Attainable superheating of liquefied gases and their solutions (ReviewArticle)V. G. Baidakov Citation: Low Temp. Phys. 39, 643 (2013); doi: 10.1063/1.4818789 View online: http://dx.doi.org/10.1063/1.4818789 View Table of Contents: http://ltp.aip.org/resource/1/LTPHEG/v39/i8 Published by the AIP Publishing LLC. Additional information on Low Temp. Phys.Journal Homepage: http://ltp.aip.org/ Journal Information: http://ltp.aip.org/about/about_the_journal Top downloads: http://ltp.aip.org/features/most_downloaded Information for Authors: http://ltp.aip.org/authors
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Attainable superheating of liquefied gases and their solutions (Review Article)
V. G. Baidakova)
Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, ul. Amundsen 106,620016 Ekaterinburg, Russia(Submitted March 25, 2013)
Fiz. Nizk. Temp. 39, 835–862 (August 2013)
This review addresses the kinetics of spontaneous boiling-up of liquefied gases and their solutions.
It discusses the theories of thermal and quantum nucleation in metastable liquids. The experimental
methods for studying the nucleation kinetics are outlined. The experimental data on the attainable
superheating temperature for cryogenic liquids and solutions of liquefied gases are presented in a
wide range of pressures, including negative ones. The properties of new-phase nuclei near the
boundary of attainable superheating are discussed. The kinetics of initiated and heterogeneous
nucleation is considered. The experimental data on detection of quantum tunneling of nuclei
are presented. The experimental data are compared with the theories of thermal and quantum
nucleation. VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4818789]
1. Introduction
One of the manifestations of intermolecular interactions
is that the first-order transitions can occur in systems con-
taining large number of particles. Since the discontinuous
nature of the transition makes simultaneous converting of
the entire bulk of the substance impossible, the phase transi-
tion begins at certain separate “points” in a homogeneous
system forming the nucleation centers of a new phase.
Formation of the new phase with significantly different prop-
erties leads to the appearance of the surface energy, so the
nucleation centers which are too small in size are energeti-
cally unfavorable. Viable nucleation centers should exceed a
certain critical size, then the growth of the new phase is
accompanied by lowering the thermodynamic potential. This
means that superheating and supercooling are possible dur-
ing first-order phase transitions due to the absence of such
nucleation centers. Each phase may exist, at least as a meta-
stable one, on both sides of the transition point.
If a homogeneous system is not exposed to any external
stimuli, the nucleation centers of the new phase arise sponta-
neously due to thermal (T> 0) or quantum (T ’ 0) fluctua-
tions in the environment. This process is called homogeneousnucleation. Typically, the phase transition begins on the
walls of a container or on impurities and requires much lesser
supersaturation (heterogeneous nucleation). The spontaneous
formation of a large mass of a new phase has a negligible
probability, so a metastable system is capable of prolonged
existence under constant external conditions.
When entering deeply into the region of metastable
states, the stability of the phase drops not only with respect
to nucleation, but also with respect to infinitesimal (continu-
ous) fluctuations of the state variables. If the phase is unsta-
ble with respect to such fluctuations, it cannot exist for any
extended period of time. Any small perturbation will be
amplified by the response of the system itself until it transi-
tions into a qualitatively new state. The line in the phase dia-
gram, separating the regions of stable and unstable states of
the homogeneous bulk, is called spinodal.As a general property of first-order phase transitions,
metastability manifests itself in the systems of a very differ-
ent nature: nuclear matter, electron-hole liquid, during the
processes of evaporation, condensation, crystallization proc-
esses, etc.
The problem of phase metastability covers a wide range
of issues related to the kinetics of nucleation, determination
of thermodynamic and kinetic properties, the boundaries of
the thermodynamic stability, etc.
This review focuses primarily on the kinetics of nuclea-
tion in superheated liquids and their solutions. To familiarize
with the subject of the review and the sequence of presenta-
tion it is sufficient to browse through the table of contents.
The subject of the discussion here is liquified gases. The
defining role of interparticle interaction law in first-order
phase transitions highlights the compounds the molecules of
which have a spherically symmetrical or similar shape and
exhibit a simple dispersion type of the intermolecular inter-
actions. Such (simple) substances are used as a touchstone in
the theoretical studies of condensed-matter problems.
Superheated fluid is a convenient system for implement-
ing homogeneous nucleation. Low viscosity of a superheated
liquid ensures rapid relaxation of the structure that is not
always the case for supercooled liquids. In the superheated
liquids, it is also easier to exclude the centers of heterogene-
ous nucleation as compared to, for example, the condensa-
tion of steam. This is due to the fact that most of the liquids
wet the glass and other hard surfaces well. Besides that, the
effect of heterogeneous nucleation centers can be lowered by
reducing the volume of the liquid under study or the transi-
tion time into a highly superheated state. Also registering the
onset of the phase transformation in a stratifying liquid rep-
resents a more complex technical challenge as compared to
the case of a superheated liquid. System purity is not a nec-
essary prerequisite to obtain a large superheating. Using
high-power heat release, it is possible to achieve the boiling-
up regimes at which, despite the presence of readily avail-
able nucleation centers in the liquid, the main role is played
by spontaneously generated vapor bubbles.
The phenomenon of metastability in liquids was first
mentioned in the second half of the XVII century, when the
existence of water and mercury at negative pressures was
experimentally registered by Huygens and Boyle. The physi-
cal interpretation of this phenomenon became possible only
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after the appearance of the famous van der Waals (1873)
equation. First reliable quantitative data on the superheating
temperatures for a number of liquids have been obtained by
Wismer and his colleagues (1922–1927). Even earlier
(1878), Gibbs1 explained the phase transformations as a pro-
cess involving nucleation of the seeds of a new phase.
Gibbs’ ideas formed the basis of the classical theory of
nucleation through thermal fluctuations that has been formu-
lated in the works by Volmer and Weber,2 Farkas,3 Becker
and D€oring,4 Zeldovich,5 and Frenkel.6 Being based on ther-
modynamics, this theory is universal and can be applied to
various types of phase metastability. Its main ideas in rela-
tion to the phenomenon of superheating of single-component
liquids and their solutions are outlined in Sec. 2.
Systematic experimental study of the phenomenon of
superheating started in 1960 by the school of Prof. Skripov,
member of the Academy of Sciences.7 The experiments
were preceded by the development of methods for studying
the boiling kinetics of liquids which could measure the
nucleation rate in the range from 107 to 1030 s�1 m�3.
Section 3 focusses on the analysis of the experimental data
on the nucleation kinetics in liquefied gases and their solu-
tions and verification of the thermal fluctuation theory of ho-
mogeneous nucleation. The studies encompass the regions of
both positive and negative pressures, and the vicinity of the
critical point. The data on the limits of superheating the solu-
tions with complete and partial component solubility are
presented.
Near absolute zero temperature, nucleation through ther-
mal fluctuations is not possible and the phase separation has
to be achieved by quantum leaking of a heterophase fluctua-
tion through the activation barrier. Quantum nucleation
theory has been developed by Lifshitz and Kagan.8 In recent
years, a series of experimental studies on detection of the
quantum nucleation mechanism in liquefied gases and
their solutions, as well as in other systems. Achievements
and challenges in this field of research form the content of
Sec. 4. The main results on comparison of the theory of
homogeneous nucleation and experiment are presented in
Conclusion.
2. Theory of homogeneous nucleation
2.1. Thermodynamics of nucleation
The boiling-up of a superheated liquid is a thermody-
namically irreversible process. In its early stage, the nuclei
of a new (vapor) phase are formed. The minimal work that
must be expended to form a nucleus (bubble) of a given
size depends on the mechanism and process conditions. Let
us consider homogeneous formation of a phase in a super-
heated binary solution under constant external pressure p,
temperature T and concentration of the liquid phase c.
Nucleation in a single-component liquid is a special case of
such consideration (c¼ 0). The fact that p, T, and c are con-
stant means that the chemical potential of the liquid l is
constant as well.
The difference in the Gibbs free energy DU between the
initial state, which is a uniform metastable liquid, and the
final state, which includes a vapor bubble of the radius R and
the metastable liquid, determines the work of nucleation of
the vapor phase,
W ¼ DU ¼ ðp� p00ÞV þ rAþX2
i¼1
ðl00i � liÞN00i : (1)
Here r is the surface tension, N00i is the number of molecules
of the i-th component in the bubble, V ¼ 4=3pR3 and
A ¼ 4pR2 are the volume and surface area of the bubble,
respectively. All these values are defined according to the
position of the tension surface. Double prime indicates that
the parameter is related to the vapor phase, the values with-
out primes are related to the liquid.
In equilibrium, the function DUðV; p00; c00Þ has an
extreme, which corresponds to an unstable equilibrium of
the bubble with the surrounding solution. The equilibrium
conditions can be written as
p00� � p ¼ rdA
dV
�����¼ 2r
R �; (2)
l00i ðp00� ; c00�; TÞ ¼ liðp; c; TÞ: (3)
Here and below the symbol “*” refers to the equilibrium
(critical) bubble, the concentration c is the mole fraction of
the second component in the solution.
Substituting Eqs. (2) and (3) in Eq. (1) gives the work
of formation of a critical nucleus,
W� ¼1
3rA � ¼
1
2ðp00� � pÞV� ¼
16pr3
3ðp00� � pÞ2; (4)
which has the same form for both single-component liquid
and solution.
In the case of a small overheating far away from the line
of critical points, the liquid solution can be assumed to be
incompressible and the vapor inside the bubble can be
considered as a perfect gas. Then the equilibrium conditions
(2) and (3) allow us to express the pressure p00� and composi-
tion c00� in the critical bubble in a binary solution through the
directly measurable values9
p00� � p ’ ðps � pÞ 1� t1s
t00s� c00s
ðt2s � t1sÞt00s
" #; (5)
c00� ’ c00s ; (6)
where ps is the saturation pressure, ti is the partial specific
volume of the i-th component, and the index “s” refers to a
planar interphase boundary.
Considering the vapor mixture in the bubble as a perfect
gas and replacing the chemical potential of the liquid in
Eq. (1) with the equal chemical potential of the vapor in the
equilibrium bubble, and taking into account Eqs. (2) and (3),
we obtain the expression for the second differential of DU at
the extremum point,10
ðd2DUÞ� ¼ �2rA�9V2�ðdVÞ2 þ V�
p001�ðdp001Þ
2 þ V�p002�ðdp002Þ
2
¼ � 2rA�9V2�ðdVÞ2 þ V�
p001�ðdp00Þ2
þ p00� V1
1� c00�þ 1
c00�
� �ðdc00Þ2; (7)
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where p00i is the partial pressure of the i-th component in the
bubble.
The absence of cross-terms in the quadratic form (7) fol-
lows from the statistical independence of the variables V, p001,
p002 and V, p00, c00. Different signs of the coefficients at the
squares of the differentials V and p001, p002 (or p00, c00) indicate
that the surface of the Gibbs thermodynamic potential in the
neighborhood of the extremum point is a hyperbolic parabo-
loid, and the point itself is a saddle point. The equilibrium of
the critical bubble and metastable liquid with respect to the
variables p001, p002 (or p00, c00) is stable, while the equilibrium
with respect to the variable V is unstable.
The considered approach to the definition of W* has
been developed by Gibbs.1 He has proposed to take the work
of formation of a critical nucleus or its dimensionless analog
G� ¼ W�=kBT, where kB is the Boltzmann’s constant, as a
measure of stability of the metastable phase. On the satura-
tion line p00� ¼ p ¼ ps and W� ! 1. Increase of the liquid
superheating is accompanied by a lowering of the activation
barrier due to the factors ðp00� � pÞ�2and r3 (Eq. (4)). In
Gibbs’ method, the surface tension is a function of the nu-
cleus size. The explicit form of rðRÞ dependence can only
be obtained by the statistical analysis of this problem.
Another approach to the calculation of the work of for-
mation of a critical nucleus which is different from that used
by Gibbs has been proposed by Cahn and Hilliard.11 This
approach does not require introducing such a parameter of
nucleus as surface tension. It is based on the van der Waals
capillarity theory.12 Within the framework of the gradient
approximation of the van der Waals capillarity theory, the
change in the Helmholtz free energy of a two-component
system associated with heterophase density fluctuations
dqðrÞ ¼ qðrÞ � q is written as*)11
D F½q� ¼ð
D f þXi;j¼1
ji;jrqirqj
� �dr; (8)
where
D f ¼ f ðqÞ � f �Xi¼1
½qiðrÞ � qi� li; (9)
f ðqÞ is the free energy per unit volume of the homogeneous
solution with the local density of components
qðrÞ � fq1ðrÞ; q2ðrÞg, rqi is the density gradient of the i-thcomponent, ji,j
is the symmetric matrix of the influence
coefficients, and li is the chemical potential of the i-th com-
ponent of the metastable phase.
The density distribution in the heterophase fluctuation
corresponding to the extreme value of DF [q] can be found by
solving the system of Euler equations for the functional (8).
If we neglect the dependence of the influence coeffi-
cients on q and only consider spherically symmetric inhomo-
geneities, the system of Euler equations takes the form
2Xj¼1
jij
d2qj
dr2þ2
r
dqj
dr
� �¼@Df
@qi
¼liðqÞ �li; i¼1;2 (10)
with the boundary conditions
qiðrÞ ! qi for r !1;dqiðrÞ
dr¼ 0 for r ¼ 0 and r !1; i ¼ 1; 2 : (11)
It can be shown that the solution q�ðrÞ of system (10) in
the metastable region with the boundary conditions given by
Eq. (11) corresponds to a saddle point of the functional (8),
i.e. the critical nucleus. Then
W� ¼ min max DF½q��: (12)
In the vicinity of the phase equilibrium line, the effec-
tive nucleation radius is large compared with the thickness
of the transition layer and the integration of Eq. (8) yields
the Gibbs expression for the work of nucleation, Eq. (4).
In the vicinity of the spinodal line, the energy barrier
separating the metastable and stable phases is small, and, to
find the work required to form a critical nucleus, it is suffi-
cient to consider only the density fluctuations of the ampli-
tude jqsp � qj. Expanding the function Df in Eq. (8) in series
in powers of the density perturbations and limiting the series
by cubic terms, we can show that upon approaching to the
spinodal line, the characteristic size of the critical nucleus Lgrows indefinitely (L � jqsp � qj�1=2
), while the work of for-
mation of a nucleus W* and the density in its center q(r¼ 0)
decrease as jqsp � qj3=2and jqsp � qj, respectively.8,11
2.2. Kinetics of nucleation through thermal fluctuation
In a clean system, the nucleation centers emerge sponta-
neously by fluctuations. The probability of the fluctuation-
driven nucleation can be related to the work W*.2
The classical theory of homogeneous nucleation considers
the fluctuation growth of subcritical bubbles as a diffusion pro-
cess in the space of variables q � qðx; y; :::; zÞ, which deter-
mine their state, in the field of thermodynamic forces. The
main kinetic equation of the theory describes the time- depend-
ent evolution of the bubble distribution with respect to the vari-
ables q. It has the form of the multiparametric Kramers-
Fokker-Planck equation for the distribution function P(q)5,13,14
@PðqÞ@t¼ @
@qPeD
@
@q
P
Pe
� �� �¼ @
@qD@P
@q� DFP
� �: (13)
Here, D is the generalized diffusion tensor in the space of
variables q, F is the thermodynamic force, Pe(q) is the equi-
librium distribution function for which the principle of
detailed balance between the transitions of the bubbles of
neighboring classes is fulfilled. The function Pe(q) is related
to the work of bubble formation W(q) by expression
PeðqÞ � exp½�WðqÞ=kBT�: (14)
The expression in square brackets in Eq. (13) has the mean-
ing of the flux of nuclei in the q-space taken with the oppo-
site sign. If the state of initial phase is time-independent, a
stationary distribution Ps(q) is established in the system and
the net flux of nuclei from the region of heterophase fluctua-
tions into the two-phase region determines the frequency of
nucleation J.
*In the shorthand notation the index-free (matrix) notation is used. Vectors
are denoted in lowercase bold, second-rank tensors are capitalized.
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The solution of the stationary nucleation problem is
reduced to determining the landscape of the excess of the
Gibbs thermodynamic potential W(q), calculating the gener-
alized diffusion tensor D, and the subsequent transition from
multi-parameter Eq. (13) to a single-parameter equation for
a single dedicated (unstable) variable.
In the case of a binary solution, the state of a bubble is
determined by its volume V, pressure p00 inside the bubble
and the composition of the vapor phase c00. The landscape of
the excess of the Gibbs thermodynamic potential in the vi-
cinity of the saddle point is given by Eqs. (4) and (7).
To calculate the generalized diffusion tensor, it is nec-
essary to know the forces acting on the system, and the
speed at which it moves under the influence of these forces.
In the Zeldovich formalism,5 the force F ¼ �rWðqÞ,while the speeds are found from the phenomenological
equations of motion of the bubble in variables q. If we
neglect the heterogeneity of pressure and temperature in
the bubbles and assume them to be spherically symmetric,
the complete system of the equations of motion includes:
the Navier-Stokes equation, the equations describing the
change in the number of molecules of each of the compo-
nents in the bubble, and the diffusion equation. Generally,
the generalized diffusion tensor contains cross terms. This
means that the nucleus variables q in the kinetic Eq. (13)
cannot be separated. After we have switched to the system
of principal axes of the tensor DG, the diagonal compo-
nents of which are defined as the roots of the characteristic
equation
det ðDG � kIÞ¼ 0 ; (15)
multi-parameter Eq. (13) is reduced to the one-dimensional
Zeldovich equation, the solution of which has the form15
J ¼ qz0jk0jkBT
2pjg0j
� �1=2
exp � W�kBT
� �¼ qz0B expð�G�Þ:
(16)
Here, as in Eq. (15), G is the matrix of second derivatives of
the Gibbs thermodynamic potential, normalized by kBT, I is
the unit tensor, q is the number of particles per unit volume
of the metastable phase, z0 is the factor that corrects the nor-
malization of the equilibrium distribution function over an
unstable variable—the size of the nucleation center, k0 is the
negative root of the characteristic Eq. (15), g0 is the second
derivative of the thermodynamic potential with respect to an
unstable variable in the saddle point of the activation barrier,
and B is the kinetic factor.
The stationary nucleation process in a superheated
single-component liquid has been first considered by
D€oring.16 In this paper, the condition of mechanical equi-
librium given by Eq. (2) is assumed to be fulfilled for sub-
critical bubbles. Thereby the boiling liquid is assumed to
be perfect and inertialess, and the growth of vapor bubbles
is determined only by the rate of evaporation-condensation
processes. In these approximations, the expression for
the nucleation rate known in literature as D€oring-Volmer
equation has been obtained.7 This equation has the form of
Eq. (16), where z0¼ 1 and the kinetic coefficient is given
by
B � B1 ¼6r
p mð3� bÞ
� �1=2
: (17)
Here m is the molecular mass, b ¼ 1� p=p00� ’ 1� p=ps.
According to Eq. (17), for large negative pressures when
b! 3, the kinetic coefficient and, accordingly, the frequency
of nucleation tend to infinity. This non-physical result is a
consequence of the instability of the mechanical equilibrium
in near-critical bubbles at b� 3.
All the major factors limiting the growth of near-critical
bubbles in superheated pure liquids, namely, the viscosity
and inertia of the liquid, the evaporation rate of molecules
into the bubble, and the heat transfer rate to the bubble, have
been taken into account in the work by Kagan.17 In this case
the characteristic Eq. (15) takes the form of a cubic equation.
Neglecting the inertia terms, it reduces to a quadratic one,
the solution of which gives the following expression for the
kinetic factor:
B �B2 ¼2xr
b
rkBT
� �1=2
� xþ 3þ b
b
� �þ xþ 3� b
b
� �2
þ 4x�1=2
" )�1
;
8<:
(18)
where
x¼ 3
2
attgrð1þ dÞ ; d¼ adlttR�
4KkBT; (19)
a is the condensation coefficient, tt ¼ ð8kBT=pmÞ1=2is the
average thermal velocity of molecules, l is the heat of vapor-
ization, g is the viscosity, K is the thermal conductivity coef-
ficient, and d ¼ dps=dT.
From Eq. (18), several limiting cases can be derived to
determine B in the situations when the factors limiting the
bubble growth are the viscosity (x 1, x 3/b), the evapo-
ration rate of the liquid (3/b� 1x, d 1), and the heat
supply (3/b� 1x, d 1). If the temperature effects on
the boundary of the bubble (d¼ 0) are neglected, Eq. (18)
coincides with the kinetic factor in D€oring-Volmer theory,
Eq. (17), up to the term 3/b. In Kagan’s approach,17 the fac-
tor z0, which adjusts the normalization of the equilibrium
distribution function, is unity.
The problem of the fluctuation-driven boiling-up of a
superheated single-component liquid in the case when the
inertia forces and the relaxation of heat at the bubble-liquid
interface are neglected has been also considered in the work
by Deryagin, Prokhorov, and Tunitsky.18 For the kinetic fac-
tor B � B3, Eq. (18) with d¼ 0 has been obtained. The nor-
malization constant for the equilibrium distribution function
of the bubble size z0 has been calculated using a large Gibbs
ensemble and found equal z0 ¼ q=q00� .19
In a binary solution the kinetic factor also depends,
besides the rate of heat supply to the bubble, the evaporation
rate of the molecules into the bubble, the fluid viscosity and
inertia, on the rate of diffusion processes in the liquid phase.
The general solution of this problem has been obtained in
the works by Baidakov.10,15 The characteristic Eq. (15) in
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this case is the equation of the fourth order. Depending on
which factors are crucial in the growth of a nucleus, the
order of the algebraic equations can be reduced.
The ratio between the free-molecule and diffusive sup-
ply of material to the growing bubble is determined by the
factor20
ci ¼aittiR�p
00�
4kBTDiq; (20)
where ai, tti; and Di are the condensation rate, the average
thermal velocity of molecular motion, and the diffusion
coefficient for the component i, respectively.
At positive pressures when the solution can be consid-
ered non-viscous and non-inertial, the fourth-order equation
for k0 becomes a quadratic one. In the solution of this equa-
tion, two limiting cases should be considered. If
a1tt1R�p00�2
4kBTD2 q c 1� 3R�p
00�1
2r; (21)
then all is determined by the solvent volatility (i¼ 1) and for
the kinetic coefficient we obtain
B � B4 ¼1
2
a1t t1
ð1� b=3Þr
kBT
� �1=2
; (22)
which, if b/3 is neglected with respect to unity, coincides
with the Kagan solution for a single-component fluid.17
Otherwise, when the solvent is non-volatile and the solute is
supplied only by diffusion, we have20
B � B5 ¼D2qc
ðb� 1=3ÞkBT
r
� �1=2
: (23)
Other limiting cases of the solution of Eq. (15) are con-
sidered in Ref. 15.
2.3. Quantum nucleation
Near the absolute zero temperature, the nucleation by
thermal fluctuations in condensed matter is impossible. In
this case, the nuclei of the new phase are formed by the
quantum tunneling of heterophase fluctuations through the
energy barrier. The kinetics of quantum nucleation during
first-order phase transitions has been considered by
Lifshitz and Kagan8 in the approximation of an incom-
pressible fluid and the absence of energy dissipation dur-
ing the evolution of nucleation centers. Neglecting the
dissipation means that new excitations are not formed dur-
ing the quantum underbarrier decay of the metastable
state, and all the parameters of the system are adiabati-
cally adjusted to a particular single parameter—the radius
of the new-phase nucleus.
The quantum decay of the metastable phase is signifi-
cantly different from that driven by thermal fluctuations.
While in the case of nucleation by thermal fluctuations in
the absence of fluid inertia, the dynamics of nucleus
growth determines only the kinetic factor (see Eq. (16)),
in the case of quantum tunneling, the kinetic energy of
the heterophase fluctuation is directly included in the
exponent in Eq. (16).
In the quasi-classical approximation, the probability of
formation of a nucleus is given with exponential precision
by the expression
Pe � expð�I�=�hÞ ; (24)
where I* is the effective action I along the extreme trajectory
R(s), s is the imaginary time.21,22 The period of the extremal
i¼ �h/kBT. The action I is determined per period. At a low
degree of metastability in the absence of dissipation, the
Lagrangian of the fluid-nucleus system is a function of the
radius R and the velocity of the interface _R. Then
I ¼ð12i�1
2i
1
2MðRÞ _R
2 þWðRÞ� �
ds; (25)
where
WðRÞ ¼ 4p R2r� 4
3pR3q00Dl ; (26)
MðRÞ ¼ 4pR3qeff is the added mass of a nucleus, qeff
¼ ðq� q00Þ2=q is the effective density of the added mass,
and Dl ¼ l� l00 is the difference of chemical potentials.
The integral is taken over the imaginary time s from the
entry point to the exit point from under the barrier.
There are two types of trajectories leading to the
extremum of the effective action I. The first one is a clas-
sical trajectory which is time-independent and passes
through the potential energy maximum W(R) (Fig. 1).
This trajectory zeros the kinetic energy (the first term in
the integrand in Eq. (25)) and leads to the classical ther-
mal fluctuations-driven nucleation, I� ¼ �hW�=kBT. The
second trajectory is explicitly time-dependent and
describes the quantum leakage through the barrier. In this
case, the formation of a nucleus of critical size R* is not
sufficient for the decay of the metastable phase. The
relaxation of supersaturation begins only after a nucleus
of the size Rr (Rr>R*) has appeared, that corresponds to
the exit point from under the energy barrier (see Fig. 1).
If the tunneling occurs from the zeroth energy level,
which is realized at T¼ 0, then
FIG. 1. Two mechanisms to overcome the activation barrier for nucleation.
Low Temp. Phys. 39 (8), August 2013 V. G. Baidakov 647
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Rr ¼ Rc ¼3
2R �: (27)
The extreme trajectory R(s) at the zero temperature can
be found by solving the equation22
MðRÞ _R2
2�WðRÞ ¼ 0: (28)
Substituting this solution into Eq. (25) gives the value of
the extreme action
I� ¼5p64ð8prMR4
cÞ1=2 ¼ 5
ffiffiffi2p
p2
16
qeff
q� q00ðqrÞ1=2R 7=2
c : (29)
Equation (29) retains the same form also at T> 0,8 so
the probability Pe does not depend explicitly on the tempera-
ture up to the transition to the regime of thermally activated
nucleation. The temperature of this transition is determined
by equating the exponents for the classical (Eq. (14)) and
quantum (Eq. (24)) nucleation
T� ¼128�h
135pkBsc¼ 128
ffiffiffiffi2p
�h
135pkB
ðq� q00Þqeff
rqR3
c
� �1=2
: (30)
Analogously to the nucleation through thermal fluctua-
tions, it can be written
J ’ CB expð�I�=�hÞ; (31)
where C is the number of virtual nucleation centers per unit
volume of the metastable phase, C ’ q, B is the kinetic fac-
tor proportional to the frequency of zero-point oscillations8
BT¼0 ¼12p3=2
C 14
� " #4=7
ð4prÞ5=7R6=7c
�h3=7M2=7: (32)
Here C(x) is the gamma function. The order of magnitude of
the kinetic coefficient in the case of quantum tunneling is
close to the value of B in the case of nucleation through ther-
mal fluctuations (for liquid helium BT¼0 ’ 2:8� 1012 s�1
[Ref. 8]). At 0< T<T*, BT¼ 0 contains some additional
factors.
The influence of viscous friction on the probability of
the under-barrier tunneling of nuclei has been examined by
Burmistrov and Dubovsky.21 In the case of an incompressi-
ble quantum fluid
I ¼ I0 þ DIdiss ; (33)
where I0 is given by Eq. (25). Internal friction reduces the
probability of quantum nucleation and leads to an explicit
dependence of the exponent in Eq. (24) on temperature.
In the case of weak dissipation, the term DIdiss in
Eq. (33) can be regarded as a perturbation. Compared to the
non-dissipative kinetics, the temperature of transition to the
thermal activation regime T* is reduced by DT* proportional
to DIdiss*/I*.
In the case of strong dissipation, the underbarrier tunnel-
ing is fully determined by the internal friction. Dissipation
affects not only the exponent in Eq. (31), but also the kinetic
factor, which now contains the shear viscosity. This leads to
a strong temperature dependence B(T).
3. Attainable superheating of single-component liquids andsolutions
3.1. Experimental methods for superheated liquids
Emerging of a nucleus of a new phase through fluctua-
tions in a metastable system is a random event. A steady flux
of such independent events is described by a distribution
function and its moments.
Experimental study of the kinetics of steady-state nucle-
ation involves determining the rate of nucleation J as a
function of temperature and pressure. The attainable super-
heating temperature Tn usually refers to the temperature cor-
responding to the experimentally recorded value of the
nucleation rate. If the sample under investigation is small,
the high growth rate of vapor bubbles in highly superheated
liquids (far away from the critical point) gives evidence of
the appearance of a critical nucleus by observing an onset of
the sample boiling-up.
Experimental methods for studying the kinetics of spon-
taneous boiling of superheated liquids can be divided into
quasi-static and dynamic (pulse). A characteristic feature of
the quasi-static methods is that the conditions of homogene-
ous nucleation are ensured at the preparation stage of an
experiment. Besides a “pure” sample, it is essential to have a
measuring cell containing no moving parts and with low
roughness and good wettability of the walls. These require-
ments are best met by glass cells.
Good wettability (in some cases close to complete wett-
ability) of glass and other hard materials by liquid gases,
their low content of foreign gases and low emission of par-
ticulate matter make these systems the most convenient for
studying the kinetics of spontaneous nucleation.
Dynamic methods do not impose strict requirements on
the “purity” of the system. The liquid is superheated by a
powerful heat release, well above the heat sink capacity of
the readily existing vaporization centers (shock boiling-up
regime).7,23 Due to the finite growth rate of the vapor phase,
it is always possible to find such a heating rate of a liquid at
which the homogeneous nuclei dominate in the observed
boiling-up feature. It can be straightforwardly expressed in
the form of a certain inequality.23
In studying the kinetics of nucleation in superheated lique-
fied gases, the methods of continuous isobaric heating,24–27 ris-
ing droplets,28–30 average lifetime31–33 (quasi-static methods)
and the methods of pulse heating and stretching34–39 (dynamic
methods) have been employed.
In the methods of continuous heating and rising droplets,
a liquid is superheated in the process of isobaric heating at a
constant rate _T . In the first method, the superheating typi-
cally occurs in glass capillaries, while in the second one it
occurs upon raising the liquid droplets in some surrounding
fluid. For given p¼ const, _T ¼ const, a number N¼ 30–50
of liquid boiling-up temperatures is recorded, and a histo-
gram distribution of the boiling-up events n is plotted vs.temperature. From the histogram, the most probable boiling
temperature Tmax which corresponds to the distribution max-
imum and the half-width of the distribution dT1=2 are deter-
mined. The temperature Tmax is taken as the attainable
648 Low Temp. Phys. 39 (8), August 2013 V. G. Baidakov
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superheating temperature Tn. These data are used to calculate
nucleation rate
JðTnÞ ¼2:44 _T
dT1=2V: (34)
Here V is the volume of superheated liquid.
In the method of the average lifetime, the metastable
region is usually entered by a sudden pressure drop in the
thermostated liquid. At given values of p and T using series
of N¼ 50–100 measurements of the boiling-up time delay s,
the average lifetime of the metastable state �s can be found,
which is related to the nucleation rate by the expression
J ¼ 1
V �s: (35)
Among the dynamic methods that have been applied to
the study of explosive boiling-up of liquefied gases, we
should mention the methods of pulse superheating of a liquid
on thin platinum wires,23,39 on single bismuth crystals in
magnetic fields,34–36 and the method of focusing acoustic
fields.37,38 A platinum wire and single crystal of bismuth
immersed in the liquid under investigation serve as both a
heater and resistance thermometer. Warm-up is carried out
by an electrical current pulse. The start of an explosive
boiling-up is detected by a sharp change in the heater resist-
ance. The nucleation rate is determined by solving a heat
problem using the recorded oscillogram of a boiling-up
event.39 In the study of cavitation,37,38 an acoustic field in
the liquid is focused by a hemispherical piezoelectric trans-
ducer. The transducer is excited by an electrical current
pulse with the duration of approx. one microsecond. The
onset of cavitation is recorded using a helium-neon laser.
3.2. The experimental results on superheating of classical fluids
In contrast to conventional liquids, the cryogenic liquids
exhibit a significantly lower attainable superheating
DTn ¼ Tn � Ts. While the limiting superheating of water at
atmospheric pressure is estimated at 205 K,23 the values for
propane and liquid nitrogen are DTn ’ 96 K [Ref. 40] and
DTn ’ 23 K [Ref. 41], respectively. The differences are due
to the relatively weak intermolecular interactions in cryo-
genic liquids.
First experiments at cryogenic temperatures have
involved liquid argon.24,31 In subsequent studies, the limiting
superheating temperatures of liquid nitrogen,27,41,42 oxy-
gen,27,42,43 methane,42,44 xenon and krypton25,45 have been
investigated. The experimental data have been obtained over
a wide pressure range using the methods of continuous iso-
baric heating and lifetime measurement. At atmospheric
pressure, the superheating of liquid nitrogen and oxygen has
been studied by Nishigaki and Saji.27 The results of this
work are consistent within 0.3 K with the data from earlier
studies.41,43 Sinha, Brodie, and Semura46 have measured the
superheating temperature of liquid nitrogen using the
method of pulse heating of metallic wires. These data refer
to the frequency of nucleation 1018–1020 s�1m�3. The
authors of Ref. 46 note that the value of superheating was
not affected (60.3 K) by the length of the wire, its location
(vertical or horizontal) and the material (platinum,
constantan, manganin). At atmospheric pressure, the super-
heating of liquid hydrocarbons (ethane, propane, n-butane,
and isobutane) has been studied using the method of rising
droplets.28–30
The results of experiments on determining the attainable
superheating temperatures for a range of liquified gases at
atmospheric pressure are collected in Table 1. It also shows
the saturation temperature Ts, the effective values of the
nucleation rate J, and the experimental method employed.
Fig. 2 shows the attainable superheating temperatures
for liquid methane, oxygen, and nitrogen obtained using the
continuous isobaric heating method and lifetime measure-
ments. Data from these experiments are related to the differ-
ent values of the nucleation rate. The attainable superheating
decreases with increasing pressure and vanishes at the criti-
cal point. The figure also shows the spinodal line for the
studied liquids—the boundary of the thermodynamic phase
stability, where ð@p=@vÞT ¼ 0.
Table 1. The attainable superheating temperature Tn for liquefied gases at
atmospheric pressure, the saturation temperature Ts, the logarithm of the
nucleation rate J, s�1m�3 and the experimental method employed: P—
pulsed, CH—continuous isobaric heating; LT—lifetime measurement,
RD—rising droplets; the data marked by “þ” are related to pressure
p¼ 0.0135 MPa.
Substance Ts, K lgJ
Tn, K
Method ReferenceExperiment Theory
Helium-3 1.81 20 2.46 2.50þ P 57
Helium-4
4.21 20 4.55 4.56 P 36
4.21 7 4.45 4.50 CH 58
4.21 6 4.58 4.50 LT 59
Neon 27.09 11 38.0 38.64 CH 52
27.20 7 38.0 38.40 CH 55
Argon 87.29 8 130.5 131.0 LT 31
87.29 11 130.8 131.5 CH 24
Krypton 119.78 7 181.0 182.0 LT 45
119.78 11 181.5 182.7 CH 25
Xenon 165.03 7 250.6 252.0 LT 45
165.03 11 251.9 253.3 CH 25
Hydrogen 20.38 11 27.8 28.05 CH 56
20.4 7 28.1 28.06 CH 55
Nitrogen
77.35 7 109.7 110.3 LT 41
77.35 11 109.9 110.7 CH 42
77.3 7 110.0 109.8 CH 27
77.35 17 110.3 111.1 P 46
Oxygen
90.19 7 134.0 134.8 LT 43
90.19 11 134.2 135.3 CH 42
90.1 7 134.1 134.2 CH 27
Chorine 8 366.5 367.8 LT 48
Methane 111.66 7 165.1 166.0 LT 44
111.66 11 166.0 166.6 CH 42
Ethane 184.95 7 267.4 LT 49
185.0 12 269.2 269.7 RD 40
Propane
231.1 7 327.2 327.4 LT 30
231.1 12 326.2 328.5 RD 40
231.1 12 329.2 RD 29
Isobutane
261.3 7 361.2 361.7 LT 40
261.4 12 361.0 360.9 RD 30
261.5 12 359.2 RD 29
n-Butane 272.7 12 378.4 378.6 RD 28
272.7 12 376.9 378.3 RD 30
Low Temp. Phys. 39 (8), August 2013 V. G. Baidakov 649
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The temperature and pressure dependences of the nuclea-
tion rate have been determined in the experiments on lifetime
measurement. The results of the experiments with liquid ar-
gon31 and liquid xenon45 are shown in Fig. 3. The regions of
sharp increase in Tn (the boundary of spontaneous boiling-up)
correspond to maximum superheating. The cryogenic liquids
are characterized by a very strong nucleation rate dependence
on temperature and pressure. While superheated liquid n-
pentane exhibits d lg J=dT ’ 2 K�1 at p¼ 0.1 pc, where pc is
the pressure at the critical point, and J¼ 107 s�1m�3,7 liquid
nitrogen shows d lg J=dT ’ 12 K�1.41
At low superheating, the experimental curves (Fig. 3)
exhibit characteristic inflection points due to the initiating
influence of the radiation background7,31,47 and weak spots
on the measuring cell walls.33 In cryogenic liquids the
initiating effect occurs at the nucleation rates J � Ji�
¼ ð1:5� 2:5Þ � 106 s�1m�3.15,33 The greatest resistance to
ionizing radiation is exhibited by xenon, which has Ji�
¼ 2:2 � 104 s�1m�3 [Ref. 45] (Fig. 3(b)).
3.3. Overheating of quantum liquids
At sufficiently low temperatures, the physical properties
of macroscopic systems are determined to a large extent by
quantum effects. Besides helium, the quantum effects are no-
table in the properties of the isotopes of hydrogen and, to a
lesser degree, neon. When the length of the thermal de
Broglie wave kT ¼ ð2p�h2=mkBTÞ1=2becomes larger than the
characteristic nucleus size R*, the thermal activation mecha-
nism of nucleation is replaced by quantum tunnelling of
nuclei8 (see Sec. 1). As estimated in Ref. 51, the crossover
temperature corresponding to the change in the nucleation
regime T� ’ 0:3 K in liquid helium and 3.2 K in hydrogen.51
In liquid neon T� ’ 1:5� 2 K.52 At T>T*, the quantum
phenomena may have an indirect effect on the kinetics of
boiling-up of a superheated liquid through the physical pa-
rameters and some special, unknown in classical systems,
nucleation sites: vortex lines and rings in superfluid he-
lium,51,53 electron and positron bubbles in liquid neon,
hydrogen, and helium,15,51,54 etc.
The measurements results of the attainable superheating
temperatures in neon,52,55 hydrogen,55,56 and helium iso-
topes36,57,58 are presented in Table 1.
The data on the attainable superheating temperatures in
neon obtained in Refs. 52 and 55 using continuous isobaric
heating in glass capillaries are in a good agreement with
each other. At pressure p¼ 0.1 pc, the attainable superheat-
ing was 8.0 K (J¼ 1011 s�1m�3).52 On the spinodal line,
DTsp ¼ Tsp � Ts ’ 9:9 K. The stability boundary of an elec-
tron bubble in neon corresponds to the superheating which
exceeds the experimentally achieved one by 0.3 K.52
First studies of the superheating of liquid hydrogen have
been carried out for the development of liquid hydrogen
FIG. 2. Line of attainable superheating Tn, spinodal line Tsp and binodal line
Ts for methane (a), oxygen (b), and nitrogen (c) obtained using the method
lifetime measurements, J¼ 107 s�1m�3 (1) and the method of continuous
heating, J¼ 1011 s�1m�1 [Refs. 41–44] (2). Dashed line shows the homoge-
neous nucleation theory, J¼ 1011 s�1m�3, C is the critical point.
FIG. 3. Nucleation rate in superheated liquid argon31 (a) and xenon45 (b) at
different pressures p, MPa: 0.19 (1), 0.36 (2), 0.81 (3), 1.10 (4), 1.40 (5) (a);
0.24 (1), 0.55 (2), 0.99 (3), 1.48 (4), 1.98 (5) (b). Dashed lines show the ho-
mogeneous nucleation theory calculated according to Eqs. (16) and (17);
thin solid lines show same theory calculated using Eqs. (16) and (18); dash-
dotted lines show the same theory calculated using Eq. (16) (B�B3, Ref.
18). In all the cases the work W* was defined in the approximation r¼r1.
650 Low Temp. Phys. 39 (8), August 2013 V. G. Baidakov
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bubble chambers and cryogenic pumps.60,61 Fig. 4 shows the
lower boundary of the liquid hydrogen sensitivity region to
high-energy particles that was obtained in the bubble chamber
of JINR in Dubna (Russia). In the study published by Hord
et al.,61 liquid hydrogen was overheated in a glass dewar
(V ’ 1 l) by a sharp pressure drop in the vapor cavity. The
instant of boiling-up was detected by a pressure surge. At pres-
sure drop rates faster than 7 MPa/s, the limiting stretching of
liquid hydrogen was similar to that obtained in the experiments
with continuous isobaric heating56 (see Fig. 4).
In Refs. 55 and 56, a significant spread in the boiling-up
temperatures of the normal liquid hydrogen has been observed.
For instance, at a heating rate of 0.03 K/s and pressure of
0.15 MPa, the boiling-up of liquid nH2 occurred within the
temperature range 26.0–28.1 K. With increasing the heating
rate the half-width of the boiling-up events distribution
decreases, remaining however larger than the theoretical value
dT1=2 ’ 0:03 K. The highest boiling-up temperatures regis-
tered in the experiment was taken as the attainable superheat-
ing temperature of hydrogen in Refs. 55 and 56.
The first measurements of the attainable superheating
temperature of liquid helium (4He) have been carried out
using the method of pulsed heating on single bismuth crys-
tals35,36 (see Table 1.). It has been noted that the superheat-
ing temperature Tn does not depend on the surface quality of
the bismuth thermometer/heater, external magnetic fields, or
X-ray radiation.
In the subsequent measurements by Nishigaki and
Saji,58 the limiting superheating of helium was measured by
continuous isobaric heating; the obtained values of Tn are in
good agreement with the data by Brodie’s group35,36 (Fig. 5)
if the differences in the recorded nucleation rates are
accounted for.
In the experiments on determining the average lifetime
of superheated 4He,59 a dependence �sðTÞ, qualitatively dif-
ferent from other liquids has been observed: �s has been
found to be almost temperature-independent throughout the
entire range from the saturation line to the boundary of spon-
taneous boiling-up. The limiting value of the initiated nucle-
ation rate in helium Jj� ’ 5� 105 s�1m�3 is less than in
argon, but higher than in xenon. The origin of this behavior
is still unclear. It should be noted that the limiting superheat-
ings of liquid helium at p> 0.08 MPa registered in Ref. 59
are significantly higher than those obtained in Ref. 36,
although they are related to a much lower nucleation rates
(Fig. 5).
Superheating of the superfluid phase of 4He has been
investigated in Refs. 62 and 63. The decay of the metastable
state in steady heating mode has been found to result in ei-
ther a sharp increase or decrease in temperature of the fluid
cell (V ’ 0:72 cm3). The first scenario corresponds to the
boiling-up of liquid helium, while the second is due to
the phase transition He II–He I. On the p, T–diagram (Fig. 6)
the points of He II–He I transition form a smooth line, which
is a continuation of the k–line into the region of metastable
states.
In the experiments on pulsed heating of 3He on bismuth
single crystals, the attainable superheating temperature could
be determined only at pressures less than 0.5 pc [Ref. 57]
(see Table 1).
3.4. Cavitation strength of liquefied gases
Liquid can exist as a metastable phase also at negative
pressures. There are no fundamental difference between the
superheated (p> 0) and stretched (p< 0) liquid. Both
boiling-up and cavitation at temperatures far from the abso-
lute zero must be described by the classical theory of nuclea-
tion by thermal fluctuations.
In the study of cavitation strength of liquefied gases both
quasi-static64–69 and dynamic37–39 methods have been
employed. Misener et al.64,65 have studied the cavitation
strength of nitrogen and superfluid helium. The liquid under
investigation was placed inside metal bellows and trans-
ferred into a metastable state by stretching the bellows. The
onset of boiling was registered visually. The measurement
results are shown in Table 2. No dramatic cavitation effect
FIG. 4. Limiting superheating of liquid hydrogen: experimental data from
Ref. 56 (1), experimental data from Ref. 61 obtained at different pressure
drop rates _p, MPa/s: 1.8 (2), 3.3 (3), 8.0 (4); the limit of radiation sensitivity
(5);9 the limit of stability of electron bubbles (6); Tn is the boundary of spon-
taneous boiling-up (J¼ 1011 s�1m�3); Tsp and Ts are the spinodal and bino-
dal lines, respectively; C is the critical point.
FIG. 5. Attainable superheating of liquid 4He: experimental data from
Ref. 36 (1), Ref. 58 (2), Ref. 59 (3). I and II show the homogeneous nuclea-
tion theory (Eqs. (16) and (17)) at J¼ 107 s�1m�3 and J¼ 1020 s�1m�3
respectively; psp and ps are the spinodal and binodal lines, respectively.
Low Temp. Phys. 39 (8), August 2013 V. G. Baidakov 651
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was found in the experiments with He II. The liquid was
found to detach from the walls of the bellows almost imme-
diately after the tension was applied.
Experiments to study cavitation in cryogenic fluids were
continued by Beams.66,67 In Ref. 66 a centrifugal method
has been used. The tensile strength of nitrogen, oxygen, ar-
gon and helium has been investigated using inertial loading
created by decelerating a U-shaped glass tube immersed in a
Dewar flask with liquid under study.67 All Beams’ attempts
to establish whether the initial rupture occurs at the liquid-
glass interface or in the bulk of stretched liquid have not
yield an unequivocal result.
To investigate the cavitation strength of liquid helium
also the methods employing the “fountain effect”68 and
osmotic pressure69 have been used. A range of studies of the
tensile strength of superfluid helium has been made by
acoustic methods.70,71 All these papers have yield stretching
values which are much lower than the theoretical values as
well as the results attained in quasi-static experiments.64–69
The results of Refs. 64–71 most likely indicate the heter-
ogeneous mechanism of the observed cavitation. All
attempts to eliminate from the system under study the readily
available and eaily activated boiling-up centers have not led
to positive results. Thus, the subsequent studies adopted a
strategy to achieve a homogeneous nucleation mechanism
not by removing the possible cavitation centers from the sys-
tem, but by neutralizing them in the shock mode of phase
transition. To achieve that the methods of focusing the
acoustic fields37,38 and pulse overheating of liquid on a plati-
num wire in a negative pressure wave39 have been used. The
first method has been used in the experiments with helium,
while the second one was employed in studies of argon and
its solutions
The main difficulty of acoustic experiments is to deter-
mine the fluid pressure at the cavitation onset. In Ref. 37, the
pressure was calculated from the data on the radiated power,
the absorption coefficient and the geometry of the piezoelec-
tric transducer. It was also determined by light diffraction on
the ultrasonic lattice generated by acoustic vibrations. Below
3 K, both methods yielded consistent results. The uncertainty
in determining the pressure amplitude has been estimated as
610%. The effective nucleation rate was 1018–1020 s�1m�3.
In the studies performed by Brodie’s group,37,72 the
acoustic cavitation in liquid helium has been investigated in
the temperature range of 1.6–4.2 K (Fig. 7). On the lower
boundary of this temperature interval, the maximal tensile
pressure amounted to �0.85 MPa. Xiong and Maris73 have
repeated the experiments done by Brodie’s group using a
similar hardware, but lowering the lower limit of the temper-
ature range down to 0.8 K. The maximal tension of super-
fluid helium73 did not exceed 0.3 MPa. Petterson, Balibar
and Maris38 tend to see the reason for such a strong mis-
match between the results of Refs. 37 and 72 and those of
Ref. 73 in the difficulties of estimating correctly the cavita-
tion onset pressure. In their own experiments, they studied
the statistical regularities of cavitation as a function of the
voltage magnitude applied to the piezo transducer. This
allowed for a more detailed comparison of the data with the
results of the homogeneous nucleation theory and helped to
verify the limiting tensile strength of helium (Fig. 7). In the
subsequent works Maris,74 Caupin and Balibar75 have deter-
mined the limiting tensile strength of liquid helium at tem-
peratures 0.2–1.2 K. The difference between the two
approaches to the definition of the cavitation pressure was
0.25 MPa (Fig. 7). When crossing the temperature of the k-
point of liquid helium, an inflection on the line of limiting
tensile strength has been observed.38,72
The method of acoustic field focusing has also been
used to determine the cavitational tensile strength of normal
Table 2. Ultimate tensile stresses in cryogenic liquids.
Liquid T, K
p, MPa
ReferenceExperiment Theory
Helium-3 1.0 �0.23 78
Helium4 2 �0.03 65
1.85 �0.014 �0.4 66
1.9 �0.016 �0.44 67
1.2 �0.003 69
2.09 �0.12 71
2.0 �0.21 73
2.0 �0.4 74
2.0 �0.6 37
2.0 �0.6 38
1.0 �0.9 75
Argon 85 �1.2 �19.0 67
117 �7.1 39
Nitrogen 71 �0.35 �6.0 64
75 �1.0 �14.0 67
Oxygen 75 �1.5 �35.0 67
FIG. 6. k-line of 4He: experimental data from Ref. 62 (1) and Ref. 63 (2).
ps is the line of liquid-gas phase equilibrium.
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liquid 3He.76–78 Caupin and Balibar78 have measured the
upper pn,max and lower pn,min limiting pressures in the cavita-
tion zone (Fig. 8). The effective nucleation rate was esti-
mated at J ’ 1022 s�1m�3. Extrapolation of the data on
pn,max and pn,min to T¼ 0 yielded the values of �0.305 and
�0.24 MPa, respectively.
To study the kinetics of the liquid argon boiling-up at
negative pressures the combination of two dynamic methods,
pulse stretching with the reflected compression wave and
pulse superheating on a platinum wire.39,79 Tensions down to
�10 MPa have been created by reflection of the compression
wave from the liquid-vapor phase boundary. At the time of
the passage of a tensile wave across the platinum wire, it was
heated up by a current pulse and its temperature was deter-
mined by measuring its resistance at the instance of boiling-
up. The effective nucleation rate was J ’ 1026 s�1m�3. The
results are shown in Fig. 9.
3.5. Nucleation in the vicinity of the critical point
The critical point is the point where the boundary of
essential instability (spinodal line) enters into the region of
stable states. In its vicinity both the correlation length n and
the fluctuation relaxation time are large. Large value of nleads to the fact that many irregularities and defects on the
wall of the measurement cell and foreign inclusions with the
size smaller than n in the bulk of the liquid have no effect on
the nucleation of the new phase. At the same time, due to
slowing down the heat transfer processes and reducing the
density difference of the coexisting phases, the boiling-up of
the metastable liquid loses its explosive character. The decay
time of the metastable phase sg becomes much larger than
FIG. 7. Limiting tensile stress in 4He: experimental data from Ref. 37 (1),
Ref. 73 (2), Ref. 38 (3), Ref. 74 (4), Ref. 75 (5). A, B, and D show the theory
of homogeneous nucleation in the approximation of r¼r1 at J¼ 10�9,
108, and 1021 s�1m�3, respectively. Dashed-dotted line shows the spinodal
line, ps denotes the binodal line, C is the critical point.
FIG. 8. Limiting tensile stress and spinodal line in 3He: experimental data
from Ref. 57 (1), the upper and lower boundaries of the limiting tensile
stress according to the data from Ref. 78 (2). A and B show the results of
the homogeneous nucleation theory in the approximation r¼r1 at
J¼ 2.5� 1010 and 1020 s�1m�3, respectively. Dotted line shows the theory
of homogeneous nucleation and density functional theory for
J¼ 1010 s�1m�3.122,125 Dashed-dotted line shows the spinodal line accord-
ing to Ref. 125. Line I is the spinodal line approximated according to
Ref. 75, ps denotes the binodal line, C is the critical point.
FIG. 9. The boundary of the attainable superheating of liquid argon (points),
ps and psp denote the binodal and spinodal lines, respectively.39 The symbols
(�) and (�) refer to the experiments carried out at different times with dif-
ferent amount of the substance under investigation filled into the measure-
ment cell. Dashed lines show the calculation results for the limiting tensile
stress according to the homogeneous nucleation theory in the macroscopic
approximation (r¼r1) at the nucleation rate J¼ 1020 s�1m�3 (upper line);
J¼ 1025 s�1m�3 (lower line); C is the critical point.
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the average delay time before appearance of a critical nu-
cleus �s. In this case, the volume fraction of the vapor phase
g obtained at the time s after the critical nucleus has been
formed can serve as the characteristic parameter of the phase
transition. In the initial stage of decay (gðsÞ 1) the inter-
action between the growing nuclei can be excluded from the
consideration. Then80
gðsÞ¼ðs0
Jðs0ÞVðs� s0Þds0 ; (36)
where V(s� s0;) is the volume which the bubble originated
at the time s0; reaches at the time s. Near the critical point
where the vapor phase growth is limited by heat supply and
the nonstationary effects are still small, Eq. (36) can be
expressed in the form81
g sð Þ¼ 8p15
DTbdT
tTc
� �3=2
Js5=2 ; (37)
where DT is the temperature conductivity coefficient, b is the
critical index of the phase coexistence curve, dT¼ Ts(q)� Tis the extent of liquid supercooling in an isochoric process.
The first results on the attainable superheating of a
single-component liquid in the vicinity of the critical point
have been obtained by Dahl and Moldover82 in the experi-
ments on isochoric heat capacity of 3He. Systematic studies
of phase separation kinetics in the vicinity of the critical
point of carbon dioxide and xenon have been carried out in
Refs. 81 and 83. The liquid was superheated in glass cells
placed in a calorimeter. The collapse of the metastable state
was detected by the thermal effect of a phase transformation.
Experiments were carried out in isochoric15,83 and isobaric81
conditions at the temperature change rate in the range
0.05–5.0 K/s. Using the thermograms obtained from the sets
of 5-10 measurements, the most probable temperature of the
phase transformation Tb has been determined. The deviations
of this temperature obtained for liquid xenon in an isobaric
heating process ( _T ¼ 0:2 K/s) from the attainable superheat-
ing temperature Tn calculated from the homogeneous nuclea-
tion theory are shown in Fig. 10. The x-axis shows the
values of the reduced temperature along the boundary curve
t0 ¼ 1� Ts(p)/Tc. Fig. 10 shows the attainable superheating
temperature of liquid xenon Tn obtained in the experiments
on determining the average lifetime45 alongside with the
data of Ref. 81
At t0< 10�2 the phase transition temperature Tb recorded
in the experiment is substantially higher than the temperature
Tn calculated from the homogeneous nucleation theory. For
t0 ’ 10�3 the value Tb� Ts is about twice as large as the the-
oretical value of Tn�Ts. As already noted, this is due to the
fact that the temperature Tn corresponds to the instance of
appearance of the first critical nucleus, while the temperature
Tb is recorded when some fraction of vapor phase is already
formed in the cell and the phase transition is registered.
Microphotography of the sample revealed that the phase
transition signal corresponds to g� ’ 0:2, and the character-
istic time of phase decay s* at the experimental conditions
�1 s. Due to the very strong dependence of J on T, Eq. (37)
is not sensitive to the choice of g*. Substituting the values s�g
and g* into Eq. (37) allows determining J and Tn. The results
of such a calculation are shown in Fig. 10 as line B. The
good agreement between the experimental data and calcula-
tion shows that the adopted model correctly reflects the main
features of the observed phenomenon, at least until
t0 ’ 10�3. At t0 < 10�3 and J> 1017 s�1m�3 the processes of
diffusion growth of the vapor phase and coalescence are not
separated in time any longer and the approximation of free
growth, Eq. (36), is not applicable.
3.6. Superheating of solutions with complete solubility of thecomponents
Liquid solutions are characterized by a variety of phenom-
ena associated with phase transitions. Due to the emergence of
new thermodynamic degrees of freedom upon dissolution,
there are exist, alongside with the liquid-vapor critical point,
the critical points of higher order (tricritical point, the final crit-
ical point, etc.), and, in addition to the instability with respect
to the vapor phase formation, the instabilities related to the gas
release (decomposition of the liquids supersaturated with gas)
and phase separation are possible.
First measurements of the attainable superheating tem-
perature have been performed in binary and ternary solutions
composed of methane hydrocarbons (ethane, propane, bu-
tane) using the method of rising droplets.30,84 The experi-
mental data have been obtained at atmospheric pressure and
the nucleation rate J¼ 1011 s�1m�3. The experimental error
in determining the attainable superheating temperature has
been estimated at 6(0.5–1.0) K.
With the respect to their thermodynamic properties, the
studied systems are nearly perfect. For binary solutions, the
dependence of the attainable superheating temperature on
the molar fraction is linear within the experimental error.
For the ternary system ethane–propane–n-butane, the super-
heating temperature calculated following the additivity rule
based on the data on the Tn of pure liquids and their molar
fractions in the mixture
Tn ¼Xi¼1
ciT0ni ; (38)
is close to the experimental value.
FIG. 10. Discrepancy between the experimental data for the superheating
temperature of liquid xenon and those calculated according to the homoge-
neous nucleation theory in the approximation r¼r1 at J¼ 107 s�1m�3:
experimental data from Ref. 45 (1), Ref. 81 (2). A shows the calculation
results obtained using Eqs. (16) and (17); B shows the calculation results
obtained using Eq. (37) for sg¼ 1 s.
654 Low Temp. Phys. 39 (8), August 2013 V. G. Baidakov
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The attainable superheating temperature in the solutions
of cryogenic liquids (Ar–Kr,85 O2–N2 [Refs. 15 and 86]) has
been studied in a wide range of pressures and nucleation
rates using the method of lifetime measurements. Fig. 11
shows the temperature dependence of the nucleation rate in
Ar-Kr solution at several concentrations of krypton in a
semi-logarithmic scale. The vertical size of the symbols cor-
responds to the statistical error in determining the rate of
nucleation. Under the conditions of constant pressure and
concentration, the dependences J(T) and �sðTÞ in solutions
are similar to those of pure liquids. All the experimental
curves exhibit regions of sharp increase in the nucleation
rate. Increasing the concentration of krypton in the solution
leads to a monotonous decrease in the crossover rate Ji*
from initiated to spontaneous nucleation.
The attainable superheating temperature in the solution
as a function of concentration at the fixed nucleation rate
J¼ 107 s�1m�3 at pressure values p¼ 0.5 and 1.0 MPa is
shown in Fig. 12. Unlike the system Ar–Kr, wherein the
maximum deviation of the superheating temperature from
the additivity rule is up to 4.3 K (c¼ 0.5) at pressure
p¼ 1.0 MPa, the solution O2–N2 exhibits a significantly
lower deviation of ’0.75 K (c¼ 0.3).
Similar to the line of attainable superheating, the spinodal
line of O2-N2 solution is, to a first approximation, a linear func-
tion of the concentration (Fig. 12). At pressures up to 0.5 pc in
the entire range of concentrations, the ratio of the attainable
superheating DT¼ Tn� Ts to the value DTsp¼ Tsp�Ts is
0.75–0.8.
3.7. Nucleation in solutions of sparingly soluble components
Depending on the chemical nature of the materials and
thermodynamic parameters of state, gas solubility in liquid
can vary within a relatively wide range. Large solubility is
characteristic for carbon dioxide in liquid hydrocarbons of
the methane series. Attainable superheating of liquid pro-
pane and butane saturated with carbon dioxide has been
investigated using the rising droplet technique by Mori
et al.29 The experiments were conducted at pressures ranging
from atmospheric to p ’ 1.0 MPa and at carbon dioxide con-
centrations up to 35 mol. %. Dissolution of CO2 shifts the
boundary of the attainable superheating towards lower tem-
peratures. The attainable superheating temperature of liquid
isobutane (p ’ 0.1 MPa) is reduced by 25 K upon dissolving
of 33 mol. % CO2 in it.
Solutions of helium in cryogenic liquids belong to the
class of weak solutions. Solubility of helium in oxygen and
nitrogen at pressures below 5.0 MPa does not exceed
5.2 mol. %. Interestingly, helium in such systems behaves as
a surfactant, lowering the surface free energy at the liquid-
gas interface.87
To study the nucleation kinetics in the solutions of
O2–He,88 N2–He,89 Ar–He,90 the method of lifetime meas-
urements has been used. The temperature dependence of the
nucleation rate in the superheated gas-saturated liquid is sim-
ilar to that of a solution with complete solubility of the com-
ponents (Fig. 13). At small superheatings, the experimental
isobars exhibit characteristic flat areas, which give way to
the areas of sharp increase in the nucleation rate.
FIG. 11. Nucleation rates in the superheated argon-krypton solutions at
p¼ 1.6 MPa and various concentrations c: 0 (1), 0.109 (2), 0.382 (3), 0.428
(4), 0.708 (5), 0.938 (6), 1 (7). Dashed lines show the results of homogene-
ous nucleation theory, Eqs. (16) and (18) at r¼r1; the same calculation
using Eqs. (16) and (18) at r¼r(R*) is shown by dashed-dotted lines.
FIG. 12. Attainable superheating (1, 2), binodal (10, 20), and diffusion spino-
dal (100, 200) lines for the nitrogen-oxygen solutions at pressure p¼ 0.5 MPa
(1, 10, and 100) and 1.0 MPa (2, 20, and 200).
FIG. 13. Temperature dependence of the nucleation rate in the oxygen-
helium solution at pressure p¼ 1.667 MPa and different concentrations c,
mol. %: 0 (1), 0.08 (2), 0.14 (3), 0.20 (4). Dashed line shows the homogene-
ous nucleation theory at r¼r1 for c¼ 0; dotted line shows the same calcu-
lation at r¼r(R*) and the concentrations shown above.
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Dissolving� 0.1 mol. % of helium in liquid oxygen lowers
the attainable superheating DTn¼ Tn�Ts in the region of
spontaneous boiling-up by approx. 10%. At the same time,
in the region of initiated nucleation, the average lifetime of
the metastable solution is reduced three- to four-fold.
The concentration dependence of the attainable super-
heating temperature of gas-saturated solutions is close to lin-
ear. When equal amounts of helium are dissolved in nitrogen
and oxygen, the ratio [Tn(0) – Tn(c)]/[Tn(0) – Ts(0)] is lower
in N2–He system compared with O2–He system. This corre-
lates with the effect of helium on the surface tension of liqui-
fied gases.87 The lower the ratio of the critical temperatures
of the dissolved gas and solvent, the higher is helium adsorp-
tion in the surface layer.
The behavior of the pressure and concentration depend-
ences of the attainable superheating temperature in gas-
saturated liquids stays the same if the solute has a higher
critical temperature compared with helium and, conse-
quently, a higher solubility. For all the gas-saturated liquids
studied up to date the increase in the concentration of the
dissolved gas always leads to a decrease in the superheating
temperature of the solvent. Fig. 14 shows the concentration
dependence of Tn for ethane-methane solution.91 At the
same reduced temperature and solvent pressure, the solubil-
ity of methane in ethane is almost an order of magnitude
higher than that of helium in oxygen.
The first study of the ternary solutions of cryogenic liquids,
N2-O2-Ar and N2-O2-He, has been presented in Ref. 15.
3.8. A comparison of theory and experiment. The surface tensionof vapor-phase nuclei
Let us summarize the results of the experimental studies
of the kinetics of spontaneous boiling-up of liquefied gases
and compare the obtained data with the thermal fluctuation
theory of homogeneous nucleation. The experiments on
monitoring the spontaneous boiling-up have been carried out
by different research groups for many liquids in a wide range
of temperatures and pressures. The obtained data show good
reproducibility and practically do not depend on the method
of superheating the liquid (the method of measurement Tn).
The results of investigations at atmospheric pressure are
assembled in Table 1, where, alongside the experimental
data, we indicate the value of Tn, calculated using the theory
of homogeneous nucleation in the “macroscopic approx-
imation” (r¼r1), i.e., without taking into account the de-
pendence of the nucleus surface tension on its size. Figs. 2–5
and 7–15 display, along with the experimental data, the theo-
retical values of Tn. Depending on the material and the
method of investigation, the discrepancy between theory and
experiment ranges from a few hundredths of Kelvin to 1.5 K.
When making such comparison, we should be aware of not
only the measurement error in Tn, but also inaccurate values
used in the theoretical calculations.
The right-hand side of Eq. (16) contains no adjustable
parameters which would change from one substance to
another or with pressure and temperature. However, different
versions of the homogeneous nucleation theory16–18 differ in
the value of the kinetic coefficient B. These differences can
reach two orders of magnitude and are entirely transferred to
the calculated value of J. At pressures close to atmospheric
pressure, two orders of magnitude change in J corresponds
to the change in argon temperature of 0.23 K, which is just
0.5% of the superheating DTn.
For all the substances studied, as a rule, Texpn < Tcalc
n (see
Table 1). This may indicate that the theory underestimates
the value of B or overestimates the work W*, for instance,
due to the fact that it does not take into account the depend-
ence of the nucleus properties on its size. If the mismatch
between theory and experiment is entirely attributed to the
dependence of the surface tension of the critical bubble on
its size, then, by substituting the experimental values of J in
Eq. (16), the “microscopic” surface tension r(R*) can be
found. At positive pressure and the nucleation rates
J< 108 s�1m�3, the value of r(R*) for the critical bubble is
by 5%–7% lower than that for a flat interface. It should be
FIG. 14. Concentration dependences of the attainable superheating tempera-
ture for the ethane-methane solutions at pressure p¼ 1.0 (1) and 1.6 MPa (2)
and the nucleation rate J¼ 107 s�1m�3. The values of Tn for pure ethane are
given according to Ref. 49. Dashed-dotted line shows the calculation results
obtained using the homogeneous nucleation theory in the approximation
r¼r1. Dashed lines show the additive approximation for the attainable
superheating temperature.
FIG. 15. Temperature dependence of the nucleation rate in argon at
p¼ 0.5 MPa (Ts¼ 105.9 K): the results obtained by the method of lifetime
measurements at J¼ 106–108 s�1m�3 [Ref. 31] (1); by the method of contin-
uous isobaric heating J¼ 1012 s�1m�3 [Ref. 25] (2); by the method of pulse
superheating on a platinum wire, J¼ 1026 s�1m�3 [Ref. 39] (3). Solid line
shows the results of the nucleation theory in the approximation r¼r1.
656 Low Temp. Phys. 39 (8), August 2013 V. G. Baidakov
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noted that changing B by two orders of magnitude can be
compensated by adjusting the surface tension by ca. 2%.
To verify the theory, studies of J as a function of super-
heating are of great importance. The theory links the deriva-
tive d lg J/dT to the temperature dependence of the Gibbs
number G*
d ln J
dT’ � dðW�=kBTÞ
dT¼ �GT : (39)
Fig. 15 shows the temperature dependence of the nuclea-
tion rate in liquid argon at pressure p¼ 0.5 MPa. This figure
displays the data obtained by the methods of average life-
time,31 continuous isobaric heating,25 and pulse superheating
on a platinum wire.39 Solid line shows the calculated values
of J obtained using Eq. (16). This theoretical curve correctly
represents the temperature dependence of J, and hence the
work of formation of a critical bubble. It should be noted
that in the experiments on lifetime measurements, the value
of the derivative d ln J/dT is typically lower than its theoreti-
cal value. The error in determining d ln J/dT can reach up to
20%–35%.
Differentiating the logarithm of the nucleation rate
(Eq. (16)) with respect to pressure (T¼ const), and taking
into account Eq. (4), we obtain
d ln J
dp’ � dðW�=kBTÞ
dp¼ �Gp ¼ �
V�kBT
; (40)
i.e., the volume of the critical bubble V* determines the slope
of the pressure dependence of J.
For different liquids, regardless of the pressure, the values
ln J¼ const correspond to almost identical Gibbs numbers.
This indicates the existence of some sort of a thermodynamic
boundary of attainable superheating Tn (J, p). In the reduced
variables p/pc, Tn/Tc at J¼ const, the values of T/Tc for differ-
ent substances satisfy a single-parameter law of corresponding
states.92
At negative pressures, the issue of agreement between
the homogeneous nucleation theory and experiment is not so
straightforward. A large uncertainty in determining the limit-
ing tensile pressure and the nucleation rate using the acoustic
techniques does not permit such a detailed comparison, as in
the case of positive pressures. The method of the pulsed fluid
superheating in a tensile wave is more informative in this
regard. An experiment with different pressure pulse duration
allows determining the temperature dependence of ln J,
and hence the derivative GT¼ dG/dT. Fig. 16 shows the
values of GT for argon obtained using such approach. At
J¼ 1026 s�1m�3 the value of GT is weakly dependent on pres-
sure. In the pressure range from �8 to 0 MPa, the theoretical
value of GT calculated in the macroscopic approximation
(r¼r1) is systematically below the experimental value. In
liquid argon at T/Tc¼ 0.95 (T ’ 143 K) and J¼ 107 s�1m�3,
the critical bubble radius R* is equal to 6.4 nm, while at
J¼ 1026 s�1m�3 it has the value R* ’ 3.4 nm. At T/Tc¼ 0.6
(T ’ 90 K) and the same values of the nucleation rate, the
critical bubble radius is then 1.4 and 0.9 nm, respectively.
The vapor density at low temperatures (T/Tc ’ 0.5–0.6) is
low therefore the critical bubbles in highly stretched liquid
are essentially empty.
The effect of nucleus curvature on the nucleation kinetics
can be accounted for by using the van der Waals capillarity
theory for calculation of W*.12 In addition to W*, this
approach also allows us to obtain information on the surface
tension of the nucleus.41 The calculated dependences r(R*)
for nitrogen, oxygen and methane bubbles at T¼ 0.95 Tc in
the region of spontaneous nucleation are shown in Fig. 17.
The surface tension of the bubbles with the radiuses R* of 5
and 2.5 nm are 4% and 15% lower than that of flat interfaces,
respectively.41,42 Accounting for the dependence r(R*) can
reconcile the theory and practical experience within their
total error margin both for positive and negative pressures. At
the triple-point temperature and J¼ 1026 s�1m�3, the discrep-
ancy between r1 and r(R*) amounts to 15%.
Basic laws of the nucleation kinetics which manifest
themselves in single-component fluids are also common for
FIG. 16. Derivative GT as a function of pressure at J¼ 1010 s�1m�3. The
height of the rectangles corresponds to the error in determination of GT.
Dashed line shows the calculation results obtained using the homogeneous
nucleation theory in the macroscopic approximation r¼r1; solid line
shows those with the dependence r¼r (R*) taken into account.
FIG. 17. Reduced surface tension of critical bubbles as a function of the
reduced radius of surface curvature: the van der Waals capillarity theory
(1-3);42 obtained from the data on the attainable superheating (4-6); flat inter-
face boundary (7-9) for nitrogen (1, 4, 7), oxygen (2, 5, 8) and methane (3, 6, 9).
The fitting parameters are the critical parameters pc, qc, Tc of the substances
under study.
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binary solutions. A fundamentally new phenomenon here is
that the dissolving liquid or gas in another liquid commonly
leads to convergence of the results of the classical theory and
experiment, and, for a number of solutions, the experimental
values of Tn even surpass the theoretical ones (O2-N2, C2H6-
CH4).85,86 This is not observed for the solutions of helium
since its solubility in cryogenic liquids is very small. This
phenomenon, as well as the effect of “undercooling” single-
component liquids below the theoretical values Tn, can be
interpreted in terms of the van der Waals model93–95 as a
result of the dependence r(R*).96 Fig. 18 shows the depend-
ence of the surface tension of a critical bubble in argon-
krypton solution on the surface curvature for several values
of concentration. For pure components, the surface tension is
a monotonously decreasing function of the curvature of the
interface. This means that the work required to create a criti-
cal bubble in a single-component system is always less than
that calculated within the macroscopic approximation
(r¼r1) of the classical homogeneous nucleation theory. In
the case of a solution, the dependence r(R*) exhibits a differ-
ent behavior. Rising the concentration of either one or
another component, leads to a maximum in the curve r(R*)
and a lower difference between the values of surface tension
on a planar interface and critical bubble r(R*). This concili-
ates the experimental and theoretical results if the theory
takes into account the dependence r(R*). Thus, abandoning
the macroscopic approximation within the classical homoge-
neous nucleation theory can reconcile theory and experiment
for both single-component systems and binary solutions.15
3.9. Initiated and heterogeneous nucleation
Phase transitions tend to occur in the presence of exter-
nal influences (radiation, acoustic, electromagnetic, and
other fields) and foreign substances, for instance, a gas dis-
solved in the liquid or adsorbed in the wall cracks and areas
with a lower wettability on suspended solid particles or
surfaces in contact with the liquid. The main difficulty in
describing such processes is the fact that it is absolutely
impossible to account for the whole variety of uncontrollable
factors influencing the process of nucleation. In order to
approach the solution of this problem, it is reasonable to
study the response of a “pure” system to a given factor ini-
tiating the boiling-up of a liquid.
At low nucleation rates (J< 2�106 s�1m�3), the experi-
mental curves deviate from the theoretical lines (Figs. 3, 11,
and 13). The reason for such discrepancy is related to the
influence of ionizing radiation.7 The initiating effect of irra-
diation can be explained by heat spikes arising due to decel-
eration of secondary electrons.9 The liquid is constantly
exposed to cosmic rays and background radiation. The inten-
sity of the background radiation along with the radiation re-
sistance of liquid determines the rate Ji* at which the
inflection of the experimental curves starts. The results of
the experiments on the influence of gamma-radiation on a
superheated liquid7,31 allow us to understand the cause of
premature (as compared to the predictions of the homogene-
ous nucleation theory) boiling-up of superheated fluids at
relatively long average lifetimes. The similarity of the J(T)
curves obtained in natural conditions and under c-radiation
(Fig. 19) indicates that in both cases the boiling-up is due to
the same initiating factor, but of different intensity.
While boiling-up of a superheated liquid upon the influ-
ence of radiation is the result of local heat release, nucleation
in acoustic fields is initiated by strong tensile forces arising
in the liquid when pressure is lowered. In addition to sponta-
neous nucleation, the acoustic wave can initiate the activity
of readily existing centers on various foreign inclusions.51 In
Refs. 97 and 98 using the method of lifetime measurements,
the boiling-up kinetics of superheated liquid xenon, oxygen
and argon in weak acoustic fields with the frequency of
�0.7–1 MHz has been studied. The liquid was superheated
in glass cells and acoustic waves were excited by an external
piezoelectric transducer after the liquid was brought into the
metastable state.
Switching on the acoustic field increases the probability
of nucleation (Fig. 20). The boundary of spontaneous
boiling-up of liquid is shifted towards lower temperatures
upon increasing the amplitude of oscillation in accordance
with the homogeneous nucleation theory. The mechanism
of acoustic cavitation at temperatures far from the border of
spontaneous boiling-up is more complex. Here the effect of
FIG. 18. Dependence of the reduced surface tension of critical bubbles in the
argon-krypton solution on the curvature of the interface at T¼ 145 K and var-
ious concentrations c: 0 (1), 0.1 (2), 0.3 (3) 0.5 (4), 0.7 (5), and 1.0 (6) [85].
FIG. 19. Temperature dependence of the average lifetime of superheated liquid
argon at p¼ 1.0 MPa in normal conditions (1) and under c-irradiation (2).31
658 Low Temp. Phys. 39 (8), August 2013 V. G. Baidakov
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the acoustic field depends on the radiation resistance of the
fluid.97
In the studies of liquid 4He superheating on bismuth
single-crystals, a strong initiating effect of light emission has
been observed.99,100 If liquid helium was superheated above
the 4.49 K (the attainable superheating temperature of he-
lium at atmospheric pressure is 4.55 K [Ref. 36]), a light
pulse of duration from 2 ls to 1 ms and intensity of 1–200
mW�cm�2 resulted in a sharp decrease in temperature of the
single crystal. Increasing the light intensity up to I¼ 200
mW�cm�2 was accompanied by increasing the cooling
effect. At I> 200 mW�cm�2 and the light wavelength above
420–450 nm the effect of the single crystal cooling was not
registered. In order to explain the initiating influence of light
exposure on the boiling-up of liquid helium, a photoelectron
model has been proposed.100
Heterogeneous nucleation is a more common case than
homogeneous. At present, the heterogeneous nucleation
theory is built along the same lines as the homogeneous one.
In the steady state, the net flux of nuclei of critical size is
equal to the sum of fluxes of nuclei that arise on different
parts of the surfaces of the impurity particles and walls of
the vessel. In the case of nucleation on smooth clean wall,
all the liquid molecules in contact with it can be considered
as the potential centers of new phase nucleation, so we can
write7,15
Jget ¼ qgetBget expð�WW�=kBTÞ; (41)
where qget is the number of molecules in contact with the
wall per unit volume of liquid, Bget is the kinetic coefficient,
which, to the first approximation, can be taken equal to its
value for homogeneous nucleation, W is the factor which
takes into account the reduction of the formation work for a
nucleus formed on the wall, Wget ¼ WW�.Real solid surfaces are never perfectly clean and smooth.
Nucleation of the vapor phase typically occurs in the micro-
cracks, crevices and other defects of the rough surface.
Cleanliness and good wettability of solid materials by
liquefied gases brings hope that a high superheating can be
achieved even in the cells with a rough surface.
Experimental confirmation of this has been obtained in Refs.
101–103 during the studies of lifetime of liquid oxygen,
nitrogen, and xenon in stainless steel and copper tubes. The
surface area of the tube in contact with the superheated
liquid was 4.25 cm2 and its volume was 95 mm3. An average
size of microheterogeneities on the inner wall of the tube
was of the order of 5 lm. The temperature dependence of the
average lifetime of superheated nitrogen in metal and glass
tubes is shown in Fig. 21.
In the experiments with metal tubes, three characteristic
regimes of the superheated liquid boiling-up have been dis-
cerned.101 The initial stage of an experiment (regime I, run-
ning-in) is characterized by significant dispersion of the
values �s and irreproducibility of the data upon changing the
thermodynamic state of the liquid. After several hundred
boiling events, the average lifetimes reproduced well on one
and the same tube upon varying p and T (regime II). The ex-
perimental data obtained in regime II form a smooth curve.
Reproducibility is maintained upon replacing a portion of
the liquid in the tube. During the long-term measurements a
point in time was always observed, after which a sharp
increase in the boiling-up delay time occurred (the transition
to regime III). In regime III, the values of �s are reproduced
within the statistical errors in all subsequent experimental
runs, even after replacing the cells. Increase in pressure has
been found to lead to a reduction in the running-in time of
the cell.
The temperature dependences �s obtained in metal and
glass tubes are qualitatively similar (Fig. 21). At the inflex-
ion points of experimental isobars, the averaged lifetimes of
a superheated liquid in metal tubes exceed the values of �sobtained in glass tubes three- to five-fold. This may be due
to somewhat higher intensity of background radiation in the
glass.
At the border of the spontaneous boiling-up of liquid,
the measurement results obtained in metal and glass tubes
match each other within the experimental error, both in
terms of superheating temperature Tn and the derivative
FIG. 20. Temperature dependence of the average lifetime of superheated
liquid oxygen at static pressure p¼ 1.171 MPa in normal conditions (1) and
in the applied ultrasound field with the voltage amplitude on the piezo emit-
ter Um, V: 10 (2) and 15 (3).97 Dashed line shows the calculation results
obtained using the homogeneous nucleation theory in the macroscopic
approximation (r¼r1)
FIG. 21. Average lifetime of superheated liquid nitrogen measured in cop-
per (1) and glass (2) tubes at different pressures p, MPa: 0.5 (A); 1.0 (B); 1.5
(C).102 Dashed line shows the calculations made using the homogeneous
nucleation theory in the macroscopic approximation r¼r1. Dashed-dotted
line shows those with the dependence r¼r (R*) taken into account.
Low Temp. Phys. 39 (8), August 2013 V. G. Baidakov 659
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d ln �s=dT, and, in view of the discussion in Sec. 3.8, also the
results of the homogeneous nucleation theory. This may con-
firm reachability of homogeneous nucleation conditions
upon superheating the liquid not only in the glass cells but
also in the metal ones.
In these experiments only regimes I and II belong to het-
erogeneous nucleation. During the process of running-in the
metal tubes, removal and dissolution of the gases adsorbed
in the microrecesses of the solid surface takes place. At high
values of superheating, the size of critical nucleus is in the
range of several to tens of nanometers. For such small bits of
a new phase, many micro-inhomogeneities can be consid-
ered as a flat smooth wall. However, when calculating the
factor W, it is now necessary to consider the effect of line
tension.102
The external fields can not only initiate the existing
boiling-up centers in a superheated liquid, but also generate
new ones, such as electron and positronium bubbles or vor-
tex lines and rings in superfluid helium. Cavitation on elec-
tron bubbles in liquid 3He has been investigated in Ref. 104.
Electrons were injected into the liquid helium and electron
bubbles were formed. In the experiment, the pressure pe at
which the electron bubbles lose their stability was deter-
mined by focusing acoustic waves. At T¼ 0, the value
pe¼�0.7 MPa. In the temperature range 1.07–2.5 K, in
order to relate the the data obtained experimentally with the
results of the theory,105 the gas pressure in the electron bub-
ble should be taken into account. A possible influence of
quantum vortices in liquid helium on the nucleation process
has been discussed in Ref. 106.
4. Nucleation kinetics near absolute zero temperature
4.1. Limiting supersaturation of 4He-3He solutions
Supersaturated superfluid 4He-3He solutions represent
one of the few metastable systems in which the manifesta-
tions of quantum tunneling of nuclei in liquid phase can be
expected. These solutions remain liquid down to absolute
zero temperature, wherein the solubility of 3He in 4He is lim-
ited. The phase separation occurs when the molar fraction of
the light isotope reaches 0.064.
Significant level of supersaturation in 4He-3He solutions
was first observed in the experiments where their physical
properties were studied.107,108 Brubaker and Moldover111
investigated the limiting supersaturation of superfluid solu-
tions of helium isotopes and have obtained the values of
approximately one order of magnitude lower than those pre-
dicted by the classical thermal-fluctuation nucleation theory.
For studying the spontaneous nucleation kinetics in the
solutions of helium isotopes, the methods of continuous con-
centration variation,109–112 depressurization,111 and simulta-
neous variation of pressure and concentration.113–115 The
onset of the phase transition was detected through changes
in the speed of first sound,110–112 amplitude of the NMR sig-
nal,111 and dielectric constant of the liquid.110–115
In the method of continuous concentration variation, the
phenomena specific for superfluid solutions, termo-osmosis
and fountain effect, are used.110 Supersaturation is created in
the measuring cell (V ’ 6.17 cm), where the light isotope of
helium flows from the control cells through a connecting
capillary. The rate of the concentration change is controlled
by varying temperature of the control cell.
The method of depressurization is based on the depend-
ence of the phase equilibrium parameters on pressure. If a
pressurized solution is near the phase equilibrium line, then,
after pressure drop, it will be in a metastable state. In Ref.
111 by reducing pressure ( _p ’ 1:5 � 10�4MPa/s), 4He-3He
solution was first brought to the part of the metastable region
where its lifetime amounted to several hours and then heated
up ( _T ’ 15� 30 lK/s) to initiate nucleation.
The results of the measurements of limiting supersatura-
tion in the superfluid phase of 4He-3He solution at pressure
p¼ 0.05 MPa are shown in Fig. 22.110–112 For temperatures
T> 50 mK, the limiting supersaturation values obtained
using the methods of depressurization and continuous con-
centration variation are in good agreement with each other.
At T< 50 mK the depressurization method gives higher val-
ues of supersaturation.
Before we proceed to the kinetics of phase separation in4He-3He solutions at very low temperatures, we compare the
data of Refs. 110–112 with the results of the thermal-
fluctuation nucleation theory. This theory has been extended
to superfluid solutions by Lifshitz et al.116 Neglecting the
energy dissipation during nucleation, the kinetic coefficient
can be expressed as
BðT > T�Þ ¼ BðT¼0Þ2
3
pM
r
� �1=2kBT
�h� 1
" #�1
: (42)
Here, B(T¼ 0) is determined by expression (32).
The work of formation of a critical nucleus has been cal-
culated by Eq. (4), where p00� � p ¼ ðl00� � lÞq00� , l00� and q00�are the chemical potential and density of 3He nucleus at
given temperature and pressure. For small levels of supersa-
turation we can assume Dl ¼ l00� � l ’ ð@l=@cÞðc� csÞ or
Dl ¼ ðls � lÞð1� cs=cnÞ.The calculation results for the limiting supersaturation
in 4He-3He solutions according to Eq. (16) and using
FIG. 22. Limiting values of supersaturation in the superfluid phase of4He-3He solutions. Experimental data: the method of continuous concentra-
tion variation110 (I); the method of depressurization111 (II). Thermal-
fluctuation theory of nucleation: solid lines show the results obtained by using
the approximation Dl ¼ ð@l=@cÞðc� csÞ for calculation of W*; dashed lines
show the same calculations but in the approximation of Dl¼ ðls � lÞð1� cs=cnÞ for different values of J, s�1m�3: 10�10 (1); 0 (2) 1010
(3). Tl and Tsp denote the phase equilibrium and spinodal lines, respectively.
660 Low Temp. Phys. 39 (8), August 2013 V. G. Baidakov
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Eq. (42), z0¼ 1 for three different values of the nucleation
rate and two approximations for Dl are shown in Fig. 22.
In the regime of continuous increase of 3He concentra-
tion at the rate _c, the effective nucleation rate is given by
JðcnÞ ¼2:44 _c
dc1=2V: (43)
At T¼ 90 mK, _c¼ 4� 10�6 s�1, and Dcn¼ 1.1� 10�2,
the half-width of the distribution of the phase transition
events was dc1/2¼ 1.5� 10�3.112 Then from Eq. (43) we
obtain J¼ 1.1� 103 s�1m�3. Fig. 22 indicates significant
discrepancy of the experimental and theoretical values of
Dcn at this nucleation rate. Such a discrepancy is likely due
to the influence of external forces or readily available nucle-
ation centers.
To study the limiting supersaturation in 4He-3He solu-
tions at ultralow temperatures (from 400 lK to 160 mK),
Satoh et al.113–115 have used the method of simultaneous
variation of concentration and pressure. The measuring cell
had the volume V ’ 77 cm3. The moment of phase separa-
tion was detected by a jump in concentration.
To compare the results of the homogeneous nucleation
theory and the experimental data,113–115 the latter must be
converted into a certain metastability degree which is
“isomorphic” to single-component system. Burmistrov
et al.117 proposed to use ðDlq00Þ� as such a reference super-
saturation. Due to essential temperature independence of the
interfacial tension in the range of parameters studied, this
choice of “isomorphic” parameter is equivalent to taking the
critical nucleus size as the reference supersaturation.
Below T0� ’ 10 mK, the value of ðDlq00Þ� is temperature
independent within the limits of experimental error, Fig. 23.
An increase of ðDlq00Þ� in the temperature range T0� � T00� is
related, according to the authors of Ref. 117, to the dissipa-
tion phenomenon in quantum tunneling of nuclei. In the
absence of dissipation, the transition temperature from
thermal-fluctuation to the quantum regime of nucleation is
estimated at 5 mK.115 This temperature is close to T0� and
approximately an order of magnitude lower than the value
T00� . The temperature at which the parameter ðDlq00Þ� in
the thermal-fluctuation nucleation regime is equal to its
value in the regime of quantum tunneling of nuclei
(ðDlq00Þ�; I ’ 0:024 MPa) as calculated from Eq. (16) is
T000� ’ 200 mK, in line with the experimental value T000� ’150 mK.
4.2. Quantum cavitation in liquid helium
4He and 3He remain liquid at absolute zero temperature
if static pressure does not exceed 2.5 MPa for 4He and
3.5 MPa for 3He. In this regard, it can be expected that at
negative pressure and very low temperatures, the homogene-
ous nucleation in a liquid 4He and 3He will occur through
the quantum tunneling nucleation mechanism. First esti-
mates of the temperature T* and pressure of p* at which the
crossover from the thermal fluctuation nucleation mecha-
nism to quantum tunneling occurs in He II have been made
by Akulichev.51 According to Ref. 51, T* ’ 0.31 K and
p*¼�1.46 MPa. Akulichev’s results have been later revised
by Maris,74 and Maris and Xiong,118,119 who approximated
the spinodal line of liquid 4He and obtained at T¼ 0 the
value psp¼�0.952 MPa. The value of psp (0) has been fur-
ther refined several times.120,121 According to the calculation
results for the work required to form a critical nucleus, the
authors of Refs. 74 and 122 have obtained the following val-
ues of the quantum crossover parameters in He II: T*¼ 200
mK, p¼�0.9 MPa.
The first attempts to realize the regime of quantum tun-
neling of the vapor nuclei were attempted by Lambar�eet al.123 in the experiments on acoustic cavitation in super-
fluid 4He. They used the method of focusing the acoustic
field.75,76 The experiments were carried out in the tempera-
ture range 65–750 mK. The statistical regularities of the cav-
itation process have been studied. Given the impossibility of
direct pressure measurement, the onset of the cavitation pro-
cess was determined by the voltage on the piezoelectric
transducer U* at which the probability of cavitation was
equal to 50%. At temperatures below 600 mK, the value U*
has been found to be independent on temperature within the
limits of experimental error.
If we assume T*¼ 600 mK as the temperature of the
quantum crossover, then equating the rates of thermal fluctu-
ation and quantum nucleation, we obtain for the extremal
action
I� ¼ lnqBV �s
ln 2
� �¼ �hW�
kBT: (44)
According to Ref. 74, qB ’ 2� 1037 s�1m�3. The effec-
tive volume of the cavitating liquid V and the characteristic
lifetime of the liquid under negative pressures �s can be esti-
mated using the length and period of the acoustic wave. At
V �s¼ 3.1� 1012 s�m3, we obtain from Eq. (44): I*¼ 32�h,
p¼�0.927 MPa. The value of p* is below the spinodal pres-
sure of 4He by only 0.035 MPa at T¼ 0.118,119
Later Caupin and Balibar120,124 have refined the data of
Ref. 123, assuming that the temperature of the liquid in the
focal point of the piezoelectric emitter differs from the tem-
perature of the surrounding liquid. They have estimated that
the temperature is approximately three fold lower, i.e., the
value T* for the superfluid 4He is about 200 mK, which is in
good agreement with theoretical predictions.74,122
Burmistrov et al.126 do not agree with this interpretation
of the experimental data by Lambar�e et al.,123 and Caupin
and Balibar.120 If nonlinear effects are negligible and theFIG. 23. Temperature dependence of ðDlq00Þ�.116
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average length of the phonon-phonon scattering is less than
the wavelength of the acoustic wave (lph k), the tempera-
ture variations in the acoustic wave must follow the pressure
variations. The experiments in Refs. 120 and 123 were car-
ried out at the frequency of �1 MHz. The characteristic
times of overbarrier motion are s0¼ 10�10 s. Thus, xs0< 1
and nonlinear effects are expected to occur. The length of
phonon-phonon scattering lph in He II at 0.7 K is 1.3 nm; it
increases with decreasing temperature as T�7 and reaches
the value of 15 mm at T¼ 0.5 K. At k ’ 0.1 mm the varia-
tions of local temperature do not follow the changes in pres-
sure and the temperature in the focal point of the
piezoelectric emitter cannot differ much from the tempera-
ture of the liquid in the cell.
The regime of quantum tunneling of vapor phase nuclei
in normal liquid 3He has been discussed in Refs. 76–78.
According to the estimates by Maris,74 the transition temper-
ature from the thermally activated regime to quantum tunnel-
ing is 120 mK in normal liquid 3He and the spinodal
pressureis is �0.31 MPa at T¼ 0. The limiting tensile stress
in 3He achieved at T< 100 mK is close to the spinodal
values76–78 (Fig. 8).
In the temperature range 40–1000 mK, 3He is the normal
viscous Fermi liquid. In Ref. 74, the viscosity of 3He has
been ignored when assessing T*. According to Ref. 126, the
mean free path of quasiparticles lF in normal liquid 3He is
’ 0.05 nm at T¼ 1 K, which is significantly less than the
characteristic size of a vapor phase nucleus Rc ’ 0.1 nm.
Upon lowering the temperature, lF grows as T2 [Ref. 127]
and reaches 5 nm at T¼ 0.1 K, which exceeds than the criti-
cal nucleus size. In this case it is necessary to take into
account the energy dissipation also for estimating the quan-
tum crossover temperature. The calculations reported in Ref.
126 have yielded the value T* ’ 2 mK for normal 3He. This
value is below the lower boundary of the temperature range
in which the limiting tensile stress has been studied in3He.76–78
5. Conclusion
The tasks of intensifying the heat transfer in cryogenic
equipment, designing the transport and storage systems for
large quantities of cryogenic liquids, bubble chambers, cryo-
genic cooling systems for microwave devices and molecular
masers have put forward a number of fundamentally new
scientific problems, including the problem of rapid phase
transitions accompanied by deviations from the state of
phase equilibrium. The relief of superheating in liquids leads
in many cases to major hydraulic shocks. Preventing the
superheating is one of the possible ways to improve the reli-
ability of cryogenic equipment. On the other hand, sustain-
ing phase metastability of a liquid for extended periods of
time is desirable to ensure the cavitation-less functioning of
cryogenic pumps. Strong metastability is also necessary for
functioning of bubble chambers. Superheated liquids repre-
sent not only an interesting object for research, but also hold
potential for new technological solutions.
Near the line of phase equilibrium, the fluctuations in
liquid phase are insignificant. This leads to the situation in
which the appearance of a new phase in a homogeneous sys-
tem is impossible unless certain “weak” spots are present.
Boiling-up at low superheating temperatures is caused by
heterogeneous nucleation.
A more interesting case is the homogeneous nucleation.
Being a manifestation of fluctuations in metastable systems,
the homogeneous nucleation does not have any physical
restrictions. The idea of overcoming the activation barrier,
which decreases with increasing the degree of supersatura-
tion of the phase, via fluctuation is clear from physical per-
spective. Numerous experiments with liquefied gases and
their solutions presented in this review argue in favor of the
feasibility of vapor phase nucleation by fluctuations under
real-world experimental conditions. At positive pressures,
the experimental data on the attainable superheating temper-
ature of liquefied gases and their solutions are in satisfactory
agreement with the results of thermal-fluctuation theory of
homogeneous nucleation in a broad range of nucleation rates
from 106 to 1023 s�1m�3. A small systematic “undercooling”
below the theoretical values of Tn. has been found for all
single-component liquids. A better agreement between
theory and experiment for pure fluids can be achieved by
assuming that the surface tension of critical vapor bubbles is
lower than that on planar interfaces. Dissolution of one liq-
uid in another reduces the mismatch between theory and
experiment. For certain solutions the experimental values of
Tn exceed the theoretical value.
At negative pressures, the experimental data are limited
both in terms of the systems studied and in terms of the state
parameters and the nucleation rate. At present, there are no
experimental data on the temperature (pressure) dependence
of the nucleation rate in cavitating liquid. All this does not
allow reaching such clear-cut conclusions as in the case of
positive pressures. At least in single-component liquids it
was observed that the discrepancy between the classical
nucleation theory and experiment with regards to the limit-
ing tensile pressure increases with decreasing temperature.
The sources of heterogeneous nucleation do not lend
themselves to simple physical identification or statistical
description. The experimental data suggest that at high
supersaturation the nucleation of vapor bubbles on them has
fluctuation character with a lower nucleation work as com-
pared to the case of the homogeneous mechanism.
The studies of quantum nucleation kinetics in the liq-
uid phase are still in the stage of accumulation of the ex-
perimental data. The experiments on determining the
limiting supersaturation of superfluid mixtures 4He-3He and
cavitational strength of 3He and 4He in the vicinity of abso-
lute zero temperature reveal a certain temperature range in
which the nucleation rate is temperature independent. The
upper boundary of this temperature range is, in some cases,
close to the theoretical estimates obtained using the quan-
tum model of nucleation. Such an agreement is observed if
the energy dissipation is not taken into account.
Accounting for the dissipation leads to quantitative dis-
agreement between theory and experiment. We should be
also aware of the fact that, according to the third law of
thermodynamics, upon approaching absolute zero tempera-
ture, the derivative dp/dT tends to zero not only on the spi-
nodal line but also on the line of limiting tensile strength
(superheating). This will inevitably lead to the emergence
of a certain interval near T¼ 0 in which the derivative dpn/
dT is close to zero.
662 Low Temp. Phys. 39 (8), August 2013 V. G. Baidakov
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The obtained experimental data on the kinetics of spon-
taneous boiling-up of liquefied gases and their solutions indi-
cates that the agreement between theory and experiment is
not accidental, but reflects, in essence, the correspondence
between the actual process and the physics of the Gibbs-
Zeldovich-Frenkel model, if not in detail, but in the ultimate
outcome. The microscopic aspects of such a model are pres-
ently under intensive investigation by computer simulation
(Monte Carlo and molecular dynamics).128–130
This work was supported by the Russian Foundation for
Basic Research (Project No. 12-02-90413_Ukr) and the Ural
Branch of the Russian Academy of Sciences (Project No.
12-P-2-1008 of the Program of the Presidium of RAS and
Project No. 12-S-2-1013 of the Program for collaborative
research between the Ural and Far East branches of RAS).
a)Email: [email protected]
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Translated by L. Gurevich
664 Low Temp. Phys. 39 (8), August 2013 V. G. Baidakov
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