Nucleation in ethane–nitrogen solutions. II. Heterogeneous and initiated nucleation

International Journal of Heat and Mass Transfer 86 (2015) 936–942

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Nucleation in ethane–nitrogen solutions. II. Heterogeneous and initiated nucleation V.G. Baidakov ⇑, A.S. Pankov Institute of Thermophysics, Ural Branch of the Russian Academy of Sciences, Amundsen street 107a, Ekaterinburg 620016, Russia

a r t i c l e

i n f o

Article history: Received 26 January 2015 Received in revised form 17 March 2015 Accepted 19 March 2015 Available online 16 April 2015 Keywords: Nucleation Superheating Heterogeneous nucleation Method of continuous pressure lowering Ethane–nitrogen solution

a b s t r a c t The method of continuous pressure decrease has been used to investigate limiting (attainable) stretches of ethane–nitrogen solutions. Experiments were conducted at temperatures of 260, 266, 270, and 276 K and at nitrogen concentrations in the solution of 0.5 and 1.5 mol%. Pressure decrease rates were taken in the range 0.01–0.1 MPa/s. Distribution functions of liquid boiling-up events in dependence on pressure have been obtained allowing one to determine the most probable (attainable) pressures of liquid boiling-up and the nucleation rate. The results are interpreted in terms of nucleation thermodynamics and kinetics in liquids on a solid surface containing inclusions that initiate the boiling-up processes. It is shown that for realizing homogeneous nucleation it is not necessarily required that the wettability is close to the ideal one. The reasons for the discrepancy between experimental data and predictions of homogeneous nucleation theory at nucleation rates J < 3  106 s1 m3 are discussed. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The present paper is a continuation of Ref. [1] devoted to homogeneous nucleation kinetics in superheated liquid ethane saturated with nitrogen. The method of continuous pressure decrease is used here to measure the limiting stretches of solutions which have to be reached to initiate intensive nucleation. In particular, the dependence of the value of the limiting stretches on the rate of pressure decrease and on the concentration of the solutions is studied. Experiments are conducted in the transition region between homogeneous and initiated nucleation. The paper examines the kinetics and thermodynamics of heterogeneous nucleation on defects of solid surfaces that are in contact with superheated liquids. The probabilities of homogeneous and heterogeneous nucleation in the conditions of the experiment conducted are compared. In the experiments, the liquid solutions are superheated in a glass capillary. The equilibrium angle of wetting of glass by ethane does not exceed 10° [2]. At a good wettability of glass by the liquid under investigation the internal surface of the capillary is not ideally homogeneous and smooth. It may have ‘‘weak’’ places, i.e. nucleation centers in the form of smallest cavities filled with an undissolved gas and poorly wetted separate isolated sections of the surface. In such places the adhesion of the molecules of the ⇑ Corresponding author. Tel.: +7 343 267 8801; fax: +7 343 267 8800. E-mail address: [email protected] (V.G. Baidakov). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.03.065 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

liquid and the solid is usually weakened, which leads to a decrease in the work of formation of a critical vapor nucleus and initiates in this way the heterogeneous mechanism of formation of the vapor phase. Nevertheless, even in the presence of such ‘‘weak’’ places, the probability of nucleus formation inside the bulk of the liquid may exceed the probability of its appearance at the solid surface since the number of molecules in the volume of the liquid exceeds by far the number of molecules being in contact with the inner surface of the capillary [3]. A superheated liquid near the boundary of spontaneous boilingup is sensitive to ionizing radiation [4] which may also catalyze processes of bubble formation. The initiating effect at irradiation can be explained by a local heat release caused by radiation, which may trigger the formation of viable bubbles. At conventional experimental conditions it is difficult to get rid of both the effects of the background radiation of cosmic particles and the natural radioactivity of materials, therefore the value of the nucleation rate registered in experiment will also contain the rate of the initiating action of the mentioned factors, which may be both lower and higher as compared to the rate of homogeneous nucleation. The paper is structured as follows. In Section 2, theoretical aspects of nucleation in a liquid at a continuous pressure decrease are analyzed. Section 3 presents the results of experimental investigations of limiting (attainable) stretches of ethane–nitrogen solutions. Section 4 is devoted to the thermodynamic and Section 5 to the kinetic description of formation of a vapor phase at a contact of superheated liquids with solid surfaces of different roughness. A

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Nomenclature Latin letters A area of the nucleus surface, m2 AS area of the nucleus surface contacting the wall, m2 B kinetic factor, s1 G reduced work of nucleus formation h height of a bubble in a critical recess, m H height of a cavity in the wall, m J nucleation rate, s1 m3 kB Boltzmann constant, J K1 n number of boiling-ups in the liquid in the pressure range from p to p + Dp N number of measurements NS number of liquid molecules contacting the wall NV number of molecules in the liquid volume p pressure, MPa pg pressure of the gas phase, MPa R nucleus radius, m T temperature, K v nucleus volume, m3 V volume of the measuring cell, m3 W work of nucleus formation, J x concentration of the second component in solution, mol%

summary of the results and their discussion (Section 6) complete the paper. 2. Liquid boiling-up at a continuous pressure decrease The density of probability x of formation of the first critical nucleus in a superheated liquid in dependence on pressure p under an isothermal pressure decrease performed at a constant rate p_ ¼ dp=ds ¼ const may be presented as [5]

JðpÞV

x ðpÞ ¼  _ p

" exp 

V p_

Z

p

# JðpÞdp ;

ð1Þ

p0

where p0 ¼ pðs ¼ 0Þ is the pressure at the beginning of observations, V is the volume of the system, J(p) is the nucleation rate depending on pressure. If an observation is carried out starting from the saturation pressure ps, then at the beginning of the process J(ps) = 0 holds. The minus sign in the right-hand side of Eq. (1) is caused by the choice of the direction of the pressure change. In the experiments considered by us p_ < 0 holds resulting, as required, in x ðpÞ > 0. From the condition of maximum of x(p) we have

_ p; Jðpn ÞV ¼ pG

ð2Þ

where pn is the pressure at the limiting (attainable) liquid stretch Gp ¼ ð@ ln J=@pÞT;x . In experiments the value of Gp ðpn Þ may be determined through the half-width of the distribution function of x(p) [5]. Eq. (2) makes it possible to find the nucleation rate that corresponds to the value of pn at the given volume of the liquid, V, and _ According to Eq. (2) the characterthe rate of pressure decrease, p. istic time scale in the regime of a continuous pressure decrease is _ s1 1 ¼ pGp . At homogeneous nucleation close to the boundary of the attainable superheating the derivative Gp varies only slightly with pressure. Going over in Eq. (1) from integration with respect to pressure to integration with respect to nucleation rate, we obtain

Greek letters 2a angle at the vertex of conical recess in the wall, 0 h wetting angle, 0 r surface tension at the liquid–gas interface, J m2 surface tension at the liquid–solid interface, J m2 rLS surface tension at the gas–solid interface, J m2 rVS s expectation time for liquid boiling-up, s W parameter determining the magnitude of the decrease in the work of nucleus formation on the wall x density of probability of appearance of the first nucleus in the liquid, MPa1 Subscripts hom homogeneous nucleation het heterogeneous nucleation in initiated boiling-up l liquid phase s phase equilibrium line v vapor phase ⁄ critical nucleus

JðpÞV

x ðpÞ ¼  _ p

  V JðpÞ : exp  p_ Gp

ð3Þ

The value of Gp may be calculated based on the nucleation theorem as Gp ¼ ve =kB T, where ve is the volume of a critical bubble when the equimolecular surface is chosen as the dividing surface [6]. Alternatively, it may be evaluated via the half-width of the distribution of x(p) via

dp1=2  2:44=Gpn :

ð4Þ

The half-width of this distribution does not depend on the rate of pressure decrease and the volume of the liquid being superheated. At heterogeneous and initiated nucleation the nucleation rate has a more complex temperature, pressure, and concentration dependence as compared to homogeneous nucleation. For the latter case, J as a function of pressure was investigated for liquid ethane in Ref. [7]. The temperature dependence of the homogeneous nucleation rate in ethane–nitrogen solutions was studied in Ref. [1]. As shown there, at nucleation rates J < 3  106 s1 m3 experimental curves deviate from the lines calculated by classical homogeneous nucleation theory. Here we present the nucleation rate registered in experiments as follow:

n o b J ¼ J 0 exp ½ðG  G0 Þ=a ;

ð5Þ

where G⁄ = W⁄/kBT is the reduced work of formation of a critical-size nucleus (Gibbs’ number). The parameters J0, G0, a, and b have been obtained by fitting of the experimental data [1,7]. The value of J0 is determined by the value of the nucleation rate at which the bending of the experimental curves begins. In an ethane–nitrogen system the parameters J0, G0 and b do not depend within the experimental error on pressure, temperature, and concentration and are equal to J0  3:0  106 s1 m3, G0  71, b = 1.53. The parameter a is a function of pressure and concentration of nitrogen. It can be expressed as

a ¼ expð5:503  0:3712p  0:5754xÞ:

ð6Þ

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30

ω (p), MPa-1

8

4

2

0

1

3

1.4

1.6 1.8 p, MPa

2.0

Fig. 1. Density of probability of liquid ethane boiling-up in the regime of _ 1 – continuous pressure decrease at T = 277 K and rates of pressure change jpj: 0.01 MPa/s, 2 – 0.1, 3 – 1.0.

The nucleation rate is an additive quantity, therefore, the value of J registered in the experiment is the sum of the rates of homogeneous Jhom and initiated Jin nucleation, i.e.

J ¼ J hom þ J in :

ð7Þ

Since the temperature (or pressure) dependence of Jhom is very strong, the interaction of the two mechanisms of nucleation will manifest itself only in the vicinity of the kink point of the dependences J(p) or J(T). At superheatings and stretches below this point J ’ J in . Fig. 1 presents the results of numerical calculation of the probability density for the boiling-up of superheated ethane employing Eq. (1) at different rates of pressure decrease. In these computations, Eq. (5) is employed for the description of the rate of initiated nucleation and classical nucleation theory is utilized for the specification of Jhom [1]. _ < 1 MPa/s) the main At small rates of pressure decrease (jpj contribution to the probability of liquid boiling-up is made by _ ¼ 0:01 MPa/s the function x(p) has initiated nucleation. For jpj one sufficiently wide maximum (Fig. 1). The half-width of this distribution increases with increasing rate of pressure decrease. In the region where the nucleation mechanism is changed a second maximum appears on the curve of the dependence x(p), which is caused by the peculiar step-like shape of the curve J(p). Since jð@ ln J hom =@pÞT;x j  jð@ ln J in =@pÞT;x j, the second maximum has a considerably smaller half-width as compared to the first. _ > 1 MPa/ When the rate of pressure decrease is high enough (jpj s), initiated nucleation has no time to make any appreciable contribution to the total effect, and boiling-up is only a result of homogeneous nucleation. The function x(p) has in this case one narrow maximum, whose half-width remains approximately constant. A further increase in the rate p_ of pressure variation changes the character of the dependence x(p) only slightly, shifting the position of the maximum of x(p) into the region of lower values of p. 3. Experimental set-up and results In measuring limiting stretches of ethane–nitrogen solutions use was made of the method of continuous lowering of pressure on a thermostatted liquid. Experiments were conducted on the same experimental set-up by which the lifetimes of the solution under study were measured in an earlier investigation [1]. The upper value of the pressure ph, from which its further decrease started, was chosen in the range from 3.0 to 3.5 MPa. The rate of

pressure decrease p_ varied from 0.01 to 0.1 MPa/s and remained constant in the process of its depressurization. At every given temperature of the liquid in a glass capillary, 20–40 measurements of the pressure p were made at the moment the liquid was boiling up. The results of the measurements were employed to construct a histogram of distribution of boiling-up events and to determine the most probable value of the pressure of boiling-up pn (the pressure of attainable stretching). Experiments on the determination of the attainable stretching for an ethane–nitrogen solution were carried out at temperatures T = 260, 266, 270, and 276 K and two values of nitrogen concentration in ethane x = 0.5 and 1.5 mol%, respectively. The characteristic form of the histograms obtained is illustrated in Fig. 2. It shows the number of boiling-up events of the solution n in the pressure range _ The p, p + Dp in a series of N measurements at given T, x, and p. smooth curve shows the results of calculation of n = Nx(p) Dp by Eq. (1). The maximums on the histograms (Fig. 2) determine the most probable boiling-up pressures of the solution. At the parameters indicated in Fig. 2a only initiated nucleation manifests itself. A _ (Fig. 2b) leads to spontaneous liquid boilfourfold increase in jpj ing-up being registered along with initiated nucleation. However, the probability of observing spontaneous nucleation here is about an order of magnitude lower than the one of observing initiated nucleation. At T = 276 K, x = 1.5 mol%, and a rate of pressure _ ¼ 0:01 MPa/s (Fig. 2c) the probability of registering decrease jpj spontaneous nucleation in an experiment is only 0.005. Fig. 3 presents in pressure–temperature coordinates data on the pressure of the attainable stretch for pure ethane [7] and ethane–nitrogen solutions [1] obtained by the method of lifetime measurement and pertaining to a nucleation rate J = 107 s1 m3. In this figure also the pressures of stretching of ethane–nitrogen solutions under initiated nucleation are shown obtained by the method of a continuous decrease of pressure on a thermostatted liquid and related to a rate of initiated boiling-up in the range of 4.8  105 to 1.7  106 s1 m3. 4. Thermodynamics of heterogeneous nucleation Let us consider the formation of a vapor–gas bubble in a liquid– gas solution which is in contact with a solid wall. The temperature of the liquid is T, the pressure, p, and the concentration of the gas component in the liquid, x. Far from the line of critical points, when a liquid solution may be considered as incompressible and the vapor–gas mixture in a bubble as ideal, the work of formation of a bubble of volume v with a surface area A, whose part AS is in contact with the wall, will look like

W ¼ ðp  ps Þð1  qv =ql Þv þ rðA  AS Þ þ ðrVS  rLS ÞAS :

ð8Þ

Here ps is the pressure of the saturated vapors of the solution, ql and qv are the liquid and vapor densities on the line of phase coexistence, r, rVS and rLS are the surface tensions at the liquid–gas, gas–solid, and liquid–solid interfaces. The surface of tension was chosen here as the dividing surface. The first term in Eq. (8) expresses the work against the pressure forces WV, the second and the third terms, the work of formation of the interfaces WA. With the use of Neumann’s equation [8], which relates the surface tensions r, rVS and rLS to the wetting angle h, the last two terms in Eq. (8) take the form

W A ¼ rA½1  ðAS =AÞð1  cos hÞ;

ð9Þ

The value of WA depends on both physical (the wetting angle h), and geometrical (the relation AS/A) properties of the surface. At fixed h and A the work of formation of interfaces will be the smaller, the larger is the part of the bubble surface that is in contact

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8

2.5

(a)

2.0 p, MPa

n

6

4

2

-1 -2 -3 -4 -5

ps

1.5 1.0 0.5

0 0.6

0.8

1.0

1.2

1.4

0.0 258

1.6

263

5

n

pn

1

1.6

1.8 p, MPa

2.0

10

(c) 8

v¼

n

6

0 1.9

1 1 cos3 u p R3 ð2  3 sin u þ sin3 uÞ þ p R3 3 3 tga

A ¼ 2p R2 ð1  sin uÞ þ 2p R2

4 2

278

for the initiation of vaporization. If the liquid wets the solid surface partially (h < p/2), the conditions of appearance of a nucleus at the solid surface prove to be qualitatively the same as in the bulk of the liquid in the sense that an activation energy is required for the initiation of a phase transition. A viable nucleus forms then only in the presence of a certain sufficiently large liquid superheating. There are a number of expressions for the work of formation of a critical nucleus in recesses of different geometry [10–12]. For a general analysis the most convenient model is that of a conical recess. In this case during the formation of a vapor–gas bubble with an interface curvature radius R inside a spherical conical cavity with a vertex angle 2a [12]

2

0 1.4

273

Fig. 3. Limiting stretch pressure for pure ethane and ethane–nitrogen solutions at homogeneous (Jhom = 107 s1 m3) (1–3) and initiated (Jin = 4.8  105 to 1.7  106 s1 m3) (4, 5) nucleation at nitrogen concentrations in ethane x = 0 (1), x = 0.5 mol% (2, 4), x = 1.5 mol% (3, 5). ps is the saturation line. Solid lines, approximation of experimental data on homogeneous nucleation, dashed lines, approximation of experimental data on initiated nucleation.

(b)

4 3

268 T, K

p, MPa

AS ¼ 2 p R 2

pn 2.1

2.3

2.5

p, MPa Fig. 2. Histograms of experiments on stretching of ethane–nitrogen solution in the _ ¼ 0:01 regime of continuous pressure lowering: (a) T = 270 K, x = 0.5 mol%, jpj MPa/s, N = 20, pmax = 1.34 MPa, Jin = 7.2  105 s1 m3; (b) T = 276 K, x = 0.5 mol%, _ ¼ 0:04 MPa/s, N = 20, pmax = 1.80 MPa, Jin = 1.7  106 s1 m3, pn = 1.455 MPa, jpj _ ¼ 0:01 MPa/s, N = 20, Jhom = 3.5  107 s1 m3; (c) T = 276 K, x = 1.5 mol%, jpj pmax = 2.18 MPa, Jin = 9.9  105 s1 m3, pn = 1.95 MPa, Jhom = 3.6  106 s1 m3. Smooth curves show results of calculations with the use of Eq. (1).

with the solid phase. Therefore, elements of surfaces in the form of recesses and cavities are more probable places of the appearance of bubbles than planar parts or ridges. The work of formation of interfaces depends very strongly on the local wettability. At h = 0 the value of WA is equal to rA, and Eq. (8) transforms to the well-known expression for the work of formation of a vapor–gas bubble in the bulk of a liquid solution [9]. Otherwise (with absolute non-wettability) h = p, the energy barrier is absent, and no liquid-phase superheating is required

cos2 u sina

cos2 u ; sina

ð10Þ

ð11Þ

ð12Þ

where u = h  a. By taking the derivative of Eq. (8)it is easy to verify that to an equilibrium state of the nucleus in the liquid medium corresponds the extreme (maximum) value of W equal to

W het ¼ W  Wða; hÞ:

ð13Þ

Here W⁄ is the work of formation of a critical nucleus inside the bulk of a homogeneous solution [9]. The function

W¼

  1 cos3 u 3 2  3 sin u þ sin u þ 4 tga

ð14Þ

plays the role of a correction factor to W⁄ reflecting the effect of the solid surface with the considered shape on the nucleation process. As in the case of formation of a nucleus in the bulk of the liquid, during the formation of a critical bubble in a conical cavity the work against the forces of surface tension WA and the volume work against the pressure forces WV are related by the simple relation W V ¼  23 W A . According to Eq. (14), with an increase in the wetting angle the work of formation of a critical bubble decreases. For every value of h there exists a certain angle a at which the value of W has a

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minimum. From Eq. (14) it follows that bubbles are most easily activated on conical cavities with sufficiently small angles at their vertices. If the depth of the cavity H where a vapor–gas bubble is forming exceeds considerably the nucleus height

ð15Þ

then at the time when the vapor phase leaves the cavity the change in the thermodynamic potential is already mainly determined by the work against the pressure forces WV. In addition to factors caused by the roughness of the surface which contacts the superheated liquid, the nucleation process may be affected by impurities in the form of surface-active substances. The inhomogeneity of the material of the walls, mechanical stresses in some places of the surface, which lead to considerable changes in the conditions of wettability, will also facilitate the formation of nuclei. The magnitude of superheating of a liquid depends on the presence in it of an undissolved gas and noncondensed vapor. Such an undissolved gas may exist in a liquid permanently only if special conditions are fulfilled. The most obvious places where the gas may be retained are microscopic and submicroscopic cracks in the vessel walls. The liquid, penetrating into a wettable cavity filled with a gas, forms a spherical meniscus and evaporates into the space below it. The process will stop with the establishment of thermodynamic equilibrium. In this case the gas pressure in the cavity will be equal to p0g : If the amount of the gas in the cavity does not change under quasi-static changes of the pressure in the liquid p, the resulting force per unit of the meniscus area is [13]

FðRÞ ¼ p0  p þ p0g

 3 R0 2r  : R R

2

ð16Þ

Δ p, MPa

h ¼ R½ð1  sin uÞ þ cos u=tga;

3

W W (2)

1

ΔW

0

W

R -1

0

1

(1) 0

(1)

RK 2

R 0(1)

R (2) 0

R 0(2)

3 4 R, nm

5

6

Fig. 4. Cavitation strength of a vapor–gas bubble as a function of its size. The insert shows the dependence of the work of formation of a vapor–gas bubble at Dp > 0 ð1Þ ð2Þ (dashed line) on its radius. Rk is the radius of a critical bubble. R 0 and R 0 are the radii of a stable and an unstable vapor–gas bubble, respectively.

DpK ¼

2 3

3=2



3=2 2r 1 2 2r ¼ 3=2 3 R0 R ð1  qv =ql Þ K ð1  q = q Þ p01=2 g v l

ð19Þ

for the critical stretch (F(R) = 0). The value of DpK determines the limiting stretch at which the non activational growth of a vapor– gas bubble in a metastable liquid is possible. If the bubble is filled with vapor (pg = 0), then at D p > 0 it is possible to observe only unstable equilibrium states, which are realized at R = R⁄. The critical radius of a vapor bubble is related to the critical radius of a bubble that contains vapor and gas simultaneously (Eq. (18)) by the relation R⁄ = 3/2 RK. 5. Heterogeneous nucleation kinetics

Here R0 is the equilibrium radius of curvature of the meniscus at a pressure p 0g . Using the relation for the partial vapor pressure in the bubble p0 ðRÞ with the saturation pressure ps [3] and introducing the notation Dp ¼ ps  p, we have

FðRÞ ¼ Dp þ p0g



R0 R

3 

  2r q 1 1 v : R ql

ð17Þ

When mechanical equilibrium is established between the vapor–gas mixture in the cavity and the liquid, the radius of curvature of the meniscus is determined by the condition F(R) = 0. In this case the dependence of Dp on R has the form shown in Fig. 4. At Dp < 0 the equilibrium in a bubble is always stable provided diffusion is inhibited. If the pressure in the liquid is lower than that of the saturated vapors (Dp > 0), then in the cavity equilibrium configurations of a vapor–gas bubble may be formed differing in the value of the curvature of the interface. Vapor–gas inclusions, for which @F=@R < 0 holds, will be in a state of stable equilibrium. The equilibrium state of bubbles fulfilling the condition @F=@R P 0 is unstable. The possibility of existence of such two kinds of equilibrium bubbles differing in their stability is retained up to a certain value of pK, which corresponds to the critical value of the radius of curvature of the meniscus RK. If the radius of curvature of the meniscus in the cavity exceeds RK, the bubble will become unstable and increase further unhindered. Taking the derivative of F(R) with respect to R and setting it equal to zero, we obtain

R2K ¼

3p0g R30 ð1  qv =ql Þ: 2r

ð18Þ

After substituting Eq. (18) into Eq. (17), we have arrived at the following relation

Heterogeneous nucleation theory may be developed following the same scheme as in homogeneous nucleation theory. In a stationary process the resultant flow of critical-size nuclei is equal to the sum of the flows of nuclei that originate at different sites of the surfaces of impurity particles and the vessel walls. In the general case different sections of the surface are characterized by different values of the wetting angle h, the roughness parameter a, and, as a result, by a different value of the correction factor W in the work of formation of a critical nucleus (Eq. (13)). If each of these different surface sections has a characteristic size exceeding the size of a critical nucleus, the heterogeneous nucleation rate may be written as a sum of the contributions from the different surface sites as

J het ¼

X Nhet;i Bhet;i expðW  Wðhi ; ai Þ=kB TÞ:

ð20Þ

i

Here Nhet,i is the number of liquid molecules near to the surface section under study participating effectively in heterogeneous nucleation, and Bhet,i is the kinetic factor at heterogeneous nucleation of boiling-up centers at the ith surface section with a wetting angle hi and a roughness parameter ai. Eq. (20) contains two parameters Nhet and W which are not easy to determine. As for the kinetic factor Bhet, it is limited from above by the value of B at homogeneous nucleation, Bhet 6 B. According to [3,14] the value of Bhet differs from B by a factor less than two, which is insignificant as compared to the uncertainty in Nhet. If liquid boiling-up takes place at a homogeneous smooth solid surface, then the stationary heterogeneous nucleation rate may be presented retaining only one of the terms in Eq. (20) as

J het ¼ Nhet Bhet exp½W  WðhÞ=kB T;

ð21Þ

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The value of W(h) is determined then by Eq. (14), where a = p/2 holds. Since WðhÞ < 1, the exponential factor in Eq. (21) is larger than for homogeneous nucleation (Eq. (1) [1]). This property leads to domination of heterogeneous nucleation at low superheatings. However, the pre-exponential factor NhetB in Eq. (21) is much smaller than the one occurring in Eq. (1) [1] since the number of molecules in the liquid volume NV is considerably larger as compared to the number of molecules of the liquid NS contacting the wall. The temperature dependence of the rate of heterogeneous nucleation can be shown to be weaker than of homogeneous nucleation. Therefore, even with an incomplete wetting of a solid surface, beginning with a certain superheating, the probability of spontaneous formation of a nucleus in the bulk of the liquid may turn out to be much larger than the probability of boiling-up on a smooth wall. Bulk nucleation will be determining in spontaneous nucleation kinetics if the inequality

WðhÞ P 1  ðW  =kB TÞ1 lnðNV =NS Þ

ð22Þ

is fulfilled. In writing Eq. (22) it is assumed that Bhet  B holds approximately. If W⁄  70 kBT, which is the characteristic value of the work of critical bubble formation in experiments on the measurement of lifetimes of superheated liquids in glass capillaries, and it is further assumed that the wall layer of the liquid, whose thickness is equal to the diameter of a critical bubble, takes an active part in nucleation, from Eq. (22) we arrive at WðhÞ P 0:86. This value corresponds to a wetting angle of h 6 580 . Thus, with the assumptions made above, the requirement of a complete wetting of the vessel walls contacting a superheated liquid is too rigid as a restriction for the possibility of homogeneous nucleation. At high superheatings homogeneous nucleation is a competitive process as compared to heterogeneous nucleation even for cases when only partial wettability of the solid wall is assumed. Assuming that the deviation of experimental curves from those calculated by homogeneous nucleation theory observed in Ref. [1] at J < 3  106 s1 m3 is caused by heterogeneous nucleation, we may evaluate from experimental data the attenuation factor W and the pre-exponential factor N het Bhet in Eq. (21). In our case for all pressure values and solution compositions investigated the pre-exponential factor is equal to about (1–3)  107 s1 m3, and the value of W changes in the range 0.012–0.028. For a smooth wall this value corresponds to a wetting angle h of 145–1500. As has already been mentioned, such large angles are not characteristic of the solutions under investigation. At T = 265 K, p = 1.0 MPa and x = 1.5 mol% the value of B in the case of homogeneous nucleation is 2.7  1011 s1. Assuming Bhet = B, from N het Bhet  1.5  107 1 m3 we have Nhet  6  105 m3. Such values imply a practical absence of readily activated centers in the measuring capillary. Even if the internal surface of the capillary has an outstandingly poor wettability (h  1500) heterogeneous nucleation on it of a vapor phase with the rate observed in experiment is hardly probable. Thus, most likely, the bends of experimental curves (see Fig. 1 [1]) are caused by some other initiating factor. The internal surface of the measuring capillary is not ideally smooth. It has microdefects, poorly wetted sections of the wall surface, etc. We will examine the formation of a vapor-phase nucleus at a solid surface with conical cavities. Let the depth of the cavities be H = 5  108 m, the apex angle 2a = 30°. The radius of the cavities at the outlet exceeds the radius of a critical bubble. The density of filling the surface with cavities is close to the maximum one and equals 1.4  1015 m2. In this case the relation of the number of liquid molecules in the capillary volume to the number of molecules in the wall cavities is NV/NS  1.9  104. From Eq. (22) at W⁄/kBT= 70 we have W P 0:86. According to Eq. (14) for W = 0.86

and a = 15° the wetting angle h  430 . The requirements for the realization of homogeneous nucleation become more rigid than in the case of a smooth wall, but insignificantly more rigid. If a bubble of a noncondensed gas (nitrogen) was initially present in the wall cavities, then, as it is shown in Section 4, such a bubble, when the liquid is stretched, may be in stable equilibrium up to a certain limiting pressure p = pK. The value of pK depends on the bubble size at pressure ph, from which its decrease is realized. For T = 269 K the bursting pressure for pure ethane pn = 0.335 MPa (J = 107 s1 m3) [7]. If at ph = 3.5 MPa in the cavities of the capillary walls there are bubbles with radii of curvature of the separating surface exceeding 1.3 nm, all of them with a decrease in the pressure to p = pn will reach the critical value of R = RK and in the course of ‘‘the run-in’’ of the surface the gas in them will be dissolved in the liquid. Bubbles with initial radii smaller than 1.3 nm may be retained to the boundary of attainable stretching of the liquid. If the initial radius of a gas bubble was 1 nm, then, as it is evident from Fig. 4, at T = 269 K and Dpn = 1.821 MPa the equilibrium ð1Þ

radius of a stable bubble will be R0 ¼ 1:38 nm, and a bubble which is in unstable equilibrium with the liquid has a radius ð2Þ

R0 ¼ 4:15 nm. The latter size is close to the critical radius of a vapor bubble R⁄  4.5 nm. At the expense of fluctuations in the sysð1Þ

ð2Þ

tem a bubble of radius R0 can ‘‘grow’’ to the size R0 overcoming a ð2Þ WðR0 Þ

ð1Þ WðR0 Þ

 and continue its potential barrier of height DW ¼ further growth already in accordance with macroscopic evolution laws. The lifetimes of a viable bubble in a conical cavity with the dimensions given above can be evaluated in the following way.  According to transient state theory, the mean expectation time s for such a bubble in a singular cavity will be determined by the equation [15]

s1 ¼ B3 expðDW=kB TÞ:

ð23Þ ð1Þ

ð2Þ

For the work of formation of bubbles with radii R0 and R0 in a volume of the liquid we have:

ð1Þ WðR0 Þ

’ 8kB T,

ð2Þ WðR0 Þ

’ 72kB T. The

value of B3 is of the order of B3 ’ 2  1012 s1 (see Eq. (10), Ref. [1]). From the condition of equality of expectation times for a viable nucleus at a homogeneous nucleation mechanism and nucleation on a completed center in a conical pore we have Wða; hÞ ’ 0:328. At a = 15° the angle of wetting of the cavity walls by the liquid must be of the order of 64°. Thus, the performed analysis of the possibility of realizing the mechanism of heterogeneous nucleation in our experiments shows its extremely small probability, if the value of the wetting angle h does not depend on the size of the forming nucleus. Taking into account the effect of line tension and introducing the notion of a microscopic wetting angle results in more complex expressions for the correction factor in Eq. (14) determining the work of formation of a critical nucleus [16]. 6. Conclusion Homogeneous nucleation is a certain limiting case of boiling-up of a superheated liquid, which determines the upper stability boundary of a metastable phase with respect to finite changes in state parameters. Since in actual conditions it is practically impossible to exclude completely the effect of external actions and foreign inclusions in the liquid and on the walls of the vessel that surrounds it, in practice heterogeneous nucleation is realized much more often. In the case of homogeneous nucleation the source of nuclei is a chaotic thermal motion of molecules, with respect to which every element of the volume of a homogeneous system is in no way distinguished. At high superheatings heterogeneous nucleation, as well as homogeneous, is a fluctuation-induced

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phenomenon. The impossibility of taking into account the whole possible variety of factors that influence the nucleation process is the main difficulty in developing heterogeneous nucleation theory. Even if the heterogeneous nucleation mechanism has been established in the course of an experiment, it proves to be a rather complicated task to determine its source. In Ref. [1] we investigated the temperature, pressure, and concentration dependences of the nucleation rate in an ethane– nitrogen solution. It is shown that at J P 3  106 s1 m3 data on the attainable superheating of a solution may be interpreted in the framework of classical homogeneous nucleation theory. In the present work we analyze the role of possible centers of heterogeneous nucleation in conditions of our experiments and examine possible reasons for the disagreement between theory and experiment at low nucleation rates. First, we used the method of continuous pressure decrease to investigate the attainable stretches of an ethane–nitrogen solution in the range of nucleation rates where the disagreement between theory and experiment starts to become significant. Two nucleation mechanisms are in competition here. The first is homogeneous nucleation localized in a narrow range of state parameters (T, p, x), the second one is initiated nucleation with a very wide range of occurrence. We presented a statistical description of the process of initiated nucleation relating the value of the limiting (attainable) liquid stretch to its most probable value. The data obtained are in good agreement with the results of the method of lifetime measurement [1]. Second, for the specification of the mechanism of initiation and the role of heterogeneous nucleation in the experiments conducted we examined the thermodynamics and kinetics of boiling-up of a superheated liquid at the site of its contact with the capillary wall. It is shown that if the conditions of wetting of the wall do not change, and are also independent of the size of the bubble that forms in it, then at superheatings close to the boundary of spontaneous boiling-up of the liquid the probability of nucleus formation in the body of the liquid is higher than at both a smooth and a rough surface. If the pores of the wall of the capillary in filling it with the liquid under investigation retain undissolved gas, then at the initial stage of an experiment in the process of ‘‘runningup’’ of the capillary it may be effectively removed by ‘‘pressurizing’’ the liquid with an excess pressure with its subsequent reduction. Third, it is shown that at J < 3  106 s1 m3 a systematic ‘‘underheating’’ of the solution predicted by classical nucleation theory cannot be connected with bubble nucleation on the wetted wall of the capillary. We arrive at the conclusion, first made in papers by Sinitsyn and Skripov [3,17], that the main reason for the deviation of experimental curves from theoretical lines is nucleation initiated by cosmic radiation and the natural radiation background. In this case the potential barrier that inhibits the boiling-up of a superheated liquid is overcome at the expense of the energy from outside, which increases considerably the energy level of thermal fluctuations in the system.

On the average, one ionizing particle of secondary cosmic radiation passes through 1 cm2 of the Earth surface every minute [18]. Interacting with the substance atoms, high-energy particles knock out of them electrons, which can effectively turn into heat and give rise to the formation of critical nuclei. The absence of a microscopic theory of such type of initiation of nucleation does not allow us yet making direct quantitative evaluations of nucleation on such centers. The authors are grateful to A.M. Kaverin for his help in conducting experimental investigations. Conflict of interest None declared. Acknowledgements The work has been performed with a financial support of the Russian Foundation for Basic Research (project No. 15-0803399_a) and the Complex program of fundamental research of the Ural Branch of the Russian Academy of Sciences (project No. 15-1-2-6). References [1] V.G. Baidakov, A.S. Pankov, Nucleation in an ethane–nitrogen solution. I. Homogeneous nucleation, Int. J. Heat Mass Transfer. [2] V.A. Grigoriev, Yu.M. Pavlov, E.V. Ametistov, Boiling of cryogenic liquids, Energiya, Moscow, 1977. [3] V.P. Skripov, Metastable liquids, Wiley, New York, 1974. [4] D.A. Glaser, Some effects of ionizing radiation on the formation of bubbles in liquids, Phys. Rev. 87 (1952) 665. [5] V.P. Skripov, V.P. Koverda, The spontaneous crystallization of superheated liquids, Nauka, Moscow, 1984. [6] J.W.P. Schmelzer, Comments on the nucleation theorem, J. Colloid. Interface Sci. 242 (2001) 354–372. [7] V.G. Baidakov, A.M. Kaverin, I.I. Sulla, Attainable superheat of liquid ethane, Teplofyz. Vys. Temp. 27 (1989) 410–412. [8] J.S. Rowlinson, B. Widom, Molecular theory of capillarity, Clarendon Press, Oxford, 1982. [9] V.G. Baidakov, Explosive boiling of superheated cryogenic liquids, Wiley, Weinheim, 2007. [10] D.A. Labuntsov, Heat transfer in nucleate boiling of liquids, Teploenergetika 12 (1959) 19–26. [11] E.I. Nesis, Boiling of liquids, Nauka, Moscow, 1973. [12] N.M. Kottowski, O.E. Dwyer, Mechanism of nucleation, superheating, and reducing effects on the activation energy of nucleation, Progr. Heat. and Mass 7 (1973) 299–324. [13] R.T. Knapp, J.W. Daily, F.G. Hammit, Cavitation, McGRAW – Hill Book Company, New York, 1970. [14] M. Volmer, Kinetik der Phasenbildung, Steinkopff, Dresden – Leipzig, 1939. [15] P. Hänggi, M. Talkner, Borkovec, Reaction-rate theory: fifty years after Kramers, Rev. Modern Phys. 62 (1990) 251–341. [16] R.D. Gretz, The line-tension effect in heterogeneous nucleation, Surf. Sci. 5 (1966) 239–251. [17] V.P. Skripov, E.N. Sinitsyn, P.A. Pavlov, G.V. Ermakov, G.N. Muratov, N.V. Bulanov, V.G. Baidakov, Thermophysical properties of liquids in the metastable (superheated) state, Gordon and Breach Science Publishers, New York, London, Paris, Montreux, Tokyo, Melburn, 1988. [18] V.A. Akulichev, Cavitation in cryogenic and boiling liquids, Nauka, Moscow, 1978.