2144
J . Phys. Chem. 1984, 88, 2144-2149
These results prompt a reexamination of the results of the study of the field-dependent relaxation rates in these system^.^^* If one can assume the correlation time determinations made in those measurements remain approximately valid even though these results arise from a mixture of components, the rotational rates can be used to calculate the values of d,, the distance between the interacting spins in the complex. The results for CF3CH20H are d,(OH) = 0.28 nm, dr(CH2) = 0.46 m, d,(CF3) = 0.52 nm, and for C6F,0H, dr(OH) = 0.25 nm. This trend in interaction distances is far more acceptable for a nonlinear complex formed between the nitroxide group of the spin probe and O H proton on an unbranched alcohol that is free to rotate, than the initial results reported previously. These suffered from the inability to separate the two contributions to the dipolar relaxation because the mechanism had to be deduced solely from the shape of the spectral density curve. These data for each nucleus were analyzed previously as if it were all of one mechanism or the other. An additional interesting comparison to make involves the different translational rates for the nuclei studied. Clearly the rate is largest for the O H protons on each of the proton donors, with the fluorine rate being considerably smaller. The larger O H proton rates may arise from the proximity of the O H protons to the unpaired spin probe electron as the complex is broken and the two species begin to diffuse apart, an observation consistent with previously hypothesized increased dipolar coupling in samples with significant scalar component^.'^ This would not account for the decreased fluorine component. However, though the smaller yz term for fluorine nuclei has a limited effect, the reduced accessibility of the electron-rich site on the spin probe to the larger CF, group may result in the diminished rate in this case, or the preferred orientation of the alcohol in solvating the spin probe may inhibit the ability of the F to approach closely. (19) E. H. Poindexter, P. H. Caplan, B. E. Wagner, and R. D. Bates, Jr., J . Chem. Phys., 61, 3821 (1974).
Conclusion Several important points result from the work described in this paper. First, the use of variable mole fractions proves to be an effective technique by which to separate rotational and translational components of dipolar interactions between unpaired electron and nuclear spins. The most effective means by which to characterize the nature of the transient complex between the two species is in samples of low proton donor mole fraction; the translational component is more accessible in samples of high donor mole fraction. Care in selecting the inert cosolvent is necessary, with a solvent of similar viscosity most appropriate to minimize changes in correlation times and thus position on the spectral density curve as the solvent mixture is varied. For the OH protons on both CF3CH20Hand C6F50H, the rotational motion of the complex is clearly the dominant dipolar relaxation mechanism. The CH2 protons and the fluorines on the alcohol have much smaller rotational components, a feature ascribed to the much longer interaction distances in the transient complex, such that the translational motion of the free solvent and solute species becomes the dominant dipolar component in solutions of high alcohol mole fraction. This method of separating the major components of the motions responsible for the paramagnetic species-induced relaxation of solvent nuclei when used in conjunction with studies of these rates as a function of magnetic field promises to be an important means by which to characterize the molecular dynamics of solvent-solute interactions. Acknowledgment. The authors gratefully acknowledge the partial support of this work by N I H Grant 1-R01-GM23670 and by N S F instrumentation Grants CHE76-05887, GP-8541, and GP-29184. The authors thank Gerry Franklin for helpful discussions and Ruthann Bates for reading the manuscript. Registry No. CF,CH20H, 75-89-8; C6F50H,77 1-61-9; tempone, 2896-70-0.
Sphere-Rod Transition of Surfactant Micelles and Size Distribution of Rodlike Micelles Shoichi Ikeda Department of Chemistry, Faculty of Science, Nagoya University, Chikusa, Nagoya 464, Japan (Received: July 15, 1983; In Final Form: September 26, 1983)
A theory of the salt- or temperature-inducedsphere-rod transition of surfactant micelles and the reversible linear aggregation of surfactant into rodlike micelles has been developed on the basis of a simple treatment of the law of mass action, and its implication is given in terms of statistical thermodynamics. The sphere-rod conversion is described by the introduction of an initiation parameter, which is strongly dependent on the salt concentration and temperature near their threshold values. The stepwise association of surfactant molecules or ions beyond the spherical micelles is analyzed by considering the configurational degeneracy of rodlike micelles. The association constant of monomer addition to the rodlike micelles is found to be inversely proportional to the aggregation number, in contrast to the equal association constant for isodesmic association. The present theory leads to the Poisson distribution of length of rodlike micelles, whose maximum is located at an aggregation number, independently of the surfactant concentration.
Introduction Surfactant molecules or ions aggregate reversibly into micelles above a certain critical concentration in aqueous solutions. Ionic surfactants generally form spherical or globular micelles in solution, but they can further aggregate into rodlike or wormlike micelles with increasing surfactant concentration, when a simple salt is present beyond a certain threshold concentration.’ In the presence of excess concentrations of salt, the electrostatic effect
of charged micelles would be sufficiently suppressed. Indeed, nonionic surfactants having a weakly hydrophilic group or a highly hydrophobic group can also form rodlike rnicelle~,23~ possibly above a certain threshold temperature. Spherical micelles are formed cooperatively at the critical micelle concentration; the hydrophobic groups of molecules or ions are incorporated into a micelle in such a way that they are exposed to aqueous media to the least e ~ t e n t . The ~ ~ spherical ~ micelle
(1) S. Ikeda, “Surfactants in Solution”, Vol. 2, B. Lindman and K. L. Mittal, Eds., Plenum Press, 1983, p 825.
(2) P. Elworthy and C. B. M. Macfarlane, J . Chem. SOC.,907 (1963). (3) D. Attwood, J . Phys. Chem., 72, 339 (1968).
0022-3654/84/2088-2144$01.50/0
0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 10, 1984 2145
Surfactant and Rodlike Micelles would be fairly monodisperse in size, because of its geometrical requirement. Rodlike micelles should be subject to a similar geometrical requirement in their cross section, but their contour length must be determined in a statistical manner, because the free energy of a rodlike micelle is proportional to its length if the end effect is negligible. Formation of rodlike micelles has been treated theoretically as a "linear aggregation" of surfactant molecules or ions, assuming an isodesmic or a quasi-isodesmic association, depending on whether the critical micelle concentration is absent or present.6-'0 This theory predicts considerable polydispersity for the rodlike micelles. However, we have experimentally demonstrated that the rodlike micelles are rather monodisperse in length. 1~11-14 In the present work we give a theory of the reversible linear aggregation of surfactant on the basis of the successive association model and show that the linear aggregation should lead to rather homogeneous aggregates, Le., monodisperse rodlike micelles. We assume, for simplicity, that a solution is ideal, except for the association of solute species.
Association-Dissociation Equilibria of Micelles For a stepwise aggregation of a molecular species, D, a series of chemical equilibria D,-l
+ D s Di
i = 2, 3,
...
(1)
should hold,6 the equilibrium constants of which are given by ci --
C,-ICl
- K,
i = 2, 3,
...
(2)
where Ci and Cl are the molar concentrations of the i-mer, D,, and monomer, D, respectively. In the isodesmic association the equilibrium constants are equal, independently of the association step, i:
(
Ki = K = exp -LT)
(3)
where g is the standard free energy of stepwise association, R is the gas constant, and T is the temperature. In order to adapt the scheme of stepwise aggregation to a micellar solution of surfactant, D, we consider the micelle formation in three stages: formation of spherical micelles, sphererod conversion, and growth of rodlike micelles. At a given salt concentration and temperature, spherical micelles are formed above the critical micelle concentration, and their number generally increases with increasing surfactant concentration. If the salt concentration or temperature is higher than the threshold value, rodlike micelles are also formed above the critical micelle concentration, instead of the number of spherical micelles simply increasing, and the number of rodlike micelles increases with increasing surfactant concentration. The observed sphere-rod conversion appears as a transition when the salt concentration of temperature is altered, and it occurs as a shift of association equilibrium when the surfactant concentration is varied, since the monomer concentration remains constant."-I8 Theoretically, the (4) H. V. Tartar, J . Phys. Chem., 59, 1195 (1955). (5) C. Tanford, J. Phys. Chem., 76, 3020 (1972). (6) J. M. Corkill, J. F. Goodman, and T. Walker, Trans. Faraday Soc., 63. --, IS9 .. (1967). ,-(7) P. Mukerjee, J . Phys. Chem., 76, 565 (1972). (8) R. J. M. Tausk and J. Th. G. Overbeek, Biophys. Chem., 2, 175 ( 1 974) (9) P. J. Missel, N. A. Mazer, G. B. Benedek, C. Y. Young, and M. C. Carey, J . Phys. Chem., 84, 1044 (1980). (10) G. Porte and J. Appell, J . Phys. Chem., 85, 2511 (1981). (1 1) S.Ikeda, S. Ozeki, and M. Tsuncda, J . Colloid Interface Sci., 73, 27 (1980). (12) S. Hayashi and S. Ikeda, J. Phys. Chem., 84, 744 (1980). (13) S. Ikeda, S. Ozeki, and S . Hayashi, Biophys. Chem., 11,417 (1980). (14) S. Ikeda, S.Hayashi, and T. Imae, J . Phys. Chem., 85, 106 (1981). I
\--
'
sphere-rod conversion of micelles should be treated in terms of a series of stepwise increases in micelle size, Le., a micelle growth?P8 We will follow this approach. The formation of a spherical micelle can be expressed by mD
~ r D, !
(4)
where the m-mer, D,, is the spherical micelle, whose aggregation number is mainly determined by the geometry of the surfactant molecule or ion and by the balance between hydrophobic interaction of hydrocarbon parts and electrostatic repulsion or hydration of polar head groups. The equilibrium constant is given by (5)
where g, is the standard free energy of micelle formation per mole of surfactant. The sphererod conversion of micelles and the growth of rodlike micelles are both represented by a series of chemical equilibria D,,
+ D + D,
r
>m
(6)
but their equilibrium constants must be considered separately. We assume that an embryo or a nucleus of rodlike micelles is formed by the cleavage of a spherical micelle into two hemispherical parts and the following insertion of some definite number of molecules or ions. The number of such nucleation steps would be equal to the number of molecules or ions in a cross-sectional layer or two of a rodlike micelle, which would be about 30-50 for usual surfactants, and it will be designated as t. Then, the equilibrium constant of the initiation of sphere-rod transition is written
--c, - uK, Cr-lCt
m C r Im
+t
(7)
where K, is the growth or propagation constant of a rodlike micelle, as will be defined below. The initiation or nucleation parameter, u, is generally smaller than unity, since the cleavage of a spherical micelle is much more difficult than the growth of rodlike micelles, and it is assumed to be independent of the nucleation step, r. The initiation parameter is a function of salt concentration and temperature and sharply changes near their threshold values: it is very small below the threshold values but largely increases across them. The equilibrium constant of the successive growth of rodlike micelles is given by
- -Cr CPlCI
-K,
r>m+t
(8)
If the end effect is not taken into account, the intrinsic free energy of stepwise association should be equal for all the steps. When the addition of a monomer takes place at an end or both of an (r - 1)-mer, the condition of equal free energy for monomer addition is equivalent to that of equal association constant. This is the isodesmic association, eq 3, for the micellar growth beyond the spherical micelle. The quasi-isodesmic association of micelles leads to a broad size distribution of rodlike micelles, which reduces to the most probable or random distribution in the limiting case of the isodesmic condition. This mechanism has been proposed by many ~ ~ r k e r ~and~ prevails - ~ ~ *over ~ ~the, aggregation ~ ~ mechanism of surfactants in rodlike forms and of globular proteins in fibrous forms.
.I'
(15) S.Ozeki and S . Ikeda, J . Colloid Interface Sci., 87, 424 (1982). (16) P. Elworthy and C. McDonald, Kolloid 2.2.Polym., 195, 96 (1964). (1 7 ) R. R. Balmbra, J. S.Clunk, J. M. Corkill, and J. F. Goodman, Trans. Faraday SOC.,60,979 (1964). (1 8) R. H. Ottewill, C. C. Store, and T. Walker, Trans. Faraday Soc., 63, 2796 (1967). (19) F. Oosawa and M. Kasai, J . Mol. Biol., 4, 10 (1962). (20) F. Oosawa, J . Theor. Biol., 27, 69 (1970).
2146 The Journal of Physical Chemistry, Vol. 88, No. 10, 1984
It proves, however, that the addition of a monomer to an end or both of the rodlike micelles is not a unique mechanism of its growth. Reversible aggregation of rodlike micelles takes place by the insertion of a monomer at any site of an (r - 1)-mer, either between two units or at both ends, with an equal intrinsic free energy. A rodlike micelle consists of two parts, one being the two hemispherical end caps and the other being the central cylindrical part. Each end cap would have a structure similar to that of spherical micelles, so that they may not have any more room available for incorporating extra molecules or ions. Then, the micellar growth should take place by the addition or insertion of molecules or ions at the central part of the rodlike micelles and cause an increase in the length of that part. We assume that each hemispherical end cap contains m/2 molecules or ions. Then, the central cylindrical part of an (r 1)-mer consists of r - m - 1 molecules or ions, and it has r - m sites capable of incorporating a molecule or ion, if they are arranged linearly. We may imagine that the central part consists of a helical array or of a circular bundle of linear arrays of molecules or ions. In the former, the number of available sites is r - m. In the latter, the number of linear arrays will be about 15-25 for usual surfactants, and then the number of available sites would be r - m less this number. The incorporation of a molecule or ion into any of r - m sites on an (r - 1)-mer leads to an indistinguishable r-mer, and each addition reduces the free energy per mole equally by g,, independent of the position of the site. (Here, g, has a much larger negative value than g,.) That is, there are r - m different ways of monomer addition leading to an r-mer, or there are r - m distinguishable (r - 1)-mers having a single vacant binding site, each having an identical intrinsic association constant. Then, each (r - 1)-mer has an identical intrinsic association constant for monomer addition, and the concentration of such a distinguishable (r - 1)-mer is equal to one another and is C Jr(., - m). This situation is similar to the final binding of a small molecule to a macromolecule having r - m identical binding sites, of which r - m - 1 sites have already been occupied by the small molecules .21,22 If the intrinsic association constant of monomer addition is expressed by k, = exp( -
5)
(9)
Ikeda this work we will be only concerned with the stage of growth of rodlike micelles: s = t or r 1 m t . Since the total molar (equivalent) concentration of surfactant, C, is given by
+
we have the relation
C - C1 = [ ( l - uf)m+ ur(m + k,Cl)ek~c~]C, (13) Thus, the molar concentration of spherical micelles is written
c, = m(1 +c u- )c+1 k,CI (e-kcCI + u )
(14)
where
Substituting eq 14 into eq 11, we have the Poisson distribution of molar concentration of rodlike micelles
c - c1
c, = m(l + u ) + k,C1
e-kcC1(kcCL)rm (r - m)!
r l m + t
(15)
which has a maximum at rmax= m
+ k,C1
(15a)
If k,C1 is larger than 1, it is seen from eq 8 and 10 that a maximum molar concentration should exist at r,,,, satisfying the condition k,Cl/(r - m) = 1, which is in accord with eq 15a. We can see that the aggregation number for the maximum micellar concentration does not change with the surfactant concentration, if the monomer concentration is constant. This condition is fulfilled at surfactant concentrations exceeding the critical micelle concentration. As k,C1 becomes larger, the maximum is sharper in the distribution of molar concentration of rodlike micelles. Then, the Poisson distribution, eq 15, can be well approximated by the Gaussian distribution
c - c1 1 cr = m(1 + u ) + k,C1 (2ak,C1)1/2 exp[-(
1
r - m - k,C1)2 2k,C1
independently of r, then the association constant of growth of rodlike micelles is given by
(16) which is centered at rmaxand has a half-width of (kcC1)1/2.23 The number- and weight-average aggregation numbers of micelles, including spherical micelles, are obtained as
where k, has a large value corresponding to a large negative value of gc. Equation 10 can also be derived by kinetic reasoning. The forward rate constant of reaction 6 is mainly determined by the diffusion coefficient of the monomer and is, therefore, independent of r. However, the reverse rate constant of reaction 6 should be proportional to r - m, since the reverse reaction can occur by the removal of a molecule or ion from any of r - m sites of the r-mer, and its rate is then r - m times faster than the deletion of a molecule or ion from a single site. At equilibrium, the forward rate is equal to the reverse rate, which leads to the association constant inversely proportional to r - m or to eq 10. Solving eq 8 and then eq 7 successively, we have the molar concentration of rodlike micelles (k,CJrrn C, = 8C, (r - m)! where s = r - m for m 5 r 5 m
r 2 m
+ t and s = t for r 2 m + t .
In
(21) I. M. Klotz in “The Proteins”, Vol. I, Part A, H. Neurath and K. Bailey, Eds., Academic Press, New York, 1957, Chapter 8, p 748. ( 2 2 ) C. Tanford, ‘Physical Chemistry of Macromolecules”, Wiley, New York, 1961, Chapter 8, p 532.
M” ukcC1 _ - m + k,C1 - -
M1
- = m + k,C1 M
W
Ml
l + U
(mu m(l
- l)k,C1
+ u ) + k,C1
(17b)
respectively, where M,, and M , are the number- and weight-average molecular weights of micelles, and M I is the molecular weight of monomer or surfactant molecules. If k,C, is large, only the first two terms have a significant value, leading to the monodispersity of micelles: M w / M n= 1. Size Distribution of Micelles Below the threshold salt concentration or temperature, a surfactant can exist either as a monomeric form or as spherical micelles. The molar concentration of spherical micelles is obtained from eq 5 and 14 with k, = 0. Above the threshold salt concentration or temperature, a surfactant can exist in three forms: as monomers, spherical micelles, or rodlike micelles. The molar concentration of spherical (23) S . Chandrasekhar, Reu. Mod. Phys., 15, 1 (1943).
The Journal of Physical Chemistry, Vol. 88, No. 10, 1984 2147
Surfactant and Rodlike Micelles I
I '
'
t
tween spherical and rodlike micelles. For the latter (b) we have chosen the condition in such a way as to give C, = C, for which ul = (27rm(n - l))1/2e-m(w1) holds. In this case the relative distribution of micelles changes with total concentration, as given by eq 14 and 15, or, for r = mn, by eq 5 and 20. With an increase in surfactant concentration beyond the critical micelle concentration, the sphere-rod equilibrium shifts from the spherical micelle side to the rodlike micelle side.
I
i
a
I!
1.2
Partition Functions of Micellar Solutions In order to elucidate the mechanism of "linear aggregation" of surfactant molecules or ions and the implication of the present results being different from the previous ones,7-10,19,20 we consider the micellar equilibria, eq 4 and 6, by statistical thermodynamics. At a given temperature T, there are N surfactant molecules in an aqueous solution of volume V, among which N 1 molecules are as monomers, m N , molecules are as N , spherical micelles, and rN, molecules are as N , rodlike micelles, where ( r ) > m . Assuming that surfactant molecules are distinguishable, the number of their configurations in solution is found to be25-27
t
0.4
ll
h bl
0.8
Q(Ni,Nm,Wrl) = N! N , ! (m!)NpN,!n(r!)NrN,!
r!
I4
r Figure 1. Relative size distribution of micelles (C,/C, vs. log r ) for m = 100 and n = 10: (a) ut = 1 or u = 0; (b) ut = 7.52 X or u = 0.0085 and Clo= Clm.
micelles is given by eq 5 and 14, and that of rodlike micelles can be calculated by eq 15 or 16. Experimentally, we have demonstrated that the micellization of surfactant and the sphere-rod conversion of its micelles proceed as
D,
(1Sa)
nD, s D,,
(18b)
mD
F?
when the salt concentration or temperature exceeds the threshold ~ a l u e . l ~ l ~ Below - ~ ~ ~the * ~ critical micelle concentration, only monomers exist. Above the critical micelle concentration, both spherical and rodlike micelles, D, and D,, are present, while the monomer concentration remains essentially constant and is equal to the value of critical micelle concentration. The observed monodispersity of rodlike micelles is consistent with the Poisson distribution of rodlike micelles derived here: the sphere-rod equilibrium of micelles, eq 18b, can substitute for a series of stepwise association equilibria of micelles, eq 6. Then, the value of mn is identified with rmaxto give
where we have taken into account the structure of rodlike micelles postulated above. The total number of surfactant molecules is given by N = N1
+ mN, + C r N r
(22)
I4
A molecule has a lower energy in micelles than in monomers. The energy of a spherical micelle is assigned a value me, relative to the free molecules. A rodlike micelle has three contributions of energy: the energy for each initiation step of the sphere-rod transition, 7,the energy of a molecule in hemispherical end caps, e, and the energy of a molecule in the central cylindrical part, e? (Here 0 > em > et.) An embryo or a nucleus of rodlike micelles is composed of s or t molecules intervened between two hemispherical end caps. During the formation of such an embryo or a nucleus, the extra energy, 7,is required for incorporating a molecule into each step, s, where 1 I s It . Then, the total energy of surfactant molecules in solution is given by E(N,,N,,{N,)) = mNmem+ C N , [ s y + me,
+ ( r - m)ecl
Irl
(23)
with s = r - m for r - m I t and s = t for r - m L t. The canonical partition function of aqueous solution of surfactant can be expressed by
kcC1 = m(n - 1) (19) Since mn is usually very large, k,C1 should be very large. The association constant for the sphere-rod equilibrium, eq 18b, is found to be
_ Cmn - K,, Cm"
=
[ m ( n - l)]!
(
(20)
!$),-l
For the moment, we can not estimate the values of d at any salt concentration and temperature above the threshold values, either experimentally or theoretically. Figure 1 illustrates relative size distribution curves of micelles, C,/C, vs. log r, for m = 100 and n = 10, calculated by eq 14 and 16, assuming (a) d = 1 or u = 0 and (b) u' = 7.52 X 10-389.6 or u = 0.0085. For the former (a) we have a negligibly low concentration of spherical micelles: surfactants exist either as monomers or as rodlike micelles. Such a situation is not experimentally true, since light scattering results show the presence of a concentration-dependent equilibrium be(24) T. Imae and
S. Ikeda, Colloid Polymer
Sci., in
press.
where the division by N! corrects the number of configurations for indistinguishable molecules and k is Boltzmann's constant. Here, f l , f,, andf, are the statistical weights of a molecule, a spherical micelle, and a rodlike micelle, respectively, and can be written
fi = qtransqrotqintv f m
= ftrans,mfrot,mqintmv
f r = Lrans,Lot,rqin(V
(25a) (25b) (25c)
where qtrans,ftranS,,, and fva,, represent the translational partition functions of a molecule, a spherical micelle, and a rodlike micelle, qrol,frot,,,andf,,, stand for the corresponding rotational partition (25) C.A. J. Hoeve and G. C. Benson, J . Phys. Chem., 61, 1149 (1957). (26) D. Poland and H. A. Scherage, J . Phys. Chem., 69, 2431 (1965). (27) R. Nagarajan and E. Rwkenstein, J . Colloid Interface Sci., 60, 221 (1977).
2148
The Journal of Physical Chemistry, Vol. 88, No. IO, 1984
functions, and qintis the internal partition function of a molecule other than the rotational part. If we take the maximum term of the canonical partition function, we can obtain the distribution of molecules and micelles, Le., the most probable values of N,, N,, and INr). Instead of doing this, we suppose that the maximum term approximation has been done, but we keep using the same symbols for N , , N,, and IN,). Then the canonical partition function is expressed by Z(T,V,N) =
-n-
elN1 emNm
-
Nl! N,!
QrNr
(rl
(26)
N,!
where Q,,Q,, and Qr are the partition functions of molecules, spherical micelles, and rodlike micelles in solution, respectively, and are (274
QI =fi
r
Qr
=
Jr
e-[sy+mr,+(rm)r,]
JkT
[(m/2)!I2(r - m)!
+
(27c)
+
withs = r - m for r 5 m t and s = t for r L m t . Statistical thermodynamics gives the law of mass action in the form that the equilibrium ratio of numbers of particles is equal to the corresponding ratio of their partition functions. Thus, we have Nm -.=Ni"
Qm
Ikeda solution. The isodesmic association such as represented by eq 3 or the quasi-isodesmic association for rodlike micelles cannot be true for the reversible aggregation, unless any specific interaction favors the monomer addition to either end or both. The statistical weight, eq 21, or the partition function, eq 27c, reveals that the configurational degeneracy of rodlike micelles, [ ( r- m)!]-', is important for the reversible aggregation. Without this factor, the equal energy for the stepwise addition of monomer is equivalent to the equal association constant for each monomer addition. This situation is discussed more clearly in the Appendix. Furthermore, eq 27c leads to the standard chemical potential of rodlike micelles, as given by plo
= R T In [(m/2)!I2
+ NAme, + NAsy + NA(r - m)e, R T In Lfr/(r - m)!] (35)
. Equation 35 differs in having the last term from the corresponding expression for the energy ladder with an equal rung spacing9 derived from the isodesmic association. The present theory predicts the Poisson distribution for rodlike micelles, while the previous theory7-I0yields polydisperse rodlike micelles. Although it is difficult to obtain the size distribution of rodlike micelles experimentally, we have evidence for their monodispersity by deriving the values of Mw/Mnfrom the concentration dependenceI4 and angular dependencez4 of light scattering in different surfactant solutions. To the monodisperse approximation, eq 18b or 19, the aggregation number of rodlike micelles is determined as + qinte-dkT
mn = m (28)
Qim
for the micellization in spherical forms, eq 4, and
+
with Qpl = Q, for r = m 1, for the sphere-rod equilibria and the growth of rodlike micelles, eq 6. Converting the number of particles into the molar concentration, Ci = NiV/NA, by introducing Avogadro's number, N A , we can derive the association constants
and the initiation parameter
By means of eq 25, the association constants are readily estimated in terms of molecular parameters. The association constant for spherical micelles reduces to
In the present theory the effect of solvent molecules has not been explicitly taken into account, but it is supposed to be implicitly included in the energy terms. The introduction of the effects of solvent molecules in the statistical treatment will be better made by means of the quasi-lattice model rather than of the present gas model. In the present treatment the spherical micelles have been assumed to be monodisperse. However, some deviation from the moncdispersity would exist for the size of spherical micelles. Then, the size distribution of rodlike micelles is a superposition of the corresponding Poisson distribution, but it is likely that the distribution is also of the Poisson type. Finally, it will be relevant to note that the present theory of reversible linear aggregation is general and can be applied for systems other than rodlike micelles of surfactant. They may include micellization of oligopeptides in rodlike forms,28formation of colloidal particles in cigarlike shapes,z9 and even association of globular proteins in fibrous forms. However, the flexibility of linear aggregates might be responsible for the exchangeability of constituent units.
Appendix Consider N distinguishable units which are divided into N1piles having one unit, Nz piles containing two units, ...,N, piles composed of i units, etc. The total number of units are defined by N =
For rodlike micelles, it can be seen that ft,a,,,r a r3JZ,and furthermore,j& a r3or 3, depending on their shape, either rigid rods or unperturbed random coils. Then, the ratio, f r / f r l , is approximately equal to unity. This means that eq 31 is in accord with eq 10, and the intrinsic association constant for rodlike micelles can be expressed by
k, =
NA
qtransqrot
e-t,JkT
(34)
Discussion We have shown that the equilibrium constant for the stepwise addition of monomer, eq 10, is fundamental for the reversible linear aggregation of surfactant molecules or ions in rodlike micelles in
(36)
i
iN,
(A- 1)
but the total number of piles is not necessarily definite. If the units are nonlocalized in each pile, the number of their configurations is N! a= (A-2) rI(i!)h'iNi! i
Equation A-2 corresponds to the case in which interchange of units in each pile can occur, and therefore, it describes the reversible aggregation if the units are arranged linearly in each pile. The association equilibria, eq 1, can be attained through the insertion of a monomer into an aggregate and its deletion from the ag(28) T. Imae, K. Okahashi, and S. Ikeda, Biopolymers, 20, 2553 (1981). (29) Y. Maeda and S. Hachisu, Colloids Surf., 6, 1 (1983).
J. Phys. Chem. 1984,88, 2149-2155 gregate. For such a reversible aggregation, the total number of piles is not specified, and the canonical ensemble of the systems defined by temperature, T, volume, V, and number of units, N , is more suitable for the statistical treatment. This has been pursued in the text and derives the Poisson distribution of aggregate size. On the other hand, if the units are localized in each pile, the number of their configurations is given by30 Q=-
N! Ni!
n:
(‘4-3)
i
Equation A-3 corresponds to the case in which interchange of units no longer occurs within a pile, once they are incorporated into it. This describes the irreversible polymerization if the units are arranged and linked linearly in each pile. Then, the polymerization equilibria, eq 1, must be reached only through the addition of a monomer to an end or two of a polymer and its deletion from there. In this case the extent of reaction, p , is defined, which proves to be equal to KC1 and eventually specifies the total energy of the system: (30) W. H. Stockmayer, J . Chew. Phys., 1 1 , 4 6 (1943).
2149
where c is the localization energy of a unit in a pile with i > 1 or the bond energy between two units in a pile. If the total number of piles is expressed by M = CNi 1
(‘4-5)
then the total energy of the system is given by E=(N-M)c
(‘4-6)
Thus, the constant total energy is equivalent to the constant total number of piles, because of eq A-1. Then, it is ready to see that M = N(l - p )
(A-7)
E = Npc (A-8) For such an irreversible polymerization, the total number of piles, M , is kept constant, and the microcanonical ensemble of the systems defined by total energy, E , or total number of piles, M , volume, V, and number of units, N , is more convenient for the statistical treatment. Such a treatment yields the most probable or random distribution of aggregate size and the isodesmic association, eq 3, for the stepwise monomer addition.
Model for Thermodynamics of Ionic Surfactant Solutions. 1. Osmotic and Activity Coefficients Thomas E. Burchfield* US.Department of Energy, Bartlesville Energy Technology Center, Bartlesville, Oklahoma 74005
and Earl M. Woolley*+ Department of Chemistry, Brigham Young University, Provo, Utah 84602 (Received: July 18, 1983; In Final Form: October 17, 19831
Equations have been derived for activity coefficients and osmotic coefficients of aqueous ionic surfactant solutions. The equations are based on a mass-action model with a single micellar aggregate species. The micellar solution is treated as a mixed electrolyte by using the Guggenheim equations. A shielding factor is included to account for screening of the micellar charge. The equations have been applied to experimental data for aqueous solutions of ionic surfactants. Osmotic and activity coefficient data from 0 to 50 OC and from 0.001 to 4 mol kg-I have been correlated with the equations using a maximum of two ion-ion interaction parameters. Data for several surfactants have been described up to 1 mol kg-’ using no ion-ion interaction parameters.
Introduction Mass-action models have been applied to surfactant aggregation phenomena for ionic and nonionic surfactants with some success. While a multiple-equilibrium model is recognized as being a more nearly correct description of the micellization process, the simplicity of the single-equilibrium model has made it more useful in constructing models to describe the properties of micellar solutions. Development of a model to describe the wealth of thermodynamic data for micellar solutions requires some compromises. The model must be close to realistic but not be so complex as to be mathematically intractable. We have derived expressions for thermodynamic properties of micellar solutions based on a mass-action model of micelle formation. The present paper describes application of the model to osmotic and activity coefficients of aqueous solutions of ionic surfactants. In the following paper we describe the application Faculty Research Participant through the Associated Western Universities, Inc.
of the model to apparent molar enthalpies, heat capacities, and volumes of surfactant solutions. The model is consistent with experimental data for a variety of surfactants over wide ranges of concentration and temperature. The theory of osmotic and activity coefficients is well established for strong electrolyte solutions and for simple mixtures of strong electrolyte^.'-^ However, applications to electrolyte solutions that include a reacting component are not nearly so well established. Long ago, Van Rysselberghe4 formally attempted to include equilibrium effects for simple weak electrolyte solutions, such as aqueous acetic acid.s More recently, others have derived ex(1) Harned, H. S.; Owen, B. B. “The Physical Chemistry of Electrolytic Solutions”, 3rd ed.; Reinhold: New York, 1958. (2) Pitzer, K. S.; Brewer, L. “Thermodynamics”, 2nd ed.; McGraw-Hill: New York, 1961 (revision of: Lewis, G. N.; Randall, M. “Thermodynamics”, 1st ed.). (3) Robinson, R. A.; Stokes, R. H. “Electrolyte Solutions”; 2nd ed.; Butterworths: London, 1959. (4) Van Rysselberghe, P. J . Phys. Chew. 1935, 39, 403.
This article not subject to U S . Copyright. Published 1984 by the American Chemical Society
