Article pubs.acs.org/Langmuir
The Mass-Action-Law Theory of Micellization Revisited Anatoly I. Rusanov* Mendeleev Center, St. Petersburg State University, 199034 St. Petersburg, Russian Federation ABSTRACT: Among numerous definitions of the critical micelle concentration (CMC), there is one related to the constant K of the mass action law as CMC = K1−n (n is the aggregation number). In this paper, the generalization of this definition for multicomponent micelles and the development of the mass-action-law theory of micellization based on this definition and the analysis of a multiple-equilibrium polydisperse micellar system have been presented. This variant of the theory of micellization looks more consistent than the earlier one. In addition, two thermodynamic findings are reported: the stability conditions for micellar systems and the dependence of aggregation numbers on the surfactant concentrations. The growth of the monomer concentration with the total surfactant concentration is shown to be a thermodynamic rule only in the case of a single sort of aggregative particles or at adding a single surfactant to a mixture. The stability condition takes more complex form when adding a mixture of aggregative particles. For the aggregation number of a micelle, it has been deduced a thermodynamic rule obeying it to increase with the total surfactant concentration. However, if the monomer concentration increases slowly, the aggregation number increases much more slowly and the more slowly the more pronounced is a maximum corresponding to a micelle on the distribution hypersurface (curve in the one-component case). This forms grounding for the quasi-chemical approximation in the mass-action-law theory (the constancy of aggregation numbers).
1. INTRODUCTION Soon after the discovery of micelles, which occurred just a centennial ago,1 Jones and Bury2 suggested that an abrupt change in a solution property observed at micellization was “a natural consequence” of the mass action law with a great aggregation number. Mathematically, this was related to the fact that the aggregation number stands as an exponent at concentration, and every quantity smaller than unity becomes negligible whereas a quantity greater than unity becomes huge with such a large exponent. Grindley and Bury3 took ci/K1/n (ci is the monomeric concentration, K is the mass-action-law constant, and n is the aggregation number) for such a quantity (which was not quite correct; we will see below that ci should be replaced by the total surfactant concentration c). Since that time, the mass-action-law theory occupied a leading position in explaining micellization and was used practically in every publication on the subject. A detailed thermodynamic grounding of the mass-action-law theory was formulated by the author.4−7 Meanwhile, in 1962, Shinoda and Hutchinson8 published their interpretation of micellization, the pseudophase separation approach that became an alternative to the massaction-law theory. They considered micellization as a first-order phase transition with a fixed chemical potential above the critical micelle concentration (CMC). In reality, the surfactant chemical potential continues to grow above the CMC, and if the phase interpretation of micellization is needed, it should be rather a second-order phase transition.9−11 As a result, the priority of the mass-action-law theory against the phase separation approach was already stated in 1975.12 The analysis7 of the phase separation approach shows it to be contradictory © XXXX American Chemical Society
and inconsistent as a theory (with a wrong interpretation of the Krafft point in addition). However, it does not mean that, with some corrections, the pseudophase approach cannot be used for calculations as an approximation (there are examples of a successive work in this direction13−15). Permission for this should be given by the mass-action-law theory after the verification that the monomeric concentration (as an indicator of the chemical potential) only slightly changes above the CMC. Besides, some theoretical problems, such as changing composition in mixed micelles, can be considered within the frames of the phase model.12,14,15 For micelles, the mass action law originates from the condition of aggregative equilibrium
μM =
∑ niμi1
(1)
i
that was deduced by Gibbs for a chemical reaction and is also rigorously valid for the aggregation process. Here, μ is chemical potential, M is the chemical symbol of a micelle, the subscript i indicates the sort of aggregative particles, and the additional subscript 1 refers to the monomeric form of particles (since a monomeric chemical potential is equal to the overall chemical potential of the same species at equilibrium, the additional subscript 1 will be omitted at μi but will be always used at concentration below). A general expression for the particle chemical potential given by statistical mechanics reads Received: September 23, 2014 Revised: November 10, 2014
A
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μ = μ0 + w + kT ln(c Λ3f )
The scheme of the mass-action-law theory is the following. Equation 6 is completed by the material balance condition for each sort of particles ci = ci1 + nic M (7)
(2)
where μ0 is the chemical potential of a single particle in a fixed position in a vacuum, w is the work of transfer of the particle to a fixed position in a pure solvent, kT is of the ordinary meaning, c is concentration (the number of particles per unit volume), f is the activity coefficient, and Λ is the de Broglie wavelength. Taking an ideal solution of standard concentration cs for a standard state, we can rewrite eq 2 as
where ci is the total concentration of particles of sort i. Substituting eq 6 in eq 7 yields ci = ci1 + niK ∏ (ci1)ni
μ = μs + kT ln(cf )
(3)
c1 = c11 + n1K ∏ (ci1)ni i
αi ≡ nic M /ci ,
c1α1 = n1K ∏ cini(1 − αi)ni
i
i
(4)
ci = bic1
(11)
(12)
eq 11 takes the form n1Kc1n − 1 = α1/∏ bini(1 − βα )ni i 1 i
(13)
where n ≡ Σini is the total aggregation number, αi = βiα1, and βi ≡ ni/n1bi (obviously, b1 = 1 and β1 = 1). At given ni, bi, and βi, the set of eqs 8 is replaced by single eq 13 from where α1 is found as a function of c1. If micelles are composed of a single sort of particles, eq 13 is simplified to
nKc n − 1 = α /(1 − α)n
(14)
The dependence of α on c is determined explicitly only at given n and K. Specification is trivial for n but problematic for K. For this reason, a definition of the CMC is given irrespective of K. Among 13 various definitions,7 the definition of the CMC can be formulated, for example, as a point of the most pronounced change of α where the condition is fulfilled ∂2α/∂c2 = 0 or ∂2α/ ∂(ln c)2 = 0 Using eq 14, the former condition yields the value of the critical micellization degree (CMD) αm as
(5)
similarly to a chemical reaction with constant stoichiometric coefficients. Micellization typically occurs at small concentrations. If, in addition, the cause of micellization is not the interaction between monomeric particles but the solvophobic effect of a solvent, a micellar solution can be considered as an ideal mixture of aggregates and monomers with f M = 1 and f i = 1. This simplifies eq 4 to i
(10)
If concentrations ci are firmly related as (which is typical for the mass-action-law theory to avoid the phase problem of micelle composition, bi are constants)
where Ka is a constant of the mass action law in terms of activities. Concerning a possible dependence of Ka on concentration via the aggregation numbers ni, one can say the following. Equations 1 and 4 are applicable not only to micelles but also to any particle aggregates. The most general point of view is considering a micellar solution as a polydisperse system with aggregates of all sizes and compositions.7,18 Then ni simply are ordinary numbers, and Ka is a true constant. If, however, a micellar system is considered as including only micelles and monomers, the aggregation number becomes a function of concentration. Then the constancy of the aggregation number is an additional assumption that is called “quasi-chemical approximation” with the interpretation of the aggregation process (Bi are chemical symbols of monomeric particles)
c M = K ∏ ci1ni
1 − αi = ci1/ci
eq 9 becomes
c MfM = K a ∏ (ci1fi1 ) ,
i
(9)
Introducing the micellization degree αi for every sort of particles as
ni
⎡⎛ ⎤ ⎞ K a ≡ exp⎢⎜⎜∑ niμis − μMs ⎟⎟ /kT ⎥ ⎢⎣⎝ i ⎥⎦ ⎠
(8)
The number of eqs 8 is equal to the number of particle sorts, and the set of eqs 8 has a unique solution at given ci and ni. For the first particle sort (usually of the highest surface activity), eq 8 is of the form
where μs = μ0 + w + kT ln(csΛ3) is the standard chemical potential and c is actually the dimensionless quantity c/cs with cs = 1 (say, one particle per cubic meter). It is of note that using concentration is more typical for statistical thermodynamics,16 and taking mole fractions instead of concentrations is widely spread in the theory of micellization.16−20 However, the use of mole fractions is quite inconvenient in the mass-action-law theory based on the joint consideration of the mass action law and material balance equations. Applying eq 3 to all chemical potentials standing in eq 1, the mass action law follows as
∑ ni Bi = M
(i = 1, 2, ...)
i
αm =
n/2 − 1 (αm = 0.061 at n = 100) n−1
(15)
And the latter yields (6)
αm =
where K is a constant of the mass-action law in terms of concentrations. In practice, eq 6 can often serve as a good approximation also for systems with interactions near the CMC. For example, the difference between ln K and ln Ka amounts to only 1% near the CMC for the SDS solution,7 which is a system with strong Coulomb forces.
1 (αm = 0.091 at n = 100) n +1
(16)
after that the CMC cm = c(αm) is to be found from eq 14. The definition of that kind is attractive by estimating the CMD only via the aggregation number, but this is valid only for onecomponent micelles whereas there are additional disadvantages: (a) the definition is given in the quasi-chemical approximation B
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⎛ ∂μi ⎞ ⎛ ∂μ ∂ci ⎞ =⎜ i >0 ⎜ ⎟ ⎟ ⎝ ∂Ni ⎠T , p , N ⎝ ∂ci ∂Ni ⎠T , p , N
(the derivative is taken at a constant n); (b) the definition is dependent on the manner of expression of concentration; (c) the definition is isolated from the mass-action-law constant. At the same time, the mass-action-law theory explains the abruptness of micellization18,19 in a way similar to that of Grindley and Bury.3 Considering the stepwise growth of a micelle with steps j = 1, 2, ..., n−1 and elementary constants Kj to be equal to each other (a rough approximation), we can set K = Kjn−1 to rearrange eq 14 to the form7 n(Kjc)n − 1 = α /(1 − α)n
0
⎛ dci ⎞ N ⎛ dV ⎞ 1 − civi 1 = − 2i ⎜ ≡ ⎜ ⎟ ⎟ d N V d N V V ⎝ i ⎠T , p , N ⎝ i ⎠T , p , N 0
⎛ ∂μi ⎞ >0 ⎜ ⎟ ⎝ ∂ci ⎠T , p , N
which shows that increasing the concentration of a single species i causes an increase in its chemical potential. In aggregative systems, μi coincides with the chemical potential of monomers of sort i. Then eq 24 means that increasing the total concentration of species i also causes the growth of the chemical potential of monomers of sort i and, as a consequence, the growth of the monomer concentration. We now imagine that some particles are present in a system in strictly determined mutual proportions. This can be a given mixture of nonionics, a mixture of ionics and nonionics, or even a single ionic surfactant whose ions are interrelated by stoichiometry. The concentrations of such particles are related with eq 12, and, dealing with eq 20, we have to set dNi = bidN1. At constant T, p, N0, and amounts of species not participating in eq 12, eq 20 takes the form
that, in contrast with the above definitions, is directly related to the mass-action-law constant. The mass-action-law theory based on such a definition of the CMC (generalized for a multicomponent micellar systems) has not yet been developed in detail, and closing this gap is the goal of this paper. The consideration will be given for a polydisperse system with multiple aggregative equilibriums, and some thermodynamic findings will be also presented.
2. GENERAL RELATIONSHIPS As a preliminary, we have to formulate the thermodynamic background, which includes conditions of equilibrium and stability and their consequences. Dealing with an aggregative system, we imagine it as a population of monomers and various particle aggregates. The aggregate composition and size are characterized by the set of aggregation numbers {n} ≡ n1,n2,..., and the condition of aggregative equilibrium reads (cf. eq 1)
dN1 ∑ bi dμi > 0
and the stability condition can be written as ⎛ ∂μ ̃ ⎞ >0 ⎟ ⎜ ⎝ ∂N1 ⎠T , p , N 0
where the chemical potential of monomers of sort i, μi, coincides with the chemical potential of the whole species i in the system. For a set of solutes i, the general condition of stability was formulated by Gibbs, irrespective of the aggregation process, as follows dT dS − dp dV + dμ0 dN0 +
∑ dμi dNi > 0 i
0
0
(26)
where μ̃ ≡ Σi≥1biμi looks like the chemical potential of a hypothetical substance with stoichiometric coefficients bi. Replacing N1 by c1 yields ⎛ dμ̃ ⎞ >0 ⎜ ⎟ ⎝ dc1 ⎠T , p , N 0
(20)
(27)
In this case, increasing concentration c1 means increasing all concentrations of the complex of particles. However, an increase in μ̃ following from eq 27 does not mean that every individual chemical potential of the complex will become larger (we will return to this problem in section 4). On the other side, every independent change occurring with some particles of the complex should obey eq 24. Let us now see how the equilibrium condition, eq 19, influences the aggregation numbers of micelles. Considering a polydisperse micellar system with an equilibrium distribution of aggregates in aggregation numbers c{n} = c{n}(n1,n2,...), we recognize micelles as maximum points on the hypersurface c{n}(n1,n2,...). The points of a minimum are also interesting because they represent critical embryos (germs) of micelles and determine kinetics of micellization. For this reason, we will pay attention to both the kinds of extremes. We assume the
where T is temperature, S is entropy, p is pressure, V is volume, μ is chemical potential, and N is the amount of matter (the number of particles in our case), with subscript 0 being introduced for a solvent. It follows from eq 20 (Gibbs’ result too) ⎛ ∂μi ⎞ >0 ⎜ ⎟ ⎝ ∂Ni ⎠T ⊕ S , p ⊕ V , μ ⊕ N
(25)
i≥1
(19)
i
(23)
(24)
0
(18)
∑ niμi
0
where vi is the partial particle volume. The solvent typically prevails in micellar solutions so that |civi| ≪ 1. This makes the derivative on the left-hand side of eq 23 positive, and, using eq 22, we arrive at the stability condition
In the process of its growth, the concentration passes from states with Kjc < 1 (when the left-hand side of eq 17 is very small) to states with Kjc > 1 (when the left-hand side of eq 17 becomes very large). The condition for the intermediate state Kjc = 1 suggests itself as a definition for the CMC18,19
μ{n} =
(22)
Using the concentration definition ci ≡ Ni/V, we have
(17)
cm = 1/Kj = K1 − n
0
(21)
where the circled plus is of the meaning “either, or”. For us, it is more convenient to deal with concentration. We could just set dV = 0 for replacing N by c, but this would not correspond to the laboratory conditions where temperature and pressure are constant. For this reason, we start from the condition (with a fixed amount of a solvent) C
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The next step is joining a particle of sort i to the aggregate B2 (B2 + Bi = B2i) at
concentration of micelles (and, moreover, of all other aggregates, especially germs) to be not high to omit the activity coefficient in eq 3, which we now write as μ{n} = φ + kT ln c{n} ,
φ ≡ μ{n}s
c 2i = K 2ic 2ci1 ,
(28)
i
ie
c 2i = Kc112ci1 ,
∂ 2 ln c{n} dni
2
=−
∂ 2φ ∂ni2
(29)
(30)
We now return to eq 29 and differentiate it again but this time with respect to nie, which means a change in the physical state (and also in the form of distribution). Now μi is not a constant, and we obtain ∂ ⎛ ∂φ ⎞ = ⎜ ⎟ ∂nie ∂nie ⎝ ∂ni ⎠
n1
K=
Referring eq 30 to an extreme point, we have the same righthand side in eqs 30 and 31 to lead to a remarkable thermodynamic relationship
ie
i>1
K 2 ≡ exp(2μ1s − μ2s )
(36)
(37)
which suggests the definition of the CMC for a mixture of a given (by coefficients bi) composition as Kc̅ 1m = 1,
(32)
c1m = K̅ −1
(38)
in terms of the concentration of particles of the first sort (this is of no principal significance, since all concentrations are related to each other by eq 12). Equation 38 corresponds well to eq 18 for one-component micelles. The definition of the CMC given in eq 38 allows one to write the mass-action-law constant in the form
relating the shape of the distribution hypersurface at an extreme point to a physical effect expected. Since an extreme is a maximum for micelles, the right-hand side of eq 32 is positive. Accounting also for the stability condition expressed in eq 24, we can formulate the following rule: increasing the concentration of one of species in a micellar system leads to increasing the aggregation number of the species in micelles. Looking at eq 32, we can also conclude that the more pronounced the micelle maximum in the distribution the slighter is a change in the aggregation number. As for micellar germs corresponding to a minimum on the distribution hypersurface (then the right-hand side of eq 32 is negative), the above rule is inverted predicting diminishing the aggregation number of a given species at increasing its total concentration. Note that eq 32 is a quantitative relationship and allows not only predictions but also calculations. We now return to the CMC definition expressed in eq 18 and generalize it for multicomponent micelles. Let us find the expression for the mass-action-law constant K in eq 6 through the constants of multiple elementary equilibriums in a polydisperse micellar system. The first step in formation of a micelle is the coupling of two monomeric particles of the first (the most surface-active) sort: B1 + B1 = B2. The aggregative equilibrium condition for this act is μ2 = 2μ1, and the corresponding formulation of the mass action law is c 2 = K 2c112 ,
k=1
α1/(1 − α1)n1 ∏ (bi − α1ni /n1)ni = n1(Kc̅ 1)n − 1
∂μi
i
i>1
where we have introduced the geometric mean K̅ of all elementary constants (we remind, n is the sum of all aggregation numbers). The substitution of eq 36 in eq 13 yields
(31)
⎛ ∂ 2 ln c ⎞ {n} ⎟⎟ = −kT ⎜⎜ 2 ∂nie ∂ n ⎝ ⎠n = n i
(35)
ni
∏ K1k ∏ ∏ K ik ≡ K̅ n− 1 k=2
∂μi
ni = nie
K ≡ exp(2μ1s + μis − μ2is )
Comparing eqs 33−35, we obtain K = K2K2i, which mirrors the known rule that the summary constant of equilibrium is equal to the product of elementary constants. Continuing our reasoning, we arrive at the following result. If the aggregation number of particles of the first sort in a micelle is n1, there are, all in all, n1 − 1 acts of joining a monomer to an aggregate and, therefore, n1 − 1 elementary equilibrium constants K1k(k = 2, 3,..,n1). Then the contribution of the first sort of particles to the total micellar constant K will be Πkn1= 2K1k. For all other sorts of particles, the number of elementary constants Kik coincides with the aggregation number, and we can mirror the contribution of each of them as Πkn=i 1Kik. Finally, we arrive at the expression
Here nie is the coordinate of an extreme on the hypersurface of distribution where d ln c{n}/dni = 0. The second differentiation yields (for any point on the distribution hypersurface)
kT
(34)
If, alternatively, we consider the one-stage formation of B2i (B1 + B1 + Bi = B2i), we have
Substituting eq 28 in eq 19 and differentiating eq 19 with respect to ni along the distribution hypersurface at a fixed physical state (then μi play the role of constants), we get ∂ ln c{n} ⎛ ∂φ ⎞ ∂φ + kT =⎜ ⎟ μi = ∂ni ∂ni ⎝ ∂ni ⎠n = n
K 2i ≡ exp(μ2s + μis − μ2is )
K = c1m1 − n
(39)
which makes it a quantity directly determined from experiment. For better understanding, we below illustrate the use of this definition of the CMC by examples.
3. THE CASE OF A SINGLE SORT OF AGGREGATIVE PARTICLES This case corresponds to the presence of a single nonionic surfactant in solution. For the process of micellization nB = M, the equilibrium condition, eq 1, takes the form μM = nμ (40) where μ is the chemical potential of monomers and of the surfactant as a whole. The stability conditions expressed in eqs 24 and 27 coincide in this simple case and become ⎛ dμ ⎞ ⎜ ⎟ >0 ⎝ dc ⎠T , p , N 0
(41)
which means that the surfactant chemical potential can only increase with the total surfactant concentration in a system.
(33) D
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negative, we obtain a different kind of behavior as compared with micelles. When the chemical potential and concentration of a surfactant increase, the number of molecules in a germ diminishes. This means that the work of formation of the germ (that is the energy barrier in the kinetics of micellization) decreases, and the kinetic barrier becomes easier to overcome. In short, increasing concentration promotes micellization. The situation resembles ordinary phase transitions, although micelles themselves (not to speak about their embryos) could be termed phases only in the microscopic meaning since they have no macroscopic analogue. Let us return to the definition of the CMC expressed in eq 38 and see how it changes the mass-action-law theory. If there is only one sort of aggregative particles in a system, eq 37 is reduced to eq 17, which, at setting K̅ c = 1, produces an equation for the CMD αm:
Since the chemical potential of a surfactant as a whole is the same as the monomer chemical potential, eq 41 also means that the monomer concentration should increase with the total surfactant concentration. Equation 41 is well-known in the thermodynamics of solutions and was multiply confirmed in experiment. It remains only to verify whether micellization can violate this condition. Considering micelles and monomers as different components, we can write the integral fundamental equation for Gibbs energy as G = μM NM + μN1 + μ0 N0
(42)
where N is the number of particles and the subscripts refer M to micelles, 1 to monomers, and 0 to a solvent. If a system is in equilibrium, putting eq 40 returns eq 42 to it original form (N = N1 + nNM) G = μN + μ0 N0
(43)
αm/(1 − αm)n = n
which was before micellization. The same will be with other thermodynamic equations and with all consequences following from them. In particular, stability conditions following from eq 43 (for example, eq 41 for monomers) should be also valid for a micellar system. It is of note that thermodynamic stability conditions act irrespective of ideality or nonideality of a system. Thus, we can conclude that a symbate change of the monomer concentration and the total surfactant concentration in a single nonionic solution is a strict thermodynamic requirement. Since high-sensitivity electrochemical methods are applicable only to electrolytes (see below), there are no experimental data on the nonionic monomer concentration in micellar solutions up to the present time. Turning to computer simulations, one can meet a single case exhibiting a decrease in the monomer concentration at increasing the total surfactant concentration.21 At first sight, this contradicts eq 41. However, eq 41 is only valid for a true-equilibrium state, whereas attaining equilibrium is a main problem of a computer simulation (too much time is needed). For the case of a single sort of aggregative particles, eq 32 acquires the form ⎛ d2 ln c ⎞ dμ n ⎟ = −kT ⎜ 2 dne ⎝ ∂n ⎠
It is seen that the advantage of estimating the CMD only from the aggregation number shown in eqs 15 and 16 is completely retained in the new variant of the theory. Equation 46 is solved numerically for each n-value. For example, n = 100 corresponds to αm = 0.070, which is close to the result shown in eq 15. For the case of a single sort of particles, the joint formulation of the mass-action law and the material balance condition shown in eq 8 becomes c = c1 + nKc1n
c ̃ = c1̃ + nc1̃ n
dne
⎛ d 2 ln c ⎞ n ⎟ = −⎜ 2 ∂ n ⎝ ⎠n = n
e
(48)
where a tilde marks that a concentration is measured in the CMC units. Equation 48 establishes a relation between the total surfactant concentration, the concentration of monomers, and the aggregation number. It is often assumed that c̃1 ≈ 1 in a micellar solution. We now see from eq 48 that this condition is attained strictly at c̃ = n + 1, that is, by two or three orders of magnitude higher than the CMC. At the CMC itself, eq 48 takes the form (equivalent to eq 46) c1m ̃ + nc1m ̃ n=1
where cn is the concentration of molecular aggregates containing n molecules. In this case, the micelle composition is fixed, and we only have the curve of the distribution of micelles in size. According to eq 44, increasing the solution concentration makes micelles larger. Ascribing the chemical potential μ to monomers and using eq 3, eq 44 can be written d ln(c1f1 )
(47)
where subscript 1 refers to monomers. Putting eq 39 in eq 47, we obtain
(44)
n = ne
(46)
(49)
and relates the monomer concentration at the CMC to the aggregation number. This dependence is plotted in Figure 1. It is seen that the greater n is, the closer c̃1m is to unity, and the wider is the interval, inside which c̃1 becomes exactly unity. In
(45)
where f1 ≈ 1 provided the monomer concentration is small. If the micellar maximum in the distribution curve is sharp, the derivative dne/d lnc1 is small. Then we have the following picture: the monomer concentration c1 only slightly changes above the CMC (see below); ln c1, naturally, changes still more slightly, and the micellar aggregation number ne changes more slightly than ln c1. In other words, ne is almost constant, which makes a reliable grounding for the quasi-chemical approximation in the mass-action-law theory. Turning now to germs (equilibrium critical embryos) of micelles, for which the right-hand side of eqs 44 and 45 is
Figure 1. Dependence of the monomer concentration on the aggregation number at the CMC. E
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c11m ̃ + n1c11m ̃ n1c 21m ̃ n2 = 1
other words, the greater n is, the weaker the monomer concentration (and, therefore, the surfactant chemical potential too) depends on the surfactant concentration above the CMC. This is important for the model of pseudophase separation as an alternative to the mass-action-law theory. Returning now to the definition of the CMC expressed in eq 38, we write eq 39 as ln K = −(n − 1)ln cm
We also use the thermodynamic relationship
c 21m ̃ + n2c11m ̃ n1c 21m ̃ n2 = b
Using eqs 57, it is possible to compute monomeric concentrations at the CMC at given n1, n2, and b. For example, setting n1 = 50, n2 = 40, and b = 1 yields c̃11m = 0.924 and c̃21m = 0.939, and setting n1 = 100, n2 = 80, and b = 1 results in c̃11m = 0.954 and c̃21m = 0.963. The dependence of c̃11m and c̃21m on n1 at n2/n1 = 0.8 and b = 1 is presented in Figure 2 for the sake of illustration.
(50) 7
ΔGs = −(RT /n)ln K
(57)
(51)
where ΔGs is the standard Gibbs energy of micellization per 1 mol of a surfactant (RT is of the ordinary meaning). From eqs 50 and 51, we obtain ΔGs =
n−1 RT ln cm n
(52)
which seems to be the most known relationship of the theory of micellization often used (in the approximation (n − 1)/n ≈ 1) for the estimation of the standard micellization work. We see that the definition of the CMC expressed in eq 38 immediately reproduces eq 52. In this formalism, eq 52 appears as an exact relationship whereas it was deduced with some approximations in the general theory.7
Figure 2. Dependence of the monomeric concentrations c̃11m and c̃21m on the aggregation number n1 at n2/n1 = 0.8 and b =1.
4. THE CASE OF TWO SORTS OF AGGREGATIVE PARTICLES We now consider an example when micelles form with participation of two sorts of particles. It may be a mixture of two nonionics taken in a certain proportion or a single ionic surfactant producing two ions. With sorts of particles 1 and 2, we write eq 12 as c 2 = bc1
For two sorts of particles and using the definition of the CMC expressed in eqs 38 and 39, eq 13 for micellization degrees becomes α1/(1 − α1)n1 (b − α1n2 /n1)n2 = n1c1̃ n − 1
At the CMC (c̃1 = 1), eq 58 acquires the form of an equation α1m/(1 − α1m)n1 (b − α1mn2 /n1)n2 = n1
0
for calculating the CMD α1m at given n1, n2, and b. For instance, by setting n1 = 100, n2 = 80, and b = 1, we obtain α1m = 0.0459. In view of the relationship cĩ 1m = 1 − αi m (60)
(54)
it is easy to verify that the above value of α1m corresponds to the value of c̃11m calculated. Moreover, eq 59 itself can be deduced from the set of eqs 57. To explore the monomer concentrations for extremes, we have to differentiate eqs 56. At a given ratio of the two sorts of particles, one may use the quasi-chemical approximation (the constancy of n1 and n2). Besides, for the sake of compactness, we can use eqs 10 for micellization degrees, which, for this case, take the form
from where it is seen that, taken separately, chemical potentials μ1 and μ2 (and, therefore, also the concentrations of monomers of the first and second sorts) unnecessarily are increasing functions of c1. We here meet a principal difference from the above case of a single sort of particles. However, both of the chemical potentials cannot decrease simultaneously: if one decreases with increasing c1, the second should increase. Below, we will return to this problem in particular calculations. For the case of two sorts of particles, the set of eqs 8 is written as (additional subscript 1 refers to monomers)
α1 = n1c11 ̃ n1c 21 ̃ n2 /c1̃ ,
c 2 = c 21 +
(61)
With it all, differentiating eqs 56 yields (55)
(1 − α1 + n1α1)
and, after substituting eq 39, takes the form c1̃ = c11 ̃ + n1c11 ̃ n1c 21 ̃ n2 c 2̃ = c 21 ̃ + n2c11 ̃ n1c 21 ̃ n2
α2 = n2c11 ̃ n1c 21 ̃ n2 /bc1̃ ,
α1/α2 = bn1/n2
c1 = c11 + n1Kc11n1c 21n2 n2Kc11n1c 21n2
(59)
(53)
The stability condition expressed in eq 27 takes the form ⎡ d(μ + bμ ) ⎤ 1 2 ⎢ ⎥ >0 dc1 ⎣ ⎦T , p , N
(58)
n2α1 (56)
d ln c11 d ln c 21 + n1α2b =1 d ln c1 d ln c1
d ln c11 d ln c 21 + (1 − α2 + n2α2)b =b d ln c1 d ln c1
(62)
The solution of this set of two algebraic (with respect to the derivatives) equations is
At the CMC (c̃1 = 1, c̃2 = b), eqs 56 become F
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dodecyl sulfate (SDS), the concentration of free dodecyl sulfate ions passes through a maximum whereas the concentration of free sodium ions increases monotonically. Such behavior was predicted for an ideal micellar system on the basis of the massaction-law theory long ago.22 Considerably later, this effect was discovered in experiment23−30 and was explained in a general case.7,31 In the case when a surfactant is a strong electrolyte, the coefficient b is given by the stoichiometric condition b = ν2/ν1 (νi is the number of ions of sort i formed at the dissociation of one molecule of the surfactant). Then quantity μ̃ in the stability condition expressed in eq 26 can be replaced by the chemical potential μ of the electrolyte as a whole
d ln c11 1 − α2(1 − n2 + n1b) = d ln c1 (1 − α1)(1 − α2) + n1α1(1 − α2) + n2α2(1 − α1) d ln c 21 1 − α1(1 − n1 + n2 /b) = d ln c1 (1 − α1)(1 − α2) + n1α1(1 − α2) + n2α2(1 − α1) (63)
Let us explore these expressions for extremes. Of course, we can primarily expect for a maximum since both the derivatives are positive at small c and α. If there is a maximum of c11, the following conditions should be fulfilled (the supplementing inequality follows from the requirement α2 < 1) α2 = (1 − n2 + n1b)−1 ,
b > n2 /n1
(64)
μ = ν1μ1 + ν2μ2
In the case of a maximum of c21, we have α1 = (1 − n1 + n2 /b)−1 ,
b < n2 /n1
(67)
Using eq 67, we can rewrite eq 26 as ⎛ dμ ⎞ > 0, ⎜ ⎟ ⎝ dc1 ⎠T , p , N
(65)
The inequalities in eqs 64 and 65 are mutually exclusive. This means that a maximum for the monomer concentration can arise only for one of the two sorts of particles. If the particles of different sorts are taken in equal amounts (b = 1), the maximum should refer to particles with higher surface activity (the first sort of particles according to our numeration) when n2/n1 < 1. At n1 = 100, n2 = 80, and b = 1, eq 64 yields the value of α2 corresponding to an extreme of c11 as α2e = 1/21. Then α1e = 5/84 ≈ 0.0595 in accordance with eq 60. Comparing this with α1m = 0.0459, we see that α1e > α1m, that is, the maximum of the monomer concentration of the surface-active sort of particles, lies to the right of the CMC on the concentration axis at defining the CMC according to eq 38. It is of note that, using the CMC definition from the condition ∂2α/∂(ln c)2 = 0 for the same system, the CMD α1m was estimated as about 0.08.7 It follows in this case that the maximum lies to the left of the CMC, although the realization of the maximum is related to the existence of micelles. Thus, one can say that the theory looks more consistent at defining the CMC according to eq 38. Although the said above equally referred to an electrolyte and a mixture of nonelectrolytes, the case of an electrolyte deserves further comment. In a micellar solution, the electroneutrality condition acts c Mz M = −c11z1 − c 21z 2 (66)
0
⎛ da± ⎞ >0 ⎟ ⎜ ⎝ dc1 ⎠T , p , N 0
(68)
where a± is the mean activity of the electrolyte as a whole. Thus, thermodynamics requires that not the activity or concentration of ions taken separately but the mean electrolyte activity should increase with the electrolyte concentration in micellar system. At first sight, a decrease in the concentration of one ion with increasing the total electrolyte concentration seems to be absurd and quite impossible. This is why, in spite of the thermodynamic character of this prediction, the experimental aspect of the problem has acquired an especial importance and is worthy of more detailed consideration here. In the case of an aqueous solution of an ionic surfactant, such as SDS, the activities of surface-active ions a1 and counterions a2 can be measured via the electromotive force (EMF) of an element with a suitable membrane electrode. Among the measurements accomplished in this direction, the work of Cutler et al.29 seems to be most reliable and detailed, and we take it as an example. The SDS solution below and above the CMC was investigated. The data on the activities of the dodecyl sulfate ion (a1) and the sodium ion (a2) above the CMC given in a tabular form29 are presented as a plot in Figure 3. It is seen that the activity of the sodium ion increases whereas the activity of the dodecyl sulfate ion decreases with increasing the total SDS concentration. Since activity of an ion as a whole and of its monomeric form is the same, and activity and concentration always change in the same direction, the behavior of the ionic monomeric
where zM is the charge number of a micelle and zi is the charge number of the monomer of sort i, with 1 referring to surfaceactive ions and 2 to counterions. The left-hand side of eq 66 displays the total charge of micelles, and the sign of this charge coincides with the charge sign of the surface-active ions. We will term the right-hand side of eq 66 as “solution charge” whose sign is evident to coincide with the charge sign of the counterions. The number of micelles grows with increasing the total surfactant concentration, and the total charge of micelles increases too. According to eq 66, the solution charge increases to the same extent, and since the solution charge sign is determined by the counterions, it follows from here that prevailing free counterions over free surface-active ions becomes more pronounced. Thus, only the electroneutrality condition leads to the conclusion that if the concentration of at least one sort of ions should increase, this should be the counterion concentration. As for surface-active ions, their concentration can pass through a maximum and decrease subsequently. Just this situation is realized in the case of a single surfactant that is a 1−1-electrolyte.18 For example, in the case of sodium
Figure 3. Experimental values of log a1, log a2, and log a± for the SDS solution above the CMC.29 G
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concentrations should be similar to the behavior of the ion activities shown in Figure 3. Indeed, calculations from experimental data29 yield an increase of the sodium ion concentration and a decrease of the dodecyl sulfate ion concentration. To verify eq 68, we have to calculate the mean activity a± ≡ (a1a2)1/2 or, in the logarithmic form, log a± = (loga1 + log a2)/ 2. The result of this calculation is depicted as a middle line in Figure 3 and shows a slight increase from −2.115 to −2.0675. A slight increase of the mean activity was also confirmed by directly measuring the EMF.29 This corresponds to the thermodynamic requirement expressed in eq 68. Concluding this section, we can say that existing experimental data are in agreement with the thermodynamic predictions of an unusual behavior of the concentrations of monomeric ions in a micellar system. Touching again the problem of interrelation between concentration and chemical potential (section 3), we have to emphasize that decreasing the concentration of dodecyl sulfate ions is accompanied by decreasing its chemical (electrochemical) potential (or activity27) so that the symbate run of concentration and chemical potential is not violated even in this case.
μi = μis + kT ln(cifi A )
(69)
where ci is the total concentration of species i. On the other side, the chemical potential in an aggregative system is determined by the monomer concentration ci1 so that
μi = μis + kT ln ci1
(70)
Comparing eqs 69 and 70 relates the aggregative activity coefficient to the micellization degree: fi A = ci1/ci = 1 − αi
(71)
Since we formulated the theory in terms of αi, we accounted for the activity coefficient f iA in all our relationships. An additional (Lewis) activity coefficient, f i, accounting for interactions in an aggregative system, was omitted in our relationships for the sake of simplicity. This is rather practical for aqueous solutions where the hydrophobic effect suppresses other factors. However, in other cases or at high concentrations, the massaction-law theory should include activity coefficients f i and start from the mass action law expressed in eq 4. The logic remains the same, but, as a preliminary, the dependence of activity coefficients on concentrations should be determined. After the substitution of f M and f i1 as functions of concentrations cM and ci1 in eq 4, cM will be found as a function of all monomeric concentrations. The subsequent substitution of cM in eqs 7 (the material balance equations) yields a set of equations (similar to eqs 8) having a unique solution at given ci and ni. Of course, accounting for activity coefficients inserts a certain complication in calculations. As for accounting activity coefficients in the analysis of behavior of micellar aggregation numbers, the problem is reduced to the influence of interactions between aggregates on the form of their distribution, which was already discussed in the literature.7
5. CONCLUDING REMARKS The above results show the oldest theory of micellization to be also capable of development. As the mass-action-law theory includes studying both very rapidly and very slowly changing quantities with respect to the total surfactant concentration, it is important that the progress concerns slowly changing quantities: the monomer concentration and the aggregation number. Both of them are of principal significance for the theory of micellization: the slowness of changing the monomer concentration allows us to use the approximation of the pseudophase separation for calculations, and the slowness of changing aggregation numbers justifies the use of the quasichemical approach (the constancy of aggregation numbers) in the mass-action-law theory. In this paper, we have formulated equations calculating the monomer concentration as a function of the total surfactant concentration and established the hierarchy of slowness: although thermodynamic requires the growth of the aggregation number, it proceeds much slower as compared with a change in the monomer concentration. The thermodynamic requirement of growing the aggregation number with the total surfactant concentration causes polymorphic transitions in micelles: since the packing space is restricted in spherical micelles, they rearrange their structure. The growth of aggregation numbers has one more interesting aspect. Since it can be perceived as increasing the aggregate size and the equilibrium monomer concentration can be interpreted as the aggregate solubility, it follows from eq 45 that the solubility increases for micelles and decreases for germs with increasing the aggregate size. Remembering the famous Kelvin equation of the theory of capillarity predicting the growth of solubility with diminishing the drop size, we can say that micelles as microscopic particles do not obey the Kelvin equation (which is an asymptotic relationship for large drop radii) whereas germs, although being still smaller than micelles, do obey. Some comments are needed for deviations from ideality. The process of aggregation itself is a deviation from ideality and can be quantitatively characterized by a special activity coefficient, fA, which can be termed as the aggregative activity coefficient and corresponds to the expression (cf. eq 3)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■ ■
ACKNOWLEDGMENTS This work was supported by the Russian Scientific Foundation (Grant 14-13-00112). REFERENCES
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