Should the Gibbs Analysis Be Revised? - ACS Publications

May 23, 2011 - tically in Figure 1, shows a plateau at the level γ0 of pure water. (Region-A) for .... In our view, although the GAI in principle can...
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Should the Gibbs Analysis Be Revised? ABSTRACT: Recently, some arguments were published that cast doubt on the validity of the Gibbs adsorption isotherm. The doubt was on whether the often visible linearly declining part in the surface tension versus logarithm of concentration plot of a surfactant solution, just before the critical micelle concentration, really represents a situation of constant adsorption. Those published arguments are partly of a conceptual nature and partly based on experimental evidence. The conceptual arguments appear to be based on a misunderstanding of the theory, while the arguments based on experimental evidence stem from an inaccurate treatment of these data. Our conclusion is that none of the relevant arguments put forward are valid. The experimental evidence, if properly treated, is in line with the Gibbs theory.

’ INTRODUCTION The Gibbs adsorption isotherm (GAI), a theory invented in the second half of the nineteenth century, has been widely used and accepted.1,2 However, this acceptance seems not general, as can be seen from two recent papers by Menger, Shi, and Rizli,3,4 published in two different reputed journals, in which those authors question the “Gibbs analysis of surface tension”. Additionally, such questioning has been judged worthwhile of mention in publications by three other research groups.57 Although we disagree with Menger et al., it is too simple to just voice disapproval about their statements on the basis of the general acceptance of the GAI. Menger et al. claim to falsify the GAI by proving that it is incompatible with specific experimental evidence. Although falsification is an important step in the advancement of science,8 the purpose of the present paper is to demonstrate that this questioning by Menger et al. is based on incorrect arguments. The arguments put forward by those authors, being conceptual in nature, as well as based on their experimental findings, are commented upon in that order. ’ CONCEPTUAL ASPECTS The starting point of Menger, Shi, and Rizli (“the authors”) is their Figure 1 in both ref 3 and ref 4 which depicts the typical surface tension γ versus logarithm of concentration c curve of an aqueous surfactant (S) solution (unless otherwise stated, we will follow their implicit assumption that individual S molecules behave ideally in the aqueous bulk phase, ignoring the difference between c and the activity a). Such a curve, reproduced schematically in Figure 1, shows a plateau at the level γ0 of pure water (Region-A) for low values of c. At a higher c, it bends downward toward a steep, almost linear decline (Region-B), and then, it levels off abruptly (Region-C) at a point that is usually considered the critical micelle concentration (CMC). The authors question3,4 “the assumption in the conventional Gibbs theory” that the surface adsorption Γ is saturated throughout region-B, r 2011 American Chemical Society

Figure 1. Typical plot of surface tension versus log [surfactant].

and they are surprised3 (reiterated in ref 4) that, according to that theory, Γ depends more sensitively on log c beyond the point of surface saturation than before that point. In their view, the adsorption in Region-B is increasing up to the abrupt transition to region-C, with that transition indicating the saturation point. They try to provide an alternative theoretical explanation using a Frumkin’s type adsorption (i.e., with “cooperative” adsorption) because that had been found9 to work better than Langmuir’s model for adsorption in a couple of cases. In the middle section of the sigmoidal shape of the Γ versus c curve of the Frumkin model, Γ rises very pronouncedly with c. Because this sigmoidal shape has “an obvious mirror correspondence” to the γ versus log c plot, it is their opinion that “the model implies that the air/water interface is not saturated in Region-B and that, therefore, the commonplace Gibbs calculations of molecular area (dependent upon the fortuitous linear section of the surface tension plots) are misdirected”. As a first comment, if these graphical shapes are judged correctly, i.e., comparing them on similar horizontal scales rather than, as the authors do, comparing one on linear with another on logarithmic Received: January 13, 2011 Published: May 23, 2011 7958

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scale, no correspondence is found whatsoever. Even if there were a graphical “correspondence” between these curves when using comparable scales, this would still have been too weak a scientific basis for extending Frumkin’s theory to a γ(c) relationship. More importantly, it is not possible to use Frumkin’s theory as an alternative to that of Gibbs. Actually, these theories are not interchangeable at all. While the latter leads to a thermodynamically based relation that reads Γ ¼  ðdγ=dμÞT

ð1Þ

where μ is the thermodynamic potential of S at temperature T, Frumkin’s theory leads to a constitutive relation that links Γ(c) to interaction energies. These theories are typically used in the following way. Experimentally determined γ(c) data are converted into a Γ(c) curve, according to eq 1. Such a Γ(c) curve can be compared with theoretical predictions of Γ(c) as based on, e.g., the isotherm of Langmuir or Frumkin to obtain insight into the nature of the interactions involved in adsorption. Isotherm theories such as those of Frumkin and Langmuir only predict Γ(c), and from them one cannot conclude anything on γ(c) without using Gibbs’ theory. However, the authors do draw such a conclusion when they speak of “a continuously increasing occupancy of the interface in Region B that corresponds smoothly to the decline in surface tension”. In fact, the confusion appears to stem from a misunderstanding of what happens in Region-B. Although the adsorbed amount in the linear Region-B hardly varies, it will not be perfectly constant. A simple picture is that the adsorbed molecules in Region-A represent a 2-D gas that is compressed on raising c to such an extent that, in Region-B, it can, for whatever reason, almost not be compressed any further (i.e., Γ hardly rises any more) on increasing the surface pressure γ0γ (for the sake of simplicity, ignoring any phase transition). This is exactly what is indicated by the expression obtained from the GAI, eq 1, in which μ = μ0 þ nRT ln c is substituted, where n represents the number of kinetically active particles within a S molecule, e.g., for an ionic surfactant, the number of dissociated ions originating from one S molecule. The constant slope in Region-B inevitably implies that Γ is constant. Thus, γ0γ rises considerably in Region-B because of the rise in c, not by the rise in Γ. In fact, the statement by Rosen,10 quoted by the authors,3 that the large change of γ in Region-B is due to “increased activity of the surfactant in the bulk phase rather than at the interface” is not “rather vague” as stated by the authors but is simply incorrect. Because the chemical potential of S is the same in the bulk and at the surface, the surface activity of S rises together with its bulk activity. We guess that Rosen10 wanted to express just our view on Region-B but by mistake used the word ”activity” instead of “concentration”. The authors start the paper4 by stating that “the Gibbs analysis ... postulates that the air/water interface is entirely saturated with absorbent throughout region B”. The paper3 speaks in this respect about “Conventional theory assumes...”. Indeed, saturation is the most obvious interpretation of most experimental data sets that have a Region-B. However, we note that this is neither part of the analysis by Gibbs himself11,12 nor a postulate.

’ EXPERIMENTAL ASPECTS The authors discuss four types of experimental data that, according to them, provide evidence for their already cited view. It involves (i) surface tension measurements of solutions of typical headtail ionic surfactants and, for a 2-component

surfactant system, measurements of (ii) surface tension, (iii) bulk electrical conductivity, and (iv) bulk diffusivity by NMR. We will discuss them in order. Single-Component Surfactant System. Results by the authors for the surface tension reduction γ0γ as a function of the area R per molecule at the surface were presented for typical headtail ionic surfactants that were slightly water-soluble3 or virtually insoluble.4 With respect to the latter type, they note that “The surface tension remained constant at the water value of 72 mN/m all the way from 60 Å2/molecule down to 28 Å2/molecule. However, a variety of single-chained surfactants have, as derived by the Gibbs approach, areas above 60 Å2/molecule (..). This means that when the Gibbs method is applied to the steeply declining surface tension plot, the area thus obtained corresponds, according to our data, to a zero surface tension change, an obvious inconsistency.” In our view, although the GAI in principle can be applied to any equilibrium surface enrichment, it is usually not practical to apply it to monolayers of virtually insoluble surfactants for a couple of reasons. (i) The GAI supposes equilibrium between the surfactant activities in the surface and the bulk. With the extremely small solubility of such surfactants, it is doubtful whether equilibrium will ever be established between bulk and surface within accessible time frames. (ii) Even if such equilibrium were established, bulk concentrations would probably be too low to be determined accurately, making an assessment of the slope dγ/d ln c cumbersome. (iii)In all “practical” experiments, if equilibration had been established, the bulk would have been saturated making the bulk concentration constant, thus complicating any evaluation of dγ/d ln c. (iv) The fundamental argument is as follows. Suppose a monolayer on top of a liquid where equilibrium between the monolayer and a large amount of bulk can be realized and that γ is always measured after such equilibration. In that case, compression will inevitably lead to diffusion of surfactant molecules from the surface to the bulk until the surface concentration (activity) will match again with that of the bulk, i.e., until the original surface tension will have been reestablished. Thus, there will be zero difference between γ before and γ after compression, definitely different from the monolayer behavior reported by the authors. Thus, in their experiment there was no such equilibrium and the GAI is not applicable. Monolayers of virtually insoluble surfactants like the authors investigated have peculiar behavior beyond the bulk saturation point, as already studied in 192613 and discussed by, e.g., Hiemenz14 in a crisp, clear way in terms of a (p, V) phase diagram of a twodimensional (2-D) system. Typically, in the monolayer gas-like and liquid-like domains coexist. The surfactant concentration in such a gas phase is very small with a correspondingly very low value of the surface pressure γ0γ, i.e., γ is smaller than γ0 by only a very small amount. On sufficient surface compression, in a Langmuir trough the gas phase will vanish, and on further compression, in the remaining 2-D liquid phase (with its much smaller compressibility) an appreciable drop in γ is observed. It is this drop, related to the compressibility of and any transitions between 2-D liquid phases, which the authors seek to interpret with the GAI in order to support their view. 7959

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Langmuir One of the arguments for the authors' study of “insoluble” monolayers to make their point against the GAI was to avoid that “molecules under compression will simply depart from the air/ water interface and enter the bulk water phase”.3,4 We just argued that insoluble monolayers cannot well be used to support/ disprove the GAI. On the other hand, monolayers of slightly soluble molecules (typical “surfactants”) have been widely studied for this purpose. The paper by Eastoe, Nave, Downer, Paul, Rankin, Tribe, and Penfold15 is exemplary for the care that is needed to obtain a well-defined surfactant system which allows a straightforward interpretation with the GAI. The authors of that paper stress the need for a comparison of GAI-based Γ data with data obtained by any of a range of other surface techniques mentioned by them, among which neutron reflection (NR) at the water/air interface appears the most popular. In the case of ionic surfactants, very small traces of multivalent metal ions, at concentrations as low as 106 M,16 have a large impact on the γ(c) curve as they affect the prefactor in the GAI.15 In the case of nonionic surfactants, the lack in monodispersity of the lengths of the building blocks easily leads to apparent anomalies in surface excesses as derived from γ(c) curves.22 The presence of very small amounts of hydrophobic building blocks has been known since Mysels in 197017 to lead to severe deviations in the γ(c) curve. The most extensive systematic work on this was carried out by Lunkenheimer18 who developed a special device for this purpose, which was later improved.19 Lunkenheimer stresses the importance of “surface”-purification instead of bulk purification like the authors performed. When all kinds of precautions are taken, generally speaking, good agreement is found between GAI based Γ(c) curves and those obtained with techniques like NR, both for ionic surfactants15,20,21 and for nonionic surfactants.22 Two-Component Surfactant System. Effect of Second Component on the cmc. For a two-component aqueous surfactant system, the authors discuss their previous data23 in order to support their conceptual view. That system is a mixture (referred to as DTABþ2) of a cationic surfactant and an anionic species, dodecyltrimethylammonium bromide (DTAB) and naphthalene to which three sulfonate groups were attached (together with Naþ counterions), at a fixed ratio of the organic ions of 15:1. The γ versus ln cDTAB plot (Figure 3 in ref 3) is essentially a clear combination of Region-B and Region-C, with a nice, sharp kink at 2.9 mM that is usually interpreted as the CMC. They also determined the CMC of that system using two alternative “bulk” methods (electrical conductivity and diffusion by NMR) which led them to conclude that the CMC is actually at 14 mM. They conclude that the kink in the γ versus ln cDTAB plot does not represent the CMC but they interpret it as an indication of surface saturation: “The point here is that saturation of the interface (i.e., where the surface tension levels off unimpaired by micelle formation) is now detectable, and it appears in Region-C rather than in Region-B as assumed in the Gibbs analysis”. Note that the authors compared these data with data on the DTAB/water system for which all three methods employed indicate a CMC at ∼14 mM. With respect to the data for the two-component surfactant system, in our opinion the authors use a non-allowed interpretation of results for this not-well understood DTABþ2 system in order to support their questioning of the GAI. Rather than using eq 1, the more general form of the GAI should have been used, the latter one being applicable to systems with more than one

COMMENT

surface active species  dγ ¼

∑Γi dμi

ð2Þ

where the summation is over all the surface active components. According to the authors,23 the second component (“2”) in the DTABþ2 mixture is not surface active up to a concentration of 15 mM when added to pure water. However, this trivalent anion may well interact with the monovalent DTAB cation and as a consequence acquire (much more) surface activity (also the authors speak of a “mixed monolayer” at the air/water interface12). Thus, linking the CMC of DTAB at 14 mM to the saturated adsorption of DTAB and “2” at 2.9 mM by using the GAI according to eq 1 is not allowed. We will now propose a simple model that is much more compatible with experimental evidence than the model of the authors is, while at the same time, our model is in accordance with the GAI. Our claim is not that details of our model cannot be disputed but that it is very well possible to interpret data in line with the GAI, which means that disputing the GAI has no grounds. Suppose that two DTAB ions form a complex with a single “2” ion. Such a monovalent complex ion (“C”) can be expected to have a high affinity to the surface, much more than DTAB itself has. This is supported by both the fact that the γ(c) curve of the DTABþ2 system, when compared to the DTAB system, has its kink at a concentration by a factor of 5 lower and the fact that it has a large linear Region-B (as based on the GAI, suggesting surface saturation in Region-B). Thus, adsorbed DTAB ions will be driven out of the surface by adsorbed C-ions leading to ΓDTAB ≈ 0. The GAI in this case becomes RTΓC = dγ/d ln cC. The kink in the γ(c) curve at cDTAB = 2.9 mM can be explained in the usual way by the occurrence of micelles (presumably composed of aggregated C ions): on adding the two-component surfactant, beyond 2.9 mM any additionally formed C ions aggregate into micelles, leading to a constant activity of the C ions and thereby also to a constant γ. With a fixed mixing ratio DTAB to “2” of 15/1, the concentration of molecularly dissolved DTAB ions beyond the point 2.9 mM in the graph will be lower than the nominal one by approximately a factor 13/15 (if all ions of component 2 are complexed). This would mean that only at ∼16 mM in the graph is the actual concentration of free DTAB ions high enough, 14 mM, to form DTAB micelles. Thus, beyond 16 mM in the graph two types of micelles can be expected, those of C and those of DTAB. Our model will of course require supporting evidence to prove the model. The assumed ratio 2:1 for the components in the complex is somewhat arbitrary, but a ratio 1/1 or 3/1 will qualitatively lead to similar conclusions. Irrespective of this uncertainty, when compared to the interpretation by the authors our model explains the lower concentration at the kink better and does not use too simple a version of the GAI. The conclusion is that the DTABþ2 surface tension results do not cast any doubts on the validity of the GAI. Note that our model is also much more in agreement with supporting evidence, as will be shown below. Electrical Conductivity in the Bulk. The authors presented electrical conductivity measurements in aqueous solutions of DTAB and DTABþ2 to support their view. Their data as a function of c of DTAB (originally plotted as conductance versus c in Figure 3 in ref 23) are replotted in Figure 2 as molar conductance κ versus log c. Below the CMC, one expects mainly nonaggregated surfactant ions, thus with a constant κ. On surpassing the CMC, the conductivity is determined by the constant 7960

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Figure 2. Molar conductance of single (DTAB) and two-component (DTABþ2) surfactant system, from ref 23, as function of the DTAB concentration.

COMMENT

be constant below the CMC and to decline gradually beyond the CMC. For DTAB, the plateau in D values ends at 13 mM, fully in accordance with the γ-based CMC. For DTABþ2, no plateau is visible down to concentrations of 1 mM. Two effects of our model may affect D around the CMC at 2.9 mM. The first effect is that micelle formation at 2.9 mM will lead to a transition at that point (at left a plateau, at right a decline), similar to that of DTAB at 14 mM. The decline right of the 2.9 mM position is clearly visible. Such a decline between 2.9 and 14 mM is not compatible with the conclusion by the authors about what happens in the bulk of the DTABþ2 system when compared to the DTAB system: “Thus, “2” binds to the 3-D micelles (...) but the CMC is hardly affected”.23 The other effect is that the complexation of ions below 2.9 mM will cause some reduction in D as well and may explain why no plateau can be noticed below 2.9 mM (note that for solid conclusions on this point more measuring points are needed between the two lowest concentrations tested, 1.0 and 4.0 mM). This implies that the NMR diffusion data do not rule out the presence of a CMC at 2.9 mM and also suggest a CMC at 14 mM. Thus, the NMR data do not provide any argument against the validity of the GAI. In conclusion, a very substantial amount of literature data, obtained with carefully prepared systems, support the validity of the Gibbs adsorption isotherm. The arguments put forward by the authors to question the Gibbs analysis appear to be based on a misunderstanding of the theory and a misinterpretation of experimental data. Their experimental data, if properly interpreted, support the Gibbs analysis rather than question it. Jozua Laven* and Gijsbertus de With

Figure 3. Diffusivity of single (DTAB) and mixed (DTABþ2) surfactants in solution, from ref 23 (Supporting Information), as a function of the DTAB concentration.

amount of molecularly dissolved surfactant molecules together with an increasing number of micelles, which normally leads to a gradual decline in the molar conductance when increasing c beyond the CMC. Such a transition, in accordance with the γ data, can indeed be noticed for DTAB at ∼16 mM (the fact that the slope of the κ = f (ln c) curve below the CMC is not zero but slightly negative, less than 10% for a variation in concentration by a factor 4.5, may be due to the general concentration dependence of the diffusivity D as addressed by Kohlrausch’s law24). For DTABþ2, the authors notice with right a transition at ∼16 mM. The accuracy of the data at low concentrations, as deduced from the original graph, is not very high. Nevertheless, the conductivity data also indicate a transition from a plateau to declining levels at ∼2.4 mM, which compares well with the value of 2.9 mM from the γ data. Thus, the data provide some evidence for the existence of two CMC’s, at 2.9 and at 16 mM, as suggested by our model, which is in accordance with the GAI. In order to deduce from the conductivity data, any arguments against the GAI would require that the authors demonstrate the absence of a CMC at 2.9 mM. The case of a single CMC, at 16 mM, would require a plateau left of that point, which definitely is absent. Thus, the conductivity data of the authors do not provide arguments against, but rather provide some evidence in favor of the GAI. Diffusivity in the Bulk. Their NMR data (the diffusivity D of the surfactant ions) were originally plotted as D versus c1 (Figure 1 in Supporting Information to ref 23). The authors claim an intersection of curves at 14 mM which they interpret as CMC (“the curves intersecting at a CMC of ∼14 mM”). However, such an intersection does not exist at all, as can clearly be seen in Figure 3 where the original data are replotted as D versus log c. With respect to the magnitude of D, like with conductivity data, D can be expected to

Laboratory for Materials and Interface Chemistry, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

’ AUTHOR INFORMATION Corresponding Author

*Jozua Laven, e-mail: [email protected].

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