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Surfactant Aggregation in Nonionic Polymer Solutions Eli Ruckenstein Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received April 19, 1999. In Final Form: July 9, 1999 The aggregation of surfactant molecules in the presence of macromolecules is treated by considering that the latter modify the microenvironment of the former by changing the interfacial tensions between the hydrocarbon core of the micelle and solvent and between the headgroups of the surfactant molecules in the micelles and solvent. Conditions are identified under which the critical micelle concentration of the free aggregates, formed in the volume of solution free of macromolecules, is smaller, greater, or equal to the critical micelle concentration of the aggregates bound to the macromolecules. The model considers cases free of specific interactions as well as cases in which specific interactions do occur and leads to the conclusion, among others, that the lack of change in the critical micelle concentration when a nonionic polymer is added to the solvent does not exclude the formation of aggregates bound to the polymer chains.
Introduction Surfactant molecules self-assemble in dilute aqueous solutions, if the concentration is greater than the critical micelle concentration, to segregate their hydrocarbon moieties from water. The presence of water-soluble polymers can have a significant effect on the surfactant aggregation. Small-angle neutron scattering experiments1,2 demonstrated that aggregates of surfactant molecules similar to free micelles can be bound to the polymer strands and that the radius of gyration of the polymer molecules remains comparable to that of the free polymer. Regarding the size, aggregates bound to coiled macromolecules are smaller than those formed outside the volume occupied by the macromolecules. The size and charge of the surfactant headgroup affect in a major way the formation of the bound aggregates. The association between surfactant aggregates and polymers was studied in some detail for the following nonionic polymers: poly(ethylene oxide) (PEO), poly(propylene oxide) (PPO), poly(N-vinyl pyrrolidone) (PVP), and poly(vinyl methyl ether) (PVME). Aggregation of the anionic surfactants, such as sodium dodecyl sulfate and homologous compounds,3-6 is facilitated by the presence of the polymer, that is, the critical micelle concentration is decreased. The aggregation of the cationic surfactants, such as the alkylammonium salts, is affected much less or not at all,3-5 and that of nonionic surfactants is considered to be totally indifferent to the presence of polymers. Brackman et al.,7,8 however, provided evidence that a critical micelle concentration unchanged by the presence of macromolecules does not exclude the formation of aggregates bound to the polymer. They concluded, on the basis of microcalorimetry, that a complexation between n-octylthioglucoside (OTG) and PPO occurred, even though the critical micelle concentration remained unchanged
by the addition of PPO. This complexation was additionally supported by (i) the change in the turbidity of the PPO solution at the CMC of OTG, (ii) the perturbed clouding behavior of PPO, and (iii) the reduced Kraft temperature of OTG. The goal of the present paper is to show that our previous model,9 which considered that the change of the atmosphere around the micelle (expressed in terms of interfacial tensions between the hydrocarbon core of the micelle and the solvent and between the latter and the headgroup) is responsible for the changed patterns of aggregation, can explain the observations of Brackman et al. Two models have been suggested to explain the changes in the aggregation patterns caused by the presence of macromolecules. Nagarajan,10,11 who pioneered the area, considered that segments of the macromolecules penetrate the interfacial region of the micelles, thus decreasing the area of the hydrocarbon core of the micelle which is exposed to water. However, this penetration also increases the area excluded to the headgroup, and, as a result, increases the repulsion among them because it decreases the entropy. Competition between the two effects is responsible for some of the experimental observations. In our treatment, the change in the microenvironment of the micelle was considered to affect aggregation, because the interfacial tension between the hydrocarbon core of the micelle and the solution becomes smaller and that between the headgroup and the solution greater in the presence of polymers. In a previous paper,9 the emphasis was on the cases free of specific interactions between surfactant and polymer. Here, the discussion about the cases free of specific interactions is broadened and extended to cases in which such interactions do occur. This will allow us to explain the experimental observations of Brackman et al. Theory
(1) Cabane, B.; Duplessix, R. J. Phys. 1982, 43, 1529. (2) Cabane, B. In Solution Behavior of Surfactants; Mittal, K. L., Fendler, E. J., Eds.; Plenum: New York, 1982; Vol. 1, p 661. (3) Turro, N. J.; Bonetz, B. H.; Kwo, P. L. Macromolecules 1984, 17, 132. (4) Shinahama, K.; Ide, N. J. Colloid Interface Sci. 1976, 54, 450. (5) Witte, F. M.; Engberts, J. B. F. N. J. Org. Chem. 1987, 52, 4767. (6) Cabane, B. J. Phys. Chem. 1977, 81, 1639. (7) Brackman, J. C.; van Os, N. M.; Engberts, J. B. F. N. Langmuir 1988, 4, 1266. (8) Brackman, J. C.; Engberts, J. B. F. N. Langmuir 1991, 7, 46.
Expression for the Standard Gibbs Energy Change of a Surfactant Solution. The size distribution of micelles in a dilute aqueous solution of surfactant is given by the expression12,13 (9) Ruckenstein, E.; Huber, G.; Hoffmann, H. Langmuir 1987, 3, 382. (10) Nagarajan, R. Colloids Surf. 1985, 13, 1. (11) Nagarajan, R. Adv. Colloid Interface Sci. 1986, 26, 205. (12) Tanford, C. J. Phys. Chem. 1974, 78, 2469. (13) Ruckenstein, E.; Nagarajan, R. J. Phys. Chem. 1975, 79, 2622.
10.1021/la990460+ CCC: $18.00 © 1999 American Chemical Society Published on Web 09/08/1999
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Xi ) X1i exp(-i(µi0 - µ10)/kT)
(1)
where X1 is the molar fraction of the singly dispersed surfactant molecules, Xi is the molar fraction of the i size aggregates, µi0 is the standard chemical potential per surfactant molecule in an i size aggregate, T is the absolute temperature, and k is the Boltzmann constant. Near the CMC, the micelles are in most cases spherical and nearly monodispersed. Consequently, the largest term in the size distribution is expected to provide the main contribution. The aggregation number g can then be calculated as the value of i for which Xi (or lnXi) is maximum. Hence, the aggregation number g is given by the expression
(
)
d(µi0 - µ10) di
kT ln X1 - (µg0 - µ10) + g
i)g
)0
(2)
However,
µ10 + kT ln X1 ) (gµg0 + kT ln Xg)/g
(3)
Because Xg is comparable to X1, and g is large, and the concentration of the monomer remains almost equal to the critical micelle concentration, eq 3 can be approximated by
kT ln X1c ≈ µg0 - µ10
(
0
)
d(µi - µ1 ) di
i)g
≈0
(5)
∆µ0tr ) -(2.05 + 1.49nc)kT - (0.50 - 0.24nc)kT
(6)
where nc is the number of carbon atoms in the hydrocarbon tail. The first term represents the free energy change due to the transfer of the hydrocarbon chain from water to a hydrocarbon phase, whereas the second is a correction which accounts for the constraint imposed by the location of the polar headgroups at the micelle-water interface. ∆µ0in is given by the expression
∆µ
in
) σ(a - ap)
(8)
and ∆µ0tr will be approximated by
∆µ0el ) ge2β2/[2r(1 + κr)]
(9)
where e is the electronic charge, is the dielectric constant, r is the radius of the micelle, β is a constant correction factor and κ, the reciprocal Debye length, is related to the ionic strength I via:
κ ) 3.288 I1/2 nm-1
(10)
For the sake of simplicity, only the cases in which 1/κ , r will be considered. Equation 9 is based on the Debye Hu¨ckel approximation and on the assumption that all the surfactant molecules are dissociated. β is a factor which corrects for the above approximations;12 it will be taken equal to 0.6. More detailed calculations based on the Poisson-Boltzmann equation, which account for the partial dissociation of the surfactant molecules and for other effects, were presented in ref 15. Consequently,
∆µ0g ≡ µ0g - µ01 ) ∆µ0tr + σ(a - ap) kT ln(1 - ap/a) +
R (11) a
where
R≡
Equations (4) and (5) can be used for surfactant solutions free of polymers and for those containing polymers. First, an expression for µi0 - µ10 for a solution free of polymers will be written, which will be later modified to account for the presence of polymers. The standard Gibbs energy of aggregation will be written as the sum of the following contributions:14 (1) the standard Gibbs energy change ∆µ0tr due to the transfer of the hydrocarbon tail of the surfactant molecule from water to the hydrophobic micellar environment; the standard Gibbs energy change ∆µ0in due to the formation of the interface between the hydrocarbon core of the micelle and water; the standard Gibbs energy change ∆µ0ex due to the surface exclusion generated by the finite size of the headgroup; the standard Gibbs energy change ∆µ0el due to the electrostatic interactions. ∆µ0tr will be approximated by the expression14
0
∆µ0ex ) -kT ln(1 - ap/a)
(4)
where X1c is the critical micelle concentration. Consequently, the aggregation number g is given by the expression 0
∆µ0ex is given by the expression14
2πe2 β2 κ
(12)
Expression for the Standard Gibbs Energy Change in the Presence of Macromolecules. The presence of polymers modifies the microenvironment and this change is taken into account in our model by replacing σ(a - ap) in eq 11 by
(σ - ∆σ)(a - ap) + a′p∆σp
(13)
In eq 13, ∆σ represents the decrease of the interfacial tension between the hydrocarbon core of the micelle and the polymer solution when compared to that between the hydrocarbon core and water, ∆σp represents the increase of the interfacial tension between the headgroup and the polymer solution when compared to that between the headgroup and water, and a′p is the surface area of contact between the headgroup and solvent. It should be emphasized that a′p/ap can be large, particularly for the nonionic surfactants. When no specific interactions are present, it is reasonable to consider that ∆σ ) ∆σp Because it is approximated by its macroscopic counterpart, ∆σ can be, in general, determined experimentally or calculated on the basis of any expression for the interfacial tension between a hydrocarbon phase and a polymer solution. When specific interactions between the macromolecules and the headgroups are present, ∆σp becomes smaller than ∆σ. Consequently, in the presence of polymers,
∆µ0g ) ∆µ0tr + (σ - ∆σ)(a - ap) + ∆σpa′p - kT ln(1 - ap/a) +
(7)
R (14) a
Discussion where σ is the interfacial tension between the hydrocarbon core and water, a is the surface area per surfactant molecule at the micellar interface, and ap is the area per surfactant molecule shielded from water by the headgroup. Equation 7 implies that the surface tension between the headgroup and water is negligible.
Let us first consider the case free of electrostatic interactions and denote by a0 the optimal area per surfactant molecule in a micelle in the absence of polymer molecules and by a the optimal area in their presence. This case is relevant for nonionic surfactants when the
(14) Nagarajan, R.; Chaiko, N. A.; Ruckenstein, E. J. Phys. Chem. 1984, 88, 2916.
(15) Beunen, J. A.; Ruckenstein, E. J. Colloid Interface Sci. 1983, 96, 469. Ruckenstein, E.; Beunen, J. A. Langmuir 1988, 4, 77.
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Langmuir, Vol. 15, No. 23, 1999
Ruckenstein
Figure 1. a′p∆σ/ap vs ∆σ for T ) 298 K in the absence of electrostatic interactions, when the critical micelle concentrations of the bound and free micelles are the same.
dipole interactions among the headgroups are negligible. Equation 5 combined with either eq 11 or 14 leads, for a0 and a, to the expressions
{ (
a 0 ) ap
)}
1 1 kT + + 2 4 σap
1/2
(15)
and
{ (
a ) ap
)}
kT 1 1 + + 2 4 ap(σ - ∆σ)
1/2
() () a0 ap
(16)
The radii r0 and r of the micelles are related to the areas a0 and a via the geometrical relations
3v 3v a0 ) and a ) r0 r
much smaller. Consequently, in the absence of specific interactions and for ∆σ not too close to 50 dyn/cm, the nonionic surfactants will aggregate only in the volume of solution free of macromolecules because the standard Gibbs energy change ∆µ0g is smaller in that case. Only after the volume free of polymers becomes saturated in micelles and the critical micelle concentration in the presence of polymer is reached will micelles bound to the macromolecules form. Because only dilute surfactant solutions are considered, this scenario cannot occur in the framework of the present calculations. In the presence of specific macromolecule-headgroup interactions ∆σp < ∆σ. If for given values of ∆σ and ap, a′∆σp/ap is smaller than the critical value provided by Figure 1 (the value for which the two critical micelle concentrations become equal), then the surfactant will form aggregates bound to the polymer at a lower critical micelle concentration than the critical micelle concentration in the volume free of macromolecules. Only after the macromolecules become saturated in bound micelles and the critical micelle concentration for free micelles is reached will free micelles be generated in the system. When a′p∆σp/ap is greater than the critical value provided by Figure 1, aggregates will form only in the volume free of macromolecules. Finally, if a′p∆σp/ap is equal to the critical value, then the critical micelle concentrations of the free aggregates and of the bound ones will be the same, and free and bound aggregates will form simultaneously. Of course, their sizes will be different, those bound to the coiled macromolecules being smaller. Only after the macromolecules become saturated in bound micelles will only free micelles be generated. For ionic surfactants, the electrostatic interactions must also be included. In these cases the areas per surfactant molecule are given by the expressions
(17)
where v is the volume of the hydrocarbon chain of the surfactant molecule. Equations 15, 16, and 17 show that the radius of the micelle bound to the macromolecules is always smaller than that of the micelles formed in a free polymer solution and that the area per surfactant molecule, a, is independent of ∆σp. The critical micelle concentrations in the presence and absence of macromolecules become equal when
(σ - ∆σ)(a - ap) + ∆σpa′p - kT ln(1 - ap/a) ) σ(a0 - ap) - kT ln(1 - ap/a0) (18) Equation 18 is plotted in Figure 1 as a′p∆σp/ap versus ∆σ, for various values of ap, T ) 298 K, and σ ) 50 dyn/cm. The latter value represents the interfacial tension between saturated hydrocarbons and water. In the absence of specific interactions between the polymer and the aggregated surfactant, ∆σ ≈ ∆σp, and Figure 1 shows that, for values of ∆σ smaller and not too close to 50 dyn/cm, the two critical micelle concentrations become equal for values of ap between 3 and 10 Å2 when a′p/ap ) 1, which are too small to be physically relevant. Because, in reality, a′p/ap is large, those values of ap are
3
-
a0 ap
2
-
kT a0 R a0 R + ) 0 (19) σap ap σa 2 ap σa 2 p
p
and
() () a ap
3
-
a ap
2
-
kT R a a + (σ - ∆σ)ap ap (σ - ∆σ)ap2 ap
R ) 0 (20) (σ - ∆σ)ap2
Equation 19 is plotted in Figure 2. Because the area a is provided by an expression obtained from eq 19 by replacing σ in the latter by σ - ∆σ, it is clear that a > a0, hence, the radius of the bound aggregates is always smaller than the radius of the free aggregates. The two critical concentrations become equal when
R ) a R σ(a0 - ap) - kT ln(1 - ap/a0) + (21) a0
(σ - ∆σ)(a - ap) + ∆σpap - kT ln(1 - ap/a) +
Equation 21 is plotted in Figure 3 as a′p∆σp/ap vs ∆σ for various values of ap and two ionic strengths. These figures are similar to Figure 1 and can be interpreted in a similar way. If, for given values of ∆σ and ap, a′p∆σp/ap is smaller than the critical value provided by Figure 3, then the critical micelle concentration of the bound micelles will be the smaller one; if a′p∆σp/ap is larger, then the critical micelle concentration of the free micelles will be the smaller one. Of course, if it becomes equal to the critical
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Langmuir, Vol. 15, No. 23, 1999 8089
Figure 4. ap vs ∆σ when ∆σp ) ∆σ and a′p/ap ) 1 for various ionic strengths and T ) 298 °K.
Figure 2. a0/ap vs σ(ap/kT) for various values of R/(apkT) and for T ) 298 K.
specific interactions are absent, for various ionic strengths and a′p/ap ) 1. For ionic surfactants, the ratio a′p/ap is much smaller than for the nonionic ones, and for illustration purposes, this ratio is taken as unity. If ap is smaller than the critical value provided by Figure 4, then the critical micelle concentration for bound micelles is the smaller one. The critical micelle concentration of the free micelles becomes the smaller one when ap is greater than the critical value, and the two critical micelle concentrations become equal when ap is equal to the critical value. The fact that the critical micelle concentration is not changed by the presence of macromolecules can mean, under some circumstances, that no bound micelles are formed and, in other circumstances, that the bound aggregates have the same critical micelle concentration as the free micelles. The first scenario occurs for nonionic surfactants when dipole interactions can be neglected and the specific interactions are absent, whereas the second scenario can occur when the specific interactions are relevant. The latter scenario was identified experimentally by Brackman et al.7,8 A similar discussion can be extended to the cases in which there are macromolecule-hydrocarbon core specific interactions or both kinds of specific interactions. Conclusion
Figure 3. a′p∆σp/ap vs ∆σ for ionic surfactants, when the critical micelle concentrations of the bound and free micelles are the same and T ) 298 °K.
value, the two critical micelle concentrations become equal. There is, however, an important difference between Figures 1 and 3. Indeed, in the latter case, ∆σp can become equal to ∆σ for physically relevant values of ap. Figure 4 represents ap versus ∆σ when ∆σp ) ∆σ, hence, when
A more detailed discussion, with broader implications, of a model suggested previously9 is presented. The model considers that the aggregation pattern in the presence of polymers is a result of the change in the atmosphere around the micelle. This change in atmosphere is characterized by two quantities: (1) the decrease ∆σ of the interfacial tension between the hydrocarbon core of the micelle and solvent, and (2) the increase ∆σp of the interfacial tension between the headgroup of the aggregated surfactant molecule and the solvent. In the absence of specific interactions, ∆σ ) ∆σp. This case was treated previously but is discussed here in a more complete manner. The emphasis in this paper is, however, on the cases in which macromolecule-headgroup specific interactions are present. In these cases, ∆σp < ∆σ. Conditions are identified under which the critical micelle concentration of the bound micelles is smaller, greater, or equal to that of the free micelles. LA990460+
