Limitations in the Application of the Gibbs Equation to Anionic

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Limitations in the Application of the Gibbs Equation to Anionic Surfactants at the Air/Water Surface: Sodium Dodecylsulfate and Sodium Dodecylmonooxyethylenesulfate Above and Below the CMC Hui Xu, Peixun Li, Kun Ma, Robert K. Thomas, Jeffrey Penfold, and Jian Ren Lu Langmuir, Just Accepted Manuscript • DOI: 10.1021/la401835d • Publication Date (Web): 02 Jul 2013 Downloaded from http://pubs.acs.org on July 4, 2013

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Limitations in the Application of the Gibbs Equation to Anionic Surfactants at the Air/Water Surface: Sodium Dodecylsulfate and Sodium Dodecylmonooxyethylenesulfate Above and Below the CMC Hui Xu,† Pei Xun Li,† Kun Ma,† Robert K.Thomas,∗,† Jeffrey Penfold,‡,§ and Jian Ren Lu¶ Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford, OX1 3QZ, UK., Rutherford-Appleton Laboratory, Chilton, Didcot, Oxon., UK., and School of Physics and Astronomy, Schuster Building, Brunswick Street, Manchester, M13 9PL, UK.

E-mail: [email protected]

Abstract This is a second paper responding to recent papers by Menger et al. and the ensuing discussion about the application of the Gibbs equation to surface tension (ST) data. Using new neutron reection (NR) measurements on sodium dodecylsulfate (SDS) and sodium dodecylmonooxyethylene sulfate (SLES) above and below their CMCs and ∗ † ‡ ¶ §

To whom correspondence should be addressed University of Oxford ISIS University of Manchester Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford, OX1 3QZ, UK.

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with and without added NaCl, in conjunction with the previous surface tension (ST) measurements on SDS by Elworthy and Mysels (EM), we conclude that (i) ST measurements are often seriously compromised by traces of divalent ions, (ii) adsorption does not generally reach saturation at the CMC making it dicult to obtain the limiting Gibbs slope, and (iii) the signicant width of micellization may make it impossible to apply the Gibbs equation in a signicant range of concentration below the CMC. Menger et al. proposed (ii) as a reason for the diculty of applying the Gibbs equation to ST data. Conclusions (i) and (iii) now further emphasize the failings of the ST-Gibbs analysis for determining the limiting coverage at the CMC, especially for SDS. For SDS, adsorption increases above the CMC to a value at 10×CMC, which is about

25% greater than at the CMC and about the same as at the CMC in the presence of 0.1 M NaCl. In contrast, the adsorption of SLES reaches a limit at the CMC with no further increase up to 10×CMC, but addition of 0.1 M NaCl increases the surface excess by 2025%. The results for SDS are combined with earlier NR results to generate an adsorption isotherm from 2100 mM. The NR results for SDS are compared with the denitive surface tension (ST) measurements of EM and the surface excesses agree over the range where they can safely be compared, from 26 mM. This conrms that the anomalous decrease in the slope of EM's σ − lnc curve between 6 mM and the CMC at 8.2 mM results from changes in activity associated with a signicant width of micellization. This anomaly shows that it is impossible to apply the Gibbs equation usefully from 68.2 mM, i. e. the lack of knowledge of the activity in this range is the same as above the CMC (8.2 mM). It was found that a mislabelling of the original data in EM may have prevented the use of this excellent ST data as a standard by other authors. Although NR and ST results for SDS in the absence of added electrolyte show that the discrepancies can be rationalized, ST is generally shown to be less accurate and more vulnerable to impurities, especially divalent ions, than NR. The radiotracer technique is shown to be less accurate than ST-Gibbs in that the four radiotracer measurements of the surface excess are neither consistent with each other nor with ST and NR. It is

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also shown that radiotracer results on Aerosol-OT are likely to be incorrect. Application of the mass action (MA) model of micellization to the ST curves of SDS and SLES through and above the CMC shows that they can be explained by this model and that they depend on the degree of dissociation of the micelle, which leads to a larger change in the mean activity, and hence the adsorption, for the more highly dissociated SDS micelles than for SLES. Previous measurements of the activity of SDS above the CMC were found to be semi-quantitatively consistent with the change in mean activity predicted by the MA model, but inconsistent with the combined ST, NR and Gibbs equation results.

Introduction This is a second article responding to recent papers by Menger et al.

1 2 3 4, 5

which have

questioned the use of the Gibbs equation for analysing surface tension data to obtain the surface excess, and to two attempts to refute these papers issues in the introduction to our previous article also recently been added by Mukherjee et al.

9

8

6. 7

We have outlined the main

and some comments on these issues have

The conclusion of our previous article was

that the indirect surface tension (ST) - Gibbs method and the direct neutron reection (NR) method consistently agree in their determination of the surface excess of a wide selection of nonionic systems, including polymers and polydisperse systems, both around the critical micelle concentration (CMC) and at lower concentrations away from the CMC. Because the main diculty in obtaining an accurate value of the surface excess from the ST data seems to be that of determining the limiting slope of the

σ − lnc

curve close to the CMC, the

consistency of the two methods suggests that saturation of the surface is complete before the CMC. This makes the

σ − lnc plot linear just below the CMC. Since the saturation issue is a

point of disagreement between Menger et al. and Bermúdez-Salguero and Gracia-Fadrique,

7

the results in our previous article support the arguments of the latter authors. In this paper we examine results for anionic surfactants and show that the situation is reversed.

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We gave an outline of the strengths and weaknesses of the ST and NR methods in our previous paper and do not review them again here. There are however some important differences between ionic and nonionic systems. NR measures only the excess of the surfactant ion because the signal from the counterion is usually negligible. The Gibbs equation ensures that ST measures both.

Since the surface is overall neutral from a thermodynamic point

of view, this normally means that there is a factor in the analysis, which is two for a 1:1 ionic surfactant if it is the counterion that exclusively neutralizes the surfactant ion. Pethica

10

pointed out that, because of this, the Gibbs equation cannot in principle be applied

to an aqueous solution of ionic surfactant in the absence of excess electrolyte because of the ambiguity in the role of the water ions.

Apart from this fundamental problem, impurities

are generally a much more serious problem for ionic than for nonionic systems.

This is

basically because ionic systems are less surface active than their nonionic counterparts and nonionic impurities in the ionic surfactant then often dominate the surface behaviour below the CMC. However, NR and radiotracer methods also show that ionic impurities are at least as important and their eect on the ST is often large, although less obvious. The consistency of NR and ST from our rst paper validates both techniques as reliable and accurate for nonionic surfactants. However, for ionic surfactants NR almost invariably gives a higher value of the surface excess at the CMC, is often large.

ΓCM C ,

than ST and the discrepancy

The consistency of the direction of this discrepancy suggests that there is

a systematic error in one or other method when it comes to ionic systems. this discrepancy the NR value of

ΓCM C

When there is

is the one that is generally closer to expectations

based on simple packing arguments, e. g. adsorbed amounts at the solid/liquid interface and the packing in lamellar phase liquid crystals or hydrated solids, which suggests that the systematic error is in the ST measurements or their analysis.

The possible causes are (i)

impurities, (ii) aggregation and/or ion association below the CMC, and (iii) the behaviour of the activity close to the CMC, which is connected to the concentration width of the micellization process. In principle the eects of impurities are well known and both ST and

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NR are aected but the combination of the two techniques allows impurities to be explored more precisely. NR is not aected by either (ii) or (iii). Of these, preaggregation is known to occur in some systems, particularly double chain systems, identied by methods such as conductivity and NMR, e. g.

11

and can be qualitatively

12 , 13 but is more or less impossible

to handle quantitatively within the framework of the Gibbs equation because of a lack of the necessary detailed information about the species in the bulk solution. Again, the combination of NR and ST has allowed some exploration of this.

14

The width of the micellar transition

appears to have been systematically examined only by Al Sou et al.,

15

although there are

many incidental references to this width, particularly in connection with the mass action model of micellization, and several papers have observed gradual changes in concentrations through the CMC that demonstrate the nite width

16 17 . 18

Apart from the case of SDS

considered by Al Sou et al. the possible consequences of a nite micellization width on the ST-Gibbs analysis do not seem to have been considered at all. In part this is because of the large number of diculties surrounding the ST method, particularly the fact that it cannot be used eectively above the CMC where it requires activities rather than concentrations and these are dicult to measure. However, the important paper by Elworthy and Mysels,

19

in combination with NR measurements, points to a fundamental weakness in the ST method for the study of ionic surfactants, which stems from the nite width of micellization, though it was not considered explicitly in this way in the original paper. Since one of the key issues in the six papers mentioned in the opening paragraph is whether or not adsorption reaches a plateau at the CMC we have used NR to determine the adsorption of sodium dodecylsulfate (SDS) above the CMC with and without added NaCl.

We note that Elworthy and Mysels

19

and later Mysels

20

have already argued from

the variation of the sub-CMC ST with concentration that adsorption of SDS does not reach saturation by the CMC, but many might nd direct measurements above the CMC more convincing. SDS continues to adsorb above the CMC in the absence of added electrolyte.

21

On the other hand, in the related sodium dodecylmonoethoxy sulfate (SLES) we nd that a

5


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limit is reached at the CMC, although addition of electrolyte increases the adsorption. We explore these dierences in the context of the mass action model of micellization and show that the signicant width of micellization, already observed by EM, undermines the determination of the limiting adsorption at the CMC for SDS. In the course of this discussion we attempt to establish a denitive adsorption isotherm for SDS from NR and we show that the correct EM isotherm, partly incorrectly displayed in the original paper, should be considered at least for the time being as the denitive surface tension isotherm.

The consequences of

ionic impurities for measurements on SDS and SLES are examined indirectly in terms of the ST/NR results on other surfactants, results that have already been published

22 23 24 . 25

Experimental Details Perdeuterated SDS was prepared as described previously by the reaction of perdeuterated dodecanol with chlorosulfonic acid in dry ether at 0

o

C

26 . 27

Rehfeld started his preparation by

purifying the dodecanol by distillation to remove other chain length impurities but it seems that the current quality of the dodecanoic acid that was our starting point is such that this is unnecessary. Perdeuterated dodecanol was prepared by reduction of perdeuterated dode-

4

canoic acid with LiAlD . The main impurities after the sulfonation are residual dodecanol and some dodecanoic acid produced by oxidation. Neutralization by sodium hydroxide can introduce calcium as an impurity and this can be minimized by using semiconductor grade NaOH for the neutralization.

2

Following removal of ether and freeze drying, nearly all the

4

salt (NaCl and Na SO ) was removed by an initial crystallization from ethanol. Residues of NaCl were then removed by recrystallization from n-butanol.

For most but not all of the

samples used in our present and previously reported work on SDS the SDS at this stage was dissolved in water and dodecanoate was removed by acidifying the solution and washing with heptane or ether. After a second neutralization with semiconductor grade NaOH and further feeeze drying a nal recrystallization from ethanol was used to remove sodium sul-

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phate and any remaining dodecanol. We note that in one of the samples used for the older NR experiments

28

the SDS had been further puried by foam fractionation.

Chain deuterated SLES, C

29

12 D25 OC2 H4 OSO3 , was prepared in two stages.

First the non-

ionic surfactant C 12 D25 OC2 H4 OH was made by the Williamson reaction of bromododecane with ethylene glycol using potassium tert-butoxide as catalyst at 110

o

C.

30

After neutraliza-

tion of the mixture and extraction in ether/water the surfactant was puried using medium pressure liquid chromatography (silica with a varying ether/acetone mixture) and was then reacted with chlorosulfonic acid using the same procedure as for SDS above. Following neutralization and freeze drying SLES was then recrystallized three times from isopropanol. SLES suers from similar impurities as SDS and these could not be completely eliminated as judged by ST measurements, although the SLES would seem to be at least as good as any other samples discussed in the literature and the value of the CMC is in good agreement with the literature value.

31

Surface tension measurements were made using a Kruss K10 tensiometer using a Pt/Ir ring, as described previously.

30

The NR measurements were made on the reectometers

SURF and INTER at ISIS. Both instruments have been described previously

32 , 33

as has the

method of analysis for the simple situation of deuterated surfactants in null reecting water (NRW).

30

Adsorption Plateaus at the Air/Water Interface It is helpful for the discussion of the dierences between the adsorption of SDS and SLES above the CMC to examine the interplay between an isotherm and the CMC. Three schematic examples are shown in gure 1.

In this gure the black line shows the variation of activ-

ity/monomer concentration with overall concentration.

The two are approximately equal

below the CMC giving a straight line of slope unity. Above the CMC the activity/monomer concentration is constant in (a) and (b) and the slope of the black line changes sharply to

7


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Figure 1: The eect of the abrupt change in monomer concentration on adsorption for three dierent circumstances (a) the plateau of the adsorption isotherm is approximately reached before the CMC, (b) the CMC cuts o the adsorption before the true plateau is reached and no more adsorption occurs above the CMC, (c) the CMC cuts o the adsorption but a slowly increasing activity above the CMC allows some further adsorption.

The theoretical

adsorption for continuously increasing concentration (i. e. no CMC) is shown as a dashed line and the actual adsorption as a continuous line. The eect of the CMC on adsorption in (b) and (c) means that the adsorption cannot conform to any simple model isotherm.

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zero.

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The adsorption isotherm will be a function of the activity and it must reect this

abrupt change at the CMC. To assess the eects of the discontinuity caused by the CMC we use a model isotherm to calculate the adsorption which is also shown in gure 1 (in red). The model used is not important for assessing the main features of the gure and we have used a Frumkin isotherm. In (a) the plateau of the isotherm is approximately reached before the CMC and this is what seems to be the general situation in nonionic surfactants.

In

(b) adsorption has not reached the limit determined by the isotherm at the CMC. Here the activity of the surfactant is constant above the CMC and if the driving force for adsorption

does not change at the CMC no further adsorption should occur above the CMC. This is a plateau but it is not the saturated adsorption determined by the parameters of the adsorption isotherm. Micellization forces a plateau on the adsorption and the overall shape of the isotherm will then not correspond to the original model isotherm. When this occurs the commonly used procedure of tting a model isotherm to the ST data below the CMC with its adsorption limit at the CMC will not be a valid procedure. In several ionic systems the situation is more complicated in that the mean activity is known to increase slowly with concentration above the CMC and in (c) we show how the adsorption would then vary assuming that the concentration dependence of the adsorption is unchanged. Thus increasing adsorption above the CMC will often be associated with an increasing mean activity. Since the Gibbs equation requires that an increasing activity leads to a decreasing surface tension above the CMC, decreasing surface tension above the CMC will generally be associated with increasing adsorption, although this is not a thermodynamic requirement.

The main mechanism for the increasing activity is thought to be the

diering concentrations of surfactant and counterions resulting from partial dissociation of counterions from the micelle.

34

Previous results have shown that for C

14 TAB

35

and SDS

21

adsorption increases slowly above the CMC, i. e. unless there is a change in the mechanism of adsorption above the CMC, these conform to pattern (c) in gure 1. This means that the adsorption pattern is interrupted at a point in the isotherm where the variation of coverage

9


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with concentration is signicant. C

That the resulting awkward shape of the adsorption for

14 TAB generated by the type of behaviour in gure 1(c) is probably the main cause of error

in the analysis of the ST data is indicated in the paper of Simister et al. in their comparison of NR and ST for C

14 TAB.

35 The problems in the ST experiment were found to be associated

partly with wetting but mainly with the varying curvature of the approach of tting a tangent at the CMC gave a value of 60 Å with the NR values of 48 and 49 Å

2

for two isotopes.

2

σ − lnc plot.

The standard

for the ST result compared

The ST Gibbs analysis could only

be made to match the NR result when a Pt plate was used rather than a ring and when the curve was tted with a quartic over a range of 31 mN m to tting a quadratic to the last 10 mN m

−1

.

−1

below the CMC, as opposed

Given that one cannot devise an objective

criterion for either the choice of polynomial or the range to be tted, the ST result in that case cannot be regarded as a truly objective measurement.

Eects of Ionic Impurities in Anionic Surfactants Nonionic impurities are generally regarded as the most serious impurity problem in ST measurements on ionic surfactants and the standard example is the minimum in the surface tension around the CMC of SDS in the presence of dodecanol. This has been very thoroughly discussed by many authors and has even been quantitatively modelled by Krakchevsky et al.

36

Here we focus on the neglected problem of ionic impurities.

The adverse eects of

ionic impurities on the ST-Gibbs analysis are clearly revealed by NR, although the rst identication of the eects of Ca for SDS using radiotracers.

2+

on an anionic surfactant was made by Cross and Jayson

24

NR results in combination with ST have shown that it is exceedingly dicult to eliminate the eects of divalent ions on the surface behaviour of the surfactant, sodium diethylhexyl sulfosuccinate (AOT). The ST of AOT is shown in gure 2. Taking Ca

2+

as the key divalent

ion we can expect that Ca(AOT) 2 is more surface active than the sodium form (this was

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shown directly by Li et al. Ca

2+

37 ).

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At concentrations well below the CMC the eect of impurity

ions will be that the NR experiment will register an incorrectly high surface excess

and the ST experiment will register an incorrectly low surface tension.

As the concentra-

tion increases towards the CMC both measurements will tend towards their correct values. Unfortunately, the outcome of the ST measurement still depends on the values measured at low concentrations and the average slope will be less than the true value even if the value of the ST at the CMC is the true one, i. e. ST will underestimate the true coverage.

On

the other hand, if the true surface adsorption occurs at the CMC, the NR measurement will give the true value. Unlike ST, each NR measurement is independent. However, there is a further problem with the ST in that it measures both surfactant ion and counterion. the same coverage of surfactant ion the slope of the Ca(AOT)2 .

σ − lnc

For

plot is dierent for NaAOT and

Assuming that Ca(AOT) 2 occupies approximately twice the area occupied by

an NaAOT molecule, the Gibbs equation for Ca(AOT) 2 is

dγ ≈ −

3RT ΓAOT dlncS 2

(1)

If, as is probable, the packing of AOT fragments does not depend very strongly on the counterion, the NR coverage will be approximately the same for NaAOT and Ca(AOT) 2 . Thus if Ca(AOT) 2 dominates the surface behaviour, the ST result for thirds of the directly measured value. If the presence of the Ca

2+

ΓAOT

will be two

impurity is not suspected

the apparent result is that ST grossly underestimates the coverage. This is borne out by the experimental results. The measured NR areas at the CMC are 78 and 68 Å for pure NaAOT and pure Ca(AOT) 2 respectively, whereas ST gives 100 Å Ca

2+

2

2

per AOT unit

2

and 68 Å . If the

ion is sequestered by the addition of EDTA the ST result comes into line with that from

NR. These results suggest that for normal AOT, even after purication, the ST result will always be incorrect because AOT binds calcium so strongly that it cannot be removed except by unusual means (the molecular area of commercial AOT cannot be measured directly by

11


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NR because it is not deuterated).

Note that in the absence of the correct value from NR

there is no indication from just the measurement of the ST of NaAOT that the value of 100 Å

2

is incorrect except that it is unreasonably high when compared with the packing of AOT

in the lamellar liquid crystalline phase

38

or with the packing of AOT at the hydrophobic

solid/water interface, which was found to be

80 ± 5

Å

2

by NR.

39

However, the error in the

ST measurement could easily have been demonstrated by measuring the eect of various electrolytes on the ST of NaAOT, as was done by Li et al. obtained by Eastoe et al.

22

Parallel results have been

40 for sodium dihexylsulfosuccinate (NaHexAOT) except that these

authors did the additional consistency verication by successfully tting their nal data for the pure material to the integrated Gibbs equation, following the procedure used for Figure 3 in paper 1.

8 Although the ST measurements with the standard Gibbs analysis give completely

the wrong result for NaAOT and NaHexAOT, neither shows an ST minimum. ST and NR agree perfectly for Ca(AOT) 2 , where there is no eect of impurity at all.

37 Overall, unless the

surfactant system under study is completely free of surface contaminants, NR will usually give the more reliable value of the surface excess at the CMC. At lower concentrations, NR will give a direct value of the total coverage but this will be the sum of the surfactant and impurity excesses at this concentration. Under the same conditions, ST will be dicult to assess but will probably give too low a value of the limiting surface excess. The surface excess of the closely related sodium dioctylsuccinate was one of the rst systems to be studied by radiotracer measurements surfactant ion and sodium were measured.

41 . 42

The surface excesses of both the

According to the abstract of the paper on the

surfactant ion only, a similar result was obtained as in NR, i. e. that the direct measurement (radiotracer) gives nearly double the adsorption obtained by ST. Menger et al.

4

have

commented that this result may be because the interface was not saturated in the linear region whose slope was used to obtain a Gibbs area for the sulfosuccinate.

The alterna-

tive explanation given in the previous paragraph was demonstrated directly by Li et al. and conrmed by using the eects of dierent added electrolytes on the ST, which also showed

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Figure 2:

Page 14 of 59

The surface tension behaviour of AOT in three dierent forms, Ca(AOT)

2

(tri-

angles), puried NaAOT (circles) and puried NaAOT to which EDTA has been added to sequester any Ca of the

σ − lnc

++

ions in the system (squares) . The straight lines are the limiting slopes

plots at the CMC. The values of the corresponding NR limiting areas per

molecule at the CMC are 68, 78 and 78 Å

2

respectively. Data are from Li et al.

13


22

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that calcium (or magnesium) contamination is the correct explanation, independently of the

saturation question .

Menger et al. are, however, right that a linear t to the

σ − lnc

plot

was used by Salley et al. to analyse their ST data and this was used over an inappropriately long range of concentration (3 decades!).

We have reexamined this data and nd that the

data of Salley et al. is a less curved version of those of our standard sample shown as circles in gure 2, i. e. it was probably contaminated by Ca

2+

to a greater degree than our sample.

This is dramatically emphasized by their measurements on the adsorption of the Na

+

ion

which was found to be much smaller than the adsorbed surfactant ion at low concentrations and to rise only slowly towards the surfactant ion with increasing concentration. The radiotracer measurements, however, give a more complicated behaviour of the surfactant ion than suggested in the original abstract. The surface excess of excess) was reached at the very low concentration of up to a value of more than

5 × 10−6

mol m

−2

2.4 × 10−6

4 × 10−5

mol m

−2

(double the ST

M and continued to increase

at the CMC of 1.8 mM. Apart from this value

of the CMC being signicantly lower than the accepted value (

2.3 − 2.5

mM), this means

that these radiotracer results reach the CMC level of adsorption, as measured by NR, at an improbably low concentration, and they give an implausibly high coverage at the actual

2

CMC, corresponding to a molecular area of 33 Å . This would give a cross-sectional area per

2

chain of only 16 Å , which is impossibly low for such a monolayer. This would suggest some multilayer formation below the CMC, a possible indicator of even higher calcium contamination. Multilayer formation has been demonstrated for AOT at the air-water interface, but at higher concentrations.

43

Whatever the explanation, there are some serious inconsistencies in

these radiotracer results and no conclusion can be reached about saturation or otherwise at the CMC. Very recently, Mukherjee et al. have attempted to answer the saturation question for AOT with their own ST measurements

9

and they concluded that the surface was 96.3%

saturated at the CMC, but this was based on an evidently impure sample since the appar-

2

ent limiting area per molecule at the CMC was found to be 107 Å , which we have shown above to be both incorrect and an indication of a strong calcium ion eect on the ST. The

14


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

radiotracer measurement for Na

+

Page 16 of 59

adsorption was one of the results that led Pethica to point

out that the Gibbs equation strictly cannot be applied to ionic surfactants in the absence of salt because the counterion at the surface may be either the true counterion or one of the water ions depending on the dissociation of water at the interface, about which we have no information.

10 Pethica's analysis is correct but in this case and a similar one discussed below

it turns out to be ionic impurities that are the source of the problem.

Figure 3: The surface excesses by ST (crosses) and NR (inverted triangles) of the peruorooctanoate ion (PFO) in CsPFO solutions under dierent conditions (a) variation of the total surfactant concentration, (b) variation of the total surfactant concentration in the presence of EDTA, (c) variation of the caesium ion concentration at constant PFO (5 mM) and (d) variation of the PFO concentration at constant caesium (10 mM). The surface tension data points are calculated from polynomial ts to the experimental isotherms and NR errors are

±7%.

The data have been replotted from An et al.

23

As a second example of the eects of impurities, we examine ST and NR results for peruorooctanoates with dierent counterions, including the free acid (MPFO, HPFO). These

15


Page 17 of 59

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Langmuir

systems were thoroughly studied by ST by Shinoda et al.

25

who used variation of the coun-

terion concentration at constant surfactant ion concentration to show that the apparent surface concentration of the counterion was mostly not much more than half the expected amount, i. e. the surfactant has an apparent degree of dissociation at the surface of around 0.3.

This analysis was conrmed more rigorously by Hall including allowance for activity

eects

44 . 45 The apparent dissociation leads to a prefactor in the Gibbs equation of about 1.5.

However, the two sets of authors were not able to nd an explanation for these observations. An et al.

23

conrmed the ST results of Shinoda et al. and compared them with NR results

(uorinated surfactants give an NR signal comparable with deuterated surfactants).

The

apparent absence of counterion from the surface was conrmed by comparison of NR and ST in that anomalously low prefactors were observed for variation of the total surfactant coverage (nominal prefactor = 2) and variation of counterion concentration at constant surfactant (nominal prefactor = 1).

They seemed to be further conrmed by there being no discrep-

ancies between the two techniques for the variation of surfactant ion at constant counterion (nominal prefactor = 1). In all of these comparisons full corrections for activity were made in tting the ST data. Since surface neutrality must be maintained the loss of counterions from the surface must be made up by a dierent coion present in the solution. The obvious possibility is that this is H

+

, i. e. acid adsorbs preferentially at the surface (this had also

been suggested earlier as a possible explanation of the AOT discrepancy described above). However, An et al. showed that the pure acid exhibits exactly the same discrepancies, which shows that the missing ion must also be an impurity for the acid system and one with a greater anity for the PFO at the surface than H and AOT, the obvious candidate is Ca to sequester the Ca

2+

2+

+

.

Given the discussion above for SDS

. An et al. showed that once some EDTA is added

ion the ST and NR results agree over the whole concentration range.

Figure 3(a) and (b) show respectively the original discrepancy and its removal by EDTA and gure 3(c) and (d) show that the discrepancy is present when the counterion concentration is varied at constant surfactant ion but not when the surfactant ion concentration is varied

16


Langmuir

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at constant counterion.

Page 18 of 59

This last variation is not normally chosen for ST measurements,

although it is the one least aected by ionic impurity.

Once again the combination of NR

and ST shows that apparent misbehaviour of the Gibbs equation is driven by impurities and that in these circumstances only NR can give the correct value of

ΓCM C

unless a thorough

examination of the eects of dierent electrolytes on the ST is also made. Again, alternative explanations in terms of the ionization of water could have been applied here but the eects of the various salts demonstrate that it is entirely an eect of an ionic impurity. Finally, we note that the added EDTA not only greatly increased the coverage from the surface tension but it only slightly reduced the value of the coverage from NR. However, at lower concentrations well away from the CMC the eect on the NR was more signicant, in line with the discussion earlier in this section. The implication of these two sets of results is that divalent ions are potentially an impurity that can have a strong eect on the surface behaviour of all anionic surfactants. The source of divalent ions can be the original sodium source, e. g. the NaOH used to neutralize the dodecylsulfonic acid that is the intermediate in SDS preparation, the water used for solutions, and even the glassware or container surfaces used in the measurements. explicitly demonstrated this for SDS by using radiotracers

24

Cross and Jayson

and in the current experiments

we have also found that the addition of EDTA does decrease the coverage of SDS measured by NR at concentrations well below the CMC in exactly the same way as for NaAOT and NaPFO above.

Results Sodium Dodecylsulfate

SDS has been by far the most studied of all surfactants, despite its impurity problems. We start by examining its behaviour in added NaCl.

An advantage of measurements in

electrolyte with a common counterion is that the eects both of nonionic and ionic impurities

17


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Langmuir

are greatly reduced.

Ionic impurities are swamped by the excess of counterions.

If the

nonionic impurities emanate from the surfactant, as is usually the case, the typical tenfold reduction of the CMC for 0.1 M electrolyte, increases the surface activity of the surfactant relative to the impurity by a similar factor. This is borne out in the case of SDS/dodecanol by tting of mixture data by Kralchevsky et al.

36 Figure 4 (a) shows our new NR data above

the CMC and all the previous radiotracer results and gure 4(b) shows surface tension data from Tajima for SDS in 0.115 M NaCl

46 . 47

We have selected Tajima's ST results because he

used these data to compare with his own radiotracer measurements in gure 4(a). derived a value of

ΓCM C

of

4.3 µmol m−2

Ta jima

from a Gibbs analysis of his ST data and reanalysis

of his data indicates that this could only be achieved by using a straight line plot to t the last ve points before the CMC, shown in gure 4(b).

Any other plot using a non-linear

t gives a signicantly higher value, e. g. tting all the points (not all shown in gure 4) to a quadratic gives a good t to all the points shown in red in gure 4 and gives a value of

ΓCM C

of

5.3 µmol

m

−2

, closer to Nilsson's radiotracer results.

48

We also show in gure

4(a) (line) and (b) the t to the ST data that gives the best t to the NR data, which gives a value of

ΓCM C

of

4.8 µmol

m

−2

.

There is nothing to choose between the various ts to

the ST curve and this makes it clear just how large are the errors in tting slopes to ST data.

If the ST data are to stand on their own it would seem that the value that agrees

with NR is close to the middle of the range and the quoted error for the ST limiting excess should then be this value

±15%.

Ta jima's radiotracer and ST results are mutually consistent

if analysed as described above but in Tajima's radiotracer measurements saturation occurs well below the CMC, whereas this is not observed in NR nor in the radiotracer measurements of Nilsson.

48

This and the dierence between the Nilsson and Tajima results suggest that

there is considerable uncertainty in the radiotracer measurements. This is reinforced by the even more dierent results obtained by Matuura et al.

49

and Cross and Jayson et al.,

24

and

suggests that there are some methodological problems in the radiotracer method that have not been overcome.

18


Langmuir

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Page 20 of 59

The methodology of the NR experiment is well established by the results on nonionic surfactants and, leaving aside the question of the eects of impurities, the NR results should be reliable with a typical error of

±5%.

Gurkov et al. have analysed the eect of a range

of concentration of added NaCl on the ST of SDS and obtained a limiting coverage of

µmol m−2

at 0.1 M (no error given).

5.2

50 This result is closest to Nilsson's result, but as we will

show below Nilsson's results are probably too high. If one estimates the reasonable error of 10% for the limiting value of Gurkov et al. and takes the usual error of about 5% for the NR measurements, these newer ST results and the NR result agree adequately.

Gurkov et

al. make the point, now conrmed here with NR, that this adsorption is much higher than the coverage of SDS at its CMC in the absence of salt, i. e. the surface has not reached a packing limit at the CMC. We now turn to the behaviour of SDS in the absence of salt. Although dierent authors seem to have their favourite set of ST data for SDS, there is general agreement that the data of Elworthy and Mysels

19

and Mysels

20

(all referred to as EM in the following) are

the benchmark against which others should be assessed.

There are good reasons for this.

The experimental precautions taken by EM were exceptional with respect to the purity of the SDS (in situ foam fractionation), the purity of the water (redistilled from potassium

o

persulfate with the vapour passing through a tube at 700 C with a stream of oxygen), the in situ

measurement of the concentration by conductivity, the special design of the

hollowed-out Wilhelmy plate for the surface tension measurements and the consistency checks for reversibility, equilibration and time dependence.

Any comparison with these data is

undermined by the fact that the higher range EM data was misplotted in the original paper. In the original paper the EM data are plotted in two separate diagrams (EM gure 1 below the CMC and EM gure 4 above the CMC) with both concentration scales as log scales. EM's gure 4 for the higher range labels the rst three concentrations as 7, 8 and 9 mM, but it is clear that the correct labelling should be 8, 9 and 10. This is suggested by the fact that the consecutive gradations on the concentration axis are labelled 7,8,9,20,30,40 etc whereas

19


Page 21 of 59

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Langmuir

Figure 4:

(a) The surface excess of SDS in salt as measured by NR, radiotracers and ST

and (b) the ST of SDS in 0.115 M NaCl from Ta jima are those of Nilsson (red inverted triangles),

48

46 . 47

The radiotracer measurements

Matuura et al. (black crosses),

49

Cross and

24 all using 0.1 M NaCl, and Ta jima et al. using 0.115M NaCl Jayson (red lled circles), 46 (turquoise triangles). The present NR results using 0.1M NaCl are the open blue circles with error bars and are the only results available. Errors in the radiotracer experiments were only given by Matuura and are not shown for simplicity. The line in (a) is the t to the ST data in (b) that gives the best t to the NR data.

Other lines in (b) are optimized to t

the Nilsson data (red) or the Ta jima data (turquoise). The limiting coverages in from the NR and two of the radiotracer experiments are marked in (a).

20


µmol

m

−2

Langmuir

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Page 22 of 59

the true series of consecutive integers would be 7,8,9,10,20,30,40, i. e. there is an incorrect jump from 9 to 20. This is conrmed by plotting the separation between the points on the horizontal axis against the log of the expected integers, which shows that there is a gap. Thus the correct range must either be 760 or 870 mM. Figure 5 conrms that it must be 870 mM for three reasons.

First, EM's points from their gure 1 above the CMC are

plotted as crosses using 870 mM and as open squares using 760 mM but only the former are consistent with the points below the CMC (EM gure 4) plotted as open triangles. Luckily, in their insert to their gure 3 EM give a set of points in the range 7 - 9 mM and when these are added to the plot in gure 5 as open circles it can be seen that the correct labelling of their gure 4 should have been 870 labels. Finally, Mysels and Florence

51

in a later paper

plot the EM data up to about 10 mM, which further conrms the interpretation we have given here. Figure 6 and gure 7 show the correct combined data from EM gures 1 and 4 with the correct relabelling and with some additional points from their insert to their gure 3. We have only added points from gure 3 that help to ll the gap. Figure 6 further compares a selection of measured ST curves in the literature with the full and correct EM data for the rst time.

We note here that all the replotted data have

been obtained by capturing original diagrams on the largest possible scale as a .jpg le and using the software GetData

57

to scale and digitize the points.

The accuracy is high but

can be limited by small original diagrams. The original EM data was, however, reproduced on the maximum scale and should be of high accuracy.

Possibly as a result of the error

in the labelling of the original data there are almost no comparisons of anyone's data with those of EM. The only direct comparison has, as far as we can nd, is one by Nikolov and Wasan above.

58

over a limited range near the CMC and this is unquestionably aected by the error

Chang and Franses

attempt any comparison.

59

replotted some of the EM data below the CMC but did not

Figure 6 is therefore the rst comparison of the EM data with

those of other authors and it is very revealing. Dodecanol and calcium ions are both known to lower the surface tension at low concentrations and both generate a dip just as the CMC is

21


Page 23 of 59

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Langmuir

Figure 5: Plots of the surface tension (ST) below the CMC (open triangles), the two alternative interpretations of the ST points above the CMC (870, black crosses) and (760, open squares), and points from the EM insert to their gure 3 (open circles).

22


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 6: The comparison of a number of dierent measurements of the ST of SDS without salt with the results from Elworthy and Mysels (EM). The EM data are combined from gures 1, 3 and 4 of the original paper with a correction for mislabelling of some of the

19 The other data are replotted from Rosen, 52 Gilanyi et al., 53 26 55 (data taken at short times and at Rehfeld, Addison and Hutchinson 56 and Hines. 29 The data are shown on two dierent plots for clarity. et al.

coordinates in their gure 1. Sasaki et

o

54 al.,

15 C), Ta jima

The CMC from EM at 8.2 mM is plotted as a vertical line.

23


Page 24 of 59

Page 25 of 59

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Langmuir

reached

60 36 . 24

Thus, the higher the ST measured in the low concentration range the better

the purity of the solutions. useful discussion in Hines.

29

This point is fully discussed by Mysels

20

and there is also a

The Rosen, Tajima, Sasaki and Rehfeldt data (upper diagram)

all fail this test by a signicant margin.

The Gilanyi data (lower diagram) are better and

the Hines data are better still. The short time Addison data are signicantly better at low concentrations. The Tajima and Addison data both show a distinct dip below the EM data as the CMC is approached, indicating residual amounts of dodecanol (or dodecanoate) as impurity and the Hines data also dip away from the EM curve, though less strongly than the other two. The Hines data is interesting in that it comes closest to verifying the convex region of EM's ST curve but the slight dip in the Hines data leads to a lower value of the CMC (7.9 mM). The purication of the Hines sample was by foam fractionation just as in the EM paper but unlike the latter it was not used in situ. A signicant problem with the Addison

o

data is that it was taken at 15 C, which is strictly not comparable with the EM data because of the known temperature dependence Krat point.

62

61

but more importantly is uncomfortably close to the

Addison and Hutchinson's data on owing systems does, however, illustrate

how the measurement of ST at short times may be able to circumvent impurity problems (see also Mysels and Staord

63 ).

The Hines data is the closest of those shown here to the

original EM measurements and this is important because a sample prepared by the same author and means was used for one set of NR data to be described below. The eect of the dips in these curves is to reduce the decline of the surface tension above the CMC so that it seems to disappear and one or two authors have questioned the EM data because of this common observation

64 . 65

However, the extreme precautions taken by EM were specically

aimed at measuring this decline accurately and the comparisons shown here tend strongly to support the EM measurements. For the purposes of tting models to SDS surface tension data the comparison above shows that, in the absence of any newer measurement that takes comparable precautions to establish surface purity, the EM data should be the standard. The ST curve of SDS has been measured by authors other than those in gure 6.

24


Earlier

Langmuir

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Page 26 of 59

data were compared with part of the EM data in the paper by Mysels and Florence.

51

Later

data not shown in gure 6 have been omitted because either the scale of the original diagrams was too small to read the data suciently accurately or they are not the ones commonly cited for comparison. We have not seen ST data for any surfactant of a quality that is likely to rival the EM data, although as we shall now see, the dicult questions raised by the EM paper make such measurements something of a priority.

Finally, we note that in later

measurements Mysels stated that he had obtained values of ST that were consistently 0.5 mN m

−1

higher than the original EM data and had no explanation for the discrepancy. We

have not included this correction in the data shown in gure 6. Apart from the two diagrams giving direct ST measurements in the EM paper, there is a third diagram where the gradient of the

σ − lnc

curve was determined directly from 2 -

50 mM by using pairs of surface tension measurements separated by small

∆lnc

intervals.

These data are entirely separate measurements from those in gure 6. These data, shown as gure 5 of the EM paper are reproduced in gure 7 but, for a more direct comparison with NR below we have converted the slope to an apparent surface excess

Γ0

dened by

1 dσ dlnf =− Γ =Γ 1+ dlnc 2RT dlnc 0

(2)

Otherwise the data are as originally published. Figure 7 compares these gradient data with the ST plot and the comparison can be used to classify dierent regions of the

σ − lnc curve,

which we label quadratic, linear and convex, where quadratic and linear refer to how the ST curve behaves with

lnc

and convex refers to how the ST faces the

assumed that below the CMC the the

lnc

σ − lnc

lnc

axis.

It is normally

curve is always concave (or at) with respect to

axis on the grounds that the surface excess always increases or is level up to the

CMC. This is denitely not the case with the EM data. The value of the CMC was determined by EM as the point of maximum curvature in gure 7 and it is marked by the vertical line at 8.2 mM. There is a maximum in

25


66 65

Γ0

at

Page 27 of 59

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Langmuir

Figure 7: (a) The comparison of the ST of SDS without salt as measured by Elworthy and Mysels

19

with their own dierential data. The ST data are combined from gures 1 and 4 of

the original paper with a correction for mislabelling of some of the coordinates in gure 1.

19

The dierential data is replotted from EM's gure 5 but the vertical axis has been converted to apparent surface excess, as dened by equation 2. The behaviour of the dierential data

σ − lnc plot can be divided into the marked regions where σ is a quadratic function of lnc and concave to the lnc axis, a short region where it is linear, and a region where it is convex towards the lnc axis. (b) Shows the same dierential data as in (a) except that a best linear t (in lnc) to the points below 6 mM is shown extrapolated

shows that the curvature of the

to the CMC.

26


Langmuir

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Page 28 of 59

around 6 mM, signicantly below the CMC. If one neglects any contribution of the activity coecient term to of

3.6 µmol

m

−2

Γ0

up to 6 mM the maximum slope at this point leads to a surface excess

2

, equivalent to an area per molecule of 46 Å , at the concentration of 6

mM but not at the CMC . EM commented that such a limiting area per molecule would be surprisingly high.

However, in this paper and in a subsequent paper,

20

this is the highest

value of the surface excess and the highest concentration for which EM or Mysels quote a value of the surface excess, apart from Debye-Huckel activity corrections.

The reason is

that if one were to obey the normal convention that the activity coecient can be taken as approximately unity up to the CMC the value of (

> 80

Å

2

Γ0

would plummet to below 2

µmol

m

−2

per molecule) at 8 mM, still below the CMC, but clearly a nonsensical value for

the surface excess. EM's Figure 5 is remarkable because it shows that the activity coecient is undergoing a dramatic change between about 6 mM and the CMC and it is not a change that can be explained by Debye-Huckel type ionic interactions. It can only be explained by the micellar transition for SDS having a signicant width. What is almost more remarkable, however, is that, as far as we have been able to nd, EM's gure 5 has never been referred to in any of the 280 current citations of EM's paper. Indeed, only Sasaki et al. and Nikolov and Wasan

lnc

58

seem to have commented that EM's ST data is convex with respect to the

axis. Chang and Franses,

59

in replotting the EM coverage data, have even ignored the

diagram altogether and show EM's coverage as varying linearly with

µmol

m

−2

at the CMC, much higher than the 3.6

µmol

m

−2

lnc

up to about 4.1

limit in the original paper.

Rusanov, in discussing the mass action model of micellization, has made it clear that the mass action model requires the

σ − lnc plot to be convex in the region of the CMC, although

he seems not to have noticed EM's very direct evidence for it.

65

We discuss the probable

explanation of this behaviour below in the context of other measurements but the behaviour itself clearly undermines the determination of the limiting surface excess at the CMC by means of the Gibbs equation.

If analysed in the conventional way, any points used in the

range from 6 - 8.2 mM will have the eect of lowering the apparent limiting coverage.

27


If

Page 29 of 59

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Langmuir

this behaviour were true for all surfactants the ST-Gibbs analysis would always tend to give an incorrectly lower value of the limiting surface excess.

Put dierently, in the absence of

any data or model for the activity variation in the region below the CMC the only ob jective analysis is that the surface excess reaches a limiting value at 6 mM of 3.6

µmol

m

−2

(apart

from a small upward adjustment for interionic contributions to the activity coecient). Any variation of surface excess between 6 and 8.2 mM cannot be determined without knowing the variation of the activity coecient associated with the preliminary stages of aggregation. The behaviour of the surface excess below 6 mM is evidently linear in

lnc.

It might then

not be unreasonable to assume that it would have continued to be linear up to the CMC if there had been no unusual activity coecient behaviour, i. e. the apparent coverage would have varied linearly with

lnc

up to 8.2 mM as shown by the straight line in gure 4(b).

4.2 ± 0.3

Neglecting any Debye-Huckel contribution this would have led to a surface excess of

µmol

m

−2

at 8.2 mM. This is eectively what was done by Chang and Franses,

59

as already

mentioned. However, the discussion of gure 1 above was that adsorption should vary with the mean activity rather than concentration and since this will lag behind the concentration the limiting coverage at the CMC should be less than the extrapolated limit of m

−2

. Thus,

ΓCM C

4.2 µmol

can only be determined by direct methods, to which we now turn.

We note that many of the early measurements of the surface excess of SDS have been tabulated by Ikeda,

70 although his table is notable for its omission of the EM result.

primarily on EM's ST results, some foam results, all our previous NR results concentrations.

27 61 21 67 , 28

68

We focus

all the radiotracer (RT) results

48 56 24 , 69

and the present new set of NR measurements at high

All our previously measured NR data are shown in gure 8(a) and, apart

from the new measurements, used samples for which the step of removing dodecanoic acid from the SDS was taken, as well as the usual measures for minimizing calcium impurities and removing dodecanol.

One of the samples (Hines et al.)

was additionally puried by

foam fractionation using the identical procedure as used for the sample used for the Hines ST curve in gure 6(b). Since the Hines ST curve was closer than any other to the EM ST

28


Langmuir

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Figure 8:

Page 30 of 59

(a) The surface excesses by measured by NR on six separate occasions (points

with error bars, Lu et al.,

27

Marrocco et al.,

61

Purcell et al.,

21

Cooke et al.,

and this paper) tted by least squares to separate functions linear in the CMC (blue lines).

into

Hines et al.

28

below and above

Two red lines show the apparent surface excess from ST using the

gradient data of Elworthy and Mysels

68 foams.

logc

67

20 and the average surface excess measured by depletion

Two thin black lines show the CMC of 8.2 mM, as determined by Elworthy

and Mysels, and the limiting surface excess in the presence of 0.1 M NaCl from gure 7. (b) Surface excesses measured by four separate radiotracer measurements (points, Nilsson, Ta jima et

56 al.,

Cross and

24 Jayson

and Matuura et

69 ) al.

48

compared with NR (tted blue

lines from (a)), ST (red line from (a)) and depletion into foam (red line from a).

29


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Langmuir

curve, this should give the most reliable results.

The new measurements were made on a

sample that did not include a specic step that removed dodecanoic acid and for this sample we therefore show only measurements at high concentrations (greater than twice the CMC). We assume that any impurity has been solubilized at these higher concentrations. the adsorption from this sample is higher at the CMC and just above (about m

−2

) than at

In fact

4.5 ± 0.3 µmol

2× and 5×CMC, and drops down sharply at about 1.4×CMC, consistent with

the presence of impurity and with its removal into micelles. These values are not shown in gure 8. The general increase in the coverage above the CMC for this sample is consistent with the single earlier measurement at 100 mM.

21

The EM results are plotted as a thick

red line over the range of their gradient experiment and the average result of Wilson et al. for 10 - 28 mM (from experiments on foams) is also shown as a thick red line.

Note that

the slope of the EM line would be steeper if Debye-Huckel corrections were included.

The

NR results at or near the CMC were 4.1 at 9 mM and 3.75 at 8 mM (Lu et al.), 3.8 at 7.8 mM (Marrocco et al.), and 3.9 at 8 mM (Hines et al.), all in average value of

3.9 ± 0.2 µmol

high quality sample. of

4.1 ± 0.2 µmol

concentrations.

m

m

−2

µmol

m

−2

. This results in an

, exactly the same as obtained by Hines et al. using a

Two values at the higher concentration of 10 mM lead to an average

−2

and are consistent with the general increase shown at much higher

These two values at the CMC and at 10 mM are also identical with the

values obtained using depletion from solution into foams by Wilson et al. , who obtained and

4.1 µmol

m

−2

3.9

respectively for these two concentrations.

The NR, ST and foam data, all as lines, are compared with the four radiotracer results in gure 8(b). Statistical errors were not given for the radiotracer results except for those of Matuura et al. and we have omitted these for simplicity. We have also omitted some foam results from Weil value.

71

because his value of the CMC of SDS is out of line with the normal

In the vicinity of the CMC none of the radiotracer results comes close to the NR

and ST or foam results.

There is also disagreement as to whether adsorption increases or

stays constant above the CMC and the values are widely dierent from each other. As was

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Page 32 of 59

apparent in gure 4 there is a worrying lack of reproducibility between the radiotracer data obtained by dierent groups and it falls well outside the range of reproducibility of the NR data. The conclusion is that, basically none of the existing radiotracer data can be used to make arguments that bear on the issue of the absolute accuracy of either ST or NR data. Reasons for the problems in accuracy and/or reproducibility of radiotracer measurements have been discussed by Lu et al.

72

However, there is good agreement between the eect of

added 0.1M NaCl on the relative adsorption determined by Tajima et al.

Nilsson and NR

and this suggests that systematic errors are dominant in the radiotracer measurements. Even though the ST-Gibbs method cannot be extended as far as the CMC, the agreement between ST, NR and the foam experiment in the range CMC can regard the value of denitive value for

3.9 ± 0.1 µmol m−2

ΓCM C

±40% is

(area per molecule of

good enough that one

42.6 ± 1 Å ) as a reasonably

for SDS at 298 K. That the coverage continues to increase from 6

mM upwards implies that the mean activity also continues to increase. This was explained by EM in terms of the mass action model and we discuss this in detail below.

Whatever

the driving force for adsorption in this higher concentration region the surface excess has reached more or less the same value at

10×CMC

as that in 0.1 M NaCl, a value about

25%

higher than at the CMC. At low concentrations the ST and NR results look as though they diverge. Hines et al. compared the NR and ST results using the integrated Gibbs equation. We do not reproduce that result here but the agreement was reasonable down to about 2 mM. In our current experiments we found that the addition of EDTA was to decrease the excess measured by NR, suggesting that at low concentrations the presence of Ca

2+

ions in the solution becomes important, as speculated by Hines et al. We have established above that Ca

2+

ions are more or less impossible to eliminate from AerosolOT (AOT) and

peruorooctanoates (MPFO) and in both cases they caused the ST and NR results to diverge as observed here

22 . 23 Cross and Jayson have also established that Ca 2+ has a strong eect on

24

The measurement of surface excess of anionic surfactants at concentrations

SDS solutions. of

0.2×CMC

is clearly not easy but we leave that as a problem distinct from that being

31


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Langmuir

considered here. Given the behaviour of the ST between 6 and 8 mM observed by EM and given that the Hines ST curve is close to that of EM the successful tting of the integrated equation of the Hines data is surprising. However, the integration can start from any point in the vicinity of the CMC and the eect observed by EM would cause the apparent CMC to shift to a lower value than the true one. This is exactly what Hines et al. observed in that they used a value of 7.9 mM for the CMC rather than 8.2 mM. At this point it is worth examining the possibility of using other experimental methods for determining coverage.

Two such methods have been applied to SDS. Bain et al. have

used ellipsometry and external reection Fourier transform infrared spectroscopy (ERFTIR) to examine SDS at the air/water surface conditions and in 0.1 M NaCl.

73 . 74

Both measurements were made under ow

In the ER-FTIR case the results were directly scaled to

the neutron value for saturated equilibrium adsorption, unfortunately to the value in the absence of salt, which is signicantly smaller than the value now reported above for 0.1 M NaCl. The ellipsometry results were eectively directly scaled to parallel NR results on the owing system.

Thus neither technique can yet be regarded as an absolute measurement

and therefore do not add anything to the above discussion. However, for anionic surfactant systems, where there are issues concerning coverage above the CMC and possibly serious problems with impurities at low concentrations and near the CMC, there is considerable scope for the application of alternative methods, even if they are only relative methods. Addison and Hutchinson's results for SDS illustrate the value of short time measurements, which would be easy with ellipsometry, for circumventing impurity problems.

The other

issue that would be usefully studied by ellipsometry is the lower concentration behaviour of the surface excess. Ellipsometry is highly sensitive to the counterion and if, as has been argued above for SDS and other anionic surfactants, the Ca

2+

ion gradually replaces Na

+

at low concentrations, one would expect signicant divergence to start to occur between NR and ellipsometry at lower concentrations. Present results indicate no signicant divergence between the two techniques,

73

although NR appears from gure 4 to be diverging from the

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Page 34 of 59

ST results of Elworthy and Mysels.

Figure 9: Comparison of two measurements of the mean activity of SDS solutions above the CMC (point) with values calculated from ts to EM ST data and current NR data (line). A linear t in

lnc

was used for both ST and NR, which leads to some errors just above the

CMC.

A further test of the consistency of the results for the surface behaviour of SDS is to apply the Gibbs equation above the CMC. In this region the Gibbs equation couples three sets of data, the ST data, the NR adsorption isotherm above the CMC, and the variation of the activity above the CMC. Any two of these may be used to determine the third. The activity has been measured by Cutler et al. while Sasaki et al.

75

in both a single cell and in separate half cells,

54 measured it using just the separate half cells.

Since there has been some

argument as to the reliability of surfactant sensitive electrodes, the most sensible approach is to use the ST and NR data to calculate the activity as was done by Cutler et al. using

1 ∆lna = − 2RT

Z

σ

σCM C

dσ Γ

33


(3)

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Langmuir

Cutler et al. did originally compare their results with values calculated from EM assuming a constant coverage above the CMC and concluded that the surface coverage would have to increase substantially above the CMC for theirs and EM's data to be consistent. There are two dierences in our calculation. First, judging from their graphical comparison, Cutler et al. were not aware of the mis-labelling of the high concentration ST data of EM. Secondly, they had to assume constant coverage. Using the present NR data and EM's surface tension data, we have plotted the result of applying equation 3 numerically, using the observed linear variation of the coverage with

lnc

from NR, in gure 9.

Although the eect of the

increasing coverage is to reduce the disagreement between the dtata of Cutler et al. and the EM data, the use of the corrected ST data makes it worse, especially at the lower end of the concentration range, where the sharply varying ST at and just above the CMC contributes a distinct hump in

loga± .

The agreement with the less direct measurements of Sasaki et al.

is better, although Cutler et al. pointed out some diculties with the less direct method of using two separate half cells.

In contrast to Cutler et al. Sasaki et al. also found a much

lower value of the mean activity coecient at the CMC than expected from the Debye-Huckel theory, 0.68 compared with the Debye-Huckel value of 0.9. Interestingly, this is qualitatively what would be expected from the EM results plotted in gure 6(b).

The range of values

consistent with the EM results would be from about 0.3 to 0.7. It might be thought that this disagreement would impact negatively on the EM results. However, the signicant diculty of creating surfactant specic electrodes and validating their performance is a topic that has been neglected since the measurements of Cutler et al. and Sasaki, which suggests that serious diculties still remain. It should also be noted that both Sasaki et al. and Cutler et al. used SDS as received and took no special purication steps.

Sodium Dodecylmonoethoxysulphate, SLES

In contrast to the behaviour of SDS above the CMC, SLES showed no variation in adsorbed amount with concentration above the CMC, as shown in gure 10(a). The limiting coverage

34


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 10: (a) The surface excess of SLES below and above the CMC as determined by NR on the alkyl chain deuterated SLES in null reecting water. The surface excess is constant above the CMC. (b) The surface tension data for SLES with (triangles) and without 0.1M NaCl (circles) and the surface excess above the CMC. The t to the surface tension data below the CMC is quadratic in

lnc

and is constrained to give the correct NR excess at the

CMC.

35


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Langmuir

was

4.0 ± 0.3 µmol

m

2

but the value at the CMC is slightly higher at

4.2 ± 0.3 µmol

2

m ,

indicating a weak maximum to correspond with the weak minimum observed in the surface tension. The adsorption of SLES is similar to SDS, as might be expected from a combination of two opposing factors, a larger head group, which should make the limiting excess smaller than SDS, and reduced repulsion because of the stronger binding of the counterions, which should make the limiting excess larger (the micelles of the SLES are signicantly less dissociated than those of SDS). The only value available from the literature is that of Barry and

2.4 µmol

Wilson from ST measurements which is

2

m ,

31

very dierent from the NR value.

The variation of the NR surface excess below the CMC, also shown in gure 10(a), is once again linear with

lnc.

Using the resulting quadratic variation of the ST with

lnc

and con-

straining the curve to t the NR surface excess at the CMC leads to the t to the surface tension of SLES shown in gure 10(b).

This is not the optimum t to the surface tension

data, i. e. not what would be obtained from an ob jective t of either a quadratic or any other polynomial function, but it is an acceptable t given the errors in the measurements. The variation of the surface excess of SLES belongs to the category of gure 1(b), i. e. there is no signicant increase in the activity above the CMC and so there is no further adsorption. However, just as found by EM for SDS, the quadratic variation with

lnc

below

the CMC indicates that the surface has not reached a true plateau of saturation at the CMC. Measurements in the presence of 0.1M NaCl conrm this in that NaCl increases the surface excess of SLES up to

4.65 ± 0.3 µmol

m

2

as shown in gure 10(b). This is a similar relative

change to that observed for SDS. The small decrease in ST above the CMC in the absence of salt shows that there is little change in the activity above the CMC. For a constant coverage above the CMC and a linear variation of

σ

with

lnc

(the linear t is shown in gure 10(b))

the change in activity is given by equation 3. This gives a value of the CMC and

10×CMC,

equivalent to an increase in

a

∆loga = 0.018

between

of 4% over this concentration range.

This is much lower than for SDS. As would be expected the change in the presence of salt is smaller again. The results for SLES therefore illustrate a situation where not only is surface

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Page 38 of 59

saturation apparently not reached before the CMC but adsorption is brought to a more or less abrupt halt at the CMC. Predictably, under these circumstances, any objective tting of the surface tension data will underestimate the limiting coverage at the CMC, as conrmed by the analysis of Barry and Wilson, whose ST data were no better or worse than ours.

Discussion We are concerned in this paper primarily with the tting of the ST data. The combination of the NR and EM results for SDS show that the nite width of micellization prevents an accurate t of the Gibbs equation to the ST data. However, ST may not be the most sensitive parameter to the width of the micellization because the Gibbs equation makes it depends on both surface and bulk properties.

Thus the extra steepness of the ST curves of SDS

as the CMC is approached is usually attributed to extra adsorption of impurities (usually dodecanol) rather than to anything unusual happening in the bulk.

Most other methods

respond to the bulk properties and are independent of the surface behaviour. al.

Al-Sou et

15 have proposed an empirical model to t dierent types of data and have given particular

attention to SDS, using ST, NMR and conductivity data.

They comment that the width

from ST was not well dened and the value they obtained from the ST data was narrower than suggested by the EM measurements, but they unfortunately chose data from Watanabe et al.,

76

which were not obtained under circumstances anything like as rigorous as for any of

the data shown in gure 6. However, Al-Sou et al. 's own conductivity data and especially the NMR experiments of Cui et al.

18

(chemical shifts and self-diusion), which Al-Sou et

al. t to a global width, give widths that are of the same order as observed in the EM data. The NMR data of Cui et al. suggests an onset of what they label as premicellization at 5.96 mM, the same onset within experimental error as in the ST slope data of gure 7. However, it is important to note that Cui et al. and Al-Sou et al. used as-received SDS and it is dicult to regard their analysis as denitive, although the NMR method would seem to be

37


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Langmuir

very suitable for studies of the micellization process and the general approach of Al-Sou et al. is also valuable. These are two relatively recent studies but there are a large number of non-ST measurements on SDS which do not really lead to any uniform conclusion that would denitively conrm or otherwise the EM observations on the micellization width.

Most of

these have also been performed on as-received SDS samples and the question then arises as to how impurities might aect the micellization process.

This is especially important

in the case of the activity measurements performed by Cutler et al. as described above. The mass action model described below would suggest that impurities that lead to higher aggregation numbers and lower degrees of micellar dissociation will narrow the micellization. Dodecanol could easily do this to SDS. On the other hand, there will be many impurities that favour smaller aggregates or a more gradual aggregation process and these will broaden the transition. We have been careful not to use anything other than experiment to dene the surface behaviour apart from some qualitative guidance from gure 1. However, EM in their original paper used their result of the decrease in ST with concentration above the CMC to argue that the mass action model of micellization gives a better description of the micellization process than the pseudophase model. To do this they calculated the mean activity of SDS above the CMC using numerical solution of the mass action model together with the known degree of ionization and aggregation number of the micelles and the assumption of a constant coverage above the CMC. At the time, there was no direct experimental evidence that the partial dissociation of micelles leads to diering concentrations of surfactant ion and hence an increasing mean activity above the CMC. This has since been well established not just by the two activity measurements described above

75 54

but in several other papers

16 17 . 77

In

some of these experiments there is also evidence for a nite width of the CMC in that the concentrations of surfactant and counter ion change gradually rather than abruptly through the CMC. In performing the mass action calculation EM were also able to t their surface tension curves not just over the range above the CMC but down to concentrations below the

38


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Page 40 of 59

CMC, i. e. they showed that the mass action calculation also accounts semi-quantitatively for the behaviour of the surface tension through the CMC. It is interesting to explore just how quantitatively this behaviour can be explained using the additional data on the adsorption isotherm from NR. In addition, although the mass action model has been thoroughly reviewed from a theoretical point of view by Rusanov,

65

there seem to be almost no attempts to use

it to model the behaviour of experimental surface tension data in the vicinity of the CMC, other than that of EM, and such modelling may cast some light on the dierence in behaviour between SDS and SLES and hence on the whole issue behind the use of the ST-Gibbs analysis for determining surface excess. The mass action model of micellization is one of the basic models of micellization.

34

The micelles are assumed to be monodisperse and their formation is governed by the single equilibrium constant

K

which is

K=

where

xn (x

m + pm)(n−p)

(4)

c is the total concentration of surfactant, x is the concentration of monomer, m is the

concentration of micelles, dissociated micelles

78 . 66

n

is the aggregation number and

p

is the charge on the partially

The overall concentration of surfactant is

c = x + nm

(5)

but it is more convenient to do calculations in terms of the free counterion concentration,

y,

78

given by

y = x + pm Rewriting equation 4 in terms of

y

and

x

(6)

and taking logs gives

nlnx = ln(y − x) − (n − p)lny − lnpK

39


(7)

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Langmuir

which can be solved for

x

by iteration for a given

As noted by Shanks and Franses

59

y

by using the Newton-Raphson method.

the iteration can become unstable for large

n close to the

CMC and they introduced a system of coupled dierential equations to match the problem to their software package. However, we found that the instability in the iteration was easily removed by increasing the precision of the iteration and accordingly we have used equation 7 for which we wrote our own software using Java. required to determine the dierential function

The much more demanding accuracy

(1 + dlnf /dlnc)

(see equation 2 for the de-

nition) could also be attained by the use of a higher precision for the iteration and a careful choice of the initial guess for done using

K=1

x.

As pointed out by Mysels

79

the calculation is conveniently

and the concentrations must all then be multiplied by a factor

b=

b

given by

1

(8)

K 2n−p−1

where we have corrected a sign in the original paper. If in addition we use the value of the

K

CMC as the basic concentration unit, the value of

required to t any real data scaled to

the CMC is then usually close enough to 1 that the factor

b

becomes approximately 1 and

no correction is necessary. The partial dissociation of the micelles imposed by taking

p≤n

leads to dierent and varying concentrations of surfactant and counterion when micelles are present and EM neglected contributions from ionic interactions and assumed that the mean activity is given by 1

a± = (xy) 2 We make the same assumption in the following discussion.

(9)

Using this model, we examine

some features of the mass action calculation which are relevant to the behaviour of the ST in the vicinity of and above the CMC. Some of the features of this calculation can be seen in the two calculations we do here, one for SDS (gure 11) and the other for SLES (gure 12). However, we note the following. First, the width of micellization in this model is signicant once the aggregation number is below

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Page 42 of 59

Figure 11: (a) Fits of curves calculated using the mass action model with the linear variation of the surface excess ( constant value of

σ0

3.8 − 4.8 µmol

(for the ST curve only).

exactly compensated by a change in

β

n = 64, β = 0.24,

from NR, and an arbitrary

(b) The corresponding changes in surfactant

ion (blue), counterion (red) and mean activity (green). that a change to a constant coverage of

m

−2

The sensitivity of the ST is such

−2

3.8 µmol m rather 0.24 to 0.265.

from

41


than the variable one can be

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Langmuir

σ − lnc curve calculated using the n = 54, β = 0.02 (values from small angle neutron scattering) and −2 above the CMC of 4.0 µmol m from NR, and an arbitrary constant

Figure 12: (a) Fit of experimental data for SLES and a mass action model with a xed surface excess value of

σ0 .

(b) The corresponding calculated changes in surfactant ion (blue), counterion

(red) and mean activity (green). The function

(1 + dlnf /dlnc) is plotted in both (a) and (b)

to show the calculated width of the micellization transition.

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Page 44 of 59

about 100, i. e. in the usual range of micelle size. The width increases rapidly as and is already signicant for SDS (

n ≈ 60),

n decreases

although not as large as observed in gure 11.

However, for larger surfactant molecules such as the double chained gemini surfactants the small values of

n

will lead to much larger micellization widths and could be an important

factor in the failure of the Gibbs-ST analysis for these systems. Clearly, for a given degree of dissociation and a given aggregation number, the larger the molar adsorption the greater the slope of the the slope.

σ − lnc

plot and an increase in adsorption above the CMC will further increase

However, the

σ − lnc

plot is less aected by the typical change in adsorption

than it is by the range of micelle aggregation number and charge.

The most important

factor determining the change in ST above the CMC is the degree of dissociation. A small degree of dissociation leads to only a small asymmetry in the concentration of surfactant and counterion monomers and hence the change in mean activity is small (zero in the limit of no dissociation). This is the main cause of the large dierence in slope of the

σ − lnc

plot

between SDS and SLES, which can be seen in gures 11 and 12. The uncertainty in the exact values of the degree of dissociation and aggregation numbers is generally too large for it to be possible to deduce anything useful about the magnitude or concentration dependence of the adsorption above the CMC. In gure 11 we attempt to t both EM's ST data and their ST gradient data (gure 7(a)) using the mass action model. EM did not attempt to t their gradient data and they assumed a constant coverage from 6 mM upwards in tting their ST data. Here we use the coverage data determined by NR (gure 8(a)) in the form of a linear dependence from the onset of the activity anomaly to

10×CMC. Otherwise we use values of the aggregation number and degree

of dissociation in the accepted range determined by a number of experiments. Conductivity and activity data give values of the degree of dissociation of the micelles in the range 0.20.3

59

and neutron small angle scattering at the lowest concentrations gives 0.3.

80

The only

truly adjustable parameter is an arbitrary additive component for the ST (the integration constant in the integrated Gibbs equation). In principle this could be determined absolutely

43


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Langmuir

by integrating from pure water but the uncertainties would be considerable. The t to EM's ST data is deceptively good, especially in the vicinity of the CMC where the convexity of

σ − lnc

the

curve is well explained. However, the width of the CMC is more apparent in the

gradient plot and the mass action model predicts too narrow a width with the accepted values of

n

and

p.

The general insensitivity of the ST above the CMC is illustrated by the range

of values of the three main parameters, curve. Thus, although

µmol m−2

n, p

and

Γ

that will give an acceptable t to the ST

Γ varies from 3.8−4.8 µmol m−2 a change to a constant coverage of 3.8

can be compensated by an increase of the degree of dissociation from 0.24 to 0.265

or by a decrease in the aggregation number from 64 to 43. This is outside the error of the determination of

n

but well within the uncertainty in the degree of dissociation. An overall

change of 10% in

Γ

can similarly be compensated. The mass action model accounts well for

several of the features of EM's ST data from just below the CMC upwards but its failure to account for the full width below the CMC can only be caused by polydispersity on the low concentration side of the transition. Thus, to extend the sharp decrease in activity down as far as 6 mM (about

0.75×CMC)

dissociation higher than 0.3. dimerization occurs in SDS.

81

requires aggregation numbers below 30 and/or degrees of

We note that Mukerjee et al. had previously suggested that Note that the function

(1 + dlnf /dlnc)

is shown calculated

down to a concentration where it just reaches the value of 1.0, i. e. where micellization just starts. This is the limiting upper value of the concentration for application of the usual form of the Gibbs equation. The CMC is dened here as the point where

(1 + dlnf /dlnc) = 0.5.

This is not exactly the same as the denition given originally by Phillips fully by Rusanov

65

66

and explained

but it is close enough for our present purposes.

The results shown earlier for SLES can also be accounted for by the mass action model using the NR limiting surface excess of 4.0

µmol

m

−2

and additional information from our

own small angle neutron scattering on the aggregation number and charge of the micelle. At the lowest concentration at which we could measure these two quantities we obtained and

p=1

n = 54

(degree of dissociation = 0.02) and these values ensure that the variation of the

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Page 46 of 59

mean activity and hence the surface tension above the CMC is only slight as shown in gure 12.

Conclusions Our primary purpose in this paper was to examine the application of the Gibbs equation to ST data from anionic surfactants by comparing the results with directly determined coverages. It should now be clear that radiotracer measurements are not reproducible enough to provide an adequate test for ST-Gibbs. In contrast, NR, which was shown in our previous paper to be methodologically sound and accurate for nonionic surfactants, gives results that are reproducible and consistent with the high quality ST measurements of Elworthy and Mysels in the concentration range around the CMC. We have shown previously that impurities present special diculties for both ST-Gibbs and NR but their impact is generally though not exclusively greater on ST-Gibbs, with the result that the apparent surface excess at the CMC,

ΓCM C ,

will be incorrectly low when determined by ST-Gibbs. At low concentrations

the situation may be dierent but this is not the concern of the present work.

Here we

have focussed on the situation when impurity eects have been minimized. In particular we have examined how the

σ − lnc

curve behaves just below the CMC because this is the part

that has most inuence on the value of

ΓCM C

obtained by ST-Gibbs. If adsorption reaches a

plateau before the CMC then the analysis is straightforward and can be reasonably accurate. However, both direct measurements at higher concentrations and less direct measurements using added electrolyte indicate that adsorption of anionic surfactants has generally not reached a packing limit at the CMC, i. e. it is necessary to t a polynomial to the data below the CMC and because this is abruptly terminated at the CMC there are no eective objective criteria for choice of that polynomial. In general, both ST and NR indicate that varies quadratically with

lnc

that this is a general rule.

σ

in this concentration range but there is no reason to suppose

This means that the errors in the ST-Gibbs determination will

45


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always be larger than in NR, except for unusual eects of impurities.

We emphasize that

this conclusion does not in any way challenge the validity of the Gibbs equation. It is purely a question of the dicult task of making measurements of the necessary accuracy. The above conclusion was the result we expected to obtain when we started this work. However, in the case of SDS, the only results in the literature that seem to have been performed to a high enough standard to use the Gibbs equation show that it cannot be used to determine

ΓCM C

because the width of micellization makes it impossible to use the

equation as the CMC is approached. Micellization introduces a change in the activity which cannot be modelled and which extends over a range of some 25% of the CMC down to lower concentrations.

Part of this result is not new because it was shown clearly by Elworthy

and Mysels in 1966, although they made little comment about the implications.

However,

its consequences are made much clearer by having the NR results for the isotherm from 6 mM upwards, which establish unequivocally that the behaviour that Elworthy and Mysels observed is an activity eect. In terms of the controversy that prompted the present work, the Gibbs equation cannot be used to determine

ΓCM C

because the information about the

activity variation below the CMC is not available (it is generally accepted that one cannot use the Gibbs equation above the CMC because one does not know the activity).

If the

behaviour of SDS is typical at least of anionic surfactants then our response to the papers by Menger et al. and the related discussion is that the surface excess of an anionic surfactant at the CMC can not generally be reliably determined from the application of the Gibbs equation to ST data. Impurities, especially ionic ones, are a particular problem for anionic surfactants and already make the determination of

ΓCM C

a signicant challenge.

overcome there remains the awkward problem of the choice of function to t the

If this is

σ − lnc

curve, but in the end the EM results show that it is basically impossible to t any function in the region from about

0.75×CMC - CMC (for

tended to rely on to obtain

ΓCM C .

SDS ). This is the region that researchers have

Menger et al. are therefore justied in their misgivings

about the ST-Gibbs analysis.

46


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Page 48 of 59

For systems where there are no impurities NR measures the surface excess at any concentration with about a 5% accuracy.

The analysis is a simple surface measurement and

is largely independent of any complexities in the solution below.

Strong scattering from

micelles can cause additional background but this is only a problem at concentrations of about 20 mM upwards, and can be dealt with.

The limitation is that the surfactant must

be at least partially deuterated, contain uorine atoms, or have a higher proportion of oxygen atoms than most simple surfactants.

For systems with impurities we have shown that

NR will mostly give more accurate results that ST because each measurement at a given concentration is independent, it measures only the surfactant ion, and the saturation coverage of surfactant ion will usually only have a secondary dependence on the counterion. At concentrations well below the CMC the last condition does not hold and NR may become as unreliable as ST. The only occasions when ST meaurements can be expected to be more reliable than NR is when a higher surfactant purity can be achieved in the ST experiment. We believe that SDS is a rare example of such a case.

This is because in situ purication

is more or less essential for SDS purity because of hydrolysis in the solution. It is often argued that NR experiments are expensive in that they usually require deuterated materials. However, the success of the EM experiments shows that the real cost may lie in much better purication procedures. There are many surfactants, e. g. gemini surfactants, that undergo ion association and dimerization well below the CMC. The Gibbs equation cannot be applied to ST data on this type of system without independent quantitative measurements of the equilibria in the solution.

14

At present, there seem no way of doing this and therefore ST measurements

should not be used to determine the limiting surface excesses in this type of system. NR is at present left as the only available method. A by-product of the investigation here is that it draws attention to the remarkable results obtained by EM. Bearing in mind that their measurements on SDS are probably the most accurate and reliable measurements made on any surfactant, and that SDS is a reasonably

47


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representative surfactant, one would expect that the widths of the micellar transition are similar in surfactant systems of comparable surface activity and comparable micellar aggregation characteristics. The EM experiment suggests that to see such eects requires purity levels higher than are normally used for ST and other measurements as well as experiments of higher accuracy and resolution.

Acknowledgement The authors thank the neutron facilities ISIS and ILL for the neutron beam time.

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Figure 13: TOC Graphic

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Limitations in the Application of the Gibbs Equation to Anionic

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