IMPACT OF PHASE TRANSITIONS IN CONDENSED PHASES ON

IMPACT OF PHASE TRANSITIONS IN CONDENSED PHASES ON THE
GEOMETRY OF VAPOR NUCLEATION RATE SURFACES
M.P. Anisimov,1,3,4 V.I. Kosyakov,1,2 P.K. Hopke1,3
1
Aerosol Technologies Laboratory at Novosibirsk State Technical University. 20 Karl Marx Prospect,
630073 Novosibirsk, Russia. E-mail: [email protected]
2
Institute of Inorganic Chemistry SB RAS, 5 Ac. Lavrentyev Street, 630090 Novosibirsk, Russia.
3
Clarkson University, Box 5708, Potsdam, NY 13699-5708. E-mail: [email protected]
4
Technological Design Institute of Scientific Instrument Engineering SB RAS, 41 Russkaya Street,
630058 Novosibirsk, Russia.
Keywords: nucleation, phase transitions, phase equilibria diagrams
INTRODUCTION
Homogeneous nucleation represents the generation of new phase embryos and their subsequent growth
within the supersaturated initially homogeneous mother phase. Nucleation is a non-equilibrium
thermodynamic phenomenon. A full understanding of nucleation is critical for accurate nucleation theory
development, for new material design, for its role in atmosphere physical chemistry, etc [1-4].
Nucleation theory such as vapor/liquid, is currently insufficient to explain known phenomena. For
example, it cannot deal with the existence of two-channel nucleation of vapor-to-liquid and vapor-tosolid near the triple point. Each nucleation channel is represented by an exponential nucleation rate
surface of J over the P-T plane, where Т is temperature and Р is pressure [3, 4]. The multiplicity of the
nucleation channels must be represented in the nucleation rate surface geometry, which is the sum of the
individual phase exponential surfaces [3]. Measurement methods for the empirical study of vapor/liquid
nucleation kinetics have been developing for more than a century. They have sufficient accuracy to
detect the impact of phase transitions of the first or second order in a condensed phase on the vapor
nucleation rate surface geometry [1-3].
Knowledge of the actual topologies of the nucleation rate surfaces is useful, for example, for
development of the new material design technologies. It is currently not possible to have a common
solution of the problem of creating nucleation rate surfaces. Thus, semi-empirical design of these
surfaces can be extremely valuable [1-4]. The impact of phase transitions in a condensed phase on the
geometry of the vapor nucleation rate surfaces is discussed in this study.
METHODOLOGY
A simplified case of nucleation kinetics can be presented for supersaturated single vapor nucleation.
Nucleation can only occur when the mother phase is far enough from equilibrium, i.e. a vapor should be
supersaturated. The chemical potential of the vapor phase, vapor, exceeds the condensed phase, cond, i.e.
vapor – cond > 0. The nucleation rate, J, can be expressed by the function: J = B exp(-ΔG*/kT), where
μcond and μvapor are the chemical potentials of the bulk phases; B is pre-exponential factor; ΔG* = n*(μcond
– μvapor) + f; where n* is number of molecules (or atoms) in the critical embryos of the new phase; f is the
excess Gibbs energy of the critical cluster of the new phase containing n* molecules as compared with
the same number of molecules in a macroscopic sample of the mother phase, k is Boltzmann constant,
and T is temperature.
The values of μcond and f (equation 1) within the embryos have a discontinuity in the first derivative with
respect to temperature at constant pressure in the mother phase. This discontinuity can be seen easily
along the line of constant nucleation rate. That kind of nucleation rate surface singularity is detectable in
the experimental data as described, for example, by Anisimov et al. [5].
Fig. 1. The nucleation rate surface at the triple point vicinity of a single system.
J is rate of nucleation; T is temperature, P is pressure, t is the triple point, line, tc, is vapor-liquid equilibria; line, kt,
is vapor-solid equilibria. The equilibria lines a continued to the conditions of metastable states (pieces rt and at).
Line, cd, presents the vapor sinodal. Dark grey surface shows the vapor to droplet transformation and light grey is
presenting the vapor – solid transitions. Line, Z, illustrate schematically the isotherm for case of second order phase
transition.
It is known [4] that isotherms of single real gases have metastable areas in pressure - specific volume
diagrams. These metastable areas are bounded by the bimodal and spinodal lines. The critical point, i.e.
a phase transition of the second order, c, lies along the critical isotherm. A P-T diagram generally has
several lines of phase equilibria. Each equilibrium line can produce a surface for that phase transition.
These surfaces can cross each other as can be seen on Fig. 1. The Р-Т plane provides the base for a threedimensional nucleation rate surface near the triple point. These nucleation surfaces cross each other
along the line et, where the first derivative for constant T or P is discontinuous. That singularity shows
the impact of the first order phase transition in the condensate on the nucleation rate surface. The curve,
v, shows schematically the nucleation rate surface isotherm for the case of a phase transition of the
second order. This kind of isotherm can be initiated by various second order phase transitions such as
critical points (lines, etc), superconductivity, super fluidity, etc. [6-8]. In this case it is seen that vapor
nucleation has two nucleation channels resulting in liquid droplets and solid particles [6].
The total nucleation rate, J = Jl + Js. where the individual nucleation rates are measured for droplets, Jl,
and solid particles, Js. Each nucleation rate surface is represented by an exponential function. One
should observe the resulting surface, J(P, T), with a fold-like singularity. A phase transition can occur
within the growing clusters. Linear relationships of the melting temperature with the reciprocal of the
radius of small clusters were reported by Thompson [9]. However, several researchers have reported
nonlinear relationships for these parameters [10, 11]. Computer modeling of the Тт against R-1
relationship for Lennard-Jones clusters was performed by Samsonov et al. [12]. The computer
experiments results were in agreement with experimental data and with independent thermodynamic
considerations. We hypothesize that similar behavior can be found for nano-sized cluster melting or
freezing when they have small values of the enthalpy of fusion. These singularities should generate
additional singularity folds on the nucleation rate surfaces. Some phase transitions within small clusters
cannot be observed because of the small quantity of the initial phase that is insufficient to permit new
embryo formation.
Fig. 2. Schematic presentation of a chemical potentials, , on vapor activity (a) for the condensed phases of 1 and 2 (left
panel) and Pressure (P) -Temperature (T) - partial volume (v) diagram for a single component system with the
metastable areas which are painted by gray (right panel).
T = 323.17 K
T = 320.92 K
T = 325.29 K
T = 327.63 K
T = 329.88 K
T = 332.31 K
T = 351.90 K
T = 349.75 K
T = 347.54 K
T = 345.48 K
T = 343.16 K
T = 340.84 K
T = 338.62 K
T = 336.45 K
T = 334.25 K
Sulfur hexafluoride – 1,3- propanediol
7
6
log J
5
4
3
Р = 0.30 МPа
2
0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76
log a
Fig. 3. Gap from a phase transition at nucleation temperature T=334 K for sulfur hexafluoride – 1,3- propanediol
nucleation at pressure of 0.30 MPa [13].
CONCLUSIONS
The phase transitions of the first and second order within a solid and liquid condensate have a clear
impact on nucleation rate surface topologies for any system. These phase transitions produce
singularities in the nucleation rate surfaces where there are discontinuities in the first or second
derivatives. To detect these phase transitions, surface continuity and monotony can be used as suggested
by Anisimov et al. [5]. A data base of phase equilibria diagrams and the nucleation rate surfaces that are
related to these diagrams along with the algorithm for the design of these surfaces will provide a reliable
basis for the understanding of a wide spectrum of technological, chemical, physical, material design, etc.
problems. Knowledge of the actual nucleation rate topologies can be extremely useful in the
development of new technologies and optimization of existing ones to produce new materials with phase
transitions, including technologies that involve catalysis, etc. This conceptual framework can be further
extended to all energetic barrier systems.
AKNOWLEDGEMENTS: This research was supported by a grant from the Russian Ministry of Science
and Education under Contract № 14.Z50.31.0041.
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