17437
J. Phys. Chem. 1995,99, 17437-17441
Micellization Model for Multivalent Ionic Surfactants Costas S. Patrickiost 27 Homer Street, Ayios Nicolaos, Limassol3095, Cyprus Received: August 8, I995@
The micellization of surfactants comprising a multivalent ionic head and a dodecyl tail was modeled using Tanford's approach, which involves the description of the Coulombic repulsive forces via the Debye-Huckel approximation and the estimation of the hydrophobic attractive forces through the use of an empirical equation. The model predicts that a low surfactant charge (valence), a high salt concentration, and a high degree of neutralization of the surfactant charge by counterions favor micellization. It was also found that an increase in the surfactant charge leads to a decrease in the micellar aggregation number and to an increase in the critical micelle concentration (cmc). More specifically, it was determined that the aggregation numbers scale as the inverse first power of the surfactant charge, while the logarithm of the cmc's scales linearly with the surfactant charge. The predictions of our model compare favorably with theoretical and experimental findings on the micellization of amphiphilic charged block copolymers.
Introduction There is an intense interest in amphiphilic charged block copolymers that form micelles in aqueous and nonaqueous media. Numerous reports on the synthesis and characterization of such molecules go all the way back to 1970, with the pioneering work of Gallot and colleagues.'-6 More recently, the groups of Webber and Eisenberg have presented extensive studies on polyelectrolytic amphiphiles both in aqueous and organic solvent^.^-]^ An exotic example of amphiphilic block polyelectrolytes is ABC triblock polyampholytes recently synthesized and currently being characterized by the group of Hatton.I4-l7 There are very few models on the micellization of these molecules in solution. Extending their pioneering work on the microphase separation of charged block copolymers in the melt,18,'9Marko and RabinI9 recently modeled the aggregation of these amphiphiles in solution. They determined the dependence of the aggregation number and critical micelle concentration (cmc) on the system variables, but in contrast to their work in the melt, they did not study salt effects in detail. Wittmer and JoannyZ0more recently, following Marko and Rabin's work,I9 determined the micellar phase boundary for the case of weak charging (no counterion binding). Dan and TirrellZ1 developed a scaling model to investigate the micellization and adsorption of strongly charged block copolymers at high salt concentrations where electrostatic effects are no longer dominant. A model developed by Gao and Eisenberg22investigates the effect of the length of the hydrophobic block on the cmc only and not on the micellar aggregation numbers; the effect of the length of the polyelectrolytic block is not studied at all. On the other hand, work done by the group of Webber23 provides experimental evidence for the effect of the length of the charged block on the aggregation number without addressing the effect on the cmc. There is, therefore, opportunity for improving on these models by studying more systematically the effects of polymer charge, hydrophobic length, and salt concentration on the micellar phase boundary, aggregation numbers, and cmc's of amphiphilic block polyelectrolytes. Present address: School of Chemistry and Molecular Sciences, University of Sussex, Falmer, Brighton, E. Sussex BN1 9QJ, U.K. Abstract published in Advance ACS Abstracts, November 1, 1995.
As a fust step toward this direction, we present in this work the results of a simple model appropriate for amphiphilic molecules bearing a dodecyl hydrophobic tail and a multiply charged head comprising 1- 12 charges. Our approach involved the expansion of Tanford's seminal work24,25 on the micellization of monovalent ionic surfactants to surfactants that bear more than one charge per chain. We used, therefore, an empirical expression to calculate the hydrophobic driving force of micellization and applied the Debye-Hiickel theory to estimate the electrostatic repulsion between the surfactants within the micelle considering the partial neutralization of the micellar charge by bound counterions. It is worth mentioning that Tanford's model has already been expanded, refined, and applied for the description of the micellization of nonionic, zwitterionic, and ionic surfactants and surfactant mixtures by Nagarajan and R u ~ k e n s t e i n ~and ~ - ~Blankschtein ~ and colleague^.^^-^^
Theory According to Tanf0rd,2~,~~ the distribution of micelles is given by the equation
In X , = -m-
Agm
RT
+ m In X , + In m
where the activity coefficients of micelles and free surfactant chains were set equal to 1, X, is the mole fraction of surfactant chains within micelles of size m, XI is the mole fraction of free surfactant chains, R is the gas constant, T is the absolute temperature, and Ag, is the free energy of transfer of a single amphiphilic molecule from the monomeric state to a micelle of size m, more commonly called the free energy of micellization.30,33The latter consists of a hydrophobic part, Agmh9and an electrostatic part, Agmel:
The hydrophobic part originates primarily from the favorable free energy of transfer of the alkyl tail from the aqueous phase to the micellar phase and, secondarily, from the unfavorable residual interfacial contact of water with the tails within the micelle. T a n f ~ r d ~gives ~ . * ~the following expression for the hydrophobic part of the free energy of micellization in cal/mol
0022-3654/95/2099-17437$09.00/0 0 1995 American Chemical Society
Patrickios
17438 J. Phys. Chem., Vol. 99, No. 48, 1995
Ag,h = -2100-700(nc
+ 25(A,h - 21)
- 2)
(3)
where the first two terms on the right-hand side describe the energy for the tail transfer to the micelle and the last term describes the energy for the interfacial water-tail contact within the micelle. nc is the number of carbon atoms in the alkyl chain, and Amhis the effective hydrophobic micellar surface area per chain in A2 calculated 3.25 8, away from the hydrophobic core to account for surface undulations:25
+
A,h = (l/m)4n(rC 3.25)2
(4)
where r, is the radius of the micellar core. Since we will be focusing on dodecyl chain surfactants, nc equals 12 and r, was estimated as 11.56 Equation 3 is thus simplified to
-9625
1 + 1.15,/Cs,, + 68906 + 136.27m(1 - p)2 z2 m 1 + 6.27&
(9)
Substitution of eq 9 in eq 1 leads to an expression describing the distribution of micelles as a function of the degree of counterion binding, p, the surfactant charge, z, the salt concentration, Galt, and the mole fraction of free surfactant chains, XI, which at 25 "C becomes
In Xm = 16.250m - 116.34 1
0.23007m2(1 - /3)2 z2 1
+ 1.15,/Cs,,, + m l n X , + l n m + 6.27&
(10)
The aggregation number of the most abundant micellar species, m*, corresponding to the maximum of X,, can be determined Agmh= -9625 689061m (5) by differentiating eq 10 with respect to m and setting the derivative equal to 0. Substituting the values of Csalt,,8, and z, The electrostatic part of the free energy of micellization arises and an assumed value for XI, the resulting quadratic equation from the repulsion between the ionic heads within the micellar can be easily solved form. A tentative distribution of micelles shell. Using the Debye-Huckel approximate solution to the can be obtained by inserting the calculated value of m in eq 10. Poisson-Boltzmann equation and assuming that the charge is The correct distribution of micelles, the true aggregation number accumulated on the outer surface of the micellar core and not of the most abundant species, m*, and the correct value for the distributed over the volume of a spherical shell of finite mole fraction of free surfactant chains, XI, can be calculated thickness, the electrostatic intramicellar free energy of interaction iteratively until a certain criterion for micelle formation is per mole of surfactant chains can be calculated from24,25,34,35 satisfied. The latter is usually arbitrarily set to be the requirement that 5% of the total concentration of surfactant chains Agmel= (1 - B)2z2e2NAv(r, 4) 1+ K a reside within the micelle^:^^^^^ (6) 1 K a K(r, 4) 2eAl: m 1
+
+
+ +
+
where p is the degree of counterion binding to the ionic head,
z is the valence of the ionic head (surfactant charge), e is the electron charge, NAV is Avogadro's number, E is the dielectric permittivity of water, K is the reciprocal Debye length, a is the radius of small electrolyte ions in solution, and r, 4 is the effective radius of the micellar shell, taken slightly greater than the effective radius of the micellar core. Ame1is the average surface area of the sphere that the ionic head groups define divided by the aggregation number and calculated 4.0 8, away from the hydrophobic core:25
+
+
Ame' = ( 1/m)4n(rc 4)'
(7)
The utilization of the Debye-Huckel approximation in the above derivation restricts the applicability of eq 6 to situations of low micellar electrical potential, i.e. low aggregation numbers and moderate to high salt concentrations. This limitation is partially raised by the introduction of the degree of counterion binding, which is a well-known phenomenon in ionic surfactants and polyelectrolyte^^^ and leads to a decrease in the micellar electrical potential and renders the Debye-Huckel approximation more reasonable. For the dodecyl surfactants of interest, and for an average hydrated radius of mobile ions of 3.5 8, at 25 "C, eq 6 becomes
':gA
= 136.27m(1 -
z2 1
+ 6.27&
(8)
where Csdtis the molar concentration of uni-univalent electrolyte and Agmelis in cal/mol. When eqs 2, 5 , and 8 are combined, the total free energy of micellization can be expressed as
cx, 2
1
= -x, 19
Furthermore, the cmc is calculated (in m o a ) as the total surfactant concentration under the conditions at which eq 11 is satisfied: m
cmc = 5 5 . 5 6 c X m 1
where 55.56 m o a is the inverse molar volume of water, which is the main component of the system. The above theory was programmed in GWBASIC, and the calculations were performed on a PC.
Results and Discussion Figure 1 shows the micellar phase diagram in the chargeionic strength space for dodecyl tail surfactants whose charge has been neutralized 61% by the binding of counterions. The degree of neutralization of the micellar charge by the binding of oppositely charged ions is always high, and it can reach 90%.35 The diagram suggests that a low surfactant charge (valence) and a high salt concentration favor micellization because they suppress interchain intramicellar electrostatic repulsion. Wittmer and Joanny20 produced a similar micellar phase diagram for amphiphilic block polyelectrolytes in the absence of counterion binding where the micellar phase shrank with the 0.75 power of the length of the charged block. Lowering the salt concentration below 0.001 M does not change appreciably the salt concentration factor in the electrostatic free energy expression (eq 8), and therefore, it has little effect on the micellar phase behavior. Surfactants with a charge of 4 or lower form stable micelles which do not break up even in the absence of salt. Surfactants
Micellization Model for Multivalent Ionic Surfactants
J. Phys. Chem., Vol. 99, No. 48, 1995 17439
& n E
2
unimers
(a)
100 :
C
.-0 m
4
micelles
3
107
io*
icr5
104 IOJ IO-*
.j
. ,, . . . .
10-1 100
UJ
g!
U J
3
10
1
101
Salt Concentration (M) Figure 1. Micellar phase diagram of surfactants with a dodecyl tail and a multivalent head at three different degrees of counterion binding.
with a valence of 5 , 6,7, or 8 form solutions of either micelles or unimers (free surfactant chains), depending on the salt concentration. Surfactants with a charge of 9 or greater cannot form micelles at any salt concentration because the electrostatic repulsion is always stronger than the hydrophobic attraction. The predictions of Figure 1 for micellar destruction in aqueous solutions ’of multivalent surfactants at low ionic strengths are consistent with the experimental findings of Patrickios et al.,I4 who could not detect micelles in solutions of their lowmolecular-weight ABC triblock polyampholytes at extreme pH, and with those of Selb and Gallot? who could only detect unimers in low salt concentration solutions of their amphiphilic polycations. The broken and dotted lines in Figure 1 define the boundaries between unimers and micelles for degrees of counterion binding of 85% and 0%, respectively. When the surfactant charge is extensively (85% binding) neutralized by counterions, the micellar region grows at the expense of the unimer region (compared with the case of 61% counterion binding) because the electrostatic repulsion, which opposes micellization, is reduced. In contrast, when no ions (0% binding) bind to the surfactant molecules, the unimer region expands because the electrostatic repulsion is very pronounced. The resulting balance of attractive and repulsive energies in the absence of counterion binding allows only univalent surfactants to form micelles over the whole range of salt concentration and permits divalent and trivalent surfactants to form micelles at elevated salt concentrations. This is consistent with the behavior of sodium dodecyl sulfate, a monovalent surfactant, which is known to form micelles over the whole range of salt concentration^.^^ Figure 2a shows the effects of salt concentration and surfactant charge on the aggregation numbers of the surfactants that do form micelles in Figure 1. The degree of counterion binding is again assumed to be 61%. Greater aggregation numbers are favored by a high salt concentration and by a low surfactant charge. The curves corresponding to the three lowest salt concentrations are almost indistinguishable from each other because, as discussed above, the electrostatic free energy of micellization is constant for ionic strengths lower than 0.001 M. Figure 2b shows the effects of the degree of counterion binding and surfactant charge on the micellar aggregation numbers at a salt concentration of 0.1 M. Greater aggregation numbers are favored by a high degree of counterion binding and by a low surfactant charge. A very important finding here is that the slope of the straight lines in the double logarithmic plots of Figure 2a,b is equal to - 1, which implies a hyperbolic dependence of the aggregation number upon the surfactant charge (see the Appendix for analytical derivation). This compares very favorably with the experimentally determined exponent of -0.8623 and the exponents of Marko and Rabin’s model’g which ranged from -1.33
10 M salt 0.1 M -
10; 10: 10’M
10
Valence L
al n
s
100
z
10 10
1
Valence
Figure 2. Effect of the head charge on the aggregation numbers of dodecyl tail surfactants (a) at different salt concentrations and at a 61% degree of counterion binding and (b) at different degrees of counterion binding and at a 0.1 M salt concentration.
r 10‘
100 h
5,
10’
0
10 2
5
salt
103
104 1
0
U
5
1
1
1
,
1
1
,
a Valence
1 2 3 4 5 6 7
,
_i I
r
I
,
9101112
10’
100
E
10’
0
10’
5
10’ 104
10
5
a 9101112 Valence Figure 3. Effect of the head charge on the critical micelle concentrations (cmc’s) of dodecyl tail surfactants (a) at different salt concentrations and at a 61% degree of counterion binding and (b) at different degrees of counterion binding and at a 0.1 M salt concentration. I 2 3 4 5 6 7
(low charge density in the polyelectrolyte block) to -3 (high charge density in the polyelectrolyte block). Parts a and b of Figure 3 give the cmc’s that correspond to the aggregation numbers given in parts a and b of Figure 2, respectively. As expected, Figures 2 and 3 show that the cmc’s follow a trend opposite to that followed by the aggregation numbers: A high salt concentration, a low surfactant charge, and a high degree of counterion binding favor a low cmc. The
Patrickios
17440 J. Phys. Chem., Vol. 99, No. 48, 1995
semilogarithmic plots of Figure 3a,b result in straight lines, which suggest an exponential increase of the cmc with surfactant charge (see the Appendix for analytical derivation). Marko and Rabin’s modelIg predicts a 1.33 exponent at the low charge density limit and a variable (from 0.5 to 0) exponent at the high charge density limit (more specifically, at the high charge density limit, it was determinedI9 that log cmc scales as -charge-]). Consistent with these trends, Chen et a1.I’ observed an increase in the cmc with the net charge of their ABC triblock polyampholytes. In contrast, Astafieva et a l l 3 found a decrease, or at best a constancy, of the cmc’s with the length of the charged block of their styrene-sodium acrylate block copolymers. This discrepancy may be due to the very long polyelectrolytic blocks in Astdieva’s polymers for which the theoretical exponent would be very small.lg
-16.250
1.15JCSalt 1 - - (A3) + 0.46014rn(l - /3)2 z211 ++ 6.27& rn
which can be used to eliminate XIfrom eq 10, resulting in In X, = -117.34
We have presented a quantitative theory for the micellization of multivalent surfactants (and, apparently to a good extent, for the micellization of amphiphilic block oligoelectrolytes) which utilizes the Debye-Huckel approximation and an empirical equation for the estimation of the competing forces of micelle formation. It was found that for dodecyl tail amphiphiles and in the absence of binding of counterions only monovalent ionic surfactants form unbreakable micelles; if the degree of counterion binding is as high as 61%, monovalent, divalent, trivalent, and tetravalent dodecyl surfactants form unbreakable micelles, while dodecyl surfactants with a valence of 5 , 6, 7, and 8 micellize only at elevated salt concentrations. Model predictions indicated that the micellar aggregation numbers decrease linearly with the inverse of the surfactant valence, and the cmc’s increase exponentially with the surfactant charge. These trends are in reasonable agreement with the results of more fundamental model^^^^^^ on the micellization of amphiphilic charged block copolymers in solution. Although our model should not be expected to give accurate results for the micellization of amphiphiles with very long (e.g. 1000-unit) hydrophobic or charged blocks (in which case one should resort to the more appropriate theories), it can probably be expanded successfully for the description of the micellization of surfactants carrying up to 30 charged orland hydrophobic units. Appendix: Analytical Derivation of the Dependence of the Aggregation Number, m,and the Critical Micelle Concentration, cmc, upon the Valence, z It is first assumed that the micelle size is monodisperse; this is a reasonable approximationbecause the aggregation numbers are distributed narrowly around the most abundant aggregation number.24 For example, for z = 3, Csdt= 0.1 M, and 61% counterion binding, the most abundant species has a micellar aggregation number of m* = 26; micelles with m = 22 and 30 are 10% as abundant as the most abundant species, and micelles with m = 21 and 32 are only 1% as abundant as the most abundant species. According to this assumption, eq 11 is simplified to
1 xm= -x, 19 and eq 12 becomes 55.56(Xl
+ X,)
20 =55.56X1 19
(A2)
Differentiating eq 10 with respect to m and setting the derivative equal to 0 gives the following expression for XI:
I i- 1 . 1 5 6
+ 0.23007m2(1 -
z2
1
+ 6.27&
+ In m
Equation A4 can be rewritten as 1 0.23W7rn2(1 - p)2 z21
Conclusions
cmc
lnX, =
++ 6.27& 1.15Jcsal, Xm = 117.34 + ln-
rn (A5)
The last term in eq A5 is the translational entropy of the micelles at the cmc, it is much smaller than.117.34, and for a given z, it is relatively insensitive to Galtand the degree of counterion binding. For z = 3, for example, the translational entropy term equals -16.40 if Csdt= 0.1 M at 61% counterion binding, it acquires a maximum value of -8.77 if = 10 M at 0% counterion binding, and it reaches a minimum value of -24.6 if Csdt= 10 M at 85% counterion binding. It follows, therefore, that the left-hand side term of eq A5 is approximately constant ( a 1 17.34) and consequently
m a l/z
(‘46)
which was observed in Figure 2. Substitution of eq A5 (ignoring the translational entropy term) into eq A3 gives In XI = - 16.250
+ 233.681rn
(A71
Combining eq A7 with eq A6 results in In X, = - 16.250
+ (const)z
(A8)
or, using eq A2 ln(58.48cmc) = -16.250
+ (const)z
049)
which was one of the findings of Figure 3. References and Notes (1) Grosius, P.; Gallot, Y.; Skoulios, A. Makromol. Chem. 1970, 132, 35-55. (2) Selb, J.; Gallot, Y. J. Polym. Sci., Polym. Lett. Ed. 1975,13,615619. (3) Selb, J.; Gallot, Y. Makromol. Chem. 1980, 181, 809-822. (4) Selb, J.; Gallot, Y. Makromol. Chem. 1980, 181, 2605-2624. (5) Selb, J.; Gallot, Y. Makromol. Chem. 1981, 182, 1513-1524. (6) Selb, J.; Gallot, Y . Makromol. Chem. 1981, 182, 1491-1511. (7) Cao, T.; Munk, P.; Ramireddy, C.; Tuzar, Z.; Webber, S. E. Macromolecules 1991, 24, 6300-6305. (8) Prochazska, K.; Kiserow, D.; Ramireddy, C.; Tuzar, Z.; Munk, P.; Webber, S. E. Macromolecules 1992, 25, 454-460. (9) Kiserow, D.; Prochazska, K.; Ramireddy, C.; Tuzar, Z . ; Munk, P.; Webber, S. E. Macromolecules 1992, 25, 461-469. (10) Kiserow, D.; Chan, J.; Ramireddy, C.; Munk, P.; Webber, S. E. Macromolecules 1992, 25, 5338-5344. (1 1) Desjardins, A,; Eisenberg, A. Macromolecules 1991, 24, 57795190. (12) Gao, Z.; Desjardins, A,; Eisenberg, A. Macromolecules 1992, 25, 1300-1303. (13) Astafkva, I.; Zhong, X. F.; Eisenberg, A. Macromolecules 1993, 26, 1339-1352.
J. Phys. Chem., Vol. 99, No. 48, 1995 17441
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