attainable superheating of liquefied gases and their solutions (review article)

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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LOW TEMPERATURE PHYSICS VOLUME 39, NUMBER 8 AUGUST 2013


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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)

644 Low Temp. Phys. 39 (8), August 2013 V. G. Baidakov


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.

Low Temp. Phys. 39 (8), August 2013 V. G. Baidakov 645


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



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.



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



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



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.



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.



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.



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.



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.



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.



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.



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.



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



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.



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.



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



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.



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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