J . Phys. Chem. 1989, 93, 1451-1457
TABLE 111: Slow Correlation Times Evaluated from the Two-step Model Calculations"
system DA in DABut DA in DAAc But in DABut
r:,
*
ns
3.7 1.0 3.0 f 1.0 1.5 f 0.2
"The error limits correspond to 80% confidence intervals. However, in this case the micellar self-diffusion coefficients exclude such an explanation for the observed difference between the two systems. The correlation times for the slow motions of the decylammonium and butanoate ions are shown in Table 111. As the I3C measurements only sample the high-frequency region of the spectral density function, the accuracy of the 7,1determination is not optimal, which is indicated by the large error limits in the table. The 7: value of butanoate is much better defined, due to the low-frequency 2H data collected for the counterion. It is usually assumed that both aggregate tumbling and amphiphile diffusion over the curved surface contribute to the slow motion in micellar systems. The theory for spin relaxation in systems of that geometry has recently been discussed by Halle.zo The slow correlation time for the counterion is shorter than for the amphiphile, which can be rationalized by assuming that the lateral diffusion of the butanoate ion at the micellar surface is faster (19) N h y , H.; Saderman, 0.; Canet, D.; Walderhaug, H.; Lindman, B. J . Phys. Chem. 1986, 90,5802. (20) Halle, B. Mol. Phys. 1987, 61, 963.
1451
compared to the amphiphile. Since the 7,f and S values of butanoate indicate that the ion is less firmly anchored at the micelle surface than the decylammonium ion, this appears to be a reasonable assumption. It is not plausible that the exchange of the butanoate ion between the micelle and solution should contribute to 7,". Yiv et al. have measured the dissociation rate constant of 1-pentanol in ionic micelles to be 1.8 X lo7 S-I.~' Although butanoate is less hydrophobic than pentanol this should, with regard to the micellar residence time, be more than compensated by the electrostatic attraction between the micelle surface and the counterion. Therefore, the micelle-counterion dissociation rate constant is probably considerably smaller than lo9 s-I.
5. Conclusions The evaluated correlation times and order parameters of butanoate indicate that the ionic headgroup is anchored at the micelle surface while the hydrocarbon chain is oriented into the hydrocarbon moiety of the micelle. The motional characteristics of the decylammonium ion are affected by the micellar inclusion of the counterion; the order is increased and the motion is slowed down for the chain segments closest to the hydrocarbon-water interface. It was also found that the slow correlation time, corresponding to the tumbling of the micelle and the lateral diffusion of the amphiphile a t the micelle surface, is shorter for the organic counterion as compared to the decylammonium amphiphile. Registry No. Decylammonium butanoate, 73702-94-0; decylammonium acetate, 2016-38-8. (21) Yiv, S.; Zana, R. J. Colloid Interface Sci. 1978, 65, 286.
Influences of Counterion Hydrophobicity on the Formation of Ionic Micelles Mikael Jansson* Institute of Physical Chemistry, Uppsala University, Box 532, S - 751 21 Uppsala, Sweden
and Bengt Jonsson Physical Chemistry 1 , Chemical Center, University of Lund. P.O. Box 124, S-221 00 Lund, Sweden (Received: May 1 1 , 1988)
Influences of counterion hydrophobicity on the formation of ionic micelles, formed by decylammonium surfactants, were studied both experimentally and theoretically. A theoretical model was proposed from which the distribution function of amphiphiles and counterions in the micellar system, modeled by a cell model, could be calculated. The degree of micellization of counterions, as predicted from the calculations, was found to be high even for the least hydrophobic counterion examined, acetate. The relevance of the model can be tested by calculating self-diffusion coefficients from the distribution functions of counterions and amphiphiles in the cell and comparing them to experimental self-diffusion data, measured with the NMR pulsed-gradient spin-echo technique. Good agreements were obtained between the measured and the calculated concentration dependence of self-diffusion coefficients of the ions.
1. Introduction The reduction of headgroup repulsions of micellized amphiphiles by counterions is an important aspect of micelle formation of ionic surfactants. The distribution of counterions in the vicinity of charged micelles has usually been described by the PoissonBoltzmann theory.' However, if interactions between counterions and micelles of nonelectrostatical origin, Le., specific counterion effects, are taken into account, the theory clearly has to be extended. This paper focuses on the effects of counterion hydrophobicity on the formation of ionic micelles, which have been investigated both experimentally and theoretically. The systems studied entail micelles formed by decylammonium ions and
carboxylic counterions with different chain lengths, ranging in size from acetate to hexanoate. Due to the pronounced amphiphilic nature of these counterions, the micellization of counterions, as well as decylammonium ions, has to be considered. The surface charge densities of micelles formed in the presence of hydrophobic counterion are thus lower as compared to micelles formed by surfactants with inorganic counterions. The distribution of amphiphiles and counterions between the micelle and the water part may be calculated from the expressions of the chemical potentials of the molecules in the micelle and the water part. With the combined use of the chemical potential expressions and the (1) Jonsson, B.; Wennerstrom,
*To whom correspondence should be addressed.
0022-3654/89/2093-1451$01.50/0
2119.
H.; Halle, B. J . Phys.
0 1989 American Chemical Society
Chem. 1980, 84,
1452
The Journal of Physical Chemistry, Vol. 93, No. 4, 1989
Jansson and Jonsson
TABLE I: Experimental cmc and B Values of the Decylammonium Surfactant with Different Counterions at 38 O C ion acetate propanoate butanoate pentanoate hexanoa te
cmc, mM
B
61 49 37 17 9
0.60 0.66 0.75 0.87
c
N
E
-u
020 0 15
1
1
m
0
E
n
Poisson-Boltzmann equation, the total distribution functions of both amphiphile ions and counterions in the micellar system, modeled by a cell model,* can be obthined. The relevance of the model may be tested by calculating the effective self-diffusion coefficients of the ions from these distribution functions and comparing it to experimental self-diffusion data.
000 1
I
005
I
010
I
015
, 020
025
[~urfactant] M
Figure 1.
2. Experimental Results This section contains a summary of the experimental findings of the influence of counterion hydrophobicity on the aggregation behavior of decylammonium ions. Most of the experimental data have been published but some new data are also included. The reader is referred to ref 3 for a description of the sample preparations and the details of the experimental setup. All experiments have been performed at 38 “C (the reason being that the Krafft point of decylammonium dichloroacetate, which was examined in ref 3, is above 25 “C). NMR based pulsed-gradient spin-echo (PGSE) self-diffusion measurements have recently emerged as a powerful tool for investigation of surfactant system^.^ With this technique one can simultaneously monitor the self-diffusion coefficients of several components. The observed self-diffusion coefficients are population-weighted time averages of intrinsic self-diffusion coefficients in different molecular environments over a time span on the order of 100 ms. The monitoring of the association process is based on the difference in intrinsic self-diffusion coefficients of amphiphile monomers, counterions, and micelles (the latter monitored by measuring the self-diffusion coefficient of an added micellar solubilizate, hexamethyldisiloxane). The effective translational mobility of an amphiphile or a counterion is considerably reduced when it diffuses the micelle, and the association is therefore reflected in the time-averaged self-diffusion Coefficient. A two-site model has commonly been used to quantify the degree of micellar association of molecules Deff = PDrnic + ( 1 - P)DO
(1)
where Defrrepresents the time-averaged diffusion coefficient, p the fraction of molecules associated to micelles, D ~ the c diffusion coefficient of the micelle, and Do the diffusion coefficient of the “free” species in the water part (which is measured on a sample with a concentration below the critical micelle concentration (cmc). From this simple model one can calculate the fraction of micelle associated amphiphiles, pa,and counterions, pc, respectively. The degree of counterion association, @, is defined as p c / p , . The two-site model is clearly a simplification; the continuous distribution of counterions in the vicinity of a charged micelle surface and the contribution of intraaggregate diffusion to the effective self-diffusion coefficient are not represented by the model. Nevertheless, the model works as a first approximation and has successfully described various association Table I shows the average @ values obtained in the surfactant concentration range 0.1-0.2 M. The observed variation in the measured @ values was on the order of f 0.01. Hence, the in(2) Wennerstrom, H.; Jonsson, B.; Linse, P. J . Chem. Phys. 1982, 76, 4665. (3) Jansson, M.; Stilbs, P. J . Phys. Chem. 1985, 89, 4868. (4) Jansson, M.; Stilbs, P. J . Phys. Chem. 1987, 91, 113. (5) Stilbs, P. Prog. Nucl. Magn. Reson. Spectrosc. 1987, 19, 1. (6) Lindman, B.; Puyal, M.-C.; Kamenka, N.; RymdBn, R.; Stilbs, P. J . Phys. Chem. 1984.88, 5048. (7) Lindman, B.; Karnenka, N.; Puyal, M.-C.; Brun, B.; Jonsson, B. J . Phys. Chem. 1984, 88, 53. (8) Stilbs, P.; Lindman, B.J . Magn. Reson. 1982, 48, 132.
Measured micellar self-diffusion coefficients of decylammonium acetate (0),propanoate (0),butanoate (A),and pentanoate (0) micelles versus surfactant concentration.
variance of counterion association with surfactant concentration, established for several was also observed here. The magnitude of @ increases from 0.60 for acetate to 0.87 for pentanoate, which demonstrates the pronounced influence of counterion hydrophobicity on the micelle-counterion interaction. Due to the formation of very large aggregates in the hexanoate systems, which gave rise to poor spin-echo signals, @ could not be obtained for hexanoate. It is in principle possible to determine the cmc of a surfactant from the concentration dependence of the self-diffusion coefficient of the amphiphilic ion.3 However, the precision of such determinations is not good, especially for surfactants with low cmc’s and weak NMR signal intensities. We have therefore determined the cmc from density measurements on a Kratky instrument,I* using 20 different surfactant concentrations for each counterion. The formation of micelles leads to a change in the partial specific volumes of the components, resulting in a distinct discontinuity in the concentration profile of the density. The estimated accuracy of the cmc determination is f l mM. the cmc values are listed in Table I. Since the difference in the observed cmc’s between decylammonium acetate and decylammonium hexanoate is on the order of 1 magnitude, it is obvious that the counterion has a profound effect on the association of the amphiphiles. Table I shows that there is a correlation between a high @ and a low cmc. Qualitatively, this may be attributed to more effective screening of the electrostatic repulsive interactions between decylammonium ions on the micelle surface when @ is high. In Figure 1 the measured self-diffusion coefficients of the decylammonium micelles are shown as a function of surfactant concentration. A correlation is found between large counterion sizes and low D,, values. Since the dynamics of micelles in solution also reflects intermicellar interactions, the interpretation of micellar diffusion data in terms of micellar size is quite complicated. Furthermore, the different sizes and degrees of micellar association of the counterions make it hard to extract relative differences in aggregation numbers in the investigated systems. Direct estimates of micellar aggregation numbers may be obtained from fluorescence quenching experiments.I3 For 0.15 M solutions of decylammonium acetate and decylammonium butanoate the aggregation numbers were found to be 60 and 84, respe~tive1y.l~ A counterion dependence on the aggregation number is thus indicated. A general decrease in Dhc is observed with increasing surfactant concentrations. This may partly be attributed to increasing micelle-micelle repulsions,I5 but a slight increase in micellar size (9) Zana, R. J . Colloid Interface Sci. 1980, 78, 330. (10) Khan, A,; Soderman, 0.;Lindblom, G.J . Colloid Interface Sci. 1980, 78, 217. (1 1 ) Bikingstad, E. J . Colloid Interface Sci. 1980, 73, 260. ( 1 2) Kratky, 0.;Leopold, H.; Stabinger, H. In Methods in Enzymology; Hirs, C . , Timasheff, S., Eds.; Academic: New York, 1973; Vol. XXVII. ( 1 3) Almgren, M.; Lofroth, J.-E. J . Colloid Interface Sci. 1980, 81, 486. (14) van Stam, J.; Jansson, M., unpublished results.
Influences of Counterion Hydrophobicity on Micelles
3 0
Figure 2. Schematic picture of the model used. The system consists of molecules in the micelle part and the water part. Both decylammonium ions and hydrophobic counterions contribute to the hydrocarbon part of the micelle. (For clarity, only one hydrocarbon tail of each species is indicated in the micelle.) The micellar radius, Rb, and cell radius, R,, are shown. The true proportions between Rb, R,, and the headgroup size are not depicted in the figure.
with concentration could also be expected. The dramatic decrease in D,ic found for the pentanoate system above 70 m M can only be rationalized with the formation of larger elongated micelles.
3. Theory 3.1. The Theoretical Model. It was shown in ref 16 that the distribution of amphiphilic ions between the micelles and the water part for aqueous sodium dodecyl sulfate solutions could be calculated from the expressions of the chemical potentials of the amphiphile in the micellized and the free state, together with the condition that the molecules in the two sites are in chemical equilibrium. Due to the amphiphilic nature of the counterions in the present systems, we also have to consider the distribution of counterions between the micelle and the water region. This is done in the same way as for the amphiphile, Le., by expressing the chemical potential of the counterion in the micellized state and the free state and assuming that we have a chemical equilibrium between the states. Hence, the modeling of the decylammonium/hydrophobic counterion system is similar to that of a mixed cationic/anionic micellar system. The theoretical model will be outlined in the following section. Figure 2 shows schematically how the micellar system is modeled. The system consists of molecules in the micelle part and the water part. Both the decylammonium ions and the hydrophobic counterions contribute to the hydrocarbon part of the micelle. A cell model2 has been used for representing the micellar solution. The total macroscopic system is divided into cells, whose geometry is determined by the geometry of the aggregates. The cell is in this case a sphere with a radius R,. The micelle is modeled as a sphere with a fixed radius Rb and a uniform charge density. Rb is the estimated length of the extended amphiphile (- 17 A). If larger micellar sizes are considered, Rb is calculated from the measured micellar aggregation number and the estimated volumes of the micellized decylammonium ions and counterions. The micelle is fixed within the cell and counterions and co-ions are free to move within the intervening space. In order to simplify the calculations, the distribution of aggregates and cell dimensions have been ignored and replaced by a monodisperse system. More refined versions of the cell model exist, which also take into account the polydispersity of the cell,'' but as a first approximation these effects may be neglected. The electrostatic interactions between the charged molecules in the system may be described with the Poisson-Boltzmann theory.' The electrostatic potential @ and the ion concentrations are obtained from the solution of the Poisson-Boltzmann (PB) equation (15) Mazo, R. M . J . Chem. Phys. 1965, 43, 2873. ( I 6 ) Johnson, 1.; Olofsson, G.;Jonsson, B. J. Chem. SOC.,Faraday Trans. I 1987, 83, 3331. (17) Landgren, M.; Jonsson, B., to be published.
The Journal of Physical Chemistry, Vol. 93, No. 4, 1989 1453 cocrV2@ = -FCzicio exp(-zie@/kT)
(2)
where eo€, is the permittivity of water, @ the electrostatic potential in the system, F the Faraday constant, zi the valency, cio the concentration of ion i for @ = 0, and k T the Boltzmann factor. The following boundary conditions are needed to solve the PB equations: (1) The electric field V@ is zero at the cell boundary. (2) At the surface of the micelle V@e,= - ( ~ / c ~ t ,= (eRbZ/3Va~r)(1- 2Xc/(l
- Vc/J'a)))
(3)
where e, denotes the normalized radius vector, e the unit charge, R b the micellar radius, Va the volume of the amphiphile molecule, V, the volume of the counterion, and X, the molar ratio of counterions in the micelle. The last term in eq 3 expresses the reduction in surface charge density of the micelle upon the micellization of hydrophobic counterions, which originates from both a partial charge neutralization of the surface and a dilution of the surface charge, due to the increased volume of the micelle. The free energy difference between an amphiphile ion or a hydrophobic counterion in the micellized and free state stems from differences in electrostatic interactions, mixing entropies, and hydrophobic interactions in the two states. The different contributions to the free energy have previously been derived and discussed in detail in ref 18 and 19. The equations used for the chemical potential of the various species are pi
p,
= pco
= pio
+ k T In cio
(4)
+ @A + k T In X, - V,(E,, - N k m S ( c j cio) d V + y47rR: + k T In Xmic)/Vmic(5)
pa=pao-@A+
kTIn(1 -&)-
V,(E,I-NkmJ(cj-
cio) d V + y47rRb2 + k T In X,ic)/Vmic (6) where pi is the chemical potential of the amphiphile or counterion in the water part, p, and pa are the chemical potentials of the micellized counterion and micellized amphiphile, the degree symbol (") indicates standard chemical potentials, @A is the electrostatic potential difference between the micelle surface and the boundary of the cell, X, is the molar ratio of the counterion in the micelle, V, and Va are the volumes of the counterion and the decylammonium ion, Vmicis the volume of the micelle, N is Avogadro's J/m2 number, y is the surface tension, for which the value 18 was used, R b is the micellar radius, Xmicis the molar ratio of micelles in the system, and E,, is the electrostatic energy of a micelle given by
The distribution of decylammonium ions and counterions between the micelles and the water part is obtained from the combined use of the PB equation, the expressions of the chemical potentials in eq 4-6, and the condition that the molecules in the two sites are in chemical equilibrium.
Since the solution of the PB equation also gives the distributions of counterions and amphiphiles as a function of distance from the micelle surface, the total distribution function of both amphiphile ions and counterions in the cell can be calculated. The cmc of the surfactants is also obtained from the calculations. The cmc is here defined as the geometric mean activity of free amphiphile and counterions at a total amphiphile concentration in which 1% of the amphiphiles are found in micelles.'6 (18) Jonsson, B. Ph.D. Thesis, University of Lund, 1981. (19) Jonsson, B.; Wennerstrom, H. J . Phys. Chem. 1987, 91, 338.
1454 The Journal of Physical Chemistry, Vol. 93, No. 4, 1989 100
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Figure 3. Calculated cmc versus Ap," for two cmc, values: 0.10 M (the upper curve) and 0.07 M. The experimental cmc value of decylammonium hexanoate is shown.
Figure 4. Counterion hydrophobic free energy, Awc0, versus the counterion chain length (the number of methyl or methylene groups in the counterion chain). Nowas assumed to be 50.
3.2. The Calculation Procedure. The computational procedure is described by the following four steps. 1. The aggregation number of the micelle, No, the volume of the amphiphilic molecule, V,, and the concentration of micelles, c, are input values. The molar ratio of counterions in the micelle, X,, is set to zero. By assuming a cmc value for the amphiphile in the absence of counterion hydrophobic interactions, cmc,, the difference between the standard chemical potential in the micelle and the water part of the decylammonium ion, Ahao, can be calculated. 2. The volume of the counterion, V,, and the difference between the standard chemical potential in the micelle and the water part of the hydrophobic counterion, Apco, are input values. The potential at the boundary of the cell, a,, and the concentrations of the ions where 0 is zero, co+ and co-, are guessed. 3. The PB equation is solved numerically. 4. The solution of the PB equation is checked with regard to the conditions given in eq 3-8. If these conditions are not fulfilled, new co+ and co- values are chosen by a Newton-Raphson method and the process is repeated from step 3. The volumes of the decylammonium ion and the counterions were estimated from general geometrical considerations, in line with the proposal of Tanford.,O V, was assumed to be 296 A3, and V, was taken from
is on the order of fO.ZkT, is due to the fact that the aggregation number of the hexanoate micelle is unknown.) cmcOwas in this way estimated to be 0.10 M, which has been subsequently used in the calculations. Theoretically, it is not possible to separate the different contributions to Apo from the hexanoate ion and the sodium ion, in the sodium hexanoate system. Since the nonelectrostatical interactions of the sodium ion with the micelle could be expected to be very small, this contribution has been neglected, however. As the cmc of decylammonium chloride is 0.064 mM at 25 0C,22 0.1 M might be regarded as a high value for cmc,,. However, cmc,, reflects not only the cmc in the absence of hydrophobic counterion interactions but also the cmc in the absence of all kinds of specific counterion effects that do not enter into the Poisson-Boltzmann calculations. It can be anticipated that the interaction between the micelle surface and the chloride counterion originates not only from pure ion-ion Coulomb interactions but also from, e.g., the polarizability of the chloride ion. Once the cmc, value was set, Apco of the different counterions could be extracted from the experimental cmc values, as indicated by Figure 3. 3.3. Results. The difference between the standard chemical potential of counterions in the micelle and the water part, Apco, may be regarded as the hydrophobic free energy of the counterion. Hydrophobic interactions have been studied by solubilization measurements of hydrocarbons in water.23 For sufficiently large molecules, it has been shown that the hydrophobic free energy is directly proportional to the size of the hydrocarbon chain. The increment per CH, group for a homologues series of compounds has been estimated to be in the order of 1.3-l.4kT.l8 Figure 4 shows the Apco values calculated from the experimental cmc values. The increment in Apco per CH2 group is considerably lower for the smallest counterions as compared to the more hydrophobic ones and is also lower than 1.4kT. Deviations from the linear dependence of the hydrophobic free energy on the hydrocarbon size could be expected for the smallest molecules. Measurements of the solubility in water of paraffin hydrocarbons demonstrate that a linear relation between the water solubility and the molar volumes of the hydrocarbons appears first at molecular sizes above that of butane.23 The increment in Apco per CH, group for the most hydrophobic counterions is 1.8kT, assuming that No = 50. However, the calculation of Apco from the surfactant cmc is dependent on the micellar aggregation number. Qualitatively, this may be understood from the fact that a low No value underestimates the micellar surface charge density and, therefore, also the electrostatic micelle-ion attraction. In order to get an agreement between the experimental and calculated cmc's, the hydrophobic counterionmicelle attraction is then overestimated. When the experimental aggregation number, No = 84, is used for the decylammonium
V, = 50
+ 26.9(n
-
2 ) A3
where n is the number of carbon atoms in the counterion. The results are in general insensitive to variations in V,. The aggregation number of the micelle was taken to be 50, which is reasonable for an amphiphile with the size of a decylammonium ion. Since it was indicated from fluorescence-quenching measurements that the micellar aggregation numbers were larger than 50, calculations have also been performed with larger aggregation numbers. This is not consistent with a spherical geometry of a micelle formed by decylammonium amphiphiles, but the maximum No used in the calculations, No = 84, corresponds to only modest deviations from a spherical geometry, however. Both cmcOand Apco can be estimated by comparing experimental cmc values of the decylammonium surfactants with those calculated from the model. Figure 3 shows the calculated cmc for decylammonium hexanoate as a function of Apco for two different cmcOvalues. As seen from the figure, Apco of hexanoate can be extracted from the experimental cmc of decylammonium hexanoate, but the magnitude will depend on what cmcOvalue is used. Hence, if Apco of hexanoate is known, cmcOmay be determined. Apco can be obtained from an experimental cmc value of a hexanoate solution and the application of step 1 in the computational procedure. The cmc of sodium hexanoate has been measured to be 0.73 M,,' which corresponds to a Apco value of approximately 3.8kT. (The uncertainty in the magnitude, which (20) Tanford, C. The Hydrophobic Effect; Wiley: New York, 1980. ( 2 1 ) Markina, Z.; Tsikurina, N.; Kostova, N. Kolloid Z.1964, 26, 76.
-
( 2 2 ) Kale, K. M.; Zana, R. J. Colloid Interface Sci. 1977, 61, 3 1 2 (23) McAuliffe, C. J. Phys. Chem. 1966, 70,1267.
The Journal of Physical Chemistry, Vol. 93, No. 4, 1989 1455
Influences of Counterion Hydrophobicity on Micelles
n 0.9 0 0
08
9 07
v O2
t
06
00
10
20
30
Figure 5. Calculated fraction of molecules confined within a distance L from the micelle surface versus L for the decylammonium ion (DABut) and the butanoate ion (But) in the decylammonium butanoate system and the decylammonium ion (DAAc) and the acetate ion (Ac) in the decylarnmonium acetate system. The surfactant concentrations are 0.15 M.
1“0
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1
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8
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Figure 6. Molar ratio of counterions in the micelle, X,,versus the counterion chain length (the number of methyl or methylene groups in the counterion chain).
butanoate micelles, Apco becomes -0.2kT as compared to 0.3kT. Although the aggregation numbers of decylammonium pentanoate and decylammonium hexanoate micelles were not measured, it could be expected that the increment in Akco would become smaller, and thus approach the expected value of 1.4kT, if the increase in size of micelles with counterion hydrophobicity was taken into account. The main output of the model calculations is the distribution functions of the amphiphilic ions and the counterions in the cell, from which degrees of counterion binding and self-diffusion coefficients may be calculated. Examples of these distribution functions are shown in Figure 5, where the fraction of molecules confined within the distance L from the micelle surface, represented as a function of L, are displayed for 0.15 M decylammonium acetate and 0.15 M decylammonium butanoate. The fraction of micellized amphiphiles and counterions are those at L = 0. As expected, the more pronounced hydrophobic character of butanoate results in a higher fraction of micellized counterions, p c = 0.5, as compared to pc = 0.3 for acetate. The surface charge density of the micelles is consequently lower in the presence of a more hydrophobic counterion. The fraction of counterions in the immediate vicinity of the micelle surface is, therefore, lower in the butanoate system; Le., the curve corresponding to the butanoate ion increases less steeply in the range of a few angstroms from the surface as compared to the acetate curve. The difference in cell radii between the two systems, which is indicated in the figure, is due to differences in micelle concentrations, originating from the lower monomer concentrations in the water part in the decylammonium butanoate systems. Although the fraction of micellized counterions obviously depends on the micellar concentration, the molar ratio of counterions in the micelle remains independent of this quantity. Figure 6 shows
Counterion Chain l e n g t h
Figure 7. Calculated (0)and measured ( 0 )@ values versus the counterion chain length (the number of methyl or methylene groups in the counterion chain). No was assumed to be 50.
the molar ratios of the counterions in the micelle, X,,as a function of counterion chain length. X , ranges from 0.30 for acetate to 0.48 for hexanoate. Hence, the model calculations predict a substantial amount of counterions having their hydrocarbon part in contact with the micelle hydrocarbon core, even for acetate ions. Due to the large amount of micellized counterions, the effective surface charge density of the micelles with the most hydrophobic counterions becomes very low. Since lower surface charge densities are in general related to the formation of larger aggregate^?^ this explains the formation of larger micelles in the decylammonium pentanoate and decylammonium hexanoate systems. As outlined in the Experimental Results, the degree of counterion association to micelles was estimated from experimental self-diffusion data by the application of a simple two-site model. Since the diffusion of counterions in the water part is also affected by the presence of micelles, the fraction of bound counterions obtained from self-diffusion data will not correspond to the fraction of micellized counterion. In order to qualitatively compare experimental @ values with those predicted from the model, “bound” counterions are defined as those confined within 3 A from the micellar surface. Three angstroms is commonly used as a cutoff distance in Poisson-Boltzmann calculations, and it usually gives the same magnitude of /3 as experiments doFS The self-diffusion measurements indicate that B is independent of the total surfactant concentration. This is in agreement with the results of the model calculations. For decylammonium acetate the theoretical @ increases from 0.629 to 0.635 in the interval 0.08-0.2 M, whereas @ increases from 0.827 to 0.846 for decylammonium pentanoate in the same concentration range. Thus, a small increase in @ with surfactant concentration is predicted by the model. This increase also shows a counterion hydrophobicity dependence. Since the increase in @ is very small even for the most hydrophobic counterion, the precision in the experimental determination of /3 is not sufficient to determine whether this is a real effect, however. Figure 7 shows the average experimental and calculated degree of counterion association in the surfactant concentration range 0.10-0.20 M. The agreement between the measured and the predicted @ is good, even though the increase in the theoretical /3 values with counterion length is slightly less steep than the measured. The dependence of the aggregation number on the results of the model calculations is in this case small but significant. (The increase in micelle surface charge density with the aggregation number is to some extent counteracted by the corresponding decrease in Awco, as outlined above.) Using the aggregation number 84 for the decylammonium butanoate micelles gives a /3 value of 0.76, as compared to 0.735 when No = 50 is used. Thus, the agreement between experimental and theoretical @ values (24) Israelachvili, J. Intermolecular and Surface Forces; Academic: New York, 1985. ( 2 5 ) Gunnarsson, G.; Jonsson, B.; Wennerstrom, H. J . Phys. Chem. 1980, 84, 3114.
1456 The Journal of Physical Chemistry, Vol, 93, No. 4, 1989 0.12
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Figure 8. Mean concentration of decylammonium ions (+) and counterions (-) in the water part (Le., not closer than 3 A from the micelle surface) in the decylammonium acetate and butanoate systems versus the
surfactant concentration. becomes better when the increase in size of micelles with counterion hydrophobicity is taken into account. Furthermore, the application of the Poisson-Boltzmann model to nonspherical geometries gives in general higher /3 values than for spherical g e o m e t r i e ~ .Since ~ ~ decylammonium pentanoate forms elongated aggregates, this implies that the calculated /3 value would become higher if this also was considered. Figure 8 shows the calculated concentrations of amphiphilic ions and counterions in the water part as a function of the total surfactant concentration of decylammonium acetate and decylammonium butanoate, respectively. The concentration of counterions in the water part increases above the cmc, whereas there is a significant decrease in the concentration of amphiphilic monomers. This agrees with previous experimental findings for ionic surfactant^.^-'+'^ The application of the two-site model to amphiphile and micellar self-diffusion data for the decylammonium acetate system indicates a decrease in the decylammonium monomer concentration from 0.060 to 0.039 M in the same concentration interval, which is in very good agreement with the model predictions. However, for the decylammonium butanoate system the self-diffusion data show a decrease in the monomer concentration from 0.038 to 0.032 M, while the model predicts a much larger decrease. This does not necessarily mean that the model fails in this context, but it may also be a consequence of errors induced by the two-site model, such as the neglection of the intraaggregate diffusion of the amphiphile, as outlined below. 3.4. Calculationsof Self-Diffurion Coefjcients. In the previous section the results of the model calculations were compared to the degree of counterion binding and the concentrations of amphiphilic monomers, obtained from the application of a two-site model to the measured self-diffusion coefficients. Since the two-site model obviously is a simplification, the relevance of such a comparison can be questioned. It is preferable to compare the results directly with the measured self-diffusion coefficients of the counterions and the decylammonium ions. In ref 26, which deals with self-diffusion of small molecules in a colloidal system, it was shown that the effective self-diffusion coefficient can be calculated, provided that the distribution function of the molecule in the cell is known. Since the distributions of both amphiphiles and counterions are obtained from the model calculations (examples of the distribution functions are given in Figure 5 ) , it is possible to calculate the self-diffusion coefficients of both species. The only new input values that are necessary are the diffusion coefficients of the amphiphile and the counterions at concentrations below cmc, Do, and the self-diffusion coefficient of the micelle, D,,,, which can all be obtained experimentally. An important difference between the diffusion model used here and the two-site model is that the diffusion of associated molecules on the aggregate ( 2 6 ) Jonsson, B.; Wennerstrorn, H.; Nilsson, P.-G.; Linse, P. Colloid Polym Sci 1986, 264, ll
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020
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Figure 9. Calculated DIDo values of acetate, propanoate, and butanoate versus the surfactant concentration. No was assumed to be 50 in all three
systems. The corresponding experimental DIDo values of acetate (A), propanoate ( O ) , and butanoate (0) are shown in the figure. BUI RCQ pc
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Figure 10. Calculated DIDo values of acetate, propanoate, and butanoate versus the surfactant concentration. No was assumed to be 60, 72, and 84 for the decylammonium acetate, propanoate, and butanoate micelles, respectively. The corresponding experimental DIDo values of acetate (A),propanoate ( O ) , and butanoate (0) are shown in the figure.
surface contributes to the effective self-diffusion coefficient in the former. The calculated self-diffusion coefficients relevant for the system investigated are presented in the following section. An extensive treatment of the theory and the calculation procedure is given in ref 27. In Figure 9 the calculated and experimental self-diffusion coefficients, represented as the ratio between the effective diffusion coefficient and the diffusion coefficient in the absence of micelle-counterion interactions, are compared a a function of the total surfactant concentration for decylammonium acetate, propionate, and butanoate. The aggregation number of the micelle is assumed to be 50 in all systems. Although the main features of the dependence of D/Do on surfactant concentration are reproduced by the model calculations, the calculated D/Do values are generally higher than the measured values. This is probably due to the fact that the aggregation numbers of the decylammonium micelles are larger than 50 (at least at surfactant concentrations above 0.15 M at which the fluorescence quenching experiments were performed). Increasing the aggregation number gives a slight increase in the micelle-counterion interaction, which gives lower DIDo values. If No = 60 and No = 84 are used for the decylammonium acetate and the decylammonium butanoate micelles and the intermediate value, No = 72, is used for the decylammonium propanoate micelles (with the corresponding Apco values), the agreement becomes very good, as seen in Figure 10. The D/Do values obtained for the decylammonium ion for the same systems, with No = 50, are shown in Figure 11. Only a minor dependence of D/Do on the aggregation number is found in this case. The agreement between the theoretical and the (27) Jonsson, B.; Jansson, M., to be published.
The Journal of Physical Chemistry, Vol. 93, No. 4,1989 1457
Influences of Counterion Hydrophobicity on Micelles EM Rop Ar
1°1
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Figure 11. Calculated D/Do values of the decylammonium ion in the decylammonium acetate, propanoate, and butanoate systems, respectively, versus the surfactant concentration. No was assumed to be 50 in all three systems. The corresponding experimental DIDo values of the decylammonium ion in the decylammonium acetate (A), propanoate ( O ) , and butanoate (0)systems are shown in the figure.
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Figure 12. Calculated D/Do values of the decylammonium ion and the pentanoate ion in the decylammonium pentanoate system versus the surfactant concentration. No was assumed to be 50. The corresponding and the penexperimental DIDo values of the decylammonium ion (0) tanoate ion (0)are shown in the figure.
experimental DIDo curves is quite good for the acetate and propanoate systems, whereas the calculated DIDo values are lower than the experimental values at higher surfactant concentrations for the butanoate system. The discrepancy between theory and experiments at higher surfactant concentrations becomes even
larger in the decylammonium pentanoate system, as shown in Figure 12. The underestimation in the model calculations of the self-diffusion coefficient for decylammonium ions may stem from two different sources: 1. The geometry of the aggregates is not spherical. This is of special importance for the decylammonium pentanoate system where elongated aggregates are formed. Due to the contribution of intraaggregate diffusion of monomers to the effective selfdiffusion coefficients, the theoretical value of D/Do for infinite prolates is 0.33, assuming that the magnitude of D is the same on the aggregate surface as in the water part.26 It can be anticipated that the contribution from intraaggregate diffusion is higher than what the spherical model predicts for the pentanoate system. Since the fraction of amphiphiles confined in the micelle is larger than the fraction of micellized counterions, this error source should be more important for the self-diffusion coefficient of decylammonium ions. 2. The micellar solution is polydisperse. It was shown in ref 26 that the effective self-diffusion coefficient is always larger in a polydisperse system as compared to a monodisperse one. This means that the D/Do value obtained from the model calculations is in fact a minimum value, which only applies to monodisperse systems. Since elongated aggregates are in general associated with a higher degree of polydispersity," this effect may be of importance, at least for the systems with the most hydrophobic counterions. 4. Summary
In the theoretical modeling of micellar solutions of amphiphiles with hydrophobic counterions, the micellizations of both amphiphiles and counterions have been taken into account. The hydrophobic free energies of the counterions, which were used within the model, were taken from experimental cmc values. The increment in ApCoper CH2was found to be considerably lower for the smallest counterions as compared to the more hydrophobic ones. The results of the model calculation showed that the fraction of counterion confined within the micelle is large, even for acetete. By calculating self-diffusion coefficients from the distribution functions of the counterion and amphiphiles in the cell and comparing it to experimental self-diffusion data, it was possible to the test the relevance of the model. In general, good agreements were obtained between the measured and the calculated concentration dependence of the self-diffusion coefficient for both counterions and amphiphiles. Registry No. DABut, 73702-94-0; DAAc, 2016-38-8; decylammonium propanoate, 39108-01-5; decylammonium pentanoate, 104875-22-1; decylammonium hexanoate, 110862-59-4.
