Modeling of Micelle-Solution Equilibria for Mixed Nonionic Micelles

Article pubs.acs.org/jced

Modeling of Micelle-Solution Equilibria for Mixed Nonionic Micelles with Strong Specific Interactions in Coronae: Group-Contribution Approach Alexey I. Victorov* St. Petersburg State University, Universitetsky prosp., 26, 198504 St. Petersburg, Russia S Supporting Information *

ABSTRACT: A molecular thermodynamic model of micellization is proposed that takes into account strong specific interactions in the mixed micelle and hydration in micellar corona. To describe local interactions between molecular segments in the micelle we employ a quasichemical group contribution model of the bulk phase. “Segments-and-bonds” formalism is applied and correction for the connectivity of molecular chains is derived for the chemical potential in quasichemical approximation. The proposed general method may be used in combination with other bulk phase models, particularly with SAFT. The new method is applied to describe (1) the equilibrium distribution of alkanols between the N-dodecanoyl-methyl-glucamine micelle and dilute aqueous solution, and (2) the free energies of transfer of m-ethylene glycol mono-n-alkyl ether surfactants from aqueous solution to the micelles. Comparison with experiment shows the robustness of our approach.

1. INTRODUCTION Mixed micellar systems are widely used in chemical and biochemical engineering, particularly in micelle catalysis, drug delivery, micelle chromatography, and separation. For mixed micellar systems, a key issue is the distribution of species (surfactants, cosurfactants, additives, and target substances) between the micelle and the surrounding medium. The description of the micelle−solution equilibrium is usually performed in the pseudophase approximation where micelles are considered as a separate uniform phase of a distinct average composition.1−6 However, the composition of the micelles is strongly non-uniform. Recent molecular dynamic simulations7,8 help to obtain structural details of amphiphilic aggregates including the location of complex molecules of a solute within a bilayer or a micelle. The COSMOmic model developed by Klamt et al.,9 combines molecular dynamic simulation that serves to establish the local composition of the aggregate with the COSMO thermodynamic routine for modeling the penetration of a solute molecule in various locations within the micelle and the distribution coefficients.10 The classical molecular thermodynamic micellization models11−13 are not well suited for describing the distribution of different organic molecules between the micelles and the bulk because they do not take into account the complex threedimensional structure of a micelle’s corona and a variety of interactions that arise owing to the penetration of an organic molecule in the surfactant aggregate. When a large (e.g., chainlike) molecule is transferred from the bulk solution in the micelle a certain part of that molecule may prefer to reside in the hydrocarbon core whereas its other part may prefer the hydrated corona, where it may participate in strong specific interactions with the surfactant’s heads and hydration water. To take into account this transfer into different environments when © 2014 American Chemical Society

calculating the standard free energy of aggregation we propose a technique where the molecule is first cut into pieces and its different parts are transferred separately to the core and to the corona (shell) of the micelle. We then have to estimate the free energy of bonding of molecular fragments to form a connected molecule in the micelle. This bonding contribution to the free energy is derived from the quasichemical group-contribution model of the bulk solution. This bulk-phase model is also used for describing the free energy of mixing in the core and the shell of the micelle. As soon as the standard free energy of aggregation is specified, the monomer−micelle equilibrium may be determined, exactly in the same way as in the classical molecularthermodynamic theory of micellization,11−13 from the condition of aggregation equilibrium and the material balance equations. In this method, the standard free energy of aggregation depends on the chosen location of the boundaries of the micellar shell. In the calculated examples of this work we illustrate this dependence and show that the model interpolates between two limits: (1) the pseudophase approximation, and (2) an essentially two-dimensional model of the micelle shell that is employed in the classical micellization theory. Complex interactions between different functional groups in the outer part of the micelle are described with the aid of the bulk-phase model, whereas the micelles remain nonuniform, possessing a hydrophobic core. The pseudophase model and a conventional micellization model are recovered in the limits of thick (whole micelle) and thin (surfactant heads only) shells, respectively. The shell thickness, Special Issue: Modeling and Simulation of Real Systems Received: January 29, 2014 Accepted: June 25, 2014 Published: July 3, 2014 2995

dx.doi.org/10.1021/je500103h | J. Chem. Eng. Data 2014, 59, 2995−3002

Journal of Chemical & Engineering Data

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quasichemical approximation. However, the original approach19 is notationaly heavy and inconvenient to use. In this work we give a simplified version of the “quasichemical-bonding” approach and relate it to the quasichemical group-contribution model. The free energies of mixing and bonding of segments,gseg + gbond, are derived in what follows. 2.2. Key Equations of the Quasichemical GroupContribution Model. For the chemical potentials, we have15

the only parameter of our method additional to the model parameters that describe interactions in the bulk phase, may be estimated from the structure of the surfactant’s head or regressed from experimental data. The shell contains solvent (e.g., the hydration water) and parts of molecules of the surfactant (cosurfactant). For incompressible shells, the solvent content of the shell is specified by the assumed shell thickness as explained in section 2.5. On the scale smaller than the length of the surfactant molecule it is difficult to describe local properties with the aid of a bulk phase model. The “segments-and-bonds” formalism helps to overcome this difficulty. Chainlike molecules are cut in segments and then corrections for the chemical bonding of these segments are introduced, in complete analogy with the SAFT methodology for the chains.14 We implement our approach using the quasichemical group contribution model of a bulk solution15 with interaction parameters determined previously from phase equilibria calculations.16 The chemical bonding terms for the chemical potentials are derived from this model. We apply the model for calculating the distribution of alkanols between the nonionic micelle and the aqueous solution and for calculating the free energies of transfer of the bulky heads of nonionic surfactants from aqueous solution to the micelle. Comparison with experiment illustrates the robustness of our approach.

Δμi ≡ μi − μi0 = Δμicomb + Δμi res

Here μi is the chemical potential of component i in fluid mixture, μ0i is the chemical potential of component i in pure fluid i at the temperature of the mixture (for fluids under ambient pressure considered in this work the effect of pressure may be neglected), Δμcomb is the combinatorial part that reflects differences in sizes i and shapes of molecules, Δμres i is the residual part that reflects differences in interaction energies between different chemical groups. All chemical potentials are expressed in RT-units. The combinatorial part is calculated from the Stavermann− Guggenheim expression Δμicomb = ln φi +

⎛θ ⎞ φ z qi ln⎜⎜ i ⎟⎟ + li − i 2 xi ⎝ φi ⎠

∑ xjlj j

(3)

Here xi, φi, and θi are the mole fraction, the volume fraction, and the surface fraction, respectively, of component i in the mixture; li = 1 − ri − (z/2)(qi − ri) is the so-called bulkiness factor that reflects the molecular shape and is equal to zero for a chain-like molecule and to one for a cycle; z is the quasilattice coordination number (usually z = 10); ri and qi are dimensionless molecular size and surface area in units of the standard segment.15 The residual part is calculated in the quasichemical approximation

2. COMPUTATIONAL METHOD 2.1. Nonideal Mixing in the Micellar Shell. The micelle body is divided in two parts as illustrated in the graphical abstract. The micellization free energy is represented as a sum of interfacial (gint), core (gcore), and shell (gsh) contributions. The free energy of the hydrocarbon core includes hydrophobic and deformation contributions; these contributions and gint are calculated from the standard micellization models.17,18 The description of gsh is at the focus of our approach. Part of a surfactant’s molecule located within the shell is cut into segments (functional groups). For these free segments, we calculate the free energy of transfer (gseg) from infinitely dilute surrounding bulk solution to the micelle shell. We take into account that a certain amount of solvent (water) is also present in the micelle shell and that the average fraction of surfactant in the shell is different from that in the core owing to penetration of the solvent. A contribution to the micellization free energy comes from the connectivity of the surfactant’s chain. This contribution reflects that the segments are not free, but are bonded to one another to form the portion of surfactant chain that resides in the shell. Apart from this bonding-of-free-segments contribution (gbond), an extra term gloc is added that describes anchoring of the terminal segment of the surfactant head-portion on the surface of micellar core. Thus, we have gsh = gseg + gbond + gloc

(2)

Δμi res =

∑ qsi ln s

Xs Xsi(0)

(4)

where summation is performed over all different functional groups s belonging to the molecule of component i; qsi is the surface area of groups of type s in the molecule of component i: qi = Σsqsi; Xs is the solution of a set of the so-called “quasichemical equations” Xs ∑ αt X t ηst = 1 t

(5)

Here s and t run over all different types of chemical groups in the system, ηst = exp (−Wst/RT) is the measure of interaction energies between groups s and t, Wst is the interchange energy for this pair of groups; αt = At/A is the fraction of surface of groups t in the mixture; At = ΣjqtjNj is the total surface of groups t located on all different molecules and A = ΣjqjNj is the total surface of all molecules, Nj is the number of molecules of component j. In eq 4, Xi(0) stands for solution of quasichemical equations for pure s component i. For calculations, we need parameters ri for all different components, and qsi for all types of functional groups. We also need a set of parameters Wst that describes interactions between all different functional groups in the system. 2.3. Bonding Contribution to the Chemical Potential. Consider a mixture that contains solvent (a component having small molecules) and one or more components having large molecules. We cut large molecules into fragments and then simulate chemical bonding between the fragments by introduc-

(1)

gseg describes the entropy of mixing and interaction energy of chemically different segments in the micelle shell. Considering a uniform shell of a given composition, gseg can be readily estimated from bulk-phase models of nonideality, good candidates being SAFT and the group-contribution quasichemical model. In this work we apply the quasichemical group-contribution model. To the best of our knowledge, the first attempt to describe chemical bonding between molecular segments, much in the spirit of SAFT, was made as early as 197819 based on the 2996



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whatever is the value of the bulkiness factor of the solvent, lsolv, provided that the fragmented components have chainlike molecules, namely: li = 0 for i ≠ solv. In other words, fragmentation of long molecules does not affect the chemical potential of solvent, as it should be, and it can always be calculated with the aid of eqs 2−5. We note in passing that this would not be the case if Flory’s expression were used in place of the Stavermann−Guggenheim combinatorial part of the chemical potential, eq 3. At thermodynamic equilibrium, the chemical potential of a fragmented component, say A, is given by the sum of the chemical potentials of its fragments:

ing infinitely strong specific interactions between the surfaces of cuts. We now calculate chemical potentials in the mixture of fragments. An obvious consequence of cutting molecules in fragments is the appearance of new types of surfaces and of specific interactions between the cuts. Hence there is an increase in the number of types of groups and of the total surface of all groups in the mixture. For the mixture of molecular fragments, we use tilde to denote corresponding quantities, e.g., q̃i,θ̃i, α̃ t = Ã t/Ã , X̃ s, etc., for the surface area, surface fraction, and so on. New types of groups that correspond to chemical bonds (cuts) are denoted by subscript “chem”. For all chemically nonbonding groups s, we have α̃s = As /Ã = A s /(A +

∑

Ãchem )

chem

s

(6)

where A and As are the surface areas in the initial mixture before cutting has been performed; the summation is over all different chemically bonding groups (cuts), and Ã chem is the total surface area of all cuts. Infinitely strong attraction between two sides of each cut guarantees the stoichiometric locking of fragments back into molecules. This chemical bonding must be reflected by the solution X̃ s of the quasichemical equations that gives the numbers of different contacting pairs in the mixture: Nst ∝ Ã α̃ sα̃ tX̃ sX̃ tηst. The simplest notation is to consider both sides of each cut as groups belonging to one unique type “chem”. For any pair of identical groups we have Wss = 0 and hence ηchem−chem = 1. Infinite attraction between these groups implies: ηchem−t → 0 for all t other than “chem”, which guarantees that two sides of the cut form only pairs (chemical bonds) with one another but not with other groups present in the mixture. Each cut of a molecule corresponds to a specific value of the chem-index, e.g., chem = 1,2,.... From the set of quasichemical equations, eq 5, we have 1/2 ⎛ Ã ⎞1/2 X̃s ⎛ αs ⎞ =⎜ ⎟ =⎜ ⎟ ⎝A⎠ Xs ⎝ αs̃ ⎠

+z

̃ Xchem

∑ qt solv ln t

=

∑ qt solv ln t

res = Δμsolv +

ΔμAbond ≡ z

(11)

⎛ X̃

∑ qchemA ln⎜⎜

⎞

chem ⎟ ̃ A(0) ⎟⎠ ⎝ Xchem

(12) −1

For chemically bonding segments, we have qchemA = z . When molecule A is cut into rA fragments we have ⎛ X̃ ⎞ 1 ⎟ = (rA − 1) ln ΔμAbond = 2(rA − 1) ln⎜⎜ chem A(0) ⎟ ̃ ̃ θ ⎝ Xchem ⎠ A

(13)

Thus, each chemical bond in component A contributes a term, ln θ̃A, to the chemical potential. The total surface of constitutive fragments of a molecule A is given by

(7)

∑ qs A + ∑ qchemA = qA + s

(8)

chem

2 (rA − 1) z

(14)

and hence q̃A = rA if molecules A are chain-like (lA = 0). Putting together expressions for different parts of eq 11 and performing algebra, we obtain ΔμÃ = ΔμA + ΔμAbond + (rA − 1)ln θÃ = ΔμA

(15)

where ΔμA is given by eqs 2−5. The condition lA = 0 is essential in the derivation of eq 15. This equation is not valid for cyclic Amolecules. On the contrary, the number of cuts, (rA − 1), is immaterial: eq 13 holds for one, two, or more cuts of the Amolecule. Equations 10, 13 and 15 are the main result of this section. Equations 10 and 15 prove equivalence of the “bonding-ofsegments” formalism to that of a conventional quasichemical model for a bulk fluid. Equation 13 gives the contribution of chain connectivity (chemical bonds) to the chemical potential. Below we apply “bonding-of-segments” approach to express the free energy of the micelle shell. 2.4. Quasichemical Group-Contribution for Micellar Shell. The free energy of transfer (in RT units) of amphiphilic molecules A from dilute solution (symbol ∞) to the bulk

X̃ t X̃ tsolv(0) X̃ t X tsolv(0)

(9)

Comparing eqs 2 and 3 with this formula we see that comb res comb Δμsolv + Δμsolv = Δμsolv + Δμsolv ̃ ̃res

⎞

chem ⎟ A(0) ⎟ ̃ ⎝ Xchem ⎠

chem

qÃ =

⎛ θ ⎞1/2 q ln ⎟ ∑ t solv ⎜ solv ̃ ⎠ ⎝ θsolv t

⎛ X̃

∑ qchemA ln⎜⎜

where the combinatorial contribution is calculated from eq 3 for each fragment of molecule A, the first sum runs over all such fragments, the second sum runs over all contact sites t of molecule A that do not form chemical bonds, the last sum is performed over the chemically bonding contact sites (cuts) of molecule A. Naturally, this latter sum is called the bonding contribution to the chemical potential and is given by

for every chemical bond. We consider separately the chemical potentials of the solvent (the component with small molecules that we do not cut into fragments) and those of the components with large molecules that are fragmented (surfactant, cosurfactant, or an additive). For the solvent, we write Δμsolv ̃res =

t

chem

for all nonbonding contacts, and ⎛ 1 ⎞1/2 =⎜ ⎟ ̃ ⎝ αchem ⎠

⎛ X̃ ⎞ t ⎟ A(0) ⎟ ̃ ⎝ Xt ⎠

∑ Δμscomb ̃ A (φs̃ A , θs ̃A ) + z ∑ qt A ln⎜⎜

ΔμÃ =

(10) 2997



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ΔGA0shell = ΔμAhead (∞) + ΔμAbond (∞) ̃

solution having the composition of the micelle shell (variables without superscript) may be found from eqs 2−4:

where

ΔGA0 ≡ μA − μA*

⎛ x ⎞shell ΔμAbond (∞) = ln xA∞ + ln⎜ solv ⎟ ⎝ θsolvxA ⎠

= μA − μA∞ + ln xA∞ = ln xA + ln

φsolv xsolv

+ z ∑ qt A ln t

+

⎛θ ⎞ z qA ln⎜⎜ solv ⎟⎟ 2 ⎝ φsolv ⎠

Xt X t∞

⎛ θ ̃ ⎞shell ⎛ φsolv ⎞shell xAshell z ΔμAhead + qÃ ln⎜⎜ solv ⎟⎟ (∞) = ln ∞ + ln⎜ ⎟ ̃ xA 2 ⎝ xsolv ⎠ ⎝ φsolv ⎠

(16)

where μA and are chemical potentials and μ*A is the reference chemical potential (infinite dilution) of component A, X∞ t is the solution of the quasichemical equations for dilute solution; the summation runs over chemically nonbonding contacts. It has been assumed in the derivation that li = 0 for all components. Note that for dilute solution we may write either μ∞ * + ln x∞ A = μA A ∞ 0 comb res or μA = μA + ΔμA (∞) + ΔμA (∞) depending on the choice of the reference state. The last equality in eq 16 holds only to the first order in x∞ A but this equality becomes exact in the limit of infinite dilution, xsolv → 1. Equation 16 may also be obtained considering the transfer and bonding of molecular fragments in which case ΔG0A consists of the following contributions: ΔμAbond (∞)

− (rA − 1) ln xsolv = (rA − 1) ln ̃ xA θsolv −

ΔμÃres (∞) = z ∑ qt A ln t

+ z ∑ qt Ahead ln t

ΔGA0shell

φsolv xA xsolv

+

⎛ θ̃ ⎞ z qÃ ln⎜⎜ solv ⎟⎟ 2 ⎝ φsolv ⎠

(24)

Equation 24 is the main result of this section. It gives the free energy of transfer of the surfactant subchain in terms of composition of the micellar shell (superscript shell). To check our formulas we may write explicitly the contribution from the transfer of surfactant tail to the core (rather than estimating this contribution via the hydrophobic terms)

(17)

⎛ φsolv ⎞core xAcore z ΔμAtail (∞) = ln ∞ + ln⎜ + qAtail ⎟ xA 2 ⎝ xsolv ⎠

(18)

⎛ θ ⎞core X core + z ∑ qt Atail ln t ∞ ln⎜⎜ solv ⎟⎟ Xt ⎝ φsolv ⎠ t (19)

(25)

Equations 24 and 25 show that when local compositions in the core and in the shell are taken equal to some average micelle shell mic core shell mic core shell composition (xcore A = xA ≡ xA , φsolv = φsolv ≡ φsolv, θsolv = θsolv mic core shell mic ≡ θsolv, Xt = Xt ≡ Xt , etc.) we obtain a usual pseudophaseapproximation expression for the free energy of transfer; that is,

Here rA is the number of separate fragments of molecule A and (rA − 1) is the number of chemical bonds between these fragments. Adding together eqs 17−19, we of course recover eq 16. The advantage of the “segments-and-bonds” approach can be seen for a nonuniform fluid where different segments of a molecule are transferred to different local environments. Applying expressions for segments at local concentrations makes possible the formulation of a molecular-thermodynamic model of a fluid with spatial variations of concentration on the scale that is less than the size of a molecular chain A. In the simplest version of such a model we only distinguish between two regions of the micelle: the core and the shell. Accordingly the molecule A is cut into head and tail portions with only one chemical bond. We write for the free energy of transfer ΔGA0 = ΔμAtail (∞) + ΔμAhead (∞) + ΔμAbond (∞) ̃

(23)

⎛ θ ⎞shell X shell z = qA ln⎜⎜ solv ⎟⎟ + z ∑ qt Ahead ln t ∞ 2 Xt ⎝ φsolv ⎠ t

μAcomb (∞) + rA ln xA∞ − μAcomb ̃ ̃ = rA ln

X̃ tshell X̃ t∞

Here the summation is performed over nonbonding groups in the head portion of surfactant A. Rearrangement gives a remarkably simple formula

xA∞

X̃ t X̃ t∞

(22)

and

μ∞ A

ΔμAbond

(21)

ΔGA0 = ΔμAtail (∞) + ΔμAhead (∞) + ΔμAbond (∞) ̃ = ΔμA (∞) = ln +

⎛ φsolv ⎞mic xAmic + ln ⎜ ⎟ xA∞ ⎝ xsolv ⎠

⎛ θ ⎞mic z (qAhead + qAtail) ln⎜⎜ solv ⎟⎟ 2 ⎝ φsolv ⎠

+ z ∑ (qt Ahead + qt Atail) ln t

(20)

X tmic X t∞

(26)

where qAhead + qAtail = qA and qtAhead + qtAtail = qtA. When the shell thickness is chosen approximately equal to the size of the surfactant terminal headgroup (H ∝ lseg,(rA − 1) = 0) then both the chain connectivity term and the head anchoring term (calculated in the next section) cancel out. The model becomes essentially equivalent to that of Nagarajan and Ruckenstein,17 in which interactions in the micellar corona are described by a particular form of the equation of state (e.g., twodimensional form of the van der Waals equation of state that

The free energy of transfer of tail, ΔμA tail(∞), is calculated as in standard molecular thermodynamic models of micellization:17 it is the usual hydrophobic term for a properly shortened tail (several groups adjacent to the polar head may be included in the shell). The bonding contribution is attributed entirely to the shell and is included in the free energy of transfer (from the bulk solution into the shell) of surfactant’s chain portion that contains the head 2998



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side of eq 32 is purely geometrical and may be expressed with the aid of the formula22

describes steric repulsion of heads) and for which no solvent penetration is allowed. Equation 24 applies to any surfactant (cosurfactant) in the mixture. Hence for surfactant B, we have ΔG B0shell

2 3 Vsh ν(ν − 1) ⎛ H ⎞ ν⎛ H ⎞ H = + ⎜ ⎟ + ⎜ ⎟ AcoreR c Rc 2 ⎝ Rc ⎠ 6 ⎝ Rc ⎠

⎛ θ ⎞shell X shell z = qB ln⎜⎜ solv ⎟⎟ + z ∑ qt Bhead ln t ∞ 2 Xt ⎝ φsolv ⎠ t

Similarly, for surfactant B, we have (27)

φBshell = x Bcore

For the solvent, we have 0shell ΔGsolv = Δμsolv

From eqs 24, 27, and 28 we finally obtain gseg + gbond (per surfactant molecule) for the mixed A + B micelle shell

φsolv = 1 − φAshell − φBshell

shell 0shell + xsolv ΔGsolv )/(xAshell + x Bshell)

gsh (R c , xAcore) = gseg + gbond + gloc

(29)

Equation 29 is easily generalized for a micelle that contains several surfactants and/or additives: i

ΔG loc

Vsh = ln Acorelseg

+ H) − 3R c2lseg

head portion−to tail portion volume ratios: vAhead r vBhead r = Ahead , = Bhead vAtail rAtail vBtail rBtail surface areas for all functional groups t = 1, 2, ..., NG forming surfactant head portions and solvent molecules: qtA head,qtB head,qtsolv; and matrix of interaction parameters for these functional groups: Wst, t,s = 1, 2, ..., NG. All these parameters have been estimated previously15,23 for ca. 30 different functional groups and extensively used in phase equilibria calculations. Additionally we need to specify the length of the surfactant-chain segment that is adjacent to the core, lseg, and assign a value to the shell thickness H, for example, by saying that it corresponds to all-trans conformation of the surfactant molecule head portion. The composition and the radius of the core that are needed to calculate gsh (Rc,xcore A ) are found by minimizing the free energy of aggregation.

sphere

+ H )2 − R c2 cylinder 2Rlseg (30)

where lseg has the order of length of one fragment of surfactant’s tail (e.g.,CH2−group), H and Vsh are the shell thickness and volume, respectively, Rc and Acore are the radius and the surface area of micelle’s core. Assuming that the entropy loss is about the same for surfactant A and surfactant B, we have for a mixed A−B micelle gloc ≅ ΔG loc

3. CALCULATED RESULTS AND DISCUSSION A model mixture of C16-surfactants has been considered to study the effect of the assumed location of the core/shell boundary on the calculated free energies. For both surfactants, similar-size polar heads are taken, ca. 1.5 times larger in volume than CH3groups and about 0.17 nm in diameter. This fixes the thickness of the micellar shell and the volume fraction of water in it. The geometrical parameters for groups are taken from the groupcontribution quasichemical model of a nonideal fluid.23 The values of interaction parameters for H2O and CH3, CH2-groups are estimated based on the same model,23 those for polarA and polarB groups are set to reflect strong specific interactions between these groups and with water (Table 1). Depending on the assumed size of the core (the core radius is related to the length of the hydrocarbon chain within the core, Ncore: Rc = lCH3 + (Ncore − 1)lCH2; lCH3 = 0.15 nm; lCH2 = 0.1265 nm) the hydrophobic contribution (that comes from the core)

(31)

A crucial question is how to determine the composition of the shell. This composition is related to the shell’s thickness, a quantity whose definition is arbitrary to a certain extent. Some effective size of surfactant’s head is introduced to estimate the shell’s thickness. We assume that the composition of the micelle’s shell on the solvent free basis is the same as the composition of the core (that does not contain solvent) and thus write for the surfactant’s volume fraction φAshell =

vAhead v AcoreR c nA = xAcore Ahead Vsh vAtail (ν + 1)Vsh

(36)

size of surfactant head portions and size of solvent moelcules: rAhead , rBhead , rsolv

where the subscript i ≠ solv denotes surfactant and/or additive. From this equation, the free energy of the micelle shell is calculated in the next section. 2.5. Shell Composition and Free Energy. As discussed elsewhere,20,21 the entropy loss owing to the attachment of the head portion of surfactant chain to the micelle core may be estimated as R c3

(35)

where gseg+gbond is obtained with the aid of eq 29, gloc is found from eq 30, and shell composition is estimated from eqs 32−35. The molecular parameters that are needed for performing calculations are

shell 0shell gseg + gbond = (∑ xishell ΔGi0shell + xsolv ΔGsolv )/∑ xishell

3

(34)

Using eqs 32−35, the shell free energy may be calculated as a function of the radius and of the composition of the core

gseg + gbond = (xAshell ΔGA0shell + x Bshell ΔG B0shell

⎡ (R ⎢ ln c ⎢ =⎢ ⎢ (R c ⎢ ln ⎣

vBhead AcoreR c vBtail Vsh(ν + 1)

For incompressible micelle shells, the fraction of solvent in the shell is given by

(28)

i

(33)

(32)

where vA head and vA tail are the volumes of surfactant’s head and tail portions, nA is the micelle aggregation number, Vcore is the core volume, and v = 0,1,2 for planar, cylindrical, and spherical geometry, respectively. The rightmost fraction in the right-hand 2999



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We include in micelle shell two CH2-groups adjacent to the polar head. Group-parameter values found by regressing phase equilibria data in many different fluid mixtures23 are given in Supporting Information. With these parameters the infinite dilution activity coefficients of alkanols in bulk water are predicted quite well. The predicted values of activity coefficients in the micelle shell (Figure 2) strongly depend, as it should be, on

Table 1. Interaction Parameters for a Model Mixture of Surfactants C16A and C16B groups

CH3

CH2

polarA

polarB

O (water)

H (water)

CH3 CH2 polarA polarB O (water) H (water)

0

0 0

0.1 0.1 0

0.1 0.1 −2.0 0

0.3 0.3 −1.5 −1.5 0

0.3 0.3 −1.5 −1.5 −4.6 0

and the shell contribution, gsh (Rc, xcore A ) = gseg + gbond + gloc, both vary. Figure 1 shows contributions to the total free energy of the

Figure 2. Predicted activity coefficients of PrOH, water, and MEGA-12 polar head in the micelle shell as a function of the fraction of water in the shell (that is specified by the assumed thickness of the shell). The calculated infinite dilution activity coefficient of PrOH in the bulk aqueous phase (empty circles) is shown together with the window of experimental data (vertical bar) from different authors.24

Figure 1. Dependence of contributions to the free energy (and their total) on the assumed division of the micelle in the core and shell regions. The free energy of transfer of the hydrocarbon subchains to the core (dotted) is calculated from equations of ref 11. Starting from about eight tail segments in the core, the total value of free energy is not very sensitive to the assumed core size.

the fraction of solvent in the shell. Since this fraction, in turn, depends on the estimated geometry of the MEGA-12 head, which is by no means reliable, we have used an experimental K∞ alk value for n-PrOH to obtain γM∞ PrOH = 0.44 and hence xwater = 0.775 (this mole fraction matches calculated and experimental γM∞ PrOH). Predictions in Table 2 are made with this particular value of xwater. Figure 2 shows that the infinite dilution activity coefficient of an alkanol in the micelle shell changes as we add more water to the shell. The estimated solvent fraction for the micelle of a given shape depends on the volume and the extent of stretching of the surfactant head and is sensitive to the latter; for example, for MEGA-12 (Vhead = 0.309 nm3) a shell of thickness H = 0.65 nm leads to xwater = 0.888 for the spherical micelle, whereas H = 0.76 nm gives xwater = 0.928. Such difference in xwater significantly affects the predicted values of γM∞ alk (Figure 2) and the distribution coefficient. Nevertheless the model brackets the experimental K∞ alk values and predicted trends are correct. The calculated infinite dilution activity coefficient of PrOH in the micelle reduces to that in aqueous solution in the limit xwater → 1, that is, when we consider an infinitely large micellar shell that incorporates the bulk phase. As the shell becomes dryer the limiting activity coefficient in the shell decreases signifying the predicted stronger tendency for PrOH to concentrate in the micelle. Also from Table 2 we see that this tendency increases with the increase in chain length of alkanol. Figure 2 shows that the calculated limiting activity coefficient of PrOH in the aqueous phase is sensitive to the addition of surfactant. Nevertheless such an addition on a CMC level (less than 1% of surfactant) leads to a change of the activity coefficient that is negligibly small comparing to how this quantity changes between the bulk aqueous phase and the micelle (that contains no less than 20% surfactant in the shell). This means that the

micelle calculated from eqs 36, 31, and 29 as a function of the assumed core size for spherical micelles. Starting from large enough core (Ncore ≈ 8), the total free energy is not too sensitive to the subdivision of surfactant molecule into the core/shell portions. Another set of calculations is performed for a real system and shows infinite dilution activity coefficients of alkanols, γM∞ alk , in the micelle shells formed by N-dodecanoyl-methyl-glucamine (MEGA-12). The activity coefficients γi (ln γi = Δμ̃ i − ln xi) are calculated for xalk → 0 using eqs 11,13, and 3−5. The results are given in Table 2 that also contains experimental data on the 4 limiting distribution coefficients K∞ alk of alkanols between the micelles and the surrounding aqueous solution. Table 2. Experimental and Calculated Values of Activity Coefficients of Alkanols in Water γW∞ alk and Distribution Coefficients of Alkanols between MEGA-12 Micelle and Aqueous Phase at Infinite Dilution (323.2K). Superscripts M and W: Micelle Shell and Aqueous Phase W∞ M∞ K∞ alk = γalk /γalk

γW∞ alk

γM∞ alk

alkanol

exptl4

calc

exptl24

calc

EtOH n-PrOH n-BuOH

5.3 34.3 102

4.4 34.3 286

3.44−4.74 12−17 41−50

4.4 15 54

calc (xwater = 0.775) 0.99 0.44 0.19 3000



Article

Table 3. Set of Model Parameters for m-Ethylene Glycol Mono-n-alkyl Ether Surfactants (CnEm) and Their Aqueous Mixtures16 groups

CH2

CH2CH2O

O (H2O)

2H (H2O)

O (gly)

H (gly)

CH2 rCH2 = 1.09 lCH2 = 1.0 CH2CH2O rCH2CH2O = 2.50 lCH2CH2O = 1.0 O (H2O) rO = 1.19 lO = 1.0 2H (H2O) rH = 0.0 lH = 0.0 O (gly) rO = 1.01 lO = 0.9 H (gly) rH = 0.0 lH = 0.5

0

0.0 0

0.0345 −0.2099 0

0.0345 −2.7472 −4.988 0

0.0165 0.0 0.0 −5.60 0

0.0165 −3.032 −5.60 0.0 −4.844 0

4. CONCLUSIONS A new molecular thermodynamic micellization model has been formulated in which a micellar shell is described as a nonideal mixture of surfactant head portions. Complex interactions between functional groups in the micelle corona are treated with the aid of the quasichemical group-contribution approach that is well developed for calculating phase equilibria in nonideal fluid mixtures. As compared to previously known micellization models the advantage of the new model is the possibility of a detailed description of strong specific interactions between different chemical groups in the micelle. The “segments-and-bonds” formalism has been combined with the quasichemical group contribution approach; correction for the connectivity of molecular chains (chemical bonding of groups) has been derived for the chemical potentials. The model has been applied to calculate the micelle-aqueous solution equilibria for micelles of nonionic surfactant Ndodecanoyl-methyl-glucamine in the presence of alkanols. For CnEm of varying chain length, we calculated the free energy of transfer from aqueous solution to the micelle. The agreement with experiment shows that the new model may be used for describing the effect of strong specific interactions and nonidealty of the swollen micelle corona. Although calculated examples in this work are limited to the mixtures that contain two surfactants or a surfactant and a cosurfactant (alkanol) the proposed method is general and may be applied to mixed micelles composed of several surfactants and a number of penetrating solutes/cosurfactants. Our general approach to describing specific interactions in the shell is not limited to the quasichemical approximation. An attractive alternative is SAFT. Free energy bonding corrections from SAFT are well-known. SAFT takes into account the compressibility of the fluid and may help to describe the effects of pressure in micellar systems. Modeling micelle-solution equilibrium for a wide range of mixed micellar systems, application of SAFT, and modeling shape transitions are among the prospects for future work.

presence of surfactant monomers at CMC may not significantly change the limiting value of the distribution coefficient. This is no longer true when the solvent contains an additive in a large proportion. The general method of this work is still viable but the activities of the additives and of the solvent must be employed (instead of mole fractions) together with the condition of the monomer−aggregate equilibrium.11 For testing our model, we have also used experimental CMC (critical micelle concentration) for individual nonionic surfactants having identical tails and different heads. For such surfactants, the difference in the logarithms of CMC gives an estimate of the free energy increment, ΔA/B, of transferring two different polar heads A and B from dilute aqueous solution into the micelle shell. The model makes possible direct calculation of the transfer free energy for polar surfactant head from dilute aqueous solution to the micelle shell, eq 24; hence, the difference for surfactants R-A and R-B can be computed. Calculations have been performed for the CnEm surfactant series. Model parameters (Table 3) have been estimated previously from vapor−liquid equilibrium data.16 One CH2 group adjacent to the polar surfactant head has been included into the shell that contains [−CH2(OCH2CH2)mOH] surfactant molecule portions and water. Estimated volume of this portion for m = 8 (V = 0.552 nm3) is close to the estimate given in the literature25 (V = 0.550 nm3). For CnEm spherical micelle with the aggregation number of 34 (and H ≈ 1.6 nm), this gives ca. 2600 water molecules in the micelle shell. An experimental estimate of oxyethylene group hydration number in micelles (7.8−9.5 water molecules per group26) gives an average of 2540 hydration water molecules per micelle, in good agreement with our estimate. Both values lead to a water mole fraction of 0.987 in the shell. Table 4. Differences between the Transfer Free Energy of Surfactant Head from Water to a CnEm Micelle for Polar Heads of Different Size. Experimental Values Are Estimated from CMC for Two Surfactants with Same n and Different m (m = i and m = j) n = 10 i/j ΔEi/Ej

calc exptl25,27

n = 12

■

6/4

8/6

8/4

9/7

0.59 0.69

0.13 0.31

1.5 1.0

1.0 0.4

ASSOCIATED CONTENT

* Supporting Information

n = 16

S

Table S1 of model parameters described in the text. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +7 812 328 0526. E-mail: [email protected].

Calculated values of ΔEi/Ej (in RT units) are given in Table 4 for surfactants differing in lengths of the head portions (i and j) and tails (n). Model predictions are in reasonable agreement with experiment. The quality of description is similar to that from a purely empirical correlation based on regression of data for CnEm surfactants.27

Funding

For financial support, we thank Saint-Petersburg State University (Project 12.38.199.2014), CRDF (Project No. rco1237), and the Russian Foundation for Basic Research (Project No. 12-0300977-a). 3001



Article

Notes

(18) Andreev, V. A.; Victorov, A. I. Molecular thermodynamics for micellar branching in solutions of ionic surfactants. Langmuir 2006, 22, 8298−8310. (19) Smirnova, N. A. Lattice model for the surface region of solutions consisting of different-sized molecules with orientation effects. Fluid Phase Equilib. 1978, 2, 1−25. (20) Goveas, J. L.; Milner, S. T.; Russel, W. B. Corrections to strongstretching theories. Macromolecules 1997, 30, 5541−5552. (21) Mayes, A. M.; De La Cruz, M. O. Cylindrical versus spherical micelle formation in block copolymer/homopolymer blends. Macromolecules 1988, 21, 2543−2547. (22) Victorov, A. Curvature elasticity of a weak polyelectrolyte brush and shape transitions in assemblies of amphiphilic diblock copolymers. Soft Matter 2012, 8, 5513−5524. (23) Victorov, A. I.; Fredenslund, A. Application of the hole quasichemical group contribution equation of state for phase equilibrium calculation in systems with association. Fluid Phase Equilib. 1991, 66, 77−101. (24) Morachevsky, A. G.; Smirnova, N. A.; Balashova, I. M.; Pukinsky, I. B. Thermodynamics of Infinitely Dilute Solutions. Khimia: Leningrad, 1982. (25) Zoeller, N.; Lue, L.; Blankschtein, D. Statistical-thermodynamic framework to model nonionic micellar solutions. Langmuir 1997, 13, 5258−5275. (26) Shiloach, A.; Blankschtein, D. Measurement and prediction of ionic/nonionic mixed micelle formation and growth. Langmuir 1998, 14, 7166−7182. (27) Ravey, J. C.; Gherbi, A.; Stebe, M. J. Comparative study of fluorinated and hydrogenated nonionic surfactants. I. Surface activity properties and critical concentration. Prog. Colloid Polym. Sci. 1988, 76, 234−241.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The computational resources provided by the Computer Resource Center of Saint-Petersburg State University are acknowledged.

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REFERENCES

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Modeling of Micelle-Solution Equilibria for Mixed Nonionic Micelles

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