An ion-binding model for ionic surfactant adsorption at aqueous-fluid interfaces

ELSEVIER

Colloids and Surfaces A: Physicochemical and Engineering Aspects 114 (1996) 337-350

COLLOIDS AND SURFACES

A

An ion-binding model for ionic surfactant adsorption at aqueous-fluid interfaces V.V. Kalinin, C.J. Radke * Department of Chemical Engineering, University of California, Berkeley, CA 94720-1462, USA Received 21st February 1996; accepted 21st February 1996

Abstract A simple ion-binding model is presented to quantify the equilibrium adsorption of ionic surfactants at aqueous-fluid interfaces. The proposed model adopts a triple layer structure for the interface: a plane of adsorbed surfactants (interface plane), a plane of partially dehydrated, contact-bound counterions (inner Helmholtz plane), and a plane of hydrated counterions (outer Helmholtz plane). An analytic expression for the surface tension is obtained as a function of the physicochemical parameters of the system. It generalizes the classical results of J.T. Davies and E.K. Rideal (Interfacial Phenomena, Academic Press, New York, 1963) as well as those, more recent, of R.P. Borwankar and D.T. Wasan (Chem. Eng. Sci., 1 (1986) 199). In the ion-binding model, the surface tension depends on the electrocapacitance in the layers closest to the interface and the distances between them, in addition to the surface charges on the planes. For the limiting case of a moderate concentration of surfactant, asymptotic formulae for the surface tension are derived. On a semilogarithmic graph of surface tension versus surfactant concentration in the presence of background electrolyte, the asymptotic slope a p p r o a c h e s - k T ( M t ) , where k is Boltzmann's constant, T is temperature, and (Mt) is the surface concentration of total sites, Mr, available for surfactant headgroups in the interface, the parentheses indicating concentration. In the case of no salt added, the asymptotic slope is -2kT(Mt). The asymptotic formulae also establish the influence on the surface tension of the equilibrium constants and the lateral interaction parameter, co, in Frumkin's isotherm. The ionbinding model results are in good agreement with the surface and interfacial tension data for sodium dodecyl sulfate (SDS). Agreement with measured (-potentials is also found for SDS at the air-water boundary.

Keywords: Adsorption; Aqueous-fluid interface; Ion-binding model; Ionic surfactants; Surface tension

1. Introduction The equilibrium tension of ionic surfactants at aqueous-fluid boundaries is important for the understanding of a wide range of colloidal phenomena. Current theories adopt a surface equation of state or equivalently an adsorption isotherm, along with a neutralizing diffuse double layer. Thus Davies and Rideal [ 1 ] use a Langmuir isotherm, * Corresponding author. 0927-7757/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved P H S0927-7757 ( 9 6 ) 03 592-3

Haydon and Taylor [ 2 ] use a Volmer twodimensional equation of state, and more recently, Borwankar and Wasan [ 3 ] select a Frumkin isotherm to allow for lateral interaction among the adsorbed surfactant tails. In each approach, an electrostatic energy contribution is included in the adsorption free energy. However for all the theories to date, the simple point-ion G o u y - C h a p m a n theory is taken to be an adequate description of interface electrostatics. Dating back to the work of Stern 1-4], and in

338

v.v. Kalinin, C.J. Radke/Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 337-350

particular that of Grahame [5], it is well known that even simple electrolytes at a charged interface deviate from G o u y - C h a p m a n theory, if no more than because of their finite size. Studies on aqueous ionic micelles show the importance of ion binding (also denoted ion pairing, ion complexation or ion condensation) of the counterions at the charged surfactant-aqueous interface [ 6 - 8 ] . Recent molecular dynamics simulations of ionic aqueous micelles clearly reveal the intimate proximity of partially dehydrated counterions to the charged surfactant headgroups. Similar ion pairing is recognized in Newton black films, which occur only above a critical background electrolyte concentration [9]. X-ray reflectivity measurements [10] and molecular dynamics simulations [11] of sodium dodecyl sulfate (SDS) Newton black films show the complete expulsion of co-ions and significant complexation of the sodium counterions with the sulfate headgroups. Here we extend the previous models for equilibrium interfacial tensions of surfactants by including contact ion pairing of the counterions with the charged headgroups. Counterion binding is handled by an equilibrium reaction similar to those used in site-binding models of the solid oxidewater interface [ 12-14]. The inclusion of ion complexation demands a more detailed picture of the double layer; we turn to the triple-layer structure of Grahame [5]. Once the equilibrium profiles of the various species are calculated, the surface tension follows directly from the Gibbs adsorption equation. We succeed in deriving an analytic expression for surface tension that shows the previous models of Davies and Rideal [1] and Borwankar and Wasan [3] to be subcases of this. The prediction of electrokinetic ~-potentials follows self-consistently from the triple-layer electrostatics.

2. Ion-distribution model

Fig. 1 presents a schematic of the equilibrium ion species distributions according to the proposed ion-binding model. A 1 : 1 fully dissociated anionic surfactant, denoted as NaS (e.g. SDS), adsorbs at an air-water interface in the presence of a simple 1 : 1 background electrolyte (e.g. NaC1) at concen-

a/a (off)

wat

(inner) (outer)

Helmholtz planes Fig. 1. Schematic of the ion distributions in the ion-binding model. Plane x = 0 corresponds to the aqueous-fluid interface; Xo is a plane of adsorbed surfactant ion centers; partially dehydrated counterions are located at xa, the B-plane (inner Helmholtz); and the d-plane (outer Helmholtz) at xd defines the closest distance between fully hydrated ions and the fully hydrated adsorbed surfactant.

trations below the CMC. The extension to cationic surfactants, partial dissociation, multivalent salts, or mixtures is straightforward and is not elucidated here. None of the aqueous species is volatile, and the air solubility in water does not change with surfactant or salt concentration. Thus, the air phase is considered to be inert. By analogy, an oil phase may replace the air phase provided that the oil is inert in the same sense, thus extending the ionbinding model to equilibrium interracial tensions between oil and water. We parcel the interface into three regions separated by the o,/3, and d planes. Driven by hydrophobic and coverage-dependent tail lateral interaction forces, surfactant ions with their accompanying hydration sheaths adsorb at the o-plane located at a distance Xo from the interface. Partially dehydrated counterions contact-bind with the adsorbed surfactant ions at the /3-plane located a distance x~ - xo from the o-plane. A finite size of all the remaining fully hydrated ionic species is recognized at the d-plane, at distance Xd beyond the surface. The diffuse double

K V. Kalinin, CZ Radke/ColloidsSurfacesA: Physicochem.Eng. Aspects114 (1996) 337-350 layer commences at the d-plane. Accordingly, the charged air-water interface exhibits a triple-layer structure with the //-plane serving as the inner Helmholtz plane and the d-plane serving as the outer Helmholtz plane. Dielectric constants in the two regions between the surface and the outer Helmholtz plane are labeled as Dos and D~a, respectively. All the charge arising from the adsorbed surfactant ions is neutralized in the aqueous phase by the bound cations and by the diffuse double layer. Far from the interface, all aqueous ions approach their bulk concentration values. One way of quantifying the adsorption of the surfactant species S- is by the equilibrium exchange reaction M+S-

Ks

~MS-

(1)

where M denotes an empty surfactant site or available space for the headgroup, M S - denotes an adsorbed surfactant, and Ks is the corresponding equilibrium constant. Parentheses indicate the surface concentrations (m-2), and square brackets indicate the bulk concentrations (m -3) (when later comparing theory to experiment we use molar units for concentrations) so that Ks

(MS-) •fcoO--eg~o~ (M)[S-]exp\ kT J

(2)

where 0 is the total fractional coverage of surfactant at the interface, co is the tail lateral interaction parameter (co is negative for attractive interactions), ~o is the electrostatic potential at the o-plane, e is the electron charge, k is Boltzmann's constant, and T is the absolute temperature. As adsorption increases, the absolute values of the charge at the interface and ~o increase, making additional surfactant adsorption more difficult. Conversely, increasing the magnitude of co favors adsorption and, if large enough, can lead to a surface phase transition between a two-dimensional gas-like state and a condensed, near-unity coverage state. Also Ks =-exp(--AG°/kT), where AG ° is the standard Gibbs free energy of adsorption. When later combined with a total site balance, Eq. (2) leads to a Frumkin isotherm [ 15]. To incorporate the ion binding of the counterions, we write an analogous equilibrium reaction

339

between mobile cations far from the surface and the adsorbed surfactant KNa

M S - + N a + ~- M S - N a +

(3)

where M S - N a + represents a bound counterion at the//-plane. The equilibrium expression for Eq. (3) reads ( M S - N a +) e~g~ KNa = ( M S _ ) [ N a + ] exp kT

(4)

where KNa is the equilibrium constant and [Na + ] is the total bulk concentration of sodium ions far from the interface. As in Eq. (2), an electrostatic correction term appears in Eq. (4), where ~, is the electrostatic potential at the//-plane. Conservation of sites or, equivalently, of the adsorption area demands that (Mt) = (M) + (MS-) + (MS- Na +)

(5)

where (Mt) is the surface concentration of total sites (i.e. (Mt)-1 is the minimum area per molecule available for surfactant adsorption at the interface). The symbol (M) represents the surface concentration of unoccupied sites. The adsorption coverage 0 of the surfactant in Eq.(2) follows as 0(Mt) -- (MS-) + (MS- Na+). Because electrostatic potentials appear in the equilibrium reaction expressions, it is necessary to specify how these vary with distance through the inner Helmholtz, the outer Helmholtz, and the diffuse-layer regions of Fig. 1. In the inner and outer Helmholtz regions with Xo < x < x~ and with x a < x < x d , no charge is distributed. Hence, Poisson's equation teaches that the potential profiles gt(x) are piecewise linear functions. Gauss's theorem, written at the inner and outer Helmholtz planes, specifies the requisite boundary conditions and leads to

( x , - Xo~ ~o - ~up = qo \ e0Do ~ j

(6)

and

(xa-x'~

(7)

~Pa -- ~ d = - - q a \ eo D#d ,]

where

eo is the

permittivity of free space,

V.V. Kalinin, CJ. Radke/Colloids SurJaces A: Physicochem. Eng. Aspects 114 (1996) 337-350

340

and qo -= - e [ ( M S - ) + ( M S - N a + ) ] and q~-e ( M S - N a +) are the surface charge densities at the inner and outer Helmholtz planes, respectively. The charge density in the diffuse layer, qa, follows from electroneutrality, qo + q~ + qa = 0, so that qd = e(MS-). G o u y - C h a p m a n theory permits analytical evaluation of the diffuse layer charge density for a 1 : 1 electrolyte [16]

0.6

==

qa=

2eoDk T

eTd

e

2kT

--Ksinh--

(8)

where ~c= x / 2 e 2 [ N a + ] / k T e o D is the inverse Debye length and D is the bulk aqueous solution dielectric constant. For multivalent surfactants and/or electrolytes, Eq. (8) must be replaced by numerical solution of the Poisson-Boltzmann equation. Once the bulk concentrations of surfactant, CNaS (i.e. [ S - ] = C N a s ) and bulk electrolyte, CN,Cl (i.e. [ N a +] = CNaS+ CNaCl) are specified, Eqs. (2) and (4)-(8) provide a complete algebraic system which is solved by Newton iteration. We reduce this system and solve for the two unknowns e~gp/kT and (MS-)/(Mt), although other choices are possible. Parameters include the equilibrium constants, Ks and KNa, the total site density, ( M t ) , the lateral interaction parameter, ~o, and the two integral capacitances of the inner and outer Helmholtz regions, g 2 o t 3 - e o D o ~ / ( x p - X o ) and g2t~d - 6oDt~d/(X d -- xt~ ).

Fig. 2 shows the calculated total surfactant coverage at the interface, 0, from the ion-binding model as a function of salt concentration at a surfactant concentration of 1 0 - 4 M. The adsorption equilibrium constant, Ks, and the lateral interaction parameter, co, are fixed at typical values, while the role of the ion-binding constant, KNa, is explored. Values for the inner and outer Helmholtz region integral capacitances in the figure are 278 ~tF cm -2 and 694 pF cm -2, respectively. These values were found from assumed thicknesses of 0.25 nm and 0.1 nm for the respective Helmholtz regions and from dielectric constants equal to the bulk value. Fig. 2 clearly reveals a major effect of the so-called indifferent electrolyte in increasing surfactant adsorption. Larger ion-binding equilibrium constants demand a larger fraction of the adsorbed surfactant to be complexed with the

I/

S 0.4I.]

/

/

0.2

.....

oe, R

--- B&W

................

0.0

0.2

0.4

0.6

"

0.8

1.0

NaC1 concentration, kmol/m 3 Fig. 2. The d e p e n d e n c e of the s u r f a c t a n t coverage, 0, on the salt c o n c e n t r a t i o n in the i o n - b i n d i n g m o d e l for a series of e q u i l i b r i u m c o n s t a n t s KNa at the c o n s t a n t surfactant concentration, CNas = 10 -4 M. O t h e r p a r a m e t e r s are K s = 6000m3kmol-1; ~o=-4.5k~ (Mt)=4.5x 10-6molm-2; x~ - Xo = 0.25 nm; x a -- xo = 0.1 nm; a n d Do~ = D~d = D. The d a s h e d line c o r r e s p o n d to the t h e o r y of B o r w a n k a r and W a s a n [ 3 ] (KN~ = t2op1 = f2~-d1 = 0). The d o t - d a s h e d line c o r r e s p o n d s to the t h e o r y of D a v i e s a n d Rideal [-1] (~0 = 0).

solution counterion leading to lowered electrostatic repulsion between the headgroups, and hence to higher adsorption. Essentially, raising the bulk sodium ion concentration drives more surfactant to the interface. Dashed and dashed-dot curves in Fig. 2 correspond to the models of Borwankar and Wasan [ 3 ], and Davies and Rideal [ 1 ], respectively. Since both of these models ignore any electrostatic structure of the interface, they each predict the effect of the added salt to follow G o u y - C h a p m a n theory. The present ion-binding model reduces to that of Borwankar and Wasan when there is no ion binding (i.e. KNa = 0) and when the inner and outer Helmholtz integral capacitances approach infinity (i.e. point ions). Fig. 3 displays calculated values for the individual fractional coverages at the interface for the total surfactant, 0-- [ ( M S - ) + ( M S - N a * ) ] / ( M t )

V. V. Kalinin, C.J. Radke/Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 33~350

341

3. Surface tension /.i

o

"

/

"~ 0 . 4 ~

/

3.1. Theory

To obtain an expression for the surface tension from the ion-binding model, we utilize the classic Gibbs adsorption equation written for all molecular species

,

#

,f,'

- d y = FH2Od#a2o + Fs- d#s- + FNa+ d#h~ +

//

"

//#

#I

11

1/

0.2

00

+ Fcl- d # o -

,' NaCI = 0

10-6

10-~

104

(9)

i

10"3

10-2

NaS concentration, kmol/m 3 Fig. 3. Surfactant coverage, 0 (solid lines), and bound counterion coverage, 0ha (broken lines), as functions of surfactant concentration for a series of salt concentrations.

where 7 is the surface (or interracial) tension and #i is the chemical potential of species i. No term for the gas (or oil) species appears in Eq. (9) due to the inertness assumption. Here ~ is the Gibbs surface excess of species i chosen relative to the outer Helmholtz plane

Fi =

J Ci(x) -- oo

(solid lines), and for the counterion-paired surfactant, 0 N ~ - ( M S - N a + ) / ( M t ) (broken lines), as a function of surfactant concentration on a semilogarithmic scale. We note that a very large fraction, over 90%, of the adsorbed surfactant is ion-complexed, even with no added salt. This statement is valid over a wide range of equilibrium constants, KNa and Ks. Thus, in the ion-binding model, solution counterions play a strong role in influencing the surfactant adsorption behavior, a feature that is missing from the current point-ion treatments of the interface. For values of 09 ~< - 4 k T , the classic Frumkin isotherm shows the presence of a surface phase transition between a dilute coverage and coverage near unity. The present ion-binding model also allows a similar surface phase transition, but at a much more negative value of the constant oJ. With the typical parameters used in Figs. 2 and 3, o9 must be less than - 1 2 k T before the surface phase transition emerges. The reason for the much larger magnitude of the Frumkin parameter is the electrostatic repulsion between the charged headgroups that must be overcome to permit a net lateral attraction between the adsorbed ionic surfactants.

+

[ C i ( x ) - C~°] d x

(10)

x d

where C~(x) is the volume concentration profile and the superscript ~ indicates the bulk aqueous value. The first integral in Eq. (10) accounts for specific ion adsorption, whereas the second integral quantifies diffuse-layer adsorption. In the ion-binding model, specific adsorption occurs in the o- and fl-planes. Thus, specific adsorption of the surfactant is [(MS-) + ( M S - N a + ) ] f ( x - Xo) while that of sodium ions is ( M S - N a + ) b ( x - x a ) , where 3(x) denotes the delta function. According to Fig. 1, there is no specific adsorption of the co-ion, C1-. From Gouy-Chapman theory, the diffuse-layer adsorption of species i for symmetric electrolytes may be expressed analytically in terms of the electrostatic potential at xa [ 16]

i [Ci(x) - C~ ] d x = -2 C~° K

Xd

Eex.(

'l (11)

where zi is the ionic valence. Next, the species chemical potentials in Eq.(9) are expressed in terms of components, dpNaS= dpNa+ + d#s and d#N~O = dpNa+ + d#cl , electroneutrality of the entire interfacial region is imposed, FNa+ =

342

V.K Kalinin, CJ. Radke/Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 337-350

F o - + Fs-, and the Gibbs-Duhem equation is written for the bulk aqueous solution. When these relationships are inserted into Eq. (9), we recover the expression -d])=

Fs-

])1 - kT(Mt) ln(1 - 0)

CNaS

4kT~oD e

× /CN.c, + CN.S cosn

C ~ o Fn2o) d#N~s

+ t//FCl-

CNaCIFn2o)x d#NaCl

(12)

Note that the second term within each set of parentheses contains a concentration ratio and may be neglected provided that Cn2o >> CN~S, CN,C~. In dilute solution, where ion activity coefficients are negligible, the component chemical potentials of the surfactant and the salt become d/ANas =

and Rideal [ 1 ]

kTd In CNaS(CNaS -~- CNaCI)

(13)

-- 1_

(18)

except that the diffuse double layer potential, ~a, now appears in the argument of the hyperbolic cosine, instead of the surface potential, ~o. Both functions on the right-hand side of Eq. (18) are negative and thus reduce the surface tension from the surfactant-free value. The first is due to specific surfactant adsorption in the interface, and the second is due to the formation of the diffuse double layer. The second term in Eq. (17)

(Mt)¢_o02

and

])2 -

d#N~Cl = kTd In CN~cI(CN.s + CN~Cl)

Substitution of Eqs. (13) and (14) into Eq.(12) yields d])

(•

+

= \ CNaS "{- CNaC1 C~NaS (

FN~_+

dCN~s

FCl- "~

+ \CNas + CNaCl + ~ )

dCNac!

2

(19)

(14)

(15)

At constant salt concentration Eqs. (10), (11), and (15) reduce to

was obtained recently by Borwankar and Wasan [-3] and accounts for Frumkin lateral interactions among the surfactant tails. For typical attractive interactions, co < 0, and ])z gives a positive contribution to the surface tension and a corresponding negative contribution to the spreading pressure, as expected. Our ion-binding theory provides the new, third term in the expression for the equilibrium surface tension ])3 = -- 4kTeoD(CNao + CNas)(C2op1~ 2 .~_~r'~-dl)

_d]) = k T I 4 (cosh ~ _

1) + ( M S - N a + ) CNaS + CN~Cl

(MS-) + ( M S - N a + ) -] + ~ J dCN~s

z sinh 2 e~td

2kr

(16)

and

the

factor

KNa(CNacl d- CNaS)

If we now chose an integration path along a constant salt concentration and if we neglect any effect of salt on the surface tension of a surfactantfree interface, then after some difficult algebra (see Appendix A), we obtain three additive terms for the surface (interfacial) tension in excess of the clean interface value, V0

]) --

])0 = ])1 At- ])2 "Jr- ])3

(17)

The first term in Eq. (17) was found by Davies

(20) ~¢ is

defined by

~1=1+

exp(--e~tJkT). Similar to the

diffuse double layer contribution, the ion-binding contribution in Eq.(20) lowers the equilibrium surface tension due to formation of the inner and outer planes of the electrostatic triple layer. Clearly the ion-binding model reduces to that of Borwankar and Wasan [-3] when KNa = g2ot~1 = O~-d1 = 0, and further, to that of Davies and Rideal [1] when co = 0. It may also be easily established that the ratio of the ion-binding contribution and the diffuse layer part of the Davies and

V. K Kalinin, Car. Radke/Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 337-350

Rideal expression is of the order X(Xd--Xo). Therefore, since (Xd -- Xo) is a characteristic molecular size, the ion-binding contribution is significant whenever the Debye radius is comparable to that molecular size. The ion-binding model demands six parameters: Ks, KNa, co, (Mt), g2o#, and I2ad. Of these, (Mt) , Qoa, and g2ad have clear molecular interpretations and may be set physically. Nevertheless, three parameters remain that must be fit from experimental surface interfacial tension data. To aid in the fitting process, we have developed several asymptotic forms of Eq. (17) for moderate concentrations of surfactant such that Ks CN,s >> 1, as enunciated in Appendix B. In the case of finite salt concentrations and CN~Cl>> CN~S we find that -- ~o - _ -ln[KsCN.s(1 + kT(Mt)

fllKNaCNaCI)]

1 [ co e2(Mt)l + 2 k--T+ ~ J

+ f12

(21)

where fll and f12 are known functions of the salt concentration (see Appendix B). In the absence of added salt, CN~Cl= 0, the corresponding result is -- ~o _ -ln(KsKN.C~as) kT(Mt)

- -

1 I co

eE(Mt)l

+ 2 V-T+ kTao 3

(22)

Eq. (21) reveals that on a semilogarithmic plot of surface tension versus surfactant concentration in the presence of added salt, the asymptotic slope approaches - k T ( M t ) . Likewise, with no salt present, a similar plot approaches an asymptotic slope of - 2k T(M t). Such asymptotic analyses, if obeyed by the data, provide both strong confirmation of ion-binding theory, and a useful means for estimating the equilibrium constants, Ks and KNa. We have also derived asymptotic forms of Eq. (17) for surfactant concentrations approaching zero, but they do not prove as useful in the fitting process. 3.2. C o m p a r i s o n to data

To assess the ion-binding model, we consider the equilibrium surface tension data by Matuura

343

et al. [17], Mysels [18], and Fainerman [19] for SDS solutions at the air-water interface, and the interfacial tension data of Haydon and Taylor [2] for SDS at the petroleum ether-water interface. Model constants (Mr) , Qo#, and f2ad were chosen by physical reasoning. Thus, we adopted thicknesses of the two closest interface layers of xa - xo = 0.25 nm and Xd -- XO= 0.1 nm that correspond to reasonable ion sizes. The dielectric constants for these layers were chosen to be equal to the bulk value. The total site density was set as ( M t ) = 4.5 x 10 . 6 mol m -2, a value corresponding to the closest packing of SDS at the interface with 0.37nm 2 for each ion. Thus, three adjustable parameters have been used in the fitting procedure, namely, the equilibrium constants, Ks and KNa, and the Frumkin interaction parameter, co. Once these values are determined, the model must represent the equilibrium surface tension measurements over the whole interval of background electrolyte concentration and surfactant concentration. To ascertain the fitting parameters we utilize the limiting tension behavior portrayed in Eqs. (21) and (22). Fig. 4 compares the asymptotic formulae, as dashed lines, to the best-fit ion-binding model results in solid lines and to the experimental surface tension data of Matuura et al. [17] and Mysels [ 18 ] at moderate surfactant concentration. Indeed, the data do approach the correct limiting slopes both with and without salt present. Eqs. (21) and (22) also show that the surface tension is quite sensitive to the product of KsKNa and is not nearly as sensitive to the individual values of Ks or KNa. The physical reasoning is that the standard Gibbs free energy of the overall reaction, M + S - + Na ÷ ~ M S - N a +, dominates the adsorption process. This observation is consistent with Fig. 3 indicating the very large fraction of counterion-paired surfactant adsorption. In Fig. 5, K s K N a = 0.368 m 6 mol-2. Once the equilibrium constant product is set, minor adjustments in Ks = 6.0 m 3 mol-X and KNa = 0.0613 m 3 mol-1 are made in order to provide the best least-squares fit to the data. The second term on the right-hand side of Eqs. (21) and (22) reveals a sensitivity of the ion-binding model to a weighted sum of the Frumkin interaction parameter and the inner Helmholtz integral capacitance:

344

V. 1~. Kalinin, C.J. Radke/Colloids Surfaces A." Physicochem. Eng. Aspects 114 (1996) 337-350 'I

'

'

''''"I

'

'

''''"1

t t

55

'

'

'''"I

'

'

l

~

'

'''"'I

'

'

'''"'I

'

'

'''"'I

'

'

' .....

I

70

t t

t

~

'

''''"i

t

~

o o

60

Z

E

•

=~ 50

°~ r/3

Mysels

• ~D

o

ioL \ °o,\,

O Matuura 40

My~,,

o\

,

~o

o \

•

o\ o

35

2 - CNaC,= 0.5M 3- CNaC,= 0.1M

30

4 - eNact = 0

. '{,- ~ 10'~' ~ ~

'~t

~,

30

"1~t

4

\ i

10.6

.......

I

10.5

,

,

,t=,,,I

=

10-4

,

23-

,,,,,,I

,

,

10"2

SDS concentration, kmol/m 3

10.6

=

CNaci

=

0.5M 0.1M

=

0

- CNaCI ,

lllllll

10.3

CNaCI

|

ml....I

.

10-s

\ .

......I

.

10-4

.

......I

.

.

......I

10.3

10-z

SDS concentration, kmol/m 3

Fig. 4. Asymptotic behavior of the surface tension at moderate surfactant concentration. The broken straight lines have a slope of - k T ( M t ) in the presence of salt (Eq. (21)) and a slope of - 2 k T ( M t ) with no salt present (Eq. (22)). The solid curves correspond to the best fit of the ion-binding theory. Experimental data are from M a t u u r a et al. [17] and Mysels 1-18].

Fig. 5. Equilibrium tension of the aqueous SDS-air interface as a function of surfactant concentration. The solid curves correspond to the best theoretical fit ( K N a = 6 0 m 3 kmol-1; other parameters are mentioned in the caption to Fig. 2); the empty circles represent the data of M a t u u r a et al. [17], the filled circles represent data of Mysels [18], and the filled diamonds represent data of Fainerman [ 19].

~o/kT+ e2(Mt)/kTl2oa. Thus,

pairs at increased NaC1 concentration, there is relatively not as much preference for the dodecanol at the surface. We caution that equilibrium surfactant tension data for ionic surfactants with no salt present can readily be suspect because of the high sensitivity to possibile impurity species. Conversely, such data with salt present appear to be more reliable. Fig. 6 demonstrates the use of the ion-binding model for the equilibrium tension of SDS at the water-oil boundary. Really quite excellent agreement is found with the Haydon and Taylor data [2]. The model fitting procedure gives Ks = 100 m 3 mo1-1, KNa = 0.07 m 3 mo1-1, and ~o = -0.5kT. A smaller lateral tail interaction parameter is to be expected at the oil-water interface compared to that at air-water interface. The ionbinding model well represents both equilibrium surface and interfacial tension data.

altering the value of 12oa, by changing either xa - Xo or Dop, is compensated by an offsetting adjustment in 09. The value of ~o chosen in Fig. 5 is --4.5kT. Little sensitivity of the model to the outer Helmholtz integral capacitance is noted. Fig. 5 highlights the comparison between the proposed ion-binding model and the entire equilibrium surface tension data for SDS at the air-water interface of Matuura et al. [ 17], Mysels [18], and Fainerman [19]. G o o d agreement is seen over the available range of surfactant and salt concentrations, even up to rather large salt concentrations of 1 M. It is important to note the discrepancy between the data of Mysels at zero salt concentration and those of Matuura et al., also at zero added NaCI. The most likely explanation is that the SDS sample of Matuura et al. was slightly contaminated with the hydrolysis product, dodecanol. At a higher salt concentration, the impurity alcohol does not appear to contribute as strongly to the measured tensions. Apparently, as more of the adsorbed surfactant is in the form of counterion-contact

4. Zeta-potential It is accepted that measured ~-potentials correspond to the mobile, most inner part of the diffuse

V. V. Kalinin, CJ. Radke/Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 337-350 '''"I

'

' ' '''"I

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' ' '''"I

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' ''''"I

-270

"I

'

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' ''''"I

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345

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1 - CNaCI = 0

40 0

2 - CNaCl= 10"* kmol/m3 - ion-binding

Haydon and T a y l o r

-220

---

1I=

30

~

"~

CNacl

--

2

- CNaci

=

3

- C N a C I --

0.25M

-

-

~

-170 _1

0.1M

[]

A

.~.

4 " CNaCI = 0.05M

~

.

B &W

0.5M

2."

e- -120

o 20

n

A

o '

1o

-70

\ 1

=,,1

10-5

=

= ......

I

10-4

/

-20

. . . . . . . .

I

. . . . . . . .

10-3

I

10.2

,,I

A _ [20], pH=5.6, micro [] - [20], pH=5.6, D o m • - [21], pH=10.3, micro ,

10.6

SDS concentration, kmol/m 3 Fig. 6. The equilibrium tension of the aqueous SDS-petroleum ether interface as a function of surfactant concentration. The solid curves correspond to the best fit for the ion-binding model (KN== 70 m 3 k m o l - 1; K s = 105 m 3 k m o l - 1; tn = - 0 . 5 k T ) . The experimental data (empty circles) were obtained from Haydon and Taylor [2].

double layer. In the ion-binding model, therefore, the potential at the d-plane, ~a, in Fig. 1, corresponds to the (-potential. The solid lines in Fig. 7 report calculated (-potentials at two salt concentrations for SDS adsorbed at the air-water surface. Parameters in these curves are those obtained from fitting the equilibrium surface tension data. In the previous point-ion models for surface tension, the (-potential corresponds to the potential ~o at the o-plane. The broken lines in Fig. 7 reflect this calculation for the Borwankar and Wasan model [3] using their best-fit parameter values from SDS surface tension data. The ion-binding model predictions of the (-potential fall well below those of Borwankar and Wasan, because a large portion of the potential drop across the interface occurs through the inner and outer Helmholtz regions and not over the diffuse part of the double layer. Maxima in both the broken and solid curves are due to an ionic strength effect. Empty and filled symbols in Fig. 7 give experimental l-potential data for air bubbles in SDS solutions by Kubota et al. [20] and Yoon and Yordan [21]. Empty triangles and filled circles

, ,,,,,it

10-s

........

I

104

,

, ,,,,,,I

.

, ,,1,,,I

10.3

10-2

SDS concentration, kmol/m 3 Fig. 7. The zeta-potential (Ttd) as a function of the surfactant concentration for the aqueous SDS-air boundary. The solid lines correspond to the ion-binding theory with model constants from the surface tension data (see Fig. 5). The broken lines correspond to the point-ion model of Borwankar and Wasan [3]. Empty triangles (data from Ref. 1-20]) and filled circles (data from Ref. 121 ]) denote experiments based on microelectrophoresis. Empty squares (from Ref. 120]) represent experimental data based on the Dorn effect.

reflect microelectrophoresis potential data, while empty squares give sedimentation potential (Dorn) data. In the calculations we take pH 5.6 as equivalent to zero added salt, and pH 10.3 as equivalent to CNaC1 = 10 - 4 kmol m -3. At low surfactant concentrations, the ion-binding model does well in predicting the experimental data. At higher concentrations, it overestimates the data of Yoon and Yordan [21] and underestimates the data of Kubota et al. [20]. We stress that no adjustable constants were used in the predictions of Fig. 7. More experimental (-potential data are needed to permit a definitive comparison between the ionbinding and point-ion pictures of the interface electrostatics.

5. Conclusions

A new ion-binding theory is presented for the equilibrium behavior of ionic surfactant adsorption

346

v.v. Kalinin, CJ. Radke/Colloids Surfiwes A. Physicochem. Eng. Aspects 114 (1996) 337-350

at aqueous-fluid interfaces. The proposed theory is based on G r a h a m ' s triple-layer electrostatic structure of the interface: a surface layer of adsorbed surfactant, a plane of b o u n d counterions (inner Helmholtz plane), and a diffuse double layer c o m m e n c i n g at the plane of the hydrated co-ions and counterions (outer Helmholtz plane). The present ion-binding model is a generalization of previous point-ions models of Davies and Rideal [ 1 ] and B o r w a n k a r and Wasan [ 3 ] . An asymptotic analysis permits an explicit analytical expression for the surface tension at moderate surfactant concentrations. A linear dependence of the surface tension on the logarithm of surfactant concentration is established, b o t h with and without added salt. We find g o o d agreement between the ionbinding theory and experimental tension results for aqueous S D S - a i r and aqueous S D S - o i l interfaces. The ion-binding model also predicts the main features of measured (-potentials with no adjustable constants.

Acknowledgement This w o r k was supported by grant CTS 9307890 of the Chemical and T h e r m a l Systems DivisionInterfacial T r a n s p o r t and Separation Processes P r o g r a m of the N a t i o n a l Science Foundation.

References [1] J.T. Davies and E.K. Rideal, Interfacial Phenomena, Academic Press, New York, 1963. [2] D.A. Haydon and F.H. Taylor, Philos. Trans. R. Soc. London, Ser. A, 253 (1960) 255. [3] R.P. Borwankar and D.T. Wasan, Chem. Eng. Sci., 1 (1986) 199. [4] O. Stern, Z. Elektrochem., 30 (1924) 508. [5] D.C. Grahame, Chem. Rev., 41 (1947) 441. [6] D. Stigter, J. Phys. Chem., 79 (1975) 1008. [7] T. Yoshida, K. Taga, H. Okabayashi, K. Matsushita, H. Kamaya and I. Ueda, J. Colloid Interface Sci., 109 (1986) 336. [8] J.A. Buenen and E. Ruckenstein, J. Colloid Interface Sci., 96 (1983) 469. [9] D. Exerova, T. Kolarov and Khr. Khristov, Colloids Surfaces, 22 (1987) 171.

[10] O. Belorgey and J.J. Benattar, Phys. Rev. Lett., 66 (1991) 313. [11] Z. Gamba, J. Hautman, J.C. Shelley and M.L. Klein, Langmuir, 8 (1992) 3155. [12] D.E. Yates, S. Levine and T.W. Healy, J. Chem. Soc. Faraday Trans. 1, 70 (1974) 1807. [13] R.O. James and T.W. Healy, J. Colloid Interface Sci., 40 (1972) 65. [14] D.C. Prieve and E. Ruckenstein, J. Colloid Interface Sci., 60 (1977) 337. [15] A. Frumkin, Z. Phys. Chem., 116 (1925) 466. [16] J. Newman, Electrochemical Systems, Prentice-Hall, Englewood Cliffs, New Jersey, 1991. [17] R. Matuura, H. Kimizuka and K. Yatsunami, Bull. Chem. Soc. Jpn, 32(6) (1959) 646. [18] K. Mysels, Langmuir, 2 (1986) 423. [19] V.B. Fainerman, Colloid J. USSR, 40 (1978) 769. [20] K. Kubota, S. Hayashi and M. Inaoka, J. Colloid Interface Sci., 95 (1983) 362. [21] R.-H. Yoon and J. Yordan, J. Colloid Interface Sci., 113 (1986) 430.

Appendix A: Analytic expression for surface tension In this appendix we derive the analytic expression for the equilibrium surface tension from the p r o p o s e d ion-binding model, Eq. (17). It proves convenient to introduce the following dimensionless variables: ~bo = eTto/kT; d?~ = e ~ / k T ; q)d = e~d/kT', ~o = q o / e ( M t ) ; ~d = qd/e(Mt); a = ~/kT(Mt); C9 = tn/kT', CNa = C N a o / C , ; Cs = CNas/C.; ks = K s C . ; kNa = K N ~ C . ; 0s = ( M S - ) / ( M t ) ; and 0Na = ( M S - Na+)/(Mt), where C . is a characteristic concentration, for example, l kmol m -3. The final results are independent of the choice of C,. Eqs. (2) and (4)-(8), which describe the present ion-binding model, yield a governing system of equations in the dimensionless variables ~bo - ~a = do ¢o

(A1)

~p - ~bd = - d d ~d

(A2)

~ = - 2 A X/CN~+ CS sinh ~

(A3)

~o = - ( 0 s + 0N~)

(A4)

~d = 0s

(A5)

0s -

kscs d k s c s + e x p ( - ~bo- ~3~o)

0Na = kNa 0s(Cs + CNa ) exp(-~bt0

(A6) (A7)

347

V. V. Kalinin, CJ. Radke/Colloids Surfaces A." Physicochem. Eng. Aspects 114 (1996) 337-350

Here the function ~¢ is expressed via the dimensionless variables analogously to its definition in the text ~¢ = 1 + kN~(cs + cNa) exp(--¢a). The constants d o and Ad in Eqs. (A1) and (A2) are defined by A o = e2(Mt)/(k TI2oa), and zJa = e2(Mt)/(k Tg2aa). The dimensionless length A in Eq. (A3) is defined by A = x/2C, k TeoD/e(Mt). In dimensionless variables, Eq. (16) for the differential surface tension at constant salt concentration becomes -da=[

2A 7~-y--, ( c o s h ~ - I )

tension -do -

2A + ( 03¢0

(cosh ~ - 1) dcs 1+1 ~ ) d¢o + 2 A ~

× [~¢ d ~ b o - ( d - 1) d¢a] s i n h ~

(A12)

Then, from Eqs. (A1)-(A7) we have

1N/CNa + Cs \ ¢o -

ONa OS + ONa-] -~- CS ~ "[- CN "[- - -Cs _.1dcs

(A8)

Substitution of Eqs. (A6) and (A7) into Eq. (A4) gives ¢o = - 0 s [-1 + kNa(Cs -F CNa)exp(--Ca)] = --~¢0s

¢~ = - A o ( 0 s + 0Na) = - - A o d 0 S

(A13)

¢o -- ~d = Ao~o -- AdCd = --(ZJo~q~ -~- Z~d)0S

Substitution of Eq. (A13) into Eq. (A12) yields

d.

(

dkscs

')

+ 03¢° 1+¢o d¢o+2Ax/~CNa+cs

~¢kscs + exp(- ¢o - 03¢0)

where the first and the last identities may be rewritten as

x sinh ~ [dSd -- A o d d(d0s) - Ad dos]

(A9)

(A14)

Then, from the definition of the function d , we obtain

Since [03¢0- 1/(1 + ¢o)] d¢o = - d In(1 + ¢o) + (03/2) d(¢ 2) and since

~¢kscs(1 + ¢o) = -¢o exp(-~bo - 03¢0)

,

Cs[1 + kNa(cs -F Csa) exp(--¢~)] ¢o exp(-~bo - 03¢0) ks(1 + ¢o)

(A10) + x/~CNa+ CSsinh ~ dCd

Differentiation of Eq. (A10), while using Eq. (A9), yields, after division by d

I

1 + kNaCSexp(-¢a) 1 dcs kNa CS(Cs +

+

..[ c

(cos 1)1

Eq. (A14) may be re-expressed as

CNa)exp(-- ¢~) de B- Cs d~bo

I ¢o(1+~o) 1 o31 csd¢o

1)d.s

-da= (All)

Now using Eq. (A1 1), we can selectively eliminate the differential dcs from Eq. (A8) for the surface

03 2 - d ln(] + ~o)+ ~- d(~o) +4A d [

~

(cosh~-

+ 4A 2X/CNa+ CSsinh ~ x Z

1)]

348

V.V. Kalinin, CJ. Radke/Colloids Surfaces A." Physicochem. Eng. Aspects 114 (1996) 337-350

trations, 0Na, the governing system becomes X[Aodd(C~N~+es~Csinh~)

+ As d

(

(B1)

~bo-~b, = Ao~o

(B2)

(A15)

+ Cs sinh

Integration of Eq.(A15) from the initial point which corresponds to a zero concentration of surfactant (a = %; ~o = 0; ~bs = 0; Cs = 0) gives the desired expression for the dimensionless surface tension according to the ion-binding model of surfactant adsorption

• cs Cd = - 2A C ~ N . slnh -~-

(B3)

~o = --(0S "~- 0Na)

(B4)

~d = 0s

(B5)

kscs Os= ~kscs + e x p ( - ~bo- 05~o) ONa= kNaOSCNaexp(--~bt~)

O~ 2

a - ao = In ( 1 + Go) - ~ ~o - 4A ~

x (cosh ~ -

Cs

(B7)

where the function d is now defined as d = 1 + kNaCNaexp(--~bp). First, we write a limiting solution of the governing Eqs. (B1)-(B7) for kscs-~

l ) - 2A2(CN. + cs)

dJ x ( d o d 2 + Ad) sinh 2 , d 2

(B6)

1

(A16)

Eq. (17) emerges after dimensional variables are inserted.

~o°~ = - E 0 F ~ + 0 ~ V ] = - 1

~(o~) = ~ o ) _ / t o

(B8)

As

Appendix B: Asymptotic formulae for surface tension

For the important and common case of relatively large surfactant concentrations, just below the critical micellization point, it is possible to obtain explicit asymptotic results for the equilibrium tension in the present ion-binding model• Two cases are considered separately in this appendix: the presence and the absence of background electrolyte. In both cases, we restrict our attention to KsCs=k~cs>> 1, which is denoted as a moderate surfactant concentration. Consider first the case of added salt. In addition, we assume that the concentration of surfactant is much less than the concentration of salt, Cs <
The constant ~(o~) is the limiting value of the function d at k s c s - ~ and is related to the limiting value of the potential at the fl-plane, ~oo), by two relationships. The first of these is the definition of function d

~(~o) = 1 + kN~CN~e x p [ - - ~ ° ) ]

(B9)

and the second relationship may be obtained from Eq. (B3) 1

d t°~' _

2 ~oo)+ ~ 5

(B10)

We note that in the present limiting case, the solution of the governing system is non-trivial and, moreover, there is no limiting value of the surface tension cr as kscs--* oo because the corresponding value of the tension decreases unbounded as ~(o~) --* - 1. Therefore the asymptotic theory below is valid for relatively large, but limited values for ksc s. That is why we refer to this case as "moderate

349

K K Kalinin, CJ. Radke/Colloids Surfaces A." Physicochem. Eng. Aspects 114 (1996) 337-350

concentrations" of surfactant. The other limiting constants in Eqs. (B8) may now be obtained by the solution of two non-linear algebraic Eqs. (B9) and (B10). We seek a solution of Eqs. (B1)-(B7) as a series expansion in negative integer powers of kscs

d-plane via the constant d (~) using Eq. (B10)

=

ZO)

(Bll)

z = z ~ ) + ksc~s + "'"

where the vector z represents a set of variables: z = {~o, ~b,, ~bd, ~o, ~d, 0S, 0Na}. Upon substitution of Eqs. (B8) for the zero-order terms and the expansions, Eq. (B11), into Eq. (A16) and neglecting terms of lesser order, we obtain

F

4~~ o~ a-a°=lnksc~

2

s

~b~~

/

cosh -~ -

l+sinh 2 2

N/

1

1 -t- 4[d(oo)]2A2CNa

Substitution of this last result, as well as Eq. (B4), into Eq. (B14) allows us to obtain a final expression for the dimensionless surface tension for the case of a moderate concentration of surfactant in the presence of background electrolyte a - ao = -ln[d(°~)kscs] +

]

4A c ~ N ~ L c ° s h 2 - - - - 1 - 4 A C~N~

- 22ZcNa{Ao[s¢(~°)] 2 + Ao} sinh 2 2

1 + 4[s~/~o~)]2A2cN~ -- 1

Ad

(a15)

(B12) We now need a first-order expansion for the variable 40- It follows from Eqs. (B4)-(B7) and the definition of a / t h a t

When expressed in dimensional form, Eq.(B15) identifiess the functions fll and fiE appearing in Eq. (21) fll - exp[--~ b~°~)]

~kscs 4o = - O s d = -

x / 3 2 k TeoDCN~cl

suCkscs + exp(-~bo - c3~o)

fie = -~b~ ~ ) -

1 =

- -

1 + (1/,~¢kscs) exp(-~o -- C3~o) x

Thus, we find that ~ol) - s¢(~ ~exp [-~b~o°°)- ~ o °~)] where, as was shown, ~ o ~ ) = - 1 . Eq. (B13) and Eq. (B12) leads to a0 =

1 + 8[d~o~)]2kTeoDCN.o

Ad

1

a -

{j

e(Mt )

- In [~¢(°~ks

Cs] -

(B13) Combining

~b~o°~)

] c 5 - 4 A C~N~LCOSh~---1 F

2[_d~oo~]2

~)

-- 2A2cN~ {[su¢~°)]2 + Aa} sinh 2 ~

(B14)

Now we can express all terms containing the limiting value ~b~a ~ of the electric potential at the

1

} (B16)

In the absence of background electrolyte, CN, = 0, the governing system of dimensionless equations (A1)-(A7) have the form ~ o - ~b#= Ao~o

(B17)

~b#-~b d = --Ad4a

(B18)

~d = - 2A ~Cs sinh ~

(B19)

4o = -(0s + 0Na)

(B20)

4d = 0s

(B21)

350

V.V.Kalinin, CJ.

Radke/Colloids Surfaces A. Physicochem. Eng. Aspects 114 (1996) 337-350

Thus, we find that

ksy Os = xdkscs + e x p ( _ ¢ ° _ e54o)

(B22)

0N. = kNaOSCs exp(--¢a)

(B23)

~ ) = k--Lexp(d o + o5) where ~ ' is now s g = l + k N . C s e x p ( - - ¢ a ) . The limiting solution of the system (B17)-(B22) at kscs ~ oo is

(B27)

kNa

Since ~d = 0S

kscs =

_-

_-

=

[ 1 + kNaCs exp(--Ct0]kscs + e x p ( - ¢ o - o5~o)

0

-1 =

(B24) 1

-Ao

Analysis of the system (B17)-(B23) allows us to find the first terms of the expansion for all variables ~d = 0s

- kNaCs + "'"

we have after substitution of Eq. (B19) 1

0~sl)

_

2A ~ s

kNaCs

= ks cs + "'"

~(d3/2~) (kscs) 3/2

It follows that k3/2

4o = - 1 + (kscs) 2 + ...

¢~3/2) =

(B28) 2AkNa

AdO~s1~

From Eqs. (B27) and (B28) we obtain the final expression for the surface tension

¢~ = -- ksc----~+ " "

(B25)

~)d = (kscs)3/2 q- ...

do+Oh

a - - ao = -- ln(kskNa c]) + ~

It should be mentioned that in the case of no salt we adopt a series expansion in negative integer powers of (kscs) 1/2. Using Eqs. (B25), Eq. (16) for the surface tension may be transformed to a - ao = ln(1 + 40) - -~ 4o -- 4A~cs

cosh ~- - 1

-- 2A2cs(Ao d 2 + Ad) sinh 2 ? 4~ )

- l n kscs 2~

~ 2

2A2dokZN, k3 [¢~B/Z}]2 (B26)

where we neglect terms that are much less than unity. Our aim now is to find the terms 4~ ) and ¢~3/2). From Eqs.(B17)-(B23) we have, within the accuracy of the main terms 40 = - 1 +

e x p ( - ¢o -- ~54o) ,~¢ks c s

= --1+

exp(Ao + ~) ks kN, C~

(B29)

Two important points may be emphasized. First, the dependence in Eq.(B29) of a - % on the logarithm of the dimensionless surfactant concentration Cs in the zero salt case is a straight line with double the slope of that in Eq. (B15) when background electrolyte is present. Second, the asymptotic formula, Eq. (B29), involves only the integral capacitance of the inner Helmholtz region and the thickness of that layer. No terms involving the outer Helmholtz region appear. Moreover, in the case of moderate surfactant concentration and in the absence of background electrolyte, the factor c o s h ( ¢ a / 2 ) - 1 in the Davies and Rideal theory [ 1 ] does not influence the surface tension.