Subscriber access provided by UNIV OF NEW HAMPSHIRE
Article
Heterogeneous vapor bubble nucleation on a rough surface Alexey O. Maksimov, Aleksey M. Kaverin, and Vladimir G. Baidakov Langmuir, Just Accepted Manuscript • DOI: 10.1021/la400340y • Publication Date (Web): 18 Feb 2013 Downloaded from http://pubs.acs.org on February 18, 2013
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Heterogeneous vapor bubble nucleation on a rough surface Alexey O. Maksimov,† Aleksey M. Kaverin,‡ and Vladimir G. Baidakov∗,‡ Pacific Oceanological Institute Far Eastern Branch of the Russian Academy of Sciences, Vladivostok, Russia, and Institute of Thermophysics of the Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia E-mail:
[email protected] Abstract
and foreign particles), the knowledge and understanding of their properties is a key factor in controlling boiling of superheated liquids which is the main subject of the present study. Our attention here focuses mainly on the one factor - the influence of surface roughness. Mechanical damage is known to promote surface nucleation, revealing a strong dependence upon the degree of surface roughness. 5–7 Nucleation at rough surfaces includes in addition to geometrical factor the effect of elastic stresses, 2,8 but the specifics of the considered problem of boiling allows for ignoring this factor. The classical nucleation theory based on the assumption that macroscopic thermodynamics can be satisfactory used for small clusters of new phase has been recently used for finding the nucleation rate in nanocavities. 9–12 There exist several alternative approaches. For example, a kinetic approach to nucleation which does not use concept of surface tension was developed in Ref. 13–15 This approach has been used for analysis of condensation of liquid drop from the vapor in nanocavity. 16 In the present study, we employ classical nucleation theory. It is mainly directed to the analysis of theoretical problems. When comparing with experiment, we will focus on simple liquids, since important physical parameters, in particular, the surface tension and the tension of the contact line can be independently calculated for this model media by the method of molecular dynamics. Liquefied inert gases are one of the best studied examples of such fluid.
Vapor bubble nucleation on a micro-rough surface wetted by a volatile, incompressible liquid has been analyzed. The work of formation of a critical nucleus in a surface cavity has been evaluated. Concave sites of a surface with negative curvature can decrease the height of the activation barrier for the formation of a critical nucleus. The average density of the nucleation sites has been evaluated for a surface whose (small) deviation from a plane is specified by a Gaussian random function.
INTRODUCTION The rate at which a new phase nucleates in a metastable ambient phase is largely determined by heterogeneity of the system. Heterogeneous nucleation proceeds at preferential sites such as phase boundaries or impurities. On interfaces, thermodynamic barriers for nucleation are typically lower than the respective bulk values, causing a predominance of surface nucleation. This was first recognized by Volmer 1 in 1939. The thermodynamics of heterogeneous nucleation has rather a long history and recent reviews on this subject can be found in Refs. 2–4 Because surfacenucleation is governed by the presence of nucleation sites (e.g. tips, cracks, scratches, cavities ∗ To
whom correspondence should be addressed Oceanological Institute ‡ Institute of Thermophysics † Pacific
ACS Paragon Plus Environment
1
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Investigations of boiling-up kinetics in superheated liquid argon and krypton have shown that depending on the values of determining parameters: p – the pressure and T – the temperature, the nucleation may proceed both by the mechanism of homogeneous nucleation and by creation of nuclei on the wall of the experimental cell. 17,19 The method of lifetime measurement of superheated liquids has been used to study the role of surface nucleation in more detail in Ref. 18 To interpret the experimental data, a model of heterogeneous nucleation on a plane surface has been used and then generalized to account for the excess free energy of the boundary of a three-phase solid-liquid-gas systems i.e. the contact line tension. The aim of the present study is the consideration of the surface nucleation on a micro-rough surface. Concave sites of a surface with negative curvature can decrease the height of the activation barrier for the formation of a critical nucleus – the mechanism, which is actively studied now. 20–22 The paper is structured as follows. In Sec. II, the description of nucleation in a surface cavity is presented, and the main expressions for calculation of the work for formation of nucleus are provided. In Sec. III, the nucleation rate at specific sites corresponding to the umbilic points of rough surface is evaluated. Results and discussion are presented in Sec. IV. Finally, the conclusions are summarized in Sec. V.
Page 2 of 17
the initial state, which includes a metastable fluid, and the final state, which includes a nucleus and a metastable fluid, determines the work of formation of a bubble or a drop: ) ( ) ( W = ∆G = pα − pβ Vβ + µβ − µα Nβ ) ( + σβ α Sβ α + σβ S − σα S Sβ S + τl Γαβ S . (1) Here, µα , µβ are the chemical potentials of particles (atoms, molecules) of the system under consideration, Vβ is the volume of the bubble (drop), Sαβ and Sβ S are the surface areas of vapor/liquid and β -phase/solid interfaces, σαβ , σα S , and σβ S are the specific surface energies of the vapor/liquid, α -phase/substrate and β phase/substrate interfaces, respectively, Nβ is the number of molecules in the nucleus. The last term in Eq. (1) accounts for the contribution of the line tension, τl , along the contact line Γαβ S . For the case when an nucleus is an incompressible liquid drop, the term ∆P ≡ pα − pβ plays the role of Lagrange multiplier. 23 For spherical cluster of a radius, X, in homogeneous nucleation (HON) the volume Vβ and the surface area Sαβ have the form: Vβ = (4π /3)X 3 and Sαβ = 4π X 2 . In writing Eq. (1) we assume that changes of the state of the ambient phase do not occur and the substrate is chemically homogeneous, thus both surface tensions σα S and σβ S as well as the line tension τl are not depend on the position on the substrate. Our analysis is based on the minimization of the work of formation of a nucleus. Assuming that the thermodynamic parameters of phase are connected by the equations of state, and the surface and line tensions at the interfaces are functions only of temperature we can calculate the first differential of Eq.(1)
NUCLEATON IN A SURFACE CAVITY A first-order phase transition begins with the formation and subsequent growth of new-phase nucleus. The minimal work, W , to form an nucleus of a new phase in a metastable system depends on the mechanisms and the conditions under which the process takes place. Let us consider the process of formation of a new phase, in particular, of a vapor bubble in a superheated liquid (or a liquid drop in a supercooled vapor) located in the experimental cell at constant external pressure, p, and temperature, T . We consider a single nucleus of a β phase that is deposited onto a solid substrate S and surrounding by a bulk phase α . The difference in the values of the Gibbs thermodynamic potential for
( ) ( ) d∆G = pα − pβ dVβ + µβ − µα dNβ ( ) + σβ α dSβ α + σβ S − σα S dSβ S + τl dΓαβ S (2) In a thermodynamic equilibrium state, the function d∆G(Vβ , Sαβ , Sβ S , Nβ , Γαβ S ) has an extremum i.e., (d∆G)∗ = 0 which leads to the conditions for mechanical and diffusion equilibrium of a bubble (drop). For spherical cluster in HON these conditions have the form
ACS Paragon Plus Environment
2
Page 3 of 17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
2σαβ dSαβ = , dVβ X∗ µα (pα , T ) = µβ (pβ , T ).
eral case of the curvilinear surface, we note that the hypersurface of the Gibbs potential difference in cluster formation (1) has the property that near the extreme point the variables associated with changes in volume or shape, and the number of particles are separated. Thus, if the equilibrium conditions with respect to the chemical potential are satisfied, then the critical clusters size is determined by considering the mechanical variables. The random nature of the roughness can appear as cavities, asperities, grooves, etc. The cavities are the most probable sites for bubble nucleation. The increase of the nucleation rate due to cavity can be understood as a result of the interplay between the work gained upon evaporation (condensation) of n building units into n-sized nucleus and the work done on creating the interface between the nucleus and the ambient phases (i.e. the vapor/liquid and β -phase/substrate interfaces). We assume that cavity of a surface has the shape of spherical segment of radius R and depth D (see Figure 1). The approach to nucleation at such a geometry is analogous to the one proposed by Fletcher 26 to describe the condensation on spherical dust particles. Later it was used in 9 for crystallization inside micro/nanocavity and in 10 to describe the nucleation inside a spherical drop. We give below some details of evaluation of the nucleation barrier. This is done first, to take into account the line tension effects, and secondly, this approach allows one to obtain simpler analytical expressions for the barrier height (compare Ref. 9,26 and comments on 10 in 11 ). Since these expressions will be used in evaluating the integral characteristics of nucleation on a rough surface, significant simplification in the expression for the barrier height makes it easy to use the asymptotic methods for calculating these characteristics. Let a nucleus of β -phase having the shape of a lens settle down on a solid surface, in a cavity with a radius of curvature R. The bubble (we shall perform all calculations for this type of nucleus and discuss the case of drops only at presenting the final results) is surrounded by α -phase which is an incompressible liquid wetting well the walls of the cell. Introducing notations for the radius of a bubble – X and the heights of the segments – H and Y (see Figure 1), the variation of the Gibbs potential for the formation of a bubble can be presented by
pβ ∗ − pα = σαβ
(3)
For heterogeneous nucleation (HEN), the state of the cluster formed on the solid substrate depends, in addition to the number of molecules Nβ and the volume Vβ , on its shape. For a cap-shaped bubble on a plane substrate we have Vβ =
π 2 π H (3X − H) = X 3 (1 + cos ϑ )2 3 3 × (2 − cos ϑ ) ,
Sαβ = 2π XH = 2π X 2 (1 + cos ϑ ) , ( ) Sβ S = π 2XH − H 2 = π X 2 sin2 ϑ , H = X (1 + cos ϑ ) , (4) where H is the spherical cap height and ϑ is the so called contact angle (the angle, conventionally measured through the liquid, at which a liquid/vapor interface meets a solid surface with values from 0 to 180◦ ). As independent variables for finding the mechanical equilibrium we can choose any pair (X, H) or (X, ϑ ) or (Vβ , Sβ S ), which leads to the conditions for mechanical equilibrium – the first line of Eq.(3) and an additional (generalized) Young’s equation: ) [( ] cos ϑ∗ = ± σβ S − σα S + τl /Lβ S /σαβ , (5) where Lβ S is the radius of the contact line and the sign plus corresponds to a bubble and the sign minus to a drop. Note that this difference is due to the adopted convention that the contact angle is measured through the liquid. The value ϑ∗ is determined by the condition of mechanical equilibrium which implies that the resultant of the forces of surface and line tensions has no a component along the solid surface. The last term of this generalized Young’s equation accounts for three-phase molecular interactions at the contact line between the solid, liquid and vapor phases. The direction of this force for the positive line tension coincides with the surface tension σβ S . This equation was developed long ago 24 based on a suggestion by Gibbs. 25 Before proceeding with the analysis of the gen-
ACS Paragon Plus Environment
3
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
This geometrical constraint implies that the bases for upper and lower spherical segments are equal. The mechanical equilibrium condition that determines the shape of the liquid/vapor interface and the contact angles is achieved by finding an extremum of the function f (7) under some welldefined constraints. In all the calculations below we assume implicitly that fluctuation ripples on the bubble wall come to equilibrium (decay) over much smaller time scale, thus the bubble interface has the form of the spherical segment. The constraint set is the level set of the function c(x, y, h) = (1 − y)2 + x2 − (h − x)2 at level 1. We use the method of Lagrange multipliers to determine the extremum values of f subjected to the constraint c(x, y, h) = 1 and introduce a new variable λ , called the Lagrange multiplier, and set up the equations
α
R
H β
X
Page 4 of 17
D ϑ
Y
Figure 1: Geometry of nucleus formation in a cavity analogy with 11 in the form: [ π Y 2 (3R −Y ) π H 2 (3X − H) ] W = −∆p + 3 3 + 2π (σV S − σLS ) RY + 2πσLV XH √ + 2πτl 2Y R −Y 2 . (6)
∂f ∂c =λ , ∂x ∂x
∂f ∂f ∂c ∂c =λ , =λ , ∂y ∂y ∂h ∂h c(x, y, h) = 1; ∂f ∂f = −h2 + σi h, = (h2 − 2xh) + σi x, ∂x ∂h ∂f 1−y = y2 − 2y + σe + τi √ ; ∂y 2y − y2 ∂c ∂c ∂c = 2h, = −2(1 − y), = −2(h − x). ∂x ∂y ∂h (8)
The terms in the first line describe variation in volume energy, the second term characterizes variation of free energy on the bubble interface, the third line accounts for the contribution of the contact line. Following, 10,11 in writing (6), use is made of the spherical cap heights H and Y in addition to the radius of the bubble – X which are the variables of multidimensional heterogeneous nucleation. The difference of pressures (pV − pL ) ≡ ∆p at small degree of superheating can be approximated with a good accuracy via (pV − pL ) = (pVs − pL ) (1 − ρVs /ρLs ), 17,27 where pVs is the saturation pressure at a planar interface, and ρVs , ρLs are the orthobaric densities of vapor and liquid phases, respectively. In the dimensionless variables: x )= X/R, ( y = Y /R, h = H/R, f = W / ∆pπ R3 , σe = 2 (σV(S − σLS) ) / (∆pR), σi = 2σLV / (∆pR), τi = 2τl / ∆pR2 the work (6) takes the form
Thus, the extremum is determined by the following equations h(h − σi ) = 2λ h, 1−y y2 − 2y + σe + τi √ = −2λ (1 − y), 2y − y2 (h2 − 2xh) + σi x = −2λ (h − x), (1 − y)2 + x2 − (h − x)2 = 1. (9) Generally, it is useful to eliminate the variable λ at an early stage, since the value of this auxiliary variable is not of direct interest. Thus, from the first line of Eq.(9) we have λ = −(h − σi )/2. Substituting this value into the third line of (9), we immediately obtain x∗ = σi . Before presenting the general solution, we consider first the situation when line tension effects
[ ] f = − y2 (3 − y) /3 + h2 (3x − h) /3 √ + σe y + σi xh + τi 2y − y2 . (7) The variables x, y, and h are not independent, but are coupled by the relation 2y − y2 = 2hx − h2 .
ACS Paragon Plus Environment
4
Page 5 of 17
are negligible. In this case, the saddle point 11 of the surface (7) has the following coordinates:
100
∗
√ x ∗ = σi , = 1 − (1 − σe ) / 1 + σi2 − 2σe ) √ ( 2 0 h∗ = σi + σe − σi / 1 + σi2 − 2σe , (10)
∗
∗
60 50 40
10 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
|τi|
Figure 2: The actual contact angle as a function of the dimensionless line tension τi for different values of the ratio of the critical bubble radius to the pit curvature: x∗ = 0.5, 0.6, 0.7, 0.8. Data points (♢) correspond to τl = −4 × 10−12 J/m2 and superheat, resulting in a critical bubble with the radius of curvature X∗ = 5 nm Written in dimensional variables we get
( ) 1[ 0 f∗ = −2 + 3σe + σi3 + 2 − σe − σi2 √3 ] 1[ × 1 + σi2 − 2σe = −2 + 3x∗ cos ϑ∗0 + x∗3 3√ ] ( ) + 2 − x∗ cos ϑ∗0 − x∗2 1 + x∗2 − 2x∗ cos ϑ∗0 .
√ σV S − σLS τl + 1 − (LLV /R)2 , cos ϑ∗ = σV L σV L LLV X∗ sin ϑ∗ , (15) LLV = √ 1 − 2(X∗ /R) cos ϑ∗ + (X∗ /R)2 where LLV is the radius of the contact line. This equation is consistent with the general expression for the contact angle on heterogeneous surface 28,29 and coincides with the expression for “the actual contact angle on a spherical surface” obtained earlier (Eq.(26) in Ref. 30 ). The two differences are that the radius of the contact line LLV for the critical bubble is not an independent variable, but depends on the actual contact angle ϑ∗ and the sign of the term describing the line tension for the bubble is opposite to the one used for drop in cited studies. 28–30 An important feature of (15) is that the contribution of the contact line is determined by the geodesic curvature χ of a circle of radius LLV √ embedded in a spherical shell of radius R:
(12) In the analysis of the general situation, we use the fact that the remaining nonlinear equations (9) determine the equilibrium contact angle, so, we rewrite them in terms of the variable ϑ . This leads to the following relations (1 − σi cos ϑ∗ ) y∗ = 1 − √ , 2 1 + σi − 2σi cos ϑ∗ , (13)
use of which allows one to find the desired solution:
σe (1 − σi cos ϑ∗ ) . + τi σi σi2 sin ϑ∗
x = 0.8
20
At x = x∗ , y = y0∗ , we obtain an expression for the equilibrium contact angle cos ϑ∗0 = σe /σi = (σV S − σLS ) /σV L , that is Young’s equation. The value of the Gibbs potential at the saddle point is equal to the following expression
cos ϑ∗ =
∗
30
{ [( )2 cos ϑ = 2y − y2 − 2y − y2 ( )( ) ]1/2 } −1 − 1 + x2 2y − y2 + x2 x . (11)
1 + σi2 − 2σi cos ϑ∗
x = 0.7
∗
70
where the superscript ’0’ denotes that we ignore line tension. It follows from the geometry of the problem (see Figure 1), that the contact angle ϑ is expressed in terms of the determining parameters x, y by the following formula
σi (cos ϑ∗ − σi )
x = 0.6
80
y0∗
h ∗ = σi + √
x = 0.5
90
ϑ / degrees
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
2 /R2 /L . χ = 1 − LLV LV A large amount of experimental work has been performed in order to determine the magnitude of line tension. 31–36 The observed values for τl were found to be very different in magnitude. There
(14)
ACS Paragon Plus Environment
5
Langmuir
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 6 of 17
Gibbs function takes the form
have been reported both negative and positive values that vary with respect to their absolute from 10−12 J/m to 10−5 J/m. Theoretical predictions on the value of line tension are based on calculation of the shape of Lennard-Jones nanodrops using molecular dynamics. 37 For the Lennard-Jones drop the values of τl are always negative. Because the application of the Lennard-Jones model is most justified for cryogenic liquids, as the estimates we will use the value τl = −(2 − 4) × 10−12 J/m. 18,38 According to (15), for negative values of line tension the contact angle ϑ∗ is greater than the macroscopic limit ϑ0 . Figure 2 shows the relations between the contact angle defined as described above and the dimensionless line tension τi for critical bubbles of different radius of curvature x∗ . Calculations have been carried out for liquid argon with ϑ0 ≈ 10◦ . The range of values of the dimensionless line tension τi corresponds to the aforementioned dimensional values of this parameter τl and a degree of metastability, leading to bubbles of the critical radius X∗ ∼ 5 nm. Remind that, in this study, we investigate the nucleation in liquids that wet the surface well ϑ∗ ≤ 90◦ . Equations x = σi and (15) determine the local extremum of the Gibbs function along the given directions. Substituting the respective coordinates of the saddle point into Eq. (7), we get
W∗ = Wh∗ Ψ (x∗ , ϑ∗ ) ,
Wh∗ =
16πσV3 L
, 3 (∆p)2 1 3 cos ϑ∗ 1 Ψ (x∗ , ϑ∗ ) = + − 3 2 2x∗2 x∗ ( )( ) 1 cos ϑ∗ 1 1 2 cos ϑ∗ 1/2 − + − 2 1+ 2 − 2 2x∗ x∗ x x∗ [∗ 3 (cos ϑ0 − cos ϑ∗ ) 1 ( ) cos ϑ∗ − + −1 2 2 x∗ cos ϑ∗ − x∗ x∗ ( )1/2 ] 1 2 cos ϑ∗ , (17) + 1+ 2 − x∗ x∗ where Wh∗ is the work of formation of a critical bubble in HON. In writing Eq.(17), we have replaced (using Eq.(15)) the explicit dependence on line tension by the term proportional to (cos ϑ∗ − cos ϑ0 ). Figure 3 illustrates the dependence of the shape factor of free energy barrier to nucleation Ψ(x∗ , ϑ∗ ) on the equilibrium wetting angle ϑ∗ and the dimensionless critical radius of a nucleus x∗ . For comparison (by dashed lines), the plots of the shape factor ignoring line tension are shown. It follows from this figure that accounting for line tension diminishes the height of the energy barrier to nucleation. At the same time, the shape factor can vanish at increasing line tension which is an unphysical result. The reason for this inconsistency is wetting instability, which was first described by Widom 39 for droplets on a solid substrate, and studied in more details in subsequent investigations. 23,29,40,41 The possibility to explore this type of instability on a curved substrate has been demonstrated by Guzzardi and Rosso 42 for droplets sitting on a sphere (convex surface). A complete study of the wetting instability for nucleation in a cavity is quite cumbersome and requires to compute the second variation of the Gibbs function, account for the constraints up to the second order, minimize the second variation on a suitable set. This is beyond the scope of the present paper, but we use the fact that the values of line tension for many non-polar liquids are small and thus, the threshold value has not been reached in the present case.
1[ −2 + 3x∗ cos ϑ ∗ + x∗3 3 ] ( )√ 1 + x∗2 − 2x∗ cos ϑ∗ + 2 − x∗ cos ϑ∗ − x∗2 τi [ + −1 + x∗ cos ϑ∗ x∗ sin ϑ∗ √ ] + 1 + x∗2 − 2x∗ cos ϑ∗ . (16)
f∗ =
Effect of line tension is manifested in this expression in two ways: by introducing additional terms (cf. (12)), and by replacing the Young’s contact angle ϑ0 with the “actual” contact angle ϑ∗ . In dimensional variables, the saddle-point value of the
ACS Paragon Plus Environment
6
Page 7 of 17
we have
1.5 x∗ = 0.5
x∗ = 0.6
x∗ = 0.7
x∗ = 0.8
1 (1 + cos ϑ∗ )2 (2 − cos ϑ∗ ) 4 3 3 4 − x∗ sin ϑ∗ + (cos ϑ∗ − cos ϑ0 ) sin2 ϑ∗ 8 4 × (1 + 2x∗ cos ϑ∗ ) . (20)
Ψ (x∗ , ϑ∗ ) =
1
Ψ
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
0.5
0
0
10
20
30
40
50
ϑ∗ / degrees
60
70
80
The above result reveals the way the barrier height decreases with approaching (not too close) of the critical radius to the size of the cavity. Up to now we have considered a vapor bubble in the cavity. The basic equations undergo the following changes when droplet nucleates in the same geometry corresponding to Fig. 1. The difference of) pressures ∆ p˜ has ( ( snows the)form ∆ p˜s = pα − pβ = (pL − pV ) = ρl /ρV − 1 (pV − pV ). The dimensionless parameter σ˜ e changes the sign σ˜ e = 2 (σLS − σV S ) / (∆ pR). ˜ The right-hand side of equation (11), which expresses the contact angle ϑ˜ in terms of geometric parameters of the droplet and the cavity, also changes the sign. As a result, the generalized Young’s equation for the contact angle of the droplet on the spherical substrate takes the form ) ( 1 + σi cos ϑ˜∗ ˜ σ e cos ϑ˜∗ = . (21) − τi σi σi2 sin ϑ˜∗
90
Figure 3: The energy barrier to nucleation (solid lines) as a function of the equilibrium wetting angle and the dimensionless critical radius x∗ . Data points (♢) correspond to τl = −4 × 10−12 J/m2 and superheat, resulting in a critical bubble with the radius of curvature X∗ = 5 nm. Dashed lines show the predicted dependence when the contribution of the line tension is neglected The asymptotic behavior of Eq. (17) in the range of small values of x∗ describes of the flows Jhet = Jhet het het the contribution from nucleation sites with (R ≥ < corresponds to (R < R ). Since the Rb ) and Jhet b derived model (17) accounts only for the process of nucleation within the spherical cavity, we shall > . Evaluation of J > will analyze below only Jhet het be done in a general form for simplified models of rough surfaces - Gaussian random surfaces 43–46 and the proper choice of the specific boundary value Rb will be made at the final stage of calculations. < is determined by the contribution The flow Jhet of small pits on the random surface, which do not fit the critical bubbles. An important stage in the growth of the nucleus is the time when it reaches the pit mouth. The process by which the interface gets out of the pit mouth is an essential step for the formation of the critical nucleus and as has been shown 47 its effect on the nucleation process can be significant. In this case, the formation of such a state requires additional energy to change the position of the contact line. For this reason, they should be energetically less favorable. Following 45,46 we can calculate the average density of these umbilic points for the case where Σ is a surface whose (small) deviation from a plane is specified by a Gaussian random function. Consider a smooth undulating surface Σ specified by its deviation ξ (r) from the reference plane r′ = (x′ , y(′ ). Let ξ)(r′ ) be sufficiently small that (∇ξ )2 = ξx2′ + ξy2′ X∗ is not too large. Since only in this case the number of nucleation centers near the umbilic points will be sufficient to compete with the nucleation in the volume. In this case, the integral (36) is determined by the contribution of the neighborhood of the upper limit and we have
The density of all umbilic points on Σ is simply ∫∞
n=
Nu (R)dR = −∞
M6 . 4π M4
(33)
For an autocorrelation function with ( ) the Gaussian form ρ (|r|) = ⟨ξ 2 ⟩ exp −r2 /2L2 this gives Nu (R) =
[ 4 ( )] 3 1 2 2 √ exp −L / 4⟨ ξ ⟩R , 4π 3/2 ⟨ξ 2 ⟩R2 n = 3/(π L2 ),
M4 = 8⟨ξ 2 ⟩/L4 ,
M6 = 48⟨ξ 2 ⟩/L6 . (34)
It is obvious that half this value of n comes from regions where the curvature is positive and half from where it is negative. On substituting (26) into (24) and introducing the curvature radius of the dominant roughness ) ( 2 2 4 Rc = L / 4⟨ξ ⟩ we have > Jhet
1 = 2al
∫∞
dR Rb
S(R) 3 Bhet (R) √ 3/2 4π ⟨ξ 2 ⟩R2
] [ 2 Rc Wh∗ × exp − 2 − Ψ (X∗ /R, ϑ∗ ) , (35) R kB T where the shape function Ψ is given by (17). The surface area of a single cavity, the radius of curvature of which is less than the average distance between the cavities L is equal to S(R) = π RD ≈ π R2 κ . For larger √ values, one can use the expression S ≈ π L < ξ 2 >, which is correct to an order of magnitude. Transition to the dimensionless variable x∗ greatly simplifies the evaluation of this integral:
> Jhet
=
3
Page 10 of 17
∫1/s
S(x∗ ) dx∗ Bhet (x∗ ) √ ⟨ξ 2 ⟩X∗
4π 3/2 al 0 ] [ 2 2 Rc x∗ Wh∗ × exp − 2 − Ψ (x∗ , ϑ∗ ) , (36) X∗ kB T
> Jhet =
kB T Wh∗ dΨ/dx∗ x
∗
=s−1
Bhet (X∗ s)
[ ] S(X∗ s) 3 R2c √ exp − 2 2 × al 4π 3/2 ⟨ξ 2 ⟩X∗ X∗ s [ ] ∗ ) Wh ( −1 × exp − Ψ s (κ , ϑ∗ ), ϑ∗ . (37) kB T
because it is now clearly seen to be ( a function ) of two dimensionless parameters R2c /X∗2 and Wh∗ /kB T . In the experiments on boiling of superheated cryogenic liquids, 17–19 the parameter Wh∗ /kB T is
ACS Paragon Plus Environment
10
Page 11 of 17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
The explicit forms of the shape-function Ψ and its first derivative at the boundary value of the curvature Rb are given by the following formulae
the dominant curvature ( ∗radius Rb). The term in the first line (kB T ) / Wh dΨ/dx∗ is the interval of the dominant curvature radii – this factor naturally arises in the calculation of the integral of the exponential function with a large exponent. Note that in the neighborhood of angles cos ϑ∗ ≈ s, a more accurate evaluation of the contribution of the dominant radii is necessary which has been done by calculation of etalone integral. However, because the difference is only in the value of the preexponential factor, which does not affect the final result in the leading order of magnitude, we ignore this feature. Evaluation of the total sum of flows of nuclei from the areas of vessel walls of different curvature (23) (contribution of non-umbilic points) requires knowledge of the density N (R1i , R2i ) of surface areas with the principal curvatures R1i and R2i , which is defined by the integral of the form Eq. (31) from the product of the indicator (δ ) functions and the Jacobian. For Gaussian random surface, this density can be found in a closed form and is written in the form of a single integral. Analytical evaluation of the Gibbs potential for the formation of a bubble on a rough surface is yet unsolved problem, though wetting of rough surfaces was intensively studied 48–50 during the past decade. A special case of umbilic points, considered here, is a step to solution of the general problem.
( ) 1 3 cos ϑ∗ s2 Ψ s−1 , ϑ∗ = + − s3 2 2 ( )√ 1 cos ϑ∗ 2 − + s−s 1 + s2 − 2 cos ϑ∗ s 2 2 3 (cos ϑ0 − cos ϑ∗ ) s2 [ + cos ϑ∗ − s 2 (cos ϑ∗ − s) ( )1/2 ] + 1 + s2 − 2 cos ϑ∗ s , ( ) ( ) dΨ ∂ Ψ ∂ Ψ ∂ ϑ∗ = + , dx∗ x∗ =s−1 ∂ x∗ ∂ ϑ∗ ∂ x∗ x∗ =s−1 [ ( ) ∂Ψ 3 2 = − s 2 (cos ϑ∗ − s) ∂ x∗ x∗ =s−1 2 ( ) 1 + cos2 ϑ∗ /2 − 2s cos ϑ∗ + s2 + 1/2 (1 + s2 − 2s cos ϑ∗ ) [ (cos ϑ0 − cos ϑ∗ ) s 2 (cos ϑ∗ − s)2 − 2 (cos ϑ∗ − s) ( ) ]} 5s (cos ϑ∗ − s)2 − 2 (cos ϑ∗ − 2s) 1 − s2 − , 1/2 (1 + s2 − 2s cos ϑ∗ ) ( ) { ∂Ψ 3 2 sin ϑ∗ 2 (cos ϑ∗ − s)2 =− s ∂ ϑ∗ x∗ =s−1 2 (cos ϑ∗ − s)2 [ ( )] (cos ϑ∗ − s) (cos ϑ∗ − s)2 − 1 − s2 + 1/2 (1 + s2 − 2s cos ϑ∗ ) [ ( )] } (cos ϑ0 − cos ϑ∗ ) s2 (cos ϑ∗ − s) − s 1 − s2 , − 1/2 (1 + s2 − 2s cos ϑ∗ ) DISCUSSION ( ) ∂ ϑ∗ = (cos ϑ0 − cos ϑ∗ ) s (cos ϑ∗ − 2s) Obtaining an analytical formulation for the nucle∂ x∗ x∗ =s−1 ation on a micro-rough surface, allows one to com( )−1/2 × 1 + s2 − 2s cos ϑ∗ . (38) pare the effectiveness of this mechanism with experimental data for the nucleation rate. According The resulting expression (37) for the nucleation to the definition of the nucleation rate, for domrate admits the following interpretation. The kiinant heterogeneous nucleation, the following renetic factor Bhet (Rb ) will be of the same order of lation τ = (Jhet Σ)−1 holds between the average magnitude as in homogeneous nucleation, and, as expectation time of a critical nucleus τ , Jhet and a first approximation, one can assume Bhet ≈ B. the area wetting by metastable phase, Σ. ExperiThe exponent in the third line is the Gibbs numments 17 trace the temperature and pressure depenber, the work of formation of a critical bubble in dence of the mean lifetime of superheated liquethe cavity of dominant curvature. The second line fied gases with τ varying from 10−1 to 1.8 − 2.4 · represents the product of the concentration of nu103 s. By this reason log τ is traditionally used for cleation sites within the umbilic cavity S(X∗ s)/al the comparison of experimental data with theoretand the surface density of umbilic Nu (Rb ) with ACS Paragon Plus Environment
11
Langmuir
ical models. Thus, we have for log (Jhet Σ)
35 30
log τ = − log (Σ/al ) + log(e) ( ) ] [ 2 Wh∗ Rc S(X∗ s) × 2 2+ Ψ(κ , ϑ∗ ) − log √ X∗ s kB T ⟨ξ 2 ⟩X∗ ] [ 3Bhet kB T − log . (39) 4π 3/2Wh∗ dΨ/dx∗
Rc = 2 X∗
25 20
log τ (sec)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 12 of 17
Rc = 3 X∗
15 10 5
Rc = 4 X∗
0
The experimental setup of the majority of the discussed experiments 17–19 includes a glass or metallic cell (a capillary with an inner diameter of ∼ 1 mm and an outer diameter of ∼ 6 mm) of volume V ≈ 80 mm3 . For the experiments Ar ( with liquid ) al ≈ 0.16 nm2 , Σ = 320mm2 , Wh∗ /kB T ≈ 70, X∗ ≈ 5 nm. Substituting these values of parameters into (39) we have
−5 −10
0
10
20
30
40
50
ϑ∗ (degrees)
60
70
80
90
Figure 4: The expectation time of a critical nucleus in liquid argon as the function of the equilibrium wetting angle for the characteristic values of the mean curvature of a rough surface Rc /X∗ = 2, 3, 4. Data points (♢) correspond to τl = −4 × 10−12 J/m2
(
) Rc 2 log τ = −12.6 + 30.4 Ψ (κ , ϑ∗ ) + 0.43 Xs ( ) ( ) ∗ Bhet Rc − log − log 2πκ . (40) dΨ/dx∗ X∗
indirect data. In most glassmaking processes, thermal and chemical effects directly influence the final composition, morphology, defect density and durability of a glass surface. It is well known that glass surfaces are generally not stable against water and humidity. The effects of water in humid air or as a liquid phase are found different: characteristic pits develop in humid air, while the roughness increases in a more uniform way under water. 51 The Atomic Force Microscopy (AFM) is a peerless way of performing surface height measurements with a precision better than 0.1 nm and over lateral scales from a few tenths nanometers to 100 mm. The review 51 of using of AFM in glass research contains a wealth of information about the scale of glass surface roughness. More recent studies of surface properties of glass micropipettes with use of scanning electron microscopy (SEM) 52 evidently demonstrate the presence of the inner wall surface roughness at the scale equal or smaller than 100 nm. Glass micropipettes pulled from borosilicate glass tubes had an outer diameter of 1.5 mm and an inner diameter of 0.86 mm which are close to the parameters of glass tubes from which the acoustic cells were made. It is plausible that the acoustic cell surface roughness has the same topography as the one demonstrated by SEM stereoscopic images (Fig.
To carry out the experimental test of the theory, in addition to information on the thermodynamic properties of the liquid, the following three parameters should be independently determined for a particular pair of superheated liquid and solid surface being in contact. These are the average lifetime of a superheated liquid, the coefficient of the line tension and the contact angle. At present, there are no such measurements. Figure 4 illustrates the dependence of the expectation time of a critical nucleus on the wetting angle ϑ∗ and the ratio of the dominant curvature Rc to the radius curvature of the critical bubble X∗ . Though calculations have been performed for different values of κ the present figure corresponds to κ = 0.3. The curves show the behavior for different values of Rc (Rc = 2X∗ , 3X∗ , and 4X∗ , respectively). It follows from this figure that the proposed model can reproduce experimental data on lifetime measurements in the range of 10−1 – 2 · 103 s in the case of presence of nanoscale surface roughness of Rc ∼ 10 nm (X∗ ≈ 5 nm). Since in experiments 17,19 the degree of roughness of the walls of a glass capillary at the nano scale has not been evaluated, we shall try to use
ACS Paragon Plus Environment
12
Page 13 of 17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
10 in Ref. 52 ).The presence of nano-scale roughness on the inner surface of the metal cells 18 is even more probable. Of course, a significant limitation of the model is the assumption that the irregularities are smooth < ξ 2 > /L2 ≈ L2 can be justified because there are no physical parameters in the problem that would affect the number of nucleation sites. The presence of the AFM images of the surface provides a possibility for the independent, experimental determination of potential nucleation sites, at least, within an order of magnitude. As has been shown above, the behavior of the contact line influences significantly the process of nucleation in the nano-cavity. Accounting for line tension is a simplest way to consider three-phase molecular interactions between the solid, liquid and vapor phases. More refined methods taking into account the shape of the disjoining pressure in the film, or the use of molecular dynamics simulation, 37,53 will advance in the study of the nucleation on a rough surface. The current tendency of miniaturization in electronic components leads to strong heating in materials. It is known that the most effective cooling can be achieved by using multiphase cooling systems. All surface features such as topography and wettability down to nanometric scales are relevant and influence the heat exchange. Experimental investigation on the onset of nucleation boiling over highly polished surface 54 quantifies the effect of wettability and the role of the roughness. Roughness parameters used to characterize different samples are mean roughness amplitude, rms roughness amplitude, maximum peak to valley roughness amplitude, reduced valley depth, maximum valley height, maximum peak height. As follows from the current study, the absence of the correlation length in this list (or direct evaluation of the curvature on the base of AFM image) does not provide correct description of the preferred sites for nucleation on the rough surface. The boiling curves representing the heat flux through the polished surface versus the superheat (Fig. 4, 6 in Ref. 54 ) are stationary characteristics and the de-
rived theory can not be directly used for their interpretation. On contrary, the bubble emission frequency at a nucleation site, defined as f = 1/(τgt + τ ) which can be determined from the nucleation videos 55 allowing comparison under condition τgt
