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Stabilization of boiling nuclei by insoluble gas: Can a nanobubble cloud exist? Michal Yarom, and Abraham Marmur Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b00715 • Publication Date (Web): 29 Jun 2015 Downloaded from http://pubs.acs.org on July 10, 2015
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.
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Stabilization of boiling nuclei by insoluble gas: Can a nanobubble cloud exist? Michal Yarom, Abraham Marmur* Technion – Israel Institute of Technology, Haifa, Israel
Abstract Liquid boiling that starts off with an insoluble gas bubble is thermodynamically analyzed. This case is an idealization of very low gas solubility and very slow diffusion of this gas in the boiling liquid. The analysis is made for a spherical, freely suspended bubble as well as for a bubble attached to a solid surface. The results predict spontaneous formation of a stable, critical bubble at pressures higher than the saturation pressure. Stable critical radii are also predicted for pressures lower than the saturation pressure, but in addition to unstable, larger critical bubbles. These bubbles are affected by the presence and nature of a solid surface. The present analysis provides a basis for a feasible explanation of the long-debated, long-time stability of nanobubbles.
Keywords: Heterogeneous nucleation; boiling; nucleus stability; nanobubbles; insoluble gas
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Introduction Boiling of liquids is one of the most common events that take place in daily life and industry. The behavior of bubbles formed during boiling is of major consequences for the properties or performance of many systems and processes. Among these are, for example, systems ranging from the simple home kettle to large, industrial heat exchangers 1, cavitation in industrial as well as biological systems
2, 3
, and the highly debated phenomenon of nanobubbles (e.g.
4
and the
references within). As is well known, homogeneous nucleation, namely nucleation in absolutely pure system, without any external nuclei, is highly unlikely. In order for a liquid to boil under practical conditions, a critical radius of a vapor nucleus must be reached. This process is facilitated by the presence of pre-existing nuclei (heterogeneous nucleation)
5, 6
. The critical
radius for boiling is achieved when the liquid pressure is lowered below the saturation pressure. The critical radius decreases as either the pressure decreases (at constant temperature), or the temperature increases (at constant pressure) beyond their saturation values. As is also well known, the critical-size nucleus is unstable: if by some perturbation it decreases, then it will continue to decrease, and will eventually disappear; if it somehow grows beyond the critical size, it will continue to grow and lead to completion of the boiling process. It is important to emphasize that it is the radius of curvature that needs to get to a critical value, not the size (e.g. volume) of the bubble. The nuclei that start the boiling process enable reaching the critical radius necessary for the completion of boiling, with a practically achievable number of vapor molecules 6. Pre-existing gas bubbles are common nucleation sites for boiling. The origin of these bubbles may be, for example, entrapment in cavities during liquid flow or mixing 7, 8
. When such nuclei are considered, the bubble growing during the boiling process will be a
mixture of the gas in the initial nucleus and the vapor resulting from boiling. This mixing process may have an important role in the course of the boiling process, as will be shown below. To date, studies have focused on the case of a nucleus consisting of a gas which is soluble to some extent in the surrounding liquid, using Henry’s law. It has been shown both experimentally 9
, and theoretically
10, 11
that these bubbles are unstable, just like the case of boiling of pure
liquids or liquid-liquid solutions. The present paper describes a study of the extreme case of a nucleus that initially consists of a completely insoluble gas. This case is highly interesting, because it turns out that the critical nucleus may be thermodynamically stable. This is in marked
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contrast to the instability of a critical nucleus that initially consists of a soluble or slightly soluble gas. As a corollary of this general thermodynamic analysis, an explanation of the puzzling stability of nano-bubbles is offered.
Theory The model system consists of a single bubble in a liquid, as schematically shown in figure 1a for a bubble attached to a surface, and in figure 1b for a freely suspended bubble (in a negligible gravity situation). This model system represents also the case of many identical bubbles (a zeroth-order approximation for a multi-bubble system), as long as they are not interacting with each other. The liquid phase is composed of a single, volatile component, and is under constant pressure at a constant temperature (therefore also of constant chemical potential). The initial bubble comprises an insoluble gas only, and serves as a nucleation site for a vapor-gas bubble to form.
Figure 1. The model system: a vapor-gas bubble formed on a pre-existing insoluble-gas bubble in a pure liquid (a) attached to a surface (b) freely suspended
In the present thermodynamic model, the gas in the initial bubble is completely prevented from diffusing into the liquid and equilibrating with it. In reality, there is no completely insoluble gas, only slightly soluble ones. However, since not equilibrium processes occur within the same time scale, we can use thermodynamic analysis to predict the behavior of a system with regard to
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some of its processes. Therefore, the proposed model represents an idealized case of a gas with very slow diffusion in the liquid. Analysis of equilibrium and stability of this system, at constant environmental pressure and temperature, requires calculations of the Gibbs energy, G. The starting point for getting the expression for G is the fundamental equation of thermodynamics for the internal energy, U, of a system in general 12 = − + +
(1)
where T is the absolute temperature, S is the entropy, P is the pressure, V is the volume, is the chemical potential, n is the number of moles, σ is the interfacial tension, and A is the interfacial area. The superscript j indicates the interface (liquid-air, solid-liquid, or solid-air), and the subscript i indicates the chemical species. The present specific system consists of a liquid continuous phase, a gas (or gas/vapor) bubble, a solid surface, and several interfaces. It is a priori assumed that the temperature is uniform throughout the system. It is also assumed that the molecules adsorbed to the interfaces are accounted for within the value of ni (at equilibrium they do have the same chemical potential as those of the same species in the other phases). The total internal energy is then given by = + − − + + +
(2)
where the superscripts indicate phase: V for vapor and L for liquid. The Gibbs energy is defined by
≡ + − ≡ + + − +
(3)
where the superscript en denotes the environment, and both V and S refer to the whole system. It
is assumed that the bulk phase is a priori at equilibrium with the environment ( = and = ). Incorporating eq. (2) into eq. (3) for the Gibbs energy results in = − + + +
(4)
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To understand the nucleation process, it is more indicative to calculate the formation Gibbs energy, ΔG, of the bubble
13, 14
, defined as the difference between G of the system of a given
nucleus radius and that of the system before boiling. The Gibbs energy at any given state is 13, 14 = − + + + +
(5)
The subscripts indicate component identity: 1 for the boiling liquid and 2 for the insoluble gas (that exists only in the bubble). At the initial state, before boiling (denoted by the superscript ⁰) the bubble contains insoluble gas only, and the Gibbs energy is given by = − + + +
(6)
Taking into account the following mass conservation equations =
(7)
n ! + n " = n #
(8)
the formation energy is given by
∆ = − − − # # + − + − # + −
(9)
The first two terms on the right-hand-side of Eq. (9) are work terms which stem from the inequality of pressures inside and outside the bubble, the next two are contributions of molecular mass transfer, and the last two terms represent the contribution of the surface energy. The possible equilibrium radii of a bubble nucleus are identified by looking for
%∆& %'
= 0 . The
stability of the bubble at the equilibrium point depends on the nature of the dependence of ∆ on r. Equilibrium is unstable if ∆ has a maximum and vice versa.In order to calculate ∆) from
eq. (9), some supplementary equations are required. Having determined as a parameter of the
system and r as the independent variable, is given by the Young-Laplace equation as + and is calculated from the geometrical relationship
* '
,
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=
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1 . -) 1 + /01 2 2 − /01 2 3
(10)
where θ is the contact angle (measured on the liquid side). Once the initial bubble is defined by the number of moles in it, its volume and radius are calculated from the following set of two equations # =
4 2 + ) #
(11)
=
1 -) . 1 + /01 2 2 − /01 2 3
(12)
and
The mole fraction of the vapor in the bubble, which is required for calculating the chemical potentials, can be determined from the following equation 5 =
4 4 = 1 − = 1 − = 1− 2 + + 6 + ) 7
(13)
where In this equation, the total number of moles (assuming an ideal gas) is given by =
⁄4, and the vapor pressure is related to the pressure in the liquid by the Young-Laplace equation. The chemical potential differences are then given by − = [ − : − − : ] = − @ − A 4 :
(14)
5 5 − = @4 = A = B4 = C 2 + )
(15)
Eq. (14) uses the fact that at the saturation point the chemical potentials of the liquid, : , and
vapor, : are equal. In these equations it is assumed that the gas and vapor are ideal gases, and that the liquid is incompressible. It can also be assumed that > − : ≈ −
* '
.
The surface energy terms in eq. (9) are expressed using geometric parameters
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− = - [) − ) ]2 + 3 /01 2 − /01 . 2
(16)
Thus, the Gibbs energy of formation is calculated from Eq.(9), using eqs. (10)-(16) and reads ∆) = [) − ) ] + B4 = + 4 =
where, as before, = + ≡
4 ) . ∘
* '
4 . ) . + > 2C @ ) − A )E 4 :
−
; = +
1 -[2 + 3 cos 2 − cos . 2] 3
* 'G
(17)
and (18)
Results and Discussion
Thermodynamic analysis of heterogeneous nucleation includes finding the critical radius and studying its stability, as previously explained. The size of the critical nucleus can be determined
from the extremal points in the ∆) curve. However, a more direct alternative is given by the
classical Kelvin equation 5, as a function of the pre-fixed environmental conditions. In case of a vapor-insoluble gas bubble, this is derived by equating the liquid and vapor chemical potentials
of the soluble component, eq. (14), and using the vapor mole fraction given by eq.(13) and the volume expression given by eq.(10) K
1
= LMN @−>K1
2 2 3 02 4 A− + 4) 1 )/ 1 -)3 1 + /01 22 2 − /01 2
(19)
The Kelvin equation is convenient for direct calculation of the critical radius associated with a
given pressure, however stability analysis must be done with ∆). Therefore, both
presentations will be used. First, the case of a free suspended bubble will be discussed, to be followed by analysis of a bubble attached to a solid. Finally, the possible implications of this study to the unsolved question of the stability of nanobubbles will be presented. Freely suspended nucleus
Fig. 2 compares ∆) for a pure vapor bubble (fig. 2a) and for a bubble that starts as an
insoluble gas and grows into a mixed, vapor-gas bubble (fig. 2b). As mentioned above and
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demonstrated in fig. 2a, the “usual” ∆) curve for heterogeneous nucleation has a maximum,
implying that the critical nucleus is unstable. In surprising contrast, as shown in fig. 2b for the
present case, ∆) may have a minimum and a maximum, indicating that there may exist also a stable (or metastable) critical nucleus. From a mathematical point of view, eq. (17) for ∆)
shows a cubic dependence on r. As such, it is clear that it may have a minimum point in addition to a maximum, depending on the circumstances.
Figure 2. Formation energy and its components at T=373K (a) pure water vapor bubble (b) insoluble gaswater vapor bubble , n2v=10-16 mol (r0=0.64µm), PL/Ps=0.49.
In order to identify the source of this possible stability from a physical point of view, the contribution of each component of the energy function is presented. Fig. 2a shows the components of the formation energy of a pure, spherical, water-vapor bubble. These include the
pressure inequality term, − , the surface energy term, , and the chemical potential term, μ − μ . While all the components contribute to the value of the formation energy, it turns out that the key component in determining stability is the chemical potential difference of
the species that boils, Δµ1, since it determines the direction of mass transfer between the liquid and the bubble 6. At the critical radius, this chemical potential difference is zero, since this state is, by definition, an equilibrium state. At this point, the chemical potential difference changes signs. If a small, spontaneous increase in bubble size decreases Δµ1, more liquid will boil and the bubble will continue growing. Such is the case for an unstable bubble (maximum points in figure 2), since any small perturbation will either shrink or increase the bubble indefinitely. However, if an
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increase in bubble size increases Δµ1, the vapor will condense and the bubble will return to its initial size, rendering a stable bubble (minimum point in figure 2b). In other words, the bubble is stable if
%∆PQ %'
|'S > 0. Δµ1 is given by
∆ = 4 =
5 − :
(20)
where is constant in the present case. ∆ is therefore a function of the pressure and composition, which have competing effects. Writing this expression as a function of the radius results in
2 4 U + ) − . V ) ∆ = 4 = − :
(21)
where A is a geometric parameter previously defined. Calculating the derivative of ∆μ with
respect to r shows that if the critical radius,rX < Z increases with r, and the bubble is stable.
.[] \ ^_ `a
, the chemical potential difference
Figure 3 complements the picture by showing a graphical representation of the Kelvin equation, namely, all possible equilibrium states. The figure displays the liquid saturaion ratio
(LSR), / : , that is associated with a given critical radius for a water vapor bubble with
various amounts of insoluble gas. For = 0, the system obviously reduces to a single-
component system, for which there is only one critical radius that corresponds to each given
liquid pressure. As is usually the case, the pressures along this curve are all below the saturation pressure.
In contrast, the curves for ≠ 0 show that a mixed gas-vapor bubble may exist at a liquid
pressure above the saturation pressure. In addition, depending on the amount of the insoluble gas, the curve may have a minimum point, thus two critical points may be possible under suitable conditions. This behavior of the equilibrium curve can be mathematically explained by analyzing equation (19). When rc is small, 1/rX. is the dominant factor, resulting in a large LSR and a
descending equilibrium curve that may lead to a minimum. However, when rc increases, 1/ rc becomes the dominant factor, the LSR is lower, and the curve ascends with rc.
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Figure 4 displays energy curves for a vapor-gas bubble with given amounts of insoluble gas. These energy curves start at ro and represent different saturation ratios, marked by the horizontal lines
in
figure
3.
The
Gibbs
energy
of
formation,
under
the
pre-determined
conditions , , , is given by eq. (17), assuming the contact angle to be zero (to conform
with the case of a freely suspended bubble). Above saturation pressure (LSR>1), there exists a minimum in the ΔG plot, implying a single, stable critical radius. Below saturation pressure
(LSR
Article
Stabilization of boiling nuclei by insoluble gas: Can a nanobubble cloud exist? Michal Yarom, and Abraham Marmur Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b00715 • Publication Date (Web): 29 Jun 2015 Downloaded from http://pubs.acs.org on July 10, 2015
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.
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Stabilization of boiling nuclei by insoluble gas: Can a nanobubble cloud exist? Michal Yarom, Abraham Marmur* Technion – Israel Institute of Technology, Haifa, Israel
Abstract Liquid boiling that starts off with an insoluble gas bubble is thermodynamically analyzed. This case is an idealization of very low gas solubility and very slow diffusion of this gas in the boiling liquid. The analysis is made for a spherical, freely suspended bubble as well as for a bubble attached to a solid surface. The results predict spontaneous formation of a stable, critical bubble at pressures higher than the saturation pressure. Stable critical radii are also predicted for pressures lower than the saturation pressure, but in addition to unstable, larger critical bubbles. These bubbles are affected by the presence and nature of a solid surface. The present analysis provides a basis for a feasible explanation of the long-debated, long-time stability of nanobubbles.
Keywords: Heterogeneous nucleation; boiling; nucleus stability; nanobubbles; insoluble gas
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Introduction Boiling of liquids is one of the most common events that take place in daily life and industry. The behavior of bubbles formed during boiling is of major consequences for the properties or performance of many systems and processes. Among these are, for example, systems ranging from the simple home kettle to large, industrial heat exchangers 1, cavitation in industrial as well as biological systems
2, 3
, and the highly debated phenomenon of nanobubbles (e.g.
4
and the
references within). As is well known, homogeneous nucleation, namely nucleation in absolutely pure system, without any external nuclei, is highly unlikely. In order for a liquid to boil under practical conditions, a critical radius of a vapor nucleus must be reached. This process is facilitated by the presence of pre-existing nuclei (heterogeneous nucleation)
5, 6
. The critical
radius for boiling is achieved when the liquid pressure is lowered below the saturation pressure. The critical radius decreases as either the pressure decreases (at constant temperature), or the temperature increases (at constant pressure) beyond their saturation values. As is also well known, the critical-size nucleus is unstable: if by some perturbation it decreases, then it will continue to decrease, and will eventually disappear; if it somehow grows beyond the critical size, it will continue to grow and lead to completion of the boiling process. It is important to emphasize that it is the radius of curvature that needs to get to a critical value, not the size (e.g. volume) of the bubble. The nuclei that start the boiling process enable reaching the critical radius necessary for the completion of boiling, with a practically achievable number of vapor molecules 6. Pre-existing gas bubbles are common nucleation sites for boiling. The origin of these bubbles may be, for example, entrapment in cavities during liquid flow or mixing 7, 8
. When such nuclei are considered, the bubble growing during the boiling process will be a
mixture of the gas in the initial nucleus and the vapor resulting from boiling. This mixing process may have an important role in the course of the boiling process, as will be shown below. To date, studies have focused on the case of a nucleus consisting of a gas which is soluble to some extent in the surrounding liquid, using Henry’s law. It has been shown both experimentally 9
, and theoretically
10, 11
that these bubbles are unstable, just like the case of boiling of pure
liquids or liquid-liquid solutions. The present paper describes a study of the extreme case of a nucleus that initially consists of a completely insoluble gas. This case is highly interesting, because it turns out that the critical nucleus may be thermodynamically stable. This is in marked
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contrast to the instability of a critical nucleus that initially consists of a soluble or slightly soluble gas. As a corollary of this general thermodynamic analysis, an explanation of the puzzling stability of nano-bubbles is offered.
Theory The model system consists of a single bubble in a liquid, as schematically shown in figure 1a for a bubble attached to a surface, and in figure 1b for a freely suspended bubble (in a negligible gravity situation). This model system represents also the case of many identical bubbles (a zeroth-order approximation for a multi-bubble system), as long as they are not interacting with each other. The liquid phase is composed of a single, volatile component, and is under constant pressure at a constant temperature (therefore also of constant chemical potential). The initial bubble comprises an insoluble gas only, and serves as a nucleation site for a vapor-gas bubble to form.
Figure 1. The model system: a vapor-gas bubble formed on a pre-existing insoluble-gas bubble in a pure liquid (a) attached to a surface (b) freely suspended
In the present thermodynamic model, the gas in the initial bubble is completely prevented from diffusing into the liquid and equilibrating with it. In reality, there is no completely insoluble gas, only slightly soluble ones. However, since not equilibrium processes occur within the same time scale, we can use thermodynamic analysis to predict the behavior of a system with regard to
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some of its processes. Therefore, the proposed model represents an idealized case of a gas with very slow diffusion in the liquid. Analysis of equilibrium and stability of this system, at constant environmental pressure and temperature, requires calculations of the Gibbs energy, G. The starting point for getting the expression for G is the fundamental equation of thermodynamics for the internal energy, U, of a system in general 12 = − + +
(1)
where T is the absolute temperature, S is the entropy, P is the pressure, V is the volume, is the chemical potential, n is the number of moles, σ is the interfacial tension, and A is the interfacial area. The superscript j indicates the interface (liquid-air, solid-liquid, or solid-air), and the subscript i indicates the chemical species. The present specific system consists of a liquid continuous phase, a gas (or gas/vapor) bubble, a solid surface, and several interfaces. It is a priori assumed that the temperature is uniform throughout the system. It is also assumed that the molecules adsorbed to the interfaces are accounted for within the value of ni (at equilibrium they do have the same chemical potential as those of the same species in the other phases). The total internal energy is then given by = + − − + + +
(2)
where the superscripts indicate phase: V for vapor and L for liquid. The Gibbs energy is defined by
≡ + − ≡ + + − +
(3)
where the superscript en denotes the environment, and both V and S refer to the whole system. It
is assumed that the bulk phase is a priori at equilibrium with the environment ( = and = ). Incorporating eq. (2) into eq. (3) for the Gibbs energy results in = − + + +
(4)
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To understand the nucleation process, it is more indicative to calculate the formation Gibbs energy, ΔG, of the bubble
13, 14
, defined as the difference between G of the system of a given
nucleus radius and that of the system before boiling. The Gibbs energy at any given state is 13, 14 = − + + + +
(5)
The subscripts indicate component identity: 1 for the boiling liquid and 2 for the insoluble gas (that exists only in the bubble). At the initial state, before boiling (denoted by the superscript ⁰) the bubble contains insoluble gas only, and the Gibbs energy is given by = − + + +
(6)
Taking into account the following mass conservation equations =
(7)
n ! + n " = n #
(8)
the formation energy is given by
∆ = − − − # # + − + − # + −
(9)
The first two terms on the right-hand-side of Eq. (9) are work terms which stem from the inequality of pressures inside and outside the bubble, the next two are contributions of molecular mass transfer, and the last two terms represent the contribution of the surface energy. The possible equilibrium radii of a bubble nucleus are identified by looking for
%∆& %'
= 0 . The
stability of the bubble at the equilibrium point depends on the nature of the dependence of ∆ on r. Equilibrium is unstable if ∆ has a maximum and vice versa.In order to calculate ∆) from
eq. (9), some supplementary equations are required. Having determined as a parameter of the
system and r as the independent variable, is given by the Young-Laplace equation as + and is calculated from the geometrical relationship
* '
,
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1 . -) 1 + /01 2 2 − /01 2 3
(10)
where θ is the contact angle (measured on the liquid side). Once the initial bubble is defined by the number of moles in it, its volume and radius are calculated from the following set of two equations # =
4 2 + ) #
(11)
=
1 -) . 1 + /01 2 2 − /01 2 3
(12)
and
The mole fraction of the vapor in the bubble, which is required for calculating the chemical potentials, can be determined from the following equation 5 =
4 4 = 1 − = 1 − = 1− 2 + + 6 + ) 7
(13)
where In this equation, the total number of moles (assuming an ideal gas) is given by =
⁄4, and the vapor pressure is related to the pressure in the liquid by the Young-Laplace equation. The chemical potential differences are then given by − = [ − : − − : ] = − @ − A 4 :
(14)
5 5 − = @4 = A = B4 = C 2 + )
(15)
Eq. (14) uses the fact that at the saturation point the chemical potentials of the liquid, : , and
vapor, : are equal. In these equations it is assumed that the gas and vapor are ideal gases, and that the liquid is incompressible. It can also be assumed that > − : ≈ −
* '
.
The surface energy terms in eq. (9) are expressed using geometric parameters
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− = - [) − ) ]2 + 3 /01 2 − /01 . 2
(16)
Thus, the Gibbs energy of formation is calculated from Eq.(9), using eqs. (10)-(16) and reads ∆) = [) − ) ] + B4 = + 4 =
where, as before, = + ≡
4 ) . ∘
* '
4 . ) . + > 2C @ ) − A )E 4 :
−
; = +
1 -[2 + 3 cos 2 − cos . 2] 3
* 'G
(17)
and (18)
Results and Discussion
Thermodynamic analysis of heterogeneous nucleation includes finding the critical radius and studying its stability, as previously explained. The size of the critical nucleus can be determined
from the extremal points in the ∆) curve. However, a more direct alternative is given by the
classical Kelvin equation 5, as a function of the pre-fixed environmental conditions. In case of a vapor-insoluble gas bubble, this is derived by equating the liquid and vapor chemical potentials
of the soluble component, eq. (14), and using the vapor mole fraction given by eq.(13) and the volume expression given by eq.(10) K
1
= LMN @−>K1
2 2 3 02 4 A− + 4) 1 )/ 1 -)3 1 + /01 22 2 − /01 2
(19)
The Kelvin equation is convenient for direct calculation of the critical radius associated with a
given pressure, however stability analysis must be done with ∆). Therefore, both
presentations will be used. First, the case of a free suspended bubble will be discussed, to be followed by analysis of a bubble attached to a solid. Finally, the possible implications of this study to the unsolved question of the stability of nanobubbles will be presented. Freely suspended nucleus
Fig. 2 compares ∆) for a pure vapor bubble (fig. 2a) and for a bubble that starts as an
insoluble gas and grows into a mixed, vapor-gas bubble (fig. 2b). As mentioned above and
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demonstrated in fig. 2a, the “usual” ∆) curve for heterogeneous nucleation has a maximum,
implying that the critical nucleus is unstable. In surprising contrast, as shown in fig. 2b for the
present case, ∆) may have a minimum and a maximum, indicating that there may exist also a stable (or metastable) critical nucleus. From a mathematical point of view, eq. (17) for ∆)
shows a cubic dependence on r. As such, it is clear that it may have a minimum point in addition to a maximum, depending on the circumstances.
Figure 2. Formation energy and its components at T=373K (a) pure water vapor bubble (b) insoluble gaswater vapor bubble , n2v=10-16 mol (r0=0.64µm), PL/Ps=0.49.
In order to identify the source of this possible stability from a physical point of view, the contribution of each component of the energy function is presented. Fig. 2a shows the components of the formation energy of a pure, spherical, water-vapor bubble. These include the
pressure inequality term, − , the surface energy term, , and the chemical potential term, μ − μ . While all the components contribute to the value of the formation energy, it turns out that the key component in determining stability is the chemical potential difference of
the species that boils, Δµ1, since it determines the direction of mass transfer between the liquid and the bubble 6. At the critical radius, this chemical potential difference is zero, since this state is, by definition, an equilibrium state. At this point, the chemical potential difference changes signs. If a small, spontaneous increase in bubble size decreases Δµ1, more liquid will boil and the bubble will continue growing. Such is the case for an unstable bubble (maximum points in figure 2), since any small perturbation will either shrink or increase the bubble indefinitely. However, if an
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increase in bubble size increases Δµ1, the vapor will condense and the bubble will return to its initial size, rendering a stable bubble (minimum point in figure 2b). In other words, the bubble is stable if
%∆PQ %'
|'S > 0. Δµ1 is given by
∆ = 4 =
5 − :
(20)
where is constant in the present case. ∆ is therefore a function of the pressure and composition, which have competing effects. Writing this expression as a function of the radius results in
2 4 U + ) − . V ) ∆ = 4 = − :
(21)
where A is a geometric parameter previously defined. Calculating the derivative of ∆μ with
respect to r shows that if the critical radius,rX < Z increases with r, and the bubble is stable.
.[] \ ^_ `a
, the chemical potential difference
Figure 3 complements the picture by showing a graphical representation of the Kelvin equation, namely, all possible equilibrium states. The figure displays the liquid saturaion ratio
(LSR), / : , that is associated with a given critical radius for a water vapor bubble with
various amounts of insoluble gas. For = 0, the system obviously reduces to a single-
component system, for which there is only one critical radius that corresponds to each given
liquid pressure. As is usually the case, the pressures along this curve are all below the saturation pressure.
In contrast, the curves for ≠ 0 show that a mixed gas-vapor bubble may exist at a liquid
pressure above the saturation pressure. In addition, depending on the amount of the insoluble gas, the curve may have a minimum point, thus two critical points may be possible under suitable conditions. This behavior of the equilibrium curve can be mathematically explained by analyzing equation (19). When rc is small, 1/rX. is the dominant factor, resulting in a large LSR and a
descending equilibrium curve that may lead to a minimum. However, when rc increases, 1/ rc becomes the dominant factor, the LSR is lower, and the curve ascends with rc.
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Figure 4 displays energy curves for a vapor-gas bubble with given amounts of insoluble gas. These energy curves start at ro and represent different saturation ratios, marked by the horizontal lines
in
figure
3.
The
Gibbs
energy
of
formation,
under
the
pre-determined
conditions , , , is given by eq. (17), assuming the contact angle to be zero (to conform
with the case of a freely suspended bubble). Above saturation pressure (LSR>1), there exists a minimum in the ΔG plot, implying a single, stable critical radius. Below saturation pressure
(LSR
