Persistence Length of Wormlike Micelles Composed of Ionic

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J. Phys. Chem. B 2007, 111, 8158-8168

Persistence Length of Wormlike Micelles Composed of Ionic Surfactants: Self-Consistent-Field Predictions Y. Lauw,* F. A. M. Leermakers, and M. A. Cohen Stuart Laboratory of Physical Chemistry and Colloid Science, Wageningen UniVersity, Dreijenplein 6, Wageningen 6700 EK, The Netherlands ReceiVed: March 4, 2007; In Final Form: April 23, 2007

The persistence length of a wormlike micelle composed of ionic surfactants CnEmXk in an aqueous solvent is predicted by means of the self-consistent-field theory where CnEm is the conventional nonionic surfactant and Xk is an additional sequence of k weakly charged (pH-dependent) segments. By considering a toroidal micelle at infinitesimal curvature, we evaluate the bending modulus of the wormlike micelle that corresponds to the total persistence length, consisting of an elastic/intrinsic and an electrostatic contribution. The total persistence length increases with pH and decreases with increasing background salt concentration. We estimate that the electrostatic persistence length l0p,e scales with respect to the Debye length κ-1 as l0p,e ∼ κ-p where p ≈ 1.98 for wormlike micelles consisting of C20E10X1 surfactants and p ≈ 1.54 for wormlike micelles consisting of C20E10X2 surfactants. The total persistence length l0p,t is a weak function of the head group length m but scales with the tail length n as l0p,t ∼ nx where x ≈ 2-2.6, depending on the corresponding head group length. Interestingly, l0p,t varies nonmonotonically with the number of charged groups k due to the opposing trends in the electrostatic and elastic bending rigidities upon variation of k.

I. Introduction A surfactant molecule often consists of one hydrophobic (tail) and one hydrophilic (head) block. The selective affinities for the solvent leads to multimolecular aggregates, better known as micelles which appear when the surfactant concentration is beyond the critical micellization concentration (CMC). Various micellar structures can be formed, depending on the architecture and concentration of the surfactant. The difference in micellar topologies is characterized by the packing parameter P, which is defined as the ratio between the volume occupied by the tail groups in a micelle and the product of the tail length and the area of the head group in the micelle. Simple geometry suggests that, for spherical micelles, P ≈ 1/3, for cylindrical micelles, P ≈ 1/2, and for planar bilayers, P ≈ 1. In this article, we focus on the self-consistent-field (SCF) modeling of a cylindrical (wormlike) micelle composed of functionalized alcohol ethoxylate CnEm surfactants in an aqueous solvent. The micellization properties of CnEm surfactants have been studied extensively.1-4 The CnEm surfactants are used in various applications, e.g., as additives for antifoaming agents,5 as pesticide adjuvants,6 as colloidal stabilizers,7-8 and as solubilizers.9-10 A few years ago, we published self-consistentfield predictions for the persistence length of wormlike micelles composed of nonionic surfactants CnEm in an aqueous solvent.11 Here, we extend these investigations and focus on ionic surfactants CnEmXk, where X denotes a negative weakly charged group. The target of our calculations is to find the bending properties of wormlike micelles formed by these ionic surfactants. In general, the bending properties of a cylindrical object are characterized by the bending rigidity/modulus κ0 and/or the persistence length lp. Following the work of Helfrich,12 the free energy per unit length τ of a cylindrical micelle may be * Corresponding author.

expanded in the mean curvature J. For a wormlike micelle, the Helfrich expression is given by

1 τ(J) ) τ(0) - κ0J0J + κ0J2 2

(1)

where τ(0) is the bending energy at zero curvature and J0 is the spontaneous curvature, i.e., the curvature where an unconstrained wormlike micelle is in equilibrium. Both τ(0) and J0 are equal to zero, i.e., the equilibrium micelle is tensionless and has no preference in the bending direction. Therefore, eq 1 can be rewritten as

τ(J) )

Ω 1 2 ) κ0J l 2

(2)

In eq 2, Ω is the total bending energy in the unit kBT, where kB is the Boltzmann constant and T is the absolute temperature. The length of the wormlike micelle is denoted by l. Predictions for the total bending energy Ω are accessible through selfconsistent-field calculations. In terms of statistical thermodynamics of small systems, the bending energy is equal to a quantity which is known as the restricted grand potential. Details about the numerical scheme to obtain the translationally restricted grand potential Ω for a nonionic toroidal micellar system were documented in our previous publication.11 In the following section, the grand potential Ω for an ionic toroidal system is analyzed. The persistence length lp of a cylindrical object can be defined as the distance over which the object can be considered as rigid (nonflexible). In other words, a cylindrical micelle behaves as a rod when its length l e lp and as a flexible chain when l > lp. Landau and Lifshitz formulated the relation between the bending rigidity and persistence length as follows13

10.1021/jp071756o CCC: $37.00 © 2007 American Chemical Society Published on Web 06/20/2007

Persistence Length of Wormlike Micelles

lp )

κ0 kBT

J. Phys. Chem. B, Vol. 111, No. 28, 2007 8159

(3)

For the case of a wormlike micelle, eq 3 states that by determining its bending rigidity, the corresponding persistence length of the wormlike micelle is obtained. As in the previous publication,11 we focus on closed wormlike micelles, i.e., toroidal micelles. Such torus-shaped micelles have been observed experimentally in various systems of block copolymeric surfactants.14-17 A schematic illustration of a toroidal micelle is given in Figure 1. The size of the torus is characterized by the major radius R and minor radius which may be separated into two contributions, rr and tr. The first part (rr) of the minor radius represents the dimension of the “bare” CnEm, and the latter (tr) is related to the position of the Xk group. By definition, the curvature of the torus equals the inverse toroidal major radius J ) 1/R. The cross section of the torus has a central region where the hydrophobic group C is mostly located, which is known as the toroidal core. The core is surrounded by the toroidal corona that consists mostly of the hydrophilic ethylene oxide units E. Finally, the weakly negative-charged groups X are present at the corona perimeter. In a way, a solution of wormlike micelles composed of charged surfactants resembles a polyelectrolyte solution. The persistence length of wormlike polyelectrolyte chains has received ample attention in the literature. There exists an analytical derivation of the persistence length of a polyelectrolyte chain by Odijk, Skolnick, and Fixman (OSF).18-19 Here, the total persistence length lp,t is expected to have contributions from an intrinsic/elastic and an electrostatic effect (superposition approximation),

lp,t ) lp,0 + lp,e

(4)

where lp,0 and lp,e are the elastic and electrostatic persistence lengths, respectively. The electrostatic persistence length lp,e of a nearly rodlike polyelectrolyte within the Debye-Huckel limit is given, according to the OSF theory, as a function of the inverse Debye length κ as follows 2

lp,e )

1 σ lB 4 κ2

(5)

where σ is the effective charge density per unit length, lB ) e2/4π0rkBT is the Bjerrum length in which e denotes the elementary charge and 0 and r refer to the dielectric constant of vacuum and the relative dielectric permittivity of the solvent, respectively. The inverse Debye length κ is proportional to (lBI)1/2, where I is the total ionic strength. Expression 5 is expected to be valid for distances between charged groups la > lB. For a higher charge density per unit length, Manning counterion condensation (sometimes also called OnsagerManning-Oosawa condensation) should be incorporated. This leads to an effective charge density term σ of the order of one elementary charge per Bjerrum length.20-24 The scaling of lp,e ∼ κ-2 has received a lot of attention, particularly in relation to the intrinsic/elastic flexibility of polyelectrolytes and the incorporation of the excluded volume effects on the swelling polyelectrolyte chains. The result is not undisputed. Indeed, there are various predictions of the electrostatic persistence length lp,e ∼ κ-p where p varies from 0.3 to 2.25-35 In this article, we apply the SCF theory to evaluate the persistence length of wormlike micelles composed of CnEmXk

Figure 1. Illustration of a toroidal micelle and its cross section. The core of the torus is mostly occupied by the C block (gray line), and the corona of the torus contains the E group (black line), whereas the functional group X (gray sphere) exists at the corona perimeter. The major radius of the torus is denoted as R, and the minor radius of the torus consists of two length features: rr is the part of the minor radius attributed to the CnEm part and tr is the part from the charged Xk groups.

anionic surfactants. As in the case of the bending modulus of polyelectrolyte chains, the bending rigidity of wormlike micelles composed of ionic surfactants can be considered to originate from two sources: an intrinsic/elastic and an electrostatic contribution. The elastic bending modulus is the result of the intrinsic flexibility of the wormlike object, whereas the electrostatic contribution is due to the repulsive interactions between the charged groups on the corona perimeter and the diffuse ion cloud around the micelle. For cylindrical micelles from nonionic surfactants in an aqueous solvent, the bending rigidity or persistence length depends mainly on the hydrophobic tail group length with an additional secondary effect from the hydrophilic head group size.11 For the case of ionic surfactants, the persistence length is also a function of the pH and background salt concentration [s], as well as of the number of charged groups k per monomer. In this article, we discuss quantitatively these dependences. II. Self-Consistent-Field Modeling As explained in the previous section, the numerical selfconsistent-field (SCF) theory is used to model the self-assembly of the ionic surfactants. It is based on the discrete SCF scheme originally developed by Scheutjens and Fleer.36-37 This, socalled, SF-SCF model is used to extract the statistical thermodynamical quantities of the self-assembled micelle. In our case, the focus is on the translationally restricted grand potential Ω. The SF-SCF is a coarse-grained model which approximates a real polymer chain with a chain of segments with a given conformation. The size of each segment is of the order of a Kuhn length. Due to this approximation, results obtained from the model only describe qualitatively experimental measurements. In general, the method is explained in detail in the previous publication.11 Here, we use the same method but extend it to account for electrostatics due to the presence of charged groups in the surfactant molecules. To facilitate reading, the SF-SCF modeling is given again here with some emphasis on the electrostatic contributions. In the SF-SCF model, the toroidal shape of self-assembled ionic surfactants is imposed on a 2D-cylindrical coordinate system. This system is considered as a member of an ensemble of many small systems ({Ni}, V, T) where {Ni} is the number of components of molecule i, V is the volume of the small system, and T is the temperature of the system. In each small system, there exists a single torus such that the center of the

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Lauw et al.

torus coincides with the center of the small system. The space in the small system is discretized in rings of lattice sites in the radial direction r ∈ [0, Mr] and in lattice layers in the lateral/ orthogonal direction z ∈ [-Mz, Mz], where Mr and Mz are the number of rings in the radial and layers in the lateral directions, respectively. Each lattice site has a characteristic length size a with volume V ≈ a3. All linear sizes below are made dimensionless with this unit length a. The center of mass of the torus is located at the center of the coordinate system (z, r) ) (0, 0). The discretized space is filled with surfactant molecules, solvent, and monovalent salt. Each molecule consists of a specified sequence of segments which defines the chain architecture. A weakly charged segment type can exist in different internal states, i.e., its dissociated and undissociated form. In this article, each weakly charged segment X of surfactant CnEmXk can exist as uncharged state HX and negatively charged state X-. They are denoted as X1 and X2, respectively. In an aqueous solvent, the two internal states are in equilibrium with one another according to the reaction

HX + H2O h X- + H3O+; K ˜a )

φX-φH3O+ φHXφH2O

(6)

where φi denotes the volume fraction of segment state i. Water molecules can exist in three distinct states, i.e., as an uncharged monomer H2O, a hydroxide ion OH-, or a hydronium H3O+. The water equilibrium is given by

˜w ) 2H2O h H3O+ + OH-; K

φH3O+φOH(φH2O)2

∑B ∑k χA B (〈φB (z, r)〉 - φBb ) +

uAj(z, r) ) u′(z, r) + kBT

In eqs 6 and 7, the tilde notations symbolize the fact that dimensionless volume fractions are used. The dimensionless ˜ w can be easily converted to the equilibrium constants K ˜ a and K commonly used quantities Ka and Kw which contain a dimension (more details below).

{

exp GAj(z, r) ) 1,

{

}

-uAj(z, r) kBT

GA(z, r) )

, 0 < z < zbulk, 0 < r < rbulk z g zbulk, r g rbulk

∑j RAb GA (z, r) j

j

[ (

)]

) (

∂Ψ(z, r) ∂Ψ(z, r) ∂ 1 ∂ r (z, r) + (z, r) ) -F(z, r) r ∂r r ∂r ∂z r ∂z ∀r ∈ [0, Mr] ∧ ∀z ∈ [-Mz, Mz] (11)

where F(z, r) and r(z, r) are the charge distribution and the total relative dielectric permittivity at layer (z, r), respectively. They are obtained by the appropriate summation over all segments in their possible states, i.e.

F(z, r) )

∑A ∑j eυA φA (z, r)L(r)

where RAb j is the fraction of segments A in state j in the bulk phase and the self-consistent-field potentials are given by

j

(12) S(r)

r(z, r) )

∑A ∑j r,A (z, r)φA (z, r) j

(13)

j

where L(r) is the total number of lattice sites in radial layer r and S(r) is the contact area between radial layer r and (r + 1) (for a given z). In the cylindrical lattice, S(r) ) 2πr and L(r) ) π(2r - 1). Both of these quantities do not depend on the z-coordinate. In the cylindrical coordinate system, the averaged volume fraction 〈φBk(z, r)〉 in eq 10 can be written in a continuous form as follows

〈φBk(z, r)〉 ≈ φBk(z, r) +

(8) (9)

k

In eq 10, u′(z, r) is the potential energy with a value such that the incompressibility constraint is locally preserved. The second term of eq 10 is taking into account the nearest-neighbor interactions between segments. The Flory-Huggins interaction parameter between segment type A in the state j and B in the state k is denoted as χAjBk, and φBk(z, r) is the volume fraction of segment type B in the state k at layer (z, r). The angular brackets indicate the averaging over the neighboring layers of (z, r) to be specified below. The notation φbBk refers to the volume fraction of segment type B in the state k in the bulk phase. The third term of eq 10 is the electrostatic contribution due to the presence of the free charges where υAj is the valence of segment A in state j, e is the elementary charge, and Ψ(z, r) is the electrostatic potential at position (z, r). The fourth term is the electrostatic contribution which accounts for the segment polarization at (z, r) due to the local electric field. Here, r,Aj(z, r) is the relative dielectric permittivity of segment A in state j at layer (z, r), and P(z, r) is the spatial average of the square of the electrostatic potential.39 The electrostatic potential Ψ(z, r) is obtained by solving a discrete form of the Poisson equation in the cylindrical coordinate system

j

The self-consistent-field theory is based on a mean-field approximation where the density fluctuations within lattice sites at same coordinate (z, r) are ignored. In other words, the volume fraction of segments at a given (z, r) is uniform, whereas gradients in volume fraction can only develop along the r and z directions. In the SF-SCF model, each segment of type A at position (z, r) is statistically weighted by a segment-weighting factor GA(z, r). Following the multiple states model by Bjorling et al.,38 the overall statistical weight of segments of type A, GA(z, r) is taken to be equal to the weighted sum over statedependent segment-weighting factors GAj(z, r) which depend on the potential energy uAj(z, r) of the segment A,

k

1 eυAjΨ(z, r) - 0(r,Aj(z, r) - 1)P(z, r) (10) 2

0

(7)

j k

λ

(

)

2 2 1 ∂φBk(z, r) ∂ φBk(z, r) ∂ φBk(z, r) + + (14) r ∂r ∂r2 ∂z2

In discrete form, eq 14 can be rewritten by applying different bond-weighting factors λ-1(r), λ0(r), and λ1(r) to the volume fractions at neighboring sites in the radial direction,



〈φBk(z, r)〉 ) λ[λ-1(r){φBk(z - 1, r - 1) + φBk(z, r - 1) + φBk(z + 1, r - 1)} + λ0(r){φBk(z - 1, r) + φBk(z, r) + φBk(z + 1, r)} + λ1(r){φBk(z - 1, r + 1) + φBk(z, r + 1) + φBk(z + 1, r + 1)}] (15) where the constraint λ-1(r) + λ0(r) + λ1(r) ) 1, ∀ r ∈ [1, Mr] is fulfilled. For the cylindrical coordinate system, the bondweighting factors are given by40

S(r - 1) L(r)

λ-1(r) ) λ

〈Gi(z, r, s - 1|1)〉 ) λ[λ-1(r){Gi(z - 1, r - 1, s - 1|1) + Gi(z, r - 1, s - 1|1) + Gi(z + 1, r - 1, s - 1|1)} + λ0(r){Gi(z - 1, r, s - 1|1) + Gi(z, r, s - 1|1) + Gi(z + 1, r, s - 1|1)} + λ1(r){Gi(z - 1, r + 1, s - 1|1) + Gi(z, r + 1, s - 1|1) + Gi(z + 1, r + 1, s - 1|1)}] (21) The chain segment-weighting factor Gi(z, r, s|N) is the quantity which is related to the probability of finding a chain fragment from the last segment Ni to s such that segment s of molecule i is at (z, r). It is evaluated using the complementary eqs 20 and 21. When all segment-weighting factors are known, the volume fraction φi(z, r, s) is obtained from the composition law

S(r) λ1(r) ) λ L(r) λ0(r) ) 1 - λ-1(r) - λ1(r)

(16)

where λ is the one-dimensional bond-weighting factor which depends on the geometry of the equivalent planar lattice. Throughout this article, λ ) 1/3 is used (face centered cubic lattice). The quantities that depend on the segment types can be generalized into quantities that depend on the molecule i and segment ranking number s in the molecule,

Gi(z, r, s) ) φA(z, r) )

∑A GA(z,

A r)δi,s

∑i ∑s φi(z, r, s)δi,sA

(18)

The segment-weighting factors Gi(z, r, s) are used to analyze the statistical weight of the complete set of all possible conformations of the (surfactant) molecules. This is performed by considering the chain segment-weighting factor Gi(z, r, s|1) for segment s of molecule i at coordinate (z, r) connected by a string of s - 1 segments to the first segment. In continuous form, the chain segment-weighting factor is obtained by solving Edwards diffusion equation,41 2 ∂Gi(z, r, s|1) 1 ∂Gi(z, r, s|1) ∂ Gi(z, r, s|1) )λ + + ∂s r ∂r ∂r2 2

)

∂ Gi(z, r, s|1) ∂z2

{

Gi(z, r, 1), s)1 〈Gi(z, r, s - 1|1)〉Gi(z, r, s), s > 1

RAj(z, r) )

(20)

where the average chain segment-weighting factor 〈Gi(z, r, s - 1|1)〉 is analogous to the local average volume fraction (cf. eq 10), weighted by the bond-weighting factors:

(23)

RAb jGAj(z, r)

(24)

GA(z, r)

In eq 24, RAj(z, r) is the fraction of segment A in state j at coordinate (z, r).42 By obtaining all the segment volume fractions, the segment potentials can be evaluated with eq 10. It is important to note that the incompressibility condition ΣAΣjφAj(z, r) ) 1 leads to a unique value of u′(z, r) for all layers (z, r). Through an iteration procedure, a unique self-consistentfield solution is obtained, i.e., all uAj(z, r) and φAj(z, r) that satisfy all constraints such that the partition function can be considered to be optimized. The grand potential Ω may be expressed as43

Ω

)-

kBT

∑z ∑r L(r)Π(z, r)

(25)

where -Π(z, r) is known as the dimensionless grand potential density (and Π(z, r) is the local pressure),

1 2

whereas in a discrete form, the chain segment-weighting factors Gi(z, r, s|1) are calculated using a first-order Markov approximation

Gi(z, r, s|1) )

φAj(z, r) ) RAj(z, r)φA(z, r)

Π(z, r) )

+ Gi(z, r, s|1) ln Gi(z, r, s) (19)

(22)

where Ci is a normalization factor which depends on the number of molecules ni of type i in the system such that Ci ) ni/Gi(1|N), where Gi(1|N) ) ΣzΣrL(r)Gi(z, r, 1|N). By using eq 18, the volume fraction of each segment type can be found where the corresponding state-dependent volume fraction φAj(z, r) is calculated from

(17)

A where δi,s ) 1 when segment s of molecule i is of type A and zero otherwise.

(

Gi(z, r, s|1)Gi(z, r, s|N) φi(z, r, s) ) Ci Gi(z, r, s)

∑A ∑j φA (z, r) j

uAj(z, r) kBT

+

∑i

φi(z, r) - φbi

-

Ni

∑A ∑j ∑B ∑k χA B {φA (z, r)(〈φB (z, r)〉 - φBb ) j k

j

φAb j(φBk(z, r) - φBb k)} -

k

k

Ψ(z, r)

1

∑∑eυA φA (z, r) 2A j j

j

kBT

(26)

where φi(z, r) ) Σsφi(z, r, s) gives the overall volume fraction of molecule i at position (z, r). Finally, the grand potential Ω of a small system containing one toroidal micelle with curvature J is identified as the bending energy of the torus. In the calculation, different curvatures are achieved by varying the total number of surfactant molecules in the system. By fitting Ω(J) to the Helfrich eq 2, one obtains the total bending rigidity κ0 (thus, the total persistence length lp,t) for the limit J f 0 (in


Lauw et al.

TABLE 1: List of the Flory-Huggins Interaction Parameters χ

X

C

O

W

s

X C O W s

0 1.6 -0.35 0 0

1.6 0 2.2 1.6 1.6

-0.35 2.2 0 -0.35 -0.35

0 1.6 -0.35 0 0

0 1.6 -0.35 0 0

practice, this limiting value is reached at a sufficiently large number of surfactant molecules). III. Model Parameter In this section, we specify the necessary parameters that are needed for the SF-SCF calculation. In principle, there are four categories of input parameters for the SF-SCF scheme, i.e., molecular architecture parameters, the Flory-Huggins interaction parameters between the molecular segments, electrostatic properties of the species, and details that specify the lattice coordinate system. In the model, surfactants CnEmXk are used with linear architecture (C)n(OC2)mO(X)k where C represents CH2 or CH3 groups, O refers to an oxygen atom, and X is the weakly charged acid group. The subindex denotes the degree of polymerization of each block. As a solvent, water W is used and modeled as a single monomer that can occupy one lattice site. Added background salt ions s exist in the system as monomers. Water is partly dissociated, depending on pH, by the chosen pK ˜w ) 17.5 which corresponds to the experimental value pKw ) 14, whereas the operative dissociation constant of the X group is set to pK ˜ a ) 6 corresponding to pKa ) 4.25, a value typical for the dissociation of a carboxylic-like group.44 The conversion of the dimensionless operative variables to the experimental variables, and vice versa, depends on the molecular volume of each molecule. For monomeric species, the conversion factor 55.4 M is used, e.g., the dimensionless volume fraction of background salt φs ) 10-5 corresponds to 5.5 × 10-4 M of salt. For nonmonomeric molecules, the conversion factor of volume fractions to molar concentrations is normalized by the chain length. In this article, the experimental variables are used when we discuss the results. The set of Flory-Huggins interaction parameters used in the model is given in Table 1. Note that these parameters are taken to be independent of the internal states of the segments. To mimic the repulsive interactions between hydrophobic and hydrophilic groups, χCO ) 2.2 is used.11 The hydrophilicity of a head group and a charged block is indicated by using χOW ) χXW ) -0.35, whereas the hydrophobicity of a tail group is reflected in a rather large value χCW ) 1.6. For simplicity, the set of interaction parameters for the salt molecules s follow the corresponding parameters of water. The relative permittivity of water is set to 80. In the SF-SCF model, the operative pH ˜ ) -log(RHb 3O+φbW) is used, where b RH3O+ is the degree of protonation of water in the bulk solution and φbW is the volume fraction of the water in the bulk. Again, the pH ˜ corresponds to the experimental pH by using the conversion factor 55.4 M, that is, pH ) pH ˜ + log 55.4. The background salt molecules consist of a monomeric positively charged s+ and negative salt ion s- with valence υs+ ) 1 and υs- ) -1. The relative permittivity of this molecule is set to 10. The weakly charged groups X exists as uncharged HX and negatively charged state X- with relative permittivity 10 and corresponding valence υHX ) 0 and υX- ) -1. All calculations are performed in a cylindrical coordinate system with characteristic length of a lattice site a ) 0.3 nm.

The dimension of the small system is at least 2Mz ) Mr ) 60, where in all cases the volume fractions attain their bulk values near the system boundaries. The size of the system is also large enough such that there is no long-range electrostatic interaction between the torus and their mirror images (through the reflecting boundary conditions). This is achieved by setting Mz and Mr considerably larger than the Debye length κ-1. Since above the CMC, the concentration of the free surfactant in the bulk is almost constant, the total number of surfactants in the system can be used to control the radius of the torus. Note that, for a given number of surfactants, the toroidal radii are automatically optimized such that the bended wormlike micelle is not stretched/extended. As a result, the grand potential of the torus is solely composed of the curvature energy. IV. Results and Discussions This section starts with an explanation of results for a typical case, namely, the surfactant C20E10X1 at [s] ) 1.7 M and pH ) 8. This is followed by the discussion of the effect of pH, background salt concentration, and the surfactant architecture parameters on the persistence length. A. Self-Assembly of a Toroidal Micelle from C20E10X1 Surfactants at [s] ) 1.7 M and pH ) 8. As mentioned in the previous section, the number of surfactants in the system determines the size of the torus. Therefore, the variation in the total surfactant concentration leads to different toroidal curvatures J. More specifically, the dimensionless curvature Ja ) 1/(R/a) ) 2πN0agg/Nagg, where N0agg and Nagg are the aggregation number per unit length of a straight wormlike micelle (limit J f 0) and the aggregation number of the torus, respectively. Typical volume fraction contour plots and profiles of a cross section of the torus are shown in Figure 2. The torus has J ) 0.12 nm-1. The plot is given in the radial (r) and lateral (z) directions. Note that the unit length used here is nanometers. Figure 2a depicts the volume fraction contour plot φC where C refers to all the alkyl segments of the hydrophobic part of the surfactants, i.e., the core of the wormlike micelle. From this figure, the diameter of the core is found to be approximately 3.5 nm; the core has the same size in the z and r directions. The volume fraction map of the hydrophilic group φE is shown in Figure 2b. This takes into account all the C and O segments of the hydrophilic block. From this plot, the thickness of the corona is seen to be approximately 2 nm. The maximum volume fraction value of the E group is on the order of 0.3, which is considerably less than the maximum value of φC (more than 0.8, see Figure 2a). This shows that, in contrast to the core, the corona of the torus is relatively well-hydrated. The corresponding volume fraction profiles of two cross sections of the torus for the same system are depicted in Figure 2c and d. The density profiles of hydrophobic C group and hydrophilic E block are plotted in the z direction at r ) 9 nm and in the r direction at z ) 0 in Figures 2c and d, respectively. The core radius and corona thickness of the cross section of the torus are easily obtained from these profiles. The core radius of the torus rCr is defined as twice the first moment of φC with respect to the center of the coordinate z ) 0. Similarly, the core + corona radius rr of the CnEm diblock (cf. Figure 1) is obtained from the first moment of the E group with respect to z ) 0. Thus, the corona thickness of the torus rEr is defined by rEr ) rr - rCr . Analogously, the major radius of the torus R is calculated from the first moment of φC with respect to the center of the coordinate r ) 0. Using these definitions, for the case of the toroidal profile from Figure 2c, we find a core radius of the torus rCr of 1.7 nm and a



Figure 2. Contour plot of the volume fraction of the hydrophobic tail group φC (a) and the hydrophilic head group φE (b). (c and d) Corresponding cross-sectional profile of φC and φE in the lateral direction z at r ) 9 nm and in the radial direction r at z ) 0, respectively. All plots are based from a cross section of a torus composed of C20E10X1 surfactants at [s] ) 1.7 M, pH ) 8, and J ) 0.12 nm-1.

Figure 3. (a) Profile plots of background salt ions, i.e., φs+ and φs-, and the negatively charged group φX- in the lateral direction z at r ) 9 nm. (b) Similar profile plots in the radial direction r at z ) 0. (c) Profile plots of the electrostatic potential Ψ and the local charge density q/e in the lateral direction z at r ) 9 nm. (d) Similar profile plots of Ψ and q/e in the radial direction r at z ) 0. All plots are from a cross section of a torus composed of C20E10X1 surfactants at [s] ) 1.7 M, pH ) 8, and J ) 0.12 nm-1.

thickness of the CnEm block rr of 2.4 nm. For the case of Figure 2d, the calculated major radius of the torus R ) 8.5 nm. In the same plot, it is clearly seen that the profile of φE is not exactly symmetric, i.e., the hydrophilic group is more densily packed near the center of the torus than at the outer part.

Figure 3a and b show the density profiles of salt ions and dissociated X- groups of the surfactant in and around the torus, in the lateral direction z at r ) 9 nm and in the radial direction r at z ) 0, respectively. It is found that the weakly negatively charged groups X are dispersed throughout the corona; the salt


Figure 4. Plot of the total persistence length lp,t as a function of toroidal curvature J from a torus composed of C20E10X1 surfactants at [s] ) 1.7 M and pH ) 8. The dashed line gives the extrapolation of lp,t to J ) 0, i.e. l0p,t, as indicated by the arrow.

ions penetrate marginally into the micellar core, and the counterions are distributed around the X- groups. The thickness tr of the region where the acid groups X largely reside is found by substracting rr from the first moment of φX- with respect to z ) 0. From Figure 3a, it follows that tr ) 0.8 nm. Nonsymmetrical density profiles of φs+, φs-, and φX- in the radial direction r are presented in Figure 3b. Again, these volume fractions are slightly higher in the inner part of the torus than at the outer side. This difference is attributed to the fact that the head groups are compressed on the inside and are allowed to explore more space on the outside. The profiles of the electrostatic potential Ψ and dimensionless charge density q/e are shown in Figure 3c and d, respectively, in the lateral direction z at r ) 9 nm and in the radial direction r at z ) 0. From these figures, it is shown that throughout the entire system, the electrostatic potential is negative as expected. Furthermore, a net overall negative charge is found at the region where the X group is mostly located, whereas, inside the core and around the micelle, the net charge is positive. In the bulk solution and toward the center of the torus, the electrostatic potential and the net charge density decrease to zero. Analogous to the volume fraction profiles described above, asymmetric profiles of electrostatic potential and charge density are found along the radial direction r (Figure 3d). The relatively compressed head groups on the inside part of the torus have a higher volume fraction φX- which increases the local electrostatic potential as well as the local charge density at the inner side of the torus. By combining eqs 2 and 3, and expressing the length of the torus as l ) 2π/J, the total persistence length at a given curvature is obtained as lp,t(J) ) 1/(πJ)‚Ω/(kBT). A plot of lp,t as a function of J is depicted in Figure 4 for wormlike micelles composed of C20E10X1 surfactants at [s] ) 1.7 M and pH ) 8. From our previous study on the bending rigidity of nonionic wormlike micelles, we know that the bending rigidity is an even polynomial function of the curvature.11 From Figure 4, this dependence cannot be seen clearly since only small curvatures are considered. Here, the primary focus is on

l0p,t ≡ limlp,t(J) Jf0

where the Helfrich eq 1 is valid. This value is obtained from the extrapolation of the total persistence length to zero curvature, as indicated by an arrow in Figure 4. The extrapolation is performed using the data curve fit (lp,t(J) - l0p,t) ∼ J2. B. Persistence Length at Different pH Values and Background Salt Concentrations. From the SCF calculation, we directly obtain the total persistence length. As described

Lauw et al.

Figure 5. Total persistence length l0p,t at zero curvature as a function of pH from linear micelles composed of C20E10X1 surfactants at [s] ) 5.5 × 10-2 M.

in the introduction, the total persistence length can conceptually be divided into two parts: the elastic and the electrostatic persistence length. This separation is not straightforward and invariably involves approximations. In general, the magnitude of the electrostatic persistence length is considerably less than the elastic one except for a system at extremely low ionic strength.18 Below, we use the Ansatz that the elastic persistence length is independent of the pH and the background salt concentration; thus, it is solely a function of the degrees of polymerization of the surfactant. This allows us to identify the electrostatic contribution which is largely affected by the changing of pH and ionic strength. Again, we focus on the persistence length in the limit of zero curvature (J f 0). The change of the total persistence length l0p,t with respect to pH is shown in Figure 5. At pH ) 3, the charged group is in an almost undissociated state, i.e., the mean degree of dissociation around the torus R j X- ≈ 0, whereas at pH ) 8, the charged group is fully dissociated (R j X- ≈ 1). The persistence length at pH ) 3 can therefore be identified as barely the elastic persistence length, i.e., l0p,t ≈ l0p,0 as l0p,e ≈ 0. In this case, l0p,t ≈ l0p,0 ) 27.3 nm. As the pH is increased, the weakly charged group X in the torus becomes more dissociated and the electrostatic repulsion between the charged groups increases. This tends to increase the rigidity of the torus via the increasing value of the electrostatic bending modulus. The change of the total persistence length l0p,t with the background salt concentration [s] is shown in Figure 6a. The toroidal micelles are composed of C20E10X1 and C20E10X2 surfactants at pH ) 8, as indicated. For both systems, analogous to decreasing pH, the persistence length decreases as salt concentration is increased. This can be understood again due to a decreasing contribution from the electrostatic persistence length, provided that the elastic persistence length does not depend on the ionic strength. It is clear that the total persistence length converges to the value l0p,0 with increasing salt concentration, i.e., to the value of the total persistence length when only the undissociated state of X exists (l0p,t ≈ l0p,0 at pH ) 3). For the case of the torus composed of C20E10X1 surfactants, the total persistence length converges to approximately 27.3 nm in high ionic strength and that for the torus from C20E10X2 surfactants approaches l0p,t ≈ l0p,0 ≈ 28.9 nm. The persistence length is plotted against the inverse Debye length κ in Figure 6b and c. Here, the inverse Debye length κ is calculated as follows

κ ) 2

F2

νj2cj ∑ RT j 0 r

(27)


J. Phys. Chem. B, Vol. 111, No. 28, 2007 8165 find a scaling l0p,e ∼ κ-p where p ≈ 1.98 for C20E10X1 and p ≈ 1.54 for C20E10X2. In both cases, the scaling is slightly smaller than κ-2, which is the OSF scaling prediction. The discrepancies may be attributed to the fact that the charged groups in the corona are not spaced linearly but dispersed throughout the toroidal corona. The penetration of the background salt ions inside the corona may also lead to scaling different from the OSF prediction. A high negative local charge density of the micelle causes an enormous accumulation of salt ions in the corona that reduces the local electrostatic potential. This lowers the effective local charge density into a renormalized value. This effect is analogous to the Manning counterion condensation mechanism which is originally developed for a line charge polyelectrolyte system. Alternatively, we can argue as follows. If the OSF relation of eq 5 is applied, it is important to consider the necessity of renormalizing the charge density per unit length σ due to the possibility of counterion condensation. Counterion condensation occurs when the distance between two charged groups, la, is less than the Bjerrum length lB or when the charge density q/e per unit length is more than 1/lB ≈ 1.43 nm-1 (in an aqueous solution at room temperature). When there is no counterion condensation, the charge density per unit length of the torus can be calculated simply from σ ) N0aggR j X- where N0agg is the aggregation number per unit length and R j X- is the average degree of dissociation of the charged group X in the torus. In Figure 7a, N0agg for linear micelles composed of C20E10X1 and C20E10X2 surfactants at pH ) 8 is plotted as a function of the salt concentration. In both cases, based on N0agg values and given that all the acid groups are fully charged, the resulting charge densities exceed by far the critical value 1.43 nm-1. Indeed, consistent with our first argument, an effective charge density value σ has to be taken into account, i.e.

l0p,t

Figure 6. (a) Total persistence length at zero curvature as a function of background salt concentration [s] from linear micelles composed of C20E10X1 and C20E10X2 surfactants at pH ) 8. (b) Total persistence length l0p,t and the electrostatic persistence length l0p,e of the wormlike micelles composed of C20E10X1 surfactants as a function of inverse Debye length κ at pH ) 8. (c) Electrostatic persistence length l0p,e of linear micelles composed of C20E10X1 and C20E10X2 surfactants as a function of inverse Debye length κ at pH ) 8.

where κ is in units of inverse nanometers, F is the Faraday constant, R is the gas constant, νj and cj are the valence and the bulk concentration (in molar) of the charged species j, respectively. For an aqueous solution at room temperature, eq 27 becomes κ2 ) 5.411Σjνj2cj. In our calculation, all the charged species (in the bulk solution), i.e., the salt ions, functional group X-, hydronium and hydroxide ions, are taken into account in calculating the inverse Debye length κ. The electrostatic persistence length is calculated by substracting the elastic persistence length obtained from the (uncharged) torus system at pH ) 3 from the total persistence length. Thus, l0p,e ≈ l0p,t - 27.3 nm for the C20E10X1 surfactants and l0p,e ≈ l0p,t - 28.9 nm for the C20E10X2 surfactants. The results are depicted in Figure 6b and c. In Figure 6b, the total persistence length l0p,t of cylindrical micelles composed of C20E10X1 surfactants is plotted together with its corresponding electrostatic persistence length l0p,e as a function of κ at pH ) 8. As discussed above, at a lower salt concentration (larger Debye length), the electrostatic persistence length becomes more pronounced. The comparison of the electrostatic persistence length for C20E10X1 and C20E10X2 surfactants is given in Figure 6c. From fitting these curves, we

σ ) fN0aggR j X-

(28)

where f is the effective fraction of the charge density. By combining eqs 28 and 5, the effective fraction of charges can be estimated. The value of f for the linear micelles from C20E10X1 surfactants varies with ionic strength from 0.26 to 0.36 at the range of background salt concentration 5.5 × 10-2 to 5.5 M. Within the same salt concentration range, for the case of the micelles from C20E10X2 surfactants, f varies from 0.45 to 0.58. It can be concluded that the renormalized charge densities are needed for a highly charged wormlike micellar system in order to fulfill the OSF scaling prediction (l0p,e ∼ κ-2). In Figure 7a, it is shown that as the salt concentration increases, N0agg values for C20E10X1 micelles converge to N0agg ≈ 20.8 nm-1, whereas for the case of micelles from C20E10X2, the values approach N0agg ≈ 19.8 nm-1. The change of the aggregation number per unit length of the wormlike micelle leads to a change of the toroidal minor radius size rr and tr as depicted in Figure 7b (for micelles from C20E10X1) and in Figure 7c (for micelles from C20E10X2). In both cases, for an increasing salt concentration, rr increases and tr decreases, yet the overall toroidal minor radius rr + tr increases slightly. Apparently, the thickness of the region rich in acid groups X decreases with increasing salt concentration, yet this is compensated by an opposite effect of the swelling of the core and corona of the toroidal cross section. Comparison of Figure 7a with b and c indicates that the increase of the aggregation number is only slightly followed by the increase of the total minor radius of the torus. The change of rr follows directly the corresponding trend of N0agg. Both rr and tr, depicted in Figure 7b and c, reach


Lauw et al.

Figure 8. (a) Total persistence length l0p,t at zero curvature as a function of the degree of polymerization n of hydrophobic group C for linear micelles composed of CnE5X1 and CnE10X1 surfactants. (b) Total persistence length l0p,t as a function of the degree of polymerization m of hydrophilic group E for wormlike micelles composed of CnEmX1 surfactants where n ) 20, 21, 22, and 24. In all cases, the background salt concentration [s] ) 5.5 × 10-2 M and pH ) 8.

Figure 7. (a) Aggregation number per unit length N0agg at zero curvature as a function of background salt concentration [s] for linear micelles composed of C20E10X1 and C20E10X2 surfactants at pH ) 8 (b) Parts of the minor radius rr and tr of the cross section of wormlike micelles from C20E10X1 surfactants of the same system. (c) Similar plot of rr and tr of the cross section of wormlike micelles from C20E10X2 surfactants of the same system.

limiting values at high salt concentrations which are, based on the previous discussion, the size of the (neutral) toroidal cross section. This limiting structure determines the elastic bending modulus. For a linear micelle composed of C20E10X1 (Figure 7b), the asymptotical values are the following: rr ≈ 2.5 nm and tr ≈ 0.8 nm. For C20E10X2 in Figure 7c, they are rr ≈ 2.4 nm and tr ≈ 0.9 nm. C. Effect of the Surfactant Size on the Persistence Length. Previous SCF calculations showed that the persistence length of wormlike micelles of CnEm surfactants is only a weak function of the head group length m, but it depends strongly on the tail group length n, i.e. lp ∼ nx with x is approximately 2.4-2.9.11 It is of considerable interest to investigate whether a similar trend can be found for wormlike micelles composed of ionic surfactants CnEmXk. Here, we use a rather longer tail length n compared with ones used in the previous publication.11 The presence of negatively charged groups X- causes an electrostatic repulsion at the outer part of a micelle such that the formation of an equilibrium wormlike micelle is prevented when surfactants have a relatively short tail.

Figure 8 shows the dependences of the total persistence length l0p,t on the tail length n and head group length m. These are calculated from toroidal micelles at [s] ) 5.5 × 10-2 M and pH ) 8. From Figure 8a, the total persistence length l0p,t of linear micelles from CnEmX1 surfactants is plotted as a function of n for two values of head group size m ) 5 and 10. For m ) 5, l0p,t scales with n as l0p,t ∼ n2, whereas for m ) 10, the scaling of the total persistence length is l0p,t ∼ n2.6. These scaling exponents are in the same range as the ones found for nonionic surfactants as described above. In Figure 8b, the l0p,t of wormlike micelles composed of CnEmX1 surfactants is plotted as a function of m for a few tail lengths n ) 20, 21, 22, and 24. For all cases, the persistence length depends weakly on m. This is also in line with the results found for nonionic wormlike micelles. In general, it can be concluded that the presence of charges in the surfactants is not important for the way the persistence length changes with variations in head and tail group lengths. As the number of charged groups k is varied, the total persistence length l0p,t changes nonmonotonically, as depicted in Figure 9a. This result is found for the micelle consisting of C20E10Xk surfactants at [s] ) 5.5 × 10-1 M and pH ) 8. In this case, the persistence length varies only slightly with k as the system is at rather high ionic strength ([s] ) 5.5 × 10-1 M). With increasing k, the bending rigidity increases initially and reaches a maximum value for k ) 3. Beyond this value, the persistence length decreases with increasing k. The nonmonotonic behavior of the persistence length with k is a result of two opposing effects. At low k values, the electrostatic persistence length increases with k due to the increase of the effective charge density. This is shown in Figure 9b where N0aggk increases with k (N0agg(k) decreases weakly). At high k values, the dimensions of the core and corona of the wormlike micelle are reduced (shown by the decrease of rr with k in Figure


J. Phys. Chem. B, Vol. 111, No. 28, 2007 8167 An increase of the salt concentration lowers the persistence length l0p,t. The change of l0p,t with pH and background salt concentration is mostly due to the change in the electrostatic contribution l0p,e, and the elastic persistence length l0p,0 is argued to remain constant. The value of l0p,0 is determined by the value of l0p,t at very low pH or the asymptotical value of l0p,t at large ionic strength. The electrostatic persistence length l0p,e is calculated from the difference between the total and the elastic persistence length. Quantitatively, the electrostatic persistence length l0p,e scales with the Debye length κ-1 as l0p,e ∼ κ-p where p ) 1.98 for a torus from C20E10X1 surfactants and p ) 1.54 for a torus from C20E10X2 surfactants. In order to fulfill the OSF scaling prediction of l0p,e(κ), the total charge density σ should be renormalized. However, it is not trivial to quantify this since the charges are distributed throughout the corona. With increasing ionic strength, the aggregation number per unit length N0agg increases, which is followed directly by the increasing rr, i.e. part of the minor radius contributed from the nonionic CnEm group of the surfactant. Nevertheless, the thickness tr of the region over which the charged groups X are distributed decreases with increasing salt concentration. In total, the minor toroidal radius rr + tr increases slightly with the salt concentration. As in the case of the wormlike micelles composed of nonionic surfactant, the total persistence length is a weak function of head group length m, but it scales strongly with the tail length n as l0p,t ∼ nx where x ) 2-2.6. The total persistence length l0p,t varies nonmonotonically with the number of charged group k as a result of the opposing response of the electrostatic and elastic bending rigidity on k.

Figure 9. (a) Total persistence length l0p,t in the limit of zero curvature as a function of the degrees of polymerization k of the weakly charged acid group X for torus composed of C20E10Xk surfactants. (b) Charge density at a fully dissociated X group, N0aggk at varying k. (c) Size of the corresponding toroidal cross section, i.e., rr and tr, as a function of k. The background salt concentration [s] ) 5.5 × 10-1 M and pH ) 8.

9c). As a result, the elastic/intrinsic bending rigidity decreases with k. This shows that our procedure to decompose the overall persistence length into an intrinsic and an electrostatic contribution is only reasonable if there are not too many charges in the chain. From Figure 9c, the increase of the number of charged groups k is accompanied by an increase of the size of the region occupied by the charged group tr. This reduces the core and corona thickness rr. The total minor radius rr + tr increases slightly with k. V. Conclusions The persistence length of wormlike micelles composed of ionic surfactants CnEmXk is calculated by means of selfconsistent-field theory. The calculation yields the relevant thermodynamical quantity (restricted grand potential) as a function of an imposed curvature and thus gives access to the total bending rigidity/persistence length of the wormlike micelles. The total persistence length is evaluated in the limit of curvature J f 0 (infinite major radius of a torus) and consists of an elastic/intrinsic and an electrostatic contribution. By increasing the pH, the total persistence length l0p,t increases, as more acid groups X of the surfactant are dissociated.

Acknowledgment. The Dutch Science and Technology Foundation (STW), Aquacare, GWA, KIWA, Witteveen&Bos, and ETD&C are greatly acknowledged for financial support. Part of this work is supported by the grant EPC.5516. We thank R. J. de Vries for helpful discussion on the bending properties of polyelectrolytes. References and Notes (1) Ortona, O.; Vitagliano, V.; Paduano, L.; Costantino, L. J. Colloid Interface Sci. 1998, 203, 477-484. (2) Ameri, M.; Attwood, D.; Collett, J. H.; Booth, C. J. Chem. Soc. Faraday Trans. 1997, 93 (15), 2545-2551. (3) Zhmud, B. V.; Tiberg, F.; Kizling, J. Langmuir 2000, 16, 25572565. (4) La Rosa, M.; Uhlherr, A.; Schiesser, C. H.; Moody, K.; Bohun, R.; Drummond, C. J. Langmuir 2004, 20, 1375-1385. (5) Sawicki, G. C. Colloids Surf. A: Physicochem. Eng. Aspects 2005, 263, 226-232. (6) Krogh, K. A.; Halling-Sorensen, B.; Mogensen, B. B.; Vejrup, K. V. Chemosphere 2003, 50, 871-901. (7) Henderson, A. M. J.; Saunders, J. M.; Mrkic, J.; Kent, P.; Gore, J.; Saunders, B. R. J. Mater. Chem. 2001, 11, 3037-3042. (8) Raman, I. A.; Suhaimi, H.; Tiddy, G. J. T. J. Dispers. Sci. Tech. 2005, 26, 355-364. (9) Anderson, K.; Kizling, J.; Holmberg, K.; Bystroom, S. Colloids Surf. A: Physicochem. Eng. Aspects 1998, 144, 259-266. (10) Barriocanal, L.; Taylor, K. M. G.; Buckton, G. J. Pharm. Sci. 2005, 94 (8), 1747-1755. (11) Lauw, Y.; Leermakers, F. A. M.; Cohen Stuart, M. A. J. Phys. Chem. B 2003, 107, 10912-10918. (12) Helfrich, W. Z. Naturforsch. C 1973, 28c, 693. (13) Landau, L. D.; Lifshitz, E. M. Statistical Physics, 3rd ed.; Pergamon Press: Oxford, 1980; Vol. 5, Part 1, p 127. (14) Lin, Z.; Hill, R. M.; Davis, H. T.; Scriven, L. E.; Talmon, Y. Langmuir 1994, 10, 1008-1011. (15) Rackstraw, B. J.; Martin, A. L.; Stolnik, S.; Roberts, C. J.; Garnett, M. C.; Davies, M. C.; Tendler, S. J. B. Langmuir 2001, 17, 3185-3193. (16) Pochan, D. J.; Chen, Z.; Cui, H.; Hales, K.; Qi, K.; Wooley, K. L. Science 2004, 306, 94-97.

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Persistence Length of Wormlike Micelles Composed of Ionic

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