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Rod Formation of Ionic Surfactants: A Thermodynamic Model A. Heindl and H.-H. Kohler* Institute of Physical and Macromolecular Chemistry, University of Regensburg, 93040 Regensburg, Germany Received October 17, 1995. In Final Form: February 5, 1996X A thermodynamic model is presented describing the equilibrium structure of micelles formed from ionic surfactants in diluted aqueous solutions at varying counterion concentrations. On a molecular level the model includes hydrophobic, steric, and electrostatic interactions. The conformational energy of the surfactant chain, direct adsorption of counterions to the micelle surface, and curvature effects are taken into account. By minimizing the Gibbs energy of the aggregates, the equilibrium structure is calculated for spherical, rod-shaped, and dumbbell-shaped micelles, the latter filling the gap between spheres and rods. To avoid edges, the dumbbells and the end caps of the rods are composed of a hemispherical and a catenoidal part. The Gibbs energy of the aggregates is used to determine the size distribution at given concentrations of surfactant and salt. From the size distributions the viscosity of the micellar solution is calculated and fitted to experimental viscosity data for hexadecylpyridinium chloride, bromide, iodide, and nitrate in diluted solutions with concentrations ranging up to 10 mM. Viscosity was measured at 50 °C with a capillary viscometer. The concentration of the anion was varied by the addition of the appropriate potassium salt. It is found that the tendency of rod formation increases in the order Cl-, Br-, NO3, I . The viscosity calculated from the model is in good agreement with experimental data. According to the model, rods do not form unless the hydrocarbon/water interfacial tension increases with increasing curvature. Rod formation is favored by increasing counterion concentrations. Rods are always found to compete with dumbbells, while the contribution of spheres is negligible. The radius of the end caps of rod-shaped micelles is identical with the full chain length; that of the cylindrical part is smaller by about 0.3 nm. The head group area in the cylindrical part is only slightly smaller than that in the end cap and decreases with increasing counterion concentration. Since the Stokes radii are virtually the same for Cl-, Br-, I-, and NO3 , counterion specificity cannot be attributed to Stokes radii. Instead it is modeled as a result of direct counterion adsorption. The model is also used to calculate the counterion association coefficients in dependence of counterion concentration.
I. Introduction The formation of micelles from surfactant molecules in aqueous solution depends on the type of surfactant, on its concentration, and on temperature, pressure, pH, ionic strength, and added organic compounds. In diluted solutions the shape of the micelles can vary from sphere to rod and to bilayer or something in between. With cryogenic transmission electron microscopy micelles can be directly imaged.1-3 Indirect methods to detect micelles are viscosity4-6 and light-scattering measurements.7 In this paper we describe the micellization in diluted solutions of cetylpyridinium salts with chloride, bromide, iodide, and nitrate as counterions. A thermodynamic model of micellar aggregation is established and compared with experimental viscosity data. The model is based on the thermodynamic principles of self aggregation introduced by Tanford8 combined with a description on a molecular level. This approach has been used successfully by other authors, both for nonionic and ionic surfactants, to describe the cmc and the formation of micellar aggregates of different shapes.9-14 Various kinds of interaction have been taken into account. X
Abstract published in Advance ACS Abstracts, April 1, 1996.
(1) Talmon, Y.; Vinson, P. K. J. Colloid Interface Sci. 1989, 133, 288. (2) Talmon, Y.; Magid, L. J.; Gee, J. C. Langmuir 1990, 6, 1609. (3) Lin, Z.; Scriven, L. E.; Davis, H. T. Langmuir 1992, 8, 2200. (4) Ekwall, P.; Mandell, L.; Solyom, P. J Colloid Interface Sci. 1971, 35, 519. (5) Kohler, H.-H.; Strnad, J. J. Phys. Chem. 1990, 94, 7628. (6) Heindl, A.; Strnad, J.; Kohler, H.-H. J. Phys. Chem. 1993, 97, 742. (7) Mazer, N. A. In Dynamic Light Scattering; Pecora, R., Ed.; Plenum Press: New York, 1980; Chapter 8. (8) Tanford, C. The Hydrophobic Effect, 2nd ed.; Wiley: New York, 1980; Chapter 7.
However, the model proposed by Missel et al.11 is the only one to describe the influence of different counterions. His model describes rod formation in sodium dodecyl sulfate solutions following the pattern LiDS < NaDS < KDS < CsDS < RbDS. The authors consider hydrophobic and electrostatic contributions to the Gibbs energy of purely spherical and purely cylindrical micelles. The counterion specifity of micellization and rod formation is modeled by the inclusion of a Stern layer with a thickness equal to the Stokes radius (“stick version”) of the cationic counterion, decreasing from Li+ (0.235 nm) to Rb+ (0.126 nm). Similar to the case of sodium dodecyl sulfate solutions, we have found in our experiments that rod formation in cetylpyridinium solutions is very sensitive to counterion identity5 although the Stokes radii of the investigated anions differ only by about 5% (see Table 1). This difference is too small to account for the observed variations in rod formation. Therefore, in our theoretical treatment, we will introduce the concept of direct counterion adsorption. A problem, also met in earlier publications, is that the rods tend to grow more strongly with increasing counterion concentration in the model than in experiment.14 To solve this problem, we assume that direct counterion adsorption is dependent on the mean curvature of the micelle surface. An important new feature of our approach is that the hemispherical endings and the central cylindrical part of (9) Israelachvili, J. N.; Mitchell, J. D.; Ninham, B. W. J. Chem. Soc. Faraday Trans. 2 1976 72, 1525. (10) Mitchell, J. D.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1981, 77, 601. (11) Missel, P. J.; Mazer, N. A.; Carey, M. C.; Benedek, G. B. J. Phys. Chem. 1989, 93, 8354. (12) Blankschtein, D.; Puvada, S. J. Chem. Phys. 1990, 92, 3710. (13) Eriksson, J. C.; Ljunggren, S. Langmuir 1990, 6, 895. (14) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934.
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Langmuir, Vol. 12, No. 10, 1996 2465 vsfNA ) 1/c0
Table 1. Comparison of Different Ionic Radii of the Counterions (refs 15-17)a counterion
stick radius/nm
slip radius/nm
crystal radius/nm
hydrated radius/nm
ClBrINO3-
0.121 0.118 0.120 0.127
0.181 0.177 0.180 0.191
0.181 0.195 0.216 0.264
0.33 0.33 0.33 0.34
a For anions the slip Stokes radii agree better with the crystal radii than the stick radii, so they seem to be more realistic. The values chosen for modeling are shown in bold type.
a rod-shaped micelle are connected by a catenoidal part. Thus we avoid edges at which curvature is not defined. This is of crucial importance for a consistent determination of the curvature-dependent contributions of electrostatic, interfacial, and conformational energy to the Gibbs energy of a rod-shaped micelle. If the cylindrical part reduces to length zero, we get a new type of aggregate called a dumbbell, consisting of two hemispherical terminations and a catenoidal connection. These dumbbells, filling the gap between spheres and rods, will play a major role in the fitting procedure. As another new feature of our approach we minimize the Gibbs energy of each aggregate as a whole. Compared with the more conventional method of minimizing separately the Gibbs energy of the different micellar parts, the global minimization procedure leads to considerable changes in the size distribution. For comparison with experimental viscosity data we use the total size distribution of the micellar aggregates to calculate the relative viscosity of the solution. II. Experimental Part II.1. Materials. Because of the high Krafft point of the CpI solution (39.4 °C without added salt) all measurements were done at 50 °C. The surfactants CpBr, CpI, and CpNO3 were prepared by recrystallization of CpCl (Merck AG, Germany) from the corresponding acid. All surfactants were recrystallized twice from absolute ethanol and dried under vacuum. The potassium salts KCl, KBr, KI, and KNO3, used to vary the anion concentration, were of p.a. grade (Merck AG, Germany). II.2. Methods. The cmc’s were determined from surface tension measurements with a digital tensiometer (Lauda, Type TE1C/2, Germany). The viscosities of the surfactant solutions were measured with Ubbelohde viscometers. Capillaries of diameters of 0.36, 0.46, and 0.53 mm were used. The viscometers were thermostated to 50 ( 0.05 °C. II.3. Viscosity Measurements. Surfactant concentrations ranged from 1 to 10 mM. In this range interactions between micelles can be neglected as long as the intrinsic viscosity is smaller than 270.5 The relative viscosity of a surfactant solution, ηrel, is defined as the ratio of the viscosity of the micellar solution, η, to the viscosity of the micelle free solution, η0. According to Einstein’s viscosity law the relative viscosity of a solution of spherical micelles is
ηrel )
cˆ A η ) 1 + 2.5Φ ) 1 + 2.5 η0 c0
(1)
where Φ is the volume fraction of the micelles. Φ can be expressed as the ratio of the concentration of surfactant ions incorporated into micelles, cˆ A, to the concentration of the pure surfactant, c0. The index A denotes the surfactant ion. The total surfactant concentration is
ctot ˆA A ) cA + c
(2)
where cA is the monomer concentration, which is approximately equal to the cmc. The volume vsf of a surfactant ion is related to c0 by
(3)
where NA is the Avogadro constant. In the case of spherical micelles the plot of ηrel against cˆ A yields a straight line (cf. eq 1). Such a straight line is obtained for all surfactant solutions without added salt (see Figure 1). Various amounts of salt must be added to induce a sphere to rod transition. As soon as rods begin forming there is a strong increase in the relative viscosity and a strong deviation from linearity between ηrel and cˆ A. For rods with axial ratio f (long axis to short axis) greater than 3 or intrinsic viscosity greater than 3.5, the relative viscosity of a monodisperse solution can be described, in good approximation,5 by
f2 ηrel ) 1 + Φ·0.25 ln(0.63f)
(4)
For polydisperse rods the relative viscosity equals the volume average on the right hand side. Denoting the volume average of a quantity x by 〈x〉v, we have5
〈
ηrel ) 1 + Φ·0.25
f2 ln(0.63f)
〉
〈f 2〉v ≈ 1 + Φ·0.25 v ln(0.63〈f 3〉v/〈f 2〉v) (5)
Figure 1a shows that, in the case of CpCl, an addition of 2.5 M KCl is necessary to obtain a measurable amount of rods with a measurable deviation from linearity. The experimental error in relative viscosity is ∆ηrel ) (0.003 and never exceeds the size of the symbols. Parts b-d of Figure 1 show the behavior of the other surfactant/salt systems. The amount of salt required for a sphere to rod transition decreases distinctly in the order chloride, bromide, nitrate, and iodide, although the four counterions have the same valency and virtually the same Stokes and hydrated radii15-17 (see Table 1). The question then is how to explain the observed counterion specifity.
III. Model First, we describe the geometrical constraints and simplifications. Second, we deal with the thermodynamics of micellization. Third, we treat the interaction energies used to calculate the Gibbs energy of micellization. Fourth, we describe the minimization of the Gibbs energy. III.1. Micellar Geometry. Our calculations are based on the geometry of a micellar surface as shown in Figure 2. Surface roughness is ignored. Head group and counterion charges will be smeared out. Random motion of head groups perpendicular to the micellar surface will not be considered. Instead, we assume that the last methylene group of the C16 tail is hydrated.18,19 Since the charge of the pyridinium ring is largely fixed to the nitrogen atom,20 we assume a positive charge at the end of the surfactant tails. Approach of counterions to the hydrocarbon core is limited by their radii. Therefore we introduce a Stern layer. We assume that counterions cannot penetrate into the hydrated methylene layer but may adsorb directly to the Stern layer (outer Helmholtz surface). Direct adsorption is controlled by the head group area and the curvature of the micelle. The diffuse ionic cloud begins on top of the stern layer. Head group repulsion acts in the middle plane of the pyridinium rings. If the interface is curved, we thus obtain different micellar (15) Israelachvili, J. N. Intermolecular & Surface Forces, 2nd ed.; Academic Press: London, 1992; Chapters 16 and 17, p 55. (16) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry; Plenum: New York, 1970. (17) Pau, P. C. F.; Berg, J. O.; McMillan, W. G. J. Phys. Chem. 1990, 94, 2671. (18) Soderman, O.; Walderhaug, H.; Henriksson, U.; Lindman, B. J. Phys. Chem. 1985, 89, 3693. (19) Tabony, J. Mol. Phys. 1984, 51, 974. (20) Jacobs, P. T.; Anacker, E. W. J. Colloid Interface Sci. 1973, 44, 505.
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a
Figure 2. Surface region of a micelle.
b
c
Figure 3. Schematic representation of micellar species used in the model: (a, top left) spherical micelle; (b, top right) dumbbell; (c, bottom left) rod; (d, bottom right) infinite disc.
forms of micelles are allowed to form simultaneously. Their concentrations depend on the Gibbs energies of the aggregates. The chain length l of a surfactant tail of nc carbon atoms cannot exceed a critical value lc given by8
lc ) (0.15 + nc·0.1265) nm g l
d
Figure 1. Experimental data of the relative viscosity, ηrel, as a function of cˆ A (surfactant concentration incorporated into micelles) for different surfactant/salt systems (a-d): straight lines, spherical micelles; dots, rods.
radii and head group areas for hydrophobic, steric, and electrostatic interactions. In our model we consider four competing micellar geometries: spheres, dumbbells, rods, and bilayers (see Figure 3). It has been shown that, in diluted solutions, disks or bilayer structures formed from a single surfactant undergo a phase transition at their cmc.15 Therefore bilayers will be regarded as infinite in size. Consequently the energetic contribution of rims can be neglected. All
(6)
We assume the last methylene group of the cetylpyridinium chain to be hydrated. Thus, for all micellar forms, the maximum radius of the hydrocarbon core is equal to the critical length of a C15 chain and amounts to lc,15 ) 2.05 nm. To obtain the length of the whole surfactant ion, lsf, the length of the head group, lhead ) 0.6 nm, must be added to lc,16. The volume vch of a surfactant chain can be calculated as a function of chain length and temperature.14 At T ) 50 °C and with one hydrated methylene group, that part of a cetylpyridinium ion which is immersed into the hydrocarbon core (C15 chain) has a volume of v15 ) 0.439 nm3. The volume of the whole surfactant ion, vsf, is the sum of the chain volume and the head volume. In the case of cetylpyridinium it is given by
vsf ) v16 + vhead ) 0.47 nm3 + 0.05 nm3 ) 0.52 nm3 (7) Since lc and vch are fixed, the spherical micelle has a maximum aggregation number of nmax ) 82. For higher aggregation numbers the sphere must deform. We assume that deformation occurs in one direction, leading to a dumbbell (index d) (see Figure 3) composed of two hemispherical (index s) parts and a catenoidal (index ca) part. A catenoid is generated by rotating a catenary around its axis of symmetry (x-axis). The catenoid has mean curvature zero. We assume that the contact between
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the spherical and catenoidal parts is smooth. The volume and the area of the dumbbell thus are given by
Vd ) Vs + Vca
(
)
rca 1 + x1 - rca/rs + 1 - rca/rs 2rsx
2π 3 r 2 s
Vs ) 2
(8) (9a)
π Vca ) 2 rca3(arcosh xrs/rca + xrs/rca - 1xrs/rca) (9b) 2
µ°M(n) + kT ln
(10)
As ) 2·2πr2s (x1 - rc/rs + 1)
(11a)
Aca ) 2πr2c (arcosh xrs/rca + xrs/rca -1xrs/rca)
(11b)
The radii rs and rca refer to the hydrocarbon core of the micelle. The aggregation number of a dumbbell is defined by
nd )
Vd v15
() (
cA cM(n) ) c0 c0
βndAca As + βAca
as aca
and
(13)
ns ) nd - nca (14a,b)
The head group areas are given by
aca )
Aca nca
and
exp -
)
n(µ°n - µ°A) kT
(17)
µM(n) ) nµA + γnµB
(18a)
and
the numbers of head groups in the two end cap regions (identified with aggregation numbers) are given by
nca )
n
where µ°n ) µ°M(n)/n can be considered as the standard chemical potential of a surfactant molecule in a micelle. The law of mass action of eq 17 can be used for nonionic and ionic surfactants. As to ionic surfactants this can be shown as follows. If the association coefficient of the counterion is denoted by γ, then the chemical reaction of micelle formation is nA. + γnB a Mn. Therefore, eqs 16a and b are replaced by
(12)
We allow the head groups of the surfactant molecules to move between the catenoidal and the hemispherical parts (so that the head group areas as and aca are variable). If β is defined by
β)
(16b)
where µ°A is the standard chemical potential of a surfactant molecule and µ°M(n) is that of a micelle of aggregation number n. cA and cM(n) denote the molar concentrations of monomers and micelles, respectively. We identify the standard concentration c0 with the surfactant concentration in the condensed state (see eq 3). From eqs 16a and b we obtain
and
Ad ) As + Aca
cM(n) cA ) nµ°A + nkT ln c0 c0
as )
µ°A + kT ln
cA cB + γµ°B + kTγ ln (18b) c0 c0
where µ°B and µ˜ °M(n) are the standard chemical potentials of a counterion and a micelle, respectively. The tilde is used for formal reasons only (cf. eq 21). If we introduce µ˜ °n ) µ˜ °M(n)/n, the concentration of micelles is given by
( )( ) [
cA cM(n) ) c0 c0
n
cB c0
γn
exp -
]
(µ˜ °n - µ°A - γµ°B)n kT
Equation 19 may also be written as
As ns
(15a,b)
The aggregation number of the dumbbells is restricted, similar to that of spheres. In the case of cetylpyridinium, the maximum number of surfactant molecules in a dumbbell is about 170. Micelles containing more surfactant molecules will form rods. A rod can be considered as a dumbbell with a cylindrical part intercalated in the middle. Therefore, the aggregation number of the rod, nr, is unrestricted and is given by nr ) nc + nd. In the case of rodlike micelles the radius of the spherical end cap, rs, may be different from the radius of the cylindrical (index c) part, rc. Thus formulas for dumbbells can also be used to describe the end caps of rod-shaped micelles. III.2. Thermodynamics of Micellization. A micelle of aggregation number n can be considered as a chemical species formed from n surfactant molecules in a chemical reaction nA a Mn. The condition of chemical equilibrium is
µM(n) ) nµA
kT cM(n) 1 ) µ˜ ° (n) + ln n M n c0
(16a)
where µM(η) is the chemical potential of a micelle and µA is the chemical potential of a monomer. Assuming ideality of the solution, the chemical potentials can be written as
() [
cA cM(n) ) c0 c0
n
exp -
Defining µ°n now by
(µ˜ °n - µ°A - γµ°B)n kT
+ nγ ln
(19)
]
cB c0
(20)
(
µ°n ) µ˜ °n - γ µ°B - kT ln
)
cB c0
(21)
eq 20 becomes formally identical with eq 17. Note that µ°n, contrary to µ˜ °n, covers changes in the Gibbs free energy caused by variations of the concentration of counterion B. More generally, µ°n covers the energetic contributions of all components other than A to the process of micelle formation. If we define the standard work for the reversible transfer of a surfactant ion from the free solution into a micelle of aggregation number n by
g ) µ°n - µ°A
(22)
then eq 17 may be written as
() ( )
cA cM(n) ) c0 c0
n
exp -
ng kT
(23)
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Heindl and Kohler
We will call g the Gibbs energy of a surfactant ion and
G ) ng
1 ∂g kT ∂ ln cB
cB cmc ) [B] - γ ln ln c0 c0
(26)
where [B] is constant. For spherical micelles, with aggregation number ns and Gibbs energy gs, eq 23 can be specified to
() ( ) ns
nsgs exp kT
∆Gr ) (gj r - g∞c )nr
() ( nd
exp -
ndgj d kT
)
() ( ) (
cM(nr) cA ) c0 c0
() ( ) nr
exp -
nrgj r kT
nc gc(nr) + nsgs(nr) + ncagca(nr) nr
(28)
(29)
(30)
(31)
exp -
)
∆Gr(nr) kT
(36)
( )
(37)
(
) (
(gj r - g∞c )nr ∆Gr(nr) ) exp kT kT
)
(38)
eq 38 can be rewritten as
( )
cA cM(nr) ) c0 cmcc
nr
1 K(nr)
(39)
Note that ∆Gr(nr) and K(nr) are functions of nr. The parameter K(nr) controls rod formation in the micellar system. If K(nr) is large, much energy is needed to form short aggregates from long ones and rods tend to become long. If rc and rs, and thus gj d, gc, ∆Gr, and K ) exp(∆Gr/ kT), are assumed to be independent of nr (as is often done in the literature9-14), ∆Gr, according to eq 35, could be simply written as
∆Gr ) (gj r - gc)nr ) nc gc + ndgj d - gc(nc + nd) ) nd(gj d - gc) (40) Note that ∆Gr (eq 35) can be given an analogous form if nr approaches infinity:
∆G∞r ) lim ∆Gr ) lim((g∞r - g∞c )nr) ) nd(gj ∞d - g∞c ) (41) nrf∞
(32)
Again, the Gibbs energies are given by eq 30 and i may be c, s, or ca. It should be noticed that the geometry and the aggregation numbers of the cylindrical, spherical, and catenoidal parts depend on the aggregation number nr of the rod as a whole. Therefore, the gi’s are functions of nr. The Gibbs energy for the formation of a rodlike micelle of aggregation number nr is given by (cf. eq 24)
Gr ) nrgj r
nr
g∞c cmcc def exp ) c0 kT
K(nr) ) exp
with
gj r )
g∞c kT
With the abbreviation
where i may be s or ca. Note that the association coefficients γs and γca of the spherical and the catenoidal part may be different. For rodlike micelles (index r) of aggregation number nr, eq 23 becomes
cM(nr) cA ) c0 c0
exp -
(27)
By analogy to eqs 21 and 22, gs and gca can be written as
gi(nd) ) µ°i - µ°A - γiµ°B - kTγi ln(cB/c0)
nr
We define a critical micelle concentration, cmcc, for the formation of a purely cylindrical micelle as follows:
where gj d is the mean Gibbs energy of a surfactant ion. If we relate gs to the spherical and gca to the catenoidal part, then gj d can also be written as
nsgs(nd) + ncagca(nd) gj d ) nd
(35)
and eq 31 can be rewritten as
Accordingly, the concentration of dumbbells is given by
cM(nd) cA ) c0 c0
(34)
where ∆Gr is the Gibbs energy to form a rod-shaped micelle of aggregation number nr from nr surfactant molecules taken from an infinitely long cylindrical micelle. From eqs 33 and 34 we have
(25)
An approximate value of γ can also be obtained from cmc data21
cA cM(ns) ) c0 c0
Gr ) nr g∞c + ∆Gr
(24)
the Gibbs energy of the aggregate. If g is known, the association coefficient γ can be calculated from (cf. eqs 21 and 22)
γ)-
and can also be written as
(33)
(21) Zana, R. In Cationic Surfactants: Physical Chemistry; Holland, P. M., Rubingh, D. N., Ed.; Marcel Dekker Press: New York, 1991; Vol. 37, Chapter 8.
where gjd∞ is the mean standard Gibbs energy of a surfactant ion in the end cap of an infinitely long rod. Using the reference concentration cmcc, the concentrations of spherical, dumbbell-shaped, and rod-shaped micelles are given by
( ) ( ( ) (
cM(ns) cA ) c0 cmcc
ns
cM(nd) cA ) c0 cmcc
nd
(gj d - g∞c )nd
exp -
( ) (
cM(nr) cA ) c0 cmcc
) )
(gs - g∞c )ns kT
exp -
nr
exp -
kT
)
∆Gr(nr) kT
(42)
(43)
(44)
In the case of bilayers we consider infinite aggregates (cf.
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Langmuir, Vol. 12, No. 10, 1996 2469
Figure 4. Rod-shaped micelle (solid line) and corresponding cylinder of identical volume (dotted line).
section III.1). The occurrence of bilayers is controlled by ∞ ∆gbi ) gbi - g∞c
(45)
As long as ∆gbi is positive no bilayer structures will form. The concentration of surfactant ions incorporated into micelles is ∞
cˆ A )
nmax s
nmax d
∑ ncM(n) ) n∑)1 nscM(ns) + ∑
n)1
ndcM(nd) +
nd)nmin d ∞
s
∑
nrcM(nr) (46)
nr)nrmin
The probability p(n) of observing a micelle of aggregation number n is
p(n) )
cM(n)
(47)
∞
∑1 cM(n)
ncM(n)
np(n) )
∞
∑1
(48)
∞
∑1 np(n)
ncM(n)
The index v denotes averaging with respect to volume. The ν-th moment of a quantity q(n) is given by
〈qν(n)〉 )
∫1∞qν(n) p(n) dn
or
〈qν(n)〉v )
∫1∞qν(n) pv(n) dn
(49a,b)
The axial ratio f of a cylinder can be written as (a ) 2rc)
f(nc) )
2 Vc ncvsf b b πrc (nc) ) ) ) 2 3 a a πr (n ) 2πr (n ) 2πr 3(n ) c
c
c
c
c
(50)
c
where vsf is the volume of a surfactant molecule (cf. eq 7). Interpreting a rod as a cylinder of the same volume (see Figure 4), we will use this result for rods, too, with nr replacing nc:
f(nr) )
nrvsf 2πrc3(nr)
dg ) ac1 dJ + ac2 dK + σ da + (φ0 - φb) dqel + (µ0 - µb) dΘ (52) For convenience, a factor a is used in the coefficients of dJ and dK. The factors c1 and c2 have dimensions of force and energy, respectively, and σ is the surface tension. In the case of ionic micelles the potentials of the bulk phase, φb, and the micellar surface, φ0, may differ. Similarly the electrochemical potentials of the counterion in the bulk phase, µb, may be different from that in the micellar surface, µ0. For integration of eq 52 we start from a plane hydrocarbon/water interface with head group area a*. The Gibbs energy of this state is called g˜*. For the calculations of micellar concentrations (cf. eqs 42-44) g˜* is only a value of reference and need not be specified. Integration of eq 52 from the initial state * to the actual state in the micelle (with Gibbs energy g) can be performed in separate steps according to the following scheme
g – ~* g = a*
while the probability pv(n) of a surfactant ion being incorporated into a micelle with aggregation number n is given by
pv(n) )
Gibbs energies g. The procedure to calculate g is outlined in the next section. III.3. Gibbs Energy of Micellization. In this section we derive explicit expressions for the Gibbs energy g. At constant temperature and pressure, g is a function of mean curvature J, Gaussian curvature K, head group area a, head group charge qrel, and the degree of direct counterion adsorption Θ. Differential changes of g can be written as
(51)
Using eqs 50 and 51, we can calculate the contribution of rods to ηrel from eq 5. Since, compared with rods, the contributions of dumbbells and spheres are small, we will use eq 1 for both spheres and dumbbells. To calculate these contributions, the concentrations cM(n) of the aggregates must be known, which, in turn, depend on the
J •
c1 dJ + a*
step 1
• σ da +
c2 dK + • 0
• 0
qo •
a
K •
• a* step 2
Θ • ∆ϕ dqel + ∆µ dΘ
• 0 step 3
(53)
• 0 step 4
The actual state thus is attained in four steps. In step 1 the plane hydrocarbon/water interface with head group area a* is curved to the final mean and Gaussian curvatures, J and K. Then in step 2 the head group area is brought from a* to its final value a. In these steps the electrostatic interaction between the head groups is ignored. Then, in step 3, the curved surface is charged to one elementary charge q0 per head group. This step includes formation of the Stern layer and of the diffuse ionic cloud. In the last step counterions are allowed to adsorb up to the final degree of occupation, Θ. Equation 53 can be simply written as
∆g ) ∆g1 + ∆g2 + ∆g3 + ∆g4
(54)
Interfacial Gibbs Energy of Curved Hydrocarbon/Water Areas (Step 1). In step 1 a hydrocarbon/water interface is curved at a constant head group area. Electrostatic interactions are not yet active. To model this step we use the Helfrich equation,22,23 which has been successfully used for liquid/liquid interfaces
σ ) σ(J,K) ) σ0 +
kc (J - J0)2 + kh cK 2
(55)
σ0 is the interfacial tension at J ) J0 and K ) 0. J0 is the spontaneous curvature of a cylindrical interface (K ) 0). We assume that kc and kh c do not depend on a and that a* is so large that any dependence of σ0 on a can be neglected. (22) Barneveld, P. A.; Scheutjens, J. M. H. M.; Lyklema, J. Langmuir 1992, 8, 3122. (23) Helfrich, W.; Andelman, D.; Kozlov, M. M. Europhys. Lett. 1994, 25 (3), 231.
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Heindl and Kohler
The mean curvature J is the sum of the two principal curvatures k1 and k2, while the Gaussian curvature is the product of both.
J ) k1 + k2
and
K ) k1k2
(56a,b)
The interfacial tension σ(0,0) of the plane interface is given by
σ(0,0) ) σ0 +
kc 2 J 2 0
(57)
For aliphatic hydrocarbons this quantity can be calculated as a function of temperature and molecular weight.14 The value obtained for hexadecane/water is σ(0,0) ) 47.5 mN‚m-1 at T ) 323 K. The integrals in steps 1 and 2 of eq 53 are calculated at qel ) 0 and Θ ) 0. Thus, for determination of these integrals, the differential dg, generally given by eq 52, can be reduced to
dg ) ac1 dJ + ac2 dK + σ da
(58)
Figure 5. Effective volume Veff (grey) for movement of a (straight) surfactant chain with fixed head group, if the thickness of the bilayer, thbi, is smaller than the extended chain length, lc (thbi ) χlc).
Steric Head Group Interactions (Step 2). The dependence of surface tension on the head group area (steric repulsion) can be described by the Volmer equation24
π ) σ(J,K) - σ )
|
|
|
or
|
∂(ac2) ∂σ ) ∂a J,K ∂K a,J
c1 ) kc(J - J0)
(59a)
or
(59b)
c2 ) kh c
∫0Jc1 dJ + a*∫0Kc2 dK )
∆g1 ) a*
((
a* kc
)
)
We do not use an explicit relation to calculate the contribution of the conformational energy of the alkyl chain to the Gibbs energy of micelle formation. Instead, we assume this contribution to be covered by the curvature dependence of the alkane/water interfacial tension (eq 55). However, eq 55 is not sensitive to changes in the thickness of bilayer structures (mean and Gaussian curvature are always zero). Therefore we introduce an additional contribution to describe, in a rather approximate manner, the influence of bilayer thickness on the conformational energy: Assume that the alkyl chains are extended. If the thickness of the bilayer is greater than the length of an extended chain, the volume for movement of the chain within the bilayer (head group fixed) is that of a half sphere (Vtot). But if the thickness of the bilayer is smaller than the length of the extended chain, then the effective volume for movement of a chain within the bilayer is restricted (Veff). According to Figure 5, one obtains
q)
Veff ) Vtot
x3 x - (1 - x2) 3 3 2 3
xe1
(61)
∆gbi,conf ) Tsconf ) -kT ln q
∫a*a σ da ) ∫a*a σ(J,K) da - kT∫a*a a -1amin da
(62)
Adding this to ∆g1 in eq 60 we get the sum of interfacial and conformational Gibbs energies for bilayer structures.
a - amin (65) a* - amin
As the sum of steps 1 and 2 we get
∆g1 + ∆g2 ) aσ(J,K) - a*σ(0,0) - kT ln
a - amin a* - amin (66)
Electrostatic Gibbs Energy (Step 3). According to the definition of step 3 ∆g3 is given by
∆g3 ) gSt + gel(ν)1)
(67)
where gSt and gel are the contributions of the Stern layer and of the ionic cloud, and v ) qe/q0 ) 1 - Θ. As will be seen in the next section, we also need gel for ν * 1. Therefore formulas of gel(ν) will be derived in this section for the various geometries. (i) Stern Layer (Part 1 of Step 3). If head group and counterion charges are assumed to be smeared out, the energy stored in a Stern layer of thickness dSt can be described as that of a plate condenser (see Figure 2). For planes or catenoids (both have mean curvature zero), spheres, and cylinders, the energy stored per surfactant ion is given by11
gca St )
This amounts to an energetic contribution of
(64)
where r* is the sum of the hard sphere radii of two adjacent head groups. For pyridinium, head group surface tension measurements26 suggest amin ) 0.375 nm2. Step 2 of eq 53 thus yields
) σ(J,K)(a - a*) - kT ln
J2 h cK ) a*(σ(J,K) - σ(0,0)) (60) - J0J + k 2
x-
amin ) πr*2
∆g2 )
Integration of step 1 of eq 53 thus gives
(63)
The excluded area amin is taken as25
We assume that c1 and c2 do not depend on a. From Schwarz’s equality and eq 55, we then obtain
∂(ac1) ∂σ ) ∂a J,K ∂J a,K
kT a - amin
q02 d 2a St
(68a)
(24) Volmer, M. Z. Phys. Chem. 1925, 115, 253. (25) Aveyard, R.; Haydon, D. A. An introduction to the principles of surface chemistry; University Press: Cambridge, 1973; p 23. (26) Strnad, J. Einfluss von Gegenionen auf das Verhalten Kationischer Tenside. Thesis (Habilitation), Regensburg, 1990; p 59.
+
+
Rod Formation of Ionic Surfactants
dSt q02rSt 2aSt rSt + dSt
(68b)
q02rSt rSt + dSt ln 2aSt rSt
(68c)
gsSt )
gcSt )
Langmuir, Vol. 12, No. 10, 1996 2471
(69)
For curved interfaces the head group area aSt is not identical with the head group area attributed to the hydrophobic core. For spheres and cylinders aSt is related to as and ac by
()
rSt aSt ) as rs
rSt aSt ) ac rc
2
(70a,b)
(ii) Diffuse Ionic Cloud (Part 2 of Step 3). We use an approximate analytical solution for curved and charged surfaces derived on the basis of the Poisson-Boltzmann27 equation by Evans et al.28,29 and Hayter.30 The dimensionless surface charge s is defined by
s≡
2qe aelx8RTcB
(71)
qe is the mean effective head group charge, related to the elementary charge q0 by
qe ) q0(1 - Θ)
(72)
The dimensionless surface potential ψ0 is obtained from the surface potential φ0 by
q0φ0 ψ0 ) kT
ψ0 2
( ) ( )
(74a)
ψ0 8s [cosh(ψ0/2) - 1] + 2 κrel sinh(ψ0/2)
(74b)
s2 ) 4 sinh2
ψ0 4s [cosh(ψ0/2) - 1] + 2 κrel sinh(ψ0/2)
(74c)
0
For planes, spheres, and cylinders this gives
2 1 gel(ν) ) ψ0 - (z - 1) ν kT s 8 1 gel(ν) ) ψ0 ν kT κrels
{
x1 + w2 - x1 + κ2rel2 + ln
4 1 gel(ν) ) ψ0 ν kT κrels
(77a)
{
}
w2(1 + x1 + κ2rel2) κ2rel2(1 + x1 + w2)
(77b)
x1 + 4w2 - x1 + 4κ2rel2 + ln
}
w2(1 + x1 + 4κ2rel2) κ2rel2(1 + x1 + 4w2)
(77c)
where
z ) cosh(ψ0/2)
and
w ) κrel(z + 1)/2
(78a,b)
For spheres and cylinders the micellar radius rel is the sum of the hydrophobic core radius, the hydrated methylene layer thickness, and the radius of the counterions, rcount
rel ) rs,c + 0.1265 nm + rcount
(79)
The head group area ael is related to as and ac by
ael ) as
() rel rs
2
ael ) ac
rel rc
(80a,b)
Θ)
cBI cBI + Kdis
(75)
The electrostatic energy stored in the diffuse ionic cloud is obtained from (27) Kohler, H.-H. In Coagulation and Flocculation; Dobia´sˇ, B., Ed.; Marcel Dekker Press: New York, 1993; Vol. 47, Chapter 2. (28) Evans, F. D.; Ninham, B. W. J. Phys. Chem. 1983, 87, 5025. (29) Evans, F. D.; Wennestro¨m, H. The Colloidal Domain; Verlag Chemie: Weinheim, 1994; Chapter 4. (30) Hayter, J. B. Langmuir 1992, 8, 2873.
(81)
where Θ is the degree of counterion adsorption and cBI the concentration of counterions just in front of the surface. Kdis is the dissociation constant. For counterions of valence zB ) -1, cBI is related to the bulk concentration, cB, by
cBI ) cB exp(ψ0)
where eqs 74a-c are for planes, spheres, and cylinders, respectively. κ is the Debye-Hu¨ckel parameter with
2cBF2 RT
e
Direct Adsorption of Counterions (Step 4). To describe counterion adsorption we use a Langmuir adsorption isotherm
s2 ) 4 sinh2
κ2 )
e
(73)
ψ0 and s are related by
s ) 2 sinh
q ψ kT ψ0 dq ) (qeψ0 - ∫0 q dψ0) ∫0q φ0 dq ) kT ∫ 0 q0 q0
(76)
dSt is taken to be equal to the counterion radius. For the spherical and cylindrical geometries the micellar radius rSt is the distance of the head group charges from the micelle center
rSt ) rs,c + 0.1265 nm
gel )
(82)
Thus the adsorption isotherm relating Θ with cB is
Θ)
cB exp(ψ0) Kdis + cB exp(ψ0)
(83)
The reversible work of adsorption per surfactant ion of step 4 is given by
∆g4 )
∫0Θ∆µ dΘ
(84)
where ∆µ is the difference in electrochemical potential between the bulk and the surface
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∆µ ) kT ln
Heindl and Kohler
cBI - q0φ0 ) cB Kdis Θ kT ln + kT ln - q0φ0 (85) cB 1-Θ
Thus
∫0Θ
∆g4 ) kT
(
)
Kdis Θ ln + ln dΘ cB 1-Θ
∫0Θq0φ0 dΘ (86)
Dumbbell-Shaped Micelles. At a given composition of the solution the mean Gibbs energy gj d is a function both of the radii rs and rca and of the surface area ratio β ) as/aca introduced in eq 13:
gj d ) gj d(rs,rca,β)
For a given aggregation number nd, eq 95 can be used to determine rca as a function of rs. Thus we get gd ) gd(rs,β). Minimization of the Gibbs energy Gd ) ndgj d with respect to rs and β thus leads to the (necessary) conditions
∂gj d
The second integral yields
∫0 q0φ0 dΘ ) ∫q
qe
Θ
-
0
∂rs φ0 dq ) gel(ν) - gel(ν)1) (87)
where gel(ν)1) is the electrostatic energy of a fully charged head group. Thus ∆g4 becomes
((
∆g4 ) kT Θ ln
)
)
Kdis Θ + ln + ln(1 - Θ) + cB 1-Θ gel(ν) - gel(ν)1) (88)
But under equilibrium conditions ∆µ ) 0 and therefore
q0φ0 Kdis Θ + ln ) ln kT cB 1-Θ
(89)
|
nd,β
(90)
It is reasonable to assume that counterion adsorption is favored by large head group areas a. We further assume that adsorption is favored if the mean curvature deviates from its spontaneous value J0. Both aspects are combined to
(92)
Both Z and φ are determined by curve fitting. III.4. Minimum of the Gibbs Energy for a Given Aggregation Number. Spherical Micelles. In the case of a spherical micelle, the aggregation number ns, radius rs, and volume Vs of the hydrocarbon core are related by
(93)
Therefore, ns defines rs. For dumbbells, the situation is more involved. The volume Vd of the hydrocarbon core is a function of rs and rca:
Vd ) ndv15 ) Vd(rs,rca)
∂β
|
nd,rs
)0
(96a,b)
Gr ) nc gc(rc) + nd(rs,rc) gj d(rs,rc,β)
(94)
Evidently, this geometrical constraint is not sufficient to determine both rs and rca. An optimal combination of rs and rca, however, is obtained from the requirement of a minimum Gibbs energy Gd ) ndgj d of the aggregate. A similar situation is met in the case of rod-shaped micelles. We will treat separately the energy minimization of dumbbells and rods.
(97)
Because nr ) nc + nd this is equivalent to
Gr ) Gr(nr,rc,rs,β) ) nr gc(rc) + ∆Gr(rs,rc,β) (98) where
(99)
The optimal rod aggregation number nr therefore is subject to the three (necessary) conditions
∂Gr ∂∆Gr ) )0 ∂rs ∂rs ∂Gr ∂∆Gr ) )0 ∂β ∂β dgc ∂∆Gr ∂Gr ) nr + ) 0 (100a,b,c) ∂rc drc ∂rc
∆g3 + ∆g4 ) gSt + gel(ν) + kT ln(1 - Θ) + q0φ0Θ (91)
4 Vs ) nsv15 ) πrs3 ) Vs(rs) 3
∂gj d
From these two equations optimal values of rs and β can be determined. Rod-Shaped Micelles. The Gibbs energy Gr of a rod is given by eq 34. Since a rod is a dumbbell plus a central cylindrical part of radius rc, Gr can also be written as
Due to eq 67 the contributions of steps 3 and 4 add up to
1 ) Kads ) aZ(1 + φ(J - J0)2) Kdis
)0
∆Gr(rs,rc,β) ) nd(rs,rc)(gj d(rs,rc,β) - gc(rc))
so that
∆g4 ) gel(ν) - gel(ν)1) + kT ln(1 - Θ) + q0φ0Θ
(95)
which can be used to calculate optimal values of rs, rc, and β. IV. Results and Discussion IV.1. Fitted Parameters. Common Parameters. The parameters called common are independent of the counterion identity. They describe the dependence of the interfacial hydrocarbon/water tension on curvature. Fitting the model to viscosity data we find
J0 ) -1.94 nm-1 The best fits for the bending modulus kc and the Gaussian bending modulus k h c are
kc ) 7.12 × 10-21
J) ˆ 1.60kT k h c ) 5.70 × 10-21
J) ˆ 1.28kT
Note that the spontaneous curvature J0 is negative. Therefore, in the interesting range of radii, the surface tension of the hydrocarbon/water interface, σ, increases with increasing mean curvature. According to eq 55 the surface tension σ0 of a spontaneously curved cylindrical interface is
σ0 ) 34.1 mN·m-1
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Langmuir, Vol. 12, No. 10, 1996 2473
Table 2. Values of Z and O Obtained by Fitting the Adsorption Constant Kads to Viscosity Data counterion
Z/mol‚L-1‚nm2
φ/nm2
R/1025 cm-3
Θca/%
ClBrINO3-
2.38 × 10-4 4.17 × 10-1 46.3 1.85
0 4.034 × 10-1 0.793 × 10-2 0.078 × 10-1
36.6 47.7 71.0 37.9
0 68 84 74
a
Representative values are given.
For rs ) rc ) 2.05 nm (extended chain length) eq 55 yields σs ) 65.6 mN/m and σc ) 55.0 mN/m. Clearly, curvature is in favor of the cylindrical part of the micelle. Specific Parameters. (i) Thickness and Dielectric Constant of the Stern Layer. The thickness of the Stern layer, dSt, is identified with the radius of the counterions. In literature two values of the Stokes radii17 are used, the stick and the slip radius. The counterion radii are listed in Table 1. Compared to the crystal radius, all stick radii are too small to be realistic. Therefore we use the slip radii (except for chloride, see below), which are not so different from the crystal radii. Fitting the relative dielectric constant of the Stern layer then gives ) 20. The chloride ion seems to be a special case. If the model is used with St ) 20 and dSt ) 0.18 nm (which is the slip radius of the chloride ion) rod formation becomes much too strong. Good agreement between model and experiment, however, is obtained if the value of dSt is increased to 0.29 nm. Thus, as we want to maintain a correlation with one of the characteristic ionic radii, we identify the Stern layer thickness of the chloride ion with its hydrated radius, which is 0.33 nm15,16 (see Table 1). From a new fit of the relative dielectric constant we then obtain ) 37. In fact, one would expect an increased value of for an increased value of dSt, because, in the limit of a very thick layer, should approach the value of the relative dielectric constant of water. In summary, we equate dSt with the slip radius and use ) 20 for the counterions bromide, iodide, and nitrate, while dSt is equated with the hydrated radius and ) 37 is used in the case of chloride. We don’t have any convincing explanation for the peculiar behavior of the chloride ion. Perhaps there is some connection with the relatively high primary hydration number of this ion.16 (ii) Adsorption Constants. The fitted values of the adsorption constants Z and φ of eq 92 are listed in Table 2. As could be expected, the value of Z increases with increasing polarizability volume R.31,32 Nitrate, having a plane structure, is a special case. For chloride the value of Z is so small that direct adsorption is negligible. The actual value of Kads depends on head group area and mean curvature, which, in turn, depend on counterion concentration. However, the actual values of Θc are nearly independent of concentration so that the Θc values given in Table 2 are representative for the whole range of counterion concentrations. IV.2. Experimental Viscosities and Theoretical Results. In Figure 6 both experimental values and theoretical curves are plotted for the relative viscosity ηrel as a function of cˆ A. The parameters used for the theoretical curves are given in section IV.1. Agreement between experiment and theory is satisfactory except for the system CpI/KI, where the model evidently reaches its limitations. At cˆ A > 6 mM rods become too long to exclude overlap of the rotary movement, but for an iodide concentration of cB ) 8 mM agreement is poor just below cˆ A ) 6 mM. In (31) D’Ans/Lax Taschenbuch fu¨ r Chemiker und Physiker Band III, 3rd ed.; Springer Verlag: Berlin, 1970; p 292. (32) Petrova, M. V.; Lanshina, L. V.; Figurovskii, N. A. Russ. J. Phys. Chem. 1981, 55 (3), 430.
Figure 6. Theoretical curves (solid) and experimental values (symbols) of the relative viscosities, ηrel, as a function of cˆ A at various counterion concentrations. Rotation overlap can be neglected except for the system CpI/KI, where beginning rotation overlap is indicated by the dashed lines.
Figure 7. Concentration of surfactant incorporated into micelles of aggregation number n for the system 2.5 M KCl/10 mM CpCl.
Figures 7-9 the distribution of surfactant over spheres, dumbbells, and rods is shown versus aggregation number. In the case of small values of ηrel, short rods dominate. An example taken from the system CpCl/KCl is shown in Figure 7 (cˆ A ) 10 mM, cB ) 2.5 M), cf. Figure 6a. The relative viscosity of this system is ηrel ) 1.020. The concentration of surfactant incorporated into spheres is about 10-10 M and can be neglected. The concentration of surfactant incorporated into dumbbells is 1 mM while that incorporated into rods is 9 mM. The distribution of rods extends to aggregation numbers of 2000 and is highly
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Langmuir, Vol. 12, No. 10, 1996
Heindl and Kohler Table 3. Mean Aggregation Numbers of Dumbbells at Various Counterion Concentrations for CpCl Solutions of cˆ A ) 15 mM Clconcentration cB/M
aggregation number n (light scattering) 31°C
aggregation number n (model) 50 °C
0.02 0.06 0.43 0.73
95 117 135 137
113 125 148 162
Table 4. Approximated and Exact Values of nmax and 〈n〉v
Figure 8. Concentration of surfactant incorporated into micelles of aggregation number n for the system 0.5 M KBr/10 mM CpBr.
surfactant
c B/ M
CpCl CpCl CpBr CpI CpNO3
4.0 2.5 0.5 0.01 0.4
nmax nmax 〈n〉v 〈n〉v (eq 101) (model) (eq 101) (model) 1526 384 825 2254 1460
1500
3052 766 1650 4538 2919
820 2200 1500
2927 684 1652 4452 2894
∆G∞r
ηrel
20.47 17.71 19.24 21.76 20.38
1.382 1.020 1.155 1.481 1.426
with light-scattering measurements of CpCl/NaCl solutions.33 With decreasing counterion concentration, cB, the mean aggregation number, 〈n〉v, is seen to approach that of the sphere (aggregation number 82). For the systems CpI/KI and CpNO3/KNO3 our calculations yield similar distributions. Israelachvili,15 assuming constant ∆Gr, has derived expressions for the maximum of the distribution, nmax, and the mean aggregation number, 〈n〉v, given by
nmax )
Figure 9. Concentration of surfactant incorporated into micelles of aggregation number n for the system 0.5 M KBr/ 0.011 mM CpBr.
polydisperse, a typical behavior of rodlike aggregates.15 The mean aggregation number and the mean axial ratio are 〈n〉v ) 684 and 〈f〉v ) 8.3, respectively. If salt is added, the rods grow, leading to increased values of the relative viscosity. This behavior is seen in all systems (see Figure 6). A distribution of micelles in the system CpBr/KBr is shown in Figure 8 (cˆ A ) 10 mM, cB ) 0.5 M). The relative viscosity now is ηrel ) 1.150, and 98% of the surfactant is incorporated in rods, with only 2% in dumbbells. Again, the amount incorporated in spheres is negligible. In comparison with Figure 7 the rods are larger and the distribution shows a second maximum at an aggregation number of 820. The mean aggregation number and mean axial ratio of the solution are 〈n〉v ) 1650 and 〈f〉v ) 20.9, respectively. Generally, rod formation is favored by increasing surfactant concentration. For comparison with Figure 8, Figure 9 shows the micellar distribution for ctot A ) 0.096 mM, and that of monomers is cA ) 0.085 mM. The total surfactant concentration is ctot A ) 0.096 mM, and that of monomers is cA ) 0.085 mM. For this counterion concentration the experimental cmcc is 0.087 mM. So 90% of the surfactant is in the monomeric state and 10% is incorporated into micelles, 6% of which are dumbbells and 4% of which are rods. As above, the amount of spheres is negligible. Significant amounts of spheres appear, if the salt concentration is sufficiently small. We have calculated mean aggregation numbers at cˆ A ) 15 mM for various counterion concentrations. In Table 3 they are compared
x
cˆ A c0
{
exp -
}
∆Gr 〈n〉v ) kT 2
(101)
Identifying∆Gr with ∆G∞r (see eqs 40 and 41), these relations can be used to calculate approximate values for our distributions. Exact and approximated values are listed in Table 4. As expected agreement is excellent for high, but not so good for small aggregation numbers. In all systems the radius of the cylindrical part increases with decreasing aggregation number (r∞c ) 1.58 nm, rc ) 1.70 nm for nr ≈ 170). The radius of the cylindrical part amounts to only 77-83% of the extended chain length (lc ) 2.05 nm), while the radius of the end cap always corresponds to the extended chain length, rs ) 2.05 nm. Thus, rods are constricted. The aggregation number of the two end caps is nearly the same for all systems, amounting to about 170, with 25 molecules in the catenoidal part. Dumbbells, too, have an end cap radius of rs ) 2.05 nm, while rca while rca decreases with aggregation number from 2.05 to 1.70 nm. More micellar parameters are listed in Tables 5 and 6 for the system CpBr/KBr. Since the radius of the end cap equals the extended chain length, the mean head group area of the end cap, a j d, defined by
a jd )
nsas + ncaaca nd
(102)
is nearly independent of aggregation number. ∆G∞r (cf. eq 41) increases with counterion concentrations up to cB ≈ 1 M (see Figure 10), which reflects the increasing tendency of rod formation. Closer inspection of the model shows that the increase of ∆G∞r is mainly due to electrostatic screening (cf. eqs 77a-c). Screening is more effective in the cylindrical part with its relatively low curvature (for a given charge density the electrostatic energy is relatively high) than in the end cap, which is dominated (33) Anacker, E. W. J. Phys. Chem. 1958, 62, 41.
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Langmuir, Vol. 12, No. 10, 1996 2475
Table 5. Micellar Parameters of a Rod-Shaped Micelle with Aggregation Number 820 for the System CpBr/KBr parameter
0.4 M KBr
0.5 M KBr
ac/nm2 as/nm2 aca/nm2 rc/nm rs/nm ns nca φ0,c/mV φ0,s/mV φ0,ca/mV Θc/% Θs/% Θca/% Kads,c/M Kads,s/M Kads,ca/M
0.565 0.545 0.720 1.55 2.05 144 24 47 41 51 67.7 68.5 65.5 0.98 1.24 0.75
0.564 0.545 0.719 1.56 2.05 144 25 42 37 46 68.7 69.7 66.3 0.98 1.24 0.75
Table 6. Micellar Parameters of a Dumbbell with Aggregation Number 157 for the System CpBr/KBr parameter
0.4 M KBr
0.5 M KBr
as/nm2 aca/nm2 rs/nm rca/nm ns nca φ0,s/mV φ0,ca/mV Θs/% Θca/% Kads,s/M Kads,ca/M
0.536 0.707 2.05 1.78 134 22 44 52 68.6 65.6 1.22 1.74
0.536 0.706 2.05 1.77 134 23 42 51 69.8 66.3 1.22 0.73
by the high curvature region of the hemispherical part (lower electrostatic energy). The Gibbs energy of a surfactant ion, g, can be written as
g ) gint + gster + gel(ν) + gSt + gad
(103)
where the terms on the right hand side represent contributions from hydrocarbon/water, interfacial tension, steric interaction, ionic cloud, Stern layer, and adsorption. According to eq 66, gint and gster are given by
gint ) aσ(J,K) - a*σ(0,0)
(104)
a - amin a* - amin
(105)
gster ) -kT ln
Figure 10. Head group area in the end cap, a j ∞d , and the ∞ cylindrical part, ac , and sphere to rod transition parameter ∆G∞r of an infinitely long rod as a function of counterion concentration for the system CpBr/KBr.
Figure 11. Gibbs energies of the endcap (solid lines) and of the cylindrical part (dashed lines) as a function of counterion concentration cB for the system CpBr/KBr. The interfacial Gibbs energy, gint, is given by eq 104, while g is given by g ) gint + gster + gel + gst + gad (eq 103).
gel(ν) is given by eqs 77a-c, gSt by eqs 68a-c, and gad by (cf. eq 91)
gad ) kT ln(1 - Θ) + q0φ0Θ
(106)
In Figures 11 and 12 these contributions are shown for the cylindrical part (index c) and the end cap (index d). Figure 11 shows that addition of salt leads to a stronger reduction of the total Gibbs energy per surfactant molecule in the cylindrical part (g∞c ) than in the end cap (gj d). This is mainly due to the counterion dependence of the ∞ ∞ and gj d,int also shown in interfacial contributions gc,int Figure 11. The interfacial Gibbs energy of the cylindrical part is always smaller than that of the end cap, even if the head group area is larger. This is due to the high curvature in the hemispherical part (cf. eq 55). Figure 12 shows that steric and electrostatic repulsion both disfavor the cylindrical part at all counterion concentrations. The same is true for the energetic
Figure 12. Gibbs energies of the end cap (solid lines) and of the cylindrical part (dashed lines) as a function of counterion concentration. The steric contribution gster to the Gibbs energy (index ster) is given by eq 105, while the Gibbs energy with index el is given by eqs 77a-c. Shown for the system CpBr/ KBr.
contributions of the Stern layer and of direct adsorption of counterions, see Figure 13. In Figure 14 the differences (end cap minus cylindrical part) of the above Gibbs energies are shown. It is seen that the only contribution favoring rod formation, the interfacial Gibbs energy of the hydrocarbon/water interface, dominates over all other
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Langmuir, Vol. 12, No. 10, 1996
Heindl and Kohler Table 8. Bilayer Parameters. ∆gbi/kT abi/nm2 thbi/nm cB/M
Figure 13. Gibbs energies of the end cap (solid lines) and of the cylindrical (dashed lines) part as a function of counterion concentration. The Gibbs energy gSt is given by eqs 68a-c and the Gibbs energy gad is given by eq 106. Shown for the system CpBr/KBr.
Figure 14. Difference of Gibbs energies between the end cap and cylindrical parts as a function of counterion concentration for the system CpBr/KBr. Indices have same meanings as in Figures 10-13. Table 7. Experimental and Theoretical Values of Counterion Association Coefficient γ experimental γ theoretical γ
CpCl
CpBr
CpI
CpNO3
0.66 0.73
0.72 0.79
0.81 0.85
0.79 0.82
contributions. The smaller interfacial energy of the cylinder is mainly due to its smaller curvature (compared with that of the sphere). Figures 10-14 refer to the system CpBr/KBr. The other systems show similar behavior. IV.3. Experimental and Theoretical Values of γ. The association coefficients γ determined from experiment and model (eqs 25 and 26) are given in Table 7. In the case of CpCl/KCl the experimental values of γ are for 31 °C, and in the other cases they are for 50 °C. Calculations are for 50 °C throughout. IV.4. Bilayer Structures. As has been pointed out above, bilayers will undergo a phase transition at their cmc. Therefore it is sufficient to consider infinitely large bilayers (so that the contributions of the rims are negligible). The differences of Gibbs energies between bilayers and infinite long rods, ∆gbi (eq 45), are listed in Table 8. The calculated bilayer thickness is only about 2/3 of the length of a single extended chain (so 2/3 in eq 61). As ∆gbi is assuming a distinctly positive value, the occurrence of bilayer structures can be excluded. V. Summary V.1. Geometry of the Micelles. The model allows for micelles of different geometriessspheres, dumbbells,
CpCl
CpBr
CpI
CpNO3
0.44 0.64 1.38 4.0
0.41 0.64 1.36 0.5
0.31 0.65 1.36 0.01
0.34 0.66 1.34 0.4
rods, and bilayerssand arbitrary aggregation numbers. The dumbbell consists of two hemispherical parts and one catenoidal part, and the rod, of a cylindrical part and two end caps (which, if combined, also form a dumbbell). An important function of the catenoidal parts is to avoid edges (where curvature would be undefined). Our calculations show that the mean head group area of the end cap is approximately the same as that of the cylindrical part. Thus rod formation cannot be attributed to an energetic advantage of smaller head group areas in the cylindrical part.15 V.2. Minimum of the Gibbs Energy for a Given Aggregation Number. The optimal radius of the cylindrical part of a rodlike micelle depends on the rod aggregation number nr. This dependence, not considered by other authors,9-14 is especially important for small rods. For instance, at 0.4 M KBr, the optimal radius of the CpBr micelle, rc, increases from 1.54 nm at n ) 10 000 to 1.70 nm at n ) 165. For all systems the optimal radius of the end cap equals the length of the extended alkyl chain, rs ) 2.05 nm. V.3. Individual Contributions to the Gibbs Energy. Interfacial Energy of the Hydrocarbon/Water Interface and Conformational Energy. Conformational energies are a result of bending and/or compressing the hydrocarbon region. In our model, effects of bending are described by the curvature dependence of the hydrocarbon/ water interfacial tension σ (eq 55). This heuristic approach is useful for curved aggregates, such as spheres, dumbbells, and rods. Curve fitting shows that σ increases with increasing curvature. This increase is of crucial importance for rod formation. Entropic restrictions in plane aggregates (bilayers) are described by a separate term (eq 62). These restrictions prevent bilayer formation. More detailed numeric calculations of the conformational Gibbs energy should be attempted in future modeling approaches.12,34 Electrostatic Interactions. We use an approximate analytical solution of the Poisson-Boltzmann equation for curved surfaces.27-29 The Evans, Mitchell, Ninham28,29,35 treatment is fairly good under conditions of high screening but fails if screening is weak. Therefore we use a self-consistent approach,30 leading to better analytical results under conditions of weak screening. The thickness of the Stern layer is assumed to be given by the slip radius of the anion. An exception is the chloride ion. To get agreement with the experimental data, an increased thickness of the Stern layer must be assumed. It should be noted, however, that, quite generally, the concept of the Stern layer, though widely used, suffers from a lack of physical plausibility. This is only one reason why, in future work, the thermodynamic description of the charged interface should be improved. To get more reliable results about micelle formation image charge and eigenvolume effects, the polarizability of counterions and the dependence of the dielectric constant on composition should be taken into account. At the same time the description of electrostatic curvature effects should be improved. (34) Bo¨hmer, M. R.; Koopal, L. K.; Lyklema, J. J. Phys. Chem. 1991, 95, 9569. (35) Evans, D. F.; Mitchell, D. J.; Ninham, B. W. J. Phys. Chem. 1984, 88, 6344.
+
+
Rod Formation of Ionic Surfactants
Direct Adsorption of Counterions. Rod formation is very much influenced by the counterion species. The differences in ionic radii of the counterions used in our investigations are too small to attribute counterion-specific effects to the thickness of the Stern layer alone. Therefore direct counterion adsorption is introduced. Adsorption is considered to depend on the curvature of the micelle surface. Without such dependency, rod formation is enhanced too much by increasing counterion concentrations. VI. Conclusions The thermodynamic model proposed in this paper leads to good agreement with experimental viscosity data. The model contains a number of new features: Firstly, the introduction of the catenoidal segments to get geometrically smooth aggregates, secondly, a multifactorial minimization of the Gibbs energy of the micelle as a whole, and thirdly, the concept of direct adsorption of counterions. The approach should be flexible enough to be applied to vesicles, liposomes, and pores in membranes and to similar structures as well. Comparison with experimental data leads to the following conclusions: (1) Rod formation requires an increase of the hydrocarbon/water interfacial tension with increasing curvature. Without such an increase the cylindrical region is too unfavorable, energetically, to compete with the dumbbell or the sphere.
Langmuir, Vol. 12, No. 10, 1996 2477
(2) At concentrations just above the critical micelle concentration there are mainly dumbbells and practically no spheres. The advantage of the dumbbells is due to the catenoidal part, which, having a mean curvature of zero, is favorable energetically. (3) Rod formation will occur (at surfactant concentrations below 10 mM), if the Gibbs energy of a surfactant ion is smaller by about 0.1kT in the cylindrical part than in the dumbbell. Even in cases where rod formation is strong this difference is not greater than 0.15kT. The radius of the cylindrical part is only about 3/4 of the length of the extended alkyl chain. Thus rods are compressed. As a consequence the mean head group area in the terminal parts is nearly the same as that in the cylindrical part. This shows that rod formation is not driven by a smaller head group area in the cylinder. It is mainly due to a smaller curvature of the cylindrical part (cf. conclusion 1). (4) For a successful fit to viscosity data direct counterion adsorption has to be assumed to increase in the sequence Cl-, Br-, NO3-, I-. (5) Since the energetic differences leading to rod formation are much smaller than kT (cf. conclusion 3), the theoretical equations and approximations need further improvement, especially with regard to electrochemical and conformational effects. LA9508811
