Institute of Physics, Maria Curie-Sklodowska University, Lublin, Poland. P s bubble in liquids Bożena Zgardzińska. P s BUBBLE MODEL FOR LIQUIDS. - PowerPoint PPT PresentationTRANSCRIPT
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 pRRREdR
dPs
- positronium energy in the bubble;- surface tension;- external pressure.
positronium energy energy energy of surface of external tension pressure
),( UREPs
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) . 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.
1 8 0 2 0 0 2 2 0 2 4 0 2 6 0 2 8 0 3 0 0 3 2 0 3 4 0 3 6 0 3 8 0 4 0 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 4 0 8 0 1 2 0 1 6 0T -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 1 2 0 1 6 0T -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 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0T -T m , K
1 6
2 0
2 4
2 8
SU
RFA
CE
TE
NS
ION
, 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
massmpositroniumm
RmRE
ePs
PsPs
2
2,
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 pRRREdR
dPs
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
Liq
uid
alk
an
es
0 .2 0 .3 0 .4 0 .5R , n m
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 pRRREdR
dPs
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)
2310A
eV
3610A
eVp catmospheri
33
2
103/4
4
pR
R
Ps BUBBLE MODEL FOR LIQUIDS
04 2 RRER Ps
nm
massmpositroniummnminRwheneVRRm
RE ePsPs
Ps
166,0
21879,0
2 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
RRmR 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 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0T -T m , K
3
4
5
6
7
8
3
ns
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0T -T m , K
3
4
5
6
7
8
3
ns
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 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 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 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 .0 2 0 0 .0 2 0 .0 4 0 .0 61 /r, 1 /A
0
2 0
4 0
6 0
8 0
1 0 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)
r
d21
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
drd
rfor 2
2
The microscopic surface tension
should be greater than the
macroscopic one.
MICRO- AND MACROSCOPIC SURFACE TENSION
Ps BUBBLE MODEL FOR LIQUIDS
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0T -T m , K
3
4
5
6
7
8 3
ns
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0T -T m , K
3
4
5
6
7
8 3
ns
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0T -T m , K
3
4
5
6
7
8 3
ns
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0T -T m , K
3
4
5
6
7
8 3
ns
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 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 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0T -T m , K
3
4
5
6
7
8 3
ns
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0T -T m , K
3
4
5
6
7
8 3
ns
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0T -T m , K
3
4
5
6
7
8 3
ns
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0T -T m , K
3
4
5
6
7
8 3
ns
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 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 2 0 4 0 6 0 8 0 1 0 0T -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 8 0 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 4 0 8 0 1 2 0 1 6 0T -T m , K
3 .2
3 .6
4 .0
4 .4
3
ns
0 4 0 8 0 1 2 0T -T m , K
2 .8
3 .2
3 .6
4 .0
4 .4
3
ns
0 4 0 8 0 1 2 0 1 6 0T -T m , K
3 .2
3 .6
4 .0
4 .4
3
ns
σ·3,05
C6H14
Alkanes
Correcting
coefficient x
C6H14 3,05
C7H16 2,86
C9H20 2,9
C13H28 3,1
C19H40 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 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0T -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 ethanol C 4H 9O H buthanol C 5H 11O H penthano l C 6H 13O H hexanol C 9H 19O H nonanol C 13H 27O H tridecanol
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 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0T -T m , K
1 5
2 0
2 5
3 0
3 5
4 0
SU
RF
AC
E T
EN
SIO
N, d
yn/c
m0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0
T -T m , K
0 .3 4
0 .3 6
0 .3 8
0 .4 0
0 .4 2
R, n
m
C H 3O H m ethanol C 2H 5O H ethanol 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 tridecanol
ALKANES AND ALCOHOLS
0 4 0 8 0 1 2 0 1 6 0T -T m , K
2 .8
3 .2
3 .6
4 .0
4 .4
3
ns
alkanes a lcohols C 6H 14 C 6H 13O H C 9H 20 C 9H 19O H C 13H 2 8 C 13H 27O H
0 4 0 8 0 1 2 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
1 2 1 6 2 0 2 4 2 8 3 2, d y n /cm
0 .2 4
0 .2 8
0 .3 2
0 .3 6
n
s
1 2 1 6 2 0 2 4 2 8 3 2, d y n /c m
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 r
un
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.
1 2 1 6 2 0 2 4 2 8 3 2, d y n /cm
0 .2 4
0 .2 8
0 .3 2
0 .3 6
n
s
1 2 1 6 2 0 2 4 2 8 3 2, d y n /c m
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