Langmuir 1991, 7, 1595-1608
1595
Aggregative Adsorption of Nonionic Surfactants onto Hydrophilic Solid/Water Interface. Relation with Bulk Micellization P.Levitz CRSOCI CNRS, 1B rue de la Ferollerie, Orleans 45071, France Received October 11,1990. In Final Form: March 14,1991 The adso tion isothermsof nonionic surfactantssuch as polyethyleneglycol alkyl ether or alkylphenol ether onto a ydrophilic surface have a typical sigmoidal shape for short or medium polar chain lengths. The rising part of the isotherm is always located below but close to the critical micellar concentration (cmc). Equilibrium as well as kinetic data show that the adsorption layer is built from isolated molecules coming from aqueous solution and not by direct adsorption of micelles. The plateau in the aclsorption isotherms stems from stabilization of the surfactant chemical potential slightly above the cmc. Several experimental argumenta support the existence of extended but finite molecular aggregates at the solid/ water interface below the cmc. Furthermore the process leading to this aggregative adsorption appears to have some similaritieswith bulk micellizationtaking place above the cmc. A 2-fold theoretical analysis of the aggregative adsorption is proposed. First a thermodynamic model of the micellizationis developed and ita predictionsare comparedwith availableexperimentalresults. Second,the thermodynamicproperties of an aggregative adsorption layer in equilibrium with the bulk solution are discussed, then by we of a grand partition function, theoretical isotherms are computed and compared with experiments.
Tl
1. Introduction Reversible adsorption of nonionic surfactants, such as polyethylene glycol alkyl ether (CN,EN or alkylphenol ether (CN,PENJ, occurs onto hydrophi 'c or polar solids below the critical micellar concentration (cmc). As shown in severalexperimentalstudies,'-lOall adsorption isotherms reach a plateau around the cmc. This plateau decreases when the polar chain length increases: a step isotherm can be observed for very short polar chains, whereas for long polars chains, the adsorption isotherm appears to have a "Langmuir shape" with no clear inflection point. In between, typical sigmoidal shapes can be observed with a rising part always located below but near the cmc. Generallyspeaking,adsorption appears to be a cooperative process involving strong lateral interaction between surfactants and weak interaction with the solid surface. A first unknown concerning the adsorption layer structure is the possible occurrence of molecular associations at the solid/water interface below cmc. Several experimental arguments support the existence of such aggregative adsorption process: (i)When expressed as reduced surface coverage B versus C/cmc coordinates, the normalized isotherms of two surfactants having the same polar head and a slightly different apolar chain can be superimposed. Thisproperty, recently observed= for the adsorption of CsPElo and CgPElo on different types of polar solids (silica, quartz, kaolinite),
e'
~
(1) Partyka, S.;&ha, S.;Lindheimer,M.; BNII, B.Colloids Interface 1984,12,2&. (2) Furlong, D.N.;Aston, J. R. Colloids Interface 1982,4, 212. (3) Lovita, P. These de Docteur d'Etat es-Sciences Physiques, Universih DOrle.a~,France, 1986. (4) Zaini, S. Thm De Docteur d'Etat w-Sciences Physiques, Univenite De Montpllier, France, 1986. (5) Denoyel, R. These De Docteur d'Etat es-Sciences Physiques, Univenib De Marwille, France, 1987. (6) Van Den Boomgaard, T.; Tadroe, T. F.; Lyklema, J. J. Colloids Interface Sci. 1907,116,8. (7) hvita, P.; El Meri, A.; Keravis, D.; Van Damme, H. J. Colloid Interface Sci. 1984, SS,484. (8) Lovita, P.; Van Dunme, H.; Keravis, D. J. Phys. Chem. 1984,88, 2228. (9) Lovita, P.; Vm Damme, H. J. Phys. Chem. 1986,90,1302. (10) Lindheimer,M.;Keh, E.;Zaini, 9.;Partyke,, S. J. Colloid Interface Sci. 1990,198,83.
stronglysuggestsa local organizationof adsorbedmolecules very similar to that of the micelles in solution. (ii) Above a surface coverage of 0.1-0.3, the differential enthalpy of adsorption of nonionic surfactant on hydrophilic silica gels shows an endothermic process whose amplitude is very similar to the one observed for bulk micellization.4*5Jo (iii) Fluorescencespectroscopyall along the adsorption isotherm of a build-in chromophoreor of an extrinsic probe (pyrene) gives clear evidence that adsorbed molecules are The average involved in finite surface aggregation number in the adsorption layer below the cmc and above a surface coverage of 0.2 turns out to be in the same range as that of the micellar size in solution measured well above the cmc. Finally, the adsorption layer of C1& on a flat SiOz/water interface was recently investigated by neutron reflection.ll From low to high coverage, the localand semilocalstructure of the adsorbed phase appears to be a "fragmented" bilayer. The major point resulting from these different experimental investigationsis that the adsorption process occurs mainly below but near the cmc. It involves a surface aggregative process similar in some respect to the bulk micellization taking place in solution above the cmc. This conclusion was formerly proposed in the pionnering work of Klimenko et al.12J3 Several modeld4J6 have recently been proposed to describe nonionic surfactant adsorption. For example, use of the self-consistent field theoryls seems really appealingeven if quantitative comparisonwith experiment is difficult at this time. However to our knowledge, no model has tried to analyze the particular properties of a "fragmented" adsorption layer. The aim of the present (11) Lee, E.M.; T h o u , R. K.; Cummine, P. G.;Staplw, E. J.; Penfold, J.; Rennie, A. R. Chem. Phys. Lett. 1989, 162, 196. (12) Klimenko, N.A,; Koganovakii, A. M. Kolloidn. Zh. 1974,SS, 161. (13) Klimenko, N.A.;Kofanov, V. I.; Sivalov,E.0.Kolloidn. Zh.1981, 43,287. (14) Rudzinski, W.;Dominko, A.; Partyka, 5.;Brun, B. Adsorpt. Sci. Technol. 1986,2, 207. (15) Koopal, L.K.; Bohmer, M.R.Presented at the Proweding8of the Third Conference on Fondamentals of Adsorption, Southefen (FRG), May 1989. Bohmer, M. R.; Koopal, L. K. Longmuir ISSO, 6,1478.
0143-1463/91/2401-1595$02.50/0Q 1991 American Chemical Society
1596 Langmuir, Vol. 7, No.8, 1991
Leuitz 2. Multiple Chemical Equilibrium. Multiple equilibrium between monomers and aggregates of size n is described by a set of equationslg p,L
= nplL n = 2, ~0
(1)
where the index L stands for the liquid phase and pnL is the chemical potential of a n aggregatetreated as a distinct component. Neglecting interaggregate interactions and following Ben-NaimB for the definition of standard thermodynamicof transfer, chemicalpotentials are written as :p
'
core
/
.'
= pnL"
+ k T In [C,]
with
Figure 1. Local structure of a micellar aggregate. paper is to propose a thermodynamic model of this specific aggregative adsorption. In a first part, the internal thermodynamic properties of nonionic surfactants aggregates are presented, a simple model of micellization is proposed, and its predictions are compared with available experimental results. In a second part, we discuss the properties of an aggregative adsorption layer in equilibrium with the bulk solution. By use of a grand partition function, the theoretical adsorption isotherms are then computed and compared with experimental data. 11. Model of Nonionic Micellization 1. General Overview. The cmc of nonionic surfac-
tants are lower than those of ionic surfactants. Poly(oxyethylene) (POE) chain repulsions on the external part of micelles are much smaller than for ionic surfactants. Generally, ionic charges can be localized on a defined external surface of the aggregate. Several phenomenological models use this approximation to calculate ionic micellization properties. In the case of nonionic surfactants, POE polar chains have a typical size well above the alkyl chain dimension and fill an extended polar corona as shown in Figure 1. The average surface per ionic head must be replaced by an average volume of interaction per polar chain. During the transfer of a POE chain inside this polar corona, two important contributions should be considered steric repulsion and stretching deformation. Similar problems appear in diblock copolymer micellization.16 A nonionic surfactant can be considered as the low molecular weight limit of a diblock copolymer in which the very short alkyl chain plays the role of the solventincompatible block and the poly(oxyethy1ene)chain that of a small solvent-compatible polymer. Obviously this approximation is not strictly applicable to low polymerization index N p (Np< 8). However, recent studies17J8 using this hypothesis qualitatively and quantitatively reproduce some important features of nonionic micellization. In the following, we shall use this approach and try to extend the previous models of Tanford19 and Nagarajan and Ruckenstein.20-22 (16) hibler, L.; Orland, H.; Wheeler, J. C. J. Chem. Phys. 1988, 79, 3660. (17) Golatein, R. J . Chem. Phys. 1986,84,3387. (18) CMES, J. M.; Levitz, P.; Poirier, J. E.; Van Damme, H. Arbiter
Symposium on Advance in Mineral Processing, SMEAIME Annual Meeting, March 1986. New Orleans, LA,P 1. (19) Tanford, C. J. Phys. Chem. 1974,24, 2469. (20) Napajan, R.; Ruckenetein, E. J. Colloid Interface Sci. 1977,60,
221. (21) Nagarw, R.; Ruckenstein,E. J. Colloid Interjace Sci. 1979,71,
(3)
is the number density of the n-aggregate population (number of aggregates of size n over the average volume of the system). pnL*' and An3 are respectively the local standard potential and the momentum partition function of a n aggregate. By use of the volume fractions Cn
nUTCn and
41 = UTC, where UT is the total volume of a surfactant, eq 1can be written as 4, = n(4,)" exp[-n
U ~ / ~ T I (4)
with
L.0
(5)
The constraint concerningthe conservation of the total amount of surfactants reads OD
c = c, + n=2
n c,
or
where C and 4 are respectively the total number density and the total volume fraction of surfactants in solution. Using the new variable u = In (@I) and the function f ( u ) defined as
n-2
we can see that df(u)/du is always negative since f ( u ) is a decreasing function withf(--) = 4 andf(+=) < 0. This property permits the solution of eqs 4 and 6 by a zero crossing algorithm and the calculation of the micellar size distribution for a given value of 4. As usual, the number and the weight average aggregation number are respec-
580.
(22) Nagarqian, R. Adu. Colloids Interface Sci. 1986, 26, 205.
(23) Ben-Naim, A. J. Phys. Chem. 1978,82,792.
Aggregatiue Adsorption of Nonionic Surfactants
Langmuir, Vol. 7, No. 8, 1991 1597
tively defined by the relations
Following previous analysis given by Stellingerand BenNaimUand Goldstein,17the criticalmicellar concentration (cmc)is defined as the value of 41where analytic inversion of a Taylor series expension of eq 6 breaks down; this singular point is defined by the relation d4/d41= 0 (8) In the following, we will briefly recallI7 the resolution of eq 8 in the case of multiple chemical equilibrium. Equations 4 and 6 can be rewritten as
with g(n) = n-
Ag:
kT
- n In (&) - In (n)
(10)
Using a steepest-descentapproximation for the summation term in eq 9 and considering a weak variation of 41 above the cmc, we obtain
= 41+ n*(2*kT/g"(n*))1/2(41)n* exp[-n*Agn.Li'/kTl (11)
where g"(n) is the second derivative of g(n) and n* the solution of dg(n)/dn = 0 Generally n* and g"(n*) are found to be weakly dependent on &. By use of eq ll,eq 8 is solved in the complex plane. The different roots are distributed on a circle. From an extension of the Yang-Lee theorem, the cmc is defined as the intersection of this circle with the positive real axis kmc
--
[(n*)2(2~kT/g"(n*))1/2 e~p[-n*Ag,,.~"/k TlI '/('+)
included in the last term. We shall now develop analytic expressions for these different contributions. The aliphaticcore of a n aggregate is generally considered as a micro liquid droplet. However the accessibility of the complete volume of this droplet to the aliphatic chain is hindered by a strong localization of one end of the chain on the external part of the aliphatic core (third term of eq 13). Classically, the standard free energy of transfer of the aliphatic chain from water to micellar core is evaluated by using a slight modification of experimental dataavailable for the solubilityof hydrocarbons in water.% These data use a molar fraction scale to calculate chemical potential and include a contribution due to the delocalization of the apolar chain inside its pure liquid phase. This logarithmic contribution has to be removed and a correction term (see the right-hand side of eq 14) must be added in agreement with the conventionused in eqs 2 and 3. In the case of CN,EN surfactants and following Tanford,lo we also assume tRat the first CH2 group adjacent to the polar head has no appreciable contribution during the transfer inside the apolar core. The "local" standard free energy of transfer APnc' reads as
(12)
In the model proposed below, the estimation of the cmc using eq 12 is stable and weakly dependent on 4 (less than 5%).
3. S t r u c t u r e of t h e S t a n d a r d Free Energy of Transfer. APnL' stands for the total standard free energy of transfer of a surfactant from solution to an aggregate of size n. This function is taken to be the sum of six contributions
where uw and Ua are respectively the molecular volumes of water (30 A8) and of the normal alkyl chain. At 298 K, is given byzs u, = 54.3
+ 27.O5(Ne- 1) in AS
(15)
The aliphatic chain has less conformational freedom inside the hydrophobiccore than in the pure liquid. From recent estimates given by Nagarajan,22the free energy of transfers of the methylene and methyl groups (in molar fraction scale) are taken to be given by &CH,'
= -3.8kT = -2265 cal/mol
AgcH,' = -1.25kT = -745 cal/mol
(16)
where Agc~pDis lower than the value for transfer measured in bulk liquidlg (-882 cal/mol). The case of &,PEN surfactants is more delicate. As found elsewhere,ln*&the phenoxy group is considered to be equivalent to three methyl groups and eq 14 is used after replacement of N,, by Ne + 3. The positive free energy contribution from residual contacts between the aliphatic core interface and water is written as19
- 20)
(17) where E,* is the average interfacial area per alkyl chain measured at a distance of closest ap roach of water molecules to the aliphatic interface (1.5 ). BOis the area screened by the chemical bond between polar head and apolar tail (typically 21 Az). 52 has the character of an interfacial free energy. There are a number of model calculations where oil-water interface tension of about 50 mJ/m2 has been used. On the other hand, more direct investigations2@*a or thermodynamicmodel of amphiphilic water systems32give a 52 value ranging between 14 and 23 P~C'
Q(Ba*
s
The first two terms on the right-hand side of eq 13 describe the transfer of the alkyl chain from solution to the aliphatic core of a n aggregate. Two processes must be considered the formation of a bulk cluster (firstterm), and the generation of an interface with the solvent (second term). The third term takes into account the localization of the tail-head bond in a restricted volume close to the hydrophobic core/water interface. The following two terms are respectively associated with excluded volume interaction and stretching deformationof the POE chains inside the polar corona. The kinetic reduction associated with the replacement of n monomers by a n aggregate is (24) Stellinger, F. H.; Ben-Naim, A. 1981,1981,2510.
(25) Abraham, M.H. J. Chem. Soc., Faraday Trans. I 1984,80,163. (26) Cruen, D. W. J. Phys. Chem. 191,89,146. (27) Poirier, J. M. These De Docteur d'Etat es-Sciencw Physiques, Univereite De Nancy, France, 1984. (28) Shinoda, K.; Tamamuehi, B.; Nakagava, T.; Iaemura, T. In Colloidal Surfactants; Academic Press: New York, London, 1963. (29) Hermann, R.B. J. Phys. Chem. 1972, 76,2754. (30)Reynolds, J. A.; Gilbert,D. B.; Tanford, C. R o c . Natl. Acad. Sci. U.S.A.1974, 71,2925.
1598 Langmuir, Vol. 7, No.8, 1991
Leuitz
mJ/m2, in good agreement with the estimation of Par~ e g i a n . As ~ ~ it will be discussed in part 11-5, good accordance with experimental data was found for an intermediate value of 19 mJ/m2. In a n-aggregate, the tail/head bond must move in a restricted volume V, around the interface between hydrocarbon core and polar corona, hence
V, = (Sa- nZ& (18) where Saie the total hydrocarbon core interface and 5 is the average "roughness* of this interface measuring the "off surface* displacement of each surfactant. Experimental results indicate that the micellar surfaceroughness for nonionic93 or ionic surfactant#*% is limited to amplitudes of a few angstrom units. Our choice for [ is twice the effective C-C bond length in trans conformation (i.e. 2.52 A). The third term in the right-hand side of eq 13 therefore becomes
Translation of the n-aggregate as a whole will be defined later in the last term of the right-hand side of eq 13 and the exponent fl is the number n of surfactants minus one. The d y t i c expression for excluded volume interaction between POE in the polar corona is derived by a Flory analysis. As demonstrated by Flory,%the free energy of transfer of two distinct and noninteracting chain portions inside the same volume 6 Vcan be written for each portion as
1 G(AF)/kT= 2'1 - 2X)UhP,2bV
= u,2/4
&,e
0 1 =,(1-2x)u
N p 2
(21)
ZPLP
Equation 21 has already been used to describe monomermonomer repulsion inside the polar corona of diblock copolymer micelles.39 The contribution due to deformation of POE chains when they are transferred into a n-aggregate is given by16
(20)
with uh
of Kjellander et al. is different from that of F l o e since the equation for the integral free energyof mixing is written differently. Their interaction parameter, noted (-w)in their work, has a different meaning that the Flory interaction parameter. However ita computation from experimenta turns to be strictly equivalent in both theoretical models. Exact analytical expression for the POE-water integral free energy of mixing stays, for us,an open question even if ita variation or ita partial derivative with respect to the total number of solvant molecules agrees with the standard Flory-Huggins theory. Equation 20 assumes the validity of the Flory-Huggins integral free energy of mixing. This assumption can certainly be challenged and the complete relevance of eq 20 remains a problem especially for high POE concentrations. Nevertheless we will use this analytic expression as a first approximation. As shown in Figure 1, the average volume associated with a POE chain is approximated to the product Z&,, in which L, is the thickness of the polar corona, Z, is the average section of the POE chain computedat L,/2. From eq 20 we get
u,
where Pr is the final volume concentration in ethoxy monomers and up is the molar volume of CH2CHzO in water (63 A3 As shown by Kjellander et al.,88several ways were used to computethe Flory interaction parameter x from experiments. First, the Flory-Huggins expression of the chemical potential of the solvent in solution relative to its chemical potential in the pure liquid fita correctly vapor pressure data of Rogers and Tam.38 In the temperature range 25 O C C T C 75 OC, a good agreement between the theoretical expression and experiments is acheived for relatively low POE degree of polymerization (13 C NpC 180) and for polymer weight fraction ranging between 0 and 15 95. Second,at higher temperatures and concentrations, the Flory-Huggins expressions of the critical composition permits a direct computation of x , using lower and upper critical solution temperatures of Saeki et The x temperature dependence given by Kjellander et al. (see their Figure 12) allows us to estimate a value of 0.093 at 298 K. It must be noted that the model (31) Pmegian, V. A. Trans.Faraday SOC. 1966,62,848. (32) Jonmn, B.; Wenneratrom, J. J. Colloid Interface Sci. 1981,80, 182. (33) Podo, F.; Ray, A.; Nemethy, C. J. Am. Chem. SOC.1979,95,6164. (34) Cabane, B. J. Phys. (Paris) 1981,42, 847. (36) Cabane, B.; Duplemix, R.; Zemb, T. J. J. Phys. (Paris) 19811,46, 2161. (36)Flog, P. J. Wnci leu of Polymer Chemistry;Cornell Univereity Prea: New York, 1963; 8haptar 12. (37) Herrington,T. M.; Sahi, S. S.;Leng,C. A J. Chem. SOC., Faraday Tram. I 1986,81,2693. (38) Kjellander, R.; Florin, E. J . Chem. Soc., Faraday Trans. I 1981, 77,2063. Rogers, J. A.; Tam, T. Can. J. P h r m . Sei. 1977,12,66. Saeki, S.; Kuwahara, N.; Nakata, M.; Kaneo, M. Polymer 1976, 17,686.
in which (R2(Np)) is the end-bend distance of POE chains in the random coil conformation. As discussed by Tanford et al.,a (R2(N,))can be obtained from parameters calculated by FloryP* A good numerical approximation reads as22
(R2(N,))= 5.07(Np- 1.06)1'2 (23) Obviously, eqs 22 and 23 can only be applicable to long POE chains (N > 8). The model of chain deformation used in eq 22 is Lased on the uniform deformation model of Flory. However the assumption of uniform concentration in the nonplanar micellar corona necessitates a nonuniformstretching of the POE chains. As a part of his treatment of micodomains in block copolymer melts, Semeno+ has developed analytical expressions for the e h t i c deformation free energies taking into account the nonuniformity in the stretching of the chains. These results were adapted by Zhulina and Birshtein& and Nagarajan and Ganesh* to the situation of sphericalaggregateswhere the core and the corona are both swollen by a solvent. The main concern of a such model is the relation between the core/corona interface curvature and the elastic deformation of the polymer blocks. In nonspherical micellar aggregates, the radius of curvature is not constant and completeanalysisrequires a rather complicated calculation (39) De Gennee, P. G. Rep. Prog. Phys. 1969,92,187. (40) Tanford, C.; Nozaki, Y.; Rohde, M. F. J. Phys. Chem. 1977,81, 1556. (41) Mark, J. E.; Flory, P. J. J. Am. Chem. SOC. 1966,87, 1416. (42) Mark, J. E.; Flory, P. J. J. Am. Chem. SOC. 1966,88,3702. (43) Flory, P. J. Statistical mechanism of chain molecules; Wiley: New York, 1969; Chaptar 5. (44)Semenov, A. N. Sou. phy8. JETP 1986,61,733. (46) Wulina, Y. B.; Birahtein,T. M. Polym. Sei. USSR 1987,29,1678. (46) Nagarajan, R.;Ganeeh, K. hfacromo~ecu~es 1989,22,4312.
Aggregative Adsorption of Nonionic Surfactants of variation along radial directions. As discussed below, mean aggregation number of nonionic micelles can be found greater than the value allowing a spherical shape. The use of eq 22 is certainly a simplified approach but allows a “continuous” analysis of the elastic deformation of the POE chains versus the aggregation size. Finally A1n,k‘ is written as
Langmuir, Vol. 7, No. 8, 1991 1599 CMC
10 A
T
“
‘2
5
Y
V-
APn,k
- kT[
n
- In (At)]
0 0
or
By addingeqs 19 and 24, one can check that the quantum length A1 is exactly cancelled in the APnL” function. 4. Micellar Shape and ANnL*OMinimization. For an aggregation number slightlygreater than the value allowing a spherical shape, nonionic aggregates are generally considered as “globular”. It is difficult to decide clearly whether these micelles are rodlike or disklike. Both hypothesesare found in the experimental literature.40*47+1 For continuityreasons (phasediagram),an elongated shape is often considered to describe large aggregatesof .sz However, it is clear that any trend could be possible !or a small departure from spherical shape. Polydispersity and intrinsic shape fluctuationtP@render any definitive statement highly difficult. In this work, an ellipsoidal shape (oblate or prolate) is used to describe the aliphatic core of a n aggregate. The minor semiaxis Lamust conform to 0 < L a e La(25) where Lamu is the maximum distance compatiblewith the chemical structure of the hydrophobic tail and the compactnessof the aliphatic core. The equivalent major semiaxisLb is calculated assuming that the ellipsoid volume is equal to nVa. Let La& be the optimal extension of the apolar chains. In the case of normal alkyl chains in alltrans conformation, Lao*’ is@
Lt* = 2.1 + 1.265(Na- 1) in A (26) The spherical aggregate of radius L a bhas an aggregation number nwb above which micelles are no longer spherical. For n < nspbm, we have
La- = L:* The external periphery of the polar corona is taken to be an ellipsoid of revolution with the minor and the major semiaxes equal to (La + L,) and (Lb + L,), respectively. As explained above, the ”average sectionn Z, per POE (47)Degiorgio, V. In Physics of amphiphiles: Micelles, Vesicles and Microemubiom; North Holland Amsterdam, 1985;pp 303-335. (48)Kaweguchi, T.; Hamanaka, T.; Mitaui, T. J. Colloids Interface Sci. 1988,!X, 437. (49)N i h n , P. E.;Wenneretrom, H.;Lindman, B. J . Phys. Chem. 1988,87,1377. (50) Nihon, G.; Lindman, B. J. Phya. Chem. 1983,87,4756. (51)Brown, W.; Pu, Z.; Rymden, R.J. Phys. Chem. 1988,92,6086. (52) Mitchell, D. J.; Tiddy, G.J. T.; Waring, L.; Boetock, T., McDonald, M. P. J. Chem. SOC.Faraday. Tram. 1 ISM9, 79,975. (53)Halle, B.; Landgren, M.; J o n ” , B. J. Phys. (Paris) 1988,49, 1235. (54) Tanford, C. J . Phys. Chem. 1972,76,3020.
1
2
3
c (10-‘~0fi) Figure 2. Monomer concentration (CJversus the total concentrationin bulk (C).The computationis performed for C I & ~ at 298 K using the oblate ellipsoidal micellar aggregates. x = 0.093 and Q = 19 mJ/m*. The vertical dotted l i e indicates the cmc predicted by using eq 12.
chain is defined at the surface of an intermediate ellipsoid with semiaxes (La + L,/2) and (Lb + L,/2). The standard free energy of transfer APnLI0is a shapedependant function of three variables: n, L,, L,. The equilibrium values of L, and La characterizing the n aggregate may be found by a two-step minimization. We first solve
This equation only involvesterms associatedwith excluded volume interaction and deformation of the POE chain (APn,; and APn,;). Close inspection of equalities 21 and 22 shows clearly that the minimum of ApnL”(LaJp) is reached for the value Lp*(nJ& which is always greater than (R2(Np)).The condition Min [ A P y ( L a J p (nJa) * 11 with La E [OJa-l (28) then gives the optimal value of the minor semiaxis La. Oblate or prolate shapes are both considered. 5. Theoretical Results and Experimental Data. a. General Results. The model predicts a rather sharp micellar transition. As shown in Figure 2, the determination of the cmc by eq 12 is in good agreementwith the monomer concentration profile but appears in general just before the apparentbreak. The monomer concentration increasea very slowly above the cmc. The micelle size distribution displays a bimodal shape (monomer and n aggregates). The main peak has a symmetrical bell shape. As shown in Figure 3, the maximum of the size distribution shifts slightly to larger n with the surfactant concentration, but this evolution is very gradual. Polydispersity (expressed as the ratio ( n ) w / ( n ) , ,is) always close of unity. As shown in Table I, calculated results depend on the estimated s2 value. Computed cmc and mean aggregation number increase with s2. A linear relationship between the average micellar size and the inverse of the number of ethoxy group is generally observed (see Figure 4). This property is in good accordance with the experimental trend.&** As shown in Figure 4, a good agreement with experimental data is found for $2 = 19 mJ/m2. We present, in Table 11,how the calculated results are modified if the value for the Flory interaction parameter is altered. (55)Becher, P. J. Colloid Sci. 1961,16,49. (56)Becher, P. In Non-Ionic. Surfactants;Schick, M. J., Ed.;Dekker: New York, 1967;Chapter 15.
Levitz
Langmuir, Vol. 7, No.8, 1991 200
05
150
AS 04
100 V
03
50
02
0
0 50
150
100
AGGREGATION
NUMBER
Figure 3. Micelle size distributionof C1&2 solution at 298 K: (a) C = 5 X lo-' mol/L; (b) C = 10-8mol/L; (c) C = 10-2mol/L. The oblate ellipsoidalshapeis used to describemicelaraggregates. x = 0.093 and R = 19 mJ/m2. Table I. Influence of 0 on the Calculated Micellar Concentrations and Average Aggregation Numbers of CIZENSurfactant# in Water at 298 K ( x = 0.093) il = 19 mJ/m2 il = 25 mJ/m2 0 = 15 mJ/m2 (n), (n), (n), at 10-2 cmc, at10-2 cmc, a t W cmc, mol/L mol/L mol/L mol/L Np mol/L mol/L 8 00 3.85X 1od 102 6.70 x lo+ 188 1.15 x lo-' 12 18 23
4.55x 1od 5.30 X 1od 6.10X 1od
44 32 26
73 54 45
8.90X 1od 1.25X lo-' 1.55X lo-'
123 85 70
170X lo-' 2.70X lo-' 3.65X lo-'
Evolution of micellar properties with x is very gradual and weaker than the one observed with a. Reduction of excluded volume interactions between POE chains in the polar corona (Le. greater value for x ) increases the average aggregation number and decreases the cmc value. b. CNEN,Series. Mean results are summarized in Table 111. Apparently, the predictions are relatively ~~
~
~
~~~
(57) Brown, B.; Johnsen, R.;Stilbe, P.; Lindman, B. J. Phys. Chem. 1988,87,4648. (58)Zana, R.;Weill, C. Phys. Lett. 1986,46, L953. (59) Zulauf, M.; Wercstrom, K.; Hayter, J. B.; Degiorgio, V.; Corti, M. J. Phys. Chem. 1 9 l , 8 9 , 3411.
(80)Meguro, K.; Takasawa, Y.;Kawahaahi, N.; Tabata, Y.; Ueno, M. J. Colloids Interface Sci. 1981,83, 50. (61) Ni&ikido,N.;Shind, M.; Sugihara,G.;Tanaka, N.;Kaneshina, S . J. Colloids Interface Sci. 1980, 74,474. (62) Kawaguchi, T.; Hamanaka, T.; Mitsui, T. J. Colloids Interface Sei. 1988,96, 437. (63) Lange, H. Proc. Int. Cow. Surf. Act., 3rd 1960, I , 279. (84) Fujimatau,H.; Takagi, K.; Matsuda, H.;Kuroiwa, S. J. Colloids Interface Sci. 1988,94,237. (65) Schick, M. J. J. Colloid Sci. 1962, 17, 801. (66)Woodhead, J. L.; Lewis, J. A.; Malcolm, G.N.; Watson,I. D. J . Colloids Interface Sci. 1981, 79,454. (67) Elworthy, P. H.; MacFarlane, C. B. J. Chem. SOC.1963,907. (68) El.Eini,D. I. D.;Barry,B. W.;Rhodhea,C.T. J. Colloids Interface Sei. -. 1976. . -,54. - -,348. - - -. (69) Richtering, W. H.;Burchard, W.; Jahne, E.; Finkelmann, H. J . Phys. Chem. 1988,92,6032. (70) Crook, E. H.; Fordyce, D. B.; Trebbi, G. F. J . Phys. Chem. 1963, 67. - . , -1987. - - .. (71) Crook, E. H.; Trebbi, G. F.; Fordyce, D. B. J . Phys. Chem. 1964, 68,3592.
0.1 5
0.1
1INp Figure 4. Influence of fl on the mean aggregation number of C I ~ E homologoa N~ series in water solution. Computations are relativeto a totalsurfactant concentrationC = 10-2mol/L at 298 K, using an oblate shape. The Flory interactionparameter x = 0.093. Key: 0, R = 15 mJ/m2; 13,52 = 19 mJ/m2; +, R = 25 mJ/m2;0,experimental results for ClJ?" at room temperature.
01
0
0.05
Table 11. Influence of x on the Calculated Micellar Concentrationsand Average Aggregation Numbers of CISEN-Surfactants in Water at 298 K (0 = 19 mJ/m') x=o x = 0.093 x = 0.2 (n), (n), (a), at 10-2 cmc, at10-2 cmc, at 10-2 cmc, ND mol/L mol/L mol/L mol/L mol/L mol/L ______________~
8 12 18 23
89 64 46 38
7.60X 1od 1.05X lo-' 1.5OX lo-'
1.90%lo-'
102 73 54 45
6.70X 8.90X 1W6 1.25X lo-' 1.55X lo-'
123 89 65 54
5.70X 1od 7.20X 1od 9.75X 1od 1.20X lo-'
independent of the micellar shape. The computed cmc and mean aggregation numbers, obtained by using either oblate or prolate ellipsoids, are equivalent. As shown in Figure 5, aggregates are always spherical below n@phem. Oblate ellipsoids are found to be slightly more favorable in an intermediate regime. Prolate aggregates are stable at higher aggregation number, when repulsion and stretching contributions of the POE chains start to override the free energy cost of the polar/apolar interface. In this last domain, the minor semiaxis Laof the oblate ellipsoid starts to decrease, as shown in Figure 6. However, this relaxation is not sufficient to reduce the polar head contribution. Mean aggregation numbers, calculated well above the cmc mol/L), are in good agreement with different experimental results (see Figure 7). The average micellar size increases as the square of Nap as shown in Figure 8. This last evolution has already been observed."J8 The predicted micellar concentrations are in general slightly lower than the experimental determinations. A direct homologous comparison is shown in Figure 9 for the CN,E~ series.60 c. Series. The calculated results are presented in Table 4 for the polyethylene glycol p-tert-octylphenol ether homologous series. According to Robson (72) Ray, A.; Nemethy, G.J. Phys. Chem. 1971, 76,809. (73) Kushner,L. M. K.; Hubbard, W. D. J. Phys. Chem. 1954,68,1163. (74) Mankowitch, A. M. J. Phys. Chem. 1964,68, 1027. (75) Mankowitch, A. M. Ind. Eng. Chem. 1956,47,2175. (76) Dwiggings, C. J.; Bolen, R. J.; Dunning, H. N. J. Phys. Chem. 1960,64,1175. (77) Selinger, B. K.; Watkins, A. R. Chem. %ye. Lett. 1978,66,99. (78) Shick, M. J.; Atlas, S. M.; Eirich, F. M. J. Phys. Chem. 1962,66, 1326.
Langmuir, Vol. 7, No.8, 1991 1601
Aggregatiue Adsorption of Nonionic Surfactants
Table 111. Theoretical Computations of the Micellar Concentrations and the Average Aggregation Numbers of CN,EN, Surfactants in Water and Comparison with Experiments oblate prolate cmc, (n)n at ( n h at cmc, (n). at (n), at exptl cmc at exptl aggregation N. Np mol/L 10-2 mol/L mol/L mol/L 1W2 mol/L mol/L 25 "C,mol& no. (room temp) 12 12 12 12 12 12 12
6 8 9 12 14 18 23
5.7 x loa 6.70 X 1od 7.25 X loa 8.90 x loa 1.0 x lo-' 1.25 X lo-' 1.55 X lo-'
126 99 90 71 63 52 43
130 102 92 73 65 54 45
5.90 x 106 6.85 X 1od 7.35 x 1od 9.0 x loa 1.0 x lo-' 1.25 X lo-' 1.55 X lo-'
114 89 81 66 60 51 43
121 92 84 68 61 52 44
16 16 16 16 16 16 16 16 16 14
8 9 12 15 17 21 32 44 63 8
6.3 x 10-7 6.95 X lW7 8.90 x 10-7 1.10 x 10-6 1.25 X 10" 1.55 X 10-6 2.50 x 10-6 3.55 x 10-6 5.20 X 1od 6.60 X 10-6
167 151 117 97 87 73 51 40 31 131
170 153 119 99 89 74 52 41 32 134
6.4 x 10-7 7.0 x 10-7 8.90 X l W 7 1.0 x 10-6 1.25 X 10-6 v 1.55 X 1 2.50 X 1 v 3.55 x 1 v 5.21 X 10-6 6.70 X 10-6
160 142 111 95 87 73 51 40 31 122
164 145 113 96
6.8 X lo4 7.1 X 104
lo-'61 1.4 X lo-'69
5.5 x
loa 66
144,67 180," 1 w 123.M 120," 98!l 100." 104" 94,& 114$l 97di 81M
5lM 40,& 41ae 6X1Ods 1.75 X lo-'@ 2.1 x 10-667 2.3 X 10-6 67 3.1 x 10-687 6.6 X l0-6@ 3.9 x 10-667 1.3 X 1.6 X l o a 6 7 2 x 1 ~ 6 7 9x10-6" 9.8 x
88 74 52 41 32 126
lW
21v 152" 9gSs
7067 56a 3968 25w 11969
150
5
100
C V
50
0
0
100
200
300
400
0
20
40
69
NP n Figure 5. Relative stability of oblate and prolate ellipsoidal n aggregates for CloElz at 298 K. x = 0.093 and Sl = 19 mJ/m2. 1.00
i
Figure 7. Evolution of the mean aggregation number of CI&N* (top) and (bottom) homologous series versus Np'x = 0.093 and Sl = 19 mJ/m*. Full lies repreeent computation relative to a totalsurfactant concentration C = 10-2 mol/L at 298 K;(a) prolate shape; (b) oblate shape. Key: W, experimental results for cl&Np at raom temperature; 0, experimental results for CIZE~p at room temperature.
A. In the range of values taken for the calculation, the
0.50 4 0
100
200
300
I
400
n
Figure 6. Relaxation of the minor aemiaxis L. versus the micellar size. Comparison between oblate and prolate ellipsoidal shapes for C12E12 at 298 K. x = 0.093 and Sl = 19 mJ/m2. and the volume ua v d the extended chain length of the apolar tail are respectively equal to 363 As and 10 (79) Rolwon, J.
R.;Dennis, E. A. J. Phys. Chem. 1977,81, 1075.
oblate ellipsoid is found to be the stablest aggregate shape (Figure 10). Relaxation of the semiminor axis is never observed (i.e. La = La-). These two results can easily be explained by the "bulky" nature of the p-tert-octylphenol chain which has a short length and a large section. The cost in free energy for the polar/apolar interface is higher than for a n-alkyl chain and prevents the formation of "rodlike" aggregates. This observation is in good agreement with various experimental results in which intrinsic viscosity measurements'@or small-angle X-ray scattering dataso are more consistent with an oblate than with an equivalent prolate ellipsoid of revolution. As shown in Figures 11 and 12,good agreement between predictions of the model and experiments is observed for the cmc and the mean aggregation number. For the CSPEN,series, the experimental literature is in general concerned with surfactants having an ill-defined apolar group (branched and polydispersed nonylphenol) and direct comparison with our model is difficult. In Table (80)Paradiea, H.H. J. Phys. Chem. 1980,84,699.
1602 Langmuir, Vol. 7, No. 8, 1991
Leuitz
8.Oe-4-
B A
C
V
4.oe-4'
50 -
04
4
200
100
0
o.oe+oJ
10
0
300
20
30
I
40
50
NP
Na**2
e ation number versus the square of the n-alkyl chain leng#%fex Nl. Computationsare relative to an oblate ellipsoidalshape at '298K and C = lo"mol/L. x = 0.093 and Q = 19 mJ/m2. Key 0 , c N & m, CN&. Figure 8. Mean
Figure 11. Comparison between predicted cmc (-1
and experimental determinations ( 0 ) for the CSEN, homologous series at 298 K. x = 0.093 and Q = 19 mJ/m*.
I h
2
J
.lo'
si E w
.12-
Y
\ 04 -1"
0
'
8
10
12
14
16
"
10
"
20
30
"
40
i
D
18
Na Figure 9, Predicted (-1 cmc and experimental (0)determination@ for the CN,& homologous series. x = 0.093 and D = 19 mJ/m2. Computation are relative to an oblate ellipsoidal shape at 298 K.
NP Figure 12. Evolution of the mean egation number of CSPEN homologous series at '298 K an?!& a total concentration C = f0-0 mol/L. x = 0.093 and D = 19 mJ/m*. Key -, model; 0, experimental results.
to 387 ASand 12 A. Predictions concerning this specific surfactant follow the general trend of the C ~ E Nfamily , but the cmc is in general 3 times smaller. Comparison with available experimental determinations is acceptable. 111. Model of Aggregative Adsorption
n
Figure 10. Relative stability of oblate or prolate ellipsoidal n aggregates in the case of CSpE12.
IV, we present computation of the micellar properties of ether the polyethylene glycol 1,3,5-trimethylhexanephenol series. From the density of the liquid (0.87)and molecular models, the volume and the extended chain length of this nonylphenolgroup are taken to be respectively equal
1. Free Energy of the Aggregative Adsorption Layer. The interaction length between water molecules in solution and a hydrophilic solid surface (of silica for example) is in general small and of the order of 1nm.81 In the following, we will only discuss properties of the surfactant molecules inside the adsorption layer, even if a part of the adsorption process (especiallyfor interactions between POE chains and solid surface) results from an exchange mechanism with water. Let us consider Na' aggregates of size n distributed on a solid surface A (the superscript s stands for the adsorption layer). The free energy I"(A,n,N,,'), noted 3%for short, has a complete differential at constant temperature written as
dF = -no dA + p i dN: where IP is the bidimensional pressure and
(29) pn*
is the
~
_
_
_
(81)Fri iat, J. J.; Casea, J.; Francoie, M.; Lettelier,M. J. Colloids Interface 1~82,89,318.
lei.
_
Aggregative Adsorption of Nonionic Surfactants
Langmuir, Vol. 7, No. 8, 1991 1603
Table IV. Theoretical Computations of the Micellar Concentrations and the Average Aggresation Numbers of CWEN,and EN, Surfactants in Water and comparison with Experimentr
(4. at (n), at cmc,mol/L exDtl cmc at 25 O C . mol/L . . 10-2mol/L 10-zmol/L 8 6 2.45X lo-' 770 844 2.3 X lo-'. 2.3 X lo-' 'O 3.0 X lo-' 8 8 205 218 2.85 X l+> 2.8 X 2.5X lo-' 70 3.25 X lo-' 151 8 9 3X 2.9 X 3.4 X lo-' 47 le0 8 9.5 3.35 x lo-' 142 2.8 X lo-,' 2.4 X lo-' 72 134 8 10 3.50 x lo-' 127 121 3.35 X lo-', 3.3 X 3.2 X 3.4 X lo-' 76 8 12.5 4.15 X lo-' 82 3.8 X lo-', 2.53 X lo-','* 4 X lo-' lo 86 5.05 X lo-' 58 8 16 61 4.3 x lo-','O 7 x lo-' 10 32 8.66 X lo-' 7 x io-'! 7.7 x io-'?o 10-3 10 8 30 33 26 27 1.10 x lo-' 8.1 x io-':o 1.2 x 10-3 10 8 4 0 119 1.0 x lo-' 124 7.5 x 106,?8 9.7 x 10-666 9 10 70 1.40 X lo-' 72 9 15 1.1X 1.2 X lo-': 8.7 X 106 Ib 1.80 x lo-' 52 54 9 20 1.4 X 1.5 X lo-' 38 2.60 x lo-' 9 30 1.85 X 2.7 X lo-': 1.53 X lo-' 37
N.
N.
I
chemical potential of the n aggregate. The Legendre transform of eq 29 yields, for pn'
(30) where p l = Nt/A and T is an integration constant whose value can be obtained when Nan 0. The standard reference states are defined by
-
p t = p,'" +kT In [ q a ] (31) in which 7'. is the surface fraction of the n-aggregate population. The standard potential pnn" will be defined later, in relation with the structure of the surface n aggregate, its possible localization, and the specific interaction with the solid surface. From eqs 30 and 31 we get 7 = ~(2' kT lim In [qa] (32)
+
q.40
and
exptl aggregation no. at room temD r
,
>3w 132p 140,'a 111,'' 13616
120n
1078 400 309 276,'8 1w 80," 52ss 6278 44,78 1P
At this level, eqs 35 and 36 give one possible configuration of the adsorption layer. The next step is to find the equilibrium conditions with the liquid phase and to predict the more probable or the mean configuration of the interfacial system. 2. Grand Partition Function and Adsorption Isotherm. In general, adsorption of nonionic surfactants occurs at very low liquid concentration. In this condition the surface excess of surfactants (mainly defined by a conventional Gibbs plane near the solid surface) and the total amount of adsorbed molecules can be considered to be very close. Keeping this approximation in mind, the grand partition function of the adsorption layer is written as (37) with In [t(A,n,N;)] = -F(A,n,Nt)/kT
+ nN,lplL/kT
(38)
By applyinga steepest-descent approximation(also called maximum term methode2)to eq 37,we obtain the following equilibrium conditions
This equation is quite general and depends on the choice of the state equation Ill. For nonionic surfactants, we will consider a localized adsorption model where is given bY
(39)
(a with a. = u(n)/a
[W,n,Nt)l) an A&* = O
The first condition is the usual equality
a = A/N,'
(34) In the last expression, u (n) is the projected surface of a n aggregate and a is the 2d compactness (0.907). The equation of state (34) supposes a hard core potential between surface aggregates (this crude approximation is certainly not correct and will be further discussed in part IV). Using eqs 33 and 34,we can write
which can be written n( In [&I -
$)
-In [a] = In
[*] -
(42)
a
1 '.
with and
0,.
C(n kT + ln [a]] (36) At constant temperature and for a solid surface A, P is a function of two independent variables, n and Nan.
For any one value of 41, an infinite number of doublets (n,Nas)constitute solutions of eq 42. In the plane {n,Nan), the set of solutions is a line C which can be described by the curvilinear coordinate n. Condition 40 permits the location, on the line C, of an optimal value of In [t(A,n, (82) Hd,T.L.AnZntroduction ToStatistical7'hermodynamicn;Dover Publication: New York, 1986.
Leuitz
1604 Langmuir, Vol. 7, No.8, 1991
(2) A local maximum for t(n) exists, but
' I
/-
t
Figure 13. Four stability criteria used in the computation of the most probable configuration of the adsorption layer.
Na8)]e For doublets (n,Nan)belonging to C,we have from eq 39
-
d In [t(A,n)l - (a dn
ahr,ll+
Lt(A,nJli')l)
aN:
A,n
dn
(a ~nrt(~,n,~:)i) an
- (a In [U,n,N:)l) an
A3.m
A&#
(44)
and eq 40 can be written as
with
(45) and (n,Nt) E C (46) Let (n*,Na") be a solution of eq 45. The stability of this optimum must be discussed. In a 3D space defined by parameters n, Nan, and z, In [t(A,n,N,n)] constitutes a surface S. The line of intersection of S with a subset defined by the generic equation (n = constant) has a negative curvature since we always have
The stability of the Solution (n*,Naa') only depends on the evolution of the absolute derivative d ln tldn given by eq 45. As shown in Figure 13, four cases need to be analyzed: (1) The absolute derivative is always negative. In this case, no aggregative adsorption can be predicted. The model fails to give an average configuration of the interfacial system. The physical meaning of this situation is that the adsorption of single moleculesor the formation of small specificmolecular associationsmust be taken into account.
n*d In [t(A,n)l dn < dn In this case, (n*,Na8*)is not the more probable solution and the former conclusions apply. (3)There is a global maximum for t(A,n). The more probable solution and the mean configuration of the adsorption layer is (n*,Na8*). (4)d In [t(a,n)]/dnis always positive. A bidimensional condensation involving an infinite aggregate occurs. Three steps are needed for the computation of the adsorption isotherm. First, the monomer concentration 61 is calculated from the total concentration 4 by solving eqs 4 and 6. Second, eq 42 is solved for each value of n. Third, the evolution of d(ln [t(A,n)])/dn is analyzed and an eventualglobal maximum is calculated by a zero crossing algorithm. 3. Structure and Minimization of Agn'**, The structure of &n8*' specifies the nature and the properties of the adsorption process. Different choices are possible and several evolutionscan be predicted. In this paper, we present the particular case of a weak interaction between solid surface and adsorb+ surfactants. The standard free energy of transfer of a surfactant from bulk liquid to a surface is treated as a perturbation of what is known in solution. &nO' is then considered as the sum of two contributions = Ag,,;'
+ Agnjnters'a
(48) The first term on the right-hand side of eq 48 describes the internal properties of the surface aggregate. Its analytic expression is very close to the one given in eq 13 and discussed in part 11-3. However, localized surface clusters do not have global translational freedom. The kinetic reduction term APn,; of eq 13 must be replaced by a vibrational contribution of the surface aggregate as a whole. Three degrees of translational motion of the free micelle in bulk liquid must be replaced by three degrees of vibration. In the following we shall neglect this vibrational contribution. Nevertheless, the exponent 0 in equality 19is changed to 0 + 1 to cancel factors involving the quantum length A1 (which disappears if we use a vibrational volume of the surface aggregate). With these various simplifications, and using eqs 3, 13, 19,and 43, &nm@' becomes &d,O
- k T (49) k T l n [(2,-2&1 where 2, is the average area per alkyl chain measured onto the apolar core/polar corona interface Sa (see Figure 1). The second term on the right-hand side of eq 48 takes into account the direct interactions between the surface aggregate and the solid surface. A choice must now be made concerning the nature of this interaction. For a hydrophilicsolid such as silica,clear evidenceeSshowsthat undissociated free silanol groups are the principal sites for adsorption by hydrogen bonding to the ether oxygens of the POE chains. Two opposite cases can be discussed. In the first, there is a large number of adsorption sites per unit surface and absorption is limited by the number of ethoxy groups able to interact with the solid surface. This situation favors (83)Howard, G.J.; McConnell, P.J. phy8. Chem. 1967, 71, 2974.
Langmuir, Vol. 7, No. 8, 1991 1605
Aggregatiue Adsorption of Nonionic Surfactants
the n-aggregate polar corona and the solid surfaceis limited by the number of accessible adsorption sites and not by available ether oxygens inside the polar corona. e. is a free energy per unit of surface related to the bonding site density r b and the free energy characterizing the displacementof adsorbed water by a ethoxy group. Since a positive contribution due to the local perturbation of the polar chain in interaction with the bonding site must also be added, c, can be expressed as
up (&b
0
."
.;
6 1 9 1 0
0
2e-4
4e-4
6e-4
8e-4
le-3
C (Mol / I ) Figure 14. Influenceof t. on the adsorption isotherm of C&'Elo. T = 298 K. Key: 0, t. = -1.6 cal/Ar; 0 , t. = -3.0 cal/A2; 0 , C, = -5.0 cd/Aa. I
250 1
+ &p)rb
(51)
which is a sufficient approximation for our purpose. is considered to be the only characteristic parameter of the solid surface. The surface aggregates are assumed to be oblate ellipsoids and a(n) is
Lb is the major semiaxis of the aliphatic core and L, the average thickness of the polar corona. Minimization of &nata is performed in two steps. First, as for eq 27
is solved, yielding a value Lp*(n,LJ. Second,the optimal La value is computed by using the condition n
50/00.0
0.2
0.4
0.6
0.8
1.0
0 Figure 1s. Influenceof €,on the adsorptionof C&'EI* Evolution of the average surface aggregation number versus the surface coverage 0. T = 298 K. Key: 0, t. = -1.6 cal/A2; 0, C. = -3.0 cd/Az;
0 , t.
-5.0 cd/Az.
flat adsorption of surfactant polar chain. The possibility of an "extendedwsurface aggregation can be strongly limited when the free energy gain associated with the displacement of adsorbed water by ether oxygen is larger than kT. In the second case, a weak interaction between surfactants and solid surface can be obtained when bonding is limited by the "active" adsorption site density. A more complex extended conformation of the POE chains is now found at the solid/liquid interface. This entropic freedom makes aggregative adsorption easier where alkyl chains must group together. A realistic description of the adsorption process is probably more subtle and certainly lies between these two extreme situations. A complete analysis is beyond the scope of a simple model with few adjustable parameters. Only the second simpler scheme is analyzed in this paper, in agreementwith the hypothesis concerning a weak perturbative interaction between adsorbed surfactants and the solid surface field. Interaction between surface aggregatesand solid surface is assumed to be proportional to their projected surfaces and reads
unhb?' = e,dn)/n
(50) In other words, we consider that the interaction between
We note that this last condition is a possible way of relaxing steric constraints of the polar corona and of increasing surface interaction with the solid. Before ending this section, we shall discuss some properties of the choice made to account for the direct interaction between aggregates and solid surface. Let us assume a situation where almost complete adsorption is obtained (i.e. van a). From eqs 42, 45, 48, and 50, we get
-
Equation 55 now replaces eq 45. Let us consider a range of tp where relaxation of La is not observed (Le. La = La-). In this case, the surface aggregation number is bounded by an absolute maximum. An approximation for this value can be computed by solving eq 55 with 41 = dme. This maximal aggregation number is almost independent of e,. In short, the plateau of the adsorption isotherm will be weakly dependent on t, under former specific conditions. 4. Comparison with Experimental Data. As shown in Figures 14 and 15, the adsorption isotherms shift to lower concentrations when e, increases. This corresponds to a simultaneous decrease of the average surface aggregation number in the weak or average coverage regime. A higher interaction with the solid surface favors, at least for the first steps, the adsorption of small molecular associations. The adsorption isotherm of the C ~ E N homologous series is presented in Figure 16. The genera! trend closely
1*or7 Leuitz
1606 Langmuir, Vol. 7,No. 8, 1991 1
47
0.8
I
I
0.6-
8 0.4
0.2i
0 .
2e-4
4e-4
6e-4
~
0.0 0.0
8e-4
C ( Mol I I ) Figure 18. Computed adsorption isotherms of the CQEN homologous series at 298 K. c, = -5.0 cal/Az. Key 0,N = A,Np= 10;0,Np= 12; 0 ,Np= 16; A, Np= 30. P r e d i d cmc are indicated by vertical arrows.
n
. . 0.2
0.4
0.6
0.8
1.0
1.2
ClCMC Figure 18. Influence of the polar chain length on the adsorption isotherm in the B versus C cmc coordinates. T = 298 K. c. * -5.0 cal/Az. Key: 0, CQ a; m, CQE12; 0, CQE16
h
“‘“1
I 9
*
t
i P
I 0.0
0.2
0.4
0.6
0.8
1.0
e Figure 17. Evolution of the aver e surface gregationnumber
versus the surface coverage 6, for% CaPENyomologousseriea. T = 298 K. c. = -5.0 cal/AP. Key: 0,N = 8; A,Np= 10; m, N = 12; A, Np= 30. Predicted bulk micefiar sizes at mol/{ are indicated by horizontal dotted lines and vertical arrows. T w o particular trends can also be deduced from the follows the experimental results already p u b l i ~ h e d . ~ ~ ~ *model. ~ J ~ When the adsorption isotherms are expressed in All along the adsorption isotherm, the relaxation of the reduced B versus C/cmc coordinates, the adsorption isotherms shift to low values of relative concentration as semiaxis L, is never observed. The size of the surface aggregates formed on the solid below the cmc increases the polar chain length increases; this property, shown in with the surface coverage, as shown in Figure 17. In the Figure 18, is clearly observed in different experimental intermediate surface coverage regime, size variation is r e s ~ l t s . ~The - ~ adsorption isotherms of surfactants having always in the range of the bulk micellar aggregation number the same POE chain and slightly different apolar chains are almost superposed, as shown in Figure 19. This calculated well above the cmc (10-2mol/L). The surface aggregate size diverges slightly near the plateau. This property is approximately verified when an octyl and a effect is more pronounced with short polar chain surfacnonylphenol poly(oxyethy1ene) are compared. Better tants (Np< 12). For surfactants with a long polar chain agreement is reached for the CN,EN, homologous series. (Np> 201,the model is not suited to the computation of Interestingly enough, each grouphas its own master curve. the first part of the adsorption isotherm, even though the The superposition is not as close and depends on the apofirst or second stability criteria are found to be true. These lar chain architecture. results clearly point to a low degree of surface association We have computed the apparent surface per molecule of these large polar chain surfactants. Experimental (a,) at the plateau of theoretical isotherms. A direct evidence pointing to ‘Langmuir type” isothermslee or the comparison with experimental determinations on silica low solubility of hydrophobicprobe inside the adsorption gel is shown in Figures 20 and 21. For the CS~EN,wries, laye$ point to isolated or small molecular associations of good agreement is observed and the linear variation of the adsorbed surfactants in a large part of the adsorption experimental cross section with Np is almost eatbfied. isotherm. Such entities are not correctly handled by the Available experimental up are relatively scattered in the model, which considers micellar-like surface clusters. case of the CaEN,series (see Figure 211,but the theoretical However, stability criteria give clear limits for the applipredictions give the right average evolution. cability of this hypothesis. The adsorption isotherms and surface aggregation
Langmuir, Vol. 7, No. 8, 1991 1607
Aggregative Adsorption of Nonionic Surfactants
300
h
I
4.0-
/
I *! . .... ... ,
m
3.0 -
** *
200
Y
G loo
I P 0
0.0 10
20
30
40
50
2e-5
4e-5
6e-5
8e-5
le-4
C(Mol/l)
NP Figure 20. Apparent surface per surfactant at the plateau of the adsorption isothermsof CaEN, homologous series. T = 298 K. e. = -5.0 cal/AZ. Key: -, model; m, experimental determinations from refs 3-5,9, and 10.
Figure 22. Computed adsorption isotherms of the C&N homologous series at 298 K. e. = -5.0 cal/As. Key: 0 , Np= d m, Np= 8; 0 ,Np= 12. Predicted cmc are indicated by vertical BrrOW8.
400
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&
G 100 I
n 0
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NP
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-
Figure 21. Apparent surface per surfactant at the plateau of the adsorption isotherms of WEN,homologous series. T 298 K. e. = -5.0 cal/A*. Key: -, model;0, experimental results from ref 2; experimental resdts from ref @A, experimental results from refs 1,4, and 5.
variations of C12&, C12E8, and C12E12 are respectively shown in Figures 22 and 23. Large aggregatesare observed for C12&. All along the adsorption isotherm, the semiaxis La is smaller than the axis computed for equivalent micellar aggregates in liquid (see Figure 24). Small variations of the plateau are also observed with The L,relaxation process can be considered as an intermixing of the n-alkyl chains inside the aliphatic core. Ita magnitude increases with These results are reminiscent of recent experimental data concerning the interdigitated structure of the C12& discontinuous adsorption layer on quartz.ll The evolution of the cross section up is shown in Figure 25. The two experimental points’ were obtained on the same silica gel used for the adsorption of C$EN, series and a similar c. value is chosen.
IV. Discussion and Conclusion The aggregativeadsorption model proposed in this work predicta some important features of adsorption of nonionic surfactantsat the hydrophilic solid/water interface. T w o stages are necessary to rationalize this adsorption process. First, a simple description of the bulk micellization is
0.2
0.4
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8 Figure 23. Evolution of the averagesurface gregationnumhr versus the surface coverage 8 for the CISEN, omologous series. T = 298 K. c. = -5.0 cal/A*. K e y 0, Np= 6;m, Np3 8; 0,N = 12. Predicted bulk micellar sizes at 10-5 mol/L are indica& by horizontal dotted lines and vertical arrows.
3
proposed. It allows a phenomenological analysis of an isolated aggregate independently of the exact nature of the surfactant. Only structural parameters such as N,, Np, UT, and La& are needed to predict the micellar properties of a specific surfactant. In a second step, a basic model of the aggregative adsorption involvinga single adjustable parameter, namely, cap describes how a weak interaction between monomers with a hydrophilic surface can generate a set of surface aggregates at equilibrium concentration below but close to the cmc. It is clear that some hypotheses used in this attempt are oversimplified. This, for instance, is the case with our assumption concerning the interaggregate potential (hard core). For average or long POE chains,some polar corona overlapping of adjacent surface aggregates should be taken into account. On the other hand, and for short polar chain surfadanta, coalescence of the surface clusters can possibly occur near the plateau of the adsorption isotherm. At this time, we do not known any state equation able to account for these properties which, when included in eq 33, will yield the correct adsorption isotherm. Similarly, the early stage of the adsorption is not correctly handled by the model. As discussed recently by Partyka and Keh,l* isolated and small molecular associations appear in the firat step of the
Leuitz
1608 Langmuir, Vol. 7, No.8,1991
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" -
0
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.................... ...... . . . . . . . . . &AA..U-
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Oa601 0.504 0.0
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8 Figure 24. Relaxationof the surface aggregate minor semiaxis L. versus the surface coverage 8. Comparison with the bulk micellar aggregates having the same size. T = 298 K. e, = -5.0 cal/A2. Key: 0,CI& surface aggregates;0,C I A bulk micellar bulk micellar aggregates; A, CUESsurface aggregates; A, CI~ES aggregates.
adsorption (e < 0.2). These associations would have hydrophobic tails oriented toward the bulk solution and could explain a slight hydrophobic character of the adsorbed layer at low coverage. Global thermodynamic description of isolated or tiny molecular associations is not straightforward, owing to their specific nature. For example, conformation of the POE chain of an adsorbed isolated surfactant depends on its length and on the solid surface field. This conformation, especially for long POE chains, can range from a flat to a "polymer-like" configuration where only a part of the available ethoxy groups are in interaction with the solid. We have not attempted a possible extension of the model taking into account this kind of adsorbed entities. It is clear that self-consistent field theory can be very useful to give some insight on the
0
I
0
5
10
15
NP Figure 25. Apparent surface r surfactant at the plateau of the adsorption isothermsof C , ~ F homologous N~ series. T = 298 K. c. = -5.0 &/AS. Key: -, model; 0,experimentalreaulta from ref 1.
first steps of the adsorption, but it seems to us that the early stages of the adsorption must weakly perturb the "extended" surface aggregation occurrence, except in two case: (i) when the interaction between solid surface and polar chains is strong; (ii) for surfactants with long POE chains as already discussed. In conclusion, the thermodynamic model proposed in the present work predicts some general trends of the adsorption of nonionic surfactants on weakly interactive hydrophilic solid. Some interesting questions concerning precursor states of the adsorption are raised. A more complete description of the interaction between a solid surface and surface aggregates is certainly needed. These different problems will certainly be addressed by forthcoming experimental or theoretical work. Acknowledgment. It is a pleasure to thank J. J. Fripiat and R. Setton for very stimulating discussions.
