YflK 544.7+536.7

BecTHHK Cn6ry. Cep. 4. 2013. Bun. 1

U. Dorn, Ph. Schrader, S. Enders

AGGREGATION AND PHASE BEHAVIOR OF NONIONIC SURFACTANTS (CiEj) IN AQUEOUS SOLUTION*

Introduction. Surface active agents consist of hydrophilic groups and hydrophobic tails combining by a covalent bond. Due to their amphiphilic structure surfactant containing systems exhibit complex aggregation and phase behavior. In the region of the critical micellization concentration (cmc) surfactants self-organize in micellar aggregates consisting of a variable number of molecules and forming different shapes. Size and shape of micelles depend on composition of the system, temperature and the length of the hydrophilic and hydrophobic tails [1]. An excellent overview regarded to the aggregation and phase behavior of surfactant mixtures in aqueous solutions was given by Smirnova [2]. The widespread use of surfactants in practical applications in the field of pharmaceutical, cosmetics, and environmental protections motivates the development of predictive theoretical approaches to improve fundamental understanding of the behavior of these complex self-assembling systems and to facilitate the design and optimization of new applications.

The most predictive and accurate theoretical models of surfactant micellization and micellar solubilisation implement what is known as the molecular thermodynamic modeling approach. From a statistical thermodynamic point of view Nagarajan and Ruckenstein [3-9], as well as Blankschtein and coworkers [10-16], developed a micellar formation model able to describe the aggregation behavior of aqueous surfactant solutions. The combination of the micelle formation [5, 16] and phase separation model was applied to various sugar surfactants solutions [17-22].

Widely used surfactants belong to the class of nonionic surfactants named poly(oxy-ethylene) alkyl ethers with the general chemical structure H^H^^OCH^H^jOH, which is simple denoted by CiEj where i refers to the number of carbon atoms in the hydrocarbon chain and j to the number of oxyethylene units in the molecule. These surfactants in aqueous solution are studied extensively. In literature cmc data e.g. [23-29] are reported for different chemical composition and the influence of the temperature is examined, where the experiments were performed with different methods, like surface tensiometry [25, 26, 28, 29], fluorescence spectroscopy [31] and microcalorimetry [23, 35]. The cmc of nonionic surfactants passes through a minimum as function of temperature [1, 29]. The average size of micellar aggregates can be obtained by fluorescence quenching [37, 40, 41], dynamic light scattering (DLS) [42-44], small angle neutron scattering (SANS) [45-48], pulsed-field-gradient-NMR (PFG-NMR) [49-52] and cryo-transmission electron microscopy (cryo-TEM) [40, 53]. The shape of micelles depends on the tail lengths, surfactant concentration and temperature. Mostly, at low surfactant concentration spherical micelles are present. At higher surfactant concentrations a transformation to worm-like micelles [42, 44, 47, 53] or bilayer vesicles [54] are observed, experimentally. With increasing surfactant concentration over 20 wt % lyotrophic phases exist e.g. [55]. Dilute aqueous solutions of nonionic amphiphiles usually

Udo Dorn — graduate student of Technical University Berlin, Germany.

Philipp Schrader — graduate student of Technical University Berlin, Germany.

Sabine Enders — Chair of Thermodynamics and Thermal Separation Science, Technical University Berlin, Germany; e-mail: sabine.enders@tu-berlin.de

* We thank the German Science Foundation (DFG) project Collaborative Research Centre/Transregio 63 "InPROMPT" (sub-project A5) for financial support.

© U. Dorn, Ph. Schrader, S. Enders, 2013

exhibit a phase separation when the temperature is raised above a value which depends on the amphiphilic concentration. The minimum of the curve, where the demixing temperature is plotted versus the surfactant concentration, is a critical point, or more precisely, a critical point with a lower critical solution temperature (LCST) e.g. [56-65]. With respect to the chemical composition of the surfactant the LCST can be adjusted in a wide temperature range e.g. [56-61]. For some surfactants a closed-loop behavior was measured [62, 63].

One special feature of the CiEj surfactants is the hydration of the polar head group, which is strongly temperature dependent. Mostly, this special effect is not taken into account in the thermodynamic model [1-12]. Puvvada and Blankschtein [10] analysed the experiments performed by Nilsson and Lindman [66] and figured out that hydration number decreases with increasing temperature and hence the average cross-sectional area of the head will decrease with temperature. Therefore, they [10] suggested a linear relationship between this quantity and the temperature. The necessary parameters were fitted to temperature variation of the cmc for C12E6 surfactant solutions. The aim of this work is to incorporate the temperature dependency of the cross-sectional area of the head in the thermodynamic framework using independently obtained experimental data, for instance Mandelstam—Brillouin light scattering data [67]. This type of measurements exhibit the advantage that for data analyses no model related to the micellar shape is needed.

Theory. The starting point for the thermodynamic considerations is the Gibbs free energy of a surfactant solution Gso\, which is modeled as the sum of three contributions: the free energy of formation G0, the ideal free energy of mixing Gm;x, and the free energy of interactions between the various components GE, that is [6]:

Gsoi _ Go Gmix Ge

~kT ~kT kT ~kT' u

where k is the Boltzmann constant and T the temperature. Micelles with different number of surfactant molecules are treated as distinguishable chemical species. The free energy of formation G0 summarizes the molecular interactions responsible for self-association in a dilute reference solution that lacks inter-micellar interactions and takes the following form

[5, 6, 10]:

tt

Go = Nw+ Ng, (2)

g=i

where ^0W is the free energy change of the solution when a water molecule is added to pure water, and ^0g reflects the free energy change of the solution when a single aggregate, characterized by aggregation number g and the shape, will be formed. The ideal free energy of mixing is given by:

G tt

-^ = Nw\nXw + Y,Ng\nXg, (3)

g=i

where XW is the mole fraction of water and Xg is the mole fraction of aggregates with the aggregation number g. The number of surfactant molecules, NS, reads:

tt

Ns = £ gNg. (4)

g=i

The interactions between the formed micellar aggregates, single surfactant molecules and water molecules can be modeled with the following mean-field expression [10, 16, 18]:

GB _ 1 fr t C2\ yNsXs ( .

where XS is the surfactant mole fraction. Ci, C2, and y are adjustable parameters. These parameters are obtained by fitting the binodal to the experimental cloud point curve of a surfactant water mixture. The application of Eq. (1) in combination with Eqs. (2), (3) and (5) allows the calculation of the chemical potentials using standard thermodynamics. Taking into account the principle of multiple chemical equilibrium [68] between aggregates of different sizes and monomers, that is |g = g|i, the following expression for the micellar size distribution is obtained [7, 18-20]:

The heart of the thermodynamic model is the expression for A|0g, occurring in Eq. (6), representing the difference in standard chemical potential of single dispersed surfactant, A|01, and the surfactant in the formed aggregate with aggregation number g, A|0g [5, 6]

A|loS = — M-01 = (A|los)tr + (A[los)def + (A[los)int + (A|los)ster- (7)

First, the hydrophobic tail is removed from contact with water and transferred to the aggregate core, which is like a hydrocarbon liquid. This transfer free energy of the surfactant tail (A|0g )tr is estimated from independent experimental data on the solubility of hydrocarbons in water. Second, the surfactant tail inside the aggregate core is subjected to packing constraints because of the requirements that the polar head group should remain at the aggregate-water interface. The free energy resulting from this constraint on the surfactant tail is called deformation free energy (A|0g)def. Third, the formation of the aggregate is associated with the creation of an interface between its hydrophobic domain and water. The free energy of formation of this interface (A|0g);nt is calculated as the product of the surface area in contact with water and the macroscopic interfacial tension of the aggregate core-water interface. Fourth, the surfactant head groups are brought to the aggregate surface, giving rise to steric repulsion between them. If the head groups are compact in nature with a definable hard-core area, then the steric interactions, (A|0g)ster, can be estimated as hard-particle interactions by using the van der Waals approach. More detailed information are given elsewhere [3-6, 18-20].

The molecular constants of the surfactant entering the theoretical framework are the number of carbon atoms in the hydrophobic tail, nT, and the effective cross-sectional area of the polar head group ap, which is the equilibrium area per molecule at the aggregate surface. The ap-value can also be obtained experimentally using the surface tension at different surfactant concentrations.

Puvvada and Blankschtein [41] assumed that the ap-value depends only from the size of the hydrophilic head group, j meaning that for instance C10E8 and C12E8 should have the same value. However, from the experimental point of view different values were found in the homologous series of CiE8 [1]. With other words the effective cross-sectional area of the polar head group depends also from the hydrophobic tail [9]. However, in the case of CiEj surfactants this quantity depends on temperature, because the hydration of the polar

head group is a strong function of temperature. This implies that the ap-value depends on temperature and on i and j. This situation requires the measurement of surface tension of CiEj-surfactants in water at different temperatures. On the other hand a large amount of cmc-data of these surfactants are already available in the literature e.g. [23-31]. For this reason another approach is followed in this paper. The ap-value is considered as an adjustable parameter and it is estimated using the experimental cmc-values for different CiEj-surfactants at 298 K. The dependence on the chemical structure is found to be:

ap(i, j) = Ai + Bij + CjD. (8)

The temperature dependence of the effective head group is caused by the temperature dependence of the hydration of the polar head. This is taken into account by the hydration number, xhydrat, in the following way:

The quantity xhydrat can be obtained from Brillouin—Mandelstam-scattering experiments of aqueous solutions of poly(ethylene oxide) at different temperatures [67].

The minimization of the free energy of formation yields information about the aggregate shape and the geometrical properties of the formed aggregates. Following Nagarajan and Ruckenstein [5, 6] we distinguish between four types of surfactant aggregate (spherical micelles, prolate ellipsoids, rodlike micelles and spherical bilayer vesicles), which are detailed described in the literature [5, 6, 18]. From the geometrical relations provided in literature [6, 18] it can be seen that, given the surfactant parameter ap and nT, the geometrical properties of spherical micelles or prolate ellipsoids depend only on the aggregation number g. In the case of rodlike micelles, by minimizing A|0g the equilibrium radius rc of the cylindrical part and equilibrium radius rs of the spherical endcaps of the micelle are determined. The geometrical relation yields the length of the cylinder, lc, the number of surfactant molecules in the spherical endcapes, gcap, and in the cylinder gcyi. Minimization of A|0g for spherical bilayer vesicles, similar to the rod-like micelles, results in the equilibrium inner and outer radii ri and ro. The inner and outer layer thickness t and to as well as the number of surfactant molecules in the inner and outer layer are given by the geometrical relations [5, 6, 18].

Results and Discussion.

Aggregation behavior. The molecular-thermodynamic framework, presented in the last section, enables us to predict a broad spectrum of micellar properties of aqueous surfactant solutions, if the parameters of the surfactant (ap and nT) are known. The parameter nT is given by the i-value of the studied surfactant and does not depend on temperature or concentration. In contrast, the parameter ap depends the i and j value, and additionally on temperature caused by the temperature-dependent hydration of the polar head. It should be noted that in Eq. (5) only one average value of the distribution function, namely XS, occur and hence this equation is not needed for the calculation of the aggregation behavior. Therefore no interaction parameter must be specified.

First, the equilibrium properties of the aqueous surfactant solution, including the geometrical properties as well as the aggregation distribution function, were calculated by minimization of the Gibbs enthalpy of solution (Eq. (1)) for a given surfactant concentration and an initial guess for the ap-value. Plotting the weight-average aggregation number, gW, or the surfactant concentration, XS, as function of the monomer surfactant concentration Xi yields information about the critical micellar concentration (Fig. 1). At a narrow surfactant

250 200 150 100 50 0

0.01

, 1E-3

, 1E-4

1E-5

, 1E-6

6.9

7.2 107 X,

Fig. 1. Estimation of the cmc-value using mass-average aggregation number (solid line) and mole fraction of single surfactant molecules (broken line) for C12 Es at 298 K

monomer concentration range the weight-average aggregation number gW increased from 2 to 150 for the Ci2E8 surfactant. This rapid increase can be seen closed to the point of inflection. From the mathematical point of view the cmc is calculated by putting the second derivative to zero. In Fig. 1 it can be recognized that the cmc is a small concentration range rather then a discrete value. These calculations were repeated with a different ap-value until the experimental cmc-value (Table 1) is obtained.

Table 1

Adjustment of ap(i, j) using experimental cmc data at T = 298 K taken from the literature

i j ■i^cmc Ref. 10'Map[rn2} fit 10'Map[rn2} Eq. (8)

6 3 1.80E-03 [23] 36.45 34.78

6 4 1.68E-03 [34] 35.80 35.84

6 5 1.73E-03 [34] 36.08 36.72

6 6 1.23E-03 [34] 33.57 37.49

8 1 8.83E-05 [26] 35.87 34.78

8 3 1.35E-04 [25] 39.09 38.36

8 4 1.53E-04 [35] 40.06 39.47

8 5 1.71E-04 [35] 40.95 40.40

8 6 1.78E-04 [25] 41.25 41.22

8 9 2.34E-04 [25] 43.42 43.24

10 3 1.08E-05 [25] 42.22 41.93

10 4 1.23E-05 [36] 43.22 43.09

10 5 1.42E-05 [69] 44.35 44.08

10 6 1.62E-05 [25] 45.50 44.94

10 7 1.75E-05 [37] 46.03 45.73

10 8 1.98E-05 [28, 32] 47.00 46.45

10 9 2.34E-05 [25] 48.37 47.13

12 1 4.14E-07 [38] 39.59 41.71

12 2 5.94E-07 [38] 42.36 43.97

12 3 9.37E-07 [27] 45.98 45.50

12 4 1.15E-06 [27] 47.60 46.72

12 6 1.30E-06 [39] 48.50 48.67

12 8 1.51E-06 [29] 49.78 50.28

12 12 2.52E-06 [70] 54.00 52.99

14 8 1.62E-07 [28, 39] 55.30 54.12

&45

40

50

1E-6

1E-5

1E-4

cmc

Fig. 2. Optimal ap-values for different surfactants (solid squares — CiE1; solid circles — CiE2; solid up-triangles — CiE3; solid down-triangles — CiE4; open squares — CiE5; open circles — CiE6; open up-triangles — CiE7; open down-triangles — CiE8; star — CiE9) at T = 298 K; the lines are regression lines (solid line — C8Ej; dashed line — C10Ej; dotted line — C12Ej; dashed-dotted line — C14Ej)

The obtained optimal ap-values are plotted in Fig. 2 versus the experimental cmc (Table 1) at T = 298 K of different surfactant solutions and listed in Table 1. For surfactants having the identical hydrophobic tail a mostly linear relationship between the calculated ap-values and the experimental cmc is found. The increase of the number of hydrophilic head-groups leads to an increased ap-value.

The use of the data in Table 1 allows the calculation of the coefficients in Eq. (8) and leads to:

Density measurements of C\2Ej surfactants [71] show a negligible change in volume of the polar head-group with temperature. However, the volume decrease of surfactant solutions with increasing temperature is an important effect [71]. For this reason, the volume change can be explained by hydration of the polar head group. The hydration number was measured independently using Brillouin—Mandelstam-scattering experiments of aqueous solutions of poly (ethylene oxide) at different temperatures [67]. The experimental results [67] are plotted in Fig. 3, together with a line which correlates the experimental data. The line follows the following relationship:

with xhydrat (T = 298 K). Eq. (9) in combination with Eqs. (8) and (11) allows the description of the temperature dependency of the equilibrium area per molecule at the aggregate surface for the whole class of CiEj surfactant solutions based on independently experimental data in contrast to the model developed by Puvvada and Blankschtein (10), where experimental cmc-values of C12E6 solutions at different temperatures were used for fitting the head group area as linear function of temperature.

Using the approach described above permits the calculation of other properties of the micellar solutions, for instance the shape of the formed aggregates. The minimized chemical

A = 1.708; B = 0.02603; C = 20.91; D = 0.1282.

(10)

(11)

2.7-

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2.4 -

2.1

283

Fig. 3. Experimental hydration numbers of an oxyethylene unit [67] (squares) and fit of the experimental data (line)

250

Fig. 4. Predicted difference in the standard chemical potential A|0a between a surfactant molecule in an aggregate of size g and one in the dispersed state for the CsE6-surfactant at 298 K at different aggregation shapes (solid line — spherical micelles, dashed line — prolate ellipsoids, dotted line — rodlike micelles and dashes-dotted line — spherical bilayer vesicles): the insert enlarges the transition range

potential for different aggregate shapes, A|0fl, are given in Fig. 4. Form this figure it can be seen that in solution with a very small number of surfactant molecules spherical micelles will be formed. With increasing the number of surfactant molecules in the solution the aggregates will change their shape from spherical micelles to spherical bilayer vesicles. This situation is caused by a larger energy advantage for spherical bilayer vesicles in comparison with prolate ellipsoids or rodlike micelles. The existence of spherical bilayer vesicles in the system C12E4 + water was also discussed in the literature [54]. Olsson et al. [54] performed solvent self-diffusion coefficient measurements using the 1H-NMR-Fourier transform pulsed gradient spin-echo technique. It was figured out that the water rapidly exchanged between the inside and outside of the vesicles on the experimental time scale. In the case of sugar surfactant similar theoretical results were obtained [18-21].

Applying Eqs. (8), (9), and (11) permits the prediction of the temperature dependency of the cmc values (Fig. 5 and Fig. 6). The experimental data as well as the predicted data

1E-5

1E-6

1E-7

278 288 298 308 318 328 338 348 358 T, K

Fig. 5. Comparison of experimental (open squares — C10E4 [39]; open circles — C10E8 [39]; solid circles — C10E8 [28]; open up-triangles — C12E4 [29]; open down-triangles — C12E6 [29]; open diamonds — C12E8 [29]; solid squares — C12E8 [28]; open stars — C14E8 [39]; solid stars — C14E8 [28]) and predicted (solid line — C10E4; dashed line — C10E8; short dotted line — C12E4; short dashed line — C12E6; dash-dotted line — C12E8; short-dash-dotted line — C12 E14) cmc-values as function of temperature

1E-4

1E-5

1E-6

1E-7

1E-8

Fig. 6. Comparison of experimental [1] (solid squares — CioEs; solid circles — C'nE8; up-triangles — Ci2E8; down-triangles — Ci3E8; solid diamonds — Ci4E8; solid stars — C1bE8) and predicted (lines) cmc-values of the homologous series of C'iEs as function of temperature

show a minimum as function of temperature. This minimum is clearly caused by the hydration of the polar head-group. Some small deviations occur at higher temperatures. Several cmc-values are estimated using surface tension measurements. At higher temperatures experimental errors occur with this method by the evaporation of water leading to a deviation in surfactant concentration.

With help of Eq. (6) the aggregate distribution function and hence the moments of this distribution function can be calculated. The moments are accessible by different experimental techniques, for instance static light scattering experiments [49]. Brown et al. [49] performed static and dynamic light scattering experiments of solutions containing C12E7 or C12E8 surfactants and estimated the mass-average aggregation number as function of temperature (Fig. 7). Additionally, these authors found that the micelles exhibit a large

o

400 -

□

□

200 -

300 -

Fig. 7. Experimental [49] (open circles — C10ET at Xs = 3.68 • 10~4; open squares — C12 E8 at XS = = 3.38 • 10~4) and predicted (solid line — C10E7; broken line — C12E8) mass average aggregation numbers, gw, as function of temperature

□

o

100 -

o

300

320

340

T, K

polydispersity at all surfactant concentrations. In Fig. 7 the experimental data [49] are compared with the model predictions. The experimental data as well as the theoretical predictions show an increase in the aggregation number with increasing temperature. Having in mind the experimental accuracy (AgW = ±50) caused by the difficulties of such studies it can be concluded that a satisfactory agreement is achieved. However, the agreement for the solutions containing C12E8 is slightly better then for the solution with C12E7 surfactant molecules. Large deviations occur at temperatures at approximately 325 K, closed to the cloud point temperature, where cluster formation is expected. Close to the lower critical solution temperature or to the cloud point temperature strong concentration fluctuations arise. These fluctuations have a large impact on the light scattering used to obtain the mass-average aggregation number [49].

Liquid—Liquid Equilibria. Many authors believe e.g. [55] that in the region above the cloud point curve there are no micelles because the surfactant molecules are segregated from the aqueous phase. According to this view, one phase contains singly dispersed surfactant monomers and the other is a surfactant phase containing dissolved water. Micelles could be found also in the diluted phase for several surfactant solutions (e.g. C6E3 [59], C8E4 [59, 72], C8E5 [46]). Corti et al. [59] pointed out that the low-concentration phase becomes more and more dilute and, presumably, merges into the cmc-curve, whereas the high-concentration phase may become anisotropic, depending on the nature of the nonionic amphiphile. The suggested theoretical framework could be used to calculate both, namely the cmc as function of temperature and the liquid—liquid phase split. Using both results, it can be clearly decided, if micelles are present in the diluted phase.

The calculation of the aggregation behavior of a surfactant solution forms the basic of the calculation of the demixing behavior [16, 18], where the surfactant solution splits in a high-concentrated and a diluted phase. The property of the diluted phase is discussed controversially in the literature, especially the question arises, if micelles are formed in this diluted phase or not. Maybe this question can be answered applying the micelle formation model. However, in order to describe the demixing behavior the parameters Ci, C2 and in Eq. (5) must be fixed. Unfortunately, there is no way to estimate these parameters using independent measurements. For this reason these parameters were fitted to experimental cloud-point data taken from the literature [56-59, 63-65]. Unfortunately, the experimental results present in the literature show a large scatter. The cloud-point curve can be change from batch to batch because of the effect of impurities. It is also known that impurities may be generated by oxidation processes which can take place in aqueous solutions of poly-

oxyethylene amphiphiles. Other points of discussion are the applied heating rate, which should be so small as possible, as well as the detection method (visual observations or light scattering methods). For all these reasons one should not expect a complete agreement among the results obtained in different experiments. This situation caused also certain arbitrariness for the selection of the experimental data entering the fitting procedure. The selected experimental data together with the obtained model parameters for different surfactants are collected in Table 2. During the liquid-liquid equilibrium calculations we took into account that distribution of micellar sizes is not fixed, but depends strongly upon solution conditions such as total amphiphile concentration and temperature.

Table 2

Adjusted Parameter of Eq. (5) for binary system CiEj + water

i 3 Ci c2/K Y Source of experimental data

6 2 9.766 -2754 18.547 [58]

6 3 3.755 -1085 4.256 [59]

6 4 5.823 -1922 14.486 [64]

7 3 7.919 -2321 36.927 [58]

7 4 4.62 -1427 36.368 [56]

7 5 8.969 -3020 53.622 [56]

8 3 8.179 -2316 103.316 [64]

8 4 4.753 -1472 53.861 [64]

8 5 5.39 -1782 43.678 [64]

8 6 7.681 -2644 49.669 [64]

10 4 4.386 -1276 299.426 [63]

12 4 15.75 -4400 1108.33 [64]

12 5 7.927 -2416 739.249 [64]

12 6 3.463 -1112 511.992 [57]

12 7 1.655 -543.9 291.498 [57]

12 8 0.874 -287.5 141.625 [65]

12 8 3.224 -1126 223.129 [57]

14 7 1.389 -451.6 610.259 [59]

The obtained parameter y (Table 2) increases with increasing chain length of the hydrophobic tail. Originally [73], y is the effective volume of the surfactant divided by the volume of water. Increasing of the chain length leads to an increase of the effective volume and hence should be increase. For the parameters C\ and C2 no trend can be recognized. Beside the shortcomings of the theoretical framework, also the experimental uncertainties contribute to this statement. This situation can be seen in (Fig. 8), where several experimental data coming from different sources [55, 57, 60, 61, 65] and the theoretical results using two different approaches are depicted. First of all, the results obtained with the suggested theoretical framework including the aggregate formation show a fair agreement with the experimental data. However, the model parameter in Eq. (5) depends from the chosen experimental data set. Using standard thermodynamics not only the cloud point curve, but also the spinodale line, is accessible by the model. An alternative to the numerical expensive micellar formation model is the classical Koningsveld—Kleintjens model [73] which ignores the formation of micelles completely. For comparison, the results of the calculations carried out with this simple model [65] are also plotted in Fig. 8. The incorporation of the micelle formation leads to a considerably improvement of the calculation results. The cmcline for this surfactant solution is located at very low surfactant concentration (Fig. 8) and

Fig. 8. Comparison of the experimental (solid circles — [65]; open squares — [57]; up-triangels — [55]; down-triangles — [60]; stars — [61]) and the modeled cloud point curve (solid line) for C12E8 surfactant solutions:

the dotted line represents the calculated spinodale line and the dashed line the calculated cmc; the dash-dotted line is the cloud point curve calculated using the modified Koningsveld—Kleintjens model [65]

hence micelles are present in both coexisting phases. With other words, there is equilibrium between a diluted micellar solution and a concentrated micellar solution.

The molecular based model allows also the calculation of the aggregation distribution in both coexisting phases. Fig. 9 shows these distributions for two different temperatures. The first selected temperature is the critical temperature, where only one phase exists. In this phase aggregates with a broad distribution of the aggregation number could be found. At a slightly higher temperature two coexisting phases appear where the aggregates are

Fig. 9. Aggregation number distribution at the critical point (solid line; X" = 1.91 • 10 3; Tc = 347.7 K) and above the critical point (dotted line — Xl = 2.57 • 10"4; T = 348 K; dashed line — XI = 6.27 • 10~3; T = 348 K) for aqueous solution of Ci2E8 surfactant

present in both phases. In the diluted phase smaller aggregates with an average aggregation number of approximately 300 exists. The average aggregation number in the concentrated phase shifts to higher numbers and hence the large aggregates will stay in this phase.

For the surfactant C12E8 the critical concentration is located at 0.00189 (Fig. 8), whereas the critical concentration for the C12E4 solution shifts to lower concentrations, namely to 3.16 • 10~4 (Fig. 10). At the same time, the cmc shifts only slightly to lower concentrations (Fig. 5). For this reason intersection point between the cmc-line and the diluted branch of the cloud-point curve could be found (Fig. 10). The consequence of this intersection point is that the presence of micelles in the dilute phase depends on temperature or surfactant concentration. At concentration above the intersection point micelles are present and at concentration below the intersection point no micelles can be formed (Fig. 10).

A similar picture arises for the C10E4 surfactant solution (Fig. 11). Again an intersection point between the cmc-line and the diluted branch of the cloud point curve is established.

Another possibility to describe the phase equilibria of water + alkyl polyoxyethylene mixtures is a simplified version of the statistical associated fluid theory approach (SAFT-HS) [74]. The original SAFT-LJ equation of state treats the molecules as chains of Len-nard—Jones segments while the simplified SAFT-HS equation treats molecules as chains of hard-sphere repulsive segments with van der Waals interactions. Within this approach the water molecules are modeled as hard spheres with four associating sites to treat the hydrogen bonding; the dispersion forces are treated at the van der Waals mean-field level. The surfactant molecules are modeled as chains of hard-sphere segments with two or three bonding sites to treat the terminal hydroxyl group and an additional three sites per oxyethy-lene group; the dispersion forces are again treated at the mean-field level. For appropriate choices of the intermolecular parameters, the SAFT-HS approach predicts cloud curves with both a UCST and a LCST [74]. The results obtained by the SAFT-HS approach [74] and by the micellar model are plotted together with the experimental data [63] in Fig. 12. In Fig. 12 it can be clearly seen that the modeling of the micelle formation using the association theory, like in SAFT-HS, is not sufficient for the description of the cloud point curve. Several other physical phenomena occurring in the present model play a significant role for the aggregation and for the demixing behavior.

Fig. 10. Comparison of the experimental (open circles — [64]; open squares — [57])

and the modeled cloud point curve (solid line) for C12 E4 surfactant solutions: the dotted line represents the calculated spinodale line and the dashed line is the cmc line

304 302 300 298 296 294

Fig. 11. Comparison of the experimental (symbols — [63]) and the modeled cloud point curve (solid line) for C10E4 surfactant solutions: the dotted line represents the calculated spinodale line and the dashed line is the calculated cmc

X C F

C10F4

Fig. 12. Comparison of the experimental (symbols — [63]) and the modeled cloud point curve (solid line) for C10E4 surfactant solutions:

the dotted line represents the calculated spinodale line and the dash-dotted line is the cloud point curve calculated using the SAFT-HS approach [74]

Conclusion. For the surfactant class CiEj the hydration of the polar head-group as function of temperature plays a dominant role. The incorporation of this effect in the theoretical framework via a temperature-dependent effective cross-section area of the hydrophilic part allows the modeling of the physical properties of aqueous surfactant solutions very close to the experimental data. Particular emphasis has been placed on the possible role in light-scattering measurements of critical concentration fluctuations and an eventual masking of size and shape changes of the particles.

For the thermodynamic modeling of demixing behavior of surfactant solutions made from water and nonionic surfactant CiEj with different numbers of i and j the aggregate formation must be taken into account. This statement is the outcome of the comparison between the calculation results using a detailed micelle formation model with calculation

results without micelle formation. Even the modeling of the self-association of water and surfactant and the cross-association between them, like in SAFT-HS, is not sufficient for the calculation of the cloud point curve.

The often discussed question, if the dilute phase contains micelles, can be answered by the intersection point of the cmc-line and the diluted branch of the cloud point curve. This intersection point depends strongly from the i and j values of the surfactant.

The theoretical framework contributes to fundamentally understand how the molecular structures of surfactants affect the solution properties, such as the cmc, the size and shape of aggregates, and the demixing behavior. Practically, this type of understanding can help

in choosing surfactant structures which will result in the desired performance properties.

***

This paper is dedicated to Prof. N. A. Smirnova in honor of her scientific impact. References

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Статья поступила в редакцию 15 октября 2012 г.