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Attempt at a unified interpretation of the self-association of 1-1 ionic surfactants in solvents of low dielectric constant
Attempt at a Unified Interpretation of the Self-Association of 1-1 Ionic Surfactants in Solvents of Low Dielectric Constant NORBERT MULLER Department o f Chemistry, Purdue University, West Lafayette, Indiana 47907
Received April 8, 1977; accepted June 23, 1977 On the basis o f electrostatic energies alone, 1-1 ionic surfactants in nonpolar solvents are expected to form cyclic dimers and c o m p a c t oligomers. However, approximate model calculations show that w h e n internal entropy contributions are included, strongly polar acyclic dimers and oligomers m a y b e c o m e the species o f lowest free energy. This implies that two distinct patterns of aggregation are available for these solutes, as s e e m s to be required by available experimental results. T h e calculations suggest that when the solvent has a relatively high dielectric constant, E, or the s u m of the radii of the ionic headgroups, d, is large, association is best represented by stepwise sequential formation of open-chain oligomers with approximately equal equilibrium c o n s t a n t s for the binding of additional m o n o m e r s . W h e n e and d are both small, c o m p a c t clusters are preferred. Their electrostatic binding energy increases with increasing aggregation n u m b e r , n, but their growth is eventually limited at n = nmax by the steric d e m a n d o f the h y d r o c a r b o n chains. W h e n n,n~x --> 16 most of the aggregated surfactant should be present in the form of clusters with n ~ nm~x, and then the association p r o c e s s is m u c h more nearly analogous to micellization in a q u e o u s solutions. INTRODUCTION
pairs, here called monomers, which are in equilibrium with one or more associated species. It seems to be widely agreed that the oligomers and higher aggregates owe their stability primarily to strong electrostatic interactions, and consequently rather similar patterns of self-association might have been expected for all 1-1 ionic surfactants unless very unusual geometric requirements dictated otherwise. Instead, a survey of reported results suggests that most surfactant-solvent systems fall into one o f two classes showing strikingly different aggregation behavior. Dodecylammonium propionate in benzene or cyclohexane (4) can serve as a prototype for what may be called type I aggregation, having the following characteristics: 1. The average aggregation number, h, always remains rather small, most often in the range 3 --- h --< 7. 2. There is no well-defined critical micelle concentration (CMC). 3. As the overall surfactant con-
The thermochemical properties of multicomponent solutions in nonpolar solvents are often profoundly affected by the addition of a small amount of an ionic surfactant. This includes equilibrium properties, notably the ability of the solvent to take up water or other strongly polar solutes, and kinetic properties, such as reaction rate constants. Studies of these surfactant-induced changes in rate, or "catalysis by inverse micelles," have generated considerable excitement in recent years (1, 2). As a step toward a detailed understanding of these effects, many investigations of two c o m p o n e n t s u r f a c t a n t - s o l v e n t systems have been undertaken, using a variety o f experimental approaches. The resulting body of data, much of it summarized in a recent review (3), can be understood in a general way by postulating that the surfactants exist in these solvents as ion383
Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
0021-9797/78/0632-0383502.00/0 Copyright © 1978by AcademicPress, Inc. All fights of reproduction in any form reserved.
384
NORBERT
centration increases, h progressively increases without reaching a limiting constant value. 4. The dependence of the micellar molecular weight on the concentration is well accounted for quantitatively by using a stepwise sequential multiple-equilibrium model (4) originally described by Meyer and van der Wyk (5) in a discussion of micellization in aqueous solutions. Other systems which appear to fall into this category include homologs of dodecylammonium propionate in benzene and carbon tetrachloride (6, 7) and various tridodecylammonium salts in nonpolar solvents (3, 8). Contrasting behavior is found for alkali dialkylsulfosuccinates (9) and dinonylnaphthalene sulfonates (10) in solvents of low dielectric constant. This type II self-association may be described as follows: 1. Aggregation numbers are relatively large, often in the range 12 -< h <- 30. 2. Aggregation appears to commence at a moderately well-defined CMC. 3. At fairly high surfactant concentrations, h reaches a constant limiting value which depends on the nature of the surfactant and solvent. 4. The preceding statements imply that the aggregation is better described by a single-equilibrium or monomer-n-mer model than by the stepwise sequential model. Most surfactants showing type I behavior are cationic, while type II aggregation is usually found for anionic surfactants, but it seems highly unlikely that the difference can be directly attributed to the difference in charge types. Anionic and cationic surfactants would be interconverted by a hypothetical process in which all positive charges are replaced by negative ones and vice versa, and such charge exchange could have no effect on the electrostatic component of the energy of aggregation. Moreover, in the series of alkylammonium carboxylates, varying the length of the hydrocarbon chains in both ions makes it possible to pass smoothly from a typically cationic surfactant (e.g., dodecylammonium propionate) to a typically anionic one, (e.g.,: Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
MULLER
propylammonium dodecanoate), and the available data (6, 7) suggest that the aggregation pattern is of type I for all members of this family. The literature contains only a few attempts to predict even approximately the sizes and stabilities of aggregates of this sort, and these provide n o indication that one should expect two distinct types of behavior (11-13). Because of the large variety of materials to be treated and their complexity, any effort to develop a generally applicable approach must involve the use of drastic simplifying assumptions, and many detailed criticisms can be raised about the nature of these assumptions. In the author's opinion a major advantage of the method used in Ref. (11), which is retained here, is that electrostatic energies are evaluated using models in which each ionic group is represented by a point charge at a specified location, thus avoiding the use of the point-dipole approximation for shortrange interactions between ion pairs. The specific question addressed here is: Is it possible to use such a model, with the stability arising primarily from coulombic forces, to explain in a plausible, consistent, and at least semiquantitative way the existence Of two fundamentally different modes of aggregation? As the title implies, the considerations presented below suggest that a single conceptual framework does indeed allow one to rationalize both type I and type II behavior. THE DIMERIC
SPECIES
A key conclusion of Ref. (11) which appears to need revision is that the cyclic form is always the most stable one for dimers of 1-1 ionic surfactants. Such a cyclic species closely resembles the dimers formed by the alkali halides in the gaseous state (14-17), since the four ionic groups form a tight cluster while the attached alkyl chains project outward into the surroundings and have a relatively minor effect on the sta-
SURFACTANTS IN NONPOLAR SOLVENTS bility. This suggests that methods for calculating dimerization energies of alkali halides may serve as guides in estimating analogous energies of dimeric and oligomeric surfactant clusters. These energy calculations are far from trivial, since the shape of the M2Xz dimer is in general rhomboidal rather than square, the interionic distance is not simply related to the crystal ionic radii, rM and rx, and it is necessary in principle to allow not only for i o n - i o n interactions but also for ion-induced dipole forces, dispersion forces, and closed-shell repulsions (14). H o w e v e r , the dimerization energies of a number of alkali halides are given surprisingly well by applying to both monomer and dimer the formula E = ~ 0.65zizje2/er~j,
[1]
385
FIG. 1. Structure of the symmetrically bent planar acyclic dimer, M2X2, whose Madelung energy is given by Eq. [3]. rising from the cyclic form by a gradual reduction of the angle a from an initial value of 90 ° to a value near zero. As a function of a, the Madelung energy is E ( a ) = -0.65(e2/Ed){1 + 1/(1 + 2 cos a)
i>j
where the numerical factor 0.65 is an empirical constant, and the interionic distances are taken to be d = rM + rx for the m o n o m e r and calculated for the dimer as if it had a square configuration with edge d. Taking • = 1.0 for gas-phase dimerization, one finds that with d expressed in angstroms Edime r -- 2Emonome r
= -126.5/d kcal/mole.
[2]
This gives for NaF, NaCI, LiC1, and KI dimerization energies of 55.0 (55.3), 45.8 (46.2), 52.4 (50.9) and 36.2 (36.6) kcal/mole, respectively. The numbers in parentheses are values calculated by a much more elaborate procedure (16). Because of the similarity between Eq. [1] and the formula used to evaluate Madelung energies of ionic crystals (18), energies calculated from [1] are called Madelung energies in the discussion to follow. The square or rhomboidal geometry always represents a minimum in the electrostatic energy, but this does not necessarily imply that the free energy is also at a minimum. Figure 1 suggests that linear and bent planar acyclic dimers can be considered as
- 2/(2 + 2 cos a)l/Z}.
[3]
As a decreases from 90 °, there is a rapid initial loss of stability, but E ( a ) becomes very nearly independent of o~ when ]a] < 60 °. The energy difference between the linear and square forms is 0.432 times the total estimated dissociation energy for the square dimer, and for this reason most discussions of gaseous M2X2 mention only the " m o r e stable" cyclic form, but it has been recognized that dimers of LiF are not exclusively cyclic (19). It is argued here that because the acyclic dimer has a much larger internal entropy it may, in some circumstances, become the thermodynamically favored form. Equations [1] through [3] suggest that the reaction MzXz(cyclic) ~ MzX2(acyclic)
[4]
has AH ° ~ 55/ed Kcal/mol and it is necessary next to estimate the corresponding AS °. The entropy change may be written as the sum of rotational and vibrational contributions, since the translational c o m p o n e n t is certainly zero. It is possible to evaluate ASrot reasonably accurately by taking the moments of inertia of the cyclic species to be Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
386
NORBERT
I ~ = IB = md2/2N, I c = md2/N, and using for the linear form the approximate value I = 5md2/2N, where m is the sum of the ionic masses, mM + m x , in atomic mass units, and N is Avogadro's number. The expression for I is exact when m u = m x, but its use does not lead to large errors in ASrot when this condition is not satisfied. Including the appropriate symmetry numbers, it is then found that
ASrot =
R
R
2
2 -Rln
In (rod 2)
20h \-N-]
•
[53
The six vibrational degrees of freedom of cyclic MzX2 species give rise to frequencies (16) between about 80 and 280 cm -1. Textbook tabulations show that at 300°K harmonic vibrations at 100, 150, and 200 cm -1 give entropy contributions of 3.5, 2.7, and 2.2 e.u. respectively. This suggests that the total vibrational entropy is of the order of 15 to 20 e.u. The linear dimer has seven vibrational degrees of freedom, comprising three nondegenerate parallel vibrations, and two doubly degenerate perpendicular vibrations. One of the latter, designated as v5 by Herzberg (20), involves essentially deformations of the angle a in Fig. 1 and requires special consideration. The remaining modes should have frequencies similar to, but somewhat lower than those of the corresponding cyclic dimer. Consequently the vibrational entropy, excluding the v5 contribution, should be not very different from the total vibrational entropy of the cyclic dimer, and ASvib should then be nearly equal to the entropy arising from the doubly degenerate v~ mode. The unique character of u~ arises from the fact already noted, that E(oz) is nearly independent of a when Ic~I < 60°, implying that the force constant and frequency for this vibration should be extraordinarily small. As a result, the "linear" dimers should be distributed over many excited Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
MULLER
states of the u5 modes, which is equivalent to saying that the acycli~c form consists of a mixture containing not only strictly linear molecules but also a great variety of planar and nonplanar bent species, which may be obtained by suitably exciting the perpendicular vibrations. To estimate the entropy contribution, one might begin with the lowfrequency limiting formula Svib = R - R In (hv/kT), but this has no upper bound as u approaches zero. It seems preferable instead to treat v5 approximately by using a square well rather than a harmonic potential. Taking the effective mass as m and the well width as the arc of a semicircle of radius d, the total contribution of the doubly degenerate vibration is Svib(b'5)
=
R +2Rln
(27rNkT)l/27rd --
h
[63
Using this as an approximation for A Svib, together with [5], one obtains for the ringopening reaction [4] R R AS ° = - - + - - I n (md 2) 2 2
Experimental data that would allow this result to be tested appear to be lacking. A value of 5.4 e.u. was calculated (19) for Li2F2 at 1200°K, where Eq. [7] gives 13.8 e.u. The discrepancy is almost entirely due to the use of harmonic oscillator contributions with ~ = 153 cm -1 for the mode u5 in place of Eq. [6]. The 153-cm -1 band was observed (21) in a rigid matrix, and the frequency may be considerably lower for the gaseous dimer. In any case, the discrepancy must be much smaller for heavier M2X2 molecules because the frequency of v5 falls as m and d increase. In seeking an expression analogous to [7] applicable to surfactant dimers it is necessary to consider the effect of attaching
SURFACTANTS IN NONPOLAR SOLVENTS hydrocarbon chains to the ionic groups and of transferring the dimers from the vapor to the dissolved state. Apart from increasing the effective value of m, the chains make the acyclic dimer nonlinear even when the ionic groups all lie on a line, and this should quench almost entirely the rotational contribution to AS °. If the vibrational degrees of freedom associated with the chains themselves are very similar in the cyclic and acyclic dimers, the ring-opening entropy then arises mainly from the contribution given by [6], which becomes at 300°K and with d in anstroms Svib(vs) = 6.5 + R In
(md2).
[8]
For a representative case with m = 200 and d = 4, this gives AS ° ~ 22.6 e.u. Such an entropy change seems anomalously large at first sight, but it is similar to estimated ring-opening entropies for cycloalkanes, obtained (22) from the differences between the internal entropies of pairs such as c-C,Hz, and n-C,H~n+2, which are 12.1, 14.5, and 23.6 e.u., respectively, when n is 4, 5, o r 6 . In regard to the effect of opening the ring in the dissolved rather than the gaseous state, it is significant that at the normal boiling point the entropy of vaporization of c-C6Hlz is only about 0.1 e.u. larger than that for n-C~Hx4. Thus the ring-opening entropies in the liquid and vapor states are virtually the same, suggesting that Eq. [8] should be as good an approximation in solution as in the gas phase. The entropy change given by Eq. [8] varies quite slowly when m or d are changed. Taking it to be essentially constant, the estimated standard free energy change for reaction [4] at 300°K is then AG o -~ 55/ed - 6.8 kcal/mole.
[9]
With e = 2.28, the dielectric constant of benzene, this implies that AG O<--0 when d -> 3.5 A and AG o > 0 otherwise. Equation [9] is, of course, semiquantitative at best. The empirical factor 0.65 in
387
Eq. [1] is implicitly retained in [9], and it may be that a somewhat different value would be more appropriate in treating materials with polyatomic ionic headgroups in the dissolved state. Moreover, although the solvent dielectric constant must surely have an effect on AG O, it is not certain that using the bulk solvent value for e is better than using a value somehow allowing for the presence of the alkyl chains and the adjacent ions in the cluster. H o w e v e r , it is noteworthy that in the derivation of [9] no parameters have been adjusted by making use of experimental data for surfactant solutions, and that the equation suggests that in a given nonpolar solvent the cyclic dimers may predominate for surfactants with compact headgroups where d is small, while acyclic ones may be favored when the sum of the headgroup radii is larger. The critical value of d for which the two forms have equal thermodynamic stabilities cannot be accurately forecast by these " b a c k of-the-envelope" calculations, but a value in the neighborhood of 3 to 5 A appears quite reasonable. It is estimated that d = 3.8 A for sodium dialkylsulfosuccinates and d ~ 4.6 A for alkylammonium carboxylates, and it seems at least possible that cyclic dimers form preferentially for the former and acyclic ones for the latter materials. AGGREGATION IN THE LIMIT OF LARGE d OR E It is now proposed, as a working hypothesis, that for systems where ed is so large that the dimers are predominantly acyclic the same will be true of the higher oligomers. Because of the many uncertainties involved in evaluating the required entropy terms, no compelling p r o o f for this conjecture can be offered, but it does not seem an implausible one. With Eq. [1] one finds that for a fixed value of d the ring-opening enthalpy of the cyclic trimer is about 1.4 times that of the dimer, while the tetramer can exist in a compact, cubic conformation (11) which can be Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
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NORBERT MULLER
fully opened to form a linear species at the expense of three times the dimer ringopening energy. In each case, however, the compact form is relatively rigid while the linear form has several doubly degenerate bending modes which should have very low frequencies. Thus the favorable ring-opening entropies must also be much larger for trimers or tetramers than for dimers, and the acyclic forms can become the thermodynamically favored ones. Higher oligomers are subject to similar considerations. For a system in which cyclic species are present at most to a minor extent, aggregation can be represented by the set of reactions A + A~(lin) ~ A~+x(lin);
K = [A~+a(lin)]/[A][A~(lin)].
[10]
The Madelung contributions to the reaction energies are, in units of 0.65e2/ed, -0.333, -0.367, -0.376, -0.380, and -0.383 for i = 1, 2, 3, 4, or 5, respectively, becoming more nearly independent o f / a s the degree of association increases. The association entropies likewise should vary only slowly with changing i, and then experimental data should be fitted rather well by a stepwise association model with nearly equal K~ values (4). Quantitative prediction of the association constants is not possible because, as already mentioned, Eq. [1] cannot be expected to yield accurate energy values without at least some further adjustment of parameters. Several important implications of the stepwise association model with equal K~ values have been worked out (5). The average aggregation number must increase monotonically as the overall surfactant concentration increases, and no definite CMC is found, so that aggregation follows the pattern designated as type I in the Introduction. For example, when K~ = 10, h rises from 1.09 to 3.70 as the overall molarity rises from 10-2 to 1, and the range of aggregation numbers for which i[A~] is appreJournal o f Colloid and Interface Science, Vol. 63, No. 2, February 1978
ciable extends (5) from 1 to about 15. The upper limit on aggregate size is not imposed by the existence of a geometrically favored "complete" micelle but rather by the fact that for all i, [Ai+l]/[Ad < 1, so that [Ai] gradually approaches zero as i increases. Data for type I systems have often been treated with monomer-n-mer models, but it would appear preferable to interpret them using the stepwise association scheme. The resulting K~values should be more suitable for systematic comparisons than micelle sizes and aggregation constants from single equilibrium treatments. When available, the temperature dependence of Ki can be used (4) to evaluate the corresponding AH~° and AS~°. The approach taken here suggests that in a homologous series the electrostatic contribution to the enthalpy is nearly independent of chain length, so that any appreciable variation of AH~° with chain length would point to the presence of chain-chain interaction energies. The size and structure of the hydrocarbon groups must also affect AS f , since aggregation not only reduces the number of solute particles but also restricts the internal freedom of motion of each participating ion-pair. Solvent changes should affect K~ through the dependence of the Madelung energies on e and perhaps also because of energy contributions arising from dispersion interactions between the solvent and the alkyl chains. A G G R E G A T I O N IN T H E L I M I T OF S M A L L d AND
A straightforward extrapolation of the hypothesis adopted in the preceding section is to suppose that for systems with small d and e, where the dimers are predominantly cyclic, the higher oligomers will have related compact structures. One is then led to ask whether such an assumption implies type II behavior. The discussion in Ref. (11) was intended for such systems, but was confined for the most part to dimers, tetramers, and rod-
389
S U R F A C T A N T S IN N O N P O L A R S O L V E N T S
like aggregates derived from a hexamer with structure I (where only the ionic headgroups are shown, with filled and open circles representing oppositely charged
TABLE I Summary of Madelung Calculations for Compact Oligomers a
Species
Az
species) by adding further dimers as indicated by the arrow. Such a growth pattern minimizes steric repulsions and might be preferred if the hydrocarbon chains were extremely bulky. However, electrostatic interactions in the higher oligomers strongly favor two-dimensional growth, leading for example to an eicosamer with structure II. The name "compact clusters" is used to distinguish such aggregates from their open-chain isomers. The given structure,
A3 A4 As A6 A7 A8 Aa A,0 A,I A12 A,3 A,4 AI~ A16 A18
Az0
Madelung energy
Binding energy per monomer
-2.586 -4.066 -5.824 -7.123 -9.007 -10.311 - 12.200 - 13.862 - 15.390 - 16.902 - 18.743 -20.123 -21.783 -23.612 -25.302 -28.487 -31.856
-0.293 -0.355 -0.456 -0.425 -0.501 -0.473 -0.525 -0.540 -0.539 -0.536 -0.562 -0.548 -0.560 -0.574 -0.581 -0.583 -0.593
Energies are in units of 0.65e2/ed.
which resembles that suggested by Peri (23), certainly represents an overidealization, and one cannot expect Madelung computations to yield accurate energies of formation for particular oligomers. However, this does not rule out the possibility of making meaningful predictions of relative stabilities of various clusters, when many of the inevitable errors may reasonably be expected to cancel. Madelung energies were evaluated for most of the species up to A 20, with results shown in Table I. Each cluster was assigned a structure derived from II by removal of an appropriate number of monomers, except that for A 28, a 3 x 6 rectangular array was used. For some of the odd-order species, this does not give the configuration of minimal electrostatic energy, but the best structure is only marginally more stable.
Since it will be shown that these species are present at very low concentrations, obtaining improved energy values for them is not worth the additional effort. The binding energy per monomer shows an overall increasing trend with increasing degree of association, confirming that electrostatic forces favor the formation of extended plates or sheets which may eventually stack to produce crystals. This propensity is opposed by the loss of entropy associated with any reduction in the number of independent solute particles, by the tendency of the solvent to penetrate between the hydrocarbon chains, and perhaps most of all by the steric demands of the chains. Indeed, if the formation of compact clusters is a feature of type II aggregation, the experimental data suggest that their growth does not proceed beyond a fairly well-defined maximal aggregation number, nmax, and it is proposed here that this is a consequence of the steric requirements. For a platelet An with n = pq, the headgroups form a rectangular block with apJournal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
390
NORBERT MULLER T A B L E II The Quantity V ( L ) Defined by Eq. [11], w i t h L = 7 A a n d d = 3.8]~ V(L)
-
p
q
n
(cm3/mole)
3 4 3 4 3 4 5 5 6 6 7 7 8
5 4 6 5 7 6 5 6 6 7 7 8 8
15 16 18 20 21 24 25 30 36 42 49 56 64
523 423 481 361 455 330 307 280 255 240 225 216 206 122
-
- -
~
proximate outer dimensions p d × q d × 2d. The organic portions of the surfactant molecules can then move within a roughly rectangular region whose volume is readily estimated when the length of the alkyl chains is given. The available volume per monomer falls off with increasing n, but with any reasonable asumption about the molecular geometry it remains larger than the minimal volume required for the chains until n is well over 100. Observations which suggest that nmax is often of the order of 25 cannot be explained in this way. However, the situation changes when two further geometric factors are taken into account. First, when bulky sidechains are located near the headgroup, as is the case for aerosol OT (sodium di-2-ethylhexylsulfosuccinate), the key steric parameter may not be the total chain volume but rather the bulk of the material in the close neighborhood of the core. This suggests taking a convenient distance, L, less than the overall chain length, and testing whether the volume per monomer in a shell of thickness L is large enough to accommodate the atoms which should lie within it. Second, it is necessary to recognize that chains belonging to molecules lying on the perimeter of a cluster Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
have access to a far greater portion of the surrounding volume than those belonging to interior molecules. The limit on micellar size then presumably arises from the onset of intolerable crowding near the center. For a cluster with n = p q , the number of interior monomers is (p - 2) (q - 2). Assuming that the main effect of molecules on the perimeter is to prevent chains of the interior molecules from extending into the plane of the headgroups, the space available to these chains consists of two domeshaped regions, one on each face of the cluster. The volume, in cubic centimeters per mole, accessible to atoms within a distance L of the headgroup is V(L)
= (p-
2 ) ( q - 2) ( p -
+ (p + q-
3 ) ( q - 3)2d2L
6)TrdL 2 + - 4 T r L 31 . [11]
3
On looking at a space-filling model of aerosol OT, it is obvious that only a rough approximation of its steric requirements can be made with an expression of this kind, which does not allow for details of molecular shape. Moreover, no rigorous procedure exists for picking a best value of L. In spite of these difficulties, [11] makes it possible to show that simple geometric considerations imply values of nmax which are at least similar to those obtained experimentally (9) and to suggest explanations for some trends in the reported values. As a numerical illustration, values of V ( L ) for clusters of various sizes were calculated taking d = 3.8 A and L --- 7 ~, and they appear in Table II. For aerosol OT, a molecular model suggests that the material lying within 7 A of the headgroup can be represented by the formula C14H~204. With an estimated density of 0.85 g/cm ~ the volume of such a fragment is 300 cm3/mole, and the table then shows that 25 < nmax < 30. For the n-octyl isomer, the corre-
391
SURFACTANTS IN NONPOLAR SOLVENTS
sponding fragment is CllH1704, with a volume of 250 cm3/mole, and then somewhat larger aggregates are allowed with nmax ~ 36. Although these numbers represent no more than a very crude approximation, they show that this approach provides a rationalization for the finding that sodium d i - n - o c t y l s u l f o s u c c i n a t e forms larger micelles than its branched-chain isomers. The report that the potassium salt forms larger micelles than the sodium salt is also consistent with [11], since V(L) increases with increasing d, and the radius of the potassium ion exceeds that of the sodium ion by about 0.38 .A,. This argument does not provide an explanation for the reported difference (9) between the sizes of aerosol OT micelles in benzene, where n = 13.6, and in three other nonpolar solvents, where n = 20 +_ 2. It does, however, suggest a reason for the fact that salts of this type could not be obtained as anhydrous crystals from nonpolar solvents (24), since formation of extended bilayers would require excessively low values of V(L). Table II also indicates that efforts to predict absolute free energies of aggregation can only hope to succeed if internal entropy contributions are included. The very large known entropies of fusion of the longer n-alkanes show that the internal entropy of
an unconstrained alkyl chain in the liquid state may be of the order of 10 to 20 e.u. or more, depending on the chain length. Micellization must drastically reduce the motional freedom of the chains, especially for the interior monomers, resulting in a significant entropy loss, over and above that associated with the decrease in the number of solute particles, and dependent on the cluster size as well as the chain length. As stated in the Introduction, type II aggregation often seems to be characterized by a fairly well-defined CMC. In this connection, it is most interesting that the Madelung energies in Table I can be combined to obtain approximate values o f A H ° for a large set of disproportionation reactions
2An<-~--A~+m + An_m,
[12]
where m < n and (n + m) -< 20. In any one of these reactions, the number of independent particles is constant, and the internal entropy contributions should nearly cancel, giving AS ° - 0 and AG O~ AH °. A number of values are listed in Table III. Although the " u n i t " 0.65eZ/ed varies somewhat from system to system (and contains the problematic factor 0.65), it should certainly be large enough to assure that disproportionation is at least 95% complete whenever the calculated AG 0 is more nega-
TABLE III Values of AG o in Units of 0.65e2/ed Estimated from Madelung Energies, for Selected Disproportionation Reactions of Compact Clusters ~ E v e n species Reaction
2A4 ~ A z 2A6 ~ A , 2A8 ~-A4 2Alo~A4 2Alz~<-~--A4 2 A14 ~ A12 2 As6 ~- A~2 2 Ass ~ As6
+ A6 + A8 + A12 + A16 +A20 + A16 + A~0 + A20
O d d species AG 0
Reaction
+0.055 -0.010 -0.167 -0.346 -0.194 -0.479 +0.041 -0.184
2A3 ~ A 2 2A~ ~ A 4 2A7 ~<~--AG 2A9 ~ A 6 2A9 ~ A ~ 2 As1 ~---A~0 2 As3 ~ As0 2 Ass ~- A1o 2 A~5 ~ As2
+ + + + + + + + +
AG o
A4 A6 A8 A12 A,6 As2 As6 A2o A~8
-0.278 -0.585 -0.585 -0.026 -0.164 -0.329 -0.446 -0.022 -0.006
a The quantity 0.65e2/ed is of the order of 25 kcal/mole. Journal of Colloid and Interface Science, Vol. 63, No. 2, F e b r u a r y 1978
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NORBERT MULLER
tive than -0.1 unit. Considering the whole set of species up to A20, the only lower aggregates found to be stable are A2, A4, and A16. However, the instabilities of A6 and A,5 are marginal, so that these may also be present at appreciable concentrations in an equilibrium mixture. These results suggest that when nmax --> 16, most of the aggregated material in fairly concentrated solutions will consist of only a few species with aggregation numbers close to nmax. The association process then becomes superficially similar to micelle formation in aqueous solutions, and a fairly abrupt change in properties at a critical concentration may indeed be expected. In more dilute solutions, the presence of the smaller oligomers, especially A2 and A 4 , should be more noticeable, in keeping with the findings of Eicke and Christen (13), who refer to the small oligomers as nuclei. SUMMARY OF CONCLUSIONS
The class of 1-1 ionic surfactants includes such an enormous variety of materials that one cannot expect to make accurate predictions of the thermochemical quantities involved in their aggregation using a single model with only a few independent variables. Although the discussion presented here is admittedly speculative and semiquantitative, it makes it possible to account plausibly for the existence of two fundamentally different modes of aggregation, it suggests that in a given solvent the single parameter that mainly determines which mode is preferred is the sum of the headgroup radii, and it may serve to define a basic method of approach from which more detailed and realistic theories can be developed. It also provides a starting point for the rationalization of some reported solvent and chain-structure effects. According to the model calculations, open-chain oligomers predominate when Journal of Colloid and Interface Science, Vol. 63, No. 2, February1978
the product ed is relatively large, as it may be for alkylammonium carboxylates in most solvents. Then stepwise sequential association with approximately equal K~'s should represent the aggregation process well, and t~ should increase with increasing surfactant concentration up to the saturation point, so that the pattern of behavior is of type I. Cyclic dimers and compact clusters are preferred when ed is small, perhaps for sodium sulfonate derivatives in very nonpolar solvents. Steric factors then limit the growth of the aggregates, and many of the species with 4 < n < nmax should be present at insignificantly low concentrations, especially for odd values of n. These features are consistent with type II self-association. Although the emphasis has been on seeking to explain the origin of " p u r e " type I or type II aggregation, the existence of systems with intermediate values of ed is also an obvious possibility. This should give rise to some form of mixed behavior where the preferred forms of the oligomers coexist with detectable amounts of the less-favored ones. Aggregates of aerosol OT and related surfactants, which appear essentially to follow type II behavior, reportedly (13) have fairly large dipole moments, which one would not expect for the compact clusters but which could reflect contributions from acyclic species. Conversely, dipole moments reported (8) for some dimeric and trimeric alkylammonium salts are perhaps a little smaller than would be expected for a mixture of linear and bent acyclic species, and this may point to the presence of lesser amounts of cyclic oligomers. REFERENCES 1. Fendler, E. J., Chang, S. A., Fendler, J. H., Medary, R. T., El Seoud, O. A., and Woods, V. A., in "'Reaction Kinetics in Micelles" (E. H. Cordes, Ed.), p. 127. Plenum, New York, 1973. 2. Fendler, J. H . , A c c o u n t s Chem. Res. 9, 153 (1976). 3. Kertes, A. S., and Gutmann, H., in "Surface and Colloid Science" (E. Matijevic, Ed.), Vol. 8, p. 194. Wiley, New York, 1976.
SURFACTANTS IN NONPOLAR SOLVENTS 4. Lo, F. Y., Escott, B. M., Fendler, E. J., Adams, E. T., Jr., Larsen, R. D., and Smith, P. W., J. Phys. Chem. 79, 2609 (1975). 5. Meyer, K. H., and van der Wyk, A., Helv. Chim. Acta. 20, 1321 (1937). 6. Fendler, J. H., Fendler, E. J., Medary, R. T., and El Seoud, 0. A., J. Chem. Soc., Faraday Trans. 1 69, 280 (1973). 7. Fendler, E. J., Fendler, J. H., Medary, R. T., and E1 Seoud, O. A., J. Phys. Chem. 77, 1432 (1973). 8. Levy, O., Markovits, G., and Kertes, A. S., J. Phys. Chem. 75, 542 (1971). 9. Kon-no, K., and Kitahara, A., J. Colloid Interface Sci. 35, 636 (1971). 10. Little, R. C., and Singleterry, C. R., J. Phys. Chem. 68, 3453 (1964). 11. Muller, N., J. Phys. Chem. 79, 287 (1975). 12. Eicke, H. F., and Christen, H., J. Colloid Interface Sci. 46, 417 (1974). 13. Eicke, H. F., and Christen, H., J. Colloid Interface Sci. 48, 281 (1974).
393
14. O'Konski, C. T., and Higuchi, W. I., J. chem. Phys. 23, 1175 (1955). 15. Milne, T. A., and Cubicciotti, D., J. Chem. Phys. 29, 846 (1958). 16. Bauer, S. H., Diner, R. M., and Porter, R. F., J. Chem. Phys. 29, 991 (1958). 17. Datz, S., Smith, W. T., Jr., and Taylor, E. H., J. Chem. Phys. 34, 558 (1961). 18. Seitz, F., "The Modern Theory of Solids," 1st ed., Chap. 2. McGraw-Hill, New York, 1940. 19. Snelson, A., J. Phys. Chem. 73, 1919 (1969). 20. Herzberg, G., "Molecular Spectra and Molecular Structure," Vol. II, ',Infrared and Raman Spectra of Polyatomic Molecules," p. 181, Van Nostrand, New York, 1945. 21. Abramowitz, S., Acquista, N., and Levin, I. W., J. Res. Nat. Bur. Stand. USA A 72, 487 (1968). 22. Benson, S. W., "Thermochemical Kinetics," p. 66. Wiley, New York, 1976. 23. Peri, J. B., J. Colloid Interface Sci. 29, 6 (1969). 24. Williams, E. F., Woodberry, N. T., and Dixon, J. K., J. Colloid Sci. 12, 452 (1957).
Journal o f Colloid and Interface Science, Vol. 63, No. 2, February 1978
Received April 8, 1977; accepted June 23, 1977 On the basis o f electrostatic energies alone, 1-1 ionic surfactants in nonpolar solvents are expected to form cyclic dimers and c o m p a c t oligomers. However, approximate model calculations show that w h e n internal entropy contributions are included, strongly polar acyclic dimers and oligomers m a y b e c o m e the species o f lowest free energy. This implies that two distinct patterns of aggregation are available for these solutes, as s e e m s to be required by available experimental results. T h e calculations suggest that when the solvent has a relatively high dielectric constant, E, or the s u m of the radii of the ionic headgroups, d, is large, association is best represented by stepwise sequential formation of open-chain oligomers with approximately equal equilibrium c o n s t a n t s for the binding of additional m o n o m e r s . W h e n e and d are both small, c o m p a c t clusters are preferred. Their electrostatic binding energy increases with increasing aggregation n u m b e r , n, but their growth is eventually limited at n = nmax by the steric d e m a n d o f the h y d r o c a r b o n chains. W h e n n,n~x --> 16 most of the aggregated surfactant should be present in the form of clusters with n ~ nm~x, and then the association p r o c e s s is m u c h more nearly analogous to micellization in a q u e o u s solutions. INTRODUCTION
pairs, here called monomers, which are in equilibrium with one or more associated species. It seems to be widely agreed that the oligomers and higher aggregates owe their stability primarily to strong electrostatic interactions, and consequently rather similar patterns of self-association might have been expected for all 1-1 ionic surfactants unless very unusual geometric requirements dictated otherwise. Instead, a survey of reported results suggests that most surfactant-solvent systems fall into one o f two classes showing strikingly different aggregation behavior. Dodecylammonium propionate in benzene or cyclohexane (4) can serve as a prototype for what may be called type I aggregation, having the following characteristics: 1. The average aggregation number, h, always remains rather small, most often in the range 3 --- h --< 7. 2. There is no well-defined critical micelle concentration (CMC). 3. As the overall surfactant con-
The thermochemical properties of multicomponent solutions in nonpolar solvents are often profoundly affected by the addition of a small amount of an ionic surfactant. This includes equilibrium properties, notably the ability of the solvent to take up water or other strongly polar solutes, and kinetic properties, such as reaction rate constants. Studies of these surfactant-induced changes in rate, or "catalysis by inverse micelles," have generated considerable excitement in recent years (1, 2). As a step toward a detailed understanding of these effects, many investigations of two c o m p o n e n t s u r f a c t a n t - s o l v e n t systems have been undertaken, using a variety o f experimental approaches. The resulting body of data, much of it summarized in a recent review (3), can be understood in a general way by postulating that the surfactants exist in these solvents as ion383
Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
0021-9797/78/0632-0383502.00/0 Copyright © 1978by AcademicPress, Inc. All fights of reproduction in any form reserved.
384
NORBERT
centration increases, h progressively increases without reaching a limiting constant value. 4. The dependence of the micellar molecular weight on the concentration is well accounted for quantitatively by using a stepwise sequential multiple-equilibrium model (4) originally described by Meyer and van der Wyk (5) in a discussion of micellization in aqueous solutions. Other systems which appear to fall into this category include homologs of dodecylammonium propionate in benzene and carbon tetrachloride (6, 7) and various tridodecylammonium salts in nonpolar solvents (3, 8). Contrasting behavior is found for alkali dialkylsulfosuccinates (9) and dinonylnaphthalene sulfonates (10) in solvents of low dielectric constant. This type II self-association may be described as follows: 1. Aggregation numbers are relatively large, often in the range 12 -< h <- 30. 2. Aggregation appears to commence at a moderately well-defined CMC. 3. At fairly high surfactant concentrations, h reaches a constant limiting value which depends on the nature of the surfactant and solvent. 4. The preceding statements imply that the aggregation is better described by a single-equilibrium or monomer-n-mer model than by the stepwise sequential model. Most surfactants showing type I behavior are cationic, while type II aggregation is usually found for anionic surfactants, but it seems highly unlikely that the difference can be directly attributed to the difference in charge types. Anionic and cationic surfactants would be interconverted by a hypothetical process in which all positive charges are replaced by negative ones and vice versa, and such charge exchange could have no effect on the electrostatic component of the energy of aggregation. Moreover, in the series of alkylammonium carboxylates, varying the length of the hydrocarbon chains in both ions makes it possible to pass smoothly from a typically cationic surfactant (e.g., dodecylammonium propionate) to a typically anionic one, (e.g.,: Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
MULLER
propylammonium dodecanoate), and the available data (6, 7) suggest that the aggregation pattern is of type I for all members of this family. The literature contains only a few attempts to predict even approximately the sizes and stabilities of aggregates of this sort, and these provide n o indication that one should expect two distinct types of behavior (11-13). Because of the large variety of materials to be treated and their complexity, any effort to develop a generally applicable approach must involve the use of drastic simplifying assumptions, and many detailed criticisms can be raised about the nature of these assumptions. In the author's opinion a major advantage of the method used in Ref. (11), which is retained here, is that electrostatic energies are evaluated using models in which each ionic group is represented by a point charge at a specified location, thus avoiding the use of the point-dipole approximation for shortrange interactions between ion pairs. The specific question addressed here is: Is it possible to use such a model, with the stability arising primarily from coulombic forces, to explain in a plausible, consistent, and at least semiquantitative way the existence Of two fundamentally different modes of aggregation? As the title implies, the considerations presented below suggest that a single conceptual framework does indeed allow one to rationalize both type I and type II behavior. THE DIMERIC
SPECIES
A key conclusion of Ref. (11) which appears to need revision is that the cyclic form is always the most stable one for dimers of 1-1 ionic surfactants. Such a cyclic species closely resembles the dimers formed by the alkali halides in the gaseous state (14-17), since the four ionic groups form a tight cluster while the attached alkyl chains project outward into the surroundings and have a relatively minor effect on the sta-
SURFACTANTS IN NONPOLAR SOLVENTS bility. This suggests that methods for calculating dimerization energies of alkali halides may serve as guides in estimating analogous energies of dimeric and oligomeric surfactant clusters. These energy calculations are far from trivial, since the shape of the M2Xz dimer is in general rhomboidal rather than square, the interionic distance is not simply related to the crystal ionic radii, rM and rx, and it is necessary in principle to allow not only for i o n - i o n interactions but also for ion-induced dipole forces, dispersion forces, and closed-shell repulsions (14). H o w e v e r , the dimerization energies of a number of alkali halides are given surprisingly well by applying to both monomer and dimer the formula E = ~ 0.65zizje2/er~j,
[1]
385
FIG. 1. Structure of the symmetrically bent planar acyclic dimer, M2X2, whose Madelung energy is given by Eq. [3]. rising from the cyclic form by a gradual reduction of the angle a from an initial value of 90 ° to a value near zero. As a function of a, the Madelung energy is E ( a ) = -0.65(e2/Ed){1 + 1/(1 + 2 cos a)
i>j
where the numerical factor 0.65 is an empirical constant, and the interionic distances are taken to be d = rM + rx for the m o n o m e r and calculated for the dimer as if it had a square configuration with edge d. Taking • = 1.0 for gas-phase dimerization, one finds that with d expressed in angstroms Edime r -- 2Emonome r
= -126.5/d kcal/mole.
[2]
This gives for NaF, NaCI, LiC1, and KI dimerization energies of 55.0 (55.3), 45.8 (46.2), 52.4 (50.9) and 36.2 (36.6) kcal/mole, respectively. The numbers in parentheses are values calculated by a much more elaborate procedure (16). Because of the similarity between Eq. [1] and the formula used to evaluate Madelung energies of ionic crystals (18), energies calculated from [1] are called Madelung energies in the discussion to follow. The square or rhomboidal geometry always represents a minimum in the electrostatic energy, but this does not necessarily imply that the free energy is also at a minimum. Figure 1 suggests that linear and bent planar acyclic dimers can be considered as
- 2/(2 + 2 cos a)l/Z}.
[3]
As a decreases from 90 °, there is a rapid initial loss of stability, but E ( a ) becomes very nearly independent of o~ when ]a] < 60 °. The energy difference between the linear and square forms is 0.432 times the total estimated dissociation energy for the square dimer, and for this reason most discussions of gaseous M2X2 mention only the " m o r e stable" cyclic form, but it has been recognized that dimers of LiF are not exclusively cyclic (19). It is argued here that because the acyclic dimer has a much larger internal entropy it may, in some circumstances, become the thermodynamically favored form. Equations [1] through [3] suggest that the reaction MzXz(cyclic) ~ MzX2(acyclic)
[4]
has AH ° ~ 55/ed Kcal/mol and it is necessary next to estimate the corresponding AS °. The entropy change may be written as the sum of rotational and vibrational contributions, since the translational c o m p o n e n t is certainly zero. It is possible to evaluate ASrot reasonably accurately by taking the moments of inertia of the cyclic species to be Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
386
NORBERT
I ~ = IB = md2/2N, I c = md2/N, and using for the linear form the approximate value I = 5md2/2N, where m is the sum of the ionic masses, mM + m x , in atomic mass units, and N is Avogadro's number. The expression for I is exact when m u = m x, but its use does not lead to large errors in ASrot when this condition is not satisfied. Including the appropriate symmetry numbers, it is then found that
ASrot =
R
R
2
2 -Rln
In (rod 2)
20h \-N-]
•
[53
The six vibrational degrees of freedom of cyclic MzX2 species give rise to frequencies (16) between about 80 and 280 cm -1. Textbook tabulations show that at 300°K harmonic vibrations at 100, 150, and 200 cm -1 give entropy contributions of 3.5, 2.7, and 2.2 e.u. respectively. This suggests that the total vibrational entropy is of the order of 15 to 20 e.u. The linear dimer has seven vibrational degrees of freedom, comprising three nondegenerate parallel vibrations, and two doubly degenerate perpendicular vibrations. One of the latter, designated as v5 by Herzberg (20), involves essentially deformations of the angle a in Fig. 1 and requires special consideration. The remaining modes should have frequencies similar to, but somewhat lower than those of the corresponding cyclic dimer. Consequently the vibrational entropy, excluding the v5 contribution, should be not very different from the total vibrational entropy of the cyclic dimer, and ASvib should then be nearly equal to the entropy arising from the doubly degenerate v~ mode. The unique character of u~ arises from the fact already noted, that E(oz) is nearly independent of a when Ic~I < 60°, implying that the force constant and frequency for this vibration should be extraordinarily small. As a result, the "linear" dimers should be distributed over many excited Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
MULLER
states of the u5 modes, which is equivalent to saying that the acycli~c form consists of a mixture containing not only strictly linear molecules but also a great variety of planar and nonplanar bent species, which may be obtained by suitably exciting the perpendicular vibrations. To estimate the entropy contribution, one might begin with the lowfrequency limiting formula Svib = R - R In (hv/kT), but this has no upper bound as u approaches zero. It seems preferable instead to treat v5 approximately by using a square well rather than a harmonic potential. Taking the effective mass as m and the well width as the arc of a semicircle of radius d, the total contribution of the doubly degenerate vibration is Svib(b'5)
=
R +2Rln
(27rNkT)l/27rd --
h
[63
Using this as an approximation for A Svib, together with [5], one obtains for the ringopening reaction [4] R R AS ° = - - + - - I n (md 2) 2 2
Experimental data that would allow this result to be tested appear to be lacking. A value of 5.4 e.u. was calculated (19) for Li2F2 at 1200°K, where Eq. [7] gives 13.8 e.u. The discrepancy is almost entirely due to the use of harmonic oscillator contributions with ~ = 153 cm -1 for the mode u5 in place of Eq. [6]. The 153-cm -1 band was observed (21) in a rigid matrix, and the frequency may be considerably lower for the gaseous dimer. In any case, the discrepancy must be much smaller for heavier M2X2 molecules because the frequency of v5 falls as m and d increase. In seeking an expression analogous to [7] applicable to surfactant dimers it is necessary to consider the effect of attaching
SURFACTANTS IN NONPOLAR SOLVENTS hydrocarbon chains to the ionic groups and of transferring the dimers from the vapor to the dissolved state. Apart from increasing the effective value of m, the chains make the acyclic dimer nonlinear even when the ionic groups all lie on a line, and this should quench almost entirely the rotational contribution to AS °. If the vibrational degrees of freedom associated with the chains themselves are very similar in the cyclic and acyclic dimers, the ring-opening entropy then arises mainly from the contribution given by [6], which becomes at 300°K and with d in anstroms Svib(vs) = 6.5 + R In
(md2).
[8]
For a representative case with m = 200 and d = 4, this gives AS ° ~ 22.6 e.u. Such an entropy change seems anomalously large at first sight, but it is similar to estimated ring-opening entropies for cycloalkanes, obtained (22) from the differences between the internal entropies of pairs such as c-C,Hz, and n-C,H~n+2, which are 12.1, 14.5, and 23.6 e.u., respectively, when n is 4, 5, o r 6 . In regard to the effect of opening the ring in the dissolved rather than the gaseous state, it is significant that at the normal boiling point the entropy of vaporization of c-C6Hlz is only about 0.1 e.u. larger than that for n-C~Hx4. Thus the ring-opening entropies in the liquid and vapor states are virtually the same, suggesting that Eq. [8] should be as good an approximation in solution as in the gas phase. The entropy change given by Eq. [8] varies quite slowly when m or d are changed. Taking it to be essentially constant, the estimated standard free energy change for reaction [4] at 300°K is then AG o -~ 55/ed - 6.8 kcal/mole.
[9]
With e = 2.28, the dielectric constant of benzene, this implies that AG O<--0 when d -> 3.5 A and AG o > 0 otherwise. Equation [9] is, of course, semiquantitative at best. The empirical factor 0.65 in
387
Eq. [1] is implicitly retained in [9], and it may be that a somewhat different value would be more appropriate in treating materials with polyatomic ionic headgroups in the dissolved state. Moreover, although the solvent dielectric constant must surely have an effect on AG O, it is not certain that using the bulk solvent value for e is better than using a value somehow allowing for the presence of the alkyl chains and the adjacent ions in the cluster. H o w e v e r , it is noteworthy that in the derivation of [9] no parameters have been adjusted by making use of experimental data for surfactant solutions, and that the equation suggests that in a given nonpolar solvent the cyclic dimers may predominate for surfactants with compact headgroups where d is small, while acyclic ones may be favored when the sum of the headgroup radii is larger. The critical value of d for which the two forms have equal thermodynamic stabilities cannot be accurately forecast by these " b a c k of-the-envelope" calculations, but a value in the neighborhood of 3 to 5 A appears quite reasonable. It is estimated that d = 3.8 A for sodium dialkylsulfosuccinates and d ~ 4.6 A for alkylammonium carboxylates, and it seems at least possible that cyclic dimers form preferentially for the former and acyclic ones for the latter materials. AGGREGATION IN THE LIMIT OF LARGE d OR E It is now proposed, as a working hypothesis, that for systems where ed is so large that the dimers are predominantly acyclic the same will be true of the higher oligomers. Because of the many uncertainties involved in evaluating the required entropy terms, no compelling p r o o f for this conjecture can be offered, but it does not seem an implausible one. With Eq. [1] one finds that for a fixed value of d the ring-opening enthalpy of the cyclic trimer is about 1.4 times that of the dimer, while the tetramer can exist in a compact, cubic conformation (11) which can be Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
388
NORBERT MULLER
fully opened to form a linear species at the expense of three times the dimer ringopening energy. In each case, however, the compact form is relatively rigid while the linear form has several doubly degenerate bending modes which should have very low frequencies. Thus the favorable ring-opening entropies must also be much larger for trimers or tetramers than for dimers, and the acyclic forms can become the thermodynamically favored ones. Higher oligomers are subject to similar considerations. For a system in which cyclic species are present at most to a minor extent, aggregation can be represented by the set of reactions A + A~(lin) ~ A~+x(lin);
K = [A~+a(lin)]/[A][A~(lin)].
[10]
The Madelung contributions to the reaction energies are, in units of 0.65e2/ed, -0.333, -0.367, -0.376, -0.380, and -0.383 for i = 1, 2, 3, 4, or 5, respectively, becoming more nearly independent o f / a s the degree of association increases. The association entropies likewise should vary only slowly with changing i, and then experimental data should be fitted rather well by a stepwise association model with nearly equal K~ values (4). Quantitative prediction of the association constants is not possible because, as already mentioned, Eq. [1] cannot be expected to yield accurate energy values without at least some further adjustment of parameters. Several important implications of the stepwise association model with equal K~ values have been worked out (5). The average aggregation number must increase monotonically as the overall surfactant concentration increases, and no definite CMC is found, so that aggregation follows the pattern designated as type I in the Introduction. For example, when K~ = 10, h rises from 1.09 to 3.70 as the overall molarity rises from 10-2 to 1, and the range of aggregation numbers for which i[A~] is appreJournal o f Colloid and Interface Science, Vol. 63, No. 2, February 1978
ciable extends (5) from 1 to about 15. The upper limit on aggregate size is not imposed by the existence of a geometrically favored "complete" micelle but rather by the fact that for all i, [Ai+l]/[Ad < 1, so that [Ai] gradually approaches zero as i increases. Data for type I systems have often been treated with monomer-n-mer models, but it would appear preferable to interpret them using the stepwise association scheme. The resulting K~values should be more suitable for systematic comparisons than micelle sizes and aggregation constants from single equilibrium treatments. When available, the temperature dependence of Ki can be used (4) to evaluate the corresponding AH~° and AS~°. The approach taken here suggests that in a homologous series the electrostatic contribution to the enthalpy is nearly independent of chain length, so that any appreciable variation of AH~° with chain length would point to the presence of chain-chain interaction energies. The size and structure of the hydrocarbon groups must also affect AS f , since aggregation not only reduces the number of solute particles but also restricts the internal freedom of motion of each participating ion-pair. Solvent changes should affect K~ through the dependence of the Madelung energies on e and perhaps also because of energy contributions arising from dispersion interactions between the solvent and the alkyl chains. A G G R E G A T I O N IN T H E L I M I T OF S M A L L d AND
A straightforward extrapolation of the hypothesis adopted in the preceding section is to suppose that for systems with small d and e, where the dimers are predominantly cyclic, the higher oligomers will have related compact structures. One is then led to ask whether such an assumption implies type II behavior. The discussion in Ref. (11) was intended for such systems, but was confined for the most part to dimers, tetramers, and rod-
389
S U R F A C T A N T S IN N O N P O L A R S O L V E N T S
like aggregates derived from a hexamer with structure I (where only the ionic headgroups are shown, with filled and open circles representing oppositely charged
TABLE I Summary of Madelung Calculations for Compact Oligomers a
Species
Az
species) by adding further dimers as indicated by the arrow. Such a growth pattern minimizes steric repulsions and might be preferred if the hydrocarbon chains were extremely bulky. However, electrostatic interactions in the higher oligomers strongly favor two-dimensional growth, leading for example to an eicosamer with structure II. The name "compact clusters" is used to distinguish such aggregates from their open-chain isomers. The given structure,
A3 A4 As A6 A7 A8 Aa A,0 A,I A12 A,3 A,4 AI~ A16 A18
Az0
Madelung energy
Binding energy per monomer
-2.586 -4.066 -5.824 -7.123 -9.007 -10.311 - 12.200 - 13.862 - 15.390 - 16.902 - 18.743 -20.123 -21.783 -23.612 -25.302 -28.487 -31.856
-0.293 -0.355 -0.456 -0.425 -0.501 -0.473 -0.525 -0.540 -0.539 -0.536 -0.562 -0.548 -0.560 -0.574 -0.581 -0.583 -0.593
Energies are in units of 0.65e2/ed.
which resembles that suggested by Peri (23), certainly represents an overidealization, and one cannot expect Madelung computations to yield accurate energies of formation for particular oligomers. However, this does not rule out the possibility of making meaningful predictions of relative stabilities of various clusters, when many of the inevitable errors may reasonably be expected to cancel. Madelung energies were evaluated for most of the species up to A 20, with results shown in Table I. Each cluster was assigned a structure derived from II by removal of an appropriate number of monomers, except that for A 28, a 3 x 6 rectangular array was used. For some of the odd-order species, this does not give the configuration of minimal electrostatic energy, but the best structure is only marginally more stable.
Since it will be shown that these species are present at very low concentrations, obtaining improved energy values for them is not worth the additional effort. The binding energy per monomer shows an overall increasing trend with increasing degree of association, confirming that electrostatic forces favor the formation of extended plates or sheets which may eventually stack to produce crystals. This propensity is opposed by the loss of entropy associated with any reduction in the number of independent solute particles, by the tendency of the solvent to penetrate between the hydrocarbon chains, and perhaps most of all by the steric demands of the chains. Indeed, if the formation of compact clusters is a feature of type II aggregation, the experimental data suggest that their growth does not proceed beyond a fairly well-defined maximal aggregation number, nmax, and it is proposed here that this is a consequence of the steric requirements. For a platelet An with n = pq, the headgroups form a rectangular block with apJournal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
390
NORBERT MULLER T A B L E II The Quantity V ( L ) Defined by Eq. [11], w i t h L = 7 A a n d d = 3.8]~ V(L)
-
p
q
n
(cm3/mole)
3 4 3 4 3 4 5 5 6 6 7 7 8
5 4 6 5 7 6 5 6 6 7 7 8 8
15 16 18 20 21 24 25 30 36 42 49 56 64
523 423 481 361 455 330 307 280 255 240 225 216 206 122
-
- -
~
proximate outer dimensions p d × q d × 2d. The organic portions of the surfactant molecules can then move within a roughly rectangular region whose volume is readily estimated when the length of the alkyl chains is given. The available volume per monomer falls off with increasing n, but with any reasonable asumption about the molecular geometry it remains larger than the minimal volume required for the chains until n is well over 100. Observations which suggest that nmax is often of the order of 25 cannot be explained in this way. However, the situation changes when two further geometric factors are taken into account. First, when bulky sidechains are located near the headgroup, as is the case for aerosol OT (sodium di-2-ethylhexylsulfosuccinate), the key steric parameter may not be the total chain volume but rather the bulk of the material in the close neighborhood of the core. This suggests taking a convenient distance, L, less than the overall chain length, and testing whether the volume per monomer in a shell of thickness L is large enough to accommodate the atoms which should lie within it. Second, it is necessary to recognize that chains belonging to molecules lying on the perimeter of a cluster Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
have access to a far greater portion of the surrounding volume than those belonging to interior molecules. The limit on micellar size then presumably arises from the onset of intolerable crowding near the center. For a cluster with n = p q , the number of interior monomers is (p - 2) (q - 2). Assuming that the main effect of molecules on the perimeter is to prevent chains of the interior molecules from extending into the plane of the headgroups, the space available to these chains consists of two domeshaped regions, one on each face of the cluster. The volume, in cubic centimeters per mole, accessible to atoms within a distance L of the headgroup is V(L)
= (p-
2 ) ( q - 2) ( p -
+ (p + q-
3 ) ( q - 3)2d2L
6)TrdL 2 + - 4 T r L 31 . [11]
3
On looking at a space-filling model of aerosol OT, it is obvious that only a rough approximation of its steric requirements can be made with an expression of this kind, which does not allow for details of molecular shape. Moreover, no rigorous procedure exists for picking a best value of L. In spite of these difficulties, [11] makes it possible to show that simple geometric considerations imply values of nmax which are at least similar to those obtained experimentally (9) and to suggest explanations for some trends in the reported values. As a numerical illustration, values of V ( L ) for clusters of various sizes were calculated taking d = 3.8 A and L --- 7 ~, and they appear in Table II. For aerosol OT, a molecular model suggests that the material lying within 7 A of the headgroup can be represented by the formula C14H~204. With an estimated density of 0.85 g/cm ~ the volume of such a fragment is 300 cm3/mole, and the table then shows that 25 < nmax < 30. For the n-octyl isomer, the corre-
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SURFACTANTS IN NONPOLAR SOLVENTS
sponding fragment is CllH1704, with a volume of 250 cm3/mole, and then somewhat larger aggregates are allowed with nmax ~ 36. Although these numbers represent no more than a very crude approximation, they show that this approach provides a rationalization for the finding that sodium d i - n - o c t y l s u l f o s u c c i n a t e forms larger micelles than its branched-chain isomers. The report that the potassium salt forms larger micelles than the sodium salt is also consistent with [11], since V(L) increases with increasing d, and the radius of the potassium ion exceeds that of the sodium ion by about 0.38 .A,. This argument does not provide an explanation for the reported difference (9) between the sizes of aerosol OT micelles in benzene, where n = 13.6, and in three other nonpolar solvents, where n = 20 +_ 2. It does, however, suggest a reason for the fact that salts of this type could not be obtained as anhydrous crystals from nonpolar solvents (24), since formation of extended bilayers would require excessively low values of V(L). Table II also indicates that efforts to predict absolute free energies of aggregation can only hope to succeed if internal entropy contributions are included. The very large known entropies of fusion of the longer n-alkanes show that the internal entropy of
an unconstrained alkyl chain in the liquid state may be of the order of 10 to 20 e.u. or more, depending on the chain length. Micellization must drastically reduce the motional freedom of the chains, especially for the interior monomers, resulting in a significant entropy loss, over and above that associated with the decrease in the number of solute particles, and dependent on the cluster size as well as the chain length. As stated in the Introduction, type II aggregation often seems to be characterized by a fairly well-defined CMC. In this connection, it is most interesting that the Madelung energies in Table I can be combined to obtain approximate values o f A H ° for a large set of disproportionation reactions
2An<-~--A~+m + An_m,
[12]
where m < n and (n + m) -< 20. In any one of these reactions, the number of independent particles is constant, and the internal entropy contributions should nearly cancel, giving AS ° - 0 and AG O~ AH °. A number of values are listed in Table III. Although the " u n i t " 0.65eZ/ed varies somewhat from system to system (and contains the problematic factor 0.65), it should certainly be large enough to assure that disproportionation is at least 95% complete whenever the calculated AG 0 is more nega-
TABLE III Values of AG o in Units of 0.65e2/ed Estimated from Madelung Energies, for Selected Disproportionation Reactions of Compact Clusters ~ E v e n species Reaction
2A4 ~ A z 2A6 ~ A , 2A8 ~-A4 2Alo~A4 2Alz~<-~--A4 2 A14 ~ A12 2 As6 ~- A~2 2 Ass ~ As6
+ A6 + A8 + A12 + A16 +A20 + A16 + A~0 + A20
O d d species AG 0
Reaction
+0.055 -0.010 -0.167 -0.346 -0.194 -0.479 +0.041 -0.184
2A3 ~ A 2 2A~ ~ A 4 2A7 ~<~--AG 2A9 ~ A 6 2A9 ~ A ~ 2 As1 ~---A~0 2 As3 ~ As0 2 Ass ~- A1o 2 A~5 ~ As2
+ + + + + + + + +
AG o
A4 A6 A8 A12 A,6 As2 As6 A2o A~8
-0.278 -0.585 -0.585 -0.026 -0.164 -0.329 -0.446 -0.022 -0.006
a The quantity 0.65e2/ed is of the order of 25 kcal/mole. Journal of Colloid and Interface Science, Vol. 63, No. 2, F e b r u a r y 1978
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NORBERT MULLER
tive than -0.1 unit. Considering the whole set of species up to A20, the only lower aggregates found to be stable are A2, A4, and A16. However, the instabilities of A6 and A,5 are marginal, so that these may also be present at appreciable concentrations in an equilibrium mixture. These results suggest that when nmax --> 16, most of the aggregated material in fairly concentrated solutions will consist of only a few species with aggregation numbers close to nmax. The association process then becomes superficially similar to micelle formation in aqueous solutions, and a fairly abrupt change in properties at a critical concentration may indeed be expected. In more dilute solutions, the presence of the smaller oligomers, especially A2 and A 4 , should be more noticeable, in keeping with the findings of Eicke and Christen (13), who refer to the small oligomers as nuclei. SUMMARY OF CONCLUSIONS
The class of 1-1 ionic surfactants includes such an enormous variety of materials that one cannot expect to make accurate predictions of the thermochemical quantities involved in their aggregation using a single model with only a few independent variables. Although the discussion presented here is admittedly speculative and semiquantitative, it makes it possible to account plausibly for the existence of two fundamentally different modes of aggregation, it suggests that in a given solvent the single parameter that mainly determines which mode is preferred is the sum of the headgroup radii, and it may serve to define a basic method of approach from which more detailed and realistic theories can be developed. It also provides a starting point for the rationalization of some reported solvent and chain-structure effects. According to the model calculations, open-chain oligomers predominate when Journal of Colloid and Interface Science, Vol. 63, No. 2, February1978
the product ed is relatively large, as it may be for alkylammonium carboxylates in most solvents. Then stepwise sequential association with approximately equal K~'s should represent the aggregation process well, and t~ should increase with increasing surfactant concentration up to the saturation point, so that the pattern of behavior is of type I. Cyclic dimers and compact clusters are preferred when ed is small, perhaps for sodium sulfonate derivatives in very nonpolar solvents. Steric factors then limit the growth of the aggregates, and many of the species with 4 < n < nmax should be present at insignificantly low concentrations, especially for odd values of n. These features are consistent with type II self-association. Although the emphasis has been on seeking to explain the origin of " p u r e " type I or type II aggregation, the existence of systems with intermediate values of ed is also an obvious possibility. This should give rise to some form of mixed behavior where the preferred forms of the oligomers coexist with detectable amounts of the less-favored ones. Aggregates of aerosol OT and related surfactants, which appear essentially to follow type II behavior, reportedly (13) have fairly large dipole moments, which one would not expect for the compact clusters but which could reflect contributions from acyclic species. Conversely, dipole moments reported (8) for some dimeric and trimeric alkylammonium salts are perhaps a little smaller than would be expected for a mixture of linear and bent acyclic species, and this may point to the presence of lesser amounts of cyclic oligomers. REFERENCES 1. Fendler, E. J., Chang, S. A., Fendler, J. H., Medary, R. T., El Seoud, O. A., and Woods, V. A., in "'Reaction Kinetics in Micelles" (E. H. Cordes, Ed.), p. 127. Plenum, New York, 1973. 2. Fendler, J. H . , A c c o u n t s Chem. Res. 9, 153 (1976). 3. Kertes, A. S., and Gutmann, H., in "Surface and Colloid Science" (E. Matijevic, Ed.), Vol. 8, p. 194. Wiley, New York, 1976.
SURFACTANTS IN NONPOLAR SOLVENTS 4. Lo, F. Y., Escott, B. M., Fendler, E. J., Adams, E. T., Jr., Larsen, R. D., and Smith, P. W., J. Phys. Chem. 79, 2609 (1975). 5. Meyer, K. H., and van der Wyk, A., Helv. Chim. Acta. 20, 1321 (1937). 6. Fendler, J. H., Fendler, E. J., Medary, R. T., and El Seoud, 0. A., J. Chem. Soc., Faraday Trans. 1 69, 280 (1973). 7. Fendler, E. J., Fendler, J. H., Medary, R. T., and E1 Seoud, O. A., J. Phys. Chem. 77, 1432 (1973). 8. Levy, O., Markovits, G., and Kertes, A. S., J. Phys. Chem. 75, 542 (1971). 9. Kon-no, K., and Kitahara, A., J. Colloid Interface Sci. 35, 636 (1971). 10. Little, R. C., and Singleterry, C. R., J. Phys. Chem. 68, 3453 (1964). 11. Muller, N., J. Phys. Chem. 79, 287 (1975). 12. Eicke, H. F., and Christen, H., J. Colloid Interface Sci. 46, 417 (1974). 13. Eicke, H. F., and Christen, H., J. Colloid Interface Sci. 48, 281 (1974).
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14. O'Konski, C. T., and Higuchi, W. I., J. chem. Phys. 23, 1175 (1955). 15. Milne, T. A., and Cubicciotti, D., J. Chem. Phys. 29, 846 (1958). 16. Bauer, S. H., Diner, R. M., and Porter, R. F., J. Chem. Phys. 29, 991 (1958). 17. Datz, S., Smith, W. T., Jr., and Taylor, E. H., J. Chem. Phys. 34, 558 (1961). 18. Seitz, F., "The Modern Theory of Solids," 1st ed., Chap. 2. McGraw-Hill, New York, 1940. 19. Snelson, A., J. Phys. Chem. 73, 1919 (1969). 20. Herzberg, G., "Molecular Spectra and Molecular Structure," Vol. II, ',Infrared and Raman Spectra of Polyatomic Molecules," p. 181, Van Nostrand, New York, 1945. 21. Abramowitz, S., Acquista, N., and Levin, I. W., J. Res. Nat. Bur. Stand. USA A 72, 487 (1968). 22. Benson, S. W., "Thermochemical Kinetics," p. 66. Wiley, New York, 1976. 23. Peri, J. B., J. Colloid Interface Sci. 29, 6 (1969). 24. Williams, E. F., Woodberry, N. T., and Dixon, J. K., J. Colloid Sci. 12, 452 (1957).
Journal o f Colloid and Interface Science, Vol. 63, No. 2, February 1978