Application of the Gibbs Equation to the Adsorption of Nonionic

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The Application of the Gibbs Equation to the Adsorption of Nonionic Surfactants and Polymers at the Air/Water Interface; Comparison with Surface Excesses Determined Directly using Neutron Reflectivity Peixun Li, Zhixin Li, Hsin-hui Shen, Robert K. Thomas, Jeffrey Penfold, and Jian Ren Lu Langmuir, Just Accepted Manuscript • DOI: 10.1021/la4018344 • Publication Date (Web): 12 Jun 2013 Downloaded from http://pubs.acs.org on June 19, 2013

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The Application of the Gibbs Equation to the Adsorption of Nonionic Surfactants and Polymers at the Air/Water Interface; Comparison with Surface Excesses Determined Directly using Neutron Reflectivity Pei Xun Li,† Zhi Xin Li,† Hsin-Hui Shen,†,§ Robert K.Thomas,∗,† Jeffrey Penfold,‡,k 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 Four recent papers by Menger et al. have questioned methods of analysis of surface tension (ST) data that use the Gibbs equation to obtain the surface excess (Γ) of a surfactant at the air-water interface. There have been two responses, which challenge the assertions of Menger et al. and a response from Menger et al. We use directly ∗ † ‡ ¶ § k

To whom correspondence should be addressed University of Oxford ISIS University of Manchester Department of Biochemistry and Molecular Biology, Monash University, Clayton, VIC 3168, Australia Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford, OX1 3QZ, UK.

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determined values of Γ from a range of neutron reectometry (NR) data to examine some of the issues that are relevant to these seven papers. We show that there is excellent agreement between NR measurements and careful ST analyses for a wide range of nonionic adsorbents, including surfactants and polymers. The reason it is possible to obtain good agreement near the critical micelle concentration (CMC) is that nonionic surfactants generally seem to saturate the surface before the CMC is reached and this makes it relatively easy to determine the limiting slope (and hence Γ) of the ST-log(concentration) plot at the CMC. Furthermore, there is also generally good agreement between ST and NR over the whole range of concentrations below the CMC until depletion eects become important. Depletion eects are shown to become important at higher concentrations than expected, which brings them into the range of many experiments, including techniques other than ST and NR. This is illustrated with new measurements on the biosurfactant, surfactin. The agreement between ST and NR outside the depletion range can be regarded as a mutual validation of the two methods, especially as it is demonstrated independently of any model adsorption isotherms. In the normal experimental situation NR is less vulnerable to depletion than ST and we show how NR and a single ST measurement can be used to determine the hitherto undetermined CMC of the nonionic surfactant C18 E12 , which is found to be 1.3 × 10−6 M.

Introduction This article is mainly a response to recent papers by Menger et al.

1234 5 ,

which question

the way that the Gibbs equation is used to analyse surface tension (ST) data to obtain the surface excess at the CMC,

ΓCM C .

It is also partly a response to two papers that

6 7

have sought to refute some of the arguments of Menger et al. ,

and partly an attempt to

assess when and how the Gibbs equation may safely be used to interpret ST data. The rst two papers by Menger et al. have two main topics.

The rst topic is the general issue of

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the behaviour of surfactants with respect to the Gibbs equation and the main point made by Menger et al. is that in the analysis of ST data by means of the Gibbs equation it is often assumed that the part of the

σ − ln c

plot just below the critical micelle concentration

(CMC) is linear. They argue that this would be physically unreasonable over a signicant range of concentration and is therefore not a good way to analyse such data. Hidden in this argument is a more subtle question about whether the saturation of a surfactant monolayer at the air/water interface is linked with the CMC and this is a question that has received little recent attention, partly because there is little reliable data on the surface excess above the CMC (there is an widely quoted older paper on this issue but it mainly deals with ionic

8

systems in the presence of added electrolyte at liquid/liquid interfaces ).

Menger at al.

attempt to resolve some of the issues by tting model isotherms and by comparison with the behaviour of insoluble monolayers. In the fourth paper of the series, Menger et al. t earlier radiotracer measurements of Nilsson

9

to a model isotherm of Lin

saturation of the layer is not reached at the CMC. Laven and de With

6

10

to show that

have refuted some

of the arguments used by Menger et al. but we believe that this refutation does not address the signicant number of genuine problems associated with the use of the Gibbs equation in the literature and which have mainly been exposed by neutron reectometry (NR). Also, Laven and de With quote the NR data very selectively and this is sometimes misleading, as for example when they state, on the basis of one of our papers, that in the case of nonionic surfactants, the lack in monodispersity of the lengths of the building blocks easily leads to apparent anomalies in surface excesses as derived from

γ (c)

curves. They made no attempt

to discuss the implied question of Menger et al. i. e. the relation between surface saturation and the CMC. Bermúdez-Salguero and Gracia-Fadrique argued on the basis of the Volmer isotherm that saturation does precede micellization, i. e. the ST plot does become linear just before the CMC. Mukherjee et al.

11

have also recently contributed to the discussion

by considering the question of saturation but their results are mainly on ionic surfactants, which we deal with separately.

The second topic in the rst two papers of Menger et al.

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concerns the apparent failure of the Gibbs equation to account for the surface tension of mixtures of dodecyltrimethylammonium bromide (C 12 TAB) with various sodium oligoarene sulfonates (NaPOSn) where they found discrepancies between the onset of the surface tension plateau in the

σ − lnc

plot and the CMC identied by conductivity. We have done our own

experiments on the same systems using a variety of techniques, including NR and neutron small angle scattering, to show that these systems can be accounted for perfectly well by the Gibbs equation and we have reported these results elsewhere.

12

Here we focus on the common failure of the combination of ST measurements and the Gibbs equation to match the limiting surface excesses more directly measured by NR. We have now explored a wide range of systems and observed a number of features that show that the ST-Gibbs combination may give misleading results. The issue of whether or not saturation is reached by the CMC is also a question that has not been well addressed from either a theoretical or experimental point of view.

Although a proportion of the data to

support the rst criticism and to enlighten the second issue has already been published by ourselves, we can now reexamine it in a more self-consistent way.

Discrepancies between

the two techniques are more marked for ionic surfactants but the agreement between ST and NR for nonionic surfactants and nonionic polymers is generally good over a wide concentration range except where there is depletion. This essentially validates both techniques for a wide range of nonionic systems, including polydisperse ones. The agreement at lower concentrations also makes it possible to use the two techniques in parallel to assess when and over what concentration range depletion problems occur. Depletion eects have been neglected in the literature, although our results show that they generally occur at higher concentrations than one might expect and is therefore a problem that should not be neglected in any experimental measurement, not just ST and NR. In this paper we conne ourselves to nonionic surfactants and polymers and we mostly revisit old data with a view to (i) addressing the issues raised in the seven papers identied above, (ii) validating both the ST-Gibbs and NR methods for nonionic surfactants, and (iii) demonstrating the importance

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of depletion, especially for higher molecular weight materials. We also introduce two new results to illustrate the importance of depletion at concentrations as high as

5 × 10−6 M. Ionic

surfactants present additional problems which make it dicult to use ionic data to validate the two techniques and we will therefore treat these separately in a second paper using the validation of ST and NR established here as a solid base. While model isotherms may be useful ways of explaining the behaviour of dierent systems, we believe that the use of any technique needs to be experimentally based as far as possible and so we avoid using model isotherms. There are two underlying problems in the determination of surface coverage from ST measurements. The rst is that the actual measurement of ST is often less reliable than many authors assume. It is sensitive to the technique used because of factors to do with wetting, dynamics or viscoelastic properties at the surface (see e. g. Lunkenheimer

13

). The second is

that the application of the Gibbs equation to the data requires a very precise knowledge of what is present in the bulk solution. This arises because the amount of material required to coat the surface with a monolayer is minute and hence traces of non-polar impurities and ion impurities can dominate what happens at the surface.

In addition, the state of

aggregation may also aect the activity of the adsorbing species in the bulk solution with consequences for the way the Gibbs equation is applied. Because of these limitations, the data is sometimes recognized not to be of high accuracy and it is then appealing to use a relatively simple analysis, i. e. to t a linear equation to the

σ −ln c plot to obtain the surface

excess at the CMC. This also implies a saturated surface coverage at the CMC. Although these comments put ST measurements in a negative light, the surface tension is itself an important thermodynamic characterization of an aqueous surface. NR has proved to be the only widely used experimental technique for the direct determination of surface excess at the air/water interface.

It has been applied to most of the

common surfactant systems, it is not restricted in concentration range (measurements are possible above the CMC), and it is able to analyse single components in mixtures. The oldest

5


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direct method is to measure surface pressure-area ( π

− A)

isotherms. This is particularly

the case for nonionic systems, which often have low solubilities.

At one time radiotracer

measurements were thought to be a promising means of determining surface excesses but the signicant experimental diculties and the large errors in the method have restricted its range of application. Optical techniques, especially ellipsometry, are more sensitive to the surface than NR but they still seem to more model dependent than NR and the most eective use of e. g. ellipsometry requires prior calibration with NR.

14

Other spectroscopic techniques

tend to be even more model dependent or require the introduction of chromophores, which changes the surface chemistry. We include a comparison of these direct measurements for some nonionic systems. Bain

14

has reviewed most of the optical techniques and Lu et al.

have compared the relative merits of the range of alternatives to NR.

15

The Gibbs Equation In its simplest form the Gibbs equation is written

−dσ = RT

X

Γi d ln ai

(1)

i where

σ

is the surface tension, and

Γi

and

ai

are the surface excesses and activities of all the

independent components. For measurements on a nonionic surfactant solution this generally reduces to

−dσ = RT Γd ln a There are some considerations concerning the denition of portant in the context of surfactant systems.

(2)

Γ

but these are generally unim-

For nonionic surfactants the usual practice

is to use the approximation of concentrations rather than activities except at or above the CMC. The slope of the plot of

σ

against

lnc

gives the corresponding surface excess

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Γ

at that

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point using equation 2. In the particular case of the limiting coverage at the CMC,

ΓCM C ,

it is not easy to determine accurately the slope of a curved experimental plot at the point where it terminates. Unlike ionic surfactants, where the ion concentrations above the CMC vary because of dissociation of the micelle and hence cause the activity to vary, the activity of nonionic surfactants is expected to be fairly constant as the concentration is varied above the CMC. Since adsoprtion varies with activity this means that adsorption will also be invariant with concentration. Thus adsorption reaches a limit at the CMC, which is not necessarily the saturation limit.

True saturation implies that the surface will be covered to its limit

just before the CMC, but there is no particular reason why this should be the case. The determination of the limiting slope at the CMC uses one of two methods. a model isotherm to generate a

σ -lnc

One is to use

plot which is then tted to the experimental points.

The other is to t a polynomial to a section of the experimental

σ -lnc

plot and extract the

lna

above the CMC and

slope from the polynomial. It is dicult to determine the value of

use of the Gibbs equation is normally restricted to below the CMC. Since the slope of the curve below the CMC is at its steepest at the CMC it is tempting to assume that the surface is saturated at the CMC and then tting of a model isotherm is an appealing means for analysing the data. The further assumption is then often made that saturation is reached somewhat before the CMC, in which case the last part of the

σ -lnc plot would be linear and

a simple straight line would t the data. Given the typical errors and typical spacing of the concentrations used for the measurements, a linear plot may appear to give a satisfactory t. The principal objection to this procedure is that there are no thermodynamic reasons that the surface should either be saturated at or before the same activity (concentration) as the onset of micellization. Both processes are driven by the hydrophobic eect but the packing requirements are dierent. One could argue, for example, that surfactants with large head groups would prefer to form micelles and therefore surface saturation might not be reached at the CMC, whereas surfactants with small head groups, i. e. that prefer to form lamellar structures, would form saturated monolayers below the CMC. Whether or not saturation

7


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is reached before the CMC may then vary according to the surfactant. If saturation of the monolayer is not complete at the CMC not only may the slope of the

σ -lnc

plot not have

reached its linear region but it may also be more or less strongly curved as it approaches the CMC. This creates considerable obstacles to an accurate analysis of the data. Fitting a polynomial to a curve which is cut o just where knowledge of the curvature is most needed has an element of arbitrariness and the alternative of tting the integrated Gibbs equation for a model adsorption isotherm is also hampered by not knowing the value of the saturation coverage. As we will show below the solution of this problem is straightforward for nonionic surfactants because they do seem generally to reach a plateau before the CMC but it is not straightforward for ionic surfactants where adsorption at the CMC seems to be signicantly below the saturation limit. In the examples below we examine the extent to which the surface is saturated at the CMC in some detail. However, an important problem in applying the Gibbs equation is the eect of impurities. In equation 1 the

d ln ai

term does not depend on the absolute value of

the concentration and therefore for an impurity present in the original surfactant, the Gibbs equation has the same concentration dependence as the original surfactant, i. e.

−dσ = RT [Γs + Γimp ] dlncs

(3)

The relatively low concentration of the impurity aects the surface tension only through the eect of this concentration on the adsorption. However, for a highly surface active impurity the adsorption may be sucient that the

dlncs

term, which is identical for impurity and

surfactant, makes a large contribution to the overall

σ − lncs

curve.

Neutron Reectometry In NR a collimated beam of neutrons is specularly reected at a glancing angle from the air/water interface. The reectivity is determined by the neutron refractive indices of the

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various parts of the system. The neutron refractive index is determined by the scattering length density

ρ,

which is given by

ρ=

X

bi n i

(4)

i where

bi

are the known scattering lengths of the various nuclear species

i

and

ni

is the

appropriate number density. Because of thermal motion of the molecules in the adsorbed layer the distribution of surfactant along the surface normal ( z -direction) is most closely described as a Gaussian

ρ = ρ0 exp(− where

σ is the full width at 1/e of the maximum.

4z 2 ) σ2

(5)

In the situation where the scattering length

densities of water and air are matched the reected signal is exclusively from the surfactant layer, apart from some background scattering. It can then be shown

15

that

Na2 Γ2s exp(−κ2 σ 2 ) κ R' 4 2

where

Γs

is the surface coverage,

Na

is the Avogadro number, and

(6)

κ

is the momentum

transfer, dened by

κ= where of

λ

lnκ2 R

4π sin θ λ

is the wavelength of the neutrons and against

κ2

θ

(7)

is the grazing angle of incidence. A plot

normally gives a straight line and the coverage can be deduced directly

from the intercept. Even though the Gaussian distribution may not be a completely accurate description of the layer all plausible distributions lead to the same limiting result. If the surfactant is deuterated the reected signal is large and extraction of the coverage requires only a knowledge of the chemical formula of the surfactant. The determination of the coverage is therefore robust and model independent. Errors arise from background scattering from the solution but this is typically less than 10

−2

of the signal from a saturated deuterated

surfactant layer. Thus measurement of surface coverage down to 1/10th of a monolayer or

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better is practicable. The only theoretical weakness in any analysis is that it is generally assumed that the layer is laterally uniform on the length scale of the experiment, i. e. NR sees an average refractive index of the layers rather than the average reected signal from parts of the surface that have dierent coverages. However, this can be tested and there has not yet been any evidence that this is not the case for a soluble surfactant layer. It should be emphasized that the basic features of the sensitivity of the method can also be established by experiments on insoluble monolayers, which are free from all of the complexities of soluble surfactant systems. With current instrumentation the coverage of a typical surfactant layer can be measured accurately in about 5 mins. NR is sensitive to all of the impurity, depletion and polydispersity problems outlined above for ST. However, the most important dierences are rst that it measures adsorption entirely independently of what is present in the bulk solution, whereas ST depends on both adsorption and on changes in the bulk activity. Secondly, each measurement at a given bulk concentration is independent of every other measurement, whereas ST requires measurements at at least two concentrations. Thirdly, NR measures everything that is at the surface. Thus if the impurity stems from the initial preparation of the deuterated surfactant, as is usually the case, NR will measure the total amount of both at the surface.

At rst sight this

might seem to be a disadvantage. However, by using H/D isotopic labelling it is possible to devise experiments where the composition of each component or suspected component can be followed individually. For the same reason NR can be used to follow the adsorption of each component in a binary or higher mixture. A number of experimental results indicate that the adsorption itself is less sensitive to impurity than is the tension. Thus, impurities have a relatively small eect on NR but may have a large eect on the ST. A useful test for self consistency in the NR measurement is to measure the coverage rst with just the deuterated material and then with a 50:50 mix of deuterated and protonated material.

Consistency

between these shows that the two dierent samples, if impure, must have identical quantities of the same impurity. One of the diculties with an ST-Gibbs analysis is that there is no

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easy self-consistency test.

Experimental Details We describe only the preparation of materials for which the measurements are new. Partially deuterated C 18 D37 (OC2 H4 )12 OH (C18 E12 ) was prepared in two steps. Each step has been described previously.

16

First perdeuterated bromo-octadecane was reacted with monodisperse

hexa-ethyleneglycol using potassium t-butoxide as a catalyst to give C 18 E6 . This was puried by column chromatography (silica with diethyl ether/acetone as eluent) to ensure its monodispersity. This was then converted to the tosylate and reacted with further monodisperse hexaethyleneglycol. The nal product was puried by column chromatography. The column can separate C 18 E6 from e g C18 E7 but it cannot separate C 18 E12 from e g C18 E13 . However, the second tosylate reaction uses milder conditions than the rst so the breakdown of the monodisperse hexaethyleneglycol in the reaction is slight and the nal product is narrowly polydisperse. The preparation of surfactin and three dierently deuterated surfactin samples has been described by Shen et al.

17

and the NR results have already been given in

full. However, the ST results on these samples have not previously been published. Surface tension measurements were made with a Kruss K10 using a Pt/Ir ring amed in between measurements. NR measurements were made on the SURF instrument at ISIS, UK. These methods have been fully described previously (see for example Shen et al.

17

).

Results

Adsorbed Amount For a nonionic surfactant there is generally good agreement between surface tension and NR data, partly because the prefactor in the Gibbs equation cannot be anything other than 1 (except if there is self-association) and partly because more advanced methods of

11


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purication can usually be used on nonionic surfactants (e. g. column chromatography). The main examples are the C 12 Em series (C12 H25 (OC2 H4 )m )OH) and the results for the area per molecule at the CMC,

ACM C , determined from NR and surface tension (ST) for members

of this series up to C 12 E12 are given in table 1. Although we determined our own values of the surface excess at the CMC by tting a polynomial to the

σ − lnc

plot and calculating

the slope of the tangent at the CMC, we show values determined by others using the linear approximation to the points just below the CMC

18 19 ,

except for C12 E12 , which are our own

measurements. The estimation of the error in the value from the surface tension is not easy. This is discussed in detail by Lu et al.

20

and is estimated to be about 10% at the CMC,

whereas the corresponding error in the NR experiment is about 5%. The agreement for the whole series between all both types of measurement is well within these errors. An important feature of the NR results is that in several cases the value of

ACM C

was determined using

more than one isotope, i. e. more than one independent measurement on a dierent sample. Apart from demonstrating the high reproducibility of NR this made it possible to identify occasional problems with purity, which could then be solved by further purication (see the discussion about C 12 E8 in Lu et al.

16

). It also further reduces the error quoted in table 1.

The data in table 1 show that, contrary to the statement of Laven and de With,

6

there are no

anomalies in the determination of the surface excess arising from the lack in monodispersity in the case of nonionic surfactants. The surfactants up to and including C 12 E6 were in any case monodisperse, which is reasonably easily achieved using column chromatography, and the two higher members of the series were generally more monodisperse than the commercial materials either because they were prepared using the two stage preparation described in the experimental section, or because, in the case of C 12 E8 , they can be puried to a higher degree than the commercial materials on a column. The reason for showing the fth column of

ACM C/2 values in table 1 is because the authors

used the type of linear plot criticized by Menger et al. over the range from about CMC/2 to CMC. The neutron results show that the coverage at CMC/2 is on average about 10%

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Figure 1: (a) variation of the coverage of C 12 E3 with concentration as determined by neutron reection and surface tension (using a polynomial t to the σ − lnc curve), (b) the π − A isotherm for C 12 E3 using the measured surface tension and the area per molecule determined by neutron reection. The CMC is marked in (a) with a vertical line. The data are taken from Lu et al.

23

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Figure 2: Variation of the coverage of (a) C 12 E6 and (b) C12 E3 with log(concentration) as determined by neutron reection (solid circles) and surface tension (open circles) (using a polynomial t to the

σ − lnc

curve). The data are taken from Lu et al.

used for ease of reference.

14


24 23 .

A log10 scale is

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Table 1: Adsorbed areas per molecule,

ACM C ,

Page 16 of 37

of nonionic surfactants, C 12 En , at their CMC

using neutron reectometry and surface tension and at CMC/2 using NR.

21

The data in the

fourth column were determined by tting a straight line to the points just below the CMC, a 18 b 19 c 22 Rosen et al., Corkill et al., Lu et al. surfactant

CMC mM

C12 E2 C12 E3

0.033 0.055

AN R ± 5% AST Å

2

Å

35

b

36

42

b

40 50 50

Å

33 36

AN R (CMC/2)

2

2

C12 E4

0.069

44

46

b

C12 E5

0.064

50

50

b

57

C12 E6

0.08

55

52

c

C12 E8

0.10

62

66

b

68

C12 E12

0.12

72

78

c

85

lower than at CMC, which is suciently small as to suggest that these nonionics are close to surface saturation before the CMC. It also means that the linear method would be expected to underestimate the coverage by about half the dierence between the two values, i. e. 5%, and something of this trend can be seen in the table. Overall the agreement is good and since nonionics show the smallest variation in amount adsorbed as the CMC is approached, the small extra eort of using a non-linear t would have led to even better agreement, as was done in some of the NR papers. Menger et al. 's criticism of such linear plots is therefore to some extent justied. Although the slope of the

σ − lnc

plot is not changing very rapidly

in the range from CMC to CMC/2, this variation can easily be allowed for and more or less any non-linear functional form will be suciently accurate. The results in table 1 only relate to the coverage at the CMC. Figure 1(a) shows a comparison of the results over the whole range of the ST curve for C 12 E3 .

The neutron

measurements have been extended to above the CMC in this case. The agreement between the surface coverage determined from the Gibbs equation and a polynomial t to the surface tension is better than might be expected from the experimental errors. Figure 1(b) plots the

π−A

isotherm. The surface pressure rises increasingly steeply as the layer is compressed.

There is, of course, no collapse. Instead, the surfactant dissolves when the pressure exceeds a limiting value, emphasizing the connection between surface tension and activity of the

15


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surfactant in solution.

For C 12 E3 the coverage remains constant above the CMC within

experimental error, i. e. there is a small vertical addition to the top of the

π−A

isotherm.

It is dicult to assess the comparison of the NR and ST results at low concentrations using gure 1. Figure 2 therefore shows results for C 12 E6 and C12 E3 on a log concentration scale, from which it can be seen that the ST −Gibbs and NR methods agree well for these nonionic surfactants for concentrations well below the CMC.

Figure 3: Comparison of

σ − logc plots (lines) calculated from the integrated Gibbs equation

using the surface tension at the CMC and the measured neutron surface excesses for dodecyl-

β -D-maltoside (C12 maltoside) (o) and dodecyl-N,N-dimethylamino betaine (C 12 betaine) (x). The data are taken from Hines et al.

25

An alternative test of the ST data for the air/solution properties of a surfactant is to use the NR data to obtain a functional form for the adsorbed amount, i e to measure

Γ(c) over a

range of concentration up to a particular concentration, usually the CMC, and to integrate the Gibbs equation as below

Z

σ

Z

c

dσ = −RT σ0

c0

Γ(c) dc c

16


(8)

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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

where

c0

is the reference ST point.

Page 18 of 37

The agreement of such a calculated curve with the

experimental ST curve demonstrates consistency between ST and NR. Figure 3 shows the application of equation 8 to two uncharged surfactants, i. e. ones for which the Gibbs prefactor is 1, the nonionic surfactant, dodecyl

β

-D-maltoside, and a net neutral zwitterionic

surfactant, dodecyl-N,N-dimethylamino betaine (C 12 betaine). Thus, using only one value of the surface tension at a point where it can be reliably made (here just at the CMC) and the measured surface excess at 6-8 concentration points gives good agreement with the overall measured ST curve except possibly at very low concentrations. The diagram shows clearly how the slopes of both curves change quite gently with concentration. There is, however, a subtle assumption in this test which may mean that good agreement is obtained here but not in the simpler comparison of values of the surface excess. The value of the CMC is not very accurately dened in an ST measurement and, in tting the integrated Gibbs equation the value of the CMC is eectively an adjustable parameter that can signicantly improve the appearance of the t.

Depletion and Polydispersity The examples above indicate that ST measurements and relatively simple analysis using the Gibbs equation give reliably accurate values of the adsorbed amount for a range of nonionic surfactants. However, the situation is not as simple for aqueous polymer solutions and for polymeric surfactants, and the signicant experimental diculties have sometimes been exacerbated by misapplication of the Gibbs equation. When the molecular weight of the adsorbing substance is large it has the eect that the molar adsorption at saturation is low, which leads to a shallow slope of the

σ − lnc

curve.

For example, poly(ethylene oxide) (PEO) adsorbs at the air/water interface with an area per segment of the order of 10 Å

2

so that a PEO of molecular weight 6000 already has an

2 26 area per molecule of about 2400 Å . This is a reduction of a factor of about 50 in the slope of the curves shown in gure 3. Moreover, the shallow slope of the curve also means that

17


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the ST should persist at a low value down to concentrations low enough for depletion to start to cause problems. Thus, at molar concentrations below about

10−7

M, adsorption at

the air/water surface or on other surfaces of the container may start to remove a signicant fraction of material and to deplete the bulk concentration. This has the eect that the actual bulk concentration drops more rapidly than the nominal/initial concentration. If this is not properly allowed for the concentration decreases.

σ − lnc

curve will curve upwards more rapidly as the nominal

This can give the impression of a discontinuity in the curve.

In

principle, depletion will be well dened for a particular experimental conguration and can be a genuine equilibrium phenomenon. However, diusion rates at the low concentrations involved will introduce kinetic eects and the whole depletion problem may become a complicated non-equilibrium problem for polydisperse samples. Some attention has been given to these problems

27 28 ,

although eects of this kind are observed in all aqueous polymer

solutions and all experimental techniques are susceptible.

Figure 4: Comparison of the experimental surface tension behaviour of two dierent molecular weight PEO (points) and the corresponding integrated Gibbs equations using directly determined surface excesses from neutron reection measurements over the range covered by the continuous lines. Data taken from Lu et al.

29

A log10 scale is used for ease of reference.

18


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Page 20 of 37

The rst example for which there are NR measurements is PEO is shown in gure 4. All previous ST measurements showed marked kinetic eects at low concentrations Gilanyi et al. have explored them in some detail.

28

26 30 31

and

The eects of depletion and possible

eects of polydispersity were not discussed in the original NR paper and here the results in gure 4 have been replotted on a molarity scale to emphasize the concentrations at which depletion occurs (the original the scale).

29

were in terms of wt % concentrations and had an error in

The surface excess was measured by NR for two molecular weights over the

concentration range 0.1 to 10

−4

wt% and this was then used in the integrated Gibbs equation

8 with the xed value of the ST being chosen to optimize the t to the observed data points.

The consistency of ST-Gibbs and NR is again excellent.

The segmental surface

excess is similar for the two molecular weights and the dierence in slope is therefore inversely proportional to the molecular weights as can be seen in the gure. The coverage was found to decrease slightly with concentration so the curve is slightly concave towards the concentration axis.

Depletion in the neutron experiment is expected to occur at a signicantly lower

concentration than for the ST measurement. This is because the volume/area ratio of the neutron troughs is much larger than those for ST and also because the use of teon troughs for NR should cause much less secondary adsorption. The calculated

σ − lnc curve can then

be extended to concentrations well below the point where the observed ST measurements deviate sharply upwards away from the predicted curve. The only possible explanation of this is depletion and the molar range is consistent with simple calculations based on surface area/volume ratios of the vessels used for the ST. However, the points at which the two molecular weight polymers diverge from the calculated curve are quite dierent and it is probable that polydispersity and kinetic eects are also playing a role, even though the ST measurements were found to be moderately reproducible. However, polydispersity has no adverse eect on the application of the Gibbs equation.

Laven and de With did not

explain why polydispersity would dominate the analysis of the ST data and it is dicult, in the context of these PEO measurements, to see why it might.

19


However, it should be

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noted that polydispersity covers a range of possibilities. For the alkyl ethoxylate surfactants and the polymer examples here, polydispersity mainly changes the area per molecule but does not signicantly aect the surface activity.

The situation might be quite dierent if

polydispersity caused a strong variation in surface activity. In the following two examples the onset of depletion is much higher in concentration than found for PEO, in the region of

10−6

M. In both cases the combination of this high

concentration and the reasonably reproducible nature of the onset of the sharp increase of ST with decreasing concentration give the ST curve the appearance of being associated with some sort of transition. In both examples researchers have interpreted the

σ − lnc

graph in

just this way and applied the Gibbs equation to the lower concentration branch of the curve. The rst example is a triblock copolymer of EO with poly(propylene oxide) (PO)

33 34 35 32 .

These materials are often referred to as poloxamers or in one of their commercial form as Pluronics. The upturn occurs in the region of

5 × 10−6

M and, although micellization in the

poloxamers generally occurs at the much higher concentration of around

5×10−4 M, the onset

of the upturn has been variously and incorrectly interpreted as unimer micellization, onset of a broad range of micellization associated with polydispersity, the surface structure. region below

35

5 × 10−6

Alexandridis et al.

35

34

33

the

or rearrangement of

applied the Gibbs equation to the steeply sloping

M to obtain values for the coverage of a series of Pluronics. Although

the onset of the upturn is rather higher than might be expected for depletion eects, it has been possible to use NR to conrm that the prediction of depletion by Linse and Hatton

36

for

these systems is part of the explanation. The NR results also show that the high coverages deduced from the slopes below the upturn by Alexandridis et al. are incorrect.

Figure

5(a) shows the NR surface excesses for the partially deuterated dE 23 P52 dE23 (the closest commercially available Pluronic to this is P105) and gure 5(b) shows the ST curve for two dierent types of measurement and for the calculation using the NR data and equation 8. The onset of the upturn when the ST is measured using the ring method is at about 10 M, which is unexpectedly high for simple depletion. However, this moves to about

20


32 −6

5 × 10−7

Langmuir

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Figure 5:

(a) Surface excesses measured by neutron reectometry for the poloxamer,

dE23 P52 dE23 , (b) surface tensions measured by the ring method (closed circles), the plate method (open circles) and calculated using the data from (a) and equation 8 (continuous line). Data taken from Vieira et al.

32

A log10 scale is used for ease of reference in (b).

21


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when a plate is used to measure the tension. This is a strong indication that kinetic factors are involved, but the general reproducibility of the upturn by dierent authors suggests that polydispersity acts to smooth out the kinetic factors and that these are themselves moderately reproducible for each ST technique. The surface excess was measured by NR to well below the onset of the upturn in both measured curves and the onset of depletion in the NR measurement does show up in the calculated curve, but at the lower value of 10

−7

M.

Dierences in depletion eects using dierent methods of ST measurement have also been observed by Petkov et al.

37

who obtained very large dierences between the pendant drop

and the bubble pressure method. As expected the depletion eects were much larger for the pendant drop, which has a very small volume/area ratio. The second example where depletion eects occur at surprisingly high concentrations is the biosurfactant, surfactin. Surfactin is a lipoheptapeptide in which a carboxy group at the end of an alkyl chain closes the hepapeptide chain into a ring via a lactone group.

38

The

heptapeptide ring contains two acid residues, glutamic and aspartic acid. Although this is technically an electrolyte, the eect of pH on the surface tension shows that it is not ionized until about pH 7.

In addition the experiments were done at constant pH which ensures

that the Gibbs prefactor has the same value of 1 as for a nonionic. The additional reason for including it here is to bring together in one place all the examples where depletion is a clearly identiable eect. Figure 6 shows that the ST curve, taken at pH 7.5, can be divided into three branches, A, B and C. The NR data in gure 6(b) are used in the integrated Gibbs equation with the value of the ST from the branch C-branch B intersection and a Gibbs prefactor of 1 to calculate branch B. The agreement is good and demonstrates that the CMC must be at the higher concentration of the two intersection points. Branch A has a much steeper slope and would correspond to a surface excess about 3 times greater than on branch B. However, although analysed in this way by Maget-Dana and Ptak

39

to obtain

2 an area per molecule of 55 Å , such a result is physically unlikely and the explanation must be that it is associated with depletion. Direct NR measurements at the air/water interface

22


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Page 24 of 37

Figure 6: (a) Observed surface tension behaviour for surfactin at a buered pH of 7.5 (points), calculated from the NR data and the integrated Gibbs equation (region B) and by tting a straight line to the variation of coverage (NR) with of the CMC of

1e − 5

lnc

(region B). This plot gives a value

M and region A is clearly a depletion eect. The NR data and t are

shown in (b).

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Langmuir

give a much larger limiting area per molecule of

147 ± 5 Å2

at the hydrophobic solid/aqueous interface gives

145 ± 5

and a separate NR measurement

2 17 Å . The only alternative could

be that surfactin, like many proteins and peptides, forms a complex, e. g. a trimer, at the intersection between A and B. If there is no signicant change of area per molecule, i. e. the trimer occupies three times the area that a monomer occupies, the slope would decrease to a third on crossing from A to B, as observed. However, this explanation would require that

2 the area per molecule in regions A and B would be approximately 55 Å , which is neither physically reasonable not in accord with NR, and hence depletion seems to be the only explanation. The NR measurements were performed on three dierent isotopes of surfactin for which the carbon sources were dierent materials. within the

147 ± 5 Å2

All three gave areas per molecule

quoted above, again demonstrating the high reproducibility of the NR

experiment. The upturn in the ST at just below

10−6

observed for surfactin is at a much

higher concentration than would be expected and indicates that for strongly surface active substances the losses of solute to the surfaces of the various vessels used to handle them can accumulate to make a substantial dierence. In contrast, PEO is not very surface active and the depletion eect occurs in a range that would be calculated in a simple estimate. The eect of polydispersity on the surface tension of polymers has hardly been considered

40

but An et al. devised a model to show the general eects

27

using another homopoly-

mer, poly(vinyl methyl ether), for which there is again agreement with the integrated Gibbs equation using NR data down to an upturn in the tension. In polymers the segmental adsorption is often approximately independent of molecular weight and this has the result that the smaller components will have steeper

σ − lnc

curves, as shown in gure 4. At the low

concentrations where depletion eects occur, diusion will be an important factor determining adsorption and smaller components in a polydisperse mixture will be more rapidly adsorbed. Under these circumstances, the surface behaviour will be mainly determined by the small components and the

σ − lnc

curve will be relatively steep. However, adsorption is

generally stronger for larger molecular weights and as the surface behaviour develops towards

24


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Page 26 of 37

something more representative of the average it will reect that of the larger components of the mixture and the

σ − lnc curve will then become less steep.

the curve will be convex with respect to the

lnc

The overall eect will be that

axis and the convexity may even be sharp

enough to give the impression of a break in the curve. The exact pattern of the behaviour will depend in a complex way on the characteristics of the polydispersity and the way it impacts on the depletion of its constituent fractions and, in that there is a kinetic element in this behaviour (the greater rate of diusion of the smaller species), the behaviour will also be closely linked with depletion of the individual components. An et al. showed calculations for such a model. Although any model is bound to be highly articial the real point is that where equilibrium can be established with certainty and where the concentration is not low enough to be vulnerable to depletion the Gibbs equation works well. It would, of course, be possible to devise particular instances of polydispersity where the simple application of the Gibbs equation would not work but, contrary to the comment by Laven and de With,

6

it

appears to work well in the normal situation of polydispersity, i. e. for polymers.

Determination of CMCs in the Depletion Region The eects of depletion are easily noticed for higher molecular weights because the molar coverage is low and the

σ −lnc plot correspondingly shallow, in contrast to the sharp increase

in the depletion region. The eect may be less easy to detect for non-polymeric surfactants but these cases can be identied by the combination of NR and ST, as has been shown above. More interestingly the combination can be used to determine the CMC in a range below that accessible by surface tension alone. We illustrate the method by application to the surfactant C 18 E12 , whose CMC has not been previously determined by any means. Surface tension measurements indicate a CMC at about

2 × 10−5

method) show signs of depletion eects (instability) by

M but the measurements (ring

1.5 × 10−5

determined by NR, using C 18 D37 (OC2 H4 )12 OH in the range

M. Surface coverages were

5 × 10−7

to

2 × 10−6

M and the

results are plotted in gure 6(b). Although these were made at about an order of magnitude

25


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Figure 7: (a) The determination of the CMC of C 18 D37 (OC2 H4 )12 OH (C18 E12 ) using a single measurement of the ST above the CMC and the integrated Gibbs equation (line) compared with measurements (points). (b) The surface excess measured by NR. The lower three points were tted with a straight line, which was then used for the clculation in (a). A horizontal line is drawn through the highest coverage point. The intersection of this with the sloping line gives a value of the CMC in agreement with that from (a).

26


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Page 28 of 37

lower concentration they give coverages that indicate that they are associated with the main part of the

1 × 10−6

σ − lnc

plot that includes the CMC and that adsorption is complete by about

2 M. Interestingly, the area per molecule at the upper concentration limit is 79 Å ,

within error ( ±5%) the same as the value of 72 Å

2

for C12 E12 at its CMC. In the absence

of any ST data below the CMC the only form of the integrated Gibbs equation that can be applied is to integrate from the pure water end of the curve. This was done by tting the coverage to the lowest three measured points as a linear function of

Γ=0

lnc.

From the limit of

and the ST of water the linear function was then integrated to give the curve shown

in gure 7(a).

With the reasonable assumption that the ST is constant above the CMC,

as observed for the related surfactants, the CMC is then determined to be

1.3 × 10−6

M.

This must be regarded as a maximum value because it is possible that the neutron results themselves are aected by depletion. However, the failure of the ring ST measurements is clear, as shown in gure 7(a). The disparity between the two curves is similar to those shown in gure 6. One important dierence here is that the ST becomes essentially unmeasurable for C18 E12 and this may be because the depletion eects are not masked by polydispersity. Note that depletion in cases such as this may be very dependent on the whole pattern of dilution.

Other Direct Methods Other methods can be used to determine the surface excesses of nonionic species. The simplest and seemingly most direct is from pressure-area curves for spread insoluble monolayers and this has been done for both PEO, where there is agreement of the measured coverage for the spread monolayer with a maximum applied surface pressure with that of the solution at saturated equilibrium, although NR indicates that the two structures are not the same,

29

and for surfactin, where the agreement is less good (see above). However, spread monolayer lms suer from the problem of leakage to the subphase, which cannot be assessed very eas-

27


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Langmuir

ily. There are two direct measurements in the literature that use radiotracer measurements. In the rst of these Tajima

41

used radiotracers to follow the adsorption of hexaethylene

glycol monododecyl ether, C 12 E6 . The radiotracer results showed that C 12 E6 adsorbed at a constant level from CMC/5 to 10 ×CMC with an area per molecule of 68 Å

2

(no error

quoted). This is signicantly dierent from the NR results for which the value above the CMC is

55±5 Å2

increasing to

68±5 Å2

at CMC/5. Although there seem to be no estimates

of the errors in radiotracer measurements, comparison of the spreads in radiotracer results for SDS, which has been studied by four authors least

9 42 43 44 ,

indicate that this error may be at

±20%, in which case the NR, radiotracer and ST measurements just overlap.

However,

Tajima's comments on the strong time dependence of his own surface tension measurements indicate that the surfactant may not have been completely pure. Cross and Jayson radiotracers to investigate tritiated C 12 E4 from well below the CMC to about

44

used

2×CMC.

As

with the NR results above the pattern of adsorption suggested that saturation was more or less complete at the CMC. However, the total adsorption at the CMC was much higher than found by NR and ST (see table 1) 4.9 compared with 3.75 (NR) and

3.6 µmol

m

−2

(ST).

Cross and Jayson did not purify their C 12 E4 by chromatography and in the reaction between bromododecane and tetraethylene glycol the conditions are severe enough that some degradation of the glycol occurs. of C12 E2 and C12 E3 (see table 1).

Their value of

4.9 µmol

m

−2

is more typical of a mix

The other method that has been used is ellipsometry.

Ellipsometry has a distinct advantage over NR in that its sensitivity is high, making it suitable for short time and/or small area measurements. The obvious advantage of the short time is that it is possible to study systems under dynamic conditions with a much shorter timescale than with NR. However, in the context of the present discussion, the important and subtle advantage for measuring equilibrium coverages with ellipsometry would be that pure surfaces at concentrations approaching the CMC are best generated as new surfaces in

which the lower concentration impurities have not yet had time to diuse to the surface.

45

For related reasons and as pointed out by Valkowska et al. ellipsometry in conjunction with

28


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Page 30 of 37

steady state adsorption is also potentially useful for exploring concentrations well below the CMC.

46

Valdowska et al. have compared NR and ellipsometric measurements on C 8 E4 OMe

and shown that ellipticity is a linear function of coverage, i. e. a single NR measurement is sucient to validate the set of ellipsometric measurements. However, their results and those of Binks et al.

47

for C12 E5 were not able to establish the absolute accuracy of ellipsometry,

which is what is needed to make ellipsometry a stand-alone technique. Both sets of authors agree that a uniform distribution of the surfactant components in the layer does not account correctly for the ellipticity. The data of Binks et al. show that a better t is obtained for C12 E5 by splitting the surfactant layer into an isotropic hydrocarbon tail layer containing no water and an isotropic ethylene glycol layer containing water, but this still underestimates the limiting coverage obtained by NR by about 10% at the CMC and above.

Valdowska

et al. concluded that the division used by Binks et al. with the limiting coverage having a 100% pure hydrocarbon layer for the outer layer does not work for C 8 E4 OMe. On the other hand the introduction of a small reduction in the density and some anisotropy is sucient to account for the discrepancy. Thus, while it may be possible with further work to establish a model that will enable ellipsometry to be used without the assistance of NR, it remains only an empirical relationship that has to be calibrated by other methods. The strength of the NR experiment is that it gives an accurate absolute measurement of coverage at the CMC. However, once ellipsometry has been calibrated by such a measurement it would seem to have considerable advantages for exploring adsorption at low concentrations, where NR becomes highly vulnerable to impurities because of the relatively long timescale of the experiment.

Conclusions We have shown that for a series of nonionic surfactants, C 12 Em , for a wide range of values of

m,

ST and NR agree at all concentrations up to and including the CMC. This agreement

is also shown for another simple nonionic, C 12 maltoside and for a net neutral zwitterionic,

29


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C12 betaine, and for a nonionic polymer, a biosurfactant and one member of the widely used poloxamer series. Agreement is also found for other members of the wider C n Em series and another member of the poloxamer series, although these results are not shown here. In all these cases adsorption shows either a distinct plateau or something very close to a plateau just below or at the CMC, although it does not always extend far enough below the CMC to justify the use of a linear t to the

σ − lnc

plot. Methods of purication are such that

many simple nonionic surfactants can be made monodisperse but polydispersity does not seem to have a negative impact on the application of the Gibbs equation. A sharp upturn in the ST at low concentrations indicates that depletion eects are often present and the combination of ST and NR reveals this in a particularly clear way. These eects occur at higher concentrations than might be expected (up to 10

−5

M) and, since they can be present

for any type of physical measurement, the results here serve as a guide for minimizing their eects. From the point of view of a response to the seven papers referred to in the opening paragraphs we note the following: (i) ST-Gibbs and NR results agree for a signicant number of nonionic surfactants and over a range of types, including polydisperse systems such as polymers, (ii) this agreement extends from the CMC down to low coverages as far as the depletion limit, (iii) the use of a linear plot for the Gibbs equation below the CMC may be satisfactory if the range of concentrations does not extend too far (not less than about CMC/2), but it is better to use a curved t, and (iv) a plateau of adsorption is often found experimentally for nonionics, which may be the reason underlying the success of linear plots. The majority of the results discussed in the original papers, and all of those discussed by Menger et al. concern ionic surfactants, for which these conclusions might also be expected to hold. However, aside from depletion eects, which would be expected to be comparable for ionic surfactants, more or less none of the positive results above apply to ionic surfactants. We describe these results in a subsequent paper using as starting point the close agreement of ST-Gibbs and NR established here, which acts as a validation of the underlying soundness

30


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of both techniques for measurements of surface coverage on nonionic systems.

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

References (1) Menger, F. M.; Shi, L. Electrostatic binding among equilibrating 2D and 3D self assemblies. J. Am. Chem. Soc.

2009, 131, 66726673.

(2) Menger, F. M.; Shi, L.; Rizvi, S. A. A. Reevaluating the Gibbs analysis of surface tension at the air/water interface. J. Am. Chem. Soc.

2009, 131, 1083010831.

(3) Menger, F. M.; Shi, L.; Rizvi, S. A. A. Additional support for a revised Gibbs analysis. Langmuir

2010, 26, 15881589.

(4) Menger, F. M.; Rizvi, S. A. A.; Shi, L. Relationship between surface tension and surface coverage. Langmuir

2011, 27, 1397513977.

(5) Menger, F. M.; Shi, L.; Rizvi, S. A. A. Reply to Should the Gibbs analysis be revised?. Langmuir

2011, 27, 79637964.

(6) Laven, J.; de With, G. Should the Gibbs analysis be revised?

Langmuir

2011,

27,

79587962.

(7) Bermúdez-Salguero, C.; Gracia-Fadrique, J. Analysis of Gibbs adsorption equation and thermodynamic relation between Gibbs standard energies of adsorption and micellization through a surface equation of state. J. Colloid Interface Sci.

2011, 355, 518519.

(8) van Voorst Vader, F. Adsorption of detergents at the liquid-liquid interface, part 1. Trans. Faraday Soc.

1960, 56, 10681077.

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Application of the Gibbs Equation to the Adsorption of Nonionic

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