Thermodynamics of aqueous carbohydrate surfactant solutions

Fluid Phase Equilibria 153 Ž1998. 1–21

Thermodynamics of aqueous carbohydrate surfactant solutions S. Enders ) , D. Hantzschel ¨ Wilhelm Ostwald Institute of Physical and Theoretical Chemistry, UniÕersity Leipzig, Linnestr. 2, D-04103 Leipzig, Germany Dedicated to: Prof. K. Quitzsch on his 65th birthday. Received 23 April 1998; accepted 9 September 1998

Abstract The paper deals with the application of the micelle formation theory, developed by Nagarajan and Ruckenstein wR. Nagarajan, E. Ruckenstein, Langmuir 7 Ž1991. 2934–2969x and Nagarajan wR. Nagarajan, in: K. Esumi ŽEd.., Structure–Performance Relationships in Surfactants, Dekker, New York, 1997, pp. 1–81; R. Nagarajan, Adv. Colloid Interface Sci. 26 Ž1986. 205-264x to various n-alkyl-b-D-glucopyranoside surfactants, differing in the surfactant tail length Ž n-octyl-b-D-glucopyranoside C 8 G1 , n-decyl-b-D-glucopyranoside C 10 G1 and dodecyl-b-D-glucopyranoside C 12 G1 .. The model predicts that the carbohydrate surfactant molecules assemble for energetic reasons in spherical bilayer vesicles. The critical micellar concentration as function of the temperature shows a minimum value. The formed micellar aggregates exhibit a broad distribution of sizes. It is demonstrated in this study that the thermodynamic theory in combination with phase separation thermodynamics can be used successfully to described the phase separation, which occurs for the system C 10 G1 q water and C 12 G1 q water at low surfactant concentrations. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Gibbs energy; Liquid–liquid equilibria; Surfactant solution; Mixture

1. Introduction When surfactants are dissolved in water at a concentration which exceeds the critical micellar concentration ŽCMC., these amphiphilic molecules self-assemble into microstructures known as micelles, with their polar head groups in contact with water and their nonpolar hydrocarbon tail shielded from water in the micellar interior. Micelles can form in a variety of shapes Ž spherical, globular, or rodlike micelles or spherical bilayers. and display narrow or broad size distributions )

Corresponding author. Tel.: q49-3419736476; fax: q49-3419736399.

0378-3812r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 8 . 0 0 4 1 8 - X

S. Enders, D. Hantzschelr Fluid Phase Equilibria 153 (1998) 1–21 ¨

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depending on solution conditions. It is also possible to induce macroscopic phase separation of an isotropic micellar-rich phase coexisting with an isotropic micellar-poor phase. In order to formulate a theory of the equilibrium properties of micellar solutions including phase separation phenomena, it is necessary to incorporate the unique characteristics of micellar aggregates and their size distribution. It is essential to recognize, in distinction to other multicomponent mixtures, that the micellar size distribution is dependent upon solution conditions, such as amphiphile concentration and temperature. Puvada and Blankschtein w1x, Blankschtein et al. w2x, Nagarajan and Ruckenstein w3x and Nagarajan w4,5x developed a thermodynamic treatment to describe aqueous solutions of surfactants. This treatment was applied in detail for the polyŽ oxyethylene. -type nonionic surfactant w1,3–5x and micellar solutions of dioctanoyl-phosphatidylcholine w2x. In the last decade scientific and industrial interest focus on carbohydrate-based surfactants, especially n-alkyl-b-D-glycopyranosides Žsee Refs. w6–10x., because they are nontoxic and biodegradable. The phase Že.g., Refs. w11,12x. and the aggregation behaviour Ž e.g., Ref. w13x. of this kind of surfactant was investigated in the literature. The phase diagram was modelled using group contribution methods w12x ŽUNIFAC and UNIQUAC. neglecting the self-association of the amphiphilic compound. The aim of this paper is the application of the micellar formation model w3,4x in conjunction with the phase separation thermodynamics w1,2x on various n-alkyl-b-D-glucopyranosides ŽC 8 G1, C 10 G1 and C 12 G1 ., differing in the surfactant tail length.

2. Theory The total Gibbs energy of the surfactant solution Gsol is modelled as the sum of three contributions: the free energy of formation G F , the ideal free energy of mixing G M , and the free energy of interactions between the various components G I , that is w1x: Gsol k BT

s

GF k BT

q

GM

q

k BT

GI k BT

Ž1.

where k B is the Boltzmann constant. The total Gibbs energy can then be analyzed using the methods of equilibrium thermodynamics to calculate many thermodynamic properties of the micellar solution. The ideal free energy of mixing is given by w1–4x: GM k BT

`

s NW ln X W q

Ý Ng ln X g

Ž2.

gs1

where Nw is the number of water molecules, X w is the mole fraction of water, Ng is the number of aggregates with the aggregation number g, and Xg is the mole fraction of aggregates with the aggregation number g. The corresponding entropy ŽyG MrT . describes the entropy of mixing of the formed aggregates, the monomeric surfactant molecules, and the solvent. The interactions between the formed micellar aggregates, monomeric surfactant molecules and water molecules are modelled with the following mean-field expression w1,2x:

g XS 1 s y C Ž T . NS k BT 2 1 q Ž g y 1. XS G1

Ž3.

S. Enders, D. Hantzschelr Fluid Phase Equilibria 153 (1998) 1–21 ¨

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where X S is the surfactant mole fraction, C Ž T . and g are adjustable parameters. These parameters were obtained by fitting the calculated coexistence curve to an experimentally measured coexistence curve. The parameter g is given in principle by g s V srV w , where V w and V s are the effective volumes of a water and a solute molecule w1,2x. The free energy contribution G 1 is responsible for phase separation phenomena. A particularly important consequence of Eq. Ž 3. is that intermicellar interactions do not affect the micellar size distribution, since G 1 depends only on the total amount of amphiphile NS Žor X S . and not on the specific way that amphiphiles are distributed. The free energy of formation summarizes the molecular interactions responsible for self-association in a dilute reference solution that lacks intermicellar interactions and takes the following form w1,3,4x GF k BT

s NW

m0w k BT

`

q

m0g

Ý Ng k

gs1

BT

Ž4.

where m0w is the free energy change of the solution when a water molecule is added to pure water, and mg0 reflects the free energy change of the solution when a single aggregate, characterized by aggregation number g and the shape, will be formed. In the framework of the theory the surfactant molecules are characterized by the volume of the hydrophobic tail, VS , the extended length of the tail, l S , and effective cross-sectional area of the polar head group, a p . Depending upon the type of surfactant and the solution conditions, the aggregates may by spherical, globular, or rodlike or have the structure of spherical bilayers. Following Nagarajan and Ruckenstein w3x and Nagarajan w4,5x we distinguish between four types of surfactant aggregates. Fig. 1 depicts the different aggregate shapes formed in dilute solution, schematically. The closed aggregates with hydrophobic interiors are called micelles while the spherical bilayers containing an encapsulated phase are called vesicles. The small micelles are spherical in shape ŽFig. 1a. with the micelle radius rs . When amphiphilic molecules cannot pack into spheres anymore, Žthis happens for aggregation numbers for which a spherical

Fig. 1. Geometrical variables defining the structures of surfactant aggregates considered in this paper. The different shapes of aggregates are the following: Ža. spherical micelles, Žb. prolate ellipsoids, Žc. rodlike micelles, Žd. spherical bilayer vesicles.

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aggregate will have a radius greater than the extended length of the surfactant tail l S ., and if at the same time the rodlike micelles for energetic reasons are not yet favoured, small globular aggregates that are not much larger than the largest spheres should form. For aggregates in this transition region, Israelachvili et al. w14x suggested globular shapes generated via prolate ellipsoids with the semimajor axis b and the semiminor axis rs ŽFig. 1b.. When large rodlike micelles form, they are visualized as having a cylindrical middle portion and parts of spheres as endcaps Ž Fig. 1c. . The cylindrical and the endcap regions are allowed to have different diameters rs and rc . l c is the length of the cylindrical part of the micelle. Surfactant molecules can also assemble into a spherical bilayer structure Ž Fig. 1d. , called a vesicle, that enclose an aqueous cavity. In vesicles, the surfactant tails are assembled into a hydrophobic spherical shell covered on the inside as well as on the outside by polar surfaces made up of the surfactant head groups. In the outer and the inner layers of the vesicle, the surface area per surfactant molecule and the number of surfactant molecules need not be equal to one another, and the thickness of the two layers t i and t o can also be different. The inner radius is named r i and the outer radius is called ro . For aggregates of various shapes containing g surfactant molecules, the volume of the hydrophobic domain of the aggregate, Vg , the surface area of contact between the aggregate and water, A g , are taken from literature w4x and listed in Appendix A. The standard free energy difference D mg0 , between a surfactant molecule in an aggregate of size g and one in the single dispersed state can be decomposed into a number of contributions on the basis of molecular considerations w1,3–5x: D mg0 s mg0 y g m 0l s D mg0 tr q Dm0g

ž

/

ž

/

def

q D mg0

ž

/

ster

q D mg0

ž

/

int

Ž5.

First, the hydrophobic tail of the surfactant 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 Ž D mg0 . tr is estimated from independent experimental data on the solubility of hydrocarbons in water. On this basis, the transfer free energy for the methylene and methyl groups in an aliphatic tail as a function of temperature T is given by w3,4x: D mg0

ž / ž / ž / k BT

s Ž n T y 1. tr

D mg0 k BT

ž / k BT

q tr ,CH 2

D mg0

ž / k BT

Ž6. tr ,CH 3

s 3.38 ln T q 4064rTy 44.13 q 0.02595T

Ž7.

s 5.85 ln T q 896rTy 36.15 y 0.0056T

Ž8.

tr ,CH 3

D mg0 k BT

D mg0

tr ,CH 2

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. Nagarajan and Ruckenstein w3x and Nagarajan w4x obtained the following expression for spherical micelles: D mg0

ž / k BT

s def

9p 2 rs2 P 80 NL2

Ž9.

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where P is the packing factor, rs is the core radius, L is the segment length Ž L s 0.46 nm., and N is the number of segments in the tail Ž N s l srL.. Eq. Ž 9. is employed also for prolate ellipsoids and for spherical endcaps of rodlike micelles. For cylindrical rods, the constant 9 in Eq. Ž 9. is replaced by 10, the radius rs is replaced by the radius rc of the cylinder, and P s 0.5. For spherical bilayer vesicles, the molecular packing differences between the outer and the inner layers must be accounted. When Eq. Ž9. is applied to spherical bilayer vesicles, the radius rs is replaced by the half-bilayer thickness t o for the molecules in the outer layer, and t i for the molecules in the inner layer, and the coefficient 9 is replaced by 10 and P s 1, as for lamellar aggregates. The equations for the different shapes are listed in Appendix B. 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 Ž D mgo . int is calculated as the product of the surface area in contact with water and the macroscopic interfacial tension sAgg of the aggregate core–water interface w3,4x: D mg0

ž / k BT

s int

sagg k BT

Ž a y a0 .

Ž 10.

Here, a is the surface area of the hydrophobic core per surfactant molecule, and a 0 Ž a 0 s L2 . is the surface area per molecules shielded from contact with water by the polar head group. The aggregate core–water interfacial tension sAgg is taken equal to the interfacial tension between the aliphatic hydrocarbon of the same molecular weight as the surfactant tail and the surrounding water at the actual temperature w3,4x. The equations for sAgg are given in Appendix B. Fourth, the surfactant head groups are brought to the aggregate surface, giving rise to steric repulsions between them. This steric contribution to the free energy Ž D mg0 . ster is given by w3,4x: D mg0

ž / k BT

ž

s yln 1 y ster

ap a

/

Ž 11.

where a p is the cross-sectional area of the polar head group. Eq. Ž 11. is utilized for spherical and globular micelles as well as for the cylindrical middle and the spherical endcaps of rodlike micelles. For spherical bilayer vesicles, the steric repulsions at both the outer and the inner surfaces must be taken into account w4x: D mg0

ž / k BT

sy ster

go g

ž

ln 1 y

aP go A go

/

y

gi g

ž

ln 1 y

aP gi A gi

/

Ž 12.

The glucoside head group in b-glucosides has a compact ring structure. The effective cross-sec˚ 2 w15x. The estimation of A g o and A g i is tional area of the polar head group a P is estimated to be 49 A explained in Appendix A. The molecular volume of the surfactant tail VS containing n T carbon atoms is calculated from the group contributions of n T y 1 methylene groups and the terminal methyl group w3,4x:

˚3 V Ž CH 3 . s Ž 54.6 q 0.124 Ž T y 298 . . A ˚ V Ž CH 2 . s Ž 26.9 q 0.0146 Ž T y 298 . . A where T is in Kelvin. The extended length of the surfactant tail l S is given by Tanford w16x: 3

l S s Ž 1.5 q 1.265 nT . A˚

Ž 13. Ž 14. Ž 15.

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The extended length of the surfactant tail l S and the effective cross-sectional area of the polar head group a p are treated as temperature independent. Appendix B collects all equations and parameters, which are necessary to calculate D mg0 for the different micellar shapes. Using the free energy expressions, Eqs. Ž2. – Ž4. in Eq. Ž 1. the chemical potentials of water and aggregates can be calculated. These are calculated from Gsol by differentiating Gsol with respect to Nw and with respect to Ng . Hence, the chemical potentials are given by:

mg s mw s

mg0

`

ž

q 1 q ln Xg y g X w q

kT

m0w

Ý

/

Xg y

gs1 `

q 1 q ln X w y X w y

kT

Ý

Xg y

gs1

C Ž T . g gX S Ž 2 X w q g X S . 2 Ž1 q Ž g y 1. XS . C Ž T . g X S2

2 Ž1 q Ž g y 1. XS .

2

2

Ž 16. Ž 17.

These results are identical to the results of Puvvada and Blankschtein w1x. Applying the principle of multiple chemical equilibrium w17x between aggregates of different sizes and monomers, that is mg s g m 1, the following expression for the micellar size distribution is obtained:

žž

Xg s exp g 1 q ln X 1 y

D mg0 k BT

/ / y1

Ž 18.

where X 1 is the mole fraction of monomer which can be evaluated using Eq. Ž 18. and the surfactant mass balance equation w1–4x: `

XS s

Ý gXg

Ž 19.

gs1

A theoretical definition of the critical micelle concentration ŽCMC. was proposed w18x as the concentration of the single dispersed surfactant at which the shape of the micelle size distribution function exhibits a transition from a monotonically decreasing one to a function possessing a maximum and a minimum. Therefore, the critical micelle concentration Ž CMC. can be calculated from the aggregate size distribution by constructing a plot of one of the functions Ž a. surfactant concentration X S , Žb. number-average aggregation number g N , Žc. weight-average aggregation number g w against the monomer concentration X 1. The CMC can be identified as that value of the monomer concentration at which a sharp change in the plotted function occurs. In the present paper the following condition was used:

E 2 g N Ž X1 . E X 12

s0

Ž 20 .

The separation of the micellar solution into two coexisting phases having different total concentrations X SI and X SII can be calculated using the thermodynamic equilibrium conditions:

mIW s mIIW m

I S,g s

m

II S,g

Ž 21. Ž 22.

where the chemical potentials are given in Eqs. Ž 16. and Ž 17.. Inserting Eqs. Ž 16. and Ž 17. in the thermodynamic equilibrium conditions results in two equations. Solving both equations simultane-

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ously at constant temperature yields the surfactant concentrations in both phases. During this solution procedure 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. Hence the aggregation number distribution function has to be calculate within every iteration step. The quantity D mg0 is only a function of temperature and not of surfactant concentration. For this reason it is favourable to calculate the surfactant concentrations in both phases at constant temperature. First, we determine mg0 for all g-values at constant temperature. During this determination the geometrical properties for all possible aggregates Ž spherical micelles, prolate ellipsoids, rodlike micelles and spherical bilayer vesicles. and for all values of g were calculated and the lowest value of D mg0 were saved as a function of g. Making an initial guess for the monomer concentrations in both phases X 1I and X 1II permits the calculation of the distribution of micellar sizes Xg in both phases, applying Eq. Ž18.. With the help of Eq. Ž19. the total surfactant concentration for both phases Ž X SI and X SII . can be calculated. Now it is possible to calculate the chemical potential for the water molecules. Keeping in view the principle of multiple chemical equilibrium Ž mg s g m 1 . permits the calculation of the chemical potential of the monomer surfactant molecules. Using a multidimensional Newton procedure in order to change X 1I and X 1II until Eqs. Ž21. and Ž22. are fulfilled results in the equilibrium values X 1I and X 1II. Repeating the whole procedure for the next temperature yields the phase diagram. Utilizing the Gibbs stability condition yields for the spinodal w2x: 1

C ŽT .g

1 q

XS g W

y XW

Ž1 q Ž g y 1. XS .

3

s0

Ž 23.

and for the critical point w2x: y

gz 2 X S2 g W

1 q

2 XW

y

3C Ž T . g Ž g y 1 .

Ž1 q Ž g y 1. XS .

4

s0

Ž 24.

The parameter C Ž T . fixes the kind of phase behaviour ŽLCST, lower critical solution temperature; UCST, upper critical solution temperature; hourglass-, closed-loop-shaped phase diagrams. . If C Ž T . decreases with increasing temperature then UCST-behaviour is predicted. On the other hand, if C Ž T . increases with increasing temperature then LCST-behaviour is predicted. To calculate hourglass and closed-loop shaped phase diagrams the function C Ž T . should have at least two roots with respect to temperature. Aqueous solutions of C 10 G1 and C 12 G1 exhibit LCST phase behaviour w11x. For that reason we assume the following relation for C Ž T .: C Ž T . s C1 y

C2 T

Ž 25.

The parameters C1 and C2 are obtained by fitting the theoretical to the experimental cloud point curve.

3. Results and discussion Eq. Ž5., in conjunction with the geometrical characteristics of the various different aggregates and the expression for the different contributions Eqs. Ž 6. , Ž 9. – Ž 11. , allows to calculate the standard free

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energy difference D mg0 between a surfactant molecule in an aggregate of size g and in the dispersed state ŽAppendix B.. From the geometrical relations provided in literature w3,4x and in Appendix A it can be seen that, given the surfactant parameter a P and n T , the geometrical properties of spherical or globular micelles depend only on the aggregation number g. In the case of rodlike micelles, by minimizing D mg0 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 l c , the number of surfactant molecules in the spherical endcaps g cap and in the cylinder g cyl . The minimization was carried out by solving the following equations:

E D mg0

ž / E rc

s0

Ž 26.

s0

Ž 27.

rs

and

E D mg0

ž / E rs

rc

simultaneously. The derivatives in Eqs. Ž 26. and Ž 27. were derived analytically. Minimization of D mg0 for spherical bilayer vesicles, similar to the rod-like micelles, results in the equilibrium inner and outer radii Ž r i and ro .. The inner and outer layer thickness Ž t i and t o . as well as the number of surfactant molecules in the inner and outer layer Ž n i and n o . are given by the geometrical relations w4x and in Appendix A. Fig. 2 illustrates the predicted standard free energy advantage associated with micelle formation D mg0 for the system C 10 G1 for different shapes. At small aggregation numbers the formation of spherical micelles leads to energy advantage. For steric reasons it is possible to form globular micelles and vesicle at a certain aggregation number. The model predicts that for energetic

Fig. 2. Predicted standard free energy difference D mg0 between a surfactant molecule in an aggregate of size g and one in the dispersed state for the C 10 G1-surfactant at 298 K at different aggregation shapes Žsolid line: spherical micelles, dashed line: prolate ellipsoids, dotted line: rodlike micelles and dashes-dotted line: spherical bilayer vesicles..

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reasons the formation of vesicles is preferred. At higher g-values the surfactant can also associate in rod-like micelles, but the standard free energy difference D mg0 is lower than for vesicles. These results are in contrast to the results of Nagarajan and Ruckenstein w3x and Nagarajan w4x for C 8 G1. These authors predicted that large polydispersed spherocylindrical aggregates were formed. The reason for the different results are not really clear. In the calculations are some differences. Ž1. Nagarajan and Ruckenstein w3x and Nagarajan w4x used for the effective cross-sectional area of ˚ 2. We follow Matsumura et al. w15x as well van Buuren and the polar head group a p a value of 40 A 2 ˚ . Assuming aP s 40 A˚ 2 leads in our calculations also to the formation Berensen w6x and set a p s 49 A of spherical bilayer vesicles, but with another CMC-values and another aggregation size distributions. ˚ 2 the predicted results are closer to the experimental findings. Applying a p s 49 A Ž2. In Ref. w4x on page 23 is written ‘‘The simpler approach is built on the recognition that for spherical or globular micelles and spherical bilayer vesicles, the size dispersion is usually narrow. . . . One can obtain quantitative estimates of the variance of the size distribution, also by the maximum-term method.’’ This means the distribution function of aggregate size is not completely taken into account. Our calculations show, that spherical bilayer vesicles show also a broad distribution function, similar to rodlike micelles. In our theoretical framework the complete size distribution was used. This differences can lead to the different aggregation shape. Ž3. In the papers of Nagarajan and Ruckenstein w3x and Nagarajan w4x are no hints about the quantity D mg if the formation of spherical bilayer vesicles is allowed. Ž4. In Ref. w4x on page 23 is written ‘‘In case of rodlike micelles, the size distribution, XG is monotonic and does not have a maximum. In this case, by minimizing D m ocyl for an infinitely long cylinder, the equilibrium radius R c of the cylindrical part of the micelle is determined. Given the radius of the cylindrical part, the number of molecules g cap in the spherical endcaps is found to be the value that minimizes D m0cap’’. The main difference between the calculation approach of Nagarajan and Ruckenstein w3x and Nagarajan w4x and our theoretical framework is the different equations for the size distribution function Xg . Starting from

mg s mg0 q kT ln Xg

Ž 28.

Eq. 3 in Ref. w4x and applying the multiple chemical equilibrium condition mg s g m 1 ŽEq. 2 in Ref. w4x. yields

ž

Xg s X lg exp y

mg0 y g m0l kT

/

ž

s X lg exp y

gD m0g kT

/

Ž 29.

Eq. Ž29. is in contrast to our equation for the size distribution ŽEq. Ž 18.. . The reason therefore is, that we used instead of Eq. Ž28. another expression for the chemical potential mg ŽEq. Ž17.. in order to retain thermodynamical consistence. With the help of size distribution function Ž Eq. Ž 18.. we found also a maximum in Xg in the case of rodlike micelles. Additionally, we do not assume infinitely long cylinder. In contrast to Nagarajan we estimate the geometrical properties by minimization of D mg0 as function of the radius of the cylindrical part rc and the radius of the spherical endcaps rs ŽAppendix A. . At aggregation numbers over 80 the D mg0-value is nearly constant ŽFig. 2.. Calculations for the C 8 G1 and C 12 G1 surfactant lead to a similar picture. The D mg0 decreased with increasing carbon

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S. Enders, D. Hantzschelr Fluid Phase Equilibria 153 (1998) 1–21 ¨

number in the hydrophobic surfactant tail Ž D mg0 ŽC 8 G1 .rk BT f y6.7, D mg0 ŽC 12 G1 .rk BT f y12.2.. The aggregation number n trans , at which vesicles are formed, shifts with increasing chain length to higher values Ž n trans ŽC 8 G1 . f 28, n transŽ C 12 G1 . f 56.. The model yields the variation of the geometric characteristics as a function of temperature and the aggregation number. In the case of rodlike micelles the number of molecules in the spherical endcaps n cap and the radius of the endcaps rs depend only slightly on the aggregation number g. The number of molecules in the cylindrical part n cyl of the micelle and the length of the cylinder l c are increased linear with g, while the radius of the cylindrical part rc remains constant. These findings indicate that the rodlike micelles grow by incorporating in the cylindrical part of the micelle. Increasing the surfactant concentration results in increasing of the average aggregation number. Assuming spherical bilayer vesicles the number of molecules in the outer n o and in the inner layer n i increased nearly linearly with aggregation number. The thickness of the inner layer t i increased in the aggregation ˚ to 5 f A. ˚ The cross sectional area of the number range from f 50 to f 100 rapidly from f 2 A ˚ 2 w15x. Assuming the sugar head has a spherical shape Ž A s 4p r 2 . than polar sugar head is be 49 A ˚ This means the head group fits in the inner as well in the outer the radius of the sugar head is 1.97 A. ˚ At higher g-values this thickness increased very slowly, layer, if their thickness is greater than 2 A. irrespective of the number of carbon atoms in the surfactant tail. The function of the outer layer ˚ to 8.5 A˚ for C 12 G1 . thickness t 0 shows the opposite behaviour. t o decreased first Ži.e. from 11 A rapidly and later very slowly. The outer layer thickness t o is a stronger function of the surfactant tail length than the inner layer thickness t i . The radii ro and r i grow continuously with the aggregation number. This finding indicates a swelling of the spherical bilayer vesicle. Plotting the average aggregation number as function of the monomer surfactant concentration X 1 yields information about the critical micellar concentration Ž Fig. 3. . At a narrow surfactant monomer concentration range the average aggregation number Ž number averaged or weight average. increased from 1 to f 32 for the C 8 G1 surfactant. If the g N-function is used as a criterion for the CMC

Fig. 3. Predicted number-averaged aggregation number Žsolid line. and weight-average aggregation number as function of the monomer concentration Ždashed line. for spherical micelles of C 8 G1 at 298 K. The predicted critical micellar concentrations are indicated by the arrows.

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ŽEq. Ž20.. than a lower value for the CMC is obtained as applying the g w-function as a CMC criterion. Both functions Ž g N and g w . reach the same value of g N or g w . This means that in the first stage small monodispersed spherical micelles were formed. La Mesa et al. w13x obtained experimentally an average aggregation number of 40 " 10. Our model is close to the experimental value at low surfactant concentrations. Fig. 4 gives a comparison of the experimental and the predicted critical micellar mole fraction. The experimental data were taken from the literature ŽC 8 G1 w19x, C 10 G1 w6x, C 12 G1 w20x.. Assuming spherical micelles, prolate ellipsoids or rodlike micelles the critical micellar concentration shifts only slightly to lower surfactant concentrations. Due to the greater standard free energy advantage associated with micelle formation D mg0 for bilayer aggregates the CMC shifts clearly to lower concentrations. D mg0 represents the standard free energy change when g surfactant monomer molecules self-assemble to form an aggregate of size g. At a certain aggregation number g trans Žfor instance for C 8 G1 at 298 K g trans s 28. D mg0 has for spherical bilayer vesicles a smaller value than for spherical micelles, globular micelles or rodlike micelles Ž Fig. 2. . Hence, the formation of spherical bilayer vesicles at higher g-values is connected with a larger energy advantage for the system. The chemical potential D mg0 enters the equation of the size distribution function Ž Eq. Ž 18.. . The higher value for D mg0 Žfor g ) g trans . shifts the average aggregation number to higher values and amplifies the dependence of the size distribution function on concentration. The critical micelle concentration was calculated from the aggregate size distribution by constructing a plot of the number-average aggregation number against the monomer concentration. Hence the CMC shifts to lower values. The predicted CMC-values ŽFig. 4. are in satisfactory agreement with the experimental findings, but it is not possible to decide, which aggregation shape is formed. The temperature dependence of the CMC for C 8 G1 for different aggregation shapes is shown in Fig. 5. Irrespective of the aggregation shape the CMC-values run through a minimum when the

Fig. 4. Comparison of experimental and predicted critical micellar mole fraction as function of the carbon number in the surfactant tail at 298 K for different aggregation shapes Žsolid line: spherical micelles, dashed line: prolate ellipsoids, dotted line: rodlike micelles and dashes-dotted line: spherical bilayer vesicles.. Experimental data was taken from the literature Žsquare, C 8 G1 w19x; circle, C 10 G1 w6x; triangle, C 12 G1 w20x..

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S. Enders, D. Hantzschelr Fluid Phase Equilibria 153 (1998) 1–21 ¨

Fig. 5. Temperature dependence of the critical micellar mole fraction for C 8 G1 for different aggregation shapes Žsolid line: spherical micelles, dashed line: prolate ellipsoids, dotted line: rodlike micelles and dashes-dotted line: spherical bilayer vesicles.. The squares are experimental data taken from the literature w21x.

temperature is raised over a wide range. Decreasing of the CMC with increasing temperature until 303 K is in agreement with experimental data from the literature w21x, but the slope of the curve is much steeper for the experimental points. At higher temperatures no experimental values are available in the literature. Puvvada and Blanckschtein w1x predicted for the nonionic surfactant class Ž polyoxyethylene glycol monoethers. that the CMC decreased with increasing temperature from 288 K to 318 K in good agreement with experimental data w22x, but no investigation at higher temperatures was carried out. For ionic surfactants Ž i.e., sodium alkyl sulfonate having C 10 , C 12 and C 14 as surfactant tails. it was experimentally w23x and theoretically w4x found that the CMC increased with increasing temperature in the temperature range from 293 to 393 K. The theoretical temperature dependence of the CMC for C xG1 is similar to the experimental temperature dependence of the CMC for the ionic surfactant w24x Žsodium dodecyl sulphate. . The experimental points are closer to the prediction using a spherical micelle or a prolate ellipsoid or a rodlike micelles then a bilayer vesicle as aggregation shape. Due to the different criteria of the CMC Ž g N or g w . it is not possible to decide which aggregation shape is really present in the surfactant solution. For this reason the minimum value of standard free energy difference D mg0 between a surfactant molecule in an aggregate of size g and one in the dispersed state was used in the further calculations. Fig. 6 illustrates the aggregation number distribution at constant temperature and constant total surfactant mole fraction X S for different n-alkyl-b-D-glucopyranoside. With increasing carbon number in the surfactant tail the average aggregation numbers shift to higher values and the distribution function becomes broader. At low aggregation numbers, where from the energetic point of view spherical micelles are preferred, the distribution function is practically zero. Hence mostly bilayer vesicles exist in the solution at the given solution conditions. In the further calculation the distribution function is completely taken into account. Fig. 7 depicts the variation of the aggregation number distribution as function of the total surfactant mole fraction for C 12 G1. Increasing the total surfactant concentrations leads to higher values for the

S. Enders, D. Hantzschelr Fluid Phase Equilibria 153 (1998) 1–21 ¨

13

Fig. 6. Aggregation number distribution function Xg for C 8 G1 Žsolid line., C 10 G1 Ždashed line. and C 12 G1 Ždotted line. at 298 K and a surfactant mole fraction X S s 0.15.

average aggregation numbers. Hence, the model predict a growing of the spherical vesicles. Increasing the total surfactant concentrations leads also to spreading of the distribution function. This is connected with a rice of the polydispersity of the formed aggregates. Even, at the lowest concentration Ž X S s 0.005. the portion of the spherical micelles are practically zero. The calculated distribution function shows only a slight dependence on temperature. Moller ¨ et al. w25x and Kameyama and Takagi w26x determined weight-average aggregation number, applying static light scattering, for C 8 G1. They obtained g w s 85 and found that the weight average

Fig. 7. Aggregation number distribution function Xg for C 12 G1 at 298 K for different total surfactant mole fractions Žsolid line, Xs s 0.005, dashed line, Xs s 0.05, dotted line, Xs s 0.15..

14

S. Enders, D. Hantzschelr Fluid Phase Equilibria 153 (1998) 1–21 ¨

Fig. 8. Theoretical spinodals for the system C 10 G1 qwater for different aggregation shapes. Žsolid line: spherical micelles, dashed line: prolate ellipsoids, dotted line: rodlike micelles and dashes-dotted line: spherical bilayer vesicles.. The parameters used are C1 s1.442, C2 s 350 K and g s 370.

molecular mass of the aggregates increased with the total surfactant concentration. At a certain concentration Ž w S s 0.1. the model predicts the following weight-average aggregation numbers g w : spherical micelle 31.6, prolate ellipsoids 36.9, rodlike micelles 40.9 and spherical bilayer vesicles 65. Comparison of experimental and theoretical values can point out that spherical bilayer vesicles will be formed. Another important micellar solution property of the systems C 10 G1 q water and C 12 G1 q water is the phase separation at low surfactant concentration w11,12x. For phase separation calculations adjusting of the parameters C1, C2 and g in Eqs. Ž3. and Ž 25. is necessary. Fig. 8 shows theoretical spinodal curves using an arbitrary chosen parameter set Ž C1 s 1.442, C2 s 350 K and g s 370. at different aggregation shapes for the system C 10 G1 q water. The spinodal separates the unstable region from the metastable region in the phase diagram. In sequence of spherical micelles, prolate ellipsoids, rodlike micelles and spherical bilayer vesicles the unstable region increased. The aggregate size increased in the same sequence. From thermodynamic considerations for polymer solutions it is known that the unstable region increases with the molecular weight of the polymer at constant interaction parameters. The lower critical solution point is at lower temperatures than the solid–liquid equilibrium. For cloud point calculations these parameters were fitted to the experimental cloud point data from the literature w11x. The parameters are given in Table 1. Fig. 9 shows the theoretical and experimental Table 1 Fitted interaction parameter in Eqs. Ž3. and Ž25. System

g

C1

C2 rK

C 10 G1 qH 2 O C 12 G1 qH 2 O

310 72

0.7661 0.6175

200 140

S. Enders, D. Hantzschelr Fluid Phase Equilibria 153 (1998) 1–21 ¨

15

Fig. 9. Comparison of experimental Žopen square: C 12 G1 qwater, solid square: C 10 G1 qwater. and theoretical cloud point curves Žsolid line C 12 G1 qwater, dashed line C 10 G1 qwater. for the systems C 10 G1 qwater and C 12 G1 qwater. The experimental data are taken from the literature w11x. The fitted parameters are given in Table 1.

phase diagram of both systems in semi-logarithmic plot. The model describes the concentration in the concentrate phase very accurate. The derivations in concentrations of the dilute phase are quite enlarged due do the semi-logarithmic representation. Nevertheless we can conclude that the model is applicable for phase separation calculations taking the self-association of the amphiphillic compound and the distribution function of the aggregation number into account.

4. Conclusion The applied statistical thermodynamics model of aggregation behaviour of surfactant can be used to model the behaviour of different n-alkyl-b-D-glucopyranosides. The model predicts that the carbohydrate surfactant assembles in spherical bilayer vesicles. Upon increasing total surfactant concentration the vesicles swell linearly. The predicted critical micellar concentration as a function of the temperature runs through a minimum. Combining the micellar formation model with phase separation thermodynamics permits the calculation of the critical point, the spinodal curve and the cloud–point curve. Adjusting three parameters, the model is able to describe the experimental cloud–point curve.

5. List of symbols A a C

surface area head group area interaction function defined in Eq. Ž3.

16

S. Enders, D. Hantzschelr Fluid Phase Equilibria 153 (1998) 1–21 ¨

G g k L l N n P r T t V X

Gibbs energy aggregation number Boltzmann constant segment length extended length of the tail number of molecules length of the hydrophobic tail packing factor radius temperature bilayer thickness or tail volume mole fraction

Subscripts agg B c def F g int M n l o p S ster tr trans w V z 1

aggregate core water interface Boltzmann constant cylindrical part of rodlike micelle deformation formation aggregate interface mixing number average quantity interaction or inner layer outer layer polar head group surfactant or sphere steric transfer transformation from spherical micelles to spherical bilayer vesicles water or weight average quantity volume z-average quantity monomer surfactant molecules

Superscripts 0 I II

standard state micellar-poor phase micellar-rich phase

greek letters g D

interaction parameter, defined in Eq. Ž3. difference

S. Enders, D. Hantzschelr Fluid Phase Equilibria 153 (1998) 1–21 ¨

E m s

infinite small change chemical potential interfacial tension

AbbreÕiations C 8 G1 C 10 G1 C 12 G1 CH 2 CH 3 CMC LCST UCST

n-octyl-b-D-glucopyranoside n-decyl-b-D-glucopyranoside n-dodecyl-b-D-glucopyranoside methylene group methyl group critical micellar concentration lower critical solution temperature upper critical solution temperature

17

Acknowledgements The financial support of ‘Deutsche Forschungsgemeinschaft’ and of the federation country Saxonia is gratefully acknowledged.

Appendix A. Estimation of the geometrical properties of the aggregates All equations are taken from the literature w4x. Input parameters are the molecular volume of the surfactant tail VS , the length of the surfactant tail l S , and the aggregation number g. Extended length of the surfactant tail l S :

˚ l s s Ž 1.5 q 1.265nT . A Volume of the surfactant tail VS :

˚3 V Ž CH 3 . s Ž 54.6 q 0.124 Ž T y 298 . . A V Ž CH 2 . s Ž 26.9 q 0.0146 Ž T y 298 . . A˚3 VS s V Ž CH 3 . q Ž n T y 1 . V Ž CH 2 . A.1. Spherical micelles Characteristic quantity: rs , radius of micelle. 3

rs s

(

3 gVs 4p

as

4p rs2 g

A s ag

S. Enders, D. Hantzschelr Fluid Phase Equilibria 153 (1998) 1–21 ¨

18

A.2. Globular micelles Characteristic quantities: l s , radius of semiminor axis; b, radius of semimajor axis. bs

)

3 gVs 4p

l s2

)

Es 1y

ls

2

as

b

2p l s2 g

1q

arcsin E E'1 y E 2

A s ag

A.3. Rodlike micelles Characteristic quantities: rc , radius of the cylindrical part; rs , radius of the endcaps; g cyl , number of surfactant molecules present in the cylindrical part; g cap , number of surfactant molecules present in the endcaps; l cyl , length of the cylindrical part of the rodlike micelle. Initial guess for rc and rs :

° ¢

H s rs 1 y~1 y

l cyl s

Vs g cyl

p rc2

A cyl s a cyl g cyl

)

1y

a cyl s

ž

rc rs

¶• /ß

2 p rc l c n cyl

2

g cap s

a cap s

2p Ž 4 rs3 y H 2 Ž 3rs y H . . 3Vs

g cyl s g y g cap

4p rs Ž 2 rs y h . n cap

A cap s a cap g cap

Calculation of

E D mg0

ž / E rc

rs

and

E D mg0

ž / E rs

rc

Changing rc and rs until both derivatives vanished. This procedure results in the optimal value of rs and rc . Subsequently the values of g cyl , g cap , l c , a cyl and a cap can be calculated. A.4. Spherical bilayer Õesicles Characteristic quantities: ro , outer radius; r i , inner radius; t o , outer layer thickness; t i , inner layer thickness; g o , number of surfactant molecules present in the outer layer; g i number of surfactant molecules present in the inner layer.

S. Enders, D. Hantzschelr Fluid Phase Equilibria 153 (1998) 1–21 ¨

19

Initial guess for ro and t o : 3

ri s

ti s

(

ro3 y

ž

3 g i Vs 4p

3 gVs

go s

4p

4p ro3 y Ž ro y t o . 3Vs

3

gi s g y g0

3

q r i3 y r i

A o s ao g o

/

a o s 4p ro2

a i s 4p r i2

A i s ai g i

Calculation of

E D mg0

ž / E ro

to

and

E D mg0

ž / E t0

ro

Changing ro and t o until both derivatives vanished. This procedure results in the optimal value of ro and t o . Subsequently the values of g o , g i , r i , t i , a o and a i can be calculated.

Appendix B. Summary of all equations and parameter necessary to calculate Dm0g Transfer free energy contribution: D mg0

ž / ž / ž / k BT

s Ž nT y 1 . tr

D mg0 k BT

ž / k BT

q tr ,CH 2

D mg0

ž / k BT

tr ,CH 3

s 3.38 ln T q 4064rTy 44.13 q 0.02595T tr ,CH 3

D mg0 k BT

D mg0

s 5.85 ln T q 896rTy 36.15 y 0.0056T tr ,CH 2

˚ 2 , lattice constant Žspacing between alkane Deformation free energy contribution: L s 4.6 A molecules in the liquid state.; N s l SrL number of segments. D mg0

ž / ž / k BT

s def

D mg0 k BT

s def

9p 2 rs2 240 NL2 9VSp 2 l s 80 NL2 a

spherical micelles

globular micelles

S. Enders, D. Hantzschelr Fluid Phase Equilibria 153 (1998) 1–21 ¨

20

ž ž ž ž

D mg0 k BT D mg0 k BT D mg0 k BT Dm

0 g

k BT

/ / / /

5p 2 rc2

s

cylindrical part of rodlike micelles

80 NL2

def

s

9VSp 2 rs

s

10p 2 t o2

spherical endcaps of rodlike micelles

80 NL2 a

def

spherical bilayer vesicles outer layer

320 NL2

def

10p 2 t i2

s

spherical bilayer vesicles inner layer

160 NL2

def

Interface free energy contribution: a 0 s L2 , surface area per molecules shielded from contact with water. Aggregate core–water interfacial tension sAgg at temperature T :

(

sagg s ss q sw y 1.1 ss sw where ss s 35-325M -2r3-0.098Ž T-298., sw s 72-0.16Ž T-298. , and M is the molar mass of the surfactant tail. D mg0

ž / k BT

s int

sagg k BT

Ž a y a0 .

Steric free energy contribution: D mg0

ž / ž / k BT

ster

D mg0 k BT

ž

s yln 1 y

sy

go g

ster

ap a

ž

/

ln 1 y

spherical-globular and rodlike micelles ap go A go

/

y

gi g

ž

ln 1 y

aP gi A gi

/

spherical bilayer vesicles

Standard free energy difference D mg0 : D mg0 s D mg0 tr q D mg0

ž

/

ž

/

def

q D mg0

ž

/

ster

q D mg0

ž

/

int

References w1x w2x w3x w4x w5x w6x w7x

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