Journal of Colloid and Interface Science 248, 477–486 (2002)doi:10.1006/jcis.2001.8206, available online at http://www.idealibrary.com on

Effect of Interactions between the Adsorbed Species on the Propertiesof Single and Mixed-Surfactant Monolayers at the Air/Water Interface

Stoyan I. Karakashev and Emil D. Manev

Faculty of Chemistry, St. Kliment Ohridski University of Sofia, 1126 Sofia, Bulgaria

Received May 30, 2001; accepted December 27, 2001; published online March 13, 2002

Experimental data on surface tension available from the litera-ture and generated in the present study are analyzed to estimate theapplicability of adsorption models, based on the Frumkin equation,to nonionic and ionic surfactants and their mixtures. Optimizationprograms based on the least-squares method in media of Delphi Vand Pascal VII are used. The effect of interactions between the ad-sorbed species on surface tension is considered in all cases. The re-sults are compared to those obtained with the simpler Szyszkowskiequation, employed in numerous studies of nonionic surfactants,when interactions are neglected. Cases where the Frumkin modelcan be successfully employed with ionic surfactants and mixturesare presented and the conditions of its applicability are analyzed.Related characteristic quantities (maximum adsorption, standardfree energy of surfactant adsorption, energy of interaction betweenadsorbed species, standard free energy of counterion adsorption,degree of coverage by surfactant/counterion associates) are estab-lished as a function of:

1. The number of methylene groups in the surfactant molecule;2. The position of the polar head on the hydrophobic tail;3. The type of (alkali) counterion.

The properties of an adsorption layer from a mixture of nonionicand ionic surface-active species are compared to those of the singlesurfactants. C© 2002 Elsevier Science (USA)

Key Words: adsorption monolayers; nonionic surfactants; ionicsurfactants; molecular interactions.

INTRODUCTION

The properties of the liquid films that build colloidal dispersesystems such as foams and emulsions are closely related to andstrongly dependent on the characteristics of the adsorption layerson their fluid interfaces, formed by the ever-present stabilizingsurfactant. For example, it is established that the stability of foamsharply increases at the onset of “black film formation” (1, 2)upon increasing surfactant concentration. Quantitative knowl-edge of the surfactant adsorption allows one to estimate theinfluence of surface interactions on the properties of the adsorp-tion layers and detect certain individual effects of the surfactants.Reliable adsorption isotherms are therefore needed.

While adsorption of single surfactants is extensively stud-ied, reliable data for surfactant mixtures are scarce. In previous

477

studies of foam stability (3, 4) we have established the com-bined effect of nonionic surfactants and (surface-active) organicions on the lifetime of an aqueous foam. Hence, the knowledgeof the adsorption characteristics of surfactants and/or ions is ofa particular interest in order to distinguish their effects on thefoam and film properties.

The correct description of the adsorption layer properties re-quires appropriate models. Adsorption of nonionic surfactantsis often described in the literature through the simple modelof Szyszkowsky–Langmuir, which does not account for interac-tions in the surface layer (5). For example, our own experimentalresults (6, 7) have shown that it holds well for the shorter-chained(polyethyleneglycol-) surfactants, while being insufficient forthe longer-chained ones. More complex models of wider validityhave to be employed when investigating the adsorption behaviorof long-chained surfactants, as well as surfactant mixtures.

The aim of the present study has been to outline the range ofapplicability of the more general Frumkin model and its mod-ifications (5, 8–11). Experimental data on surface tension ofaqueous surfactant solutions, available from the literature andgenerated in the present study, are used to estimate the validityof the adsorption models for nonionic and ionic surfactants, aswell as their mixtures. The properties of the mixed adsorptionlayer from nonionic and ionic surface-active species are com-pared to those of the single surfactants.

BASIC RELATIONS BETWEEN ADSORPTIONAND SURFACE TENSION

There are various expressions (adsorption isotherms), de-scribing the properties of the monomolecular surface layers(see, e.g., (5)). They all originate from the Gibbs adsorptionisotherm, which gives the relations between the surface tensionand (Gibbs’) surface excess:

dσ = −n∑

i=1

�i dµi . [1]

Here �i is the surface excess of the i th component in the system,µi is its chemical potential, and ci is its bulk concentration; σ issurface tension of the surfactant solution.

0021-9797/02 $35.00C© 2002 Elsevier Science (USA)

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V

478 KARAKASHE

With dµi = RT d ln ci (ci is the bulk concentration of the i thcomponent), Eq. [1] yields

dσ = −N∑

i=1

�i RT d ln ci . [1a]

For a single surfactant,

� = − 1

RT

dσ

d ln c. [1b]

Our study is based on the Frumkin adsorption isotherm (5,8–10), which is frequently and most successfully used:

K c = �

�∞ − �exp

(−2β�

RT

). [2]

Here K is the equilibrium constant of surfactant adsorption, �

is the adsorption of nonionic surfactant and �∞ is its maximumadsorption, β is the parameter of interaction in the adsorptionlayer, and c is the bulk surfactant concentration.

Together with Eq. [1b], Eq. [2] yields Eq. [3], where σ0 is thesurface tension of water:

σ = σ0 + RT �∞ ln

(1 − �

�∞

)+ β�2. [3]

When analyzing Eq. [3], one can detect two opposite trends.Attraction between the surfactant species causes an increase ofadsorption (see also Eq. [2]) and, consequently, should lowersurface tension. However, according to the third term in Eq. [3],surface tension should rise. From a mathematical point of viewthe effect of the second term is always stronger. This is con-firmed by our estimates, based on the experimental results (seeAppendix).

Equations [2] and [3] are valid only for nonionic surfactants;they do not account for the effect of electric charges on the sur-face that are due to adsorption of ions. The adsorbed ionic surfac-tant species, produced by the electrolytic dissociation, generatean electric field decaying toward the bulk. Counterions are at-tracted to compensate the charge, thus forming the Stern’s layerand the diffuse (Debye) tail.

The model of Frumkin for ionic surfactants can be presentedby the following set of relations (11):

(i) Modified (for ionic surfactants) adsorption isotherm:

a1S(K1 + K2a2s) = �1

�∞ − �1exp

(−2β�1

RT

). [4]

Here a1S and a2S denote the activities of surfactant ions andcounterions in the sub (-surface) layer, K1 is the (adsorption)equilibrium constant for the surfactant ions, K2 is the equilib-

rium constant for (surfactant ion/counterion) association, and �1

is the adsorption of surfactant ions.

AND MANEV

(ii) Adsorption isotherm for the counterions in the Stern’sadsorption layer:

�2 = �1K2a2s

K1 + K2a2s. [5]

�2 is the adsorption of counterions (or the adsorption of “un-charged” surfactant).

(iii) Relation between the surface potential and adsorption ofcharges in the surfactant layer (the Grahame equation (13)):

�1 − �2 = 4

κc

√a sinh

(�s

2

). [6]

�S is the dimensionless surface potential �S = Z FS/RT fora 1:1 ionic surfactant, κ is the inverted Debye length (κc =κa−1/2), and a is the bulk surfactant activity.

(iv) Equation of state of the surfactant layer:

σ = σ0 + RT �∞ ln

(1 − �1

�∞

)+ β�2

1

− 8RT√

a

κc

[cosh

(�s

2

)− 1

]. [7]

This model operates with four parameters: K1, K2, β, and �∞.We must point out that from statistical point of view this numberof parameters is too big for a good optimization. It is appropriatein such a case to reduce the number of optimization parametersby additional experiments, as it is done in our work (12).

Mixed (Ionic/Nonionic) Surfactant Systems

As in Ref. 11, we operate with two adsorption isotherms:

a1s(k1 + k2a2s) = �1

�∞ − �exp

(−2β�

RT

)[8]

KnCn = �n

�∞ − �exp

(−2β�

RT

)[9]

�1 is the adsorption of the surfactant ions; �n is the adsorptionof surfactant molecules. Respectively: � = �1 + �n.

The maximum total adsorption �∞ is given by the relation(14)

�∞ = �1∞K1a1∞ + �n∞KnCn

K1a1∞ + KnCn, [10]

where �∞1 and �n∞ denote the maximum adsorption of therespective surfactants when present alone in the surface layer,K1 and Kn are the equilibrium adsorption constants of the twosurfactants, a1∞ is the activity of surfactant ions in the bulk, andcn is the concentration of the nonionic surfactant in the bulk.

Equations [8]–[10], in combination with the Stern adsorptionisotherm and the Grahame equation, yield a relation between

INTERACTIONS BETWEE

σ and � similar to Eq. [7]. The only difference is that � nowrepresents the total adsorption of the two surfactants together:

σ = σ0 + RT �∞ ln

(1 − �

�∞

)+ β�2

− 8RT√

a

κc

[cosh

(�s

2

)− 1

]. [7a]

The optimization procedure operates with six parameters:K1, K2, Kn, β, �1∞, and �n∞.

APPLICATION OF THE MODELS FOR DESCRIPTIONOF THE ADSORPTION BEHAVIOR OF

SURFACTANT-CONTAINING SYSTEMS

Experimental data on surface tension of aqueous surfactant so-lutions, available from the literature and generated in the presentstudy, are analyzed to estimate the applicability of different ad-sorption models for nonionic and ionic surfactants, as well astheir mixtures.

Nonionic Surfactants

An example for adsorption behavior well described by theFrumkin equation are our experimental data for the surfacetension vs bulk concentration of the short-chained nonionicsurfactant tetraethyleleneglycol-octylether (C8E4), measured atT = 25◦C by the pendant drop method. We used also an op-timization procedure based on the least-squares method in amedium of Maple V release 4 mathematical software. The fol-lowing results were obtained:

equilibrium adsorption constant: K = 24.6 m3/mol;parameter of interaction: β = 0.67 J · m2/mol;energy of attraction: E = 0.29 mJ/m2;maximum adsorption: �∞= 3.42 × 10−6 mol/m2;best fit: W = 3.1 × 10−6;

and

W =N∑

i=1

(σi(exp) − σi(th)

)2[11]

(σi(exp) is the experimental value and σi(th) is the theoretical valueof surface tension at varying surfactant concentration).

If we assume β equal to zero (as in the model of Szyskowski),the fit is 3.5 × 10−6. The standard free energy of adsorption �µ0

1can be derived from its relation with K (see Eq. [14]): �µ0

1 =29.8 KJ/mol. The greatest contribution of β to σ at �∞ is ca.0.3 mN/m, while the experimental error is usually ca. 0.2 mN/m.Thus, the Szyskowski equation is a good approximation for thissurfactant, because the value of the interaction parameter is verysmall.

It must be pointed out that other (similar) surfactants may

not be described well by the Szyszkowski equation, as is, e.g.,the case with octaethyleleneglycol-nonylether (C9E8), (Fig. 1),

N ADSORBED SPECIES 479

C, mol/m3

Sur

face

tens

ion,

N/m

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.001 0.01 0.1 1

Experimental points Eq. (22)Gb=2,8.10

-6 mol/m

2

K=62m3/mol

K=250m3/mol

FIG. 1. The best fits of C9E8 to Szyskowski’s equation.

for which we have used experimental data for σ vs c from theliterature (15). Figure 2 represents an attempt to fix the product�∞ RT equal to the slope of the experimental dependence σ vsln c: (i) at low surfactant concentrations, and (ii) near (before)CMC. In both cases the result is a bad fit for a major part of theexperimental dependence.

Figure 2 shows the best fit for C9E8, based on the Frumkinequation (Eq. [2]). The equality of �∞ RT with the maximumslope in the experimental surface tension isotherm and the goodfit are indications for the applicability of the Frumkin model.Evidently, it describes the behavior of this surfactant much betterthan the Szyszkowski model.

In order to estimate more comprehensively the range (and de-gree) of the Frumkin model applicability to adsorption of suchsurfactants on air/water interface, we have exploited the system-atic experimental data for the homologs C9E8 to C15E8, availablefrom the literature (15).

Our optimization procedure was carried out in the followingorder:

1. Optimization of β and K , at �∞ estimated from the max-imum slope.

C, [mol/m3]

sig

ma

, [N

/m]

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0 0.01 0.1 1

Experimental pointsCalculated from Frumkin’s equation

FIG. 2. Frumkin’s fit on C9E8.

480 KARAKASHEV

2. Optimization on β, K , and �∞ is carried out upon findingthe best fit in step 1.

Step 1 led to reasonably good fittings; step 2 led to a slightlybetter fitting (by 5 to 20%) but with the value of �∞ RT no-ticeably different from that of the slope. After completing steps1 and 2, the value of �∞ RT was found to coincide with themaximum slope only for C9E8.

The data for C14E8 and C15E8 have yielded poor fits and theFrumkin model is hardly applicable there. The analysis of theproblems related to the optimization procedure has indicatedeffects, which are not accounted for in the Frumkin model, butare not negligible for C14E8 and C15E8.

The optimized values of the parameters are presented inTable 1. One can detect from the table that the standard freeenergy increases in average by ca. 2.5 kJ/mol for each methy-lene group added to the hydrophobic tail. However, we havepersistently obtained negative values for the parameter β (from−79.2 × 107 to −255.6 × 107 J m2/mol). This fact implies re-pulsion between the adsorbed species and contradicts the classi-cal lattice adsorption theory, which accounts for repulsion onlythrough the excluded area of the adsorbed molecules (8–10). Itis of interest to note that indications of negative sign of the inter-action parameter for certain nonionic surfactants can be found inthe literature (16–18). According to the authors there, the possi-ble reason for this unexpected effect should be sought in dipole–dipole repulsion on the interface. We have analyzed this hypoth-esis and found it unjustified (17). The decay of “repulsion” withdistance is suspiciously slow for interactions of dipole–dipoleorigin. The dependence of the interaction parameter β on the av-erage distance between adsorbed molecules for C10E8, followingthe procedure in Ref. 16, is given as an example in Fig. 3.

We shall support our conclusions here with another relation-ship between the parameter of interaction β and the averageintermolecular distance on the surface. The expression, whichwe have obtained, is (see Appendix):

β = − 2π P4

5kT (4πεε0)2r4. [13]

Here P is the dipole moment of the surfactant molecule, ε0 and ε

are the dielectric permittivities of the free space and the surfacelayer, and r is the intermolecular distance. The dependence ofthe interaction parameter β on distance, according to Eq. [13](with ε = 2), is presented in Fig. 4.

Whichever relation is used, the estimated values for the pa-rameter of interaction are much lower (at least by an order ofmagnitude) than those obtained by the optimization procedure.Moreover, if ε = 80 is assumed in Eq. [13] (dielectric permitiv-ity of bulk water), the value of β will drop ca. 103 times! We find,therefore, that the idea of negative interaction parameter due todipole–dipole repulsion is unacceptable. In a current study (19)we seek the origin of this contraction with the classical theory in

the surface aggregation, suggested by Lucassen-Reynders et al.(20). Thus, we find that negative values of β are obtained due

AND MANEV

FIG. 3. Dependence of the interaction parameter β for C10E8 on the averagedistance of the molecules according to Ref. 17.

to reduced adsorption. A mixed adsorption layer (aggregatesand single molecules) is built. Hence, the size of the elementaryadsorption cell increases with surfactant concentration, whichresults in decreased area for adsorption. The aggregation num-ber distribution is also presented in Table 1. One can see that thevalues of β and the aggregation distribution number are related.

FIG. 4. Dependence of the interaction parameter β for C10E8 on the averagedistance of the molecules according to Eq. [13].

E

INTERACTIONS BETWE

Ionic Surfactants

An optimization procedure is developed on the basis ofGibbs’s isotherm (Eq. [1]), the Frumkin isotherm (Eq. [4]), theStern isotherm (Eq. [5]), and Eq. [7] (the model proposed inRef. (11)). The parameter of interaction β, the maximum ad-sorption �∞, the equilibrium constant of adsorption of surfactantions K1, and the equilibrium constant of adsorption of counter-ions K2 are thus estimated. We have also made use of therelations

K1 = δ1

�∞exp

(�µ0

1

kT

)[14]

K2

K1= δ2

�∞exp

(�µ0

2

kT

), [15]

where K1 is the adsorption constant of surfactant ions at the onsetof the Debye tail, K2 is the equilibrium adsorption constant ofcounterions in the Stern layer, �∞ is the maximum adsorptionof the surfactant ions, δ1 is the thickness of the adsorption layer,δ2 is the thickness of the Stern layer, �µ0

1 is the standard freeenergy of adsorption of surfactant ions, �µ0

2 is the standard freeenergy of adsorption of counterions in the Stern layer, and kTis the thermal energy of the surfactant molecule. For SDS δ1 isca. 2 nm. Here δ1/�∞ has the meaning of molar volume; thevolume of one –CH2– group is ca. 25 A3 (21).

We have used experimental data from Refs. 22–24 for threeseries of surfactants to analyze adsorption of ionic species onair/water interface:

1. a series of position isomers of sodium n-hexadecylsulfate,where sulfate group is situated on second, third, fourth, fifth,sixth, seventh and eighth position (22);

2. a series of Me dodecylsulfate, Me = Li, Na, K, Rb, Cs(23);

3. a homolog series of sodium octyl-, decyl-, dodecyl-,tetradecyl-sulfate (24).

The optimized constants K1, K2, β, and �∞ for these surfac-tants are given in Tables 2–4. An optimization procedure wasperformed in medium of Pascal VII and Delphi, again on thebasis of the least-squares method.

It can be seen from the tables that the values of �∞ RT aresubstantially smaller than the maximum slope of -dσ/d ln c. Inthis case the maximum slope represents the sum of the maximumadsorption of surfactant ions plus that of counterions from thediffuse tail.

Because of the counterion binding into the adsorption layer,for a binary 1–1 ionic surfactant the Gibbs adsorption isothermacquires the form (see (25))

− 1

RT

dσ

d ln a

∣∣∣∣CMC

= n�∞, [16]

where n is a coefficient, related to the degree of coverage in

N ADSORBED SPECIES 481

FIG. 5. (a) Increase of the contact area between molecules with two hydro-carbon tails. (b) Expansion of the area occupied by the adsorbed molecules.

the adsorption layer by counterions from the Stern layer (n = 2when the surface coverage θ = 0, while n = 1 for θ = 1).

Since we do not know the exact value of the maximum ad-sorption �∞, we cannot determine the surface coverage fromthe slope −dσ/d ln c. We use the optimization procedure for thepurpose.

It is evident from the data in Table 2 that there is a tendencyin �∞ to decrease (that is, the area per molecule increases)when the sulfate group shifts toward the middle of the hydrocar-bon tail. The natural explanation is that, with this shift, the ad-sorbed species actually acquires two (shorter) hydrophobic tails,whence the area per molecule on the surface increases (Fig. 5).The average free energy of adsorption here is �µ0

1 = 3.5 kJ/molper methylene group (Fig. 5).

Another tendency, that of increasing attraction between themolecules, can be noticed as well when the hydrophilic headshifts to the middle of the tail. This fact can be explained withthe increasing area of contact between the hydrophobic tails ofneighboring molecules.

The mean value of the standard free energy �µ01 is

55.7 kJ/mol. The adsorption of counterions into the Stern layeris quite high, although no clear trend can be established whenchanging the position of the polar head. For example, in the caseof sodium hexadecyl sulfate the greatest adsorption of counter-ions is with the sulfate group in the fourth position on the hy-drophobic tail; ca. 83% of the adsorbed surfactant is associatedwith counterions. With the sulfate group in the eighth position,we find the lowest level of counterions: 43%.

Let us consider now Table 3, where the results of the op-timization procedure with data for the dodecylsulfate surfac-tant with different counterions are presented. The high degreeof ion/counterion association in the adsorption layer is evident(between 89.9 and 92.6%). There is a correlation between thehydrated radius of the counterions and �∞. Moving from largerto smaller hydrated counterions correlates with higher �∞, i.e.,smaller area per molecule. Consequently, the hydrated radiusdetermines the area occupied by the surfactant ion and the asso-ciated counterion. The possible picture is presented in Fig. 6.

This is not strictly in power for Rb and Cs, because their hy-

drated radii are practically equal, which may indicate additionaleffects.

482 KARAKASHEV AND MANEV

TABLE 1The Results of the Optimization on the Data for Nonionic Surfactants

�∞ × 106 �∞ × 106 K �µ0 β × 3.6 × 10−7 Theoretical error Aggregation No.Surfactant (mol/m2) (mol/m2) (m3/mol) (kJ/mol) (J · m2/mol) (mN/m) Ref. (19)

C9E8 2.81 2.84 8.1 × 102 35.67 −22.88 ±0.50 1.5–2C10E8 2.72 2.72 3.8 × 103 39.41 −26.41 ±1.0 2–3C11E8 3.13 3.13 8.1 × 103 41.39 −30.03 ±0.65 3–5C12E8 3.29 3.29 2.2 × 104 43.55 −35.95 ±0.70 5–40C13E8 2.97 2.99 4.1 × 104 45.09 −33.10 ±0.80 40–80C14E8 3.8 3.80 1.0 × 105 47.23 −41.45 ±1.19 80–130C15E8 5.70 5.70 4.0 × 105 50.54 −70.68 ±1.55 130–250

TABLE 2The Results of the Optimization on the Data for Isomeric Hexadecyl Sulfates

− 1RT

dσd ln C

∣∣CMC × 106 �∞ × 106 K1 K2 E �µ0

1 �µ02 Fit.106 θ2

C16(SO4)-n (mol/m2) (mol/m2) (m3/mol) (m6/mol2) (mJ/m2) (kJ/mol) (kJ/mol) (N2/m2) max

n = 2 4.96 3.21 3.0 × 106 2.62 × 103 −13 56.2 3.57 4.57 0.68n = 3 2.89 1.79 1.3 × 106 4.15 × 103 0.87 54.1 5.47 3.10 0.71n = 4 3.70 1.51 4.8 × 106 8.71 × 103 0.83 57.3 3.9 1.64 0.83n = 5 3.00 1.51 2.1 × 106 1.40 × 103 3.3 55.3 1.43 2.29 0.50n = 6 2.97 1.56 2.2 × 106 8.93 × 102 2.6 55.5 0.36 0.10 0.45n = 7 2.78 1.37 2.5 × 106 2.95 × 103 5.45 55.8 2.53 1.58 0.55n = 8 2.78 1.35 2.3 × 106 1.14 × 103 6.49 55.6 0.44 1.37 0.43

TABLE 3The Results of the Optimization on the Data for Alkyl Sulfate Homologs

− 1RT

dσd ln C

∣∣CMC × 106 �∞ × 106 K1 K2 E �µ0

1 �µ02 Fit.106 θ2

(mol/m2) (mol/m2) (m3/mol) (m6/mol2) (mJ/m2) (kJ/mol) (kJ/mol) (N2/m2) max

Octyl −4.24 2.5 1.52 3.9 × 10−3 0.56 23.3 6.16 4.1 0.76Decyl −5.25 3.0 15.1 9.98 × 10−2 −3.29 28.55 9 2.8 0.83Dodecyl −6.7 3.4 60.6 4.68 × 10−1 2.31 31.57 9.7 9 0.84Tetradecyl −7.2 3.68 1.4 × 104 2 × 10−1 15.86 44.91 — 5.2 0.22

TABLE 4The Results of the Optimization on the Data for Dodecylsulfate with Different (Alkali-) Counterions

Rh − 1RT

dσd ln C

∣∣CMC × 106 �∞ × 106 K1 K2 E �µ0

1 �µ02 Fit.106 θ2

Me+ (A) (mol/m2) (mol/m2) (m3/mol) (m6/mol2) (mJ/m2) (kJ/mol) (kJ/mol) (N2/m2) max

Li+ 3.4 −5.43 3.03 13.6 0.52 −2 28.1 13.1 0.53 0.89Na+ 2.8 −6.00 3.796 20.4 0.60 −7.16 29.2 13.5 4.78 0.90K+ 2.3 −8.60 4.584 15 0.43 −0.53 28.4 14.4 0.23 0.91Rb+ 2.3 −7.22 3.99 10.4 0.6 2.95 27.4 15.8 0.72 0.93Cs+ 2.3 −7.00 3.801 19.8 0.6 4.35 29.1 14.1 1 0.93

TABLE 5Results of the Optimization on the Data for (Single) C8E4, (Single) TPeAB, and (Mixture) C8E4 + TPeAB

�∞n �∞1 K1 µ1 K2 µ2 Kn µn β × 3.6 × 10−7 E

Surfactant( mol

m2 × 106) ( mol

m2 × 106) ( m3

mol

) ( kJmol

) ( m6

mol2) ( kJ

mol

) ( m3

mol

) ( kJmol

) ( Jm2

mol

) ( mJm2

)

C8E4 3.42 — — — — — 24.64 29.78 0.7 0.29TPeAB — 2.1 6.1 × 103 41.30 15 5.79 — — −222.5 −32.3C8E4+ 4 2.9 19.4 26.98 4 × 10−3 0.42 15.5 28.68 7 2.8TPeAB

Note. The subscripts n, 1, and 2 denote nonionic surfactant molecules, surface activeions, and counterions, respectively.

E

INTERACTIONS BETWE

FIG. 6. The Ster’s layer gets compressed with decreasing hydration radiusof the counterions.

In general, a tendency to stronger attraction upon diminishinghydration radius can be noticed. Concurrently, a slight tendencyto rising degree of association in the adsorption layer can benoticed; i,e., the smaller the hydrated ion, the greater the degreeof association. Therefore, the increased attraction may be at-tributed to the reduced electrostatic repulsion, because of thestronger association.

Table 4 presents the results of the optimization procedureon data of the alkyl sulfate surfactants homolog series. Max-imum adsorption �∞ increases, together with the increasingnumber of carbon atoms in the hydrophobic tail. Consequently,there is a tendency for increasing of the attraction forces: thestronger attraction leads to smaller occupied area by the surfac-tant ions. Besides, with the increasing length of the hydrophobictail, the degree of association in the adsorption layer rises, withthe exception of the surfactant with the longest hydrocarbonchain.

One interesting fact can be noted here. For sodium tetradecylsulfate we obtain �µ0

2 = 0 (absence of a Stern layer). Neverthe-less, the model yields degree of association (counterions cover-age) θ2 = 22%. It is strange that in the absence of counterionsadsorption (�µ0

2 = 0), about one-fifth of the adsorption layershould be occupied by counterions (in the Stern’s layer).

Mixture of Nonionic and Ionic Surfactants

The analysis of the state in the adsorption layer of mixed sur-factants is based on our experimental data. The surface tensionisotherms of: (i) single TPeAB, (ii)single C8E4, and (iii) mixtureC8E4 + TPeAB (Eqs. [5], [6], [7a], and [8]–[10] are used. Theresults of the optimization procedure are presented in Table 5.

The optimization procedure gave for the maximum adsorptionof TPeAB the value �∞ = 2.1 × 10−6 mol/m2. Using mathe-matical software for quantum calculations we obtained: �∞ =1.9 × 10−6 mol/m2. It is important to note that C8E4 is moresurface-active than TPeAB (e.g., CMC [C8E4] = 6 × 10−3 M,while CMC [TPeAB] > 0.01 M), although the value of sur-face free energy of adsorption of TPeAB is greater then thatof C8E4, because of the electrostatic repulsion between theTPeAB ions.

With surfactant mixures the attraction among molecules onthe surface is stronger. The result is compression of the adsorp-tion layer and a decrease in the area, occupied by the surfactantmolecules. Concurrently, we observe a decrease in the free en-ergy of adsorption of the TPeAB species.

Stronger attraction in the mixed adsorption layer of TPeA+

and C8E4 has its explanation in the interaction between ionic

N ADSORBED SPECIES 483

and nonionic species. Van der Waals forces give rise to attractionbetween the ions and the dipole molecules, which reduces thepower of the electrostatic field and the repulsion between theions themselves. This increases the compression in the mixedlayer. There are indications in the literature (26, 27) that whenionic and nonionic surfactants are mixed the adsorption layerbecomes more compressed.

CONCLUSIONS

The experimental data on surface tension, analyzed here, haveallowed us to estimate the validity and the limits of applicabil-ity of different models, describing the adsorption behavior ofnonionic and ionic surfactants.

As it can be judged from the literature (2, 5, 28), theSzyszkowski equation, regardless of its limitations, is exten-sively exploited for various types of nonionic surfactants. How-ever, of all surfactants analyzed in the present study, C8E4 isthe only one to fit well this equation. The short-chained C8E4,used in our study of foam stability (3), is an example of a non-ionic surfactant with weak interactions in the adsorption layer.For the series of nonionic surfactants considered here, the Gibbsfree energy �µ0 = 2.5 kJ/mol (per methylene group) was estab-lished (see Table 1), while 3 kJ/mol is reported in the literature(28), but on the base of the Szyskowski equation. The differencethus found is an indication for noticeable interactions, which areincorporated in the Frumkin model. When interactions are sig-nificant one can conclude that the more general Frumkin equa-tion (Eq. [3]) is a better tool. Nevertheless, C14E8 and C15E8

make an exception, which shows that it cannot be universallyapplied.

In contrast to our expectations (the adsorbed molecules shouldattract one another through van der Waals forces), the modelof Frumkin has yielded considerable negative values of the in-teraction parameter, which corresponds to repulsion betweenthe adsorbed molecules. Such a serious contradiction with thetheory requires an interpretation, which lies outside the scopeof the present paper. Nevertheless, we can point out here thedirections in which the explanation of this experimental find-ing must be sought. We have found the hypothesis of negativevalues of β due to dipole–dipole interaction to be unacceptable,and in a current study of ours (19) we arrive at the conclusionthat the effect can be linked with the aggregation processes inthe adsorption layer.

We have shown that the model developed in Ref. (11) canbe used to describe reasonably well the state in an adsorp-tion layer containing ionic species. It was established thatwith the increasing length of the hydrophobic tail, attractionamong the adsorbed surface-active ions becomes stronger andthe adsorption layer more compressed. Concurrently, higherrelative content of counterions in the Stern layer is obtained.Decreasing hydration radius of the counterions results in

weaker repulsion between the (surface-active ion/counterion)associates, which reduces the intermolecular distances and

484 KARAKASHEV

the occupied area, so that the van der Waals attraction be-comes stronger. Consequently, the surface charge density in-creases and more counterions are attracted into the Sternlayer.

When the hydrophilic head shifts toward the middle of thehydrocarbon chain, the attraction becomes stronger (greater areaof contact), while the maximum adsorption decreases, due tothe stronger steric repulsion (the surface-active ions acquire twoshorter hydrocarbon tails).

Changes in the basic adsorption parameters (equilibrium ad-sorption constant, energy of interaction, and area per moleculeat close packing) are observed with mixtures. The ionic TPeABexhibits strong repulsion and considerably higher free energyof adsorption than the considerably more surface-active C8E4,where weak attraction between the adsorbed molecules is de-tected. When mixing these two surfactants, the optimizationprocedure shows only attraction, which is ca. 10 times strongerthan in the case of C8E4 alone. The latter fact must be relatedto the van der Waals attractive forces between C8E4 and TPeA+

ions. The maximal adsorption �∞ also increases, because thestronger attraction reduces the average intermolecular distancebetween the adsorbed species and the area per molecule becomessmaller than in the case of a single surfactant. Besides, we ob-serve a decrease in the free energy of adsorption (especially forTPeAB), as well as smaller quantity of counterions in the Sternlayer.

In a previous study (3) we have established the effect of anonionic surfactant on the lifetime of an aqueous foam in thepresence of inorganic and organic ions. The correct descriptionof the adsorption behavior of the different species at the surface,based on an appropriately chosen models, will help to distinguishelectrostatic from molecular interactions, as well as the relativecontribution of different species to the stability of single filmsand foam systems.

APPENDIX

(i) Evaluation of the Effects of the Interaction Parameterβ on Surface Tension

The second and third terms in the Frumkin surface tensionisotherm (Eq. [3]) indicate two opposite effects of the interactionparameter β on surface tension. The estimates, using appropriatevalues for the quantities in Eq. [3], show that the second termis always significantly larger than the third one. The effects areillustrated in Fig. 7.

(ii) Analytical Expression for the InteractionParameter β (Eq. [13])

Consider two ethoxylated nonionic surfactant moleculesin parallel on the water/air interface, as presented inFig. 8.

Figure 9 is a schematic presentation of a free rotating dipole,

P , interacting with the electric field, E . The theorem of distri-

AND MANEV

FIG. 7. Contribution of the second and third terms in Eq. [3] to surfacetension as a function of adsorption of a nonionic surfactant.

bution of the energy of free rotationg dipole (13) gives

dξ (�) = exp(−w(�)

kT

)d�∫

d�. [i]

Here d� = sin θ dθ d� (θ and � are the polar and azimuth an-gles, respectively). The instant energy of interaction is givenby the relation w(θ ) = −P · E · cos(θ ), and the induced instantdipole is Pind(θ ) = P · cos θ , where P is the permanent dipole,E is the intensity of the electric field, and θ is the instant anglebetween the dipole and the direction of the field. In the generalcase, the dipole averaged over all angles is given by the relation

Pind =∫

Pcos θ dξ (θ ) = 1

�

∫�

Pcos θ exp

(P E cos θ

kT

)d�.

[ii]

In the case of maximum repulsion: θ = π . If we assume theenergy of interaction of the dipole with the field to be muchweaker than the thermal energy, Eq. [ii] can be presented in apower series and with θ = π we obtain the following relationfor the induced dipole:

Pind = P2

kTE . [iii]

FIG. 8. Schematic of nonionic surfactant molecules in parallel orientation(dipoles, P) on the water/air interface.

INTERACTIONS BETWEE

FIG. 9. Picture of interacting free rotating dipole, P , with electrical field, E .

Thus, for the fixed dipoles, the orientation polarizability is

αor = P2

kT. [iv]

Let us consider a single vertical dipole on the water/air in-terface (Fig. 10). The intensity of electrostatic field E arounda permanent dipole P1 depends on the angle θ in the followingmanner (13):

E = P1

4πεε0r3

√1 + 3 cos2 θ. [v]

Here r is the distance from the dipole P1. If we assume thepresence of a second dipole in the vicinity of P1, the induceddipole is P2 = α2 E , where α2 is its orientation polarizability.The energy of interaction between them (the permanent dipoleP1 and the induced P2) is equal to the work, performed by theelectrostatic field for creating the induced dipole:

U (r, θ ) = −E∫

0

α2 E dE = −1

2α2 E2

= − α2 P21

2(4πεε0)2r6(1 + 3 cos2 θ ). [vi]

Since both dipoles are situated in the plane of the water/air in-terface, θ = 0 and Eq. [vi] can be converted into the form

U (r ) = 2P21 α2

(4πεε0)2r6. [vii]

If α2 from Eq. [iv] is substituted in Eq. [vii], we obtain thefollowing relation for the energy of interaction of two permanent

FIG. 10. Intensity of electrostatic field, E , around a permanent dipole, P1.

N ADSORBED SPECIES 485

dipoles situated on the water/air interface:

U (r ) = − 2p21 p2

2

(4πεε0)2kT r6. [viii]

Here all the dipoles possess the same permanent (dipole) mo-ment: P1 = P2 = P:

U (r ) = − 2p4

(4πεε0)2kT r6. [ix]

The interaction parameter β (10) can be presented (for weakinteraction) in the form

β = −π

∞∫rc

U (r )r dr . [x]

Upon substituting [ix] in [x], we obtain the respective relationfor β (see Eq. [13]):

β = − 2π P4

5kT (4πεε0)2 r4.

ACKNOWLEDGMENT

The present work has been conducted with the financial support of the NationalFund for Scientific Research, Project 338/2343.

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