Entropy-Driven Micellar Aggregation - The Journal of Physical

8480

J. Phys. Chem. B 1998, 102, 8480-8491

Entropy-Driven Micellar Aggregation L. De Maeyer,† C. Trachimow,†,‡ and U. Kaatze*,‡ Max-Planck-Institut fu¨ r Biophysikalische Chemie, Am Fassberg, D-37077 Go¨ ttingen, Germany, and Drittes Physikalisches Institut, UniVersita¨ t Go¨ ttingen, Bu¨ rgerstrasse 42-44, D-37073 Go¨ ttingen, Germany ReceiVed: January 9, 1998; In Final Form: May 8, 1998

A theoretical model allowing us to calculate the size distribution of neutral and charged micellar systems from molecular parameters of the aggregating monomers is given. The derivation includes the range of oligomeric species between the monomer and the average-size micelle. An loss of entropy, resulting from the reduction in allowed orientations of solvent H-bonds in the neighborhood of nonhydrophilic parts of a solute molecule or aggregate, is introduced as the main factor influencing the size distribution of the aggregates. Equations describing the distribution in terms of known molecular quantities and only a few free parameters are given. Numerically evaluated examples are discussed. The predictions of the model are in very good agreement with the observed aggregation behavior for several classes of solute molecules in aqueous medium.

Introduction The distribution of aggregation numbers of micelles is peaked at a number corresponding to a globular aggregate. The nonaggregated monomers represent a second peak in equilibrium with the micelles. Aggregates of intermediate size are present only in much smaller concentrations. Distributions of this kind have been approximated by empirical equations, usually assuming a Gaussian distribution, but more often micelle formation is discussed in terms of an average-sized polymeric species in equilibrium with the monomer. Based on simplifying but arbitrary assumptions about the rate constants for the consecutive aggregation, analytical or numerical linearized perturbation kinetics have been used to interpret the observed relaxation behavior of such systems.1,2 Of the two relaxation processes observable by acoustic and temperature-jump relaxation experiments,3-7 the slower process (in the millisecond region) has usually been assigned to the formation/dissolution of a whole micelle; the faster one has been asssigned to the incorporation/ release of a monomer in an aggregate. Dynamic light-scattering experiments with dodecyl pyridinium iodide, however, reveal that very slow equilibration processes (taking several hours) are present in solutions where the total concentration exceeds the critical micelle concentration (cmc) only by a small amount.8 To study the detailed mechanism of micelle formation, computer simulations with the complete coupled nonlinear kinetic equations of the consecutive aggregation mechanism are needed. This requires information on both the forward and backward reaction rate constants for each aggregation step. While certain reasonable assumptions can be made for the forward process (e.g., diffusion-controlled kinetics), rate constants for the reverse process consistent with these assumptions must be obtained from a measured or theoretically derived equilibrium distribution of aggregate sizes, including the range of oligomeric species that plays an important role in the rate-determining processes. The small temperature dependence of the cmc has generally led to the conclusion that the main driving force for the aggregation is of entropic nature. Direct calorimetric measure† ‡

Max-Planck-Institut fu¨r Biophysikalische Chemie. Universita¨t Go¨ttingen.

ments9 indicate that enthalpic as well as entropic forces are at work, but for nonionic micelles the entropic term is dominant, and for nonionic as well as for ionic amphiphiles there is a temperature range where the enthalpic contribution vanishes. In the present paper, a simple thermodynamic theory for entropydriven molecular aggregation in aqueous medium is developed, aiming to predict a distribution from which a consistent set of forward and backward rate constants can be derived. In this theory, only the most relevant entropy contributions are calculated, and the total entropy is maximized. The first and important one driving the aggregation is the difference between the configurational entropy of water molecules in contact with the aliphatic chain of an amphiphilic monomer and those in bulk solvent. We attribute the loss of entropy in the hydration sheet mainly to the reduced orientation space for hydrogen bonding with neighboring water molecules. A second entropic term, also favoring the aggregation, is the gain in mixing entropy when aggregates of different aggregation number are formed. The gain in the number of distinguishable microstates, when particles of a new kind with translational degrees of freedom are formed by the aggregation, is the origin of the increase in mixing entropy. The translational entropy also contains a part opposing the aggregation. It represents the loss of entropy when several monomers in an aggregated complex loose their individual freedom of positional localization and translational momentum. It may be assumed that the entropy associated with the internal degrees of freedom of a surfactant molecule is not significantly affected by the aggregation. An aggregate, however, apart from replacing the individual translational degrees of freedom of its constituents by their joint translation also gains some internal degrees of freedom associated with relative motions of the molecules in the micelle. The outline of the paper is as follows: In the next section, we shall apply this approach to the case of an idealized uncharged surfactant, which is supposed to consist of a saturated aliphatic alkyl chain connected to an hydrophilic headgroup. The latter is assumed to fit very well in the hydrogen-bond structure of the solvent, so that aggregation induces no entropic changes in the hydration of the headgroup, which remains exposed to the water when the aliphatic chains aggregate. In Aggregation of Ionic Surf-

10.1021/jp9807367 CCC: $15.00 © 1998 American Chemical Society Published on Web 10/03/1998

Entropy-Driven Micellar Aggregation

J. Phys. Chem. B, Vol. 102, No. 43, 1998 8481

actants section, the model will be enlarged to include ionic dissociation of the surfactant molecule and charged micelles. Examples of the complete equilibrium distribution of micellar size and chage are also presented in this section. In the Results section, the results obtained are compared with experimental data and discussed. Since most experimental measurements result in data on average size or charge10 or thermodynamic data related to these averages, the distributions obtained in the previous sections must be appropriately averaged to allow comparisons. In the final section, the main differences between our approach and other current theories of micellar aggregation are presented and possible applications of our model to other forms of molecular aggregation also driven by hydrophobic forces are pointed out. An appendix describes the method used for determining the entropy loss caused by the hydrophobic chain of the monomer. This work has led to a theoretical derivation of the complete size distribution of hydrophobic aggregates, based on a simple model of solute-solvent interaction, that allows for a complete numerical simulation of the kinetics of micelle formation,11 confirming the presence of a third slow final relaxation mode corresponding to a change in the total numer of non-monomeric (micellar)entities accompanied by the equilibration of their final distribution. Aggregation of Uncharged Surfactants The following relevant entropies are taken into account in this model: Entropy of Hydration of a Hydrophobic Hydrocarbon Chain. When a water molecule engages in hydrogen bonding to its neighboring molecules, its contribution to the configurational entropy of the system decreases because the orientations of the mutually bound molecules are correlated. The entropy of a water molecule engaged in the hydrogen-bond network of the bulk water structure differs therefore by an amount sbulk from the entropy of a free water molecule whose orientation is not correlated with that of its neighbors. The immersion of a hydrocarbon chain in water restricts the range of orientations in which an adjacent water molecule can engage in hydrogenbonding interactions with its other neighbors. In bulk water, hydrogen bonds from a water molecule can extend in any direction over the whole space angle from 0 to 4π with a constant probability distribution. For water molecules neighboring the hydrocarbon chain, hydrogen bonds directed within a space angle γ are excluded. For a water molecule adjacent to a chain, this leads to a further entropy reduction:

[4π4π- γ]

δsadj ) -kB ln

(1)

under the condition that the molecule remains part of the H-bond network. Let p be the probability that the molecule remains involved in hydrogen bonding. The probability that it does not is (1 - p). In the latter case, it gains back the entropy that was lost by the orientational correlation of its connected neighbors. The difference in entropy between a water molecule in the hydrocarbon solvation layer and one in the bulk phase is then

[4π4π- γ] + (1 - p)(-s

s ) -kBp ln

bulk)

(2)

The corresponding enthalpy change is

h ) pδhadj + (1 - p)(-hbulk)

(3)

where hbulk is the average enthalpy of the incorporation in the

H-bond network, and δhadj is the difference between the incorporation of a water molecule from the solvation layer adjacent to the hydrocarbon chain and one from the bulk phase. This difference can be rather small since any decrease in the maximum number of simultaneous H-bonds in which the molecule may be involved can easily be compensated by a change in the relative probabilities of the states in which only a smaller number of bonds is present. This is possible because consecutive H-bonds to donor or acceptor sites on the same molecule influence each other and have decreasing enthalpies. For enthalpy changes to be small, either the two terms in eq 3 must compensate each other or p must nearly equal 1 and δhadj nearly 0. In the former case, since hbulk is negative, δhadj must also be negative for compensation to be possible, i.e., the H-bonds of the solvating water molecules must be stronger than those of bulk water molecules. The entropy change as well as the enthalpy change depends in the same formal way on p. The temperature dependence p(T) of this quantity may be responsible for the observed correlated changes of the hydrophobic entropy and enthalpy with temperature.9 For the present, however, we shall consider the temperature range where the enthalpic contribution vanishes (δhadj nearly zero) and assume that p is nearly 1. The total entropy reduction induced by a dissolved hydrophobic molecule or aggregate depends on the number of water molecules that are affected. This number is a function of the length of the hydrocarbon chain and of the number of chains in the aggregate. The calculation of the entropy of chain exposure to the solvent (which will be called the hydrophobic entropy of solution) for the monomer and for micelles with a different aggregation number is given in Appendix A. For a neutral surfactant; the hydrophobic entropy per particle with aggregation number i is written as

si ) f (i)s1

(4)

where s1 is the hydrophobic entropy of solution of a monomer. s1 contains the dependence on the chain length. The factor f (i) represents the effective number of monomeric chains exposed to water when the size i of the aggregate increases. While f (1) ) 1, initially f (i) increases less than linear with increasing i, since aggregating chains protect each other from full contact with water. f (i) reaches a maximum for a certain value of i and then decreases back to zero when the micelle reaches a spherical shape and all hydrocarbon chains are hidden. Only the hydrophilic headgroups are then exposed to the water, but according to our assumptions, their interaction with the solvent is not significantly changed by the aggregation. For small values of i, the factors f (2), f (3), f (4), ... must represent some averages over the different possible geometric arrangements of the monomers in dimeric, trimeric, tetrameric, ... assemblies. We approximate the factor f (i) by a function:

f (i) )

i

(i -b a)

for i g 2

(5)

1 + exp

The parameters a and b can be estimated by fitting this function to the calculated exposed surface areas of the dimeric structure, represented by touching cylinders containing the hydrocarbon chains, and of other small polymeric structures as suggested, for instance, by molecular dynamics calculations.12-14 We found that the values a ) 10 and b ) 7 give good lower limit approximations for the estimated hydrophobic entropies of such assemblies. f (i) then reaches its maximum value for i ) 12.

8482 J. Phys. Chem. B, Vol. 102, No. 43, 1998

De Maeyer et al.

Our choice of lower limit approximations is related to the increasing computation time required for calculating distributions including aggregates with values of i exceeding 100. The function f (i) can also be adapted to experimental data on the structure of micellar shapes with residual hydrophobic contacts15 or to other forms of hydrophobic aggregation (cf. the Conclusions). Entropy Associated with Translational Degrees of Freedom. The solute monomers and aggregates are free to move over the entire volume of the system. If we treat them as pointlike masses moving like an ideal gas without coupling to the solvent, the translational entropy of Ni molecules of a component in a separate volume V is given by the SackurTetrode equation:16

Sti ) kBNi

[( (( ln

) ) )

2πim1kBT h2

3/2

]

V + 3/2 - (ln Ni - 1)

(6)

In this equation kB is Boltzmann’s constant, T is the temperature, h is Planck’s constant, m1 is the mass of the monomeric molecule, and i is the number of monomers in the aggregate particle. The first term in the square brackets represents the entropy of the probability distribution over the accessible translational quantum states of a particle of mass im1 in V, if its average translational energy is equal to 3kBT/2. The last term arises because the particles of the same kind, sharing the same volume V, are indistinguishable if one averages over their other nontranslational energy states. The total translational entropy of the different components in equally large, separate volumes V is the sum of these entropies. If Ni is the number of the particles of different species i present at equilibrium, there is no change in the total entropy when the separate volumes are combined to a single common volume V, since this does not imply a change in the uncertainty about their spatial confinement. (The situation is different if the Ni particles are not confined to identical volumes V but to individual volumes Vi of Ni particles, corresponding to standard states of fixed particle densities. Their mixing then results in volume changes. The corresponding change in translational entropies is commonly known as mixing entropy). Using the ideal gas model to characterize the translational entropy of a dissolved molecule or aggregate in a dense medium may be considered a too strong approximation. It is used here, nevertheless, since we intend to include only the contributions that are absolutely necessary. It is a reasonably good approximation if the densities of the solute particles are sufficiently low, so that their mutual couplings and interactions can be neglected. These can be taken into account explicitly by additional terms, or by introducing new, chemically distinguishable species describing the result of these interactions. (This will be the case in the next section where the aggregation of ionic species will be discussed). The essential neglection is the coupling to the solvent. Instead of moving in a large volume V with reflecting walls, the particle is moving in a small potential well made up by surrounding solvent molecules. The average energy of the center of mass of the particle remains, however, equal to 3kBT/2, and its position is distributed with equal probability over the total volume V, which is the available volume for mixing, i.e., the space not filled with identical particles. Although the motion in positionmomentum phase space is different, the volume occupied in phase space will be nearly the same and therefore also the translational-positional entropy. For molecules of a pure liquid for the solvent or for solutes at high concentration, on the other hand, a correction factor for the free volume is applicable.

Excluded volume effects must be taken into account for the solute monomers and aggregates only when their concentration becomes large. Entropy of Internal Degrees of Freedom of Aggregates. The translational entropy in the previous paragraph is due to the motion of only the center of mass of the chemical species considered, i.e., the monomer respectively to the aggregates of different size. We assume that the internal degrees of freedom due to motions (rotations, vibrations) of atoms and atom groups belonging to a monomeric molecule are not affected by the aggregation. The aggregated assemblies, however, gain some additional internal degrees of freedom resulting from the motions of their composing monomers relative to the common center of mass. Within the aggregate, the mutual positions and neighboring relations between the monomers are not completely fixed, but at finite temperature, their mobility contributes to the heat capacity of the aggregate. The calculation of the entropy corresponding to these internal degrees of freedom would be very difficult if this had to be done on the basis of configurations and interaction potentials determining the accessible quantum states of an aggregate. Instead, we will use a much simpler approach for estimating the magnitude as well as the size dependence of this contribution to the entropy. An aggregate composed of i monomers can be compared to a small solid crystal composed of i atoms. According to the Dulong-Petit rule, which is valid at sufficiently high temperatures, each atom contributes 3kB to the specific heat at constant volume. This corresponds to its contributing 3 degrees of freedom for vibrational modes of phonon motion in the crystal. At very low temperatures, the rule breaks down because the vibrational modes cease to be excited when their energy becomes larger than kBT. The heat capacity, however, rises steeply within a narrow temperature range from nearly zero to its constant high temperature value. According to this model, a reasonable expression for the internal entropy of an assembly of i monomers is then

si )

c (T)

∫0T VT

dT ) 3kB(i - 1) ln

(θT)

(7)

In evaluating the integral, the s-shaped function cV(T) has been replaced by a rectangular step function that is equal to zero below a threshold temperature θ and equal to 3kB(i - 1) above the threshold. The factor (i - 1) arises because only monomers beyond the first can contribute to the internal degrees of freedom of an aggregate. The treshold temperature θ remains a free parameter that has to be empirically determined. Guided by the cV(T) behavior of atomic solids with atomic weights approaching the molecular weight of micelle-forming surfactants, we have used the value θ ) 10 K in all numerically calculated examples. The sum of all entropy terms influencing the aggregation is now written as

Stot )

[

( [(

∑i Ni f (i)s1 + kB ln

) ]

2πim1kBT h2

3/2

) ( )]

5 V + - Ni + 2

3kB(i - 1) ln

T

θ

(8)

The particle numbers Ni in this equation represent the equilibrium distribution when the entropy Stot is maximized. Using the Lagrange variational procedure for dSi/dNi ) 0 and ∑iNi ) N0, the equilibrium concentrations for each species can be



Figure 1. Distribution of the aggregate size ci of nonionic micelles at different concentrations, calculated from eq 9 with T ) 298 K, θ ) 10 K, V ) 1 m3, and f (i) as defined in eq 5. With these values, the concentration ci is given by ni/1000. The displayed curves 1-9 correspond with total concentrations of 0.1, 0.3, 1, 2, 4, 6, 7, 8.09 ()cmc), and 30 mM.

Figure 2. Monomer concentration c1 of nonionic micelles as a function of the total concentration c0. The underlaying distributions are calculated as described in Figure 1. The dotted line is the function c1 ) c0. The arrows show the concentration, for which the distributions are displayed in Figure 1.

derived from this equation in the form

ln (ni) ) i ln (n1) + (f (i) - i) (i - 1) ln

[(

) ]

2πm1kBT 2

h

s1 kB

3/2

V 3 - (i - 1 - ln i) + NA 2 3(i - 1) ln

()

T (9) θ

Here ni ) Ni/NA is the number of moles present in volume V. NA denotes Avogadro’s number. The expression is, of course, equivalent to the law of mass action. It has the advantage that the equilibrium constants are now explicitly given in terms of the relevant entropic contributions. (According to our original assumptions, all enthalpic factors have been neglected). Numerical evaluations for a desired N0 ) ∑iiNi have been obtained for different values of s1, corresponding to different hydrocarbon chain length. An example of the calculated distributions of aggregate concentrations as a function of total monomer concentration is given in Figure 1. The dip in the distribution for concentrations of small aggregates is very deep. At the minimum, the concentration is many orders of magnitude smaller than that of the monomer or that of the dominant aggregate. The distribution is skewed to larger aggregates as compared to a Gaussian distribution. The equilibrium concentration of the monomeric species as a function of total concentration is given in Figure 2. This figure shows very clearly the appearance of the critical micelle concentration (cmc), above which almost all added monomers to the system are incorporated in aggregates. Strictly, there is a concentration range for the transition. Figure 3 shows that the average size of the aggregates, defined as

∑iNi ∑Ni

for i > imin

(10)

does not change very much once the cmc is reached. The halfwidth of the distribution in Figure 1 is not large (the ordinate axis is logarithmic). It is also appropriate, for comparison with some experimentally observable quantities (e.g., light scattering, where the change in refractive index is proportional to the size

Figure 3. Average aggregate size 〈i〉 of nonionic micelles, defined in eq 10 as a function of the total concentration c0. The underlaying distributions are calculated as described in Figure 1.

of the particles), to consider the distribution of the numbers of monomers present in aggregates of different size. The percentages of surfactant molecules condensed in micelles of increasing size are given in Figure 4 for concentrations equal to the cmc and also 3.7 times the cmc. The half-width of this distribution is about one-fifth of the average aggregation number. Aggregation of Ionic Surfactants In the aggregation of ionic surfactants, enthalpic terms arising from interactions of electrostatic origin can no longer be neglected. Besides the increased number of chemically distinguishable species (dependent on their electrical charge), we will have to introduce three additional interactions. Ion Dissociation Equilibrium of Monomers. Ionic dissociation of strong electrolytes in a polar medium is driven mainly by the stronger interaction of the polar solvent molecules with separated ions as compared to that with the dipole of a contact ion pair. The enthalpy decrease from binding solvent molecules to the separated ions almost balances or sometimes exceeds the enthalpy needed to separate the oppositely charged ions in a medium with high dielectric susceptibility. In weak electrolytes additional covalent forces must be overcome. The


De Maeyer et al.

Figure 4. Relative fraction of nonionic amphiphiles ici/c0 present in micelles as a function of the aggregate size i. The full line is calculated at cmc ) 8.09 mM; the dotted line is calculated at 3.7 cmc. The fraction values at i ) 1 are 0.98 and 0.29.

sign of the heat of solution of the electrolyte may serve as a rough indication that enthalpy is larger. The entropic part of the free enthalpy decrease upon dissociation includes the gain in translational (or mixing) entropy from the creation of independently moving ionic entities, but it also includes an entropy loss from restricting some degrees of freedom of the ion-solvating solvent molecules. The first, translational, entropy part will be treated as an explicit entropic term in the same way as for neutral molecules since for the surfactant ion it also plays an important role in the aggregation equilibrium. The second, ionic hydration related part will be expressed together with the ionic separation part by including a hydrationseparation free enthalpy term µsep 0,1 . This term then represents the free enthalpy difference between, on one hand, a localized (fixed in their positions), hydrated, dissociated surfactant ion together with its localized counterion at infinite distance and, on the other hand, the localized, hydrated, contact ion pair or undissociated surfactant molecule. It will also be assumed that the hydration-separation free enthalpy µsep 0,1 has the same value for a single surfactant molecule as for one incorporated in an aggregate. We shall not try to calculate µsep 0,1 (which contains an energetic as well as an entropic term, partially compensating each other) from some a priori assumptions about the size of the ions and the electrostatic forces involved. It remains a free parameter of the model; the dependence of the aggregation equilibrium on this parameter can be evaluated from the numerical calculations. The total separation-hydration free enthalpy related to ionic dissociation of the system is then

Gsep ) N0,1µsep 0,1

(11)

N0,1 is the total number of free counterions present in the system. It should be noted that µsep 0,1 is not the standard free enthalpy of dissociation of the dissolved neutral amphiphile. To obtain this quantity, one must still add and respectively subtract, the contribution from the translational entropies of the counterion and the ionized (respectively the un-ionized) molecule at their selected standard state concentrations. In the following derivations and in later numerical evaluations, we will assume that the neutral surfactant molecule can dissociate in a monovalent surfactant ion and a monovalent counterion. Repulsive Interaction in Charged Aggregates. Building up the surface charge of an aggregate at infinite dilution can be considered as a process in which charges are brought in sequence

from infinity to an initially uncharged aggregate. We shall designate the aggregation number of an ionic micellar species by a first index and the number of counterions bound on its surface by a second index. The symbol Ni,j then represents the number of aggregates (i,j) in the system that are assembled from i monomeric species, of which i - j are dissociated. N1,1 and N1,0 represent the numbers of undissociated and dissociated monomeric molecules, respectively; N0,1 is the total number of free counterions in the system. An entity characterized as (i,i) is a neutral species. The dissociation processes (1,1) f (1,0) + (0,1) or (i,i) f (i,i - 1) + (0,1) are taken care of by the introduction of the term µsep 0,1 as explained in the previous paragraph. The process (i,j) f (i,j - 1) + (0,1) is equivalent to two composing processes (i,j) + (1,0) f (i,j - 1) + (1,1) and (1,1) f (1,0) + (0,1). On the right-hand side of the first process, a neutral particle separates from a charged one; this does not involve electrostatic work. On the left-hand side, however, two charged particles (i,j) and (1,0) carrying charges of equal sign must be brought together. It is this repulsive energy that still must be accounted for. It depends on the charge (i - j)e0 already present on the aggregate. The work wi,j done to create the total charge of an aggregate is the sum of a sequence w1, w2, ..., wi-j, where these terms represent the work done to bring the first, the second, ..., the last charge to the aggregate. w1 is equal to zero, since no work must be done to bring an elementary charge e0 from infinity to a point of zero potential (the position of the neutral particle (i,i)). It follows that the total repulsive work for charged, spherical, aggregate (i,j) is given by k)i-j

wrep i,j

)

k)i-j(k

- 1)e20

∑ wk ) k)1 ∑ 4π a k)1

0 i,j

)

e20

(i - j)(i - j - 1)

4π0ai,j

2 (12)

This applies at infinite dilution. e0 represents the elementary charge, 0 is the vacuum permittivity, is the effective relative permittivity of the solvent, and ai,j is the radius of the aggregate. Although the shape of a micellar aggregate may very well deviate from a perfect sphere, it would be unreasonable to try to account for such deviations in our simple model. The value of ai,j should be comparable to the length of the hydrocarbon chain of the surfactant molecule for full-size aggregates but could be significantly smaller for initial multimers. Nonspecific Ionic Interactions at Finite Concentrations. Apart from bound counterions (contact ion pairs on the surface of the micelle), a screening atmosphere of counterions is formed around the charged micelles. Additional ionic species, often present in micellar solutions, may contribute to the screening. Under the Debye-Hu¨ckel approximations,24 the electrical potential at the surface of a spherical ion of radius ai,j and charge q can be written as a sum of two terms:

φ(a) )

qκ q 4π0ai,j 4π0

(13)

The first term comes from the surface charge on the ion; the work done to create this charge has been considered in the two previous paragraphs of this section. The second term comes from the distribution of other ions in the vicinity. It is proportional to the screening parameter κ, which is calculated from


e20

κ ) 2


∑e

0kBT

(Nez2e /V)

(14)

Ne/V is the volume-averaged concentration of an electrically charged species e carrying ze elementary charges. The work done when the ion distribution is formed may be calculated using the gradual charging process. In the screening term both q and κ contain the elementary charge e0 as a factor. The charge is gradually built up from 0 to ηe0, with η increasing from 0 to 1. With q′ ) (i - j)ηe0, dq′ ) (i - j)e0 dη, κ′ ) ηκ, the work done to charge the counterion distribution screening the central ion is

wscr i,j )

q′κ′ dq′ ) ∫ - 4π 0

(i 4π0

j)2e20κ

∫01η2 dη )

This work represents only the interionic interaction energy; it does not account for volume change nor for a small entropy decrease resulting from the nonuniform relative spatial distribution of the ions. We neglect these contributions as well as other corrections to the Debye-Hu¨ckel treatment needed when the approximation aκ , 1 is not fulfilled at high ionic concentrations. The expression for the total free enthalpy G of the system is formed by summing all work terms of this section and adding, after multiplication with -T, the entropic terms given in the previous section. The equilibrium distribution of the concentrations Ni,j of the different species corresponds to the minimum of G and is obtained by Lagrange variation of this expression with respect to the Ni,j under two constraints: (i) for a fixed total number of monomers: i

∑ ∑iNi,j ) N0 i)1 j)0

(16)

and (ii) the system is electrically neutral: ∞

i

∑ ∑(i - j)Ni,j ) N0,1 i)1 j)0

(17)

The Lagrange multipliers are evaluated in terms of the species N0,1 and N1,0 and are eliminated. A set of equations for each Ni,j for i g 1 and 1 e j e i in terms of N0,1 and N1,0 is obtained, which can be brought in the form:

s11 ln ni,j - i ln n1,0 - j ln n0,1 ) (f (i) - i) + kB 3/2 2π(im1,0 + jm0,1)kBT V ln 2 N h A 2πm1,0kBT 3/2 V 2πm0,1kBT 3/2 V j ln i ln NA NA h2 h2

[(

[(

) ] [(

) ]

) ]

µsep 3 0,1 T (i + j -1) + 3(i - 1) ln + j 2 θ kBT

[

χk,l )

Nk,l

1 2

∞

(N0,1 +

(19)

k

∑∑Nk,l(k - l)

2

k)1 l)0

(i - j)2e20κ (15) 12π0

∞

One recognizes the mass action law in this expression. The ni,j ) Ni,j/NA are the numbers of moles of species (i,j) present at equilibrium in volume V of the system. (The volume V must be expressed in units consistent with those selected for kB, h, .... If SI units are selected for all dimensions, then ni,j/1000 represents concentrations ci,j (in mol L-1). The quantities χi,j, χ1,0, and χ0,1 in the κ-dependent terms are short-hand notations for a nonlinear concentration dependence of the screening by counterions. They are given by

e20 (i - j)(i - j - 1) (i - j)2κ (1 + χi,j) + 4π0kBT 2ai,j 3

]

iκ jκ (1 + χ1,0) + (1 +χ0,1) (18) 3 3

+

∑e

Nez2e )

Ne and ze are particle numbers and valencies of extraneous electrically charged species added to the system. χ1,0 and χ0,1 can attain a maximum value of 1/4 if foreign ions are not present and aggregation is not taking place. Equation 18, representing the distribution of the aggregation number and the charge of ionic micelles, cannot be solved explicitely for ni,j because the variable n0,1, constrained by eq 17, contains a sum of all ni,j. Solutions can be obtained by constrained optimization algorithms. Treating n0,1 as a free variable, eq 18 can be solved explicitely for ni,j. The distribution then depends on an initial selection of n0,1 and n1,0 and on the set of thermodynamic or molecular parameters m0,1, m1,0, T, θ, µsep 0,1 , s11, ai,j, κ, and V. Calculating the distribution for a given pair of n0,1 and n1,0, the value obtained for ∑i∑j(i - j)ni,j of this distribution can be compared with the value n0,1 used for the calculation. The deviation of both values is defined as i

∆(n1,0, n0,1, ...) ) n0,1 -

∑ ∑(i - j)ni,j i)1 j)0

(20)

To get a correct solution that is consistent in n0,1, the root of ∆(n1,0, ...) is determined as a function of n1,0. Thus, for the initially selected value of n0,1, a corrected value n1,0 is found and used for the next iteration. The procedure is iterated until convergence and can be repeated also with different values of n0,1 until ∑i∑jni,j converges to the desired total concentration n0. Results Calculations for the ionic model have been carried out using the function f (i) as defined in Entropy of Hydration of a Hydrophobic Hydrocarbon Chain and with the following molecular parameter values: m0,1 ) 35.5 × 1.66 × 10-27 kg, m1,0 ) 248.4 × 1.66 × 10-27 kg, s11 ) -2.023 × 10-22 J K-1, -1 ) a + T ) 298 K, θ ) 10 K, µsep i,j 0,1 ) 12kBT, a1,0 ) 5 Å, κ 20 Å. The molecular masses are those for dodecylpyridinium chloride; s11 is the hydrophobic solvation entropy calculated for an aliphatic dodecyl chain. The values of θ and µsep 0,1 are the free parameters of the model. θ influences the predicted cmc, whereas µ01 influences the average degree of ionic dissociation of multimeric aggregates. With the chosen values, the predictions for the cmc and for the degree of dissociation correspond to common experimental observations. The values selected are within the expected range for these parameters. The value for the radii a1,0 and ai,j, with

ai,j ) a1,0i1/3

(21)

have been chosen to give spherical volumes that nearly


De Maeyer et al.

Figure 5. Distribution of size of charge ci,j of ionic micelles, calculated from eq 18 with values defined in the Results. i is the aggregation number, and j is the number of counterions. The selected total concentration is the cmc ) 25.3 mM. Figure 7. Average aggregate size 〈i〉 of ionic micelles defined in eq 22 as a function of the total concentration c0, calculated from eq 18 with parameter values defined in the Results.

Figure 6. Monomer concentration c1,0 + c1,1 of ionic micelles as a function of the total concentration c0, calculated from eq 18 with parameter values defined in the Results.

correspond to the sum of the volumes of CH2 groups in the aggregate. The dependence of κ-1 on the size of the aggregate has been introduced to avoid overcompensation of the intramicellar repulsive interactions by the counterion screening. Rather, the screening is assumed to occur in a layer of constant thickness around the charged surface. The ions in this layer are assumed to be contributed by the presence of an additional salt electrolyte (about 30 mM) in the system. The distributions of size and charge of ionic micelles, shown in Figure 5, have been calculated for a total surfactant concentration corresponding to the cmc. Figure 6 shows the concentrations of neutral and ionized monomers as a function of total surfactant concentration. We define the cmc by the point of maximum curvature of the curve on Figure 6. Above the cmc, the monomer concentration decreases slightly with increasing total surfactant concentration. This is due to the increasing counterion concentration, pushing the dissociation equilibrium of the monomer toward the neutral form, which is more easily incorporated in a charged aggregate. This behavior is different from that predicted for nondissociating surfactants, shown in Figure 2. In both cases, the predictions by the model are supported by the experimental observations.25-27 For ionic micelles, the average aggregation number is defined by

Figure 8. Average degree of ionic dissociation β defined in eq 23 as a function of the ion separation-hydration free enthalpy µsep 0,1 , calculated from eq 18 with parameter values defined in the Results.

In this definition, only multimers with i > 10 are counted as aggregates because averaging over a bimodal distribution would not be physically meaningful. The average aggregation numbers as a function of total surfactant concentration for a dissociating surfactant are shown in Figure 7. The predicted average aggregation numbers are smaller than those derived from experiments. This is a consequence of our selection of the parameters a and b in the function f (i). This function determines the number i beyond which the hydrophobic entropy of aggregates approaches zero. We have deliberately chosen the parameters such that the numerical calculations do not need to be extended to values of i above 100, since otherwise very long computation times would result. The average degree of ionic dissociation of the aggregates β, defined as i

β)

∑∑ i>10 j)0

(i - j)Ni,j/i i

(23)

∑ ∑Ni,j

i>10 j)0 i

〈i〉 )

∑∑ i>10 j)0

iNi,j i

∑ ∑Ni,j

i>10 j)0

(22)

is largely determined by the ion separation-hydration free enthalpy µsep 0,1 . This is shown in Figure 8. One should remember that µsep 0,1 is not the complete standard free enthalpy of the dissociation of the dissolved ion pair state of the monomer



Figure 9. Free enthalpy (µsep 0,1 ) dependence of the cmc. The cmc is determined by a series of distributions calculated from eq 18 with parameter values defined in the Results.

since the entropic contributions from the creation of additional translational degrees of freedom by the dissociation are not included. The difference in the translational entropies ∆S°trans between the dissociated ions and the contact ion pair at 1 M standard states is obtained from eq 6. The standard free enthalpy of ionic dissociation

∆G°diss ) NAµsep 0,1 - T∆S° trans

(24)

then amounts to - 7.5 kJ mol-1 if one uses the parameters given at the beginning of this section. The experimentally observed values of β require positive values of the order of several kBT sep for µsep 0,1 . The dependence of β on µ0,1 may be verified experimentally by comparing the degrees of dissociation of an ionic surfactant with different counterions.28 In the case of dodecyl sulfates, Li+ counterions with stronger solvent interactions as compared to Na+ are less able to induce micellar clustering at high counterion concentrations.29 Below a threshold value, which is about 6 kBT (or ∆G°diss ) - 22.5 kJ mol-1) for our other parameter values, the aggregates would remain almost completely dissociated. µsep 0,1 also influences the cmc, as shown in Figure 9. The cmc falls rapidly with initial decreasing degree of dissociation until it becomes again independent when the aggregates are more than half uncharged. This behavior reflects the easier incorporation of an uncharged monomer into an aggregate. Even then, the cmc remains larger than that of nonionic surfactants because the incorporation of the fraction (1 - β) of uncharged monomers requires their ionic recombination, raising the standard free enthalpy of micelle formation of an ionic surfactant with respect to that of a nonionic surfactant with the same hydrocarbon chain by -(1 - β)∆G°diss. The difference in the cmc for partially charged vs nonionic micelles is shown in Figure 10, which indicates the almost linear dependence of the logarithm of the cmc on the number of CH2 segments in the hydrocarbon chain. The predicted slopes of δ log (cmc) per CH2 of -0.41 for nonionic and of -0.38 for ionic surfactants are within the range of experimentally observed values.30 The ratio of the ionic to the nonionic cmc remains almost constant, as required by the approximate relation RT ln(cmcionic/ cmcnonionic) ) -(1 - β)∆G°diss. Experimental values for the entropy and enthalpy changes per monomer incorporated in a micelle have been determined for the system on which our selection of the model parameters

Figure 10. Chain-length dependence of the cmc for charged and uncharged micelles. The cmc is determined by a series of distributions calculated from eq 18 with parameter values defined in the Results. The slope of log cmc vs chain-length curve is -0.41 for uncharged and -0.38 for charged micelles.

is based.31 The measured values are average quantities of the size distribution. They are valid for the reaction of 1 mol of ionized monomers with (1 - β) mol of counterions to form 1 mol of a distribution of micelles with average charge 〈i〉βe0 and average aggregation number 〈i〉, and they correspond to standard states with molar fractions of unity. They must therefore be compared to calculated averages from i

∑ ∑S°i,jni,j

〈∆S°〉 )

i>10 j)0

- S°1,0 - (1 - β)S°0,1

i

i

(25)

∑ ∑ni,j

i>10 j)0

where S°i,j is defined as

[

( [(

S°i,j ) NA f (i)s11 + kB ln

) ] ]

2π(im1,0 + jm0,1)kBT

)

h2

3/2

V° + NA

T 5 + 3kB(i - 1) ln (26) 2 θ V° is the molar volume of the selected standard state, i.e., 1 L if the standard state is the ideal 1 M solution or 1/55.5 L if the standard state is of unity mole fraction. For S°0,1 the last term must be omitted. (Instead, a term ssep 0,1 should be added, representing the entropy contribution from ion-solvent interactions contained in µsep 0,1 .) Without this addition, we obtain a calculated average molar aggregation entropy of 70.3 J K-1 mol-1 of incorporated monomer for dodecylpyridinium chloride, resulting from our model for the origin of the hydrophobic entropy. This is about 65% of the experimental value.31 The difference is partly due to the neglection of the entropic part of ionic solvation and is also a result of our selection of parameters a and b in the function f (i) used to approximate the hydrophobic entropy of oligomeric aggregates as a function of their size. Our need to select these parameters so that the numerical calculations did not have to be extended beyond i ) 100 reduces the average size and the average molar aggregation entropy. In view of the rather restrictive simplifying assumptions on which the theory is based, the correpondence between experimental and predicted values is very satisfactory.

8488 J. Phys. Chem. B, Vol. 102, No. 43, 1998 Conclusions We have shown that a single driving force, caused by the tendency of water molecules to maximize the entropy of their spational distribution of hydrogen bonds, is able to explain the aggregation of dissolved amphiphilic molecules into micellar structures above a critical concentration of the solute. This entropic driving force is still rather small for small oligomeric aggregates, such that it cannot compete with the enormous gain in translational entropy when these oligomers dissociate. In larger aggregates, the hydrocarbon part of the monomers becomes more and more shielded by the combined polar headgroups. Most thermodynamic theories on micelle formation start from assumptions about the size dependence of the standard chemical potentials of micellar aggregates. The classical approach17,21 distinguishes two classes of contributions. The first one stems from bulk effects in the interior of the micelle; the second one is derived from surface effects. The bulk interior core is usually considered to be liquidlike, its properties being obtained from corresponding hydrocarbon liquids. There are several kinds of surface contributions, some attractive and others repulsive. Surface tension at the interface between the hydrocarbon core and the surrounding water solvent yields the attractive, hydrophobic part. Repulsive terms depend on charge, shape, size, and packing constraints for the hydrophilic headgroups. Recent theoretical work18,19 also considers the influence of interaggregate interaction, due to excluded volume effects, on the distribution of the different forms and shapes of the micellar aggregates. Although these theories are able to account in some detail for the narrow size distribution of spherical micelles and for the relative stabilities of the wider distributions of spherocylindrical and other micellar shapes, they are not very appropriate to describe the distribution of the unstable oligomeric species that are, however, most important for the kinetics of aggregation equilibria2 and for the formation of micelles in the concentration range around the cmc. For oligomers, a separation in volume and surface terms and the use of macroscopic quantities, like surface tension or bulk free energy of the hydrocarbon core, are hardly applicable. The new concept, introduced in our approach, is that of a hydrophobic entropy, which is obtained quantitatively from an estimation of the number of water molecules in the solution, whose phase space of orientation of hydrogen bonds is restricted. The present treatment differs in several respects from the statistical mechanical approach by Nagarajan and Ruckenstein,20 also aiming at the calculation of the size distribution of micellar aggregates. To account in detail for the individual contributions to the various interactions, their approach requires a much larger number of estimated parameters. The two approaches are quite different in the way they treat the hydrocarbon-solvent interaction. Whereas here all changes in interactions other than hydrogen-bond orientations are considered to be of minor importance, Nagarajan and Ruckenstein attempt to account explicitly for changes in intra- and intermolecular van der Waals interactions, for configurational changes in the hydrocarbon chain, for changes in the structure of water in the neighborhood of hydrocarbon chains of single amphiphiles, and for residual aqueous hydrocarbon interfacial interactions of the aggregates. Terms for changes in translational and rotational degrees of freedom and a term for remaining but restricted translational degrees of freedom inside the micelle are also included. All this introduces several parameters that have to be estimated from empirical data, whereas in our treatment these influences are assumed to be small and more or less compensating each other.

De Maeyer et al. Their joint effect might be taken care of by adjusting the single parameter constant θ, which, of course, should remain within its physically reasonable limits. The empirical free energy parameters used in their expressions do not explicitly reveal their temperature dependence and therefore hide their mainly entropic nature. Nagarajan and Ruckenstein include the influence of ionic strength on the additional repulsive forces of charged headgroups with known average charge, but they do not calculate the two-dimensional combined distribution of charge and size from which the average charge can be derived as was done in the previous sections. Simulation of the process of self-association of amphiphilic chain molecules has been used in recent years22,23 as another approach for studying the distribution of aggregate sizes. These calculations are based on finding minimum energy configurations for an ensemble of molecules with nonbonding intermolecular interactions between their chain segments. In principle, such models can deal with oligomeric structures, but in practice, only if the interaction energies are weak and the occupation densities of the lattice model are large. Otherwise the limited number of molecules in an ensemble with reasonable computation times prohibits a reliable estimate of their contribution. The assignment of interaction energies to the different categories of next-neighbor configurations is a difficult issue. According to our model, the presence of an intermolecular chain-segment interaction corresponds to the release of a certain average number of water molecules from the solvation layer to the bulk water phase. Since a lattice model22 can take into account the different individual internal conformations of the interacting molecules, it could be used to calculate the function f (i) by similar simulation techniques. Very large numbers of iterations are required since with an increasing number of interactions on the lattice they are easily trapped in a state where transitions to neighboring states with similar energies may invlove considerable shuffling of the internal conformations of the participating molecules. As compared to simulations on a lattice, a selfconsistent field model, based on a mean-field approximation in concentric spherical layers with the thickness of a solvent molecule, has also been investigated.23 This model allows for considerable faster calculations, but the assignment of interaction energies and conformational energies of the surfactant molecules from molecular parameters remains a nontrivial problem. The replacement of the free energy change of a solvent molecule, gaining entropy of H-bond orientation, by a free energy gain assigned to neighbor interactions is an artifact that must be introduced in these theories. The concept of an entropy loss caused by restricted orientational phase space for hydrogen bonding of water molecules in the solvation layer of a solute can be applied to other categories of solute molecules. The shape of the function f (i) determines the character of the aggregation. The function defined in eq 3 is appropriate for the aggregation of the special class of amphiphilic molecules that are characterized by a large hydrophilic headgroup at one end of a non-hydrophilic chain. Single molecules of this class and small aggregates will induce a decrease in configurational entropy of the aqueous solvent, but sufficiently large aggregates are able to form structures that expose only their hydrophilic parts to the solvent, thereby avoiding the decrease in entropy. On the contrary, aggregates of completely non-hydrophilic molecules will always have a non-hydrophilic surface, which must increase monotonically with the aggregation number. Molecules of this class, e.g., aliphatic chains, can minimize the exposed hydrophobic surface by forming spherical aggregates whose surface area increases



Figure 11. Function f (i), representing the effective number of monomeric chains exposed to water for an aggregate size i, displayed for three different classes of aggregating molecules.

Figure 13. (a) Monomer concentration c1 as a function of the total concentration c0. (b) Distribution of aggregate size, for stacking aggregates, calculated from eq 9 with the function f (i) ) 1 displayed in Figure 11.

Figure 12. (a) Monomer concentration c1 as a function of the total concentration c0. (b) Distribution of aggregate size, for aliphatic chains, calculated from eq 9 with the function f (i) ) i2/3 displayed in Figure 11.

with the 2/3 root of the aggregation number. These molecules will not form a stable distribution of micellar aggregates but will separate in an aqueous phase and an oil phase, i.e., the spherical aggregates will grow until they must be considered a separate macroscopic phase. Flat polyaromatic rings with hydrophilic groups substituted at the periphery or with one or more charged heteroatoms in ring positions form still another class of partly hydrophilic and partly non-hydrophilic molecules

that are water soluble. Many aromatic dye compounds, e.g., the cyanines, belonging to this class are known to aggregate in the form of stacks.14,32 In the extreme case, when the cylindric peripheral surface of a disk-shaped molecule can be considered as hydrophilic and the upper and lower disk surfaces considered as hydrophobic, the entropy decrease induced in the aqueous solvent by a single disk or by a stack of disks remains the same. In this case the function f (i) would be a constant, equal to one. The three extreme shapes of f (i) are shown in Figure 11. The result of applying our aggregation model using the function f (i))(i)2/3, and f (i) ) 1 are shown in Figures 12 and 13, respectively. Figure 12a indicates a very sharp transition corresponding to the critical concentration of the phase transition at the solubility limit. The distribution shown in Figure 12b shows that intermediate size aggregates are extremely unstable. The aggregation limit at i ) 1000 is artificial, due to the cutoff of the numerical evaluation to avoid excessive computation time. Increasing the cutoff always shifts the most stable aggregate to the cutoff limit and also lowers the predicted cmc. The model predicts the phase separation actually observed for this class of molecules, but because of the artificial polydispersity introduced by the cutoff, it is not possible to predict the true solubility limit. It is evident that kinetic models for the aggregation and phase separation must take into account the extreme unstability of small size aggregates. Stacking aggregation occurs in a more gradual manner; it would be artificial to define a critical concentration by some particular value of the free monomer


De Maeyer et al. and, in analogy, the number nd at the cylinder base is

π ndπa2 ) πr2 4

(31)

The total hydrophobic entropy of solution of a monomer can be written as

s1 ) -(nz + nd)kBp ln Figure 14. Cross section of a CH2 chain with a radius r, surrounded by H2O molecules with a radius a.

concentration shown in Figure 13a. The size distribution shown in Figure 13b indicates that, contrary to the case of micelles, there is no dip in the distribution for small aggregates. The distribution remains unimodal. These examples for different classes of aggregating molecules with a characteristic dependence f (i) of the solute-induced entropy loss were calculated, for comparison, with the same values of the other molecular parameters m1, s1, etc. as for the micelle calculations. This does not necessarily correspond to real molecules in the case of aromatics. We conclude, however, that the model considered here is able to adequately represent the basic phenomenon of molecular aggregation of solutes driven by an induced entropy reduction that can be related to the change in H-bond orientations imposed on the H -bond-associated solvent.

)-

(

[4π4π- γ]

)

[

π 2l r r2 π 1 + + 2 kBp ln 4 a a a π - arccos

(

)

(r +a a)

]

(32)

where p is the probability that a water molecule in the solvation layer remains engaged in one or more hydrogen bond interactions. p is nearly 1, since in bulk water not more than 15% of the possible number of H-bonds are supposed to be broken at any time.21 With a C-C bond length of 1.54 Å and a bond angle of 120°, the length l of a dodecyl chain is 16 Å. The radius of the equivalent cylinder is r ) 1.485 Å for a C-H bond length of 1.1 Å. The radius of the water molecule is set to a ) 1.45 Å. With these values, eq 32 gives

s1 ) -2.023 × 10-22 J K-1) -121.8 J mol-1 K-1

(33)

References and Notes Appendix A: Hydrophobic Entropy of Solution The calculation of the space angle in which hydrogen bonds cannot be formed follows from a simple geometric consideration. In Figure 14, the large circle represents the cross section of the folded CH2 chain, and the smaller circles represent that of surrounding H2O molecules. In the direction perpendicular to the chain axis, the excluded angle is δ ) 2 arccos (a/(r + a)). Along the axis the angle is π. The excluded space angle is then γ ) 2δ ) 4 arccos (a/(r + a)). The statistical entropy of the distribution of equally probable allowed orientations of H-bonds for a water molecule in contact with the chain is

∫04π-γ4π 1- γ ln 4π 1- γ dΩ

sc ) -kB

(27)

whereas, in bulk water, it is

1 1 ln dΩ ∫04π4π 4π

sw ) -kB

(28)

The entropy change induced by the restriction of H-bond orientations in the solvation layer is then

s ) -kB on

4π 4π - γ

(29)

If a cylinder surface is covered by circles, representing water molecules in a closest quadratic packing, a surface fraction equal to (1 - π/4) remains uncovered. The number nz of molecules that can be arranged at the curved cylinder surface can then be calculated from

π 4

nzπa2 ) 2(r + a)l

(30)

(1) Aniansson, E. A. G.; Wall, S. N. J. Phys. Chem. 1974, 78, 1024. (2) Aniansson, E. A. G. J. Phys. Chem. 1978, 82, 981. (3) Kresheck, G. C.; Hamori, E.; Davenport, G.; Scheraga, H. A. J. Am. Chem. Soc. 1966, 88, 246. (4) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffmann, H.; Kielmann, I.; Ulbricht, W.; Zana, R.; Lang, J.; Tondre, C. J. Phys. Chem. 1976, 80, 905. (5) Kahlweit, M.; Teubner, M. AdV. Colloid. Interface. Sci. 1980, 13, 1. (6) Lessner, E.; Teubner, M.; Kahlweit, M. J. Phys. Chem. 1981, 85, 1529, 3167. (7) Telgmann, T.; Kaatze, U. J. Phys. Chem. B 1997, 101, 7758, 7766. (8) Trachimow, C.; De Maeyer, L.; Kaatze, U. J. Phys. Chem. B 1998, 102, 4483. (9) Paula, S.; Su¨s, W.; Tuchtenhagen, J.; Blume, A. J. Phys. Chem. 1995, 99, 11742. (10) Zana, R., Ed. Surfactant Solutions; Surfactant Science Series 22; Marcel Dekker Inc.: New York, 1987. (11) Trachimow, C. Dissertation, University Goettingen, 1997. (12) Bo¨cker, J.; Brickmann, J.; Bopp, R. J. Phys. Chem. 1994, 98, 712. (13) MacKerell, A. D. J. Phys. Chem. 1995, 99, 1846. (14) Wallqvist, A.; Covell, D. G. J. Am. Chem. Soc. 1995, 99, 13118. (15) Gruen, D. W. R. Prog. Colloid Polym. Sci. 1985, 70, 6. (16) (a) Aston, J. In A Treatise on Physical Chemistry; Taylor, H. S., Glasstone, S., Eds.; Van Nostrand: New York, 1952; p 555. (b) Sackur, O. Ann. Phys. 1911, 36, 958. (c) Tetrode, H. Ann. Phys. 1912, 38, 434. (17) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 1976, 72, 1525. (18) Gelbart, W. M.; Ben-Shaul, A.; McMullen, W. M.; Masters, A. J. Phys. Chem. 1984, 88, 861. (19) (a) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 3170. (b) Zoeller, N.; Lue, L.; Blankschtein, D. Langmuir 1997, 13, 5258. (20) Nagarajan, R.; Ruckenstein, E. J. Colloid Interface Sci. 1977, 60, 221; 1979, 71, 580. (21) Tanford, Ch. The hydrophobic effect: formation of micelles and biological membranes; John Wiley & Sons: New York, 1980. (22) Rodrigues, K.; Mattice, W. L. J. Chem. Phys. 1991, 94, 761; 1991, 95, 5341. (23) Wijmans, C. M.; Linse, P. Langmuir 1995, 11, 3748. (24) Debye, P.; Hu¨ckel, E. Physik. Z. 1923, 24, 185. (25) Corkill, J. M.; Goodman, J. F.; Tate, J. R. Trans. Faraday Soc. 1964, 60, 996.

Entropy-Driven Micellar Aggregation (26) Sasaki, T.; Hattori, M.; Sasaki, J.; Nukina, K. Bull. Chem. Soc. Jpn. 1975, 48, 1397. (27) Cutler, S. G.; Meares, P.; Hall, D. G. J. Chem. Soc., Faraday Trans. I 1978, 74, 1758. (28) Bratko, D.; Lindmann, B. J. Phys. Chem. 1985, 89, 1437. (29) Berr, S. S.; Jones, R. R. M. Langmuir 1988, 4, 1247.

J. Phys. Chem. B, Vol. 102, No. 43, 1998 8491 (30) Rusanov, A. I. AdV. Colloid Interface Sci. 1993, 45, 1-78. (31) Bezˇan, M.; Malavasˇicˇ, M.; Vesnaver, G. J. Chem. Soc. F. Trans. 1993, 89, 2445. (32) Scheibe, G. Kolloid. Z. 1938, 82, 1. (33) Wallqvist, A.; Berne, B. J. J. Phys. Chem. 1995, 99, 2893.

Entropy-Driven Micellar Aggregation - The Journal of Physical

Recommend Documents