Journal of Colloid and Interface Science 440 (2015) 109–118

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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis

Explaining the growth behavior of surfactant micelles L. Magnus Bergström ⇑ KTH Royal Institute of Technology, School of Chemical Science and Engineering, Department of Chemistry, Surface and Corrosion Science, SE-10044 Stockholm, Sweden

a r t i c l e

i n f o

Article history: Received 11 August 2014 Accepted 18 October 2014 Available online 4 November 2014 Keywords: Surfactants Micelles Spontaneous curvature Bending rigidity Saddle-splay constant Second CMC

a b s t r a c t The growth behavior of surfactant micelles has been investigated from a theoretical point of view. It is demonstrated that predictions deduced from the spherocylindrical micelle model, which considers micelles that are only able to grow in the length direction, are inconsistent with experimental measurements. Accordingly, the rise in aggregation numbers above a certain concentration, roughly corresponding to the second critical micelle concentration, appears to be much stronger than predicted by the spherocylindrical micelle model. On the other hand, predictions deduced from the general micelle model, which considers micelles that are able to grow with respect to both width and length, show excellent agreement with experimental observations. The latter theory is based on bending elasticity and it is demonstrated that the associated three parameters spontaneous curvature, bending rigidity and saddle-splay constant may all be determined for a micellar system from experimental measurements of the aggregation number as a function of surfactant concentration. The three parameters turn out to influence the appearance of a micellar growth curve rather differently. In accordance, the location of the second cmc is mainly determined by the saddle-splay constant and the bending rigidity. The shape of the growth curve, when going from the region of weakly growing micelles at low surfactant concentrations to strongly growing micelles above the second cmc, is mainly influenced by the bending rigidity. Ó 2014 Published by Elsevier Inc.

1. Introduction Surfactant molecules spontaneously self-assemble in an aqueous solvent above a certain concentration, the critical micelle concentration (cmc), to form micelles or bilayers. One well-known feature of surfactant micelles is their ability to grow in size as a response to, among other things, increasing surfactant concentration [1,2]. The extent to which micelles grow in size may differ widely among different surfactants and surfactant mixtures depending on molecular structure of the surfactant (tail length, size of head group, charge number, length of spacer in dimeric Gemini surfactants, etc.) as well as solution properties (electrolyte concentration, surfactant composition in mixed micelles, etc.) [1,3]. From a practical point of view it is very important to understand the growth behavior of micelles since it may, to a large extent, determine macroscopic properties of surfactant solutions such as viscosity, consistency, solubilization properties, etc. There are a number of techniques available to determine aggregation numbers of surfactant micelles of which, perhaps, the most important ones are time-resolved fluorescence quenching (TRFQ), static light scattering (SLS) and small-angle neutron and X-ray ⇑ Fax: +46 8 20 82 84. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jcis.2014.10.054 0021-9797/Ó 2014 Published by Elsevier Inc.

scattering (SANS and SAXS). With these techniques the aggregation number as a function of surfactant concentration have been determined for several common surfactants and the resulting growth behaviors are found to strongly depend on chemical structure of the surfactant as well as on solution properties. The growth behaviors of the cationic surfactants dodecyl trimethyl ammonium bromide and chloride (DTAB and DTAC, respectively) [4,5] with respect to surfactant concentration have been found to be very weak. The aggregation numbers of micelles formed by the anionic surfactant sodium dodecyl sulfate (SDS) [6,7] or the cationic surfactant hexadecyl trimethylammonium bromide (CTAB) [8] have been found to increase rather weakly with surfactant concentration but yet stronger than for DTAB/DTAC. The extent of growth of both SDS [9–12] and CTAB [13] micelles is found to significantly increase upon addition of salt whereas the growth behavior of DTAB micelles only slightly depends on electrolyte concentration [10]. One illuminating investigation of the changes in growth behaviors of micelles by means of modifying the chemical structure of surfactant molecules have been carried out by Lianos et al. [4] In accordance, micelles formed by cationic surfactants of the type C12H25(CmH2m+1)N(CH3)2Br (abbreviated 12, m-Br) were studied with TRFQ. It was found that micelles formed by surfactants with m = 1–4 do not grow at all with concentration within experimental

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errors. In contrast, the two surfactants 12, 6-Br and 12, 8-Br, i.e. with a longer hydrocarbon chain attached to the quaternary amine head group, exhibit completely different growth behaviors. In accordance, both surfactants were observed to grow weakly at low surfactant concentrations followed by a much stronger growth behavior beyond a certain surfactant concentration. Such a more or less abrupt increase in the extent of growth has been observed for several surfactant systems and the surfactant concentration where the aggregation numbers start to grow rapidly is often referred to as the second cmc [14–21]. Another common example of a group of surfactants that may display a second cmc is dimeric Gemini surfactants, the growth behavior of which has been found to depend on the length of the spacer group that links the two unimeric units of a Gemini surfactant. [3] In accordance, Gemini surfactants with a short spacer group have been found to grow strongly with increasing surfactant concentration whereas Gemini surfactants with a longer spacer group grow much more weakly [22–24]. In the past, the growth behavior of surfactant micelles has usually been rationalized in terms of small and strictly spherically shaped micelles that may grow in only one direction to form cylindrical rod-shaped micelles with hemi-spherical end caps, so called spherocylindrical micelles [14,19,25–30]. In accordance, the observation of a second cmc, and the transition from weakly to strongly growing micelles, has usually been interpreted as the result of a transition from spherical to rod-shaped micelles, a so called sphere-to-rod transition [31,32]. However, due to geometrical constraints, micelles can only be strictly spherically shaped below a certain aggregation number. For instance, surfactants with a tail consisting of C12 aliphatic chain has a maximum aggregation number Nmax = 56 [33]. As a consequence, already Tartar [34], more than 50 years ago, and later on Tanford [33], realized from simple geometrical considerations that micelles formed by common surfactants, that grow rather weakly with surfactant concentration, are too large for being strictly spherical and must assume some kind of non-spherical shape. Likewise, micelles showing a second cmc, e.g. 12, 6-Br and 12, 8-Br [4], usually start to grow strongly in size at an aggregation number largely exceeding Nmax, indicating that the second cmc does not correspond to a sphere-to-rod transition. The present paper is organized as follows. In Section 2.1, the micellar growth behavior according to the conventional spherocylindrical micelle model is derived from fundamental equations based on the thermodynamics of self-assembly and it is demonstrated that the spherocylindrical micelle model is inconsistent with experimental results. In Section 2.2 we make a brief outline of the basic principles and equations behind the recently derived general micelle model [35]. This theory is based on thermodynamics of self-assembly combined with bending elasticity theory. In Section 3, we propose a completely novel approach to rationalize the growth behavior of surfactant micelles. Predictions based on the general micelle model are being compared with experimentally determined aggregation numbers and their dependence on surfactant concentration. In accordance, we are able to demonstrate that the growth behavior of ordinary surfactant micelles may be rationalized in terms of the three bending elasticity constants spontaneous curvac ). ture (H0), bending rigidity (kc) and saddle-splay constant (k

2. Methods 2.1. The growth behavior of spherocylindrical micelles 2.1.1. Thermodynamics of self-assembly and the size distribution of micelles The process of self-assembling surfactant molecules to form micelles is unfavorable from an entropy of mixing point of view

and always imposes a driving force endeavoring to dissociate surfactant self-assemblies and favoring small aggregates. Moreover, it is possible to demonstrate from straightforward thermodynamic arguments that the driving force for micelles becoming smaller in size increases with decreasing free surfactant concentration [36]. As a result, the entropic driving force of dissociating micelles must increase with decreasing total surfactant concentration giving rise to the important property of micelles growing in size with increasing surfactant concentration. The growth behavior of micelles with a minimum aggregation number Ns may be rationalized from the following completely general expression for the size distribution of surfactant micelles [25,37]

/mic ¼

Z

1

eEðNÞ=kT dN ¼

Ns

Z

1

Ns

/Nfree eNDlmic =kT dN

ð1Þ

k is Boltzmann’s constant, T is the absolute temperature and /mic and /free are the concentrations of surfactant aggregated in micelles and free surfactant, respectively. Dlmic denotes the ‘‘interaction’’ free energy per molecule of forming a micelle and E(N)  N(Dlmic  kT ln /free) is the free energy of forming a single micelle out of N surfactant free monomers. More details including the full derivation of Eq. (1) is provided in the Supplementary Material. 2.1.2. The spherocylindrical micelle In Eq. (1), we have denoted quantities related to concentrations /free and /mic, where / stands for volume fraction. The use of volume fraction rather than mole fraction in the logarithmic terms in expressions for the entropy of mixing particles or molecules with different sizes is well established and was derived independently by Flory and Huggins for case of polymeric molecules dispersed in a solvent of small molecules [38–40]. The Flory– Huggins derivation treats the macromolecules as flexible species but the expression has been demonstrated from more general arguments to also be valid for rigid particles mixed with smaller solvent molecules. For instance, based on lattice statistical mechanics Guggenheim and coworkers have derived expressions for the entropy of mixing particles of various size and shape with smaller solvent molecules [41]. For all cases treated, the expression for the free energy of mixing with volume rather than mole fraction in the logarithmic terms was obtained in the limit where the solvent is treated as a continuum medium. For the special case of solute and solvent with equal molecular volumes, the two concentration quantities become identical, volume fraction may be replaced with mole fraction, and the conventional free energy of mixing expression is recovered. It is perhaps less known that the choice of volume or mole fraction in the expression for the free energy of mixing has a large and decisive impact on the predicted growth behavior of surfactant micelles. We may illustrate this with the example of rodlike or spherocylindrical micelles growing exclusively in the length direction. The free energy of a spherocylindrical micelle may be written as a sum of two contributions [25–27]

EðNÞ=kT ¼ a þ bN

ð2Þ

The free energy of forming the cylindrical part of a micelle out of free surfactants in solution (bN) is an extensive quantity that is proportional to the aggregation number, where b is the free energy per surfactant aggregated in the cylinders, whereas the free energy of forming the hemispherical end-caps out of surfactants aggregated in the cylindrical part (akT) is constant with respect to aggregation number. Inserting Eq. (2) in Eq. (1) gives the total volume fraction of spherocylindrical micelles

L.M. Bergström / Journal of Colloid and Interface Science 440 (2015) 109–118

/mic ¼

Z

1

eabN dN ¼

Ns

eaNs b b

ð3Þ

where the lower limit of the aggregation number Ns for spherocylindrical micelles corresponds to the number of surfactant molecules in the hemi-spherical end caps. The volume-weighted average aggregation number may be calculated from the size distribution function as

R1 hNi ¼

NeabN dN

RNs1 Ns

1 ¼ þ Ns b eabN dN

ð4Þ

Notably, for large aggregation numbers (hNi  Ns), the free energy parameter b  1/hNi assumes values much lower than unity (b  1) and b ? 0 in the limit hNi ? 1. Eliminating b by combining Eqs. (3) and (4) gives the following relation between surfactants aggregated in micelles and average aggregation number

/mic ¼ ðhNi  Ns Þe

aN s =ðhNiNs Þ

ð5Þ

In the limit hNi  Ns, Eq. (5) may be simplified so as to give /mic = hNiea and the following relation for the maximum extent of growth of spherocylindrical micelles

hNi ¼ ea /mic

ð6Þ

according to which hNi is proportional to the concentration of surfactant aggregated in the micelles. As a matter of fact, this result for the growth behavior of spherocylindrical micelles may be rationalized by a simple thought experiment. Consider the case of micelles that from an energetic point of view have an optimal aggregation number. In this case, the free energy of forming an additional micelle out of N free monomers is much lower than the free energy of adding an additional surfactant to N already existing micelles. In other words, adding an increasing amount of surfactant to the solution would result in an increase in the number of micelles while keeping the aggregation number unaffected and, as a consequence, the micelles do not grow at all with changing surfactant concentration. Long spherocylindrical micelles, on the other hand, differ conspicuously from this case since the free energy of adding surfactant molecules to the cylindrical part of a micelle is much lower than the free energy of forming the hemi-spherical end caps, i.e. a/Ns  b. This means that the free energy of incorporating N free monomers into existing micelles is much lower than the free energy of making a completely new micelle. As a result, by means of, for instance, doubling the surfactant concentration would cause the spherocylindrical micelles to become twice as big, in accordance with hNi / /mic, whereas the number of micelles would remain unaffected. If, on the other hand, mole fraction is employed in the logarithmic terms in the free energy of mixing expressions, a completely different growth behavior is obtained. Replacing volume fraction /mic with mole fraction xmic in Eq. (1) gives,

xmic ¼

Z

1

eabN dN

ð7Þ

Ns

For dilute solutions we may convert from mole to volume fraction as follows

/mic ¼

V mic

vw v surf ¼ vw

xmic ¼

v surf vw

eaNs b 2

b

Z

1

111

R1

N2 eabN dN 1 b2 N2s þ 2bNs þ 2 s hNi ¼ RN1 ¼ b bNs þ 1 NeabN dN Ns

ð9Þ

Eq. (9) may be rewritten (for positive values of b) as

bNs ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðhNi  Ns Þ2 þ 8Ns ðhNi  Ns Þ  ðhNi  2Ns Þ 2ðhNi  Ns Þ

ð10Þ

which may be combined with Eq. (8) in order to eliminate b. In the limit hNi ? 1, bNs (and b) approaches zero and Eqs. (8) and (9) may be combined so as to give

/mic ¼

v surf vw

ea 2

b

¼

ea v surf hNi2 4v w

ð11Þ

or

hNi ¼ 2ea=2



vw v surf

1=2

/1=2 mic

ð12Þ

This expression for the growth of cylindrical micelles on the form hNi / /1=2 mic , in the limit of large hNi, is well-known and has previously been derived by a number of authors [19,25,26,28]. Notably, the growth behavior according to Eq. (12) is much weaker than the one according to Eq. (6), consistent with using volume fraction in the expression for the entropy of mixing. However, in accordance with our arguments above, Eq. (12) is not consistent with the thermodynamics of mixing particles of unequal size. Moreover, a growth behavior on the form hNi / /1=2 mic is inconsistent with the above described thought experiment, according to which an additional amount of added surfactant must contribute to increase the aggregation number of already existing micelles without changing the number of micelles in the limit of large hNi. According to the spherocylindrical micelle model (also frequently referred to as the ladder model [26]), the growth behavior of micelles is expected to be on the same form for all surfactant systems (provided the composition of aggregated surfactant in the case of mixed micelles is fixed). In accordance, the extent of growth is expected to increase rather smoothly with surfactant concentration. The upper limit for the extent of growth is hNi / /mic, according to Eq. (6), and for the case of mole fractions being employed in the free energy of mixing expressions, i.e. Eqs. (8) and (10), the extent of growth is even weaker with hNi propor1=2 tional to /mic in the limit of large aggregation numbers. In a log–log plot the spherocylindrical micelle models display straight lines with slopes equal to 1 and ½, respectively. Notably, according to the spherocylindrical micelle mode, only the first micelle (or first few micelles) that is formed at the (first) cmc is strictly spherical in shape. As the surfactant concentration is raised above the cmc, the micelles grow extensively into a non-spherical (that is spherocylindrical) shape. The spherocylindrical micelle model was primarily set up with the intention to deal with comparatively small rigid rods that grow exclusively in the length direction and the model does not explicitly take into account effects due to flexibility of the micelles [25,26]. However, later on it has been demonstrated that micelle flexibility is not expected to influence the predicted growth laws valid for spherocylindrical micelles [28,42]. 2.2. The general micelle model

NeabN dN

Ns

ðbNs þ 1Þ

ð8Þ

where Vmic = Nvsurf is the volume of a micelle and vsurf is the volume of a surfactant molecule. The volume weighted average aggregation number then equals

Comparison with experimentally obtained aggregation numbers indicates that the spherocylindrical micelle model of micellar growth is not correct. This lack of agreement between the spherocylindrical micelle model and experimental results has also been found when geometrical shapes of micelles have been experimentally determined with small-angle neutron scattering (SANS)

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[10,43–48]. In accordance, in contrast to a simple one-dimensional growth behavior, several common surfactants have been found to self-assemble to form triaxial tablet-shaped micelles, with a distinct thickness, width and length that may grow (or shrink) with respect to both width and length. In order to rationalize the formation and growth behavior of generally shaped micelles, we have recently set up a theoretical model for the formation of triaxial tablet-shaped micelles, with distinct thickness, width and length (the general micelle model) [35]. A schematic illustration of the geometrical shape of tablets we have considered in the model is shown in Fig. 1 in Ref. [35] as well as in the Supplementary Material. The free energy of a single arbitrarily shaped micelle with interfacial area A and interfacial tension c0 can be deduced from the well-known Helfrich-expression [49], i.e.

E ¼ c0 A þ 2kc

Z

c ðH  H0 Þ2 dA þ k

Z

KdA

ð13Þ

The Helfrich expression introduces three important parameters related to different aspects of bending or curving a self-assembled surfactant monolayer, (i) the spontaneous curvature (H0), (ii) c ). The three bending rigidity (kc) and (iii) saddle splay constant (k bending elasticity constants may be interpreted as thermodynamic parameters and calculated from a suitable molecular model by means of minimizing the free energy per molecule of a surfactant interfacial layer at given values of H and K [50–52]. Evaluating Eq. (13) for tablet-shaped micelles gives the following expression for the free energy of forming generally shaped tablets [35]

Eðr; lÞ ¼ a þ dwðrÞbðpr þ lÞ þ 2rðpr þ 2lÞk kT

increases monotonously with the hydrophilic–lipophilic balance c reaches a (HLB) of a surfactant molecule, whereas both kc and k maximum at some optimal value of the HLB [52]. Similar to the spherocylindrical micelle model, the general micelle model does not explicitly take into account flexibility of the micelles. 3. Results and discussion 3.1. Rationalizing the growth behavior of surfactant micelles employing the general micelle model The growth behaviors of micelles formed by the two cationic surfactants 12, 6-Br and 12, 8-Br, as investigated by Lianos et al. [4], are shown in Figs. 1 and 2, respectively. It is seen that both surfactants display a conspicuous abrupt increase in aggregation numbers when going from a region of weakly growing micelles to a region of strongly growing micelles. This particular behavior appears to be typical for a large number of micellar systems and the concentration above which micelles begin to grow more strongly in size has traditionally been denoted the second cmc [14–16,19,20].

ð14Þ

By means of inserting Eq. (14) in the size distribution function in Eq. (1), the following important relation for the volume fraction /mic of triaxial, generally shaped, micelles as a function of the dimensionless length l  L/n and dimensionless half width r  R/n have been deduced [35]

/mic ¼

pn6 ea

v^ 2

Z 0

1

8r 2 þ 6pr þ p2 dwðrÞpbr2pkr2 dr e b þ 4kr

ð15Þ

n denotes the half thickness of the tablet-shaped micelles and v^ is the surfactant molecular volume. k  n2 cp =kT is the reduced and cp the real planar interfacial tension of the surfactant c  8nkc H0 Þ=kT þ 4pk; layer forming the micelles. a  2pð3kc þ 2k b  pkc ð1  4nH0 Þ=kT þ 2pk and d  2pkc =kT are three dimensionless parameters taking into account the bending free energy. The w-function (0 < w < 1) equals unity in the limit r ? 0 and zero as r ? 1. More details, including the complete derivation of Eq. (15), are provided in Ref. [35]. The distributions in length and width of generally shaped micelles may be deducted from Eq. (15) and, as a consequence, the average width hXi  2(hRi + n) and length hKi  hLi + hXi as functions of the concentrations of aggregated surfactants may be deduced. By means of dividing the average volume of a micelle with the surfactant molecular volume, it is thus straight forward to calculate the micelle aggregation number as a function of /mic. A micelle may be considered to be made up by a single curved monolayer and the general micelle model is generally valid for any surfactant system that is able to form a monolayer. Surfactant properties such as chemical structures of head and tail, whether the surfactant is anionic, cationic or nonionic, ionic strength, temperature, type of counter-ion, and pure as well as mixed systems do not change the validity of Eq. (15). These more specific effects are all taken into account by means of influencing the quantitative c values of each of the three bending elasticity constants H , k and k 0

c

[50–52]. Among other things, it is demonstrated that kcH0

Fig. 1. Linear (a) and log–log plot (b) of aggregation number (N) plotted against volume fraction /mic for micelles formed by the surfactant 12, 6-Br. Symbols represent experimental data obtained from time resolved fluorescence quenching (TRFQ) in Ref. [4]. The solid line represents theoretical predictions from the general c have been micelle model, where the three bending elasticity constants H0, kc and k optimized to generate best agreement between theory and data. The resulting c ¼ 8:1kT and nH = 2.0. The predictions from the spherocyvalues are kc = 0.5kT, k 0 lindrical micelle models in accordance with Eq. (5) (dash-dotted line) and Eqs. (8) and (10) (dash-dot-dotted line) have also been included. The free energy parameter a/kT was set to 6.0 (dash-dotted line) and 8.9 (dash-dot-dotted line) and the minimum aggregation number Ns = 30 in order to optimize the growth behavior with respect to experimental observations below about /mic = 0.02.

L.M. Bergström / Journal of Colloid and Interface Science 440 (2015) 109–118

Fig. 2. Aggregation number (N) plotted against volume fraction /mic for micelles formed by the surfactant 12, 8-Br. Symbols represent experimental data obtained from time resolved fluorescence quenching (TRFQ) in Ref. [4]. The solid line represents theoretical predictions from the general micelle model, where the three c have been optimized to generate best bending elasticity constants H0, kc and k c ¼ 9:5kT agreement between theory and data. The resulting values are kc = 0.5kT, k and nH0 = 2.0. The predictions from the spherocylindrical micelle model in accordance with Eq. (5) (dash-dotted line) and Eqs. (8) and (10) (dash-dot-dotted line) have also been included. The free energy parameter a/kT was set to 7.0 (dashdotted line) and 10.0 (dash-dot-dotted line) and the minimum aggregation number Ns = 30.

The micelles reach a highest measured aggregation number equal to N = 390 (12, 6-Br). The rather low values of the aggregation numbers demonstrate that both 12, 6-Br and 12, 8-Br form micelles of moderate size rather than bilayers. Small aggregates indicate a high hydrophilic–lipophilic balance suggesting that the chain with 6 and 8 carbons, respectively, is mainly located in the outer head group region. Moreover, an aggregation number equal to 400 corresponds to a cylinder length of about 200 Å, which is a typical value for the persistence length of micelles formed by surfactants with a C12 hydrocarbon chain as tail [10,11]. Hence, we may conclude that elongated micelles formed by 12, 6-Br and 12, 8-Br are shaped as rigid rods rather than flexible worms in the entire range of measured concentrations, i.e. below as well as above the second. It is seen that both the spherocylindrical and general micelle model predict a weak growth behavior at low surfactant concentrations and it is difficult to distinguish between the two models in this range of concentrations. As a matter of fact, the spherocylindrical micelle model may be regarded as a special case of the general micelle model, valid at concentrations much lower than the second cmc. As a result, the spherocylindrical micelle model has been found to be able to rather accurately describe the growth behavior of micelles with a high second cmc or in very dilute regions [26,31]. It is, however, evident from Figs. 1 and 2 that neither of the growth behaviors anticipated for spherocylindrical micelles at higher surfactant concentrations agree with experimental observations. While the micellar growth in accordance with the spherocylindrical micelle model agree well with the moderate growth behavior at low surfactant concentrations, the model does not at all take into account the progressive increase in aggregation number observed in a linear plot [cf. Fig. 1a] and the rapid rise in N typical for systems exhibiting a second cmc. In mathematical terms, the progressive behavior of the experimental growth curve is characterized by a positive second derivative of N with respect to /mic, whereas this quantity is negative according to the spherocylindrical micelle model.

113

A progressive growth behavior that appears much stronger than anticipated for spherocylindrical micelles have previously been observed for several micellar systems. These include, for instance, the cationic unimeric surfactant dodecyl ammonium chloride [5], dimeric Gemini surfactants with a short spacer group (for instance 12-3-12) [22–24] as well as the numerous number of systems where the presence of a second cmc has been reported [14–21]. It is also well-known that long wormlike micelles, in general, may display an enormous concentration-induced growth behavior, largely exceeding that predicted by the spherocylindrical micelle model [53–55]. Theoretical N versus /mic curves may be deduced from the general micelle model in accordance with Eq. (15). In Figs. 1 and 2 we have plotted the growth behaviors as predicted from the general micelle model for the surfactant 12, 6-Br (Fig. 1) and 12, 8-Br (Fig. 2). The curves were obtained by means of adjusting the three c so as to optimize the bending elasticity constants H0, kc and k agreement between theory and experiments. It is seen that the general micelle model gives excellent agreement with experimental observations in the entire range of surfactant concentrations. Notably, in conspicuous contrast to the spherocylindrical micelle model, the general micelle model is able to account for the progressive increase in aggregation numbers evident in the linear plot in Fig. 1a, and the sharp rise in N above the second cmc. The bending elasticity constants as determined from our analysis equal kc = 0.5kT and nH0 = 2.0 for both 12, 6-Br and 12, 8-Br, whereas the saddle-splay constant differs between the two surfactants, i.e. c ¼ 9:5kT for 12, 8-Br [cf. Section c ¼ 8:1kT for12, 6-Br and k k 3.3.1 below]. In Section 3.3 below, we will demonstrate that the detailed appearance of the growth curves largely depends on the actual values of the three bending elasticity constants. As a result, the general micelle model is expected to be generally valid and, in contrast to the spherocylindrical micelle model, it may account for the diversity of growth behaviors observed for different surfactants and micellar systems. As a matter of fact, by means of tuning c , thus let them the three bending elasticity constants H0, kc and k assume appropriate values, it is possible to obtain excellent agreement between the general micelle model and virtually any experimentally obtained growth curve. In Figs. 1 and 2, we have chosen to make a detailed comparison between theory and experiments provided by Lianos et al. [4] for the two surfactants 12, 6-Br and 12, 8-Br, respectively. The reason for this particular choice of experimental study is the high quality of these sets of data in the sense that they comprise several points in a wide regime of surfactant concentrations extending from the weakly growing regime at low surfactant concentrations to the strongly growing regime at high concentrations. It is not possible to determine all three of the bending elasticity constants from experimental data that do not cover the entire regime of concentrations on both sides of the second cmc. It is obvious from Figs. 1 and 2 that it is more difficult to generate a good agreement between theory and experiments as data extends into the strongly growing regime beyond the second cmc, where differences between various theoretical models become particularly evident. Although the experiments published in Ref. [4] were carried out more than 30 years ago, no one has to our knowledge, until the present work, been able to explain the appearance of the growth behaviors displayed in Figs. 1 and 2. 3.2. Correlation between micellar shape and growth behavior In order to rationalize the growth behaviors predicted by the general micelle model one has to take a closer look at the geometrical shape of the micelles and its correlation to the growth. In Fig. 2 we have plotted the average length and width, respectively,

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of generally shaped tablet-shaped micelles as deduced from Eq. (15) for the same case of 12, 6-Br surfactant micelles as shown in Fig. 1. It is seen that the micelles grow weakly with respect to both width and length at low surfactant concentrations. In this region the local curvature of the micelles must change as they grow in size which is unfavorable from an energetic point of view and, as a consequence, the extent of growth becomes inhibited. Adding an additional amount of surfactant in the regime of weakly growing micelles mainly results in an increase in the number of micelles. At higher surfactant concentrations, the micelles grow more strongly in length and the corresponding curve in Fig. 3 much resembles the growth in aggregation number as seen in Fig. 1. However, as the length begins to grow more strongly above about the second cmc, the width reaches a maximum and, as matter of fact, decreases at concentrations beyond the second cmc. The reason for this behavior may be found in the last term in the free energy expression in Eq. (14) which couples width (r) and length (l). As a result, only micelles with a sufficiently small width may grow substantially in length. Consequently, within a size distribution there is a strong correlation between width and length in the way that the width decreases with increasing length, implying that the longest micelles in a distribution are close to spherocylindrical in shape [35,56]. This also means that micelles, in order to grow substantially in length, must reduce their width as predicted by the general micelle model. As the micelles grow more strongly and become less wide, the semi-toroidal end caps become more curved, thus raising the curvature energy (terms involving a and d in Eq. (14)). As a consequence, the tendency of increasing the number of micelles upon adding more surfactant at low concentrations become weakened and eventually ceases as the concentration is further raised. Consequently, the micelles start to grow more strongly in size at higher surfactant concentrations giving rise to the abrupt rise in aggregation numbers at about the second cmc. Eventually, above the second cmc, the extent of growth may exceed what is anticipated for spherocylindrical micelles which means that the end cap energy have grown so large in magnitude that the micelles begin to decrease in number upon further increasing the surfactant concentration. 3.3. Role of the bending elasticity constants

c influence the curves constants and it is found that H0, kc and k in rather different manners. Hence, as described above, it is possible to determine the bending elasticity constants from an experimental growth curve by means of optimizing the agreement between theory and experimental aggregation numbers. Below we will discuss in some more detail how the micellar growth c , respectively. behavior is influenced by H0, kc and k 3.3.1. Saddle-splay constant According to the Gauss-Bonnet theorem,

Z

KdA ¼ 4pð1  gÞ

ð16Þ

where g represents the number of handles or holes present in a selfassembled aggregate. A micelle consists of one geometrically closed surfactant monolayer which means that g = 0. As a consequence, the c , the quantity of which does not last term in Eq. (13) equals 4pk c contributes with a sizedepend on the size of a micelle. Since k c indirectly determines the size independent term to E, the value of k c of geometrically closed surfactant aggregates. Positive values of k means that the total free energy in the system increases as one single micelle is split up to form two or many smaller micelles. Likec implies a decreasing total free energy wise, negative values of k as micelles are split up to increase their number while keeping surc factant concentration fixed. In either case, increasing values of k favors larger micelles with higher aggregation numbers. c while The influence of changing the saddle-splay constant k keeping spontaneous curvature (H0) and bending rigidity (kc) fixed c favors larger is shown in Fig. 4. It is seen that higher values of k micelles by shifting the second cmc (that is the point of transition from the weakly to the strongly growing regime) to lower surfacc the second tant concentrations. At sufficiently low values of k cmc is shifted beyond /mic = 1 which means that the micelles grow modestly in the entire range of surfactant concentrations. This kind of behavior of micelles growing weakly in a large range of concentrations has been observed in experiments for several surfactants, including 12, 1-Br (DTAB), 12, 3-Br and 12, 4-Br [4]. In Fig. 5 the very same curves are shown as normalized with respect to the second cmc, i.e. the curves have been shifted along the abscissa so as to fall on top of each other. It is seen that the saddle-splay constant has very small influence on the shape of the curves which are found to appear more or less identical to one another.

The more detailed appearance of the growth curves depends strongly on the actual values of all three bending elasticity

Fig. 3. The average length (hKi, solid line) and width (hXi, dashed line) of micelles plotted against volume fraction of surfactants aggregated in micelles formed by 12, 6-Br. The lines represent theoretical predictions using the general micelle model c set to their optimized values with the bending elasticity constants H0, kc and k giving the growth curve displayed in Fig. 1.

Fig. 4. The micelle aggregation number (N) plotted against volume fraction of surfactant micelles. The lines represent theoretical predictions from the general c ¼ 8:1kT (solid micelle model corresponding to different saddle-splay constants k c ¼ 9:5kT (dashed line) and k c ¼ 5:0kT (dotted line) at given values of line), k kc = 0.5kT and nH0 = 2.0. The solid line is identical to the one for micelles formed by the surfactant 12, 6-Br shown in Fig. 1. The dashed line corresponds to the growth behavior of micelles formed by 12, 8-Br shown in Fig. 2.

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As a matter of fact, in Ref. [4] it was found that the aggregation number versus concentration curve for the surfactant 12, 8-Br appears identical to the one for 12, 6-Br, except that it is horizontally shifted along the concentration scale [cf. Fig. 1 in Ref. [4]]. Hence, in accordance with our predictions employing the general micelle model, this means that the two surfactants have same valc . Conseues of kc and H0, but different saddle-splay constants k quently, optimizing the agreement between theory and data with respect to the three bending elasticity constants gives a somewhat c ¼ 9:5kT for the surfactant 12, 8-Br, whereas H and larger value k 0 kc are identical for the two surfactants. The growth curve corresponding to 12, 8-Br micelles shown in Fig. 2 is included as the dashed line in Figs. 4 and 5, respectively. It has been demonstrated that electrostatic effects as taken into account by the Poisson–Boltzmann mean field theory always give a negative contribution to the saddle-splay constant [50,57]. Likewise, excluded volume repulsive interactions among non-ionic c , the magnitude surfactant head groups also results in negative k of which is expected to increase with increasing head group size [52]. Moreover, the saddle-splay constant is a measure of the ability of surfactants to form a saddle-shaped interface, with positive curvature in one direction and a negative curvature in the perpenc implies that dicular direction [58]. Hence, positive values of k there must be an asymmetry in the aggregation behavior that favors different curvatures in perpendicular directions. This asymmetry may be caused by replacing a methyl group in the head c is found to group with a larger alkyl group and, as a result, k increase from very low (possibly negative) values for 12, 1-Br (–CH3) to 8.1kT for 12, 6-Br (–C6H13) and 9.5kT for 12, 8-Br (–C8H17). It is also seen from the growth curves in Figs. 4 and 5 that the general micelle model predicts a maximum aggregation number for finite sized micelles [35]. This may be interpreted as implying that an isotropic solution of finitely sized micelles is no longer stable above this concentration. As a matter of fact, it is well known that an isotropic surfactant micellar solution may frequently more or less abruptly transform into a liquid crystalline phase above a certain surfactant concentration [59–62]. 3.3.2. Spontaneous curvature The spontaneous curvature H0 represents the sign and magnitude of the preferential curvature of a single surfactant monolayer. Since smaller micelles must be more curved than larger ones, H0 is expected to influence the aggregation number of micelles. In Fig. 6 we have plotted growth curves for different values of H0 while c fixed. It is seen that micelles increase in size as keeping kc and k

Fig. 5. The micelle aggregation number (N) plotted against volume fraction of surfactant micelles (arbitrary scale). The curves are identical to the ones shown in Fig. 4 but are shifted along the abscissa so as to become normalized with respect to the second cmc.

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nH0 is lowered from 2 to 1, and the second cmc is shifted toward lower surfactant concentrations. However, further decreasing nH0 from 1 to 0.5 has the counterintuitive consequence of raising the second cmc. The reason for this is that the spontaneous curvature influences the width of the micelles, thus increasing the width as H0 is decreased. Hence, lowering H0 and increasing the width inhibits the growth in length of the micelles through the coupling between width and length as discussed above and, as a consequence, the second cmc is shifted to higher concentrations. In Fig. 6 it is also seen that the maximum aggregation number for an isotropic solution of micelles increases with decreasing spontaneous curvature. In Fig. 7 the very same growth curves as shown in Fig. 6 have been normalized with respect to the second cmc. It is seen that the main effect of changing the spontaneous curvature, in addition to influencing the second cmc, is to raise the aggregation numbers vertically along the ordinate as H0 is decreased. 3.3.3. Bending rigidity The bending rigidity kc quantifies the resistance against deviations from a uniform mean curvature H = H0 and the sign of kc determines the stability of surfactant aggregates, i.e. stable micelles may only exist for positive values of kc. Moreover, high kc-values are expected to favor self-assembled aggregates with small deviations from a homogenous curvature or geometry, i.e. rigid and monodisperse objects with a uniform shape, while geometrically more heterogeneous aggregates are favored by low values of kc [35,63]. The influence of changing kc on the micellar growth behavior, c fixed, is shown in Fig. 8. It is seen that while keeping H0 and k the bending rigidity has a large impact on the second cmc and raising kc has a clear effect of favoring large micelles with high aggregation numbers. This may be rationalized as due to the tendency of larger micelles becoming more geometrically homogenous as the fraction of surfactant molecules making up the end caps decreases with length and the micelles becoming more spherocylindrical in shape. The maximum aggregation number of finite micelles is seen to decrease with increasing bending rigidity. Fig. 9 shows the growth curves when normalized with respect to the second cmc. Unlike the corresponding curves for the saddle-splay constant in Fig. 5, the curves do not coincide but show different shapes as kc is changed. It turns out that the slope

Fig. 6. The micelle aggregation number (N) plotted against volume fraction of surfactant micelles. The lines represent theoretical predictions from the general micelle model corresponding to different spontaneous curvatures nH0 = 2.0 (solid line), nH0 = 1.0 (dashed line) nH0 = 0.5 (dotted line) at given values of kc = 0.5kT and  ¼ 8:1kT. The minimum aggregation number set to N = 30 corresponds to k c s n = 13.5 Å. The solid line is identical to the one for micelles formed by the surfactant 12, 6-Br shown in Fig. 1.

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Fig. 7. The micelle aggregation number (N) plotted against volume fraction of surfactant micelles (arbitrary scale). The curves are identical to the ones shown in Fig. 6 but are shifted along the abscissa so as to become normalized with respect to the second cmc.

Fig. 9. The micelle aggregation number (N) plotted against volume fraction of surfactant micelles (arbitrary scale). The curves are identical to the ones shown in Fig. 8 but are shifted along the abscissa so as to become normalized with respect to the second cmc.

increases with decreasing bending rigidity in the weakly growing regime below the second cmc, while the opposite holds true above the second cmc. In other words, the appearance of a second cmc becomes more obvious and appears more well-defined at higher kc-values. Low kc-values imply a reduced resistance to change the curvature of a micelle and, as a consequence, the micelles may grow easier at low surfactant concentrations as kc is decreased. Above the second cmc, on the other hand, the growth behavior is mainly determined by the end cap free energy and the parameter d in Eq. (14). As a result, increasing kc raises the end cap free energy and the micelles begin to grow more strongly in length as a response to a decrease in the number of micelles and unfavorable end caps.



3.4. Polydispersity of micelles The polydispersity as a function of surfactant concentration may be deduced for any given function of N versus /mic using the following very important (but surprisingly little known) relation

Fig. 8. The micelle aggregation number (N) plotted against volume fraction of surfactant micelles. The lines represent theoretical predictions from the general micelle model corresponding to different spontaneous curvatures kc = 0.5kT (solid line), kc = 1.0kT (dashed line) and kc = 2.0kT (dotted line) at given values of nH0 = 2.0  ¼ 8:1kT. The minimum aggregation number set to N = 30 corresponds to the and k c s thickness of the surfactant monolayer n = 13.5 Å. The solid line is identical to the one for micelles formed by the surfactant 12, 6-Br shown in Fig. 1.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d lnhNi ¼ d ln /t hNi

rN



ð17Þ

derived by Hall and Pethica [64]. The derivation of Eq. (17) is shown in detail in the Supplementary Material. In accordance, we have plotted rN/hNi (based on the volume-weighted size distribution) as a function of /mic in Fig. 10 for micelles formed by 12, 6-Br using the general micelle model fitted to experimental data as shown in Fig. 1. It is seen that the polydispersity, as expected, is rather low for weakly growing micelles at low surfactant concentrations and increases slowly in a close to linear dependence with respect to the logarithm of concentration, from about rN/hNi = 0.25 at /mic = 0.001 to rN/hNi  0.4 at /mic = 0.01. As the second cmc is reached, the polydispersity is found to increase more strongly and may become as large as rN/hNi exceeding 2. Eventually rN/hNi reaches a maximum which correspond to the point of inflexion of the growth curve, just below the concentration where finite micelles assume their maximum aggregation number. The concentration dependence of rN/hNi predicted by the general micelle model and shown in Fig. 10 differs conspicuously from that predicted by the spherocylindrical micelle model. According

Fig. 10. The relative standard deviation rN/hNi as calculated, using Eq. (17), from the general micelle model and the theoretical curve for 12, 6-Br shown in Fig. 1 (solid line). The dash-dotted and the dash-dot-dotted lines represent the corresponding curves for spherocylindrical micelles according to Eqs. (5), (8) and (10), respectively.

L.M. Bergström / Journal of Colloid and Interface Science 440 (2015) 109–118

to the latter, hNi / /mic in the limit of large aggregation numbers which, inserted in Eq. (17), gives a maximum relative standard deviation rN/hNi = 1. The growth behavior hNi / /1=2 mic , derived by using mole fraction rather than volume fraction in the expression for the entropypof ffiffiffi mixing, implies a maximum polydispersity equal to rN/hNi = 1/ 2  0.71. The latter result is inconsistent with experimental observations that rN/hNi is frequently found to be close to or larger than unity for long rodlike or wormlike micelles [10,47,48,65,66]. A relative standard deviation exceeding unity implies an extent of growth stronger than predicted by the spherocylindrical micelle model [cf. Fig. 1] and, as a matter of fact, a decrease in the number of micelles as an additional amount of surfactant is added. As mentioned above, this behavior as predicted by the general micelle model, and in contrast to the spherocylindrical micelle model, is consistent with the unusually strong growth behaviors frequently observed in several giant wormlike micellar systems [53–55]. 4. Conclusions The growth behaviors of surfactant micelles have been investigated from a theoretical point of view. It is demonstrated that experimental aggregation numbers determined in the full range of relevant surfactant concentrations are, in general, inconsistent with the conventional spherocylindrical micelle model [25,26] and the idea that micelles are only able to grow with respect to the length direction. On the other hand, the recently developed general micelle model [35], which considers micelles that are able to grow with respect to both width and length, predicts growth behaviors that agree excellently with experimental observations. In particular, the general micelle model predicts, in conspicuous contrast to the spherocylindrical micelle model and in agreement with a large number of experimental studies, a sharp rise in aggregation numbers above a certain concentration [4,14– 16,19,20,22,23,53–55]. The general micelle model takes into account bending elasticity properties in terms of the three quantities spontaneous curvature (H0), bending rigidity (kc) and saddlec ). The spherocylindrical micelle model is recovsplay constant (k ered as a special case of the general micelle model in the limit kc ? 0 as well as at surfactant concentrations much lower than the second cmc. We demonstrate that the growth curves of micelles according to the general micelle model, with aggregation number as a function of surfactant concentration in a logarithmic plot, always appear as consisting of a weakly growing regime at low surfactant concentrations followed by a strongly growing regime at higher concentrations. The surfactant concentration at the transition between the two regimes roughly corresponds to what has traditionally been denoted the second cmc [14]. The detailed appearance of the growth curves are demonstrated to be strongly influenced by the three bending elasticity constants spontaneous curvature (H0), c ) [49]. The role of bending rigidity (kc) and saddle-splay constant (k the three parameters may be briefly summarized as follows. (i) The saddle-splay constant mainly shifts the curve along the axis related to surfactant concentration. As a result, micelles with low, possibly c show weak and smooth growth behaviors in negative, values of k a comparatively wide range of concentrations. The appearance of c the second cmc is shifted toward lower concentrations as k  assumes larger values. Hence, kc is found to usually be positive for surfactants or surfactant mixtures that display a second cmc at comparatively low surfactant concentrations. (ii) The spontaneous curvature mainly shifts the curve vertically along the ordinate. Thus high values of H0 correspond to small aggregation numbers. (iii) The bending rigidity kc influences the shape of the growth curves in that way the extent of growth increases with decreasing kc at low surfactant concentrations below the second cmc, whereas

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the opposite holds true at high concentrations above the second cmc. In other words, the aggregation number rises more steeply, and the second cmc appears more well defined, at larger kc values. The general micelle model not only predicts the growth behavior in terms of aggregation numbers of surfactant micelles. It also predicts a typical change in geometrical shape of the micelles as they grow in size, i.e. both width and length increases for weakly growing micelles at low surfactant concentrations whereas the length increases while the width decreases for strongly growing micelles above the second cmc. The geometrical shape of micelles may be accurately determined by small-angle neutron scattering (SANS) measurements [10,67]. In order to further test the predictions set forth by the general micelle model, we have recently investigated the correlation between geometrical shape and growth behavior of surfactant micelles [68]. As a result, this behavior as predicted by the general micelle model has been confirmed. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.jcis.2014.10.054. References [1] R. Zana, in: D.N. Rubingh, P.M. Holland (Eds.), Cationic Surfactants, Physical Chemistry, Marcel Dekker Inc., New York and Basel, 1991, p. 41. [2] R. Zana, E.W. Kaler (Eds.), Giant Micelles: Properties and Applications, CRC Press, 2007. [3] R. Zana, J. Xia (Eds.), Gemini Surfactants: Synthesis, Interfacial and SolutionPhase Behavior and Applications, Dekker, New York, 2004. [4] P. Lianos, J. Lang, R. Zana, J. Colloid Interface Sci. 91 (1983) 276. [5] A. Malliaris, J. Lang, R. Zana, J. Phys. Chem. 90 (1986) 655. [6] E.Y. Sheu, S.-H. Chen, J. Phys. Chem. 93 (1988) 4466. [7] F.H. Quina, P.M. Nassar, J.B.S. Bonilha, B.L. Bales, J. Phys. Chem. 99 (1995) 17028. [8] B. Naskar, A. Dan, S. Ghosh, V.K. Aswal, P. Moulik, J. Mol. Liq. 170 (2012) 1. [9] N.A. Mazer, G.B. Benedek, M.C. Carey, J. Phys. Chem. 80 (1976) 1075. [10] M. Bergström, J.S. Pedersen, Phys. Chem. Chem. Phys 1 (1999) 4437. [11] L.J. Magid, Z. Li, P.D. Butler, Langmuir 16 (2000) 10028. [12] L. Arleth, M. Bergström, J.S. Pedersen, Langmuir 18 (2002) 5343. [13] T. Imae, R. Kamiya, S. Ikeda, J. Colloid Interface Sci. 108 (1985) 215. [14] G. Porte, Y. Poggi, J. Appell, G. Maret, J. Phys. Chem. 88 (1984) 5713. [15] M. Törnblom, U. Henriksson, M. Ginley, J. Phys. Chem. 98 (1994) 7041. [16] M. Törnblom, R. Stinikov, U. Henriksson, J. Phys. Chem. B 104 (2000) 1529. [17] A. Bernheim-Groswasser, E. Wachtel, Y. Talmon, Langmuir (2000) 4131. [18] O. Glatter, G. Fritz, H. Lindner, J. Brunner-Poppela, R. Mittelbach, R. Strey, S.U. Egelhaaf, Langmuir 16 (2000) 8692. [19] S. May, A. Ben-Shaul, J. Phys. Chem. B 105 (2001) 630. [20] A. González-Pérez, L.M. Varela, M. Garcia, J.R. Rodriguez, J. Colloid Interface Sci. 293 (2006) 213. [21] Y. Geng, L.S. Romsted, F. Menger, J. Am. Chem. Soc. 128 (2006) 492. [22] D. Danino, Y. Talmon, R. Zana, Langmuir 11 (1995) 1448. [23] M. In, B. Bendjeriou, L. Noirez, I. Grillo, Langmuir 26 (2010) 10411. [24] L.M. Bergström, V.M. Garamus, Langmuir 28 (2012) 9311. [25] J.N. Israelachvili, D.J. Mitchell, B.W. Ninham, J. Chem. Soc., Faraday Trans. 2 (72) (1976) 1525. [26] P.J. Missel, N.A. Mazer, G.B. Benedek, C.Y. Young, J. Phys. Chem. 84 (1980) 1044. [27] J.C. Eriksson, S. Ljunggren, J. Chem. Soc., Faraday Trans. 2 (81) (1985) 1209. [28] M.E. Cates, S.J. Candau, J. Phys.: Condens. Matter 2 (1990) 6869. [29] Y. Chevalier, T. Zemb, Rep. Prog. Phys. 53 (1990) 279. [30] R. Nagarajan, E. Ruckenstein, Langmuir 7 (1991) 2934. [31] P.J. Missel, N.A. Mazer, G.B. Benedek, M.C. Carey, J. Phys. Chem. 87 (1983) 1264. [32] P.J. Missel, N.A. Mazer, M.C. Carey, G.B. Benedek, J. Phys. Chem. 93 (1989) 8354. [33] C. Tanford, J. Phys. Chem. 76 (1972) 3020. [34] H.V. Tartar, J. Phys. Chem. 59 (1955) 1195. [35] L.M. Bergström, ChemPhysChem 8 (2007) 462. [36] L.M. Bergström, in: M. Tadashi (Ed.), Application of Thermodynamics to Biological and Material Science, InTech, Rijeka, 2011, p. 289. [37] J.C. Eriksson, S. Ljunggren, U. Henriksson, J. Chem. Soc., Faraday Trans. 2 (81) (1985) 833. [38] M.L. Huggins, J. Chem. Phys. 9 (1941) 440. [39] P.J. Flory, J. Chem. Phys. 19 (1942) 51. [40] P.J. Flory, Faraday Disc. 57 (1974) 7. [41] E.A. Guggenheim, Mixtures, Clarendon Press, Oxford, 1952. [42] D.C. Morse, S.T. Milner, Phys. Rev. E 52 (1995) 5918. [43] H. Pilsl, H. Hoffmann, S. Hoffmann, J. Kalus, A.W. Kencono, P. Lindner, W. Ulbricht, J. Phys. Chem. 97 (1993) 2745. [44] M. Bergström, J.S. Pedersen, Langmuir 15 (1999) 2250.

118

L.M. Bergström / Journal of Colloid and Interface Science 440 (2015) 109–118

[45] S. Prévost, L. Wattebled, A. Laschewsky, M. Gradzielski, Langmuir 27 (2011) 582. [46] L.M. Bergström, V.M. Garamus, J. Colloid Interface Sci. 381 (2012) 89. [47] L.M. Bergström, S. Skoglund, K. Edwards, J. Eriksson, I. Grillo, Langmuir 29 (2013) 11834. [48] L.M. Bergström, S. Skoglund, J. Eriksson, K. Edwards, I. Grillo, Langmuir 30 (2014) 3928. [49] W. Helfrich, Z. Naturforsch. C 28 (1973) 693. [50] L.M. Bergström, Langmuir 22 (2006) 3678. [51] L.M. Bergström, Langmuir 22 (2006) 6796. [52] L.M. Bergström, Langmuir 25 (2009) 1949. [53] P. Schurtenberger, C. Cavaco, J. Phys. II France 3 (1993) 1279. [54] P. Schurtenberger, C. Cavaco, J. Phys. II France 4 (1994) 305. [55] P. Schurtenberger, C. Cavaco, Langmuir 12 (1996) 2894. [56] M. Bergström, J. Chem. Phys. 113 (2000) 5559. [57] D.J. Mitchell, B.W. Ninham, Langmuir 5 (1989) 1121.

[58] S. Hyde, S. Andersson, K. Larsson, Z. Blum, T. Landh, S. Lidin, B.W. Ninham, The Language of Shape, Elsevier, Amsterdam, 1997. [59] R.R. Balmbra, J.S. Clunie, J.F. Goodman, Nature 222 (1969) 1159. [60] P. Kékicheff, C. Grabielle-Madelmont, M. Ollivion, J. Colloid Interface Sci. 131 (1989) 112. [61] R.G. Laughlin, The Aqueous Phase Behaviour of Surfactants, Academic Press, London, 1994. [62] B. Jönsson, B. Lindman, K. Holmberg, B. Kronberg, Surfactant and Polymers in Aqueous Solution, Wiley, Chichester, 1998. [63] G. Porte, J. Phys.: Condens. Matter 4 (1992) 8649. [64] D.G. Hall, B.A. Pethica, in: M.J. Schick (Ed.), Nonionic Surfactants, Marcel Dekker, New York, 1967, p. 516. [65] M. Bergström, J.S. Pedersen, J. Phys. Chem. B 103 (1999) 8502. [66] L.M. Bergström, S. Skoglund, K. Danerlöv, V.M. Garamus, J.S. Pedersen, Soft Matter 7 (2011) 10935. [67] J.S. Pedersen, Adv. Colloid Interface Sci. 70 (1997) 171. [68] L.M. Bergström, I. Grillo, Soft Matter 10 (2014) 9362.