Markov chain model for the critical micelle

Markov chain model for the critical micelle concentration of surfactant mixtures

Radomir I. Slavchov & George S. Georgiev

Colloid and Polymer Science Kolloid-Zeitschrift und Zeitschrift für Polymere ISSN 0303-402X Volume 292 Number 11 Colloid Polym Sci (2014) 292:2927-2937 DOI 10.1007/s00396-014-3337-2

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Author's personal copy Colloid Polym Sci (2014) 292:2927–2937 DOI 10.1007/s00396-014-3337-2

ORIGINAL CONTRIBUTION

Markov chain model for the critical micelle concentration of surfactant mixtures Radomir I. Slavchov & George S. Georgiev

Received: 23 May 2014 / Revised: 20 June 2014 / Accepted: 5 July 2014 / Published online: 29 July 2014 # Springer-Verlag Berlin Heidelberg 2014

Abstract An extension of the Markov chain model (MC) for micellization is proposed, which allows the distribution of the surfactants between the monomer solution and the micelles in a mixed surfactant system to be predicted. The dependence of the critical micelle concentration (cmc) on the composition of the solution is investigated. The equilibrium thermodynamic relation between cmc and micelle composition is discussed. The case of ternary mixtures is analyzed, and theoretical triangular diagram is constructed according to MC. Available experimental data for binary and ternary mixtures agree well with the new MC theory. The dependence of MC parameters on the structure of the surfactants is discussed. Comparison of MC with the simple mixture (“regular solution”) model is presented. The parameters of the MC theory are related to the interaction parameter βSM of the simple mixture model. Keywords Mixed surfactant system . Interaction parameter . Synergism and antagonism . Regular solution . Markov chain model . Ternary mixture

Introduction Micellar solutions used in practice typically consist of several surfactants. This is most often intentional since mixed Electronic supplementary material The online version of this article (doi:10.1007/s00396-014-3337-2) contains supplementary material, which is available to authorized users. R. I. Slavchov (*) Department of Physical Chemistry, Sofia University, 1 J. Bourchier Blvd., 1164 Sofia, Bulgaria e-mail: [email protected] G. S. Georgiev Laboratory of Water-Soluble Polymers, Polyelectrolytes and Biopolymers, Sofia University, Sofia, Bulgaria

surfactant systems show enhanced performance compared to a single surfactant solution. In fact, even what is called a “single surfactant” is usually a mixture of homologues and inevitably contains various surface-active impurities. For this reason, there is a constantly increasing interest toward the understanding of the mechanisms of mixed micellization and the reliable prediction of the mixed micelle properties [1–3]. Several models have been proposed in the literature for the mechanism of micellization [4, 5]. They can be divided in three main groups. The most widely used approach to the problem remains the pseudo-phase separation theories, such as the single-parameter simple mixture (SM) model [6, 7], referred often as to the regular solution model (which is not an entirely correct term, cf. Supplementary material 1). The so-called interaction parameter βSM of the simple mixture model is the most common characteristic of the mixed micellization [8–11]. The second group of models stands for the detailed molecular thermodynamic theories [4, 12, 13]. They use a more correct approach to the problem, but these models have certain disadvantages: they are rather complicated, contain many unknown or uncertain parameters, etc. The final group is for the kinetic mass action/nucleation models [14, 15], which are as a rule the most intricate but allow in principle the prediction of the time evolution of micellization. This study is developing a model for micellization based on the Markov chain (MC) mechanism proposed previously [16]. In essence, the MC model is a new mass action model, based on the assumption for the existence of relatively stable active centers (AC) in the micelle. The idea of the MC model [16] can be easily understood if we consider the linear aggregate formation illustrated in Fig. 1. Suppose that this aggregate can grow and diminish only via its active centers (~S1, ~S2), which in the considered case are the tips of the aggregate. Further, the properties of these active centers are assumed to be determined only by the last surfactant molecule attached onto the micelle (the last molecule is the AC). Each time a new molecule joins

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the micelle, it replaces the old AC and becomes the new one. The last plus the penultimate molecules form the so-called dyad active center (~SiSj). For linear aggregates, this mechanism is rather convincing. For disk-shaped micelles growing in a spire or in concentric circles, similar mechanism can be proposed: the ACs are situated at the breaks of the circle or at the end of the spire. For cylindrical micelles, again the two edges can serve as ACs. A Markov mechanism can be invented also for spherical micelles, but it must be multistage and more intricate in comparison with the other shapes. In any case, even without considering a specific mechanism, MC model is useful as a semiempirical model, especially in view of the relatively simple final results. To illustrate the MC formalism, let us consider the case of a single surfactant solution first. The surfactant monomer Si reacts reversibly with the micelle active center ~Si to form a dyad ~SiSi (Fig. 1): h i h i K ii S þ S ⇄ S ; K ¼ S S S = eSi ½Si ; i i i i ii i i e e e

ð1Þ

Kii is the equilibrium constant of the process; [Si], [~Si], and [~SiSi] are the respective concentrations (for the convenience of the reader, a list of symbols and abbreviations is given in the Supplementary material 5). Since in a single surfactant solution there is only one possibility for the penultimate molecule, for the dyad concentration one has [~SiSi]=[~Si], so that Eq. 1 becomes K ii ¼ 1=½Si ≡1=cmci ;

ð2Þ

where cmci is the critical micelle concentration of Si. Thus, in a single surfactant solution, micelles are in equilibrium with the solution only if the monomer concentration [Si] is equal to the cmci. If the total surfactant concentration C is below cmci, the equilibrium condition (2) is impossible and no micelles will occur. Previously, Markov chain model was used to relate monomer and micelle composition in binary mixtures [16]; the cmc of the mixture was not investigated. However, the most important experimental method for studying mixed micelles is the measurement of cmc with varying monomer compositions [8–11, 17], while the micelle composition is almost always calculated theoretically (either by using a micellization model [8–11] or through Duhem-Margules relations [17]). Only few Fig. 1 Scheme of the formation of a mixed linear aggregate through a Markov chain process. ~Si is the hypothetical active centers (in red), ~SiSj are the dyad active centers, and Si is a free surfactant molecule

experimental studies provide direct data for the micelle composition [18, 19]. Therefore, in this study, we derive the MC formula for the mixed cmc. The cases of binary and ternary mixtures are considered in “Markov chain model for the cmc and the composition y of micelles in binary surfactant solution” and “Ternary mixtures”. A way to relate the equilibrium constants of MC to surfactant structure is devised in “Ternary mixtures.” MC model is a kinetic model; the results from Ref. [16] refer to stationary nonequilibrium state. We consider this question in detail in “Micelle composition—equilibrium thermodynamic treatment” and derive a new expression for the micelle composition, which should be used for established equilibrium between micelles and monomers. Comparison with the simple mixture model is given in “Relation between the Markov chain and the simple mixture models”—it turns out that MC is almost equivalent to SM. A relation between MC model parameters and the interaction parameter βSM of the simple mixture is deduced.

Markov chain model for the cmc and the composition y of micelles in binary surfactant solution Prediction of the critical micelle concentration In a mixed solution of two surfactants, S1 and S2, four detailed equilibrium processes are established [16]: h i h i K 11 eS1 þ S1 ⇄ eS1 S1 ; K 11 ¼ 1=cmc1 ¼ eS1 S1 = eS1 ½S1 ; h i h i K 22 eS2 þ S2 ⇄ eS2 S2 ; K 22 ¼ 1=cmc2 ¼ eS2 S2 = eS2 ½S2 ; h i h i K 12 eS1 þ S2 ⇄ eS1 S2 ; K 12 ¼ eS1 S2 = eS1 ½S2 ; h i h i K 21 þ S ⇄ S ; K ¼ S S = S S S 2 1 2 1 21 2 1 e e 2 ½S1 : e e ð3Þ A similar set of equilibria is well-known and widely used for two-component copolymerization [20–22]. The first two homomeric reactions have the same equilibrium constant as given by Eq. 1: K11 =1/cmc1 and K22 =1/cmc2. However, the monomer concentration [Si] in mixed systems differs from cmci since now [~Si] and [~SiSi] are not equal (compared to

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Eq. 2). As there are two possible penultimate molecules, the ith active center concentration [~Si] is, in fact, a sum of two dyad concentrations: eS1 ¼ eS1 S1 þ eS2 S1 ; eS2 ¼ eS1 S2 þ eS2 S2 ð4Þ Obviously, [~SiSi]cmc2. It can be shown that if β12 β12 >−ln(cmc1/cmc2), the dependence c vs. x2 is monotonic. This is the case of the mixture of octyltrimethylammonium bromide (C 8 Me 3 NBr) and octylpentaoxyethylene ether (C 8 E 5 ), investigated by D’Errico et al. [18]. Their data are compared to Eq. 11 in Fig. 2b; the best value of the Markov chain interaction parameter is β12 =−0.73±0.06. When β12 is almost zero, the mixed surfactant system behaves as ideal (the case is illustrated in the Supplementary material 3 with the data of Osborne-Lee and Schechter [19]). Finally, if β12 is large and positive, β12 > ln(cmc1/cmc2), the mixture can have cmc larger than both cmc1 and cmc2 of the pure surfactants—a phenomenon known as antagonism. The data for mixed hexadecyltrimethylammonium chloride (C16Me3NCl) and dodecylamine (C 12NH2) reported by García-Río et al. [25] seem to display such a behavior, cf. Fig. 2c. The theoretical Eq. 11 again compares satisfactory with the experimental data, with β12 =2.1±0.6. Synergistic and antagonistic mixtures have an extremum of the cmc at monomer fraction


pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cmc2 − cmc1 cmc2 e−β12 =2 : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cmc1 þ cmc2 −2 cmc1 cmc2 e−β12 =2

ð12Þ

This value of x2 corresponds [26] to positive or negative azeotropic composition of the mixture, where the micelle and the monomer compositions are equal (the black and the green lines in Fig. 2a, c meet). If one takes the limit at β12 =0 of Eq. 11, it simplifies to 1 x1 x2 ¼ þ ; c cmc1 cmc2

ð13Þ

i.e., to the ideal mixture model [6, 23], valid when two different surfactant molecules interact with each other in the same way as two equivalent surfactants. The ideal mixture model is drawn in Fig. 2 for comparison (gray dashed-dotted lines; cf. also Supplementary material 3). In a previous work [16], the relationship between the surfactants mole fraction in the micelle y and in the monomer solution x was worked out: y1 1 þ g 1 x1 =x2 1 þ g 1 x1 =x2 −1 ¼ or y2 ¼ 1 þ ; ð14Þ y2 1 þ g 2 x2 =x1 1 þ g 2 x2 =x1 where g 1 ¼ 1=cmc1 K 12 ;

g2 ¼ 1=cmc2 K 21

ð15Þ

are the monomer reactivity ratios [16, 22]. From these equations and Eq. 8 for β12, a simple relation between the interaction parameter and the g values follows: b12 ¼ lng1 g 2 :

ð16Þ

Equation 14 was compared to experimental data for various surfactant mixtures in Ref. [16]. An example is given in Fig. 3 (black line) with the system C8Me3NBr+C8E5, analyzed by D’Errico et al. [18]. We already know the interaction

parameter for this system from the cmc data (β12 =−0.73, cf. Fig. 2b), so from Eq. 16, it follows that g1 =exp(−0.73)/g2, and now, there is only one unknown parameter left in Eq. 14: g2. It is determined from the fit in Fig. 3: g2 =14±4. Micelle composition—equilibrium thermodynamic treatment According to Eqs. 11 and 14, within the original Markov chain model, it is principally impossible to predict the micelle composition using only cmc vs. x2 data (cmc data allow the determination only of β12, so one cannot determine both g1 and g2 in Eq. 14). Let us consider the question on a more fundamental level: is it possible that two different y2(x2) functions yield the same cmc(x2) dependence? In the typical case, the monomer solution is infinitely diluted, so one can write for the chemical potentials of the monomers dissolved in the water (superscript W) the expressions: μW 1 ¼ μ10 þ RT ln

μM 1 ¼ μ10 þ RT lnγ 1 y1 ;

Fig. 2 Dependence of the cmc of three mixed surfactant systems on the total molar fraction X2 of the second surfactant. a An example for synergistic mixture—data from Ref. [24] for C12SO4Na+DHPC (points). b An example for monotonic cmc vs. X2 dependence—data from Ref. [18] for C8Me3NBr+C8E5. c An antagonistic system—data from Ref. [25] for C16Me3NCl+C12NH2. Solid black lines: Eq. 11 of MC model

μM 2 ¼ μ20 þ RT lnγ 2 y2 ;

ð18Þ

where γ1 and γ2 are the activity coefficients of a surfactant in the micelle; because of the choice of the standard state, the standard potentials μ10 and μ20 in Eqs. 17 and 18 are the same. In thermodynamic equilibrium, μ1W =μ1M ≡μ1 and μ2W = μ2M ≡μ2. In addition, the Gibbs-Duhem relation is valid for the micelles, and it yields y1

cmc [mol/kg]

cmc [mM]

X2

ð17Þ

The critical micelle concentrations cmci of the single surfactants are used as a standard state in these expressions. For a surfactant molecule in the micelle (superscript M), one can write

dμ1 dμ þ y2 2 ¼ 0; dx2 dx2

ð19Þ

(c) C16Me3NCl + C12NH2

(b) C8 Me3 NBr + C8 E5

(a) C12 SO4 Na + DHPC

x1 c x2 c ; μW : 2 ¼ μ20 þ RT ln cmc1 cmc2

cmc [mmol/kg]

x2;e ¼


X2

X2

with β12 =−2.6, −0.73, and +2.1, respectively; gray dashed-dotted lines: ideal mixed micelles (β12 =0); red dashed-dotted lines: simple mixture model, Eq. 38, with βSM =−3.3, −0.8, and +1.6, respectively. SM model predicts nearly the same cmc vs. X dependence as MC, but with different value of the interaction parameter β. Green dashed line: dependence of the cmc on micelle fraction y2—a parametric plot of Eq. 11 vs. Eq. 21


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Fig. 3 Dependence of the molar fraction y2 of C8E5 in the mixed micelles on the molar fraction x2 of C8E5 in the monomer solution. Data from Ref. [18] for C8Me3NBr+C8E5. Solid black line: MC model, Eq. 14 with g2 = 14 (the other g-parameter, g1 =0.035, was calculated through Eq. 16 using β12 =−0.73, cf. Fig. 2b). Green dashed line: equilibrium MC model, Eq. 22, no fitting parameters (β12 =−0.73 from Fig. 2b). Red dasheddotted line: simple mixture model with βSM =−0.8 as obtained in Fig. 2b. The SM and equilibrium MC graphs coincide, within the experimental error

an equation analogous to the Duhem-Margules relation [6, 17]. Substituting here Eq. 17, we obtain a strict oneto-one relation between the micelle composition y2 and the c(x2) dependence: y2 ¼ x2

x1 dc 1− : c dx2

ð20Þ

For the sake of simplicity, here we will discuss neither the question for the counterion adsorption in the case of charged micelles nor the one for the presence of water inside the micelle. The thermodynamics of ionic micelles in the presence of background electrolyte (or mixture of inorganic electrolytes) is discussed in Ref. [27]; for a mixture of surface-active ions, cf. Refs. [28, 29]. If one substitutes Eqs. 11 and 14 into Duhem-Margules relation (20), the identity will not be fulfilled, i.e., Eqs. 11 and 14 are thermodynamically incompatible and cannot be both valid at equilibrium. This is a reflection of the fact that Eq. 3 refers to local equilibrium concerning only to the molecules at the edges of the aggregate. Markov chain model assumes that once the monomer is inside the aggregate, it is no longer mobile—instead, it is in a nonequilibrium “glass” state and cannot leave the micelle for any reasonable period of time. In order for a Markov chain process to occur, it is required that the active centers are stable enough for certain time, sufficiently long to allow several consecutive steps of the micelle growth to

occur. While this requirement is fulfilled for polymerization, it is obviously wrong for micelles, where the monomer solution and the micelles are constantly exchanging surfactant molecules. The average time that a monomer spends in the micelle is of the order of 10−8–10−3 s, depending on the chain length of the surfactant molecule [30, 31]. In addition, the values of the diffusion coefficient of a surfactant molecule in the micelle [18] suggest that micelle core is liquid. This violates the assumption for a pure Markov chain mechanism of the micellization. Various side processes, leading to surfactant exchange, are able to equilibrate the micelle composition, e.g., formation and disintegration of active centers, micelle-micelle interactions, or another process aside the reactions (3). If one assumes that the reactions (3) are faster than all these secondary processes, then Eq. 11 for c will be valid (active centers control the monomer concentration). However, micelle composition y 2 will be affected, relaxing to the true thermodynamic equilibrium through the side “reactions.” If the relation (11) between c and x2 holds, and the one for y2 vs. x2 does not, we can calculate the equilibrium micelle composition y 2 through the Gibbs-Duhem equation. Substitution of Eq. 11 in Duhem-Margules relation (20) yields a new expression for y: x2 c cmc2 y1 ¼ ; x1 x2 c 2− þ cmc1 cmc2 1−

x1 c cmc1 y2 ¼ ; ð21Þ x1 x2 c 2− þ cmc1 cmc2 1−

where c is given by Eq. 11. These formulae can be rearranged in the more elegant form y1 1−x2 c=cmc2 ¼ ; y2 1−x1 c=cmc1

ð22Þ

compare to Eq. 14. Since xic=[Si]0, β23 > 0), a more complex ternary diagram will be observed (Fig. 5c), where the mixture can have either higher or lower cmc than any of the surfactants. Details for the construction of the triangular diagram are given in the Supplementary material 4. Let us now illustrate how one can use experimental data and extra thermodynamic arguments to determine Markov chain model parameters and to construct the respective ternary diagram. The conductometric cmc data by Basu Ray et al. [36] for the ternary mixture of alkyltriphenylphosphonium bromides, with alkyl- being dodecyl- (C12TPB), tetradecyl- (C14TPB), or hexadecyl- (C16TPB) is especially convenient due to the welldefined difference between the three surfactants. The system is neither synergistic nor antagonistic (Fig. 6). Since cmcn have been measured [36] (1.75, 0.57, and 0.20 mM for C12TPB, C14TPB, and C16TPB), there are four unknown parameters,

β12, β23, β13, and β123, which must be determined through optimization or estimated independently. We will now show how Traube’s rule can be used to decrease the number of the unknowns. We will first remark that the critical micelle concentrations and respectively, the homomeric constants Knn =1/cmcn follows the well-known linear dependence [37, 27]:

Fig. 5 Examples for triangular diagrams of cmc vs. x2 and x3 calculated through Eq. 26. The interaction parameters βij and β123 are so chosen that the diagram represent one of the common types of behavior: a) synergism,

b) antagonism, and c) mixed behavior (antagonism at the side x1 =0/ synergism at the side x3 =0). Lines correspond to cmc(x2,x3)=const (the values of cmc [mM] are given next to the line)

ln cmcn ¼ −ln K nn ¼ −ln K 0 −nΔμCH2 =RT ;

ð29Þ

related to Traube’s rule. Here, the index n is 12, 14, or 16; ΔμCH2 is the transfer energy of a –CH2– group from the micelle to the aqueous solution; −lnK0 is the intercept of the dependence lncmcn vs. n. Linear regression over the measured [36] cmc values yields ln(K0/[mM−1])=−7.05 and ΔμCH2/ RT=0.54 (close to the typical value 0.69 [38, 27]). On the other hand, the heteromeric constants Kij represent the transfer energy of a molecule from the monomer solution to a mixed micelle. We can assume that dependences on n similar to Eq. 29 hold for the heteromeric constants: ln K nþ2;n ¼ ln K þ2 þ nΔμCH2 =RT ; ln K nþ4;n ¼ ln K þ4 þ nΔμCH2 =RT ;

ln K n−2;n ¼ ln K −2 þ nΔμCH2 =RT; ln K n−4;n ¼ ln K −4 þ nΔμCH2 =RT:

ð30Þ



concentration be cmc0(x). Upon addition of a –CH2– to all hydrocarbon chains, the new critical micelle concentration cmc+1 of the mixture will be cmc0(x)exp–ΔμCH2/RT (i.e., all β-parameters will have the same value, while cmc will scale according to (29)). This scaling law can be called Traube’s rule for mixed surfactant micelles. Let us further assume that j is a small parameter so that lnKj can be expanded into Taylor series: ln K j ¼ ln K 0 þ k 1 j þ k 2 j2 þ O j3 ;

ð32Þ

where k1 and k2 are dimensionless coefficients. Substituting this series into Eq. 31, we obtain: β 12;14 ¼ β 14;16 ¼ −8k 2 ; Fig. 6 Comparison of Markov chain model for ternary mixtures with experimental conductometric data for cmc vs. x2 and x3 of the ternary mixture C12TPB+C14TPB+C16TPB by Basu Ray et al. [36]. The isoconcentrational lines are calculated through Eq. 26 with a single fitting parameter, k2 =0.05 (β12 =β23 =−8k2, β13 =−32k2, and β123 =−24k2, cf. Eq. 33). The deviations between theory and experimental data are denoted with peaks (for scaling, the height of the largest peak is 0.08 mM)

K −2 K þ2 K −4 K þ4 ; β 12;16 ¼ −ln ; K 20 K 20 K −2 K −2 K þ4 þ K þ2 K þ2 K −4 ¼ −ln : 2K 30

β 12;14 ¼ β14;16 ¼ −ln β 12;14;16

σ2 ð k 2 Þ ¼

According to Eq. 31, the interaction parameter βn1,n2 of two surfactants of the same polar head group and of different hydrocarbon chain lengths n1 and n2 depends only on the difference Δn=n2–n1. Similar notion is valid for the ternary parameter βn1,n2,n3: adding the same number of –CH2– to all three surfactants do not change the value of βn1,n2,n3. Equation 31 can be formulated as a scaling law, as follows. Consider any mixed binary or ternary surfactant system where the surfactants have normal hydrocarbon chains of different length but share the same head group; let its critical micelle

ð33Þ

2 cMC x2;i ; x3;i ; k 2 −cmcexp;i x2;i ; x3;i

i

; ð34Þ

N −1

where cMC is the theoretical cmc obtained by solving Eq. 26 with β-parameters given by Eq. 33, and cmcexp is the conductometric data of Basu Ray et al. [36]; i is the i-th measurement. The result from the optimization is k2 =0.05±0.02, corresponding to β12 =β23 =−0.4, β13 =−1.7, β123 =−1.3. The standard deviation is σ=0.047 mM, which is of the order of the experimental uncertainty. The result is satisfactory—Eq. 26 agrees well with the experimental data (Fig. 6). Using the Gibbs-Duhem equation, one can predict also the composition of the ternary micelles. There are two DuhemMargules relations for a ternary mixture: y1

ð31Þ

β 12;14;16 ¼ −24k 2 ;

i.e., the interaction parameters are independent of the linear term in Eq. 32. The formulae (33) leave a single unknown parameter in the MC model for the ternary mixture: the coefficient k2. We will determine it by regression over the data by Basu Ray et al. [36]. The merit function is given by the dispersion: X

For the considered ternary mixture, Kn+2,n is either K14,12 (attachment of a C12TPB monomer to C14TPB active center) or K16,14; Kn-2,n is either K12,14 or K14,16; Kn+4,n is K16,12 and Kn-4,n is K12,16. It is reasonable to assume that the same value ΔμCH2/RT=0.54 holds for all heteromeric constants Kij (the energy for transfer of a –CH2– from the liquid hydrocarbon micelle core to the aqueous solution is the same irrespectively of the micelle precise composition). However, the intercepts lnK j ( j = 0, ±2, ±4) in Eqs. 29–30 must be different. Substituting Eqs. 29–30 into the definitions (8) and (27) of the interaction parameters β gives for the considered mixture:

β12;16 ¼ −32k 2 ;

dμ1 dμ dμ þ y2 2 þ y3 3 ¼ 0 dx2 dx2 dx2

and y1

dμ1 dμ dμ þ y2 2 þ y3 3 ¼ 0 ; dx3 dx3 dx3

ð35Þ

These equations generalize Eq. 19. Using Eq. 17 for μi and then solving Eq. 35 for y2 and y3 yields x2 1−x2 d lnc x1 −x3 1− − 1−x1 2 dx2 2 x3 x1 −x2 d lnc 1−x3 1− − y3 ¼ dx2 1−x1 2 2 y2 ¼

d lnc ; dx3 d lnc : dx3

ð36Þ

Inserting here the solution of Eq. 26 for c, one obtains the composition y of the micelles as a function of the monomer composition x.


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Relation between the Markov chain and the simple mixture models The expression (9) for Markov chain’s βij parameter is analogous to the one for the interaction parameter βSM widely used in SM theory [9] under the assumption for strictly regular solution [39]: βSM ¼

H 11 þ H 22 −2H 12 ; RT

ð37Þ

Here, Hij denotes the corresponding interaction enthalpies of each surfactant couple (cf. Supplementary material 1). The similarity between Eqs. 9 and 37 leads to the question: are the interaction parameters of Markov chain and simple solution models related? Instead of Eqs. 7 and 14 of the MC model, SM offers a system of two nonlinear equations [6]: x1 c x2 c −2 βSM ¼ y−2 ln ln ¼ y ; ð38Þ 2 1 y1 cmc1 y2 cmc2 which must be solved numerically for c vs. x and y2 vs. x2. The results from the SM model are compared to the experimental data for binary mixtures in Fig. 2 (cmc) and Fig. 3 (y2 vs. x2). The single fitting parameter—βSM—was obtained from this comparison. It is seen from these figures that the cmc(X2) dependences predicted by SM and MC models are almost identical (not, however, at β12 =βSM). This allows one to treat Eq. 11 as an approximate solution of the system (38); let us now find such a value of β12, which would give the best approximation (i.e., let us “fit” SM’s Eq. 38 with the MC’s Eq. 11). Toward this, we will first find c and x according to the SM theory at one fixed value of y (y=½ is preferable since it is far from y2 =0 and y2 =1); then, we will require Eq. 11 to pass through that point by adjusting the value of β12. This means that we are making a three-point interpolation of the unknown solution of the system Eq. 38 via the function given by Eq. 11. The first two points are at the edges {y1 =1, x1 =1, c=cmc1} and {y2 =1, x2 =1, c=cmc2}; for the third point, we use {y=½, x2 at y=½, c at y=½}. A simple analytical expression for x2(y=½) and c(y=½) can be obtained from Eq. 38: x2 ðy ¼ 1=2Þ ¼

cmc2 cmc1 þ cmc2

and cðy ¼ 1=2Þ ¼

cmc1 þ cmc2 expðβ SM =4Þ: 2

ð39Þ Substituting these values in Markov chain model’s expression for с vs. x, Eq. 7, after some rearrangements, one obtains for the sought interpolation condition the following relation:

β 12 ¼ −2 ln 2e−βSM =4 −1

or

β SM ¼ −4 ln

1 þ e−β12 =2 ð40Þ : 2

Obviously, the approximation is only applicable if βSM β12 >−2, this difference is not great. However, for large negative interaction parameters, the partition coefficient Kp,2 predicted by the two models can be of different order of magnitude. This fact is essential for the whole gex(y) curve since Kp is related to the slope of gex at y2 =0 and y2 =1. Table 1 Comparison between the interaction parameters of Markov chain (β12) and simple mixture (βSM) models Mixture

β12 (fit), Fig. 2

βSM (fit), Fig. 2

βSM calculated from β12 (fit) through Eq. 40

C12SO4Na+DHPC C8Me3NBr+C8E5 C16Me3NCl+C12NH2

−2.6 −0.73 2.1

−3.3 −0.80 1.6

−3.4 −0.80 1.6



When Eq. 40 is fulfilled, MC and SM predict nearly the same results for the c(x) dependence. From Duhem-Margules relation (20), it follows that if the two functions c(x) are approximately equal, then the respective y(x) must also be the similar. For this reason, the SM and the equilibrium MC graphs in Fig. 3 coincide. As far as Duhem-Margules relation does not hold for the kinetic relation (14), SM and the original MC model [16] can predict in principle different micelle compositions y. Let us finally consider briefly the comparison between the ternary simple mixture model of the micelle and the Markov chain model. The ternary simple mixture is defined with the following expression for the excess molar Gibbs energy of the mixed micelle: gex =RT ¼ β SM;12 y1 y2 þ β SM;23 y2 y3 þ β SM;13 y1 y3 :

ð44Þ

The activity coefficients γi of the i-th surfactant in the micelle are obtained by differentiation of (n1 +n2 +n3)gex with respect to ni [6]: ln γ i ¼ β SM;ij y2j þ β SM;ik y2k þ β SM;ij þ β SM;ik −β SM; jk y j yk :

ð45Þ It is rather obvious that the 3-parametric model (44) (with parameters βSM,12, βSM,13, βSM,23) cannot be equivalent to the 4-parametric Markov chain model for ternary mixture (which contains also the ternary parameter β123). The correct full analog of the ternary Markov chain model, that we will call the “quasi-simple ternary mixture” model, is defined with the following excess Gibbs energy: gex =RT ¼ β SM;12 y1 y2 þ βSM;23 y2 y3 þ βSM;13 y1 y3 þ β SM;123 y1 y2 y3 :

ð46Þ

The respective activity coefficients are ln γ i ¼ β SM;ij y2j þ β SM;ik y2k þ β SM;ij þ β SM;ik −β SM;jk y j yk þ β SM;123 ð1−2yi Þy j yk :

ð47Þ In both cases, the three unknowns of the ternary surfactant system (c, y2, and y3 as functions of x2 and x3) are determined by a system of three nonlinear equations: γ i ðyÞyi ¼ xi c=cmci ;

i ¼ 1; 2; 3;

ð48Þ

which generalize Eq. 38. Here, the main advantage of MC becomes evident: Eq. 26 of Markov chain model is simpler than the respective quasi-simple mixture expressions (47)–(48). Otherwise, the ternary Markov chain and the quasi-simple ternary mixture models are approximately equivalent. Conditions for equivalence analogous to Eq. 40 can be imposed over β123 and βSM,123, but the final result for the relation between them is rather intricate and hardly useful.

Conclusion Markov chain mechanism for micelle growth predicts cmc and micelle composition of binary and ternary surfactant mixtures as functions of the monomer composition. The cmc follows directly from the basic equilibria (3). It is shown that the result (11) we obtained for cmc is thermodynamically incompatible with the result (14) obtained previously for the micelle composition [16]—Eq. 14 refers to a nonequilibrium glass-like state, which is able to relax to equilibrium through some undefined side processes in micellar solutions. A new expression (21) for the micelle composition y is derived through the Gibbs-Margules relation. Comparison with experimental data confirms the results—cmc vs. X2 data in Fig. 2— are well described by Eq. 11 with a single adjustable parameter (Markov model’s interaction parameter βij, Eq. 8). Once the interaction parameter is known, one can predict the micelle composition—experimental data for y agree well with Eq. 21 with no adjustable parameters (Fig. 3). Generalization for ternary systems is proposed, “Ternary mixtures.” Equation 26 allows the construction of the triangular diagram for the dependence of cmc on the monomer composition (Fig. 5). The results once again agree with the experimental data (Fig. 6). Simple scaling laws were used to determine most of the parameters involved in the Markov chain model—using a variant of Traube’s rule, we were able to decrease the number of unknown parameters for a ternary mixture from four to one. A comparison with the classical simple mixture model shows that MC and SM are almost equivalent, provided that βij of Markov model and βSM are related through Eq. 40. The cmc and y2 vs. x2 curves predicted by the two models coincide (Figs. 2 and 3); the same is valid for the excess Gibbs energy of mixing (Fig. 4). However, there is a significant difference in the predicted value of the partition coefficient of a surfactant of small concentration between the monomer solution and the micelles of another surfactant in excess, cf. Eqs. 42, 43, and 40. Markov chain model has an important advantage: it is simpler than the simple mixture model, and the final results for c and y vs. x are analytical. Markov chain model is widely used for copolymerization [20–22], and as far as many phenomena in the copolymerization are analogous to mixed micellization, some of the proposed corrections and extensions of the polymer MC theory can be adjusted to surfactants. An important point here is the possibility to estimate or predict Kij values theoretically [21] or via the measured Kij values for standard mixtures with a standard monomer (the Q-P scheme for copolymerization [40]). Doing this for mixed micelles is a task for a following study. Acknowledgments This work was funded by the Bulgarian National Science Fund Grants DDVU 02/12 and DDVU 02/43. The fruitful

Author's personal copy Colloid Polym Sci (2014) 292:2927–2937 discussions of the work with Prof. Emil Manev are gratefully acknowledged. R. Slavchov is grateful to FP7 project BeyondEverest.

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