Article pubs.acs.org/Langmuir
Rate of Bubble Coalescence Following Dynamic Approach: Collectivity-Induced Specificity of Ionic Effect Yael Katsir and Abraham Marmur* Department of Chemical Engineering, Technion − Israel Institute of Technology, Haifa 32000, Israel ABSTRACT: A simple, quantitative model is suggested to explain the specificity of ions with respect to inhibition of bubble coalescence following a dynamic approach. For the first time, the mode of thinning of the film in between the bubbles, as determined by the density of the bubble dispersion, is recognized as a determining factor. The specificity of the ionic effect is explained by a major difference in adsorption properties of ions, which is enhanced by the film thinning. This leads to charge separation that forms an electrical double layer at each interface of the thin, liquid film, and consequently to electrostatic repulsion. This effect is described by a simple theoretical model that consists of two fundamental equations: mass conservation of each ion in the film, and the Gibbs adsorption equation. In addition, we explain the rapid coalescence of bubbles in purified water under dynamic conditions, which is in contrast with the very slow coalescence under quasi-static conditions.
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INTRODUCTION Bubble coalescence in aqueous solutions is a ubiquitous phenomenon that is essential for life-supporting processes as well as for various industrial procedures. A prominent example of a natural process is the eventual coalescence with the sea−air interface of bubbles that are formed during breaking of seawater waves.1−6 This process leads to transfer of gases and ionic aerosols to the atmosphere,5−8 which is essential for marine life9 on one hand, and for the formation of clouds10 on the other. Industrial examples include mass transfer processes in bubble columns, distillation towers, gas−liquid contactors, bioreactors, electrochemical cells, and so forth.11,12 Bubble coalescence has attracted much scientific and engineering attention, because the specific, inhibiting effect of some ion combinations on coalescence is not yet well understood. In fact, the inhibition of bubble coalescence by specific ions has been a puzzling phenomenon since it was first observed in 1929.13 Moreover, the experimental observations appear to contradict the well-established DLVO (Derjaguin, Landau, Verwey, and Overbeek) theory,14 which describes the interaction between colloidal particles in general, based on the combined effects of van der Waals (vdW) attraction and electric double layer (EDL) repulsion. The effect of various electrolytes on bubble coalescence has been investigated using a variety of methods. Since bubble coalescence is actually a rupture process of a thin, liquid film (to be referred to as “thin film” for brevity), it has been studied by methods that apply to either whole bubbles or isolated thin films: pairs of bubbles,7,15−21 bubbles rising to the interface (which serves as very large, second bubble),22−24 two bubbles in atomic force microscopy (AFM),25−27 breaking waves,1−6,28 bubble columns,7,11,12,29−36 and a thin, liquid film between two gas phases.37−39 These studies have revealed the effect of © 2014 American Chemical Society
electrolyte type and concentration on the rate of bubble coalescence. The transition between very rapid or rapid coalescence (hundreds of milliseconds or seconds, respectively) and relatively slow coalescence (above a minute) occurs only with certain electrolytes, at specific concentrations (termed “transition concentrations”31). Craig et al.,31,32 using the bubble column method, empirically classified the ions of the electrolytes as type α or β. Roughly speaking, most cations and anions that form strong bases or acids, respectively, were classified as α ions. H+ and some anions that form weak acids were considered β ions. They found that αα and ββ electrolytes in aqueous solutions slow bubble coalescence, while αβ and βα electrolytes have no meaningful effect compared with purified water.31,32 This classification was completely supported by the results obtained using the thin-film method,37 and partially supported by other studies using multibubble methods (although they examined only αα electrolytes).1−7,11,12,28−30,33−36 However, observations using twobubble methods were ambiguous and, unfortunately, were mostly studied only with αα electrolytes.7,15−27 Recently, the important influence of the approach velocity of the bubble toward the second interface was elucidated, using a thin film38 and a sliding bubble.21 The coalescence behavior of the bubbles at low approach velocities of about 1−10 μm/s (quasi-static approach) was observed to be different from that above this velocity range. For the dynamic approach, NaCl solutions inhibited bubble coalescence, as had been reported in previous studies.38 However, for the quasi-static approach, the observations were completely different: slow coalescence in Received: August 22, 2014 Revised: October 26, 2014 Published: October 27, 2014 13823
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purified water (minutes to hours) and very rapid coalescence for NaCl solutions (less than a second).38 Unfortunately, these observations38 were studied only with αα electrolytes. It was later shown that the coalescence of a bubble with the air− solution interface, following the quasi-static approach, does obey the DLVO theory.40 Slow coalescence in purified water was shown to be due to the EDL repulsion originating from the speciation of dissolved CO2. Rapid coalescence in electrolyte solutions was demonstrated to result from neutralization of the carbonaceous, charged species by acids, or screening of the EDL by electrolytes.40 However, the inhibiting effect of electrolytes following the dynamic approach of bubbles is still unclear. So far, only partial explanations were given, based on surface elasticity,29,32,41 surface diffusion,42 ion partitioning at the tested liquid−air interface,43−46 hydration and structural forces,19,47,48 effective/ partial immobilization of interfaces, and deformation effects.49−52 Moreover, no explanation has been offered for the differences between the results of the two-bubble methods and those attained by other methods. In the present study, we experimentally examine the effect on the rate of bubble coalescence of the electrolyte type and concentration, and of the rate of approach, using a two-bubble apparatus.40 We analyze the results by focusing on the differences between twobubble systems and published results for multibubble or thinfilm systems (a thin-film system is a liquid film stretched between two gaseous phases, with possibility of drainage speed control; multibubbles systems are usually columns in which a high-density bubble stream flows through a liquid, under dynamic conditions). The previously obtained understanding of bubble coalescence following quasi-static approach, the difference between the two-bubble methods and the other methods, and the difference between coalescence in purified water and in electrolyte solutions all led us to simple theoretical explanations of the hitherto puzzling observations regarding inhibition of bubble coalescence.
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Figure 1. Coalescence times versus bubble approach velocity in purified water. Our data (full circles; arrows indicate coalescence times longer than 103 s); data of Castillo et al. (empty squares) using a sliding-bubble apparatus;21 and semiquantitative data of Yaminski et al.38 using a thin-film apparatus (the shaded regions represent upper and lower limits of bubble approach velocity and coalescence times, as reported by Yaminsky et al.).38 differences in the coalescence time when the approach velocity is varied; also, it is quite sufficient for comparing the present coalescence times with literature data for dynamic approach velocities.
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RESULTS AND DISCUSSION Experimental data showing the effect of bubble approach velocity on coalescence times are presented first for a bubble rising in purified water (Figure 1), then for a bubble rising in various aqueous electrolyte solutions (Figure 2). As explained below, the mechanisms underlying coalescence in purified water and in aqueous electrolyte solutions are entirely different; therefore, they will be discussed separately. Figure 1 shows coalescence times in purified water, for approach velocities that cover the whole range of interest, from ∼1 μm/s to ∼50 mm/s. The coalescence time, which is the measurable parameter representing the rate of coalescence, is defined as the time difference between the moment at which the top of the bubble is seen to contact the liquid−air interface and the last frame before coalescence.40 As previously discussed, the accuracy of this method is more than adequate for the purpose of showing major differences in the coalescence time. The data in Figure 1 come from a few different sources: (a) our present data (indicated by full circles, and marked by upward arrows at the level of 103 s when the coalescence times are longer that 103 s40); (b) data of Castillo et al. (indicated by squares), who used another kind of a two-bubble method, a sliding-bubble apparatus;21 and (c) semiquantitative data of Yaminski et al., who used a thin-film apparatus38 (shown in this figure as shaded regions that indicate upper and lower limits of coalescence times and the corresponding approach velocities). It is clear that all the data in Figure 1 are quite scattered, which is a natural outcome of the statistical nature of the phenomenon. However, over orders of magnitude of the coalescence time, all presented experimental methods suggest the same trend: bubble coalescence in purified water is very slow following a quasi-static approach (∼1 to ∼10 μm/s), but rapid (1−10 seconds) or very rapid (less than a second) following dynamic approach. Figure 2 shows coalescence times vs bubble approach velocity, for various aqueous electrolyte solutions, including αα and βα electrolytes. Using αα electrolytes solutions, our
EXPERIMENTAL SECTION
The experimental method chosen was the same one that had been used in our quasi-static study:40 a bubble is grown from a capillary toward the tested liquid−air interface (representing a second, very large bubble), by pushing air using a syringe. The production of the bubble was controlled manually by turning a micrometer, and the approach velocity of the bubble was controlled by setting the distance between the capillary tip and the liquid−air interface. The processes of bubble rise and coalescence were recorded by a video camera (17.4 frames/s) attached to a microscope. The approach velocity of the top of the bubble was directly calculated from the video frames preceding the touching of the interface, where the coalescence process/dynamics took place. In our experimental setup we managed to achieve three ranges of approach velocities: 1−10 μm/s (defined as quasistatic38,40), 50−500 μm/s, and 5−50 mm/s (see Figure 1). The two latter ranges are defined, by default, as dynamic. In addition, it should be mentioned that all experiments were performed at a temperature of 296 K and relative humidity of ∼55%. This simple system was used because of the following technical advantages: (a) the approach velocity can be easily controlled by forming the bubble at various distances from the airliquid interface; (b) it allows simple optical visualization of the coalescence process, hence enables measurement of the bubble approach velocity and the coalescence times at the required accuracy level. The uncertainty in the coalescence times, based on the standard deviation as well as on an analysis of possible errors in determining the beginning of the actual coalescence point, is about 1 s and 0.2 s regarding the quasi-static and the dynamic approaches, respectively. This is more than adequate for the purpose of showing the major 13824
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et al.,37 using a thin-film apparatus with αα electrolytes, show coalescence times of 10−100 s at about 0.1 M and above. Thus, the transition from either very rapid or rapid to slow coalescence that is induced by the αα electrolytes is observed with thin-film methods, but not with two-bubble methods. Our results, using αβ or βα electrolytes, show very rapid coalescence following a quasi-static approach, and either very rapid or rapid coalescence following dynamic approach (see Figure 2b). Unfortunately, most of the previous studies using similar two-bubble methods did not examine coalescence times in αβ or βα electrolytes (exceptions include studies by Christenson,16 who measured coalescence percentage and not time, and by Sagert,17 who did measure coalescence times, but only up to 1 s). However, using a thin-film apparatus, Karakashev et al. showed coalescence times in αβ electrolyte solutions that were similar to our results (only a few seconds).37 Thus, the slow coalescence in electrolyte solutions, using thin-film apparatus, is ion-specific, since it occurs with αα electrolytes, but not with αβ or βα electrolytes. Moreover, this inhibition and its ion specificity have been observed also with multibubble methods, in addition to thin-film methods, but not with two-bubble methods. Previous studies showed that two main parameters affect bubble coalescence: the composition of the solution, and the approach velocity of the bubbles. The present study brings into focus a third, very important parameter: the density of the system of bubbles, or, as will be explained below, the thickness of the film in between the bubbles during coalescence. The rate of bubble coalescence, as it depends on all these parameters, is qualitatively presented in Table 1. The items in this table that are emphasized by italic font are different from their counterparts in terms of either bubble system density or approach velocity. As can be seen in this table, the coalescence behavior of bubbles in purified water and in aqueous electrolyte solutions is qualitatively different. This indicates, in the discussion that follows, that the mechanisms underlying the inhibition of coalescence are different in the two cases. First, the case of water is discussed, to be followed by a discussion of the case of electrolyte solutions. In purified water, coalescence is very slow following quasistatic approach of the bubble (minutes to hours),21,38,40 but is rapid following dynamic approach (less than about 10 s).1−7,11,12,15−39 Yaminsky et al. explained the transition to rapid coalescence with increasing approach velocity by the hydrodynamic pressures in the film.38 However, another approach to the explanation is considered in this study, which is based on a simple energy balance. In our previous study40 we demonstrated that the slow coalescence following quasi-static approach is due to the potential barrier originating from the speciation of dissolved CO2.40 The hypothesis put
Figure 2. Coalescence times vs bubble approach velocity in various aqueous electrolyte solutions. (a) αα electrolytes: circles, 0.01 M NaCl; squares, 0.5 M NaCl; rhombuses, 1 M NaCl; triangles, 0.5 M KCl. Data by Castillo et al.: empty squares, 0.1 M KCl; empty triangles, 0.5 M KCl.21 Data of Yaminsky et al. (shaded regions since the results are semiquantitative).38 (b) αβ aqueous electrolyte solutions: circles, 0.5 M NaAc; squares, 0.5 M KAc. βα aqueous electrolyte solutions: triangles, 0.01 M HCl; rhombuses, 0.01 M HNO3.
results (full symbols in Figure 2a) show very rapid coalescence following a quasi-static approach.40 However, following a dynamic approach, coalescence times are somewhat scattered between very rapid and rapid coalescence (0.1−10 s). These results are in agreement with coalescence times of Castillo et al., who used another two-bubble method (empty symbols in Figure 2a).21 In contradiction, relatively slow coalescence was observed by Yaminsky et al. in solutions with sufficient concentration of NaCl, using a thin-film apparatus (these results are only semiquantitative, therefore are presented as shaded regions in Figure 2a).38 In addition, data by Karakshev
Table 1. Qualitative Summary of the Dependence of the Rate of Bubble Coalescencea quasi-static approach purified water two bubble systems multibubble systems thin-film system
21,40
Slow N/A Slow38
αα electrolytes 21,40
Very Rapid N/A Very Rapid38
dynamic approach αβ electrolytes Very Rapid N/A N/A
40
purified water
αα electrolytes
αβ electrolytes
7,15−27
7,16−27
Rapid16,17 Rapid31,32,37 Rapid31,32,37
Rapid Rapid1−7,11,12,28−39 Rapid1−7,11,12,28−39
Rapid Slow1−7,11,12,28−39 Slow1−7,11,12,28−39
(1) Composition of the solution (types of ions); (2) approach velocity of the bubble: quasi-static approach (up to about 10 μm/s), or dynamic approach (about 50 μm/s and above); (3) density of the bubble system or area of the film (depending on the experimental system). Definitions in terms of order of magnitude: slowsignificantly more than 10 s; rapid10 s and less. It should be noted that the bubble approach velocity in multibubble systems is, by the nature and size of the system, always dynamic. a
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taken in calculating the kinetic energy is that of water having the same volume as the assumed bubble. The velocity that enables the kinetic energy of this mass to acquire the same kinetic energy as the maximum EDL repulsion energy is then given by
forward in this paper is that coalescence in purified water, following dynamic approach, is rapid, because the kinetic energy associated with the bubble is sufficient to overcome the electric double layer (EDL) repulsion. In order to estimate the energy required to overcome the potential barrier between the bubble and the water−air interface (due to EDL interactions), we calculated their interaction energy through the water film. In our previous publication, regarding coalescence following the quasi-static approach, we calculated the interaction energy assuming planar double layers.40 This was justified by the observation that when the coalescence is very slow, the two interfaces have time to flatten. However, following a dynamic approach, when coalescence is rapid, the two interfaces may not have time to flatten. Hence, the interaction energy for this case is roughly assessed by considering a bubble of similar size to the one formed in the experiments (say, Rb = 1 mm) and a very large bubble (say, R∞ = 1 m) representing the air−liquid interface. This interaction energy is given by53 ⎡ 1 + e−χh AR b 2R ∞R b + επψ 2⎢ln 6h R∞ + Rb ⎣ 1 − e−χh ⎤ + ln(1 − e−2χh)⎥ ⎦
v≥
(1)
where (2)
The first expression on the right-hand side of eq 1 is the vdW attraction energy, where h is the minimal thickness of the fluid film between the bubble and the liquid−air interface, and A is the Hamaker constant. In order to calculate the appropriate A for our system, namely, for two bodies of the same material immersed in a different medium, we use eq 2,54 where A11 and A22 are the Hamaker constants of water and air, respectively. This equation accounts for all possible pairwise interactions, and since the Hamaker constant of air is practically zero, the Hamaker constant of two bubbles in water turns out to be equal to that of water in vacuum (A = A11 = 3.7 × 10−20 J). The second expression on the right-hand side of eq 1 is the EDL repulsion energy, where ε is the permittivity of water, and ψ is the surface potential of the bubble interface in purified water25 at the natural pH of 5.8 (ψ = −40 mV). The Debye length, χ−1, is given by40 χ −1 ≡
κε0kBT 2e 2NAI
(4)
The water mass turns out to be about 4 × 10−6 kg, so the velocity must be greater than 70 μm/s. This fits quite well the order of magnitude of our experimental observations (see Figure 1; rapid coalescence around 50 μm/s and above). Thus, rapid coalescence in purified water following a dynamic approach can be explained using a black-box approach, namely, by a global energy balance between the kinetic energy and the EDL repulsion energy. Such an order of magnitude estimation may be done without considering the details of the hydrodynamic phenomena. The main problem remains, however, the explanation of the inhibiting effect of electrolytes and their ion type specificity. Unfortunately, existing experimental data enable analysis of only αα and βα electrolytes, but these data cover most of the relevant cases anyway. In order to explain why αα electrolytes slow down coalescence following dynamic approach of bubbles, while βα electrolytes do not, we examined the differences between the behavior of two-bubble systems,7,15−27 multibubble systems, 1−7,11,12,28−36 and the thin liquid film system37−39 (see Table 1). The two latter systems always give the same results under dynamic conditions, which is an indication that the mechanism may be similar for both of these apparently very different methods (the multibubble methods were not used in the mode of static approach). The only major difference within the results attained by the three types of systems following dynamic approach is that bubble coalescence, in some electrolyte solutions, may be slow in the case of multibubble or thin-film systems (minutes), while it is always rapid when the two-bubble method is applied (seconds). These observations lead to the hypothesis that coalescence inhibition and its ion specificity are associated with the behavior of thin films, as explained below. Intuitively, it is clear that, in terms of thin-film behavior, multibubble systems can be treated as being in between the extreme cases of double-bubble systems and those of a macroscopic thin film. Figure 3 demonstrates this observation
E=−
A = A11 + A 22 − 2 A11A 22 = A11
2E m
= 211 nm (3)
where κ is the dielectric constant of water (80), ε0 is the free space permittivity (8.85 × 10−12 F/(m2)), kB is the Boltzmann constant, T is the temperature (293 K), e is the elementary charge, NA is the Avogadro number, and I is the ionic strength of purified water at the natural pH of 5.8 considering the speciation of dissolved CO2 (I = 2 × 10−3 (mol)(m3)).40 Hence, the maximum in energy, according to eq 1, comes out to be about 10−14 J. Now, the main energy available to overcome the EDL repulsion is the kinetic energy of the rising bubble. However, bubble coalescence is actually a process of rupture of a thin liquid film. Hence, the kinetic energy must be transferred from the bubble to the liquid. Roughly speaking, the bubble motion accelerates a similar volume of the liquid, so the mass to be
Figure 3. Qualitative presentation of the thin-film area in various systems (the thickness of the film is not to scale): (a,b) thinning process of the central part of a thin film, assuming constant volume during the process; (c) limited-size, thin film in a double-bubble system; (d) thin films in between bubbles in a dense, multibubble system. 13826
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Table 2. Calculated Values of L− and L+, Using eq 10 and Experimental Data for (dσ)/(dC)48a A: LH+ = −0.10 nm ion type +
H ↓
β
−
α α
Cl NO3− OH−
31
L− (nm) Assuming LH+ = −0.01 nm
LNa+ (nm) α
LK+ (nm) α
LLi+ (nm) α
−0.837 −0.750 −0.845 −0.810 ↓ −0.024
−0.804 −0.635 −0.730 −0.723 ↓ 0.003
−0.791 −0.701 −0.688 −0.727 ↓ 0.048
→ → →
0.112 0.195 0.010 Averages
Average LOH− (nm)
0.009 ←
LOH− (nm) from averages of the cations B: LH+ = −1.0 nm
Ion type31
L− (nm)
H+ ↓
β
assuming LH+ = −1.000
−
α α α
Cl NO3− OH−
LNa+ (nm) α
LK+ (nm) α
LLi+ (nm) α
−1.827 −1.740 −1.835 −1.800 ↓ 0.966
−1.794 −1.625 −1.720 −1.713 ↓ 0.993
−1.718 −1.691 −1.678 −1.717 ↓ 1.038
→ → →
1.103 1.185 1.000 Averages
Average LOH− (nm)
0.999 ←
LOH− (nm) from averages of the cations
The empirical (α, β) ion classification of Craig et al.31 is also listed. Left-hand side: L− values of anions, assuming a negative value for LH+ and using (dσ)/(dC) data for the various acids.48 Right-hand side: values of L+ for the cations, using the calculated L− of the corresponding anions and (dσ)/ (dC) data for the electrolytes.48 Also shown are the averages of L+, and values of LOH−, calculated from them, using (dσ)/(dC) data for the various bases.48 Section A is based on assuming LH+ = −0.10 nm. Section B is based on assuming LH+ = −1.0 nm. a
in a schematic way: thin-film methods use a film of relatively wide (macroscopic) area; in contrast, the thin-film area, in between two bubbles, is very limited; the collective thin-film area in dense multibubble systems is in between those in the above two systems. The strong similarity in the effect of electrolytes on the rate of coalescence in multibubble and thinfilm systems indicates that the former are closer to thin films than to double-bubble systems. Since a thin-film system is easier to theoretically analyze than a multibubble one, it can serve as a limiting case, from which the mechanism of inhibition can be learned. Thus, we present an analysis of the specificity of inhibition in a thin film of relatively large surface area, which, hopefully, reflects also on the behavior of a multibubble system. The above qualitative picture can be transformed into a simple, approximately quantitative model, which is based on mass balance of ions on one hand, and adsorption characteristics on the other. When the liquid film thins, toward coalescence of its interfaces, the liquid is pushed sidewise, while increasing it surface area. The ions in the thin film must rearrange between the solution and the interfaces, in accordance with the Gibbs adsorption equation. Since the film is thin and large, lateral diffusion between its thin and thick parts is very limited, except for at the edges of the thin part. Therefore, it is reasonable to treat the central part of the film as a closed system that just changes its surface area due to thinning, keeping its volume constant (see a qualitative demonstration in Figure 3a,b). Thus, when the thickness decreases, the surface area proportionally increases. Consequently, mass conservation for each ion in the central section of the thin film (of constant volume) is expressed by CioV = CiV + Γia
for cations and Ci− for anions, Γi is the surface excess of the ion, and a = V/h is the surface area of the film, which increases during thinning. By defining
Li ≡
Γi Ci
(6)
and introducing into eq 5, one gets Γi =
h Cio i + h/Li
(7)
It should be noted that Li is not necessarily a constant, although according to the data that will be used, it is approximately constant for ions. Equation 7 shows that as the film thins (h decreases) the surface excess decreases, because the surface area increases. If the anion and cation have different values of Li, then their adsorption to the interface of the film will be different, leading to the formation of an EDL. The difference in surface excess (for a 1-1 electrolyte, for example), according to the present model, is Γ− − Γ+ =
h/L+ − h/L− hCio (1 + h/L−)(1 + h/L+)
(8)
In order to calculate the difference in surface excess, we need independent measurements and theory for estimating Li. This is achieved by employing the Gibbs adsorption isotherms that enables calculating Γi (therefore Li) for the ions from data on the dependence of surface tension on the concentrations of these ions − dσ =
∑ Γidμi i
(5)
(9)
where σ is the surface tension of the solution, and μi is the chemical potential of each ion. The usual, important assumptions underlying this equation are (a) the surface excess of a solute is defined relatively to that of the solvent, and (b) the electrolyte is completely dissociated. For a single electro-
where the subscript i is the ion identity, Cio is the nominal molar concentration of the ion, V is the volume of the film that is kept constant during thinning, Ci is the instantaneous concentration of the ion in the interior of the film, denoted Ci+ 13827
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Figure 4a,b shows calculated (Γ− − Γ+) values for the αα and βα electrolytes, for the two different values of LH+. These figures
lyte, eq 9 is usually transformed into the following approximate expression55 (assuming the solution to be ideal, so dμi = RT d lnCi) −
d ln C+ 1 dσ d ln C − = Γ+ + Γ− RT dC dC+ dC − Γ Γ = + + − C+ C− = L+ + L−
(10)
The surface excess of water is eliminated from this equation, because, as mentioned above, surface excess is defined relative to the solvent. Also, because of the complete dissociation of the ions, and the salt being a 1-1 electrolyte, dC+ = dC−. Thus, we get a simple, additive dependence of the surface tension variation on L+ and L−. However, we still need to separate the contributions of the cation and the anion. To this end we employ the approach developed by Pegram et al.,44,45 but assign the reference value of Li in an entirely different way. As stated above, an assumption underlying eq 10 is that the surface excess is defined relative to that of the solvent (H2O). Therefore, it is assumed here that the contribution of water to eq 10 is zero, i.e., LH+ + LOH− = 0. Based on experimental indications that the air−water interface is negatively charged,56−59 we assumed that the hydrogen ion is partially excluded from the surface region, therefore LH+ must be
Rate of Bubble Coalescence Following Dynamic Approach: Collectivity-Induced Specificity of Ionic Effect Yael Katsir and Abraham Marmur* Department of Chemical Engineering, Technion − Israel Institute of Technology, Haifa 32000, Israel ABSTRACT: A simple, quantitative model is suggested to explain the specificity of ions with respect to inhibition of bubble coalescence following a dynamic approach. For the first time, the mode of thinning of the film in between the bubbles, as determined by the density of the bubble dispersion, is recognized as a determining factor. The specificity of the ionic effect is explained by a major difference in adsorption properties of ions, which is enhanced by the film thinning. This leads to charge separation that forms an electrical double layer at each interface of the thin, liquid film, and consequently to electrostatic repulsion. This effect is described by a simple theoretical model that consists of two fundamental equations: mass conservation of each ion in the film, and the Gibbs adsorption equation. In addition, we explain the rapid coalescence of bubbles in purified water under dynamic conditions, which is in contrast with the very slow coalescence under quasi-static conditions.
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INTRODUCTION Bubble coalescence in aqueous solutions is a ubiquitous phenomenon that is essential for life-supporting processes as well as for various industrial procedures. A prominent example of a natural process is the eventual coalescence with the sea−air interface of bubbles that are formed during breaking of seawater waves.1−6 This process leads to transfer of gases and ionic aerosols to the atmosphere,5−8 which is essential for marine life9 on one hand, and for the formation of clouds10 on the other. Industrial examples include mass transfer processes in bubble columns, distillation towers, gas−liquid contactors, bioreactors, electrochemical cells, and so forth.11,12 Bubble coalescence has attracted much scientific and engineering attention, because the specific, inhibiting effect of some ion combinations on coalescence is not yet well understood. In fact, the inhibition of bubble coalescence by specific ions has been a puzzling phenomenon since it was first observed in 1929.13 Moreover, the experimental observations appear to contradict the well-established DLVO (Derjaguin, Landau, Verwey, and Overbeek) theory,14 which describes the interaction between colloidal particles in general, based on the combined effects of van der Waals (vdW) attraction and electric double layer (EDL) repulsion. The effect of various electrolytes on bubble coalescence has been investigated using a variety of methods. Since bubble coalescence is actually a rupture process of a thin, liquid film (to be referred to as “thin film” for brevity), it has been studied by methods that apply to either whole bubbles or isolated thin films: pairs of bubbles,7,15−21 bubbles rising to the interface (which serves as very large, second bubble),22−24 two bubbles in atomic force microscopy (AFM),25−27 breaking waves,1−6,28 bubble columns,7,11,12,29−36 and a thin, liquid film between two gas phases.37−39 These studies have revealed the effect of © 2014 American Chemical Society
electrolyte type and concentration on the rate of bubble coalescence. The transition between very rapid or rapid coalescence (hundreds of milliseconds or seconds, respectively) and relatively slow coalescence (above a minute) occurs only with certain electrolytes, at specific concentrations (termed “transition concentrations”31). Craig et al.,31,32 using the bubble column method, empirically classified the ions of the electrolytes as type α or β. Roughly speaking, most cations and anions that form strong bases or acids, respectively, were classified as α ions. H+ and some anions that form weak acids were considered β ions. They found that αα and ββ electrolytes in aqueous solutions slow bubble coalescence, while αβ and βα electrolytes have no meaningful effect compared with purified water.31,32 This classification was completely supported by the results obtained using the thin-film method,37 and partially supported by other studies using multibubble methods (although they examined only αα electrolytes).1−7,11,12,28−30,33−36 However, observations using twobubble methods were ambiguous and, unfortunately, were mostly studied only with αα electrolytes.7,15−27 Recently, the important influence of the approach velocity of the bubble toward the second interface was elucidated, using a thin film38 and a sliding bubble.21 The coalescence behavior of the bubbles at low approach velocities of about 1−10 μm/s (quasi-static approach) was observed to be different from that above this velocity range. For the dynamic approach, NaCl solutions inhibited bubble coalescence, as had been reported in previous studies.38 However, for the quasi-static approach, the observations were completely different: slow coalescence in Received: August 22, 2014 Revised: October 26, 2014 Published: October 27, 2014 13823
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purified water (minutes to hours) and very rapid coalescence for NaCl solutions (less than a second).38 Unfortunately, these observations38 were studied only with αα electrolytes. It was later shown that the coalescence of a bubble with the air− solution interface, following the quasi-static approach, does obey the DLVO theory.40 Slow coalescence in purified water was shown to be due to the EDL repulsion originating from the speciation of dissolved CO2. Rapid coalescence in electrolyte solutions was demonstrated to result from neutralization of the carbonaceous, charged species by acids, or screening of the EDL by electrolytes.40 However, the inhibiting effect of electrolytes following the dynamic approach of bubbles is still unclear. So far, only partial explanations were given, based on surface elasticity,29,32,41 surface diffusion,42 ion partitioning at the tested liquid−air interface,43−46 hydration and structural forces,19,47,48 effective/ partial immobilization of interfaces, and deformation effects.49−52 Moreover, no explanation has been offered for the differences between the results of the two-bubble methods and those attained by other methods. In the present study, we experimentally examine the effect on the rate of bubble coalescence of the electrolyte type and concentration, and of the rate of approach, using a two-bubble apparatus.40 We analyze the results by focusing on the differences between twobubble systems and published results for multibubble or thinfilm systems (a thin-film system is a liquid film stretched between two gaseous phases, with possibility of drainage speed control; multibubbles systems are usually columns in which a high-density bubble stream flows through a liquid, under dynamic conditions). The previously obtained understanding of bubble coalescence following quasi-static approach, the difference between the two-bubble methods and the other methods, and the difference between coalescence in purified water and in electrolyte solutions all led us to simple theoretical explanations of the hitherto puzzling observations regarding inhibition of bubble coalescence.
■
Figure 1. Coalescence times versus bubble approach velocity in purified water. Our data (full circles; arrows indicate coalescence times longer than 103 s); data of Castillo et al. (empty squares) using a sliding-bubble apparatus;21 and semiquantitative data of Yaminski et al.38 using a thin-film apparatus (the shaded regions represent upper and lower limits of bubble approach velocity and coalescence times, as reported by Yaminsky et al.).38 differences in the coalescence time when the approach velocity is varied; also, it is quite sufficient for comparing the present coalescence times with literature data for dynamic approach velocities.
■
RESULTS AND DISCUSSION Experimental data showing the effect of bubble approach velocity on coalescence times are presented first for a bubble rising in purified water (Figure 1), then for a bubble rising in various aqueous electrolyte solutions (Figure 2). As explained below, the mechanisms underlying coalescence in purified water and in aqueous electrolyte solutions are entirely different; therefore, they will be discussed separately. Figure 1 shows coalescence times in purified water, for approach velocities that cover the whole range of interest, from ∼1 μm/s to ∼50 mm/s. The coalescence time, which is the measurable parameter representing the rate of coalescence, is defined as the time difference between the moment at which the top of the bubble is seen to contact the liquid−air interface and the last frame before coalescence.40 As previously discussed, the accuracy of this method is more than adequate for the purpose of showing major differences in the coalescence time. The data in Figure 1 come from a few different sources: (a) our present data (indicated by full circles, and marked by upward arrows at the level of 103 s when the coalescence times are longer that 103 s40); (b) data of Castillo et al. (indicated by squares), who used another kind of a two-bubble method, a sliding-bubble apparatus;21 and (c) semiquantitative data of Yaminski et al., who used a thin-film apparatus38 (shown in this figure as shaded regions that indicate upper and lower limits of coalescence times and the corresponding approach velocities). It is clear that all the data in Figure 1 are quite scattered, which is a natural outcome of the statistical nature of the phenomenon. However, over orders of magnitude of the coalescence time, all presented experimental methods suggest the same trend: bubble coalescence in purified water is very slow following a quasi-static approach (∼1 to ∼10 μm/s), but rapid (1−10 seconds) or very rapid (less than a second) following dynamic approach. Figure 2 shows coalescence times vs bubble approach velocity, for various aqueous electrolyte solutions, including αα and βα electrolytes. Using αα electrolytes solutions, our
EXPERIMENTAL SECTION
The experimental method chosen was the same one that had been used in our quasi-static study:40 a bubble is grown from a capillary toward the tested liquid−air interface (representing a second, very large bubble), by pushing air using a syringe. The production of the bubble was controlled manually by turning a micrometer, and the approach velocity of the bubble was controlled by setting the distance between the capillary tip and the liquid−air interface. The processes of bubble rise and coalescence were recorded by a video camera (17.4 frames/s) attached to a microscope. The approach velocity of the top of the bubble was directly calculated from the video frames preceding the touching of the interface, where the coalescence process/dynamics took place. In our experimental setup we managed to achieve three ranges of approach velocities: 1−10 μm/s (defined as quasistatic38,40), 50−500 μm/s, and 5−50 mm/s (see Figure 1). The two latter ranges are defined, by default, as dynamic. In addition, it should be mentioned that all experiments were performed at a temperature of 296 K and relative humidity of ∼55%. This simple system was used because of the following technical advantages: (a) the approach velocity can be easily controlled by forming the bubble at various distances from the airliquid interface; (b) it allows simple optical visualization of the coalescence process, hence enables measurement of the bubble approach velocity and the coalescence times at the required accuracy level. The uncertainty in the coalescence times, based on the standard deviation as well as on an analysis of possible errors in determining the beginning of the actual coalescence point, is about 1 s and 0.2 s regarding the quasi-static and the dynamic approaches, respectively. This is more than adequate for the purpose of showing the major 13824
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et al.,37 using a thin-film apparatus with αα electrolytes, show coalescence times of 10−100 s at about 0.1 M and above. Thus, the transition from either very rapid or rapid to slow coalescence that is induced by the αα electrolytes is observed with thin-film methods, but not with two-bubble methods. Our results, using αβ or βα electrolytes, show very rapid coalescence following a quasi-static approach, and either very rapid or rapid coalescence following dynamic approach (see Figure 2b). Unfortunately, most of the previous studies using similar two-bubble methods did not examine coalescence times in αβ or βα electrolytes (exceptions include studies by Christenson,16 who measured coalescence percentage and not time, and by Sagert,17 who did measure coalescence times, but only up to 1 s). However, using a thin-film apparatus, Karakashev et al. showed coalescence times in αβ electrolyte solutions that were similar to our results (only a few seconds).37 Thus, the slow coalescence in electrolyte solutions, using thin-film apparatus, is ion-specific, since it occurs with αα electrolytes, but not with αβ or βα electrolytes. Moreover, this inhibition and its ion specificity have been observed also with multibubble methods, in addition to thin-film methods, but not with two-bubble methods. Previous studies showed that two main parameters affect bubble coalescence: the composition of the solution, and the approach velocity of the bubbles. The present study brings into focus a third, very important parameter: the density of the system of bubbles, or, as will be explained below, the thickness of the film in between the bubbles during coalescence. The rate of bubble coalescence, as it depends on all these parameters, is qualitatively presented in Table 1. The items in this table that are emphasized by italic font are different from their counterparts in terms of either bubble system density or approach velocity. As can be seen in this table, the coalescence behavior of bubbles in purified water and in aqueous electrolyte solutions is qualitatively different. This indicates, in the discussion that follows, that the mechanisms underlying the inhibition of coalescence are different in the two cases. First, the case of water is discussed, to be followed by a discussion of the case of electrolyte solutions. In purified water, coalescence is very slow following quasistatic approach of the bubble (minutes to hours),21,38,40 but is rapid following dynamic approach (less than about 10 s).1−7,11,12,15−39 Yaminsky et al. explained the transition to rapid coalescence with increasing approach velocity by the hydrodynamic pressures in the film.38 However, another approach to the explanation is considered in this study, which is based on a simple energy balance. In our previous study40 we demonstrated that the slow coalescence following quasi-static approach is due to the potential barrier originating from the speciation of dissolved CO2.40 The hypothesis put
Figure 2. Coalescence times vs bubble approach velocity in various aqueous electrolyte solutions. (a) αα electrolytes: circles, 0.01 M NaCl; squares, 0.5 M NaCl; rhombuses, 1 M NaCl; triangles, 0.5 M KCl. Data by Castillo et al.: empty squares, 0.1 M KCl; empty triangles, 0.5 M KCl.21 Data of Yaminsky et al. (shaded regions since the results are semiquantitative).38 (b) αβ aqueous electrolyte solutions: circles, 0.5 M NaAc; squares, 0.5 M KAc. βα aqueous electrolyte solutions: triangles, 0.01 M HCl; rhombuses, 0.01 M HNO3.
results (full symbols in Figure 2a) show very rapid coalescence following a quasi-static approach.40 However, following a dynamic approach, coalescence times are somewhat scattered between very rapid and rapid coalescence (0.1−10 s). These results are in agreement with coalescence times of Castillo et al., who used another two-bubble method (empty symbols in Figure 2a).21 In contradiction, relatively slow coalescence was observed by Yaminsky et al. in solutions with sufficient concentration of NaCl, using a thin-film apparatus (these results are only semiquantitative, therefore are presented as shaded regions in Figure 2a).38 In addition, data by Karakshev
Table 1. Qualitative Summary of the Dependence of the Rate of Bubble Coalescencea quasi-static approach purified water two bubble systems multibubble systems thin-film system
21,40
Slow N/A Slow38
αα electrolytes 21,40
Very Rapid N/A Very Rapid38
dynamic approach αβ electrolytes Very Rapid N/A N/A
40
purified water
αα electrolytes
αβ electrolytes
7,15−27
7,16−27
Rapid16,17 Rapid31,32,37 Rapid31,32,37
Rapid Rapid1−7,11,12,28−39 Rapid1−7,11,12,28−39
Rapid Slow1−7,11,12,28−39 Slow1−7,11,12,28−39
(1) Composition of the solution (types of ions); (2) approach velocity of the bubble: quasi-static approach (up to about 10 μm/s), or dynamic approach (about 50 μm/s and above); (3) density of the bubble system or area of the film (depending on the experimental system). Definitions in terms of order of magnitude: slowsignificantly more than 10 s; rapid10 s and less. It should be noted that the bubble approach velocity in multibubble systems is, by the nature and size of the system, always dynamic. a
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taken in calculating the kinetic energy is that of water having the same volume as the assumed bubble. The velocity that enables the kinetic energy of this mass to acquire the same kinetic energy as the maximum EDL repulsion energy is then given by
forward in this paper is that coalescence in purified water, following dynamic approach, is rapid, because the kinetic energy associated with the bubble is sufficient to overcome the electric double layer (EDL) repulsion. In order to estimate the energy required to overcome the potential barrier between the bubble and the water−air interface (due to EDL interactions), we calculated their interaction energy through the water film. In our previous publication, regarding coalescence following the quasi-static approach, we calculated the interaction energy assuming planar double layers.40 This was justified by the observation that when the coalescence is very slow, the two interfaces have time to flatten. However, following a dynamic approach, when coalescence is rapid, the two interfaces may not have time to flatten. Hence, the interaction energy for this case is roughly assessed by considering a bubble of similar size to the one formed in the experiments (say, Rb = 1 mm) and a very large bubble (say, R∞ = 1 m) representing the air−liquid interface. This interaction energy is given by53 ⎡ 1 + e−χh AR b 2R ∞R b + επψ 2⎢ln 6h R∞ + Rb ⎣ 1 − e−χh ⎤ + ln(1 − e−2χh)⎥ ⎦
v≥
(1)
where (2)
The first expression on the right-hand side of eq 1 is the vdW attraction energy, where h is the minimal thickness of the fluid film between the bubble and the liquid−air interface, and A is the Hamaker constant. In order to calculate the appropriate A for our system, namely, for two bodies of the same material immersed in a different medium, we use eq 2,54 where A11 and A22 are the Hamaker constants of water and air, respectively. This equation accounts for all possible pairwise interactions, and since the Hamaker constant of air is practically zero, the Hamaker constant of two bubbles in water turns out to be equal to that of water in vacuum (A = A11 = 3.7 × 10−20 J). The second expression on the right-hand side of eq 1 is the EDL repulsion energy, where ε is the permittivity of water, and ψ is the surface potential of the bubble interface in purified water25 at the natural pH of 5.8 (ψ = −40 mV). The Debye length, χ−1, is given by40 χ −1 ≡
κε0kBT 2e 2NAI
(4)
The water mass turns out to be about 4 × 10−6 kg, so the velocity must be greater than 70 μm/s. This fits quite well the order of magnitude of our experimental observations (see Figure 1; rapid coalescence around 50 μm/s and above). Thus, rapid coalescence in purified water following a dynamic approach can be explained using a black-box approach, namely, by a global energy balance between the kinetic energy and the EDL repulsion energy. Such an order of magnitude estimation may be done without considering the details of the hydrodynamic phenomena. The main problem remains, however, the explanation of the inhibiting effect of electrolytes and their ion type specificity. Unfortunately, existing experimental data enable analysis of only αα and βα electrolytes, but these data cover most of the relevant cases anyway. In order to explain why αα electrolytes slow down coalescence following dynamic approach of bubbles, while βα electrolytes do not, we examined the differences between the behavior of two-bubble systems,7,15−27 multibubble systems, 1−7,11,12,28−36 and the thin liquid film system37−39 (see Table 1). The two latter systems always give the same results under dynamic conditions, which is an indication that the mechanism may be similar for both of these apparently very different methods (the multibubble methods were not used in the mode of static approach). The only major difference within the results attained by the three types of systems following dynamic approach is that bubble coalescence, in some electrolyte solutions, may be slow in the case of multibubble or thin-film systems (minutes), while it is always rapid when the two-bubble method is applied (seconds). These observations lead to the hypothesis that coalescence inhibition and its ion specificity are associated with the behavior of thin films, as explained below. Intuitively, it is clear that, in terms of thin-film behavior, multibubble systems can be treated as being in between the extreme cases of double-bubble systems and those of a macroscopic thin film. Figure 3 demonstrates this observation
E=−
A = A11 + A 22 − 2 A11A 22 = A11
2E m
= 211 nm (3)
where κ is the dielectric constant of water (80), ε0 is the free space permittivity (8.85 × 10−12 F/(m2)), kB is the Boltzmann constant, T is the temperature (293 K), e is the elementary charge, NA is the Avogadro number, and I is the ionic strength of purified water at the natural pH of 5.8 considering the speciation of dissolved CO2 (I = 2 × 10−3 (mol)(m3)).40 Hence, the maximum in energy, according to eq 1, comes out to be about 10−14 J. Now, the main energy available to overcome the EDL repulsion is the kinetic energy of the rising bubble. However, bubble coalescence is actually a process of rupture of a thin liquid film. Hence, the kinetic energy must be transferred from the bubble to the liquid. Roughly speaking, the bubble motion accelerates a similar volume of the liquid, so the mass to be
Figure 3. Qualitative presentation of the thin-film area in various systems (the thickness of the film is not to scale): (a,b) thinning process of the central part of a thin film, assuming constant volume during the process; (c) limited-size, thin film in a double-bubble system; (d) thin films in between bubbles in a dense, multibubble system. 13826
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Table 2. Calculated Values of L− and L+, Using eq 10 and Experimental Data for (dσ)/(dC)48a A: LH+ = −0.10 nm ion type +
H ↓
β
−
α α
Cl NO3− OH−
31
L− (nm) Assuming LH+ = −0.01 nm
LNa+ (nm) α
LK+ (nm) α
LLi+ (nm) α
−0.837 −0.750 −0.845 −0.810 ↓ −0.024
−0.804 −0.635 −0.730 −0.723 ↓ 0.003
−0.791 −0.701 −0.688 −0.727 ↓ 0.048
→ → →
0.112 0.195 0.010 Averages
Average LOH− (nm)
0.009 ←
LOH− (nm) from averages of the cations B: LH+ = −1.0 nm
Ion type31
L− (nm)
H+ ↓
β
assuming LH+ = −1.000
−
α α α
Cl NO3− OH−
LNa+ (nm) α
LK+ (nm) α
LLi+ (nm) α
−1.827 −1.740 −1.835 −1.800 ↓ 0.966
−1.794 −1.625 −1.720 −1.713 ↓ 0.993
−1.718 −1.691 −1.678 −1.717 ↓ 1.038
→ → →
1.103 1.185 1.000 Averages
Average LOH− (nm)
0.999 ←
LOH− (nm) from averages of the cations
The empirical (α, β) ion classification of Craig et al.31 is also listed. Left-hand side: L− values of anions, assuming a negative value for LH+ and using (dσ)/(dC) data for the various acids.48 Right-hand side: values of L+ for the cations, using the calculated L− of the corresponding anions and (dσ)/ (dC) data for the electrolytes.48 Also shown are the averages of L+, and values of LOH−, calculated from them, using (dσ)/(dC) data for the various bases.48 Section A is based on assuming LH+ = −0.10 nm. Section B is based on assuming LH+ = −1.0 nm. a
in a schematic way: thin-film methods use a film of relatively wide (macroscopic) area; in contrast, the thin-film area, in between two bubbles, is very limited; the collective thin-film area in dense multibubble systems is in between those in the above two systems. The strong similarity in the effect of electrolytes on the rate of coalescence in multibubble and thinfilm systems indicates that the former are closer to thin films than to double-bubble systems. Since a thin-film system is easier to theoretically analyze than a multibubble one, it can serve as a limiting case, from which the mechanism of inhibition can be learned. Thus, we present an analysis of the specificity of inhibition in a thin film of relatively large surface area, which, hopefully, reflects also on the behavior of a multibubble system. The above qualitative picture can be transformed into a simple, approximately quantitative model, which is based on mass balance of ions on one hand, and adsorption characteristics on the other. When the liquid film thins, toward coalescence of its interfaces, the liquid is pushed sidewise, while increasing it surface area. The ions in the thin film must rearrange between the solution and the interfaces, in accordance with the Gibbs adsorption equation. Since the film is thin and large, lateral diffusion between its thin and thick parts is very limited, except for at the edges of the thin part. Therefore, it is reasonable to treat the central part of the film as a closed system that just changes its surface area due to thinning, keeping its volume constant (see a qualitative demonstration in Figure 3a,b). Thus, when the thickness decreases, the surface area proportionally increases. Consequently, mass conservation for each ion in the central section of the thin film (of constant volume) is expressed by CioV = CiV + Γia
for cations and Ci− for anions, Γi is the surface excess of the ion, and a = V/h is the surface area of the film, which increases during thinning. By defining
Li ≡
Γi Ci
(6)
and introducing into eq 5, one gets Γi =
h Cio i + h/Li
(7)
It should be noted that Li is not necessarily a constant, although according to the data that will be used, it is approximately constant for ions. Equation 7 shows that as the film thins (h decreases) the surface excess decreases, because the surface area increases. If the anion and cation have different values of Li, then their adsorption to the interface of the film will be different, leading to the formation of an EDL. The difference in surface excess (for a 1-1 electrolyte, for example), according to the present model, is Γ− − Γ+ =
h/L+ − h/L− hCio (1 + h/L−)(1 + h/L+)
(8)
In order to calculate the difference in surface excess, we need independent measurements and theory for estimating Li. This is achieved by employing the Gibbs adsorption isotherms that enables calculating Γi (therefore Li) for the ions from data on the dependence of surface tension on the concentrations of these ions − dσ =
∑ Γidμi i
(5)
(9)
where σ is the surface tension of the solution, and μi is the chemical potential of each ion. The usual, important assumptions underlying this equation are (a) the surface excess of a solute is defined relatively to that of the solvent, and (b) the electrolyte is completely dissociated. For a single electro-
where the subscript i is the ion identity, Cio is the nominal molar concentration of the ion, V is the volume of the film that is kept constant during thinning, Ci is the instantaneous concentration of the ion in the interior of the film, denoted Ci+ 13827
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Figure 4a,b shows calculated (Γ− − Γ+) values for the αα and βα electrolytes, for the two different values of LH+. These figures
lyte, eq 9 is usually transformed into the following approximate expression55 (assuming the solution to be ideal, so dμi = RT d lnCi) −
d ln C+ 1 dσ d ln C − = Γ+ + Γ− RT dC dC+ dC − Γ Γ = + + − C+ C− = L+ + L−
(10)
The surface excess of water is eliminated from this equation, because, as mentioned above, surface excess is defined relative to the solvent. Also, because of the complete dissociation of the ions, and the salt being a 1-1 electrolyte, dC+ = dC−. Thus, we get a simple, additive dependence of the surface tension variation on L+ and L−. However, we still need to separate the contributions of the cation and the anion. To this end we employ the approach developed by Pegram et al.,44,45 but assign the reference value of Li in an entirely different way. As stated above, an assumption underlying eq 10 is that the surface excess is defined relative to that of the solvent (H2O). Therefore, it is assumed here that the contribution of water to eq 10 is zero, i.e., LH+ + LOH− = 0. Based on experimental indications that the air−water interface is negatively charged,56−59 we assumed that the hydrogen ion is partially excluded from the surface region, therefore LH+ must be
