Adsorption at liquid interfaces: The generalized Frumkin isotherm and interfacial structure

ISSN 1023-1935, Russian Journal of Electrochemistry, 2006, Vol. 42, No. 10, pp. 1073–1078. © MAIK “Nauka /Interperiodica” (Russia), 2006.Published in Russian in Elektrokhimiya, 2006, Vol. 42, No. 10, pp. 1194–1200.

1073

INTRODUCTION

Traditional models for calculation of adsorption iso-therms [1–7] are based on the assumption that surface-active compounds can substitute adsorbed molecules ofonly one solvent and cannot penetrate the interface(Fig. 1a). Although these models are useful formetal/water interfaces, recent interest has focused onthe electrochemistry of amphiphilic compounds whichcan penetrate both phases and replace adsorbed mole-cules of both solvents [8–14], for example water and oil(Fig. 1b). Amphiphilic molecules consist of two moi-eties with opposing properties: a hydrophilic polar headand a hydrophobic tail. We present here a theoreticalanalysis of the generalized Frumkin adsorption iso-therm for surface-active compounds with hydrophilicand hydrophobic groups.

In the present study, we examine the interfacialadsorption of pentafluorobenzoic acid (PFBA,

p

K

a

=1.48) in an attempt to understand the adsorptive behav-ior of non-amphiphilic PFBA molecule with pH-depen-dent hydrophobic and hydrophilic properties by consid-

ering the thermodynamics of adsorption equilibrium atthe interface between two immiscible liquids. Theadsorption of PFBA at the octane/water interface willbe discussed as a model for the adsorption of adsorbate,which penetrates the interface and substitutes adsorbedmolecules of both solvents.

MATERIALS AND METHODSA system consisting of equal volumes of octane and

water was equilibrated for 48 hours. All the solutionswere prepared with twice-distilled water. Chromato-graphic-grade octane purchased from Fluka was used inthe experiments. Reagent-grade pentafluorobenzoicacid was bought from Aldrich Chem. Co. Potassiumdihydrogen phosphate, sodium hydroxide, and sulfuricacid were “Baker analyzed” reagents.

Various quantities of PFBA were added to water sat-urated with octane and equilibrated for 48 hours with anequal volume of an octane saturated with water. Theinterfacial tension at the water/octane interface wasdetermined using the drop-weight method at

22°ë

.This method determines the weight and volume of thedrop falling from the end of a capillary under the force

Adsorption at Liquid Interfaces: The Generalized Frumkin Isotherm and Interfacial Structure*

M. I. Volkova-Gugeshashvili,

a

A. G. Volkov,

a

,

z

and V. S. Markin

b

a

Oakwood College, 7000 Adventist Blvd., Huntsville, AL 35896, USA

b

University of Texas, Dallas, 75390-9068 Texas, USA

Received December 30, 2005; in final form, January 18, 2006

Abstract

—The thermodynamics of adsorption of amphiphilic surface-active compounds at the interfacebetween two immiscible liquids is considered. At the interface, these molecules are supposed to replace a fewof the adsorbed molecules of both solvents. Classical isotherms of adsorption (Frumkin, Frumkin–Damaskin,Langmuir, Henry) were based on the model of non-penetrable interface, where an adsorbate can substitute onlymolecules of one solvent. At the interface between two immiscible electrolytes, nonpolar oil/water interfaces,and liquid membranes amphiphilic molecules can substitute molecules of both solvent and classic isothermscannot be used. The generalization of Frumkin isotherm for permeable and non-permeable interfaces, knownas the Markin–Volkov isotherm, gives the possibility to analyze adsorption in a general case. The adsorptionisotherms of pentafluorobenzoic acid at the octane/water interface at different pHs were measured by the drop-weight method. The thermodynamic parameters of pentafluorobenzoic acid (PFBA) adsorption at octane/waterinterface were determined. From the measurements of PFBA adsorption, the structure of the octane/water inter-face was determined. Substitution of one adsorbed octane molecule requires approximately three adsorbedPFBA molecules. This result shows that the orientation of solvent molecules at the interface is different fromthe bulk. Adsorbed octane molecules have a lateral orientation with respect to the interface. Gibbs free energyof adsorption equilibrium and thermodynamic parameters of PFBA adsorption show that the adsorption ofPFBA at the octane/water interface is accompanied by a reduction in the attraction between adsorbed PFBAmolecules as the pH decreases to the acidic region.

DOI:

10.1134/S1023193506100132

Key words

: Frumkin isotherm, Markin–Volkov isotherm, adsorption, liquid/liquid interface

*

The text was submitted by the authors in English.

z

Corresponding author, email: [email protected]

1074

RUSSIAN JOURNAL OF ELECTROCHEMISTRY

Vol. 42

No. 10

2006

VOLKOVA-GUGESHASHVILI et al.

of gravity. The surface tension

γ

is obtained from theequation

(1)

where

f

is a correction factor from the Harkins–Browntable [15] (it is a function of

r

/

V

1/3

and takes intoaccount the deviation of the drop shape from an idealsphere),

V

is the volume of the drop,

d

1

and

d

2

are thedensities of the immiscible liquids, and

g

is the acceler-ation due to gravity. The radius of the tip,

r

, is taken asthe radius of the outside wall when the drop covered thebottom of the tip or radius of the inside wall when theliquid exuded without wetting the bottom of the orifice.The surface tension apparatus consists of a glass capil-lary tube, 1-cm

3

syringe, micrometer, and a container.The internal capillary diameter was 0.060 cm and theouter one was 0.522 cm. The drop lifetime used toestablish adsorption equilibrium was at least 10 min-utes. If the drops form quickly, they will detach prema-turely. This early detachment causes errors in the mea-surement of interfacial tension. Thus, slow drop forma-tion is imperative prior to detachment.

The remarkable accuracy of this method arises fromthe fact that the surface excess

Γ

is proportional to thesurface activity

d

γ

/

dc

, and the relatively small changesof

γ

due to changes of the concentration of the surfaceactive compound are more important than the absolute

γV d1 d2–( ) fg

r-------------------------------= ,

values of the decreasing interfacial tension withincreasing concentration.

Two important details of the drop-weight methodshould be noted. First, it is essential to purify all of thesolvents and solutes, including water, to remove all pos-sible impurities, which can reduce interfacial tension.Secondly, the apparatus must also be assembled so thatthe influence of vibration is minimized. Otherwise, thedrops will detach too soon and erroneous values ofinterfacial tension will be obtained.

The adsorption of pentafluorobenzoic acid at theoctane/water interface is presented as the Gibbs surfaceexcess,

Γ

. This is determined from the change in theinterfacial tension using the Gibbs equation.

GENERALIZED MODEL

The interface between two immiscible liquids maybe considered as a surface solution of surfactant in aspecial kind of solvent. In order to calculate the entropyof such a solution, we will adopt a simplified latticemodel and use lattice statistics, a widely used methodfor describing surface solutions. The transition fromthree-dimensional to two-dimensional (2D) geometrymay cause errors in statistical formulas, if some pecu-liarities of 2D solutions are overlooked.

The solvent molecules do not form a monolayer, butrather a multilayer. Therefore, the transition from 3D to2D geometry should be specified. Consider moleculesof both solvents, which are substituted by a surfactant(Fig. 1b). Suppose that these molecules can be assem-bled into columns consisting of

m

o

molecules of oil and

m

w

molecules of water. Suppose that one column of oilmolecules matches the

n

w

molecules of water. Thismatch of 1 oil column and

n

w

water columns will beconsidered in what follows as a quasi-molecule of sol-vent Q. These quasi-molecules constitute a “mono-layer” of solvent. They consist of

m

o

oil molecules and

n

w

m

w

water molecules.Designate the molecules of surfactant in the bulk as

A, and in the monolayer as B. At the interface aggrega-tion of surfactant molecules can take place,

r

A

B,such as dimerization of porphyrin or pheophytin mole-cules at the octane/water interface. Let the surfactant Breplace

p

quasi-molecules at the interface. Therefore,one can write

(2)

The chemical potentials for previous reaction are

(3)

Taking the 2D solution as ideal, we have

(4)

and

(5)

pQ rA+ B p oil( ) pnw water( ).+ +=

pµQs rµA

b+ µBs pµo

b pnwµwb .+ +=

µQ µQ0 s, RT XQ

sln+=

µBs µB

0 s, RT XBs .ln+=

Phase β

non-penetrable interface

oil

water

(‡)

A

B

nW

AA

Q

Phase α

A A

H2O H2O H2O H2O H2O H2O

oil oil oil

pQ

(b)

Fig. 1. (a) Model of adsorption of a surfactant substitutingadsorbed solvent molecules at the interface without perme-ation the interface; and (b) general structure of the interfacewith adsorbed monolayer of amphiphilic surfactant B pene-trating the interface with substitution of adsorbed moleculesof both solvents.

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 42 No. 10 2006

ADSORPTION AT LIQUID INTERFACES 1075

In the bulk phase we have

(6)

(7)

(8)

In all these equations, X designates the mole ratio ofcorresponding substances. Substituting these equationsinto (2), one obtains

(9)

Using the standard Gibbs free energy of adsorption

(10)

one obtains the adsorption isotherm:

(11)

We considered the 2D solution of surfactant B in thesolvent of quasi-particles Q, in which the mole ratioswere defined as

(12)

Some authors prefer another set of definitions whenreal particles in the interface are considered. The equa-tion for this state with real particles A, O, W becomes

(13)

(14)

(15)

and we obtain

(16)

Then the adsorption isotherm can be presented in theform

(17)

In the past the adsorption isotherm was presented interms of the fraction Θ of the surface actually coveredby the adsorbed surfactant.

If we introduce η as the ratio of areas occupied inthe interface by the molecules of surfactant and oil, the

µAb µA

0 b, RT XAb ,ln+=

µob µo

0 b,= RT Xob,ln+

µwb µw

0 b,= RT Xwb .ln+

pµQ0 s, rµA

0 b, µB0 s, pµ0

0 b, – pnwµw0 b, RT+––+

×XA

b( )r

Xob( )p

Xwb( )

pnw--------------------------------ln RT

Xbs

XQs( )p

--------------.ln=

∆bs G0 µB

0 s, rµA0 b, pµo

0 b, pnwµw0 b, pµQ

0 s, ,–++–=

XBs

XQs( )p

--------------XA

b( )r

Xob( )p

Xwb( )

pnw-------------------------------- –

∆bs Go

RT------------⎝ ⎠

⎛ ⎞ .exp=

XBs NB

s

NBs NQ

s+--------------------; XQ

s NQs

NBs NQ

s+--------------------.= =

XAs NA

s

NAs NO

s Nws+ +

----------------------------------,=

XQs NQ

s

NAs NO

s Nws+ +

----------------------------------,=

Xws Nw

s

NAs NO

s Nws+ +

----------------------------------,=

XBs XA

s

XAs Xo

s+-------------------;= XQ

s X0s

XAs Xo

s+-------------------.=

XAs

XQs( )p

-------------- XAs Xo

s+( )p 1– XAb( )r

Xob( )p

Xwb( )

pnw----------------------------- –

∆bs G0

RT------------⎝ ⎠

⎛ ⎞ .exp=

mole fractions in surface solution can be presented asfollows:

(18)

The adsorption isotherm takes the form

(19)

In this isotherm, the mole fractions , , ofthe components in the bulk solution are presented. Inthe general case, they must be substituted with activi-ties

(20)

If the B molecules can interact as pairs in theadsorbed layer and the energy of each new particle isproportional to its concentration, then their chemical

potential, , instead of equation (5), should be pre-sented as

(21)

where a is so-called attraction constant. After somealgebra we obtain the isotherm [8–14]

(22)

Recall that η was introduced as the ratio of areasoccupied in the interface by the molecule of surfactantto the same of oil and p was introduced as the numberof columns of oil, which could be supplanted with onemolecule of surfactant. Therefore, p is a relative size ofthe surfactant molecule in the interfacial layer. It is rea-sonable to suppose that

η = p. (23)

If the concentration of surfactant in the solution isnot high and the mutual solubility of oil and water is

low, then we can use the approximation = = 1, sothat the general equation (20) simplifies to

XBs Θ

Θ η 1 Θ–( )+--------------------------------; Xo

s η 1 Θ–( )Θ η 1 Θ–( )+--------------------------------.= =

Θηp 1 Θ–( )p--------------------------- Θ η 1 Θ–( )+[ ]p 1–

= XA

b( )r

Xob( )p

Xwb( )

pnw----------------------------- –

∆bs G0

RT------------⎝ ⎠

⎛ ⎞ .exp

XAb Xo

b Xwb

Θηp 1 Θ–( )p--------------------------- Θ η 1 Θ–( )+[ ]p 1–

= aA

b( )r

aob( )p

awb( )

pnw--------------------------- –

∆bs G0

RT------------⎝ ⎠

⎛ ⎞ .exp

µBs

µBs µB

0 s, RT X 2aRTX ,–ln+=

Θ η η 1–( )Θ–[ ]p 1–

ηp 1 Θ–( )p------------------------------------------------- –2aΘ( )exp

= XA

b( )r

X0b( )p

Xwb( )

pnw----------------------------- –

∆bs G0

RT------------⎝ ⎠

⎛ ⎞ .exp

Xob Xw

b

1076



(24)

This is the final expression for the isotherm that we willcall the amphiphilic isotherm. It is straightforward toderive classical adsorption isotherms from theamphiphilic isotherm (24):

(1) The Henry isotherm, when a = 0, r = 1, p = 1,Θ � 1:

(25)

(2) The Freundlich isotherm [4], when a = 0, p = 1,Θ � 1:

(26)

(3) The Langmuir isotherm [3], when a = 0, r = 1, p = 1:

(27)

(4) The Frumkin isotherm [1], when r = 1, p = 1:

(28)

Therefore, the amphiphilic isotherm (24) could beconsidered as a generalization of the Frumkin isotherm,taking into account the replacement of some solventmolecules with larger molecules of surfactant. Of

Θ p p 1–( )Θ–[ ]p 1–

pp 1 Θ–( )p------------------------------------------------- –2aΘ( )exp

= XAb( )r

–∆b

s G0

RT------------⎝ ⎠

⎛ ⎞ .exp

Θ Xab= –

∆bs G0

RT------------⎝ ⎠

⎛ ⎞ .exp

Θ Xab( )r

= –∆b

s G0

RT------------⎝ ⎠

⎛ ⎞ .exp

Θ1 Θ–------------- Xa

b –∆b

s G0

RT------------⎝ ⎠

⎛ ⎞ .exp=

Θ1 Θ–------------- –2aΘ( )exp Xa

b –∆b

s G0

RT------------⎝ ⎠

⎛ ⎞ .exp=

course, the amphiphilic isotherm includes all the fea-tures of the Frumkin isotherm and displays some addi-tional ones. To elucidate them, it will be convenient to

change the variable a to the relative concentration

y = / (Θ = 0.5), where (Θ = 0.5) is the concen-tration corresponding to the surface coverage Θ = 0.5:

(29)

This equation gives the coverage fraction θ as a func-tion of relative concentration y, while a and p are theparameters of this isotherm. The first being the attrac-tion constant, and the second, the size of surfactant.These parameters play an important role because theireffect on the shape of amphiphilic isotherm is verystrong.

Amphiphilic isotherm (24) analysis can be used forthe determination of the interfacial structure. Anamphiphilic molecule, which consists of two moietieswith opposing properties such as a hydrophilic polarhead and a hydrophobic hydrocarbon tail, should beused as an analytical tool located at the interface. Pheo-phytin ‡ is a well-known surfactant molecule that con-tain a hydrophobic chain (phytol) and a hydrophilichead group. The value of p less than 1.0 indicates thatadsorbed molecules of n-octane are parallel to the inter-face between octane and water [8–14]. Substitution ofone adsorbed octane molecule requires about 4–5adsorbed pheophytin or chlorophyll molecules. Theseexperimental data are supported by molecular dynamicstudies in the systems decane/water, nonane/water, andhexane/water. The structure of both water and octane atthe interface is different from the bulk. Adsorbed at theinterface octane molecules have a lateral orientation atthe interface.

RESULTS AND DISCUSSION

Pentafluorobenzoic acid

is a surface-active compound, and hence when it isadsorbed it may considerably reduce the surface ten-sion at the octane/water interface. This property wasused to determine the surface excess of pentafluoroben-zoic acid, γ, according to Gibbs.

Figure 2 shows the dependence of interfacial tensionat the octane/water interface on the PFBA concentra-tion at different pHs. A gradual reduction of γ in thePFBA concentration range (from 10–3 to 10–2 M at the

XAb

XAb XA

b XAb

yΘ p p 1–( )Θ–[ ]p 1–

p 1+( )p 1– 1 Θ–( )p------------------------------------------------- a 2aΘ–( ).exp=

COO–

F

F

F

FF

5 10Concentration, mM

0

110

20

∆γ, mN/m

15

2

3

Fig. 2. The dependence of surface tension at theoctane/water interface on the PFBA concentration in (1)aqueous solution and in the presence of (2) 0.5 M H2SO4 or(3) 0.5 M KH2PO4.


ADSORPTION AT LIQUID INTERFACES 1077

pH = 0.1 and from 10–3 to 4×10–2 M at pH from 2 to 5)was observed.

The PFBA isotherms obtained at different pHs usingEq. (29) are shown in Fig. 3. It is apparent from Fig. 3,adsorption isotherms of PFBA at the octane/waterinterface at various pHs are quite different.

A convenient form of isotherms for comparativepurposes is obtained using the coordinates y = y(θ).These isotherms are presented in Fig. 4. The coordi-nates y = y(θ) are useful for comparing adsorption iso-therms of surfactants which have surface activity in dif-ferent regions of concentration.

The adsorption parameter p and attraction constanta can be calculated from equation (29). Theamphiphilic isotherm in Fig. 4 yields the next PFBAadsorption parameters: p = 0.301 and a = 0.124 at pH2.1, p = 0.416 and a = –0.536 at pH 0.1, and p = 0.372and a = 0.916 at pH 5. A p value less than 1.0 indicatesthat adsorbed molecules of n-octane are parallel to theinterface between octane and water. Substitution of oneadsorbed octane molecule requires approximately 3adsorbed PFBA molecules. This results and our previ-ous study of the adsorption of amphiphilic compoundssuch as chlorophyll, hydrated oligomer of chlorophyll[10, 13], and pheophytin [12] correspond to the conclu-sions of the molecular dynamic study at decane/waterinterface in [16]. At the interface, the structure of bothwater and octane is different from the bulk. Octane hasa lateral orientation with respect to the interface.

The dependence of the attraction constant a on pH isshown on Fig. 5. The positive attraction constant a atpHs higher than pK of PFBA shows attraction interac-

tion between adsorbed molecules of PFBA. The nega-tive constant a at the pH below pK indicates repulsionbetween adsorbed molecules. At pH � pK neutralPFBA molecules adsorbed at the octane/water interfaceare present as dipoles oriented in the field of the electricdouble layer in the same direction. This is the reasonwhy repulsive interactions can be observed betweenadsorbed molecules. When the pH � pK, PFBA isadsorbed at the interface as anions with COO– groups

5 10Concentration, mM

0

1

2

4

6Γ × 1010, mol/cm2

15

2

3

Fig. 3. The dependences of adsorption of PFBA atoctane/water interface on concentration in (1) aqueous solu-tion and in the presence of (2) 0.5 M H2SO4 or (3) 0.5 MKH2PO4.

2 4 Û0

1

0.5

1.0�

6

2

3

Fig. 4. Experimental (points) and theoretical (lines) depen-dences of the surface coverage θ on PFBA concentration inreduced coordinates y = c/cθ = 0.5. Medium: (1) PFBA inwater, (2) PFBA + 0.5 M H2SO4, and (3) PFBA + 0.5 MKH2PO4.

1 3 pH

–0.5

0

1.0a

5

0.5

Fig. 5. The pH dependence of attraction constant a betweenadsorbed at the octane/water interface PFBA molecules.

1078



1 3pH

20

30–∆Gads, kJ/mol

515

25

Fig. 6. The pH dependence of Gibbs free energy of adsorp-tion equilibrium of PFBA molecules at the octane/waterinterface.

0

oriented to water and screened by the electric doublelayer of inorganic cations such as H+, K+, and Na+. Ben-zene rings exhibit Van-der-Waals interaction, so it leadsto positive attraction constant. The influence of theelectric double layer can change the pK of PFBA in theadsorbed state compared to the bulk aqueous phase.When the pH is equal to the pK, either attractive orrepulsive interactions between adsorbed particles arepossible.

Using equation (24), it is possible to calculate thestandard Gibbs free energy of adsorption of PFBA at

the n-octane/water interface. The dependence of ∆on pH is plotted on Fig. 6. The absolute value of stan-dard Gibbs free energy decreases with increasing pH.As described above, the higher surface activity ofPFBA at the octane/water interface takes place atlower pH.

The Markin–Volkov or generalized Frumkin iso-therm was successfully applied for study adsorption atthe polarized and nonpolarized nitrobenzene/water,decane/water, and octane/water interfaces in the pres-ence of amphiphilic compounds and for the evaluationof the interfacial structure [8–14].

REFERENCES

1. Frumkin, A.N., Z. Phys., 1926, vol. 35, p. 792.

Gads0

2. Damaskin, B.B., Soviet Electrochem., 1967, vol. 3,p. 1309.

3. Langmuir, L., J. Am. Chem. Soc., 1918, vol. 40, p. 1369.

4. Freundlich, H., Colloid and Capillary Chemistry, Lon-don: Methuen, 1926.

5. Frumkin, A.N., Z. Phys. Chem., 1925, vol. 116, p. 466.

6. Damaskin, B.B., Petrii, O.A., and Batrakov, V.V.,Adsorption of Organic Compounds on Electrodes, NewYork: Plenum, 1971.

7. Frumkin, A.N., J. Electroanal. Chem., 1964, vol. 7,p. 152.

8. Markin, V.S. and Volkov, A.G., in Liquid–Liquid Inter-faces: Theory and Methods, Volkov, A.G. andDeamer, D.W., Eds., Boca Raton (FL): CRC, 1996,p. 63.

9. Volkov, A.G. and Markin, V.S., in Emulsions: StructureStability and Interactions, Petsev, D.N., Ed., Amster-dam: Elsevier, 2004, p. 91.

10. Markin, V.S. and Volkov, A.G., in Encyclopedia of Elec-trochemistry, Bard, A.J., Stratmann, M., Gileadi, E., andUrbakh, M., Eds., Weinheim: Wiley–VCH, 2002, vol. 1,p. 162.

11. Volkov, A.G., Deamer, D.W., Tanelian, D.J., and Mar-kin, V.S., Liquid Interfaces in Chemistry and Biology,New York: Wiley, 1998.

12. Markin, V.S., Gugeshashvili, M.I., Volkov, A.G.,Munger, G., and Leblanc, R., Thin Solid Films, 1992,vol. 154, p. 264.

13. Volkov, A.G., Deamer, D.W., Tanelian, D.I., and Mar-kin, V.S., Prog. Surf. Sci., 1996, vol. 53, p. 1.

14. Markin, V.S., Volkov, A.G., and Volkova-Gugeshash-vili, M.I., J. Phys. Chem. B, 2005, vol. 109, p. 16 444.

15. Harkins, W.D. and Brown, F.E., J. Am. Chem. Soc.,1919, vol. 41, p. 499.

16. Van Buuren, A.R., Marrink, S.-J., and Berendsen, H.J.C.,J. Phys. Chem., 1993, vol. 97, p. 9206.

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Adsorption at liquid interfaces: The generalized Frumkin isotherm and interfacial structure