Adsorption of interacting long-chain surfactant molecules: Isotherm equations

Adsorption of Interacting Long-Chain Surfactant Molecules: Isotherm Equations LUUK K. KOOPAL 1 Department of Physical and Colloid Chemistry, Wageningen Agricultural University, De Dreijen 6, 6703 BC Wageningen, The Netherlands AND

GORDON T. WILKINSON AND JOHN RALSTON School of Chemical Technology, South Australian Institute of Technology, Ingle Farm, S.A. 5098, Australia Received May 18, 1987; accepted January 15, 1988 A recently introduced lattice model for the adsorption o f long-chain flexible ionic surfactants is briefly explained and generalized to describe the adsorption of both ionic and nonionic surfactants. To obtain analytical isotherm equations the general model is simplified by considering a stepwise segment density profile and a conformation in which m of the r segments are thought to be adsorbed in the first layer, whereas the remaining r - m segments protrude in the solution phase as a tail. Segment-segment and segment-solvent interactions are incorporated using the Flory-Huggins approach. The isotherm equations derived for both types of surfactants are very similar. Apart from the adsorption energy they show the effects of train size, chain length, solvent quality, and the specific nature of the head segments. Approximate expressions are given for the conformational entropy loss upon adsorption. For chains of one segment the equations reduce to the classical models. The isotherm equation for nonionic surfactants is an improvement of Kronberg's adsorption model. © 1988AcademicPress,Inc.

INTRODUCTION

In a previous paper (1) a general model has been developed to describe the adsorption of long-chain ionic surfactants. The model is based on the polymer adsorption theory of Scheutjens and Fleer (2, 3) in which electrostatic interactions have been incorporated in a way similar to that of Van der Schee et al. (4, 5) in their polyelectrolyte adsorption model. The model is given as a series of implicit equations which can only be solved numerically. Analytical isotherm equations are derived for two specific adsorbed chain conformations, flat and end-on (perpendicular) adsorption with the charged headgroup in the first layer. These equations reduce to the regI TO whom correspondence should be addressed.

ular solution-regular monolayer model (6-8) if the surfactant molecule becomes equal in size to the solvent molecules and to the SternLangmuir model (9-11) for equal-sized solvent and adsorbate molecules which behave perfectly. A major disadvantage of the treatment given in (1) is that the adsorbed chain conformations which have been considered are rather extreme and unlikely to apply to a real situation. Flat adsorption can only occur if the surface is strongly hydrophobic and nearly unoccupied with surfactant. Perpendicular head-on adsorption of a flexible chain is even more unlikely. This conformation leads to many unfavorable segment-solvent interactions and the conformation entropy loss is so large that practically speaking adsorption will not occur. A more realistic conformation Of an ad493

JournalofColloidandInterfaceScience,Vol. 126,No. 2, December1988

0021-9797/88 $3.00 Copydght© 1988by AcademicPress,Inc. All fightsof reproductionin any formreserved.

494

KOOPAL, WILKINSON, AND RALSTON

sorbed surfactant molecule is that m of the r segments are adsorbed in the first layer adjacent to the surface, the train segments, and that the remaining (r - m) segments protrude in the solution phase. For short chains the segments protruding in the solution phase are most likely to be present in a tail. In principle such a train-tail conformation will occur over a wide range of surface coverages, but the number of segments adsorbed as train will be a function of the amount adsorbed, P. In the present paper the model introduced in (1) will be outlined and extended by incorporation of the amphiphilic nature of the surfactant chain. On the basis of a simplification of this general model, isotherm equations will be derived for both ionic and simple nonionic surfactants adsorbed in a train-tail conformation and with a specified position of the polar and apolar segments. In principle in these equations a relation between m and P can be introduced. The derived basic equations will include the conformation entropy loss and will show how the net adsorption free energy and effective surfactant-solventinteraction parameter depend on chain length and composition, train size, and adsorbed chain orientation. In the discussion the derived equation will be critically examined and compared with other treatments. OUTLINE OF THE THEORY

Free Segment Probability In the present model a surfactant molecule is considered as a flexible sequence of segments equal in size to the solvent molecules. Each molecule consists of two types of segments, polar segments constituting the headgroup and apolar segments forming the aliphatic tail of the surfactant. To be able to describe the adsorption of such a molecule, a model of a quasi-crystalline lattice with lattice layers parallel to the adsorbing surface is used. Starting from the surface the layers are numbered i = 1, 2, 3 . . . . . M, where M is a layer in the bulk of the solution (see Fig. 1). The lattice geometry is defined by the parameters Xo and Journal of Colloidandlnterface Science, Vol. 126, No, 2, December 1988

// A

B

/I

o=

~

4

-'5

•

oo

2

on // U-

"=1

2

34

5 6

7

B 9101112

M

FIG. 1. Illustration of the lattice model. Two possible chain conformationsfor an octamer with a polar headgroup are indicated,one adsorbed(A) and one in solution (B). The solventmolecules(equal in sizeto a lattice site) are not indicated, they simply flu the remaining sites. X1, where Xois the fraction of nearest neighbors in the same lattice layer and XI that in each of the adjacent layers, hence, 2X~ + ~o = 1. All lattice sites within one layer are considered to be energetically equivalent and random mixing per layer of the solvent molecules and surfactant segments is assumed (mean field approximation). In other words, the probability of finding any lattice site in a specific layer i occupied by a segment is equal to the volume fraction ~i of segments in that layer. The energy for any segment in layer i is now determined by the layer number and type of segment only, and not by the site occupied in i, or by the ranking number of that segment in the surfactant chain. Consequently, each segment can be assigned a relative weighting factor pX which depends only on the layer number i and the type of segment x; a layer in the bulk solution is chosen as a reference point. As the ranking number in the chain does not affect p~, it is also called the "free" segment probability. Since a segment can be either apolar (a) or polar (p), two free segment probabilities occur for each layer, p7 for the apolar segments andp~ for the polar segments. A free segment differs from a solvent molecule by its interactions only, therefore the relative weighting factor p [ can be given by the Boltzmann expression

(

pX=~**exp -

kT]'

[1]

where 4~° is the volume fraction of solvent

SURFACTANT

ADSORPTION

molecules in layer i, ~b° that in a layer in the bulk solution, and e x p ( - A G X / k T ) is a weighting factor which is different for both types of segments. The factor ~b°/4~° is related to the entropy of mixing. The magnitude of AG~ is determined by the external field exerted by the surface and by the segment-solvent and segment-segment interactions. The way AG x can be obtained for chains containing one type of segment has been given by Scheutjens and Fleer (2) and their procedure is followed in (1). For chains containing two types of segments the situation is analogous but three types of Flory-Huggins interaction parameters have to be used. Following Leermakers et al. (12)7 in the absence of coulombic and surface interactions the net interaction energy for a segment x surrounded by segments a and p and solvent molecules 0 can be written as -(AG~)mix/kT Xtot

= X

( x y° - x X Y ) ( ( * ~ )

- 4,Y,),

[21

y=O

where the summation is carded out over segments a and p and the solvent molecules O, x aO, Xp0 and X ap are Flory-Huggins interaction parameters, being measures of the segment a-solvent, segment p-solvent, and segment a-segment p interactions, respectively (by definition Xx~ = 0), and (~b~') is the weighted average volume fraction of y over the layers i - 1, i, and i + 1, (q~)

= XI~tY'_I JV )k0~ty" Al- )kl~tY'+l .

[31

Apart from (AGX)mix, electrostatic (coulombic) interactions and interactions with the surface also occur. Restricting the latter ones to molecules or segments present in the first layer only, the general expression for AG~ becomes (2, 3) AG x -- a~,~xf- rXyi kT Xtot

+ E (X "° - x x Y ) ( ( ( # ~ ) -

q~Y,),

[41

495

MODELS

energy for segment x, ~,i is the Kronecker delta equal to unity if i = 1 and to zero if i 4= 1, r x is the valency or charge of segment x, and I1, is the dimensionless electrostatic potential in layer i relative to that in bulk solution. The term rXYi expresses the coulombic interactions which segment x experiences due to the presence of the surface charge and that of the adsorbed charged head segments. A general expression for rXyi has been given in (1). Conformation Probability and Amount Adsorbed On the basis of the free segment probabilities, chain conformations of the surfactant molecules can be described as step-weighted random walks in the lattice. A conformation is defined as the order of lattice layers that are visited during the walk. Each step in the walk is weighted by the factor p~ and each bond by Xo (for a bond parallel to the surface) or X~ (for a perpendicular bond). Hence, the probability that a chain is in a certain conformation c can be given as (2, 3) pc=

l-Ii-i (Pi) x ,.c, i

where r xl,C is the number of segments x that conformation c has in layer i and Wcthe ratio between the number of arrangements of conformation c and that in bulk polymer (solution), 0Oc = Xg)k~ - l - q ,

[61

where q is the number of parallel bonds and r - 1 - q the number of perpendicular bonds. Pc is called the conformation probability. To find the volume fraction Cx it should be realized that the adsorbed chains exhibit many conformations and that each conformation may contribute to ~x. l e t ¢~,c be the contribution of conformation c to ~b,.x. In that case ~bi~,cfollows from (2)

y=O

where X~ is the net standard adsorption (free)

[5]

x

x

4~ixc--- 4~. r~;c p r

Journal of Colloid and Interface Science.

*c~

[7]

VoL 126, No. 2, December 1988

496

KOOPAL, WILKINSON, AND RALSTON

where r~,c/r is the fraction of segments x that a chain in conformation c has in layer i. The fraction ~x is found as

Z r-- Pc,

=

¢

[8]

where the summation is carried out over all conformations. The overall volume fraction q~ is the summation of q~[ over all types of segments

~i = Z ~x.

[9a]

Alternatively 4~i can be obtained as ~ic

q~i = q~, Z ~ Pc, r

[9b]

where ri,c is the number of segments a chain in conformation c has in layer i, irrespective of the nature of the segments. The amount adsorbed, expressed as excess surface coverage Oex, simply follows from $i, 0ex =

Z

In the following we will restrict ourselves mainly to surfactant molecules with a small headgroup consisting of one segment only. This headgroup can be either ionic or nonionic (polar), For the more typical nonionic surfactants made up of a sequence of polar segments and a sequence ofapolar segments only the most simple case will be considered. IONIC SURFACTANTS

Surface and Surfactant Oppositely Charged

x

c

SURFACTANT ADSORPTION ISOTHERMS

(q~i - - ~ * ) .

[101

i

Defined in this way 0ex is expressed in equivalent monolayers. Equations [ 1] to [ 8 ] form a set of implicit equations which in principle can be solved numerically for a certain lattice to find q~Xfor each layer using the matrix method given in Refs. (2, 3). Once the segment concentration profile {4~x } is known, other properties, such as the equilibrium probability of each individual conformation and the amount adsorbed, can be calculated. The adsorption isotherm is obtained by doing these calculations for a series of values of q~,. In the next section the set of equations outlined above will be used as a starting point for the derivation of analytical adsorption isotherm equations. Such equations can only be obtained at the expense of rigor, however, their advantage is that the prediction of trends and the comparison with older models are facilitated. Journal of Colloid and Interface Science, VoL 126. No. 2. December 1988

First an ionic surfactant adsorbing from an aqueous solution onto an oppositely charged interface will be considered. To simplify the situation the following assumptions will be made: (1) The surfactant molecule is a chain containing one charged group (valency r) and r - 1 apolar segments. As a first approximation and for the line of reasoning each CH2-group of the aliphatic chain can be considered as one segment. It should be noted that the choice of a unit cell matching in size with both the different types of segments and the solvent molecules and which allows for the correct chain flexibility is a problem inherent to the application of lattice theories to practical situations. (2) In the adsorbed state each chain of, in total, r segments has a sequence of m segments adsorbed in the first layer as a train, and the remaining r - m segments protrude in the solution as a tail. (3) Due to coulombic attraction the charged head segment of the chain has a strong electrostatic affinity for the surface, so that all adsorbed head segments are present in layer l. In other words, one of the m train segments per molecule is the charged headgroup. This assumption simplifies the calculation of the coulombic interactions in the adsorbed layer considerably. On the other hand, it restricts the applicability of the sought isotherm equation.

497

SURFACTANT ADSORPTION MODELS

(4) In a poor solvent, as water is for surfact,ants, segments tend to cluster together. The segment density distribution is therefore assumed to be homogeneous (step function) and extends over r/m layers each with a volume fraction ~bi equal to ~b1. For such a simplified segment density distribution the calculation of the free energy of mixing is relatively simple.

smaller than ~b~, the following result is obtained: H ps~ =

1

+ - - (x~ + X~xp°) + m

[111

H ~-~ (Px) rL = I-I Pff. i

x

s

On the basis of the simplifications given above, counting segments along the chain is somewhat easier than counting them per layer. In the present case Eq. [ 11 ] can be written as I-I P f = ( P sP, , ) ( P , ,aI )

( P sa) r-m •

m-I

[12]

s

Expressions for P[1, p a , and p} can be derived from the general equation [1]. Considering situations for which ~ , is considerably

0.6

¢,

(tl

i-+.

L

+ 2xa°{r-

0.4

I I~+02

(x a + Xt× a°)

-- ~ * )

2Xlm -- (XI + )k0))(~) 1 -- ~)*)1

o

[13] In a simplified notation Eq. [ 13 ] can be written as H ps~ = ,

(p:,/

exp[m~

+ 2r2'(~b, - •,) - rYl],

[141

where 2" is the weighted average net adsorption (free) energy per segment adsorbed in the first layer and 2' the effective Flory-Huggins parameter in the adsorbed layer for the given segment density and chain orientation. The expression for 2~ reads _, _1 (Xg+Xl× Xs- m -~-

1.0

i

m

Xo --2XaP(Xo + X 1 - ~ ) ( ~ 1

to,

. . . . . . . . .

m-1

+ 2XP°(Xo + Xl)(~bl - ~b,)

( 5 ) The surfactant chain is flexible both in solution and in the adsorbed state. Schematically the adsorbed layer and the segment density profile are shown in Fig. 2. For the derivation of the isotherm equation the product of free segment probabilities over all layers is required. This product can also be written as a product o f free segment probabilities by counting the segments along the chain, viz.,

exp - r Yl

s

p°)

m-1 m

(Xsa + X1xa°).

[15]

The terms X~Xp° and ),~Xa° account for the fact that an isolated adsorbed segment loses X~ of its segment-solvent contacts when it comes in contact with the surface. The weighted average Flory-Huggins parameter specific for the given segment distribution equals

i-; -1

'213'4'5'6

a..~

o

2

4

6

8

10

0

FIG. 2. Schematic representation of the adsorbed layer and the segment density profile for head-on adsorption; sites not occupied by surfactant segments contain solvent molecules.

2' = (Xo + Xl)fpx p°

-[- (1

0"~FH)faXa0--()k0"~

~.1 ---~)fpx ap. [16]

Journal of Colloid and Interface Science,

Vol. 126,No. 2, December1988

498

KOOPAL, WILKINSON, AND RALSTON

In view of the approximation inherent in the presupposed segment density distribution, { r - 2Xlm - (X1 + ~0)} in Eq. [13] is replaced by ( r - 1){1 - (0.5m/r)}. In Eq. [16],fpis the molar fraction of polar segments in the surfactant chain and f~ the molar fraction of apolar segments (in the present situation fp = 1/r andf~ = (r - 1)/r). Note that 2' differs from the weighted average Flory-Huggins parameter in the bulk solution, x , . The latter equals (13) X , = f p X pO -[-fax a0 - - f a f p X ap.

[17]

The difference between 2' and X, is due to the fact that at the interface the head segments are enriched in the first layer and the apolar segments in the other layers, whereas in bulk solution a random mixture of segments a and p has been assumed. The term r I11 in Eq. [13] is due to the coulombic interactions between head segments and the surface and between the head segments mutually. The reduced potential Y1 is related to the potential ~k~ at the plane through the center of the first layer where the adsorbed head segments are located,

Y~ = e~b~/kT.

[18]

In turn ffl is related to ~1p and the surface charge or potential. In this simplified situation evaluation of ~k~ can be done in the classical way (see, e.g., (6) or (7)) or using the model developed in (1). For low potentials (DebyeHfickel approximation) and a charge-free Stern layer the result is

ell = (cr~ + reckg/ao)/Er,

[19]

where as is the surface charge density, ao is the area of a unit cell (the molecular cross section of a segment), e is the dielectric constant equal to ~O~rwith ~0 the permittivity in vacuum and e~ the relative permittivity of water, and r is the reciprocal Debye length, r is determined by the ionic strength K2 _

e2~ zZ Ni i EkT

[20]

Journal of ColloM and Interface Science. VoL 126,No. 2, December1988

Zl and Ni are the valency and the number concentration of the various ions i in the bulk solution and e is the elementary charge. In most cases the ionic strength will be dominated by the background electrolyte. The expression for r Y~ is found by substitution of Eq. [ 19 ] in [ 18 ] and realizing that ~b~ = ¢ ~/ m:

= { e ~[r~s + (r2ecbl/mao)} T ~Y1

[21]

_

In the present case where r and as have opposite signs, the surface-adsorbate interaction is attractive. The lateral interactions between the head segments are, however, repulsive because r 2 is always positive. Equation [21] is limited to low potentials (<60 mV); an expression for rY~ for high potentials can be found in Ref. (6). On the basis of Eq. [14] the conformation probability Pc (see Eq. [ 5 ] ) can be written as

Pc = Wc

exp [ mXs + 2r2'(~bl -- q~,) -- r Y d .

[22]

For the given segment density distribution and m segments adsorbed as train segments this expression for Pc applies to all possible chain conformations with only wc a function of the conformation. Therefore, according to Eqs. [9b] and [22] we find ~l ~,=

m 4~° r _, W(r)(~**)exp[mXs + 2r2'(q~l - 4~,) - rYl],

[231

where use is made of the fact that ~c rl,c = m and W = ~ we.

[24]

c

In order to approximate the factor W more than one route can be followed. The simplest way is to assume that only the train segments are restricted in their bond positions and that, once the first layer is left, the chain is as free as in solution. This approach neglects the assumption that the segment density distribution

SURFACTANT

ADSORPTION

is a block profile. For a chain of r - 1 bonds, adsorbed with m - 1 bonds in the first layer, bond m going from layer 1 to layer 2 and the remaining bonds distributed as in the solution phase, Wwill become L(ZcXo) W m f =.

m-1

r-m-I

ZchIZ~ Z Z rc_1

= ~k~n_ 1 ~kl '

[25a1 where the subscript mf denotes that a train of m segments is placed in the first layer and that the other segments are left "free." This approximation underestimates the conformational entropy loss, - k In W, because the tail has been given more freedom than is realistic. Alternatively, W can be approximated by restricting the positions of the segments in order to satisfy the assumed segment density profile. In this case, a train of m - 1 bonds is followed by segment m pointing towards the solution and the remaining bonds distributed over layers 2 to r / m , with ( r / m ) - 2 steps fromitoi+ landr-m-(r/m)+ lsteps parallel to the surface. In this case W can be approximated as W m b -~- ~k~ -1 ~l~k I r/m)-2 ~k~-m-(r/m)+l

[ X

[(r/m) -

(r-m-l), ] = m S - ( r / m ) + 11!

2]i

or

Wmb

"~-

[

X

)k~-trIm)

~k

t rim)-1

(r-m-l),

]

499

MODELS

turn from layer i to i - 1 is not allowed, whereas in bulk solution no restrictions apply. Moreover, each chain is forced to stay within the assumed segment density profile, which is an oversimplification. In both approximations W is a relative number, hence the errors introduced by allowing backfolding are eliminated to some extent. The two approximations can be considered as extremes, the true value of W being somewhere between the two. Equation [25b] has the advantage that the limits for m = r and m = 1 are well represented and it corresponds with the stepwise density distribution used for the calculation of the interactions. Equation [25a] represents the simplest possible approximation and it cannot be applied if m approaches one. In the case of a rigid chain the direction of the first adsorbed bond is "felt" along the entire chain. Hence, the position of only one bond can be chosen independently and W (rigid) = 1 - hl. In the case of a rigid flat molecule, such as a rod-type or dye molecule, which adsorbs in the first layer, W equals Xo. In essence this situation has been considered by Prigogine and Mar6chal (14); the small entropy loss ( - k In X0) is incorporated in the affinity constant. For the present situation of a flexible chain and strong head-on adsorption 4~1 is found by substitution of Eq. [25a] or [25b] in [23]. The amount adsorbed, 0ex, is given by Eq. [ 10] and reads

[ ( r / m ) - 2-~.~---~ ---(r/m) + 11! ' 0ex = r (¢1 - ¢ , ) . m

[25b] where the subscript mb denotes that a train of rn segments is placed in layer 1 and that the position of the remaining segments satisfies a block distribution. The product of the X's is the relative number of arrangements of a conformation and the factor within brackets is the combination factor expressing the number of possible conformations relative to that in bulk solution. The approximation of W by Eq. [25b ] overestimates the loss is conformation entropy, because in the adsorbed layer the re-

[26]

Substitution of the expression for q51 results in the isotherm equation '

0ex = ¢ , W

exp [ mE'

Equation [27 ] applies to dilute solutions. For most surfactants water is a poor solvent and the maximum volume fraction ¢ , is given by Journalof Colloidand InterfaceScience, Vol. 126, No. 2, December 1988

500

KOOPAL, WILKINSON, AND RALSTON

the critical micelle volume fraction, which is always very low, Hence, for most practical sit. uations ~b, ,~ 1 so that 0~x :~ 0 and

O=(9*W(1-C~l)rexp[m~(~ + 2r2'~bt - ~'Yd.

~ ~11~ I I ] ~ 1

]~e-

1.0 i

i

[28]

Within the limits of the present approximation 2~, Y1, and R' depend on the orientation of the adsorbed chains but not on the chain conformation, provided m is constant. The equation for W depends on the supposed conformations but not on the chain orientation and for a given value of m, W is also a constant. Equation [ 28 ] applies up to moderate surface coverages depending on the surface charge. When the surface charge is "neutralized" by the specific adsorption o f head segments, the tendency to adsorb head-on i s minimized and the interfacial zone has become hydrophobic due to the presence of the aliphatic loops and tails. On such a surface, surfactant molecules will adsorb with the headgroup pointing towards the solution side. This aspect is not covered in the derivation of Eq. [ 28], it demands a specific treatment. As a rule of thumb Eq. [28] can only be applied for re4~l/mao < as.

Surface and Surfactant Similarly Charged

t°4 ~ 1

~ - -

"kf 1 2345

0

2

to.2

4

6

8

10

FIG.3. Latticemodelof the adsorbedlayerand the segment densityprofilefor adsorption with the headgroups protruding in the solution.Solventmoleculesare distributed over the sites not occupiedby surfactantsegments. 0 = ~b,W(1 " ~bl)rexp[m2* + 2r2*~b 1 " "rYt],

[291

where W can be approximated by Eq. [25a] or by a slightly modified version of Eq. [25b],

, Wmh

=

k~'-Cr'lrn)xtr'lm)

[

(F-m-I)! ] × [ ( r ' / m ) - 2 ] ! [ r ' - - - m - - - ( r ' / m ) + 1]! [301 with r' = r - 1 and the extra Xl accounts for the step from t - 1 to t. 2 " is the net adsorption (free) energy [31]

2s* -- Xsa -[- •IX a0.

For the adsorption of an ionic surfactant onto a similarly charged surface an equation comparable to Eq. [28 ] can be derived. In this case it is assumed that due to the coulombic repulsion between headgroup and surface, all charged segments are adsorbed in the layer adjacent to the bulk solution. Moreover, in this outer adsorbed layer with hydrated head segments, probably no apolar segments reside. Hence, a two-step volume fraction profile seems likely: in the layers 1 to t - 1 ( = ( r - 1) / m), q~[ = ~b~, and in layer t, q~ = 4 ~ = cb]/m. A schematic representation of the adsorbed layer and the density profile are shown in Fig. 3. Following a similar reasoning as before we find for very dilute solutions ( ¢ , ~ 1) Journal of Colloidand InterfaceScience; Vol.126,No. 2, DecemberI988

10

2" is the effective Flory-Huggins parameter in the adsorbed layer which can be approximated as 2"

= [~Xl +

p°

m]

and r Y~ can be derived from the general expression given in (1), =

e [rO's

•"r2e~b 1

exp

-,,ro(t

-

1)}

[I + exp{--2rro(t- 1)}]]

+ 2erma-----~o

j

[33]

SURFACTANT ADSORPTION MODELS where r0 is the distance between the lattice layers. The first term of the RHS ofEq. [ 33 ] is due to the surface charge, and the second to the charge of the head segments in layer t. Both terms represent a coulombic repulsion. Consequently adsorption will only occur if m2* is sufficiently large. In general this will be the case when the surface is (partially) hydrophobic. For ro(t - 1) > r -~ , that is, for intermediate salt concentrations, Eq. [ 33 ] reduces to

rYt = ~-~

1

exp{-Kr0(t - 1)} + eKmaol" [34]

For high salt concentration, i.e., ro(t - 1) >> r -x , the surface charge contribution to Yt can also be neglected. The value of 2", see Eq. [32], is strongly dominated by x s0. This is due to the fact that (1) only one of the r segments is a p segment, (2) q~~ = 4~1/ m, and ( 3 ) the number of a-p contacts is very small, each segment a in layer t - 1 having only a fraction fpX~ of its contacts with segments p in layer t. NONIONIC SURFACTANTS By putting r = 0 the equations derived in the previous section can also be applied to the adsorption of simple nonionic amphipolar molecules consisting of an aliphatic chain with a small headgroup such as long-chain alcohols and amines. In general, simple amphipolar molecules adsorb well on hydrophobic surfaces (see Fig. 3) and Eq. [29] with r = 0 can be applied,

501

rected to the surface (Fig. 2) and Eq. [28] with r = 0 is appropriate, i.e., 0 = q~.W(1 - ~l)rexp(m2 ' + 2r2'q~l), [36] where W is defined in Eq. [25], 2 ' in Eq. [15], and 2' in Eq. [16]. Adsorption on a fully hydrophilic surface will only occur ifa strong net interaction exists between the polar headgroup and the surface. The first layer adjacent to the surface will be preferentially occupied by polar segments adhering to the surface and solvent molecules. The aliphatic segments residing in the first layer will have no affinity for the surface. In this case Eq. [ 36 ] still applies, but Eq. [ 15 ] can be simplified to 1 m-1 2 ' = - - ( x ~ + ~,ix p°) +

m

m

(~lxaO).

The aliphatic chains protrude in the solution phase and at moderate surface coverages the orientation of the adsorbed molecules may reverse. Under these conditions the molecules adsorb with their headgroup towards the solution side and Eq. [36] no longer applies. It should be noted that Eqs. [ 35 ] and [ 36 ] are similar, the only difference being that the expressions for the net (free) energy of adsorption, the effective Flory-Huggins parameter, and Wdiffer. In its general form Eq. [ 35 ] also applies to the adsorption of the more common nonionic surfactants composed of a series ofapolar segments connected to a series of polar segments. For a random arrangement of the segments in the adsorbed layer (as in the solution phase) it can easily be verified that Eq. [ 35 ] applies with Wgiven by Eq. [ 25 ], 2* = X. (see Eq. [17]), and Xs* = f a ( X s a "~- )kl Xa0) "~-fp(Xsp -]- )klXP0). [37]

0 = qS.W(1 - ~bl)rexp(m2 * + 2r2*q~ 1),

[35] where Wis defined in Eq. [25a] or [30], 2* in Eq. [31], and 2* in Eq. [32]. For adsorption of simple amphipolar molecules on partially hydrophilic, partially hydrophobic surfaces the headgroups may be di-

In practice, however, preferential orientation of the apolar segments towards the surface and of the polar segments towards the solution will occur, leading to other expressions for 2* and 2 " . Such expressions can be obtained by writing down the conformation probability for the desired orientation. Journal of Colloid andlnterface Science, Vol. 126,No. 2, December1988

502

KOOPAL, WILKINSON, A N D RALSTON DISCUSSION

tion of the derived equations to practical systems is thwarted by the fact that rn is not a constant.

Variation o f m

Adsorption of a molecule in a train-tail conformation leads to a conformation entropy loss. Therefore, for a fully flat adsorption a minimum net adsorption energy, 2s(min), is required, even at very low surface coverages. At very low 0 and 2s the isotherm equations can be simplified to In 0 / ~ , = In W + m~s.

[38]

When adsorption occurs 0 > 4~,, hence for flat adsorption xs(min) is obtained for 0 = ~ , and W = ~,~-1, ~s(min)=(-lnW)/r=-

( L ~ - ~ ) l n ~0

(flat adsorption).

[39]

For a hexagonal lattice (~,1 = 0.25, X0 = 0.5) and not too short chains this leads to 2~(min) ~ 0.70kT. Hence, for values of xs >i 0.70kT, flat adsorption will occur at very low values of 0. At higher surface coverages lateral interactions between the adsorbed molecules become important and, also for 2~ I> 0.70kT, tails develop. Consequently, in general the value of m decreases as a function of 0. Theoretical calculations on homopolymers (Ref. (2a), Fig. 8) indicate that in this case m is about constant for low values of 0, but for higher values of 0, m ~ r[a + (b/O)],

[40]

where a and b are constants depending on x* (=xs + ~,1×) and r. For low values of x* ( < l k T ) and 10 < r < 20, a and b are of the order of a few tenths. For ×* > 3 k T , a can be neglected and b approaches unity. In the present situation two types of segments are present which further complicates the situation. However, as a rule of the thumb it seems reasonable to say that m is weakly dependent on riO for segments with a small X* and strongly dependent on r/O for segments with a large ×*. In general it should be realized that the applicaJournal of Colloid and Interface Science, Vol. 126, No. 2, December1988

Coulombic Interactions

In this present treatment the electrostatic interactions are restricted to coulombic interactions at low potentials. In practice high potentials may occur. In the case of head-on adsorption Eq. [21] can easily be extended (see Ref. (6)). In this case the electrostatic interactions are no longer simply proportional to ~b~. The situation for adsorption of an ionic surfactant onto a similarly charged surface is more complicated. Essentially only numerical solutions for Y, exist. In practice a first approximation is obtained when er in Eq. [33] is replaced by Kd, the integral capacitance of the diffuse layer. Ko can be calculated using the Gouy-Chapman theory: Kd=~K l n { q + ( ~ +q

1) °5} ] '

[411

where q is a dimensionless "charge" equal to zea/ErkT with a either as (first term on RHS of Eq. [33]) or at (second term on RHS of Eq. [33]). Equation [41] indicates that Kd is larger than eK and hence by using the DebyeHiickel approximation, values are obtained for the potential which are too high (note that ~b = alKd). Another complication with the calculation of the electrostatic potential is that in the adsorbed layer the relative dielectric permittivity ei is lowered by the presence of the aliphatic segments. Moreover, the aliphatic segments interact unfavorably with the ions, which can be translated in an excluded volume effect (15). Hence, the double layer will become more extended in the adsorbed layer region and K will be smaller. As a consequence of both effects, ~r will tend to be somewhat smaller in the adsorbed layer than outside this layer. To some extent this effect is counteracting the increase in Kd due to high plane charges.

SURFACTANT

ADSORPTION

Dipole Interactions

/ / 2 0 O's \

0

iE14~a1,

[42]

where iz° is the normal component of the dipole m o m e n t of the adsorbed water dipoles and E1 is the field strength in layer 1. The dipole contribution of the adsorbing segment has been neglected. According to Gauss' law E1 = crs/~0~1,

[43]

where ~1 is the average relative permittivity in the first layer. To a first approximation el may be represented as a linear combination ofpermittivities -

E I q~ 1,

[44]

where ~0 is the relative permittivity of the first layer when this layer is totally occupied with water molecules and e~ is the same when the layer is fully occupied with segments a. Substitution of Eqs. [43] and [44] in [42] shows that, at constant a s , Agdip is not simply proportional to ~b~, but

AgdiP= e0(1 -- q~) + ,~b~ "

a

[46]

Besides coulombic electrostatic interactions in the adsorbed layer, dipole interactions may be important with adsorption from an aqueous solution on a charged interface. The water dipoles initially present in the first layer adjacent to the surface may have a preferential orientation due to the presence of the surface. Upon adsorption these oriented water dipoles are exchanged for segments with often a negligible dipole moment. Therefore this exchange leads to a dipole contribution to the free energy of adsorption which can be given as (16) Agdip = - - ~

503

MODELS

[45]

In the case of hydrophobic interfaces the orientation of the water molecules in the Stern layer is largely determined by hydrophobic bonding so that t~° and E° are hardly affected by the magnitude of as. Moreover, e0 is probably much smaller than its value in bulk solution and comparable in magnitude to E~. Under these conditions ~gdip can be approximated as

This equation shows that for a hydrophobic interface Agdip is simply proportional to ~b~. The same applies for other interfaces where the water molecules are strongly oriented (for instance the AgI- and Hg-electrolyte interface). For typical hydrophilic interfaces, such as the metal oxide-electrolyte interface, a layer of chemisorbed water molecules is forming the interacting surface and the relative permittivity of the first layer of physisorbed water is close to that of bulk water (17) and much larger than e~. Moreover, both #0 and q~ will be small. Hence, in this case Agdipis small and dipole effects can be neglected. In the case of a hydrophobic surface where Agdipis given by Eq. [ 46 ], in principle a factor exp(Agdip/kT) should be included in p~. However, in practice Agdip is not very well known and its contribution to the total free energy of adsorption is generally neglected. At low surface charges this seems reasonable, but at high values of as the dipole contribution should be taken into account. For instance, taking for ~0 half the dipole m o m e n t of an isolated water molecule, i.e., 0.5 × 1.84 D (=0.5 X 6.13 × 10 -3 Cm), for the dielectric permittivity of the first layer ~0 = 10 (E0 = 8.8 × 10 -12 C V -1 m - l ) , and for as a value of 60 mC m-2, we find that Agdio is approximately -0.5da~kT, roughly indicating the order of magnitude of this term.

Comparison with Other Models The isotherm equations given in this paper are essentially extensions of the regular solution/regular monolayer models ( 6 - 8 ) in which both the configuration and conformation effects due to the size of the molecules have been incorporated. Substitution o f m = r = 1 and r = 0 in Eq. [27] leads to the regular behavior model. For low values of ~b,, m = r = 1, and = 0, Eqs. [35] and [36] reduce to the LangJournal of Colloid and Interface Science, Vol. 126, No. 2, December 1988

504

KOOPAL, WILKINSON, A N D RALSTON

muir model (¢ 1 = 0) and Eq. [28 ] to the SternLangmuir model. Comparison of Eqs. [28] and [29] with the isotherm equations for ionic surfactants derived in the previous paper (1) (fiat and headon perpendicular adsorption) reveals that in all cases the form of the isotherm equation is the same, i.e., for ~ . ~ 1, 0 = ¢ . W ( 1 - q~l)'exp[m~s + 2r~bl - r Y ] ,

[47]

where Wdepends on the adsorbed chain conformation (s), ~s and ~ are the effective adsorption (free) energy and the effective surfactant-solvent interaction parameter, respectively, and Y is the reduced potential in the plane of adsorption of the head segments. The advantages of the present derivation over that in Ref. (1) are (1) the number of train segments is an explicit parameter, (2) the contribution of both types of segments to Xs and is made explicit, and (3) the chain has retained its flexibility in the adsorbed state. The first two points need no further discussion. Comparison of the presently derived expressions for Wwith W(flat) (=~,~-1) and W(_I_) ( = X4- l ) shows that both fiat and perpendicular adsorption are rather unfavorable situations for flexible chains because of the considerable loss in conformation entropy. Nevertheless at low surface coverages and sufficiently high values of ~s, fiat adsorption is possible. However, for flexible chains, perpendicular adsorption (stretched chain) is a hypothetical limit, and this conformation can only occur for fairly rigid chains. Equation [ 47 ] not only applies to ionic surfactants, but with r = 0 it is equivalent to Eq. [ 35 ] or [ 36 ] describing the adsorption ofnonionics. Hence, when the segment density distribution in the adsorbed layer can be approximated by a block-type distribution, Eq. [ 47 ] represents a generalized isotherm equation for surfactant adsorption. Equation [ 47 ] is essentially an extension of the well-known Frumkin equation, which has been discussed in relation to the adsorption of Journal of ColloM and Interface Science, Vol. 126,No. 2, December 1988

ionic surfactants in Refs. (1, 6). As a consequence of the chain nature of a surfactant molecule the factor W(1 - q~l)r appears in [47], which is not present in the Frumkin equation. In general W(1 - ~b1)r < 1, showing that the chain effects counteract the adsorption tendency. Two forms of Eq. [47] have been examined in relation to older models for the adsorption of ionic surfactants in Ref. (1). The coulombic contribution to the interaction term can be subdivided in a surface-surfactant (attractive or repulsive) and a surfactant-surfactant (repulsive) contribution. Pseudoideal behavior occurs when chain effects (repulsive) and coulombic surfactant-surfactant repulsion are just counteracted by surfactant-solvent attraction. In the case ofnonionics, pseudoideal behavior occurs when chain effects and surfactant-solvent effects mutually compensate. The maximum adsorption predicted by Eq. [ 47 ] in principle amounts to r~ m equivalent monolayers. However, it should be realized that m is not constant and that the hydrophilic segments tend to be surrounded by solvent molecules so that ~bl '< 1, Prediction of saturation values with the present theory is in principle possible by considering the minim u m in the overall excess free energy of adsorption as a function of m and 0. This aspect will be considered in a future paper. Equation [47] is also an extension of the model developed by Kronberg et aL (18, 19), describing the adsorption of long-chain nonionic surfactants. In the present notation Kronberg's equation for q51 can be written as ln[ ~l~[~°*~ r= m× k + r X . ( 1 - 2 ¢ , ) -

rxk(1

-- 2 ~ b l ) ,

[48]

where ×k is an average segment-solvent interaction parameter in the adsorbed layer and X k = ao(yl - 3`2)/kT,

[49]

where (3,1 - 3'2) is the difference of the interfacial free energies of the pure surfactant-adsorbent and the pure solvent-adsorbent inter-

SURFACTANT ADSORPTION MODELS

505

faces. Kronberg evaluates Xk by interfacial Substitution of Eq, [ 51 ] in [48 ] gives tension and contact angle measurements. Acm ( x k + ~,IX,) cording to Kronberg Xk is different from X, :ln~{4~1][4~°~r= q~,]~ ~bo } because (1) the number of nearest neighbors to a segment in the adsorbed layer Will be dif+ 2rx,(~bl - q ~ , ) - 2m~lX,~bl. [52] ferent from that to a segment in bulk solution and (2) the adsorbed molecules are oriented For dilute solutions (q~, ~ 1) Eq. [52 ] reduces with their apolar part towards the surface and to their polar part towards the solution. In order (61 ~b,(1 - (al)rexp{m(xks + X1X,) to account for these effects an intuitive expres+2r~btX,[1-(hi/t)]}. [53] sion for Xk has been proposed (18). Evaluation of x k on the basis of Eq. [49] implicitly ac- Recalling that 0 = (r/rn)4~l, 0 is found as counts for preferential adsorption of the apolar 0 = ( r / m ) ~ , ( 1 - 01) r segments. In the derivation of Eq. [48] the adsorbed × exp[mx k* + 2rxk~bl], [54] layer has been considered as a separate phase in equilibrium with the bulk solution. The where Xk* = X~k + X1X,. Equation [ 54] should be compared with Eq. expression for the chemical potentials of solvent and solute in this phase have been based [35]. Although these equations are similar upon the Flory-Huggins extension of the reg- they differ in two respects: (1) Eq. [ 54 ] differs ular solution theory, applied in the presence by a factor W ( m / r ) from the presently derived of an adsorbing interface. Conformation en- equations and (2) the expressions for the "eftropy effects have been neglected. In other fective" xs and x parameters differ. The first words, implicitly it has been assumed that the difference occurs because Kronberg has asadsorbed phase is homogeneous. Introduction sumed that the possible bond positions of the of a near-separation of the polar and apolar surfactant in the adsorbed state, in the solusegments in the adsorbed layer is in serious tion, and in pure surfactant are the same, conflict with this assumption and application whereas in the present model the bond posiof [ 48 ] to oriented adsorbed molecules should tions in the adsorbed layer are different from be considered with some reservation. In order those in bulk solution. As a consequence conto analyze Eq. [48] we therefore first consider formational entropy effects are not accounted Xk as being different from X, due to nearest- for in Kronberg's model. The second differneighbor effects only (i.e., one type of segment ence occurs because our model did allow incorporation of a preferential orientation of the only). In a lattice model, if the lattice is kept the adsorbed molecules. It has been shown that, same throughout the whole system, the lower for dilute solutions, the expressions for the efnumber of nearest-neighbor contacts in the fective Xs and X parameters in the adsorbed layer adjacent to the surface can be acknowl- layer can be adapted to the chain composition edged by giving the segments in the first layer and orientation without changing the form of (1 Xa)Zc nearest neighbors instead of Zc. the isotherm equation. Hence, the use of efThe average coordination number in the ad- fective x and Xs parameters (orientation effects included) is also allowed in Kronberg's model, sorbed layer becomes provided Eq. [48] is reduced to [54], i.e., ~b, z , = Z c [ 1 - (Xl/t)l [501 ~b1. Application ofEq. [ 54] to experimental results is mainly thwarted by the neglect of for a one-step volume fraction profile with t the conformation entropy contribution. The = r / m , and expression for Xk as given by Kronberg et al. Xk = [1 - (X~/t)]X,. [51] (18, 19 ) is somewhat arbitrary. =

Journal of Colloid and Interface Science, Vol. 126, No. 2, December 1988

506

KOOPAL, WILKINSON, AND RALSTON

The use of Eq. [48] as presented by Kronberg et al. (18, 19) may lead to a wrong interpretation of the difference r(X, - X k). From Eq. [51] it follows that r ( X , - Xk) = m ~ X , . This term results from the fact that upon adsorption in the first layer even an isolated segment loses X1 of its segment solvent contacts. In the case of an aliphatic segment this effect promotes adsorption in exactly the same way as an increase in ×~. In Eqs. [35], [36], and [ 54 ] the factor mXl×, is therefore incorporated in X~effective. Moreover, it should be realized that this contribution to X~effective also exists for a purely apolar chain for which orientation effects are absent. In other words, the factor r ( × , - Xk) is not due to surfactant orientation or hydrophobic bonding between surfactant molecules in the surface phase, as suggested by Kronberg et al. (19). The magnitude of r n ~ X , can, however, be affected by the orientation of the adsorbed surfactant molecules if a certain type of segment is enriched in the first layer. For instance, with adsorption on a hydrophobic surface the first layer will be enriched in apolar segments and the value of mX1X, approaches mX~X a°. In the case of adsorption from aqueous solution this will lead to the maximum possible increase of Xs effective. This maximum increase amounts to about 0.5kT(X1 = 0.25, Xa° ~ 2)per adsorbed train segment. Kronberg et al. (18, 19) obtain values for r ( X , - Xk) (=m~,~X,) ranging from 4 k T to 1 2 k T under the assumption that m equals 3 or 4. Possibly these values are too high because the conformation change is not properly taken into account in Kronberg's analysis. In the first place In W is neglected, but the way in which x k is estimated does not exclude that In W is implicitly incorporated or even overestimated. It is also possible that the assumption that m is a constant leads to erroneous values of r ( X , - xk). In conclusion, the present model for nonionic surfactants has as advantages over Kronberg's model that chain conformation effects are included and that the effective adsorption parameters can be derived from the model. Journal of Colloid and Interface Science, Vol. 126,No. 2, December1988

CONCLUSIONS

(i) On the basis of a general model for surfactant adsorption, relatively simple adsorption isotherm equations have been derived for both ionic and nonionic surfactants. In order to do this a stepwise segment-density profile has been assumed. The general form of the isotherm equation is in all cases similar and reads 0 = ~b,W(1 - ~bl)r × exp[m~s + 2 r ~ 1 - r Y ] ,

[55]

where W is related to the conformation entropy, ~s is the effective adsorption energy, is the effective Flory-Huggins interaction parameter in the adsorbed layer, and Y is the electrostatic potential at the plane of adsorption of the charged headgroup. The expressions for W, Xs, x, and Ydepend on the number of train segments, the chain composition, and the chain orientation in the adsorbed layer. (ii) Older equations based on the regular solution-regular monolayer models (6-8) such as the Frumldn (1, 6 ) and the Langmuir or Stern-Langmuir models (9-11) are limiting eases of Eq. [ 55 ]. In the ease of flexible molecules, perpendicular adsorption (1) is unrealistic because of the very large entropy loss. At low surface coverages flat adsorption is possible provided ×~ effective is 0 . 7 k T or more. (iii) Comparison of the present model for nonionic surfactants with the model derived by Kronberg (18 ) shows that the main weaknesses of the latter model are that the conformation entropy loss is not accounted for and that orientation effects upon adsorption are introduced on an empirical basis. This may lead to an incorrect evaluation of the difference r ( X , - X k) (18, 19). ACKNOWLEDGMENT Professor Gerard Fleer is acknowledged for his critical comments and helpful discussions. REFERENCES 1. Koopal, L. K., and Ralston, J., J. Colloid Interface Sci. 112, 362 (1986).

SURFACTANT ADSORPTION MODELS 2. Scheutjens, J. M. H. M., and Fleer, G. J., (a) J. Phys. Chem. 83, 1619 (1979); (b) 84, 178 (1980). 3. Scheutjens, J. M. H. M., "Macromolecules at Interfaces," Ph.D. thesis, Agricultural University, Wageningen, 1985. 4. Van der Schee, H. A., and Lyklema, J., J. Phys. Chem. 88, 6661 (1984). 5. Papenhuijzen, J., Van der Schee, H. A., and Fleer, G. J., J. Colloid Interface Sci. 104, 540 (1985 ). 6. Koopal, L. K., and Keltjens, L., Colloids and Surfaces 17, 371 (1986). 7. Koopal, L. K., in "Fundamentals of Adsorption" (A. L. Myers and G. Belfort, Eds.), p. 283. Engineering Foundation, New York, 1984. 8. Defay, R., Prigogine, I., Bellemans, A., and Everett, D. H., "Surface Tension and Adsorption." Longmans, London, 1966. 9. Grahame, D C., Chem. Rev. 41, 441 (1947). 0. Fuerstenau, D. W., in "Principles of Flotation" (R. P. King, Ed.), Chaps. 2 and 3. South African

11. 12. 13.

14. 15. 16. 17. 18.

19.

507

Institute of Mining and Metallurgy, Johannesburg, 1982. Healy, T. W., J. Macromol. Sci., Chem. A 8, 603 (1974). Leermakers, F. A. M., Scheutjens, J. M. H. M., and Lyklema, J., Biophys. Chem. 18, 353 (1983). Kilb, R. W., and Bueche, A. M., J. Polym. Sci. 28, 285 (1985); Krause, S., J. Polym. Sci. 129, 558 (1959). Prigogine, I., and Mar6chal, J., J. Colloid Sci. 7, 122 (1952). Brooks, D. E., J. Colloid lnterface Sci. 43, 687 (1979). De Keizer, A., and Lyklema, J., J. Colloid Interface Sci. 75, 171 (1980). Kleyn, M. J., and Lyklema, J., J. Colloid Interface Sci., 120, 511 (1987). Kronberg, B., J. ColloidlnterfaceSci. 96, 55 (1983); Kronberg, B., and Stenius, P., J. Colloid Interface Sci. 102, 410 (1984). Kronberg, B., Stenius, P., and Thorssell, Y., Colloids and Surfaces 12, 113 (1984).

Journal of Colloid and Interface Science, Vol. 126, No. 2, December 1988