Langmuir 1994,10, 3512-3528
3512
Complexation of Nonionic Polymers and Surfactants in Dilute Aqueous Solutions Y. J. Nikas and D. Blankschtein* Department of Chemical Engineering and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received March 25, 1994. I n Final Form: June 13, 1994@ We present a molecular-thermodynamic theory ofthe complexationof nonionic polymers and surfactants in dilute aqueous solutions. The theoretical formulation combinesa thermodynamicdescription of polymersurfactant solutions with a molecular model of polymer-surfactant complexation. The molecular model of complexation is based on the “necklace model”, in which a polymer-surfactant complex is described as composed of a series of spherical micelles with their surfaces covered by polymer segments and connected by polymer strands belonging to the same polymer molecule. The theory incorporates explicitly the effects of (i) solvent quality, (ii)polymer hydrophobicity and flexibility, and (iii)specific interactions between the polymer segments and the surfactant hydrophilic moieties on the complexation behavior. Moreover, the theory also accounts for the competition between repulsive electrostatic interactions between polymerbound micelles and elastic restoring forces within the polymer chain. The theory can be utilized to predict (1)the critical aggregation concentration (CAC),which signalsthe onset ofpolymer-surfactant complexation, (2)the number of micelles bound per polymer chain, (3)the aggregationnumber of polymer-bound micelles, (4)the average distance between these micelles, and (5)the mean-square end-to-end distance ofthe complex, as well as the critical micelle concentration (CMC),which signals the onset of micelle formation in the absence of polymers. Our theoretical results indicate that the two dominant driving forces leading to the complexation of nonionic polymers and surfactants are (i) the tendency of the polymer to adsorb at the micellar core-water interface and (ii)the existence of specific attractions between the polymer segments and the surfactant hydrophilic moities. In addition, poor solvent quality and small sizes of the surfactant hydrophilicmoieties may also promote complexation. One of our central findings is that polymer-surfactant complexation proceeds via a stepwise association process. In other words, as the surfactant concentration is increased, complexes containing one micelle form first, and only when all the available polymers have one micelle bound to them do complexes containing two micelles begin to form at the expense of those containing one micelle, and so on. The theoretical predictions of the CAC, the CMC, the aggregation number of the polymer-bound micelles, and the number of surfactant molecules bound per polymer chain at binding saturation,for aqueous solutions of poly(ethy1eneoxide)(PEO) and polyvinylpyrrolidone(PVP) mixed with various sodium alkyl sulfates (with and without added salt), are found to be in reasonable agreement with the available experimental data.
I. Introduction Understanding the nature of the interactions between polymers (both synthetic and biological) and surfactants in bulk aqueous solutions, as well as a t interfaces, is of central importance in the manufacturing of a broad spectrum of industrial p r o d ~ c t s . l -These ~ include detergents, cosmetics, paints and coatings, adhesives and glues, lubricants, photographic films, and food and pharmaceutical products. Typically, many ofthese products are based on dispersion technologies which require the simultaneous presence of surfactants and polymers, where the surfactants provide emulsification capacity, interfacial tension
* To whom correspondence should be addressed.
Abstract published inAdvance ACSAbstracts, August 15,1994. (1) Goddard,E. D. In Interactions of Surfactants with Polymers and Proteins; Goddard, E. D.,Ananthapadmanabhan,K. P., Ed.;CRC: Boca Raton, FL, 1993; p 395. (2) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Marcel Dekker: New York, 1987. (3) Breuer, M. M.; Robb, I. D. Chem. Ind. 1972,13,531. Robb, I. D. In Anionic Surfactants-Physical Chemistry; Lucassen-Reynolds,E. J., Ed.; Marcel Dekker: New York, 1981; p 109. (4) Goddard, E. D. Colloid Surf. 1986, 19. 255, 301. (5) Saito, S.In Nonionic Surfactants-Physical Chemistry; Schick, M. J., Ed.; Marcel Dekker: New York, 1987; p 881. (6) Hayakawa, K.; Kwak, J. C. In Cationic Surfactants-Physical Chemistry; Rubingh, D. N., Holland, P. M., Ed.; Marcel Dekker: New York, 1991; p 189. (7) For an excellent review on recent developments in the area of polymer-surfactant interactions, see: Lindman, B.; Thalberg, K. In Interactions of Surfactants with Polymers and Proteins; Goddard, E. D., Ananthapadmanabhan, K. P., Ed.; CRC: Boca Raton, FL, 1993; p
control, and colloidal stability, and the polymers impart colloidal stability and special rheological features. The practical importance of polymer-surfactant systems has led to a significant experimental effort to study their behavior. Indeed, a variety of experimental techniques have been utilized to probe the nature of polymersurfactant interactions, including viscosity and conductivity measurements, dialysis, fluorescence spectroscopy, NMR, and neutron scattering technique^.^-^ Some of the most investigated model systems include aqueous solutions of nonionic polymers and ionic (both anionic and cationic) surfactants. These include aqueous solutions of poly(ethy1eneoxide)(PEO)and sodium dodecyl (PVP)and SDS,23-28 sulfate (SDS),s-zzpolyvinylpyrolidone and PEO and cetyltrimethylammonium bromide (CT-
@
203.
( 8 ) Jones, M. N. J . Colloid Interface Sci. 1967,23, 36.
(9) Schwuger, M. J. J . Colloid Interface Sci. 1972, 491. (10)Schwuger, M. J.; Lange, H. In Chimie Physique et Applications Practiques des Agents de Surface; Ediciones Unidas: Barcelona, 1968; Vol. 2, Part 2, p 955. (11) Shirahama, K. Colloid Polym. Sci. 1974, 252, 978. (12) Zana, R.; Lang, J.; Lianos, P. In Microdomains in Polymer Solutions; Dubin, P., Ed., Plenum: New York, 1985. (13) (a) Cabane, B. J . Phys. Chem. 1977,81, 1639. (b) Cabane, B.; Duplessix, R. J . Physique 1982, 43, 1529; Cabane, B.; Duplessix, R.; Zemb, T. J . Physique 1985,46,2161;Cabane, B.; Duplessix, R. Colloid and Surfaces 1985, 13, 19; Cabane, B.; Duplessix, R. J . Phys. (Paris) 1987, 48, 651. (14) Francois, J.; Dayantis, J.;Sabbadin, J. Eur. Polym. J . 1985,21, 165. (15) Witte, F. M.;Engberts, J. B. F. N . J . Org. Chem. 1987,52,4767;
Colloid Surf. 1989, 36, 417. (16) Brackman, J. C.; Engberts, J. B. F. N . J . Colloid Interface Sei. 1989,132, 250.
Q743-7463/94/241Q-3512$Q4.5QlQ 0 1994 American Chemical Society
Complexation of Polymers and Surfactants AB).15-17 It was observed that above a critical surfactant concentration, known as the critical aggregation concentration (CAC),some ionic surfactants can bind cooperatively to nonionic polymer^.^^^^^^ The CAC is usually much smaller than the critical micelle concentration (CMC),which signals the onset of micelle formation in the corresponding polymer-free surfactant solution.This cooperative binding results in the formation of polymersurfactant complexes in the solution. A structural model for these complexes was proposed by Shirahama29and by Cabane,13 and was confirmed by NMR13a,27 and neutron scattering experiment^.^^^,^^ In dilute solutions, the polymer-surfactant complex is viewed as composed of a series of spherical micelles with their surfaces covered by polymer segments and connected by strands of the same polymer molecule, thus resembling a “necklace of beads”-the so-called “necklace model”. Results of NMR s t ~ d i e s also ~ ~ provided ~ s ~ ~ ample evidence that in these complexes water-soluble polymers do not penetrate into the hydrophobic micellar core, but instead adsorb at the micelle surface and remain in close contact with the surfactant hydrophilic moieties (hereafter referred to as “heads”). On the theoretical side, two important models, which are of particular relevance to the present work, have been p r o p o ~ e dto ~ ~describe ! ~ ~ polymer-surfactant complexation. The first model assumes30 that polymer segments present a t the micelle surface penetrate into the region occupied by the surfactant heads and shield partially the contact area between the micellar hydrocarbon core and water. This effect is considered to constitute the main driving force for polymer-surfactant complexation. However, the tendency for polymer shielding is opposed by the increased steric repulsions between the surfactant heads and the polymer segments at the micelle surface. Whether the cooperative complexation process actually occurs is determined by the balance between these two competing tendencies. The second model assumes31that polymer adsorption at the micelle surface induces a change in the microenvironment surrounding the micelle (from pure water to an aqueous polymer solution, with the latter being more hydrophobic than the former) and that this change in the microenvironment brings about a decrease in the interfacial free energy between the micellar hydrocarbon core and the solvent. This effect is considered to be the primary driving force for the polymer binding to surfactants. However, since the surfactant heads are hydrated (17) Brackman, J. C.; van Os, N. M.;Engberts, J. B. F. N. Langmuir 1988,4,1266. Brackman, J. C.; Engberts, J. B. F. N. Langmuir 1992, 8 424 - 7
(18)Dubin, P. L.;Gruber, J. H.;Xia, J.;Zhang,H. J . Colloid Interface Sci. 1992, 148, 35. (19)Xia. J.: Dubin. P. L.: Kim. Y. J. Phrs. Chem. 1992, 96. 6805. (20)Maltesh, C.; Somasundaran, P. Laigmuir 1992, 8, ‘1926. (21)Maltesh, C.; Somasundaran, P. J . Colloid Interface Sci. 1993, em 157, 14. (22) Chari, R; Antalek, B.; Lin, M. Y.; Sinha, S. K J. Chem. Phys. 1994,100, 5294. (23) Lange, H. Kolloid 2.2.Polym. 1971,243, 101. (24)Fishman. M. L.: Eirich. F. R. J . Phvs. Chem. 1971, 75, 3135. (25) Arai, H.; Murata, M.;Shinoda, K. J . Colloid InterfaceSci. 1971, 37, 223. (26) Murata, M.; Arai, H. J . Colloid Interface Sci. 1972, 44, 475. (27) Chari, K.; Lenhart, W. C. J. Colloid Interface Sci. 1990, 137, 204.
(28) Chari, K. J . Colloid Interface Sci. 1992, 151, 294. (29) Shirahama, K.; Tsujii, K.; Takagi, T. J . Biochem. 1974,75,309. (30) (a)Nagarajan,R. Chem.Phys.Lett. 1980,76,282. (b)Nagarajan, R.; Harold, M. P. In Solution Behavior of Surfactants; Mittal, K. L., Fendler, E. J., Eds.; Plenum: New York Vol. 2, p 1391. (c) Nagarajan, R. Colloid Surf. 1985, 13, 1. (d) Nagarajan, R.; Kalpakci, B. In Microdomains in Polymer Solutions;Dubin, P., Ed.;Plenum: New York, 1985. (e) Nagarajan, R. Adv. Colloid Interface Sci. 1986,26, 205. (0 Nagarajan, R. J. Chem. Phys. 1989,90, 1980. (31) Ruckenstein, E.; Huber, G.; Hoffmann, H. Langmuir 1987,3, 382.
Langmuir, Vol. 10,No. 10, 1994 3513 and highly hydrophilic, the change of solvent from pure water to an aqueous polymer solution is assumed to result in an increase in the interfacial free energy between the surfactant heads and the solvent, thus opposing polymersurfactant complexation. In this paper, we present a theoretical description of the complexation of nonionic polymers and surfactants in dilute aqueous solutions. The theoretical approach involves combining a thermodynamic description of polymer-surfactant solutions with a molecular model of polymer-surfactant complexation. The central element in the theory involves a detailed calculation of the free energy of complexation,g,,,, which represents the change in free energy when a polymer-surfactant complex is formed from singly-dispersed surfactant molecules and a single polymer chain. In calculating gcomp, we account explicitly for the essential individual molecular-level factors associated with the polymer binding process. In so doing, we are able to delineate the dominant driving forces for polymer-surfactant complexation, as well as the effects of various polymer characteristics, including their hydrophibicity, flexibility, and interactions with surfactant heads, on the binding behavior. As a result, our model can be utilized to quantify at the molecular level the observed differences in binding behavior between commonly used surfactants or polymers, including (1) anionic versus cationic surfactants, (2) hydrophilic versus hydrophobic polymers, and (3) flexible versus “stiff’ polymers. Moreover, the theory also incorporates intermicellar interactions as well as elastic restoring forces within a polymer-surfactant complex. This enables us to predict, for the first time, the number of micelles bound per polymer chain, the aggregation number of these micelles, the average distance between polymer-bound micelles, and the mean-square end-to-end distance of the complex. The molecular parameters in our theory include the polymer segment size, the free energy of adsorption of a polymer segment at a hydrocarbon-water interface, the polymer segment-solvent interaction free energy, and the interaction free energy between a polymer segment and a surfactant head, all of which can, in principle, be obtained from independent experimental data (for details, see section V.A). The remainder of the paper is organized as follows. In section 11, we outline the thermodynamic framework for describing aqueous solutions containing polymers and surfactants. Specifically,we first present expressions for the chemical potentials of the various solution components and subsequently utilize these chemical-potential expressions and equilibrium conditions to derive explicit expressions for the aggregate (free micelles and polymersurfactant complexes) size distributions. In section 111, we briefly review a recently developed molecular model aimed a t calculating the free energy of micellization, gmic, of nonionic surfactants and its extension to the ionic surfactants case. In section IV.A,we develop a molecular model to describe the complexation of nonionic polymers and surfactants. In section IV.B, the complexation behavior predicted by the model is discussed at a qualitative level. Numerical results and a comparison with available experimental data are presented in section V. Finally, concluding remarks and discussions are presented in section VI. 11. Thermodynamic Description of Aqueous Polymer-Surfactant Solutions In this section, we present explicit expressions for the chemical potentials ofthe various solute species and derive expressions for the size distributions of free micelles and polymer-surfactant complexes.
Nikas and Blankschtein
3514 Langmuir, Vol. 10, No. 10, 1994
As discussed in the Introduction,in a solution containing nonionic polymers and surfactants, when the surfactant concentration exceeds the critical aggregation concentration (CAC), polymers and surfactants bind cooperatively to form necklace-like complexes. In addition, when the surfactant concentration exceeds the critical micelle concentration (CMC,, which is different from the CMC of a polymer-free surfactant solution, see section V.B.l), free micelles form and coexist with polymer-surfactant complexes. Experimental results ~ u g g e s t ’that ~ , ~ the ~ polymerbound micelles are of spherical shape and have a narrow size distribution, even when free micelles are predominantly cylindrical and have a wide size distribution. Accordingly, our model description will assume monodispersity in the size distribution of spherical polymerbound micelles but will allow free micelles to assume spherical and cylindrical shapes. Note that we will not consider discoidal micelles, since this morphology is not generated by the surfactants of interest in this paper. In addition, we will consider only the case of monodisperse polymers. An extension to polydisperse polymers is conceptually similar and will not be discussed here. Moreover, we will focus on relatively dilute solutions and consequently assume that interactions between complexes are negligible, and that polymer chains are not entangled. Consider a solution containing N , water molecules, N , surfactant molecules, and N p polymer molecules in thermodynamic equilibrium at temperature T and pressure P. In this ternary solution, polymer-surfactant complexes are characterized by a distribution (Np,mg), where Np,mg denotes the number of polymer-surfactant complexes containing one polymer chain and m, micelles of aggregation number g (with Np,odenoting the number of free polymer molecules), and free micelles are characterized by a size distribution {N,,),where N,, denotes the number of micelles containing n surfactant monomers (with N1 denoting the number of free surfactant monomers). Since the solution under consideration is dilute, we assume that interactions between the various species are negligible and that the free energy of mixing can be modeled using the ideal-solution approximation. Under these conditions, the chemical potentials of the surfactant monomers, the free polymers, the polymer-surfactant complexes of type m,, and the free micelles of aggregation number n are given respectively by
(3)
Specifi~ally,~~~~~~ Pn = nP1
(5)
for free micelles, and Pp,mg = Pp,o
+ m&Pl
(6)
for polymer-surfactant complexes. Substituting eq 4 for p, and eq 1for p1 in eq 5, as well as eq 1for p1, eq 3 for pp,mg,and eq 2 for pP,oin eq 6, we obtain expressions for
the equilibrium micellar and polymer-surfactant complex size distributions. Specifically,
xn = (xle-Bgmic(n))n
(7)
for free micelles, and
for polymer-surfactant complexes, where p = IIkBT, and
(9) is the free energy of micellization, representing the freeenergy change (per surfactant molecule) associated with assembling a micelle of aggregation number n from n free surfactant monomers in solution, and
is the free energy of complexation, representing the freeenergy change (per surfactant molecule) associated with assembling a polymer-surfactant complex of type m, from m g singly-dispersed surfactant monomers and one isolated polymer chain in solution. Equations 7 and 8 are supplemented by the surfactant and polymer mass-balance relations. Specifically,
for the surfactant, and
xp = NpJNtotal= xp,O + mg=l C xp,mg
(12)
for the polymer, where X, and X, are the total surfactant and polymer mole fractions. Note that in a dilute solution, that is, when N , l), and the last decrease in the slope corresponds to the CMC,, at which free micelles begin to form (in the presence of polymer). Therefore, from XI versus X, plots generated using the theoretical framework developed in sections IIIV,one can predict the values ofthe CAC, CMC, and CMC,. Figure 3 shows the predicted variation of XI with X, corresponding to the model polymer-anionic surfactant pair described above, for x = 0.45 and xp = -2.6 (dashed line), -3.0 (dotted line), and -3.4 (dash-dotted line). The solid line in Figure 3 corresponds to the pure model surfactant case (no added polymer) and exhibits an abrupt drop in its slope beyond X, % 6.2 mM, which is identified as the CMC value. In the presence of polymer, two abrupt drops in the slope of the XI versus X, curves are observed
(the one extra plateau at XI % 3.3 mM observed for xp = -3.4 is due to the formation of complexes containing two micelles, as will be discussed later). The surfactant concentration X, at which the first abrupt drop in the slope is observed is identified as the CAC, which is approximately equal to 5.3,3.4,and 1.8mM for xp= -2.6, -3.0, -3.4, respectively. Above the CAC, there is a range ofX, values (for example, from about 3.4 to 9.4 mM in the xp= -3.0 (dotted-line) case) whereXl increases very little with X,, because most of the added surfactant molecules are incorporated into polymer-surfactant complexes. The X, value at whichX1 begins to increase again corresponds to polymer binding saturation (this is less pronounced in the xp= -2.6 case). At theX, values correspondingto the second drop in slope (CMC,), which are approximately equal to 14,12, and 12 mMforx, = -2.6, -3.0, and -3.4, respectively, XI has reached a value equal to that in a pure surfactant solution atX, = CMC, and therefore free micelles begin to form and coexist with polymer-bound micelles. The value of the CMC, increases with the total amount of surfactant bound in the complexes. Figure 3 also reveals that the CAC decreases as xp become more negative, namely, as the propensity of the polymer to adsorb at the micellar core-water interface increases. The dependence of the CAC on xp and x can be seen more clearly in Figure 4, where the ratio CACICMC is plotted as a function of xpfor x = 0.3 (dashed lined), 0.45 (dotted line), and 0.5 (solid line). The polymer in this case consists of N = 400 segments. Figure 4 shows that, for a fixed value of x,the CAC decreases as xpbecomes more negative and that, at a fured value ofx,, the CAC decreases as x increases or, in other words, as the solvent quality becomes poorer. In addition, Figure 4 also indicates that as the value ofxpexceeds a certain threshold value, which is equal to -2.5, -2.3, and -2.1 for x = 0.3,0.45,and 0.5, respectively, the CAC becomes equal to the CMC of the pure surfactant solution, in which case the XI versus X, curves become indistinguishable from that corresponding to the pure surfactant solution case. In the past, the observation that the CAC is equal to the CMC was
3524 Langmuir, Vol. 10, No. 10,1994
Nikas and Blankschtein
1.o
0.5
0.4
0.3
x 0.2
'
0.0 -4.0
I
-3.5
t
-3.0
I
-2.5
I -2.0
0.1
No Complexation
XP
Figure 4. Predicted CAC/CMC ratio at 25 "C and a polymer concentration of 0.1 M (in polymer repeat units) as a function of xpfor polymers having N = 400, b = 4 A,cps= 0, and x = 0.3 (- -1, 0.45 (***), and 0.5 (-).
interpreted as an indication that the polymer and the surfactant do not form c~mplexes.~ However, our numerical results clearly show that when (CAC/CMC) = l, the value of XI at the CAC can still be smaller than that corresponding to a pure surfactant solution a t the CMC, and that in this case only polymer-surfactant complexes are present in the solution without any coexisting free micelles. This important finding indicates that the results of the experiments, which are based on direct or indirect measurements of XI, including surface tension and conductivity methods, that show no apparent decrease in the CMC in the presence of polymers should be examined very carefully, and they may not provide conclusive evidence that polymers and surfactants do not form complexes. Indeed, experiments by Brackman et a1.17have shown that, for some polymer-surfactant pairs, the failure of a polymer to reduce the CMC does not necessarily imply the absence of polymer-surfactant complexation. However, no explanation was offered17 for this finding, and indeed some questions were later raised regarding its plausible cause^.^ A careful examination of the definition of the CMC and the CAC leads one to concludethat the fundamental reason behind this apparent discrepancy is related to the fact that micellization and complexation are not true phase transitions due to the finite sizes of polymer-bound and free micelles. If the aggregation numbers of these micelles were infinite, then the XI versus X, plots would exhibit slope changes from 1to 0 a t the CAC and a t the CMC. In that case, the CAC and the CMC would be defined precisely, with the CAC necessarily being smaller than the CMC for complexation to occur. However, since all micelles have a finite size, the slope changes gradually rather than in a step-function fashion and is not zero over the concentration range where aggregation takes place. In that case, the CAC and the CMC are typically defined as the intercept of the extrapolation of the two linear parts of the X I versus X , curve. If, in the presence of polymer, the X, value at which the slope of the XI versus X , curve first deviates from 1is not much smaller than theX, value at which this occurs in the absence of polymer, then the X , range over which complexes exist without free micelles is small (not all the polymers may be able to reach binding saturation), and theX1 versusX, curves (with and without polymer)may appear to be identical.30eThe true criterion to determine whether complexation takes place is the relative magnitude of the free energies of micellization and complexation. Indeed, ifgcomp is more negative than gmic, then complexation will preceed micellization, albeit bind-
0.0 -2.8
-2.4
-2.0
-1.6
-1.2
XP
Figure 5. Phase diagram in the x-xp space, corresponding to N = 400,b =4A,cps= 0, T = 25 "C, and a polymer concentration of 0.1 M (in polymer repeat units).
o'2 w.w
t ~~
0
100
200
300
400
Number of Polymer Segments,N Figure 6. Predicted CAC/CMC ratio at 25 "C and a polymer concentration of 0.1 M (in polymer repeat units) as a function of the number of polymer segments, N , for polymers having b = 4 A, x = 0.45, cps = 0, and xp = -2.8 (.**), and -3.2 (-). ing saturation may not be reached in some cases. On the other hand, ifgcomp is greater thangmic,then complexation will not take place. To illustrate this very important point, we have calculated a phase diagram in the x - xp space, which is shown in Figure 5. It can be seen that the x - xp phase diagram is divided into two regions: (1) the region corresponding to gcomp -= gmic, where polymer-surfactant complexation occurs, and (2) the region corresponding to gcomp > gmic, where polymer-surfactant complexation does not occur. Varyingx, at a constant x value may lead from one region to the other. However, in the region where complexation does occur (where g c o m p gmic), one can further identify two subregions: I and 11, which are separated by the dotted line on which the apparent CAC's, as determined from XI versus X, plots, are equal to the CMC of the pure surfactant solution. We have also studied the dependence of the CAC on the number of polymer segments or, equivalently, on the polymer molecular weight. A plot of the predicted ratio CAC/CMC as a function of the number of polymer segments, N , is shown in Figure 6, for x = 0.45 and xp = -2.8 (dotted line) and -3.2 (solid line). Figure 6 indicates that, for a fixed xp value, the CAC remains constant for
Complexation of Polymers and Surfactants
Langmuir, Vol. 10, No. 10, 1994 3525
increasing polymer hydrophobicity. The predicted trend can be easily understood. If a polymer is more hydrophobic, then a larger portion of the area a t the micellar core-water interface is occupied by polymer segments, which, in turn, generates stronger steric repulsions with the surfactant heads present at the interface. These stronger repulsions force the surfactant heads further apart, thus leading to the formation of smaller micelles. This picture is consistent with the experimental observation15that micelles bound by PPO are smaller than those bound by PEO (since PPO is more hydrophobic than PEO). 3. Binding of Multiple Micelles. Next, we examine the variation of the average number of surfactant molecules bound per polymer molecule, Mb = cm,gmgNp,m/ Np, as a function of the free surfactant concentration, Xf -4.0 -3.5 -3.0 -2.5 -2.0 = Xl X,,an, which is a commonly used variable in XP dialysis experiments. Note that, in the absence of free Figure 7. Predicted optimal polymer-bound micelle aggregamicelles, that is, for X, < CMC,, one has Xf = XI. Figure tion number, g*, of the first micelle (mg = 1)at 25 "C,as a 8 shows the variation of Mb with Xf for polymers having function of xp for x = 0.3 (- -1, 0.45 and 0.5 (-). The N = 100 (solid line), 200 (dotted line), 400 (dashed line), molecular parameters are b = 4 A, cps = 0, and N = 400. and 800 (dash-dotted line) segments. The interaction parameters are xp= -3.4 and x = 0.45, and the polymer polymers having a molecular weight beyond a critical value concentration is 0.1 M in polymer repeat units. Figure N but increases with decreasing N for polymers having 8 reveals several interesting features. First, in all cases, N < N. This behavior was actually observed experimenMb remains equal to 0 at low surfactant concentrations tally in aqueous solutions of PEO-SDS and PVP-SDS and increases sharply at Xf % 1.8mM (note that, for the mixtures,9J2Jobut no satisfactory explanation was given. N = 100 case, the increase occurs a t Xf % 2 mM, which Within our theory, this observation can be easily underillustrates the dependence of the CAC on polymer mostood. Recall that, in calculating the CAC, we minimize lecular weight discussed above). This is a clear signature gcomp(mg = lg,yj,,nb) with respect tog, yjp, and nb, and the of the cooperative nature of polymer-surfactant comvalues of these three variables determine the optimal plexation. Second, for relatively short polymers, N = 100 structure of the complex containing one micelle. The total and N = 200, Mb reaches a saturation value of 13 and 14 number of polymer segments, Nb, bound on a micelle, is a t Xf % 3 and 2.5 mM, respectively. This Mb value equal to gnb. If is smaller than the total number of corresponds to one micelle bound per polymer coil. For polymer segments, N , then the optimal structure can be longer polymers, N = 400 and N = 800, there are sudden realized, and the free energy of complexation is given by increases in Mb at Xf 3 and 4 mM, respectively. These the minimum value of gzomp. On the other hand, if is increases inMb are due to the addition of a second micelle larger than N , then the optimal structure is unattainable, to the complex. As an illustration, in the N = 400 case and& is equal to N, which is the total number of available (dashed line), &fbincreases from 0 t o 14 within a narrow segments. This implies that the free energy of complexconcentration range (from 1.8to 2.5mM), at which point ation is greater than that corresponding to the optimal all the polymers have one micelle bound to them. As the structure, gzomp,and consequently the value of the CAC surfactant concentration is increased further, Mb increases is greater for polymers that have less than segments. very little with Xf until 3 mM, beyond which it increases The underlying physical reason for the increase in gcomp to a value of 22, where a second micelle begins to be with decreasing N, for N < is that if the number of incorporated into the already existing complexes. This is polymer segments Nb that bind onto a micelle is very small, also a cooperative process, as can be seen from the then the thickness of the polymer layer, D , must be very sharpness ofthe increase. The N = 400 polymer can bind small, and this, in turn, results in a large conformational a maximum of two micelles before free micelles begin to entropy loss when the polymer is confined within this form a t Xf % 6.2mM. This stepwise binding process is layer. An extreme case occurs when D % b, in which case also reflected in the X1 versus X, curve corresponding to the polymer molecule is confined within a two-dimensional xp = -3.4 (dash-dotted line) shown in Figure 3, where space, which renders the adsorption of a polymer molecule there is a second plateau atX1 % 3 mM. For the other two onto a micelle extremely unfavorable. Figure 6 also xp values shown in Figure 3, the polymer having 400 indicates that the critical molecular weight of the polymer segments can bind only one micelle before free micelles beyond which a constant CAC is attained depends on the begin to form. For the polymer havingN = 800 segments, value of xpL and that it decreases as xp becomes more Figure 8 indicates that the sharp increase in Mb from 28 negative (N % 200 and 100 for xp = -2.8 and -3.2, to 36 atXf= 4 mM is caused by the incorporation of a third respectively). This follows from the fact that the decrease micelle into the existing complexes which already contain in the optimal aggregation number of the polymer-bound two micelles. Note that, in the N = 800 case, one does not micelles, g*, as xp becomes more negative (see below), is observe the transition from one-micelle- to two-micellecontaining complexes. This reflects the fact that, for this faster than the concomitant increase in n;, which, in relatively long polymer chain, the first two micelles are turn, results in a decrease in the value of 6 = g*nt. sufficientlyfar away from each other that the intermicellar 2. Polymer-BoundMicelle Aggregation Number. interactions (captured ing,,) are very weak. Accordingly, The polymer-bound micelle aggregation number depends the increment ing,,,,(m,g) from m, = 1 to m, = 2 is very on the values of xp and x as well. Figure 7 shows the small, and no distinct transition is observed. optimal aggregation number,g*, of the first polymer-bound This stepwise complexation process can be visualized micelle (m, = 1) as a function of xp for x = 0.3 (dashed more clearly in Figure 9, which shows the predicted line), 0.45 (dotted line), and 0.5 (solid line). Figure 7 distribution of surfactant molecules in complexes conindicates that g* decreases as xpbecomes more negative taining different numbers of micelles (m,) as a function and as x increases, both of which are indicators of
+
A,
(..e),
x
g
x,
3526 Langmuir, Vol. 10, No. 10, 1994
Nikas and Blankschtein
si 40 L-
-E%
2
30
% Q
-u C
3 0
fn 20 B C a w
8
't
z
10
Y-
O
%
a E '3 z o
2
0
8
6
4
Free Surfactant Concentration, X, (mM) Figure 8. Predicted average number of surfactant molecules bound per polymer chain, Mb,as a function of the free surfactant for polymers having b = 4 A, xp= -3.2, x = 0.45,eps= 0, and N = 100 (-1,200 (. -1,400 (- - -1, and 800 (- -1 concentration,Xf, segments, and a polymer concentration of 0.1 M (in polymer repeat units) at T = 25 "C. Table 3' . Comparison of Predicted and Measured CACW Average Aggregation Numbers of Polymer-Bound Micelles: (g), and Binding Ratios of PEO-Sodium Alkyl Sulfate Surfactant Mixtures in Water and in 0.1 M NaCl Aqueous Solutionsc binding ratio CMC (mM) CAC (mM) @)
theo.
exp.
theo.
exp.
theo.
exp.
theo.
exp.
No Salt CloS04Na C12S04Na
C12S04Na
30.4 6.3 1.1
32.5d 8.3d
1.3'
27.8
5.5
2E1.2~ 5.5: 4 9
34 46
24f 228
0.1 0.15
0.2-0.3h
0.82
0.1 M NaCl 0.83'
42
28i
0.2
0.4k
a Some of the reported CAC values correspond to temperatures other than 25 "C, the temperature at which the theoretical predictions were made. The aggregation numbers have been predicted at the CAC, while the measured ones have been mostly obtained at surfactant concentrations higher than the CAC. The molecular parameters used in the calculations are listed in Tables 1 and 2. Reference 16. e Reference 8. f Reference 12. Reference 15. Reference 13a. Reference 18. Reference 21. Reference 11. J
of total surfactant concentration, X,, for polymers having N = 400 (solid line) and N = 800 (dashed line) segments. Figure 9 reveals that, in both cases, at low X, values, only complexes containing one micelle exist, but asX, increases, the population of complexes containing two micellesbegins to overtake that of one-micelle-containingcomplexes. This occurs at X, x 7.5 mM for the N = 400 case and at X,x 4.5 mM for the N = 800 case. For the N = 800 case, at X, = 7 mM, complexes containing three micelles begin to appear. Another interesting feature of the N = 400 plot is that two-micelle-containing complexes do not appear until the population of one-micelle-containing complexes has reached its peak (at X, x 6 mM), indicating that the population of two-micelle-containing complexes increases at the expense of that of one-micelle-containingcomplexes. On the other hand, for the N = 800 case, there is a concentration range (2-4 mM) in which the populations of two-micelle- and one-micelle-containing complexes increase a t the same time. This reflects the fact that the difference in the free energies of complexation of these two types of complexes is very small. Finally, in the N = 800 case, one observes a slight decrease in the population of complexes containing three micelles (mg= 3) for X, > 8 mM and a concomitant increase in the population of complexes containing four micelles (mg= 4). However,
the population of four-micelle-containing complexes remains relatively small due to the formation of free micelles. C. Comparison with Experimental Data. In section V.B, we presented qualitative predictions of various aspects of the complexation behavior of a model nonionic polymer-anionic surfactant system. Below, we report some quantitative predictions for aqueous solutions of PEO and PVP mixed with sodium alkyl sulfates, in the absence and presence of salt, including a comparison of the theoretical predictions with available experimental data. The predicted and measured values of CAC's, average aggregation numbers of polymer-bound micelles, (g), and the number of surfactant molecules bound on a polymer per polymer segment at saturation (referred to as the binding ratio, which is equal to M d N ) are listed in Table 3 (for PEO) and in Table 4 (for PVP). The parameters used in the calculations are listed in Tables 1 and 2. As stated earlier, the ePevalues given in Table 2 were obtained by fitting the predicted CAC values to the experimentally measured ones for aqueous solutions of PEO-SDS and PW-SDS. Subsequently, the epsvalues so deduced were used to predict various complexation characteristics for other alkyl sulfates possessing different tail lengths. As can be seen from Tables 3 and 4, the agreement between theory and experiment for the CAC and the (g)
Complexation of Polymers and Surfactants
Langmuir, Vol. 10, No. 20, 1994 3527
Table 4. Comparison of Predicted and Measured CAC'S,~Average Aggregation Numbers of Polymer-Bound Micelles,*(g), and Binding Ratios of PVP-Sodium Alkyl Sulfate Surfactant Mixtures in Water and in 0.1 M NaCl Aqueous Solutionsc CMC (mM) CAC (mM) (g) binding ratio theo. exp. theo. exp. theo. exp. theo. exp. No Salt CloSOaa 30.4 32.5d 11 13' 34 0.16 C12SOaa
6.3
C16SOaa
0.35
C14SOaa 1.3
8.3d 2.1' 0.35'
2.3 0.48 0.12
2.3' 0.38'
0.08' 0.1 M NaCl
CloSOaa 13
C11SOaa 3.9
13.38 12.9 4.g 3.6 1.1 1.5 0.9
8.09 2.e
C12S04Na
0.4s
42 41 51
31f
34 40 48
0.25 0.20 0.19
0.3'7'
0.17 0.20 0.25
0.48 0.48 0.4
a Some of the reported CAC values correspond to temperatures other than 25 "C, the temperature at which the theoretical predictions were made. The aggregation numbers have been predicted at the CAC, while the measured ones have been mostly obtained at surfactant concentrationshigher than the CAC. e The molecular parameters used in the calculations are listed in Tables 1and 2. Reference 16. e Reference 10. f Reference 12. g Reference
25.
I
E -1
mg=l
5
10
15
Total Surfactant Concentration,X,(mM)
Figure 9. Predicted distribution of surfactant molecules in complexes containing one (m, = 1) to four (m, = 4)micelles for polymers having N = 400 (-1 and 800 (- -) segments, correspondingto the same molecular parameters listed in Figure 8.
values is reasonable in all cases, but the predicted binding ratio is underestimated, which implies that the theory predicts fewer polymer-bound micelles than what is actually observed. This underestimation can be attributed to an approximation in the electrostatic theory, where all the counterions have been treated as point charges and counterion binding at the micelle surface has been ignored. Two consequences follow from this approximation. First, it tends to overestimate the electrostatic repulsions between polymer-bound micelles, which, in turn, favors fewer micelles in each complex. Second, it also tends to overestimate the number of polymer segments that are bound on each micelle, NI,, since there is no competition from the counterions for the available space near the micelle surface. Since the number of segments connecting two neighboring micelles, N,, is given by (N - m&J/(m, - l), it follows that, for a given value of N , an overestimation inNb leads to shorter polymer strands (smaller N , values) for connecting neighboring micelles or, equivalently, to smaller distances between neighboring micelles. This, in turn, leads to stronger electrostatic repulsions between polymer-bound micelles, which also favors fewer micelles.
However, it is encouraging that the theory can predict CAC's and average aggregation numbers that are in reasonable agreement with the experimental data for surfactants possessing different tail lengths using the same set of molecular parameters. This suggest that the assumption that polymer-surfactant interactions mainly take place at the micelle surface is indeed correct, and that our theory has successfully captured the essential physical factors which are responsible for polymersurfactant complexation. We have also examined aqueous solutions containing mixtures of PEO and PVP with the cationic surfactant CTAB and the nonionic surfactant C12Es. We find that if we use the parameters listed in Tables 1 and 2 and a value of cpe= 0, then the theory predicts that PEO and PVP do not form complexes with either surfactant, in agreement with the experimental findings. However, this is not due solely to the larger head area, ah, of these two surfactants, as compared to the sulfate-head case. In fact, we find that these two surfactants would actually form complexes with PEO and PVP, if instead of using a value of cps= 0 we use cpsvalues corresponding to the interactions of PEO and PVPwith a sulfate head. This is an important conclusion, which indicates that the specific interactions between polymer segments and surfactant heads can play an important role in determining the complexation behavior.
VI. Conclusions and Discussion In this paper, we have presented a theoretical study of the complexation of nonionic polymers and surfactants in dilute aqueous solutions. The central element of the theory involves the calculation of the free energy of complexation, gcomp, which represents the change in the free energy when a polymer-surfactant complex is formed from singly-dispersed surfactant molecules and a polymer coil. By considering separately the adsorption free energy and the interactions between polymer segments adsorbed at the micelle surface, we have been able to quantify and rationalize the dominant driving forces responsible for the complexation of a given polymer-surfactant pair. Moreover, we have also been able to account for the effects of solvent quality and polymer hydrophobicity and flexibility on the complexation behavior. Furthermore, by including explicitly a contribution characterizing the specific interactions between the polymer segments and the surfactant heads, we have also been able to rationalize the observed different binding behaviors of anionic and cationic surfactants with nonionic polymers. An important new element in our theory is the incorporation of repulsive electrostatic interactions between polymer-bound micelles and elastic restoring forceswithin the polymer chain, whose competition determines the number of micelles bound to a polymer molecule at a given total surfactant concentration. This has enabled us to predict, for the first time, the distribution of surfactant molecules between monomers, various polymer-surfactant complexes, and free micelles, as well as structural characteristics of the complexes. An important finding of this paper is that polymer-surfactant complexation proceeds via a stepwise association process, namely, as the surfactant concentration is increased, complexes containing one micelle form first, and as long as this process continues, the surfactant monomer concentration remains approximately constant. Only when all the available polymers have one micelle bound to them does the surfactant monomer concentration increase once again, and when it reaches a certain critical value, complexes containing two micelles begin to form, and so on. However, because the polymer samples used in most
3528 Langmuir, Vol. 10, No. 10, 1994 experimental studies are typically highly polydisperse, the predicted discrete steps on theXl versusX, curves, or the surface tension versus X, curves, have not yet been observed. Nevertheless, it has indeed been observed that the slopes in the plateau regions of the surface tension versusX, curves, where complexation occurs, deviate from zero much more than those in the micellization region. We believe that this may reflect the stepwise complexation process, in which the increase in the surfactant monomer concentration with X, is much greater than that in the micellization process. Our analysis indicates that PVPSDS mixtures consisting of highly monodisperse PVP of molecular weight in the range of20 000-30 000 are likely candidates to exhibit the predicted stepwise complexation behavior. In comparing the theoretical predictions with the available experimental data, we have found reasonable agreement for aqueous solutions of PEO and PVP mixed with various sodium alkyl sulfate surfactants. However, we would like to point out that, although significant experimental work has been carried out on polymersurfactant aqueous solutions, relatively few systematic studies using highly monodisperse polymer samples and well-characterized surfactant systems have been conducted. Although the monodispersity of the polymers is not necessary for measurements of critical aggregation concentrations, it is essential in order to observe the predicted stepwise binding process. In addition, there is very little available systematic experimental data on the dependence of the CAC, the polymer-bound micelle aggregation numbers, and the number of micelles bound per polymer chain on polymer molecular weight, polymer concentration, surfactant tail length, salt concentration, and temperature for well-characterized polymer-surfactant pairs, such as PVP-alkyl sulfate mixtures. Furthermore, in order to test the range ofvalidity and applicability of our theory, experimental data on additional polymersurfactant pairs is needed. For example, poly(viny1methyl ether) (PVME) appears to be an ideal candidate for this purpose, since it can bind both anionic and cationic ~urfactants.~'While experimental studies on the complexation behavior of aqueous solutions of PVME-CTAB and PVME-sodium decylphosphate have been carried out,16data for PVME-alkyl sulfates is not yet available. Moreover, data on the surface pressure of an oil-PVME aqueous solution are needed in order to obtain an estimation of the xp value for this polymer.
Nikas and Blankschtein As discussed in section V.A, in the absence of experimental data, the value of the molecular parameter cps, characterizing the specific interactions between polymer segments and surfactant heads, was obtained by fitting the predicted CAC value of a given polymer-surfactant pair to the experimentally measured one. At least two possible ways come to mind to experimentally deduce the value of cps. The first one involves measuring the surface tension of a polymer-surfactant solution as a function of surfactant concentration and fitting the experimental results to a theory which can predict the surface tension of such solutions using cps as an adjustable parameter. The second one involves utilizing the force measurement technique63 to measure the force between two mica surfaces, one grafted with the surfactant and the other with the polymer, where the measured force should reflect the magnitude for cps. Among the various simplifications and approximations that have been made in the development of the theory presented in this paper, future work should concentrate on improving the description of ion-size effects,which may influence both the micellization and the complexation behavior and have been neglected in our electrostatic theory. Although here we have considered only the complexation of hydrophilic nonionic polymers with surfactants in dilute aqueous solutions, our theory can be modified to study the complexation of nonionic hydrophobic polymers (such as PPO) with surfactants, as well as the complexation of hydrophilic polymers with surfactants in semidilute polymer solutions, where the polymer-bound micelles can form ordered structures in the s01ution.l~Work along these lines is in progress.
Acknowledgment. We are grateful to Dr. Tom Whitesides for illuminating discussions on the topic of polymersurfactant interactions. This research was supported in part by the National Science Foundation (NSF) Presidential Yound Investigator (PYI)Award to Daniel Blankschtein, and NSF Grant No. DMR-84-18778 administered by the Center for Materials Science and Engineering at MIT. D.B. is grateful to BASF, Kodak, and Unilever for providing PYI matching funds. (63)Israelachvili,J.N.;Adams, G. E. J.Chem. Soc.,Faraday Trans. 1 1978, 74, 975.
