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Langmuir 1997, 13, 4926-4928
Equilibrium Charge Distribution on Linear Micelles Paul van der Schoot* Max-Planck-Institut fu¨ r Kolloid- und Grenzfla¨ chenforschung, Kantstrasse 55, D-14513 Teltow-Seehof, Germany Received April 10, 1997. In Final Form: May 22, 1997
Charged rodlike micelles are often modeled as structureless end-capped cylinders with the charged headgroups distributed evenly over the micellar surface.1-12 The surface charge density is then presumed to be constant, save a possible difference in surface charge densities between the spherical end-caps and the cylindrical midsection. This is a plausible ansatz when the micelles consist of a single, (almost) fully ionized surfactant. However, when charged and uncharged molecules populate the micelles, and the ionic strength is low, the idealized picture of a uniform charge distribution along the micellar rods is not necessarily accurate. Such should be relevant when the surfactants are not fully ionized13 or when one mixes charged surfactants and uncharged cosurfactants.5,14,15 It is possibly also important for the case where oppositely charged counterions specifically adsorb onto the micellar surface.4,12,16 In this Note we theoretically study the effect of charge inhomogeneities on the self-assembly of weakly charged linear micelles in solutions of fairly low ionic strength. It is shown that the net charge has a tendency to accumulate near the ends of the micelles over a length scale comparable to the Debye length. This leads to a significant modification of the free energy associated with the ends and therefore to a potentially strong impact on the micelle growth. Our starting point is a dispersion of charged, polydisperse rodlike micelles in brine, far above the critical micelle concentration. Almost all of the aggregating material resides in the cylindrical micelles. Let φ denote the total concentration of aggregating material. The number density F(L) of micelles of length L normalizes to ∫dL LF ) φa2, with a an unimportant microscopic length scale. We (arbitrarily) choose a to be equal to the diameter of the micelles, typically around 5 nm.17 The micelles are thought to be long and slender; the mean length of the micelles therefore obeys 〈L〉 . a, where the brackets denote a number average defined as 〈‚‚‚〉 ≡ ∫dL (‚‚‚)F(L)/∫dL F(L). All electrolytes in the solution (surfactants and added salt) are of the 1-1 type. We model the presence of a (net) * E-mail:
[email protected]. (1) Nagarajan, R.; Ruckenstein, E. J. Colloid Interface Sci. 1979, 71, 580. (2) Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1981, 77, 601. (3) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 84, 1044. (4) Porte, G.; Appell, J. In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; Vol. 2. (5) Ben-Shaul, A.; Rorman, D. H.; Hartland, G. V.; Gelbart, W. M. J. Phys. Chem. 1986, 90, 5277. (6) . Odijk, T. J. Phys. Chem. 1989, 93, 3888. (7) Eriksson, J. C.; Ljunggren, S. Langmuir 1990, 6, 895. (8) Safran, S. A.; Pincus, P. A.; Cates, M. E.; MacKintosh, F. C. J. Phys. (Paris) 1990, 51, 503. (9) MacKintosh, F. C.; Safran, S. A.; Pincus, P. A. Europhys. Lett. 1990, 12, 697. (10) Odijk, T. Physica A 1991, 176, 201. (11) Odijk, T. Biophys. Chem. 1991, 41, 23. (12) Heindl, A.; Kohler, H.-H. Langmuir 1996, 12, 2464. (13) Zana, R. J. Colloid Interface Sci. 1980, 78, 330. (14) Porte, G.; Marignan, J.; Bassereau, P.; May, R. J. Phys. (Paris) 1988, 49, 511. (15) Zana, R. Adv. Colloid Interface Sci. 1995, 57, 1. (16) Imae, T.; Kohsaka, T. J. Phys. Chem. 1992, 96, 10030. (17) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869.
S0743-7463(97)00368-5 CCC: $14.00
charge on the assemblies phenomenologically by letting an as yet unknown number of charges distribute themselves freely over the individual micelles, which are viewed as quasi one-dimensional objects. The linear charge density may (and will) vary with micelle size. On each single micelle the charges need also not be distributed evenly. The charge distribution, expressed as the number of charges per unit length on a micelle of length L, is described by a linear charge density νL(l) with a coordinate l measuring the distance from one of the ends. Charges on the same aggregate are assumed to interact via a Debye-Hu¨ckel potential, implying that the linear charge density should not be too high. (See below.) Implicit also is that the Debye length λD is much larger than the micelle diameter a, because otherwise our approximate description of the charged cylinders as line charges would break down.6 Finally, we assume the solution to be sufficiently dilute so as to be able to ignore interactions between charges on different micelles.18 Within density functional theory, the grand potential of our system of self-assembled line charges may now be written as17
Ωa/VkBT )
∫0∞dL F(L)(ln(F(L)a3) - 1 + E - µsL + Ωc(L)) (1)
Here, V denotes the volume of the system, kBT the thermal energy, and µs a chemical potential regulating the total amount of aggregating material in the solution. In the first two terms between the brackets of eq 1 one recognizes an ideal entropy of mixing. E is the so-called end-cap energy, measuring (in units of kBT) the free energy penalty associated with the two hemispherical end-caps, typically between 5 and 25 (kBT).17 In eq 1, the end-cap energy contains only contributions from local interactions and packing constraints. The long-range effects of charge are described in the mean-field approximation by the (dimensionless) semigrand partition function of an inhomogeneous one-component (screened) plasma19
Ωc(L) )
∫0Ldl νL(l)(ln(νL(l)a) - 1 -µc) + Eel[νL]
(2)
The number of (net) charges on the micelles is regulated by µc, a suitable reference chemical potential. Ignoring the possibility of micellar flexibility, the electrostatic selfenergy of the rodlike association colloids is given by
Eel(L) ) L 1 Q dl 2 0
∫ ∫0Ldl′ νL(l) νL(l′) |l -1 l′| exp(-|l - l′|/λD)
(3)
in units of thermal energy. Here, Q ) q2/DkBT denotes the Bjerrum length, with q the elementary charge and D the permittivity of the medium. Q = 0.71 nm in water at room temperature. In solutions at an excess salt concentration csalt the Debye length is λD ) (8πQcsalt)-1/2. Notice that the integrand of eq 3 contains a divergence at l ) l′. Divergences of this kind will, as usual, be dealt with by the introduction of a short-distance cutoff. (18) Interactions between the charged rods can be neglected when the second virial correction to the free energy is small. For homogeneously charged rods this is the case when φ〈L〉aλD ln(2πQ〈ν〉2λD) > λD >> a. (Q is the Bjerrum length and 〈ν〉 the mean linear charge density defined elsewhere in the text.) See: Odijk, T. J. Chem. Phys. 1990, 93, 5172. Note that the rods overlap when φ〈L〉2a >> 1. This implies that there is a regime where the particles are strongly entangled yet do not interact significantly. (19) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: London, 1986.
© 1997 American Chemical Society
Notes
Langmuir, Vol. 13, No. 18, 1997 4927
The equilibrium micelle size distribution optimizes the grand potential. Setting δΩ/δF) 0 gives
ln(F(L)a3) ) -E + µsL - Ωc(L)
(4)
The chemical potential µs < 0 may be eliminated through the normalization condition of F(L). When Ωc ) 0, i.e., for single-component uncharged linear micelles, we obtain µs ) -1/〈L〉, leading to the well-known mean-field growth law 〈L〉 ∝ φ1/2 exp(1/2E).17 To obtain the growth law for the charged case, we first have to evaluate the optimal charge distributions on each of the micelles. A minimization of Ω with respect to νL gives
ln(νL(l)a) ) µc -
∫0Ldl′ νL(l′) |l -1 l′|exp(-|l - l′|/λD)
Q
(5)
where as before a small region around l′ ) l is implied to be excluded from the integration to prevent the integral from diverging. We have not been able to solve this highly nonlinear integral equation exactly. However, since we assume the Debye-Hu¨ckel approximation to be valid, the condition νLQ < 1 must hold, allowing us to attempt a low charge density expansion ∞
νL(l) ) a-1 exp(µc)[1 +
fL,n(l)(Qa-1 exp(µc))n] ∑ n)1
(6)
insert this in eq 5, expand terms where appropriate, and then solve order by order in Qa-1 exp(µc) < 1. The first two terms are
∫0
fL,1(l) ) fL,2(l) )
1 2 f (l) 2 L,1
L
Having obtained a limiting form of the equilibrium charge distribution, we can assess the impact of the equilibrium charges on the growth of the linear micelles. Inserting eqs 6 and 7 in eq 2 gives a complicated expression for Ωc, which simplifies to a sum of terms linear in L and terms independent of L in the regime where L >> λD >> a (i.e., at low, but not too low, ionic strength). The terms linear in L may then, in eq 4, be absorbed in µsL. The remaining terms “renormalize” the end-cap energy E, leading to an effective end-cap energy Eeff. Thus we write F(L)a3 ) exp(-Eeff + µsL), where to second order in the mean charge density
Eeff ) E - QλD(a-1 exp(µc))2 +
1 dl′ exp(-|l - l′|/λD) |l - l′|
2Q2λD(a-1 exp(µc))3[E1(a/λD) + E1(2a/λD)] + ‚‚‚ (9)
∫0Ldl′ fL,1(l′) |l -1 l′| exp(-|l - l′|/λD)
(7)
The first-order correction to a uniform charge distribution can be expressed in terms of the exponential integral20 E1(‚) as
() ( )
fL,1(l) ) fL,1(L - l) ) E1
Figure 1. Charge density along a rodlike micelle of average length L ) 〈L〉 relative to the mean value as a function of the scaled position according to eqs 6 and 7 to linear order in the charge density. The parameters are chosen such that Q ) 0.71 nm, a ) 5 nm, λD ) 50 nm, 〈L〉 ) 500 nm, and Q〈ν〉 ) 0.2. The mean charge density 〈ν〉 was calculated using eq 10.
l L-l + E1 - 2E1(a/λD) λD λD (8)
where we have now explicitly introduced the cutoff length a and where a e l e L - a. It is immediately clear that the charge distribution must be inhomogeneous, because fL,1(L/2λD) ≈ 2fL,1(a/λD). (Recall that a 0 diminishes the charge accumulation near the ends, correcting for the excess. We come back to the range of validity of our analysis below. (20) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover Publications: New York, 1965. (21) Berghold, G.; Seidel, C. To be published.
for a 1 limits the extent to which this happens. Notice that the effective end-cap energy depends (weakly) on the volume fraction of aggregating material through its dependence on 〈L〉 in eq 11. This could, in principle, lead to a nontrivial dependence of the mean length of the micelles on the concentration of self-assembling material. Indeed, eq 11 shows not only that (i) growth does not conform to the power law expected for highly screened (or uncharged) micelles3 but also that (ii) the mean length increases more slowly with concentration than 〈L〉 ∝ φ1/2. It is at this point useful to briefly review the main approximations made to arrive at eq 11. (i) The DebyeHu¨ckel approximation, in combination with the expansion in eq 6, restricts the validity of the analysis to low charge densities. From a comparison with the computer simulations of Berghold and Seidel,21 we estimate the perturbation expansion in eq 6, truncated after the first-order correction in Q〈ν〉, to become inaccurate when Q〈ν〉 > 0.2. Since we have gone to second order in Q〈ν〉 in our calculation of the effective end-cap energy, eq 11 should be expected to remain accurate for mean charge densities
somewhat higher than 0.2Q-1. One might get more accurate results for values of Q〈ν〉 approaching unity by applying a suitable variational treatment22 based, e.g., on the first-order solution to eq 5. At high charge densities, when Q〈ν〉 and the Debye-Hu¨ckel approximation breaks down, one would have to self-consistently deal with an inhomogeneous counterion condensation10,11 and an inhomogeneous charge distribution on the rods. This is a formidable problem that we did not attempt to address. (ii) The linear micelles are modeled as infinitely rigid. This is a less crude approximation than it seems, because in the dilute regime the cylindrical micelles normally do not grow to sufficient length for long-distance interactions along the chains to become appreciable.17 Provided the persistence length of semiflexible micelles is quite larger than the Debye screening length and the mean micelle length does not exceed a few persistence lengths, eq 11 should approximately hold. (iii) A mean-field approximation has been used to establish the charge density profile on the linear micelles, which one expects to be reasonable when many charges interact simultaneously.23 The number of (net) charges per Debye length is therefore to be greater than, say, unity. The condition 〈ν〉 also ensures that the continuum description of the charge distribution is valid.6 We realize that fluctuations may still be significant, because of the (quasi) one-dimensional character of the assemblies. The computer simulations of Berghold and Seidel21 show, however, that fluctuations do not suppress the charge accumulation near the ends, so the physical mechanism leading to the modification of the effective end-cap energy is left intact. Fluctuations will presumably only modify the precise form of eq 11. (iv) Cylindrical assemblies have been modeled as onedimensional objects, the tacit assumption being that when λD >> a, universal features dominate the self-assembly. This expectation has turned out not to be completely justified: the nonlinear corrections to the end-cap energy eq in 11 depend, albeit only weakly, on the microscopic cutoff a. (These corrections are nevertheless well-behaved in the limit where we let the cutoff go to zero, a/λD f 0.) We would still put forward that the influence of inhomogeneities over a length scale of the order of the micelle diameter5 is swamped by the effects on the much larger length scales, λD >> a, studied here and that in that sense the behavior is universal. In conclusion, we have shown that when the ionic strength of linearly aggregating surfactants is fairly low, the growth of the micelles is strongly influenced by an inhomogeneous charge distribution on the aggregates. Although the theory presented in this Note applies only to systems with a low net charge, it seems reasonable to expect similar effects on highly charged systems.8-10 It would be interesting to test the theory on suitable experimental systems, which presumably include weakly ionized surfactants, such as cetyltrimethylammonium tosylate,24 or cationic surfactants, the net surface charges of which have been dimished by strongly adsorbed salicylate counterions.16,25 Acknowledgment. I thank Carina van der Veen for assistance with the presentation of the results and Julian Shillcock for discussions and a careful reading of the manuscript. Financial support from the Max-PlanckGesellschaft is also gratefully acknowledged. LA970368I (22) Ullner, M.; Jo¨nsson, B. Macromolecules 1996, 29, 6645. (23) Lieb, E. H.; Mattis, D. C. Mathematical Physics in One Dimension; Academic Press: New York, 1966. (24) Narayanan, J.; Manohar, C.; Langevin, D.; Urbach, W. Langmuir 1997, 13, 398. (25) Oizumi, J.; Kimura, Y.; Ito, K.; Hayakawa, R. J. Chem. Phys. 1996, 104, 9137.
