Large Clusters in Cesium
Vapor not far from the Critical
Point
D.I. Zhukhovitskii
Joint Institute for High Temperatures, RAS
XIXth Research Workshop Nucleation Theory and Applications
BLTP, Dubna, Russia, April 1–30, 2015
Motivation:
1. Development of the universal thermal and caloric equations of state
based on the minimum information concerning the substance.
2. Search for new exactly solvable thermodynamic problems.
3. Study of thermodynamic properties of a dense cesium vapor near the
critical point.
4. Investigation of the structural transition in the metal “hot” light clusters.
5. Analysis of the rate of homogeneous nucleation in the transitional range.
Outline:
1. The notion of the structural transition in the “hot” lightest clusters.
2. The compressibility factor, the heat capacity, etc. for an ideal mixture of
such clusters.
3. Extension of the model to arbitrary-size clusters.
4. Assessment of the structural transition temperature for the cesium
clusters.
5. Homogeneous nucleation of cesium vapor in the transitional region.
“Hot” lightest clusters
(k = 5)
and an embryo-size cluster
(k > 30)
structural transition
“Cold” clusters:
Partition function of a light cluster
Above the transition temperature, a light cluster has the minimum number of bonds k – 1
and
Then the compressibility factor of a mixture of the lightest clusters is
Compressibility factor for cesium vapor (isotherms)
1 – 1100 K, 2 – 1500 K, 3 – 1900 K, 4 – 2500 K
Heat capacity and velocity of sound
The internal energy per one molecule of an ideal gas of k-atom clusters is
then its heat capacity is
The entropy of an ideal mixture of k-atom clusters is
whence it follows that
Isobaric heat capacity for cesium in the gaseous state at the isobar 1.5 MPa
5
cp /N
lightest clusters
atoms and dimers
reference data
4
3
1500
1700
1900
T, K
2100
Isobaric heat capacity for cesium at the isotherm
5
cp /N
lightest clusters
atoms and dimers
reference data
4
3
0.00
0.01
0.02
, g/cm
3
0.03
Velocity of sound for cesium in the gaseous state at the isobar 1.5 MPa
4.7
cs, 104 cm/s
lightest clusters
ideal gas
reference data
4.2
3.7
3.2
1500
1700
1900
T, K
2100
Velocity of sound for cesium in the gaseous state along the saturation line
lightest clusters
ideal gas
reference data
cs , 104 cm/s
3.8
3.4
3.0
1000
1200
1400
T, K
1600
Velocity of sound for mercury in the gaseous state along the saturation line
lightest clusters
ideal gas
reference data
3.2
cs , 104 cm/s
3.0
2.8
2.6
2.4
800
1000
1200
T, K
1400
1600
Heat capacity ratio for argon along the isobars
0.3 MPa,
0.3 MPa,
0.6 MPa,
0.6 MPa,
1.0 MPa,
1.0 MPa,
cp /cv
1.78
calculation
reference data
calculation
reference data
calculation
reference data
1.70
120
170
220
T, K
270
Arbitrary size cluster in the “layer over the core” model
The size distribution of the clusters partial pressures is
The equation with respect to p1 is
Thus, the compressibility factor is
The average number of atoms in a cluster is Z–1 ~ 2.
Compressibility factor for cesium in the gaseous state along the saturation line
0.9
0.8
Z
arbitrary clusters
lighters clusters
atoms and dimers
Vargaftik et al.
Kozhevnikov
0.7
0.6
0.5
1000
1200
1400
T, K
1600
1800
Compressibility factor for cesium in the gaseous state along the isobars
1.00
0.95
Z
0.90
0.85
2 MPa, calc.
2 MPa, exp.
1 MPa, calc.
1 MPa, exp.
0.4 Mpa, calc.
0.4 MPa, exp.
saturation line
0.80
0.75
1100
1300
1500
T, K
1700
Assessment of the structural transition temperature
The ratio of the probabilities to find a light cluster in the chain-like and the solid-like state is
The characteristic temperature of the structural transition is defined by the equality Pvc = Psol:
At large k, we have
This equation has two roots if
.
The width of the temperature transitional range DT is defined by the condition
. Hence,
, where
Structural transition temperatures for the Lennard-Jones and cesium clusters
For the L-J cluster, a/r0 = 6 and e = 1, then we have
For cesium clusters, we estimate a/r0 = 6 as follows:
Thus, (r0/a)2 = 0.0747. Since for
, the binding energy of a solid-like cluster
can be estimated as
, where Dk is the binding energy for a bond, we
obtain e = 0.409 and arrive at
Homogeneous nucleation rate of cesium vapor
For the critical supersaturation ratio
We used the same l = 2.98 as for the calculation of compressibility factor.
The cluster critical size varies from 21 to 57.
Critical supersaturation for the nucleation in cesium vapor
size correction
CNT
experiment
Scr
100
10
300
350
400
450
T, K
500
550
Conclusions
1. Thermal and caloric properties of a dense cesium vapor and of
vapors of other substances are satisfactory reproduced by
proposed model.
2. For cesium clusters, the structural transition temperature is well
above the melting temperature (in contrast to the L-J clusters).
3. Dense cesium fluid in the vicinity of the critical point is a mixture of
clusters heavier than dimers.
4. Peculiarity of the homogeneous nucleation of cesium vapor is
accounted for by the cluster structural transition.
Thank you for the attension!
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