Langmuir 1992,8, 36-42
36
Composition Fluctuations Promoting the Formation of Long Rod-Shaped Micelles Magnus Bergstrom and Jan Christer Eriksson* Department of Physical Chemistry, Royal Institute of Technology, S-100 44 Stockholm, Sweden Received July 9,1991 Because of composition fluctuations,the number of accessiblestates is generallymuch larger for a mixed surfactant aggregate than for the corresponding pure aggregate. In particular, for a rod-shaped, ionic surfactant micelle, the number of additional states due to adding a long-chain alcohol increases notably with the length of the micelle. By means of model calculations for sodium dodecyl sulfate/dodecanol/salt systems we demonstrate this circumstance to have interesting consequences as to the genesis and size distribution of rod-shaped micelles. For pure rod-shaped micelles, the size distribution is exponentialand it broadens significantly when the salt concentrationis increased, but the position of the maximum remains essentiallyunaffected. However, when both the long-chainalcoholand salt concentrationsare comparatively high, the size distribution becomes distinctly nonexponential, and the maximum is shifted to well above the minimum aggregation number. A rod-shaped micelle is a small thermodynamic system, and the ratio between the composition fluctuation entropy contribution to the free energy and the average excess free energy is 0.15-0.25 at low aggregation numbers N and decreases slowly at higher N . For very large N , i.e., for macroscopically long, rod-shaped micelles, the composition fluctuation contribution eventually becomes negligible compared with the average free energy of a micelle.
Introduction In the thermodynamics of small systems derived by Hill,l fluctuations are taken into account explicitly. A surfactant micelle is a rather typical small system, and among micelles of different shapes, the effects of fluctuations are most prominent for rod-shaped micelles.2 Thus, it is wellknown that rod-shaped micelles usually have a rather broad size distribution. For ionic surfactants this holds true, particularly for high salt concentrations. It is largely because of the much broader size distribution that higher volume fractions of rod-shaped than of spherical micelles ever can be attained. Calculations for pure sodium dodecy1sulfate micelles were carried out previously which show this e f f e ~ t . ~However, ?~ the fluctuation contribution is anticipated to be even more pronounced for mixed rodshaped micelles due to the larger number of states involved. The occurrence of size fluctuations implies that the chemical potential of aggregated monomers in apure micelle with aggregation number N (which is defined as p~ = aG/aN) is exactly equal to the monomer chemical potential in the surrounding solution, pz, only for micelles of equilibrium size, i.e., for those micelles with an aggregation number such that G has a minimum. Here G denotes the Gibbs free energy of the micellea4 It is convenient to introduce the excess free energy in the Qpotential sense, N C Nof , a surfactant aggregate defined by NcN = G - Np2
(1)
The dispersion equilibrium conditions then yield the relations NcN + k T In 4N = 0
( N = Nmin, ...,N,,,)
(2)
proportionally to N for rods, the volume fraction of rodshaped micelles decays exponentially with the aggregation number. However, when mixed rod-shaped micelles are treated, the situation becomes rather different, as we will show below. Our calculations to be presented in this paper concern the effect of composition fluctuations for rod-shaped micelles composed of sodium dodecyl sulfate (SDS) mixed with dodecanol (DOH), i.e., of two different monomers with identical C12 hydrocarbon chains. In the following, SDS is denoted as component 2 and DOH as component 3. The calculations have been carried out at 25 "C and at an alcohol mole fraction in the solution, x 3 , equal to 2.449 X that roughly corresponds to the solubility limit in water.
Excess Free Energy c(N&) of a Mixed, Spherical or Rod-Shaped Aggregate Let us first more precisely define the Gibbs free energy excess of a single (noninteracting) SDS/DOH micelle as follows, using the hydrocarbon/water contact interface as the dividing surface: G = F + p,V
- Flnl - p2+n; - pin,-
- p3n3
(3)
where peis the external pressure acting on the hydrocarbon core volume V of the micelle and nl, n2+, n2-, and n3 are the micellar excesses of water, counterions, co-ions, and alcohol, respectively,outside the hydrocarbon core,relative to the solution. We assume that the surfactant solution is dilute with respect to micelles and monomers and can hence employ the following standard expressions for the chemical potentials of the ions:
where 4~ is the volume fraction of aggregates composed of N monomers. As the excess free energy, N ~ Nincreases , ~_______
(1)Hill, T. L. Thermodynamic of Small Systems; Benjamin: New York, 1963-64;Vol. 11. (2)Eriksson, J. C.;Ljunggren, S. J . Chem. Soc., Faraday Trans. 2 1985,81, 1209. (3)Eriksson, J. C.;Ljunggren, S. Langmuir 1990, 6 , 895. (4)Eriksson, J.C.;Ljunggren, S.; Henriksson, U. J. Chem. Soc., Faraday Trans. 2 1985,81, 833.
0743-7463/92/2408-0036$03.00/0
where c2+ = c y = cp are the single ion concentrations in solution. Moreover, in eq 3, the excess of the uncharged alcohol, n3, is likely to be negligible. 0 1992 American Chemical Society
Langmuir, Vol. 8, No. 1, 1992 31
Formation of Long Rod-Shaped Micelles
_-___
The excess free energy C(NZfl3)per chain in a mixed micelle composed of N = N2 N3 monomers is then given by
+
t---I
i i i i
Nt(N2fl3) = G - N 2 ~ 2- N 3 ~ 3 (6) where N Z and N3 denote the number of aggregated surfactant and alcohol monomers, respectively, and pa = pzand p3 are the chemical potentials of the same monomers in solution.
i i
i
Rc !
Dispersion Equilibrium Next we consider a set of dispersion equilibrium conditions, one for each N z , N3 complex, an4 obtain the following equations for the manifold of compositional states occupied by the micelles: (7) N d N z f l 3 ) + kT In d(N2fl3) = 0 where 4(Nzfl3) is the volume fraction of an Nz, N3 aggregate. Adding up all the volume fractions, $(NZfl3), obviously yields the following "partition function"
4=
5
e-N4N~sVd/kT
Nz
(8)
3
that accounts for the overall concentration of aggregates in solution. Mixed Rod-Shaped Micelle We assume a rod-shaped micelle to be a geometrically composite aggregate made up of a cylindrical central part with a fixed radius R, and two hemispherical end caps of a size corresponding to the radius R, of a spherical equilibrium micelle (Figure The rod-shaped aggregates will generally have a rather broad size distribution when the important free energy parameter, 0, is much less than 1. 0 is defined as the work per monomer in kT units needed to form the central cylindrical part of a rod-shaped micelle out of monomers in the surrounding solution. In analogy with eq 6, the expression for the free energy of formation a t constant chemical potentials of a whole rod-shaped micelle from monomers in the solution is written l).293
Nt'(N2fl3) = G' - NpC, + M[pC,- (1- x & ) P ~ - x",p3I + W[pC,- ( 1 - X ~ H ) & - X b H / ~ b 3 ]= G' - N p i + M ( X ~ H X ~ H ) ( . U~ CLJ
+ NhC, - (1- XbH)& - XbHp31
(9)
Here 2bH and X& denote the mole fraction of alcohol in the cylindrical and the hemispherical parts, respectively, of micelle, i.e. = N",/(W2+ N",)
= %/(%+ q)(10) and pC_ is the free energy per mixed monomer unit, that is, of ( 1 - xbH) DS-monomers and xbH DOH monomers, in an infinite, cylindrical micelle. As before, pz and p3 are the chemical potentials of surfactant and alcohol, respectively, in the solution. If we now set XbH
CY
XbH
+ 6(N) = [G'- NpC, + W ( X &- ~ & 4 ) ( ~-3p J l / k T
(11)
and
P = [CC;- (1 - &)PZ
-x & P ~ I / ~ T
(12)
we obtain the following expression for the overall excess free energy of a rod-shaped micelle:
+
Nt'(N2JV3)/kT= a + NP 6(N) (13) where a is to be interpreted as the work in kT units of
Figure 1. Modified spherocylindrical micelle model with R, # R, and a junction zone between the end cap and the cylinder
part, where the head group concentration gradient is counterbalanced by stresses in the hydrocarbon core. In the present calculations the rod-shapedmicelle is treated as a straight cylinder with two equilibrium-sized,hemispherical end caps. The endto-end repulsion arising when the junction zones overlap is taken into account by a cutoff function 6(m.
forming the end caps out of monomers located in the cylindrical part of the micelle. 6 represents the work to form the cylindrical part out of monomers in solution, and it depends on the aggregate composition and the state of the solution. To correct for the end-to-end repulsion caused by the overlap of the junction zones between the end caps and the cylindrical middle part, the positive function 6(N) is introduced that is supposed to increase abruptly when the rod length is reduced below a certain critical length. Analogously with the approach taken earlier for pure SDS micelles, this feature is dealt with by means of cutting off the micelle size distribution at NO= 175, implying that in our model calculations Ner/kT has a minimum equal to a + 175P at N = NO = 175.3 Referring to our model description of a rod-shaped micelle as being composed of a cylindrical middle part of equilibrium thickness and two hemispherical end caps of equilibrium size, with adjacent junction zones, we now write
+
a = ( t s / k T0.01)(M2+ M3) (14) where ( N 8 N !)e8 is the excess free energy needed to form a spherically shaped aggregate of the radius R, from monomers in solution. M2 and M3 are the corresponding numbers of surfactant and alcoholmonomers,respectively. Note that ea and PkT are both excess free energies per monomer relative to the solution state. The last term in our a-expression accounts for the misfit free energy in the junction zones, which is estimated to be roughly (M2 + M3)0.01kTper micelle. Accordingly, from eqs 13 and 14 we obtain the following final expression for Ne'/ kT: Ne'/kT = ( t s / k T -6 + O.01)(Wz+ W3) + NO ( N L No) (15) The total volume fraction of rod-shaped micelles then becomes 4= e-(tl/kT-B+O.Ol)(NI1+N.s)-NB (16)
+
NzrN3&&
where the number of monomers, W2+ M3,in the two end
Bergstrom and Eriksson
38 Langmuir, Vol. 8, No. 1, 1992
caps is fixed for geometrical reasons. We have assumed, in fact, that
W2+ W3= 71
(17)
because the aggregation number of a spherical equilibrium micelle was found earlier to be 71 in the mixed SDSI DOHcaseS5Thus, the total volume fraction of rod-shaped micelles becomes $=
(18)
e-(EJkT-B+O.O1)(N;+~)-NB Nd3&
where the sum extends over all the different N2, N3,
Ivs3 complexes which are compatible with the subsiduary condition given by eq 17.
Contributions to the Overall Excess Free Energy from the Hemispherical End Caps According to Tanford? the work of bringing W2C12 chains a t the mole fraction 2 2 and W3C12 chains at the mole fraction 2 3 in water into a pure hydrocarbon bulk phase is given by the expression (W, + Ns3)c&,/kT = -[19.96o(Y2 + Ivs3) + W2In x 2 + W3In x31 (19) This contribution, of course, strongly favors aggregation. An important counteracting effect arises due to concentrating the charged head groups and their counterions. Using the Poisson-Boltzmann approximation, Evans and Ninham7 have derived the following approximate expression for the electrostatic free energy of a charged spherical surface:
the surface area per charge, simply a = 4rR:,/%. The hydrocarbonlwater contact is, of course, unfavorable from a thermodynamic point of view and yields a contribution equal to Yhc/w W2alkT,YhcJw, denoting the hydrocarbon1 water interfacial tension. As before, the macroscopic value of 50 mJ/m2 for Yhcjw has been employed in the present
calculation^.^^^ Inside the micelle core there are conformational restrictions on the aggregated hydrocarbon chains which tend to raise their free energy. A quantitative theory of this contribution was elaborated by Gruen and de Lacey8 and, accordingly, we set ezonf/kT= 7.612 - 0.9272 X lO'OR, + 0.03233 X 1O2'RS2- 6.688 X 1025R,3 (25) for the spherically shaped part of the micelle. When the contributions associated more specificallywith the sulfate head groups of the SDS monomers are considered, severalless known effects have to be accounted for: (i) shielding of the hydrocarbonlwater contact by the comparatively large sulfate head group, (ii) asymmetric hydration of the ionic species in close proximity of the hydrocarbon core, and (iii) repulsive local interactions between the hydrated head groups, which have not been included in the electrostatic free energy as the PB approximation assumes a smeared-out surface charge. Unfortunately, there are no quantitative estimates of these effects available at the present time. We have therefore introduced in our calculations a head group parameter, cig, for which we have used the constant value cig = -1.310kT
4% KRe,sIn
- [
1
+
G
] (20)
2
Rei, the radial distance to the smeared-out surface charges, was assumed to be
Rel = R, + 3.0 X 10-l' m
(21)
for the SDS sulfate head groups where R, is the radius of the hydrophobic core, viz. 3
R, = d 3 ( %
+ N",)v147r
(22)
v
denoting the volume of a single C12 hydrocarbon chain that is equal to 351 A3. The dimensionless, reduced charge parameter S introduced above is defined as S = 0 / d 8 ( ~+2 ~,)tot$T
whereas
K,
(26)
which was found to yield a correct cmc value for SDS when applied to solutions with and without added salt. Moreover, it is worth noting that employing a constant head group parameter, tpg, is supported by similar calculations on the surface tension of dodecylammonium chloride solutions and comparison with experimental d a h g Analogously, we have to introduce a parameter associated with the alcohol head group, e&. Due to the fact that this head group is comparatively small and chemically rather similar to a water molecule, it is reasonable to set cgH = 0. The free energy of mixing the different monomers in a spherical aggregate is obtained by means of the combinatorial expression
(23)
the inverse of the Debye length, is given by K
= d 2 ( c 2+ c , ) ~ A e 2 / t o t $ T
(24)
N Adenotes the Avogadro number. c2 is the SDS monomer concentration, and c, is the concentration of added monovalent salt. The surface charge density is defined by u = eta, e denoting the charge of the surfactant head group and a ,
This contribution may differ as much as 2kT units for the comparatively small end caps from the usual kTCNi In X i expression, based on the Stirling approximation. Adding all these different contributions yields the following expression for a pair of end caps, i.e., for a whole spherical micelle.
( 5 ) Ljunggren, S.; Eriksson, J. C. Prog. Colloid Polym. Sci. 1987, 74,
38.
(6) Tanford, C. The Hydrophobic Effect; Wiley: New York, 1980; Chapter 7. (7) Evans, D.; Ninham, B. W. J. Phys. Chem. 1983,87, 5025.
( 8 ) Gruen, D. W. R.; Lacey, E. H. B. Surfactants in Solution; Mittal, K., Lindman, B., Eds.; Plenum: New York, 1984; Vol. I, p 279. (9) Eriksson, J. C.; Ljunggren, S. Colloids Surf. 1989, 38, 179.
Formation of Long Rod-Shaped Micelles
Langmuir, Vol. 8, No. 1, 1992 39
(W2+ W3)es/kT= -19.960(W2 + ha3)- W, In x 2 -
increase of the cylinder radius from 14.6 to 14.85 A when alcohol is added, the head group area decreases somewhat to ac = 47.3 A 2 .
Contribution to the Overall Excess Free Energy from the Cylindrical Part To generate an expression for /3 for the cylindrical part, we will have to make the following changes compared with eq 28 for eOlkT.As in our previous calculations the head group parameter of the DS- ion was set equal to
tig= -0.5897kT
(29) which was obtained by comparing with available data on the micellar growth in SDS solutions caused by raising the salt concentration. The following conformational Gruen expression has to be used:5
+
t&,/kT = 0.555 2.288 X 10-3R, + 2.404 X 10-2Rm24.482 x I O - ~ R ,-~ 1.482 x ~ o - ~ R+, 1.902 ~ x ~ o - ~ R+ , ~ 2.112 x ~ o - ~ R (30) ,~
where R, = R, - 11.04 A and R,, as before, is the radius of the cylinder part of the rods. Due to the fact that the solution is saturated with respect to DOH ( x 3 = 2.449 X and the aggregates contain a considerable amount of alcohol monomers,R, was set equal to 14.85A in accordance with previous calculation^.^ The electrostatic expression is changed into %ezl/kT = 2AF,[ln (S +
G) -V K 2% KR,,s In
- [
1
Degeneracy Factor gN In accordance with eq 18, the following expressions define a "degeneracy" factor, gN:
Ce-N'(@/kT+O.O1)
e-NcB
= gNe-(ae+BJv) (35)
4 Nfi& implying that the degeneracy factor gN is actually defined as gN 4N/4h (36) where 4; is the volume fraction of N-micelles with equilibrium composition given by (37) In other words, gN accounts for the increase of the volume fraction of N-micelles due to the composition fluctuations around the average composition.
Fluctuation Entropy Factor Summing eq 7 for a fixed N we obtain
- 1 1 + 2
q (31)
where the last curvature-dependent term is half of that for a spherical surface. Here, the free energy of mixing term can be quantified by the conventional expression In x O H + (1 - x O H ) In (1 - xOH) (32) because of the comparatively large aggregation numbers in the cylindrical part. Concerning the alcohol head group parameter, COH, we first assumed it to be the same for both geometries. But the consequences of such an assumption were rather drastic. Instead, the conjecture that the difference between the head group parameters of the two different monomers remained constant for the various geometries was introduced; i.e., we have assumed that
where the average value of the excess free energy of a micelle is defined by WEN)
=
C
~ N ~ f l ~ q ~ e N =~ , N ~ q
N2fl3&
tkix = xOH
cPg - eOH = -1.310kT
(33)
Hence, in the case of cylindrical geometry = 0.7062kT (34) The variation of the head group parameters with geometrical shape that shows up here might well indicate that there is, in fact, a curvature dependence of the hydrocarbonlwater interfacial tension and that Yhc/w is actually smaller than 50 mJ/m2 for oillwater interfaces curved toward the oil. It has been proposed previously that Yhc/w should be in the range of 40mJ/m2for asphericalshaped surface with a radius of about 20 A.l0 For a pure cylindrical micelle consisting of SDS monomers only, the area per head group was found to be aC= 48 A2 at the free energy minimum., However, due to the e&
(IO) Puwada, S.;Blankschtein, D. J . Chem. Phys. 1990,92,3710.
and where the fraction of N2, N3, W3aggregates is given by 'lv2fl3Sv',
= ~N,JV~&N.,/~N
(40)
Thus 4N = SNe-(Nw)/kT where the expression
(41)
SN= e-~PNds#"'df& (42) defines the fluctuation entropy factor SNand -C 'N2fl3fl In 'N2fl34 is the fluctuation entropy that arises due to the composition fluctuations. Discussion There are three effects contributing to making mixed rod-shaped micelles more stable than the corresponding pure micelles: (i) lowering of the electrostatic free energy, e,l, due to lowering the surface charge density, (ii) the negative free energy of mixing, emix, and (iii) the larger number of accessible states. Primarily, it is the last one of these factors that causes the size distribution to become nonexponential. This is illustrated in Figure 2, where the broadening of the potential curves with increasing aggregation number is a direct consequence of the increasing number of states. If these potential curves instead had
Bergstrom and Eriksson
40 Langmuir, Vol. 8,No. 1, 1992
4
1
36
8 1.2 1’4 h . 0 M
%
1
1.0
1 t
24
‘\
zn
,
, 1
1
n
ion
50
200
150
0’
loon
-. -- -3000
5000
7000
Aggregation number
Aggregated alcohol monomers
Figure 2. Overall excess free energy of a rod-shaped micelle, NCN,plotted as a function of the number of aggregated alcohol monomers in the cylinder part, W3,for different aggregation numbers at the surfactant monomer concentration c2 = 0.2501 mM and the salt concentration cI = 1.0 M. The mole fraction of alcohol monomers in the spherical end caps was fixed to the equilibrium value, x&., = 0.139. The total volume fraction 4 of rods was close to 0.1.
!
0.5 M
6 p
1
1
I
I
400
800
1200
1600
2000
Aggregation number
Figure 3. Quotients 4N/4175 (solid line) and 4&/4;75 (dashed line) as functionsof the aggregation number N , at the surfactant monomer concentrationcz = 0.3780 mM, salt concentrationc. = 0.5 M,and alcohol mole fraction x 3 = 2.449 X yielding a total volume fraction 4 = 0.1. The free energy parameters, Be for the cylinder part and ale for the end caps, equal 3.64 X and 21.7, respectively, for the equilibrium micelle. been identically shaped, the degeneracy factor gN would have been independent of N (cf. eqs 35-37) and the size distribution would have remained exponential. This is also evident from the circumstance that the free energy parameter 0depends on the aggregate composition only but is independent of the total number of monomers N, i.e., that /3 is an intensive thermodynamic parameter. For pure SDS, the volume fraction 4~ of rods decreases exponentially with the aggregation number N. This is so because Nwlk T increases linearly with N at a rate determined by the parameter /3. An exact analogy to the pure rod case is obtained for the mixed case by considering micelles of equilibrium composition only, with xLH and x$H fixed. This is illustrated in Figures 3 and 4, where &,/&, is plotted vs N (dashed lines) for two different solution states. 4\ denotes the volume fraction of Nmicelles of equilibrium composition. The composition fluctuations about the equilibrium micelle gain in weight with increasing aggregation number.
Figure 4. Quotients (fJN/&75 (solid line) and 4&/4;,5 (dashed line) as functionsof the aggregation number N , at the surfactant monomer concentrationc2 = 0.2501 mM, salt concentrationc8 = 1.0M, and alcohol mole fraction x 3 = 2.449 X yielding a total volume fraction 4 = 0.1. The free energy parameters, p, for the cylinder part and for the end caps, equal 1.19 x and 25.8, respectively, for the equilibrium micelle. (re
The number of complexions of aggregated monomers involved increases dramatically with the overall aggregation number and, consequently, the number of numerically significant terms in the fluctuation entropy expression, - CpN,Jv4 In p N s ~ ’increases. The composition fluctuations d t e r the s:zgd$tributions of rod-shaped micelles as is shown in Figures 3 and 4, and hence, they promote the formation of long rod-shaped micelles. In Figures 3 and 4, the ratio @N/4175 is plotted against the aggregation number N, at a total volume fraction 4 of about 0.1, at two different concentrations of added salt (solid lines). At low salt concentrations the functions are decreasing in the whole range of aggregation numbers, and the most frequent aggregation number of the rods is thus 175. However, at salt concentrations higher than about 0.5 M there occurs a maximum at N > 175; the higher the total volume fraction 4 is, the higher the aggregation number where the maximum is found. At 1.0 M salt content and 4 equal to about 0.1, it is located a t approximately N = 500. It is the difference between (N ~ N ) and kT In E+” that determines the volume fraction 4~ (cf. eq 38). The maxima of the graphs in Figures 3 and 4 occur where this difference has a minimum. Despite the fact that the excess free energy, N ~ Nis, always smaller for the smaller aggregatesthan for the larger ones, the volume fraction of long rods may well exceed the volume fraction of the smaller rods as is evident from Figure 4. This is entirely due to the larger number of accessible states at higher N . Another way of illustrating this effect is seen in Figure 2, where NCNis plotted as a function of the number of aggregated alcohol monomers in the cylindrical part, q.The curves broaden when N increases, which implies that more states contribute significantly at higher N. The weight-average aggregation numbers increase due to this fluctuation effect, as seen from Table I. From Figures 5 and 6 it appears that the degeneracy factor gN, as well as the fluctuation entropy factor SN, increases considerably with the aggregation number. The only difference between these two factors is that SN is larger by a factor exp[((NeN) - NENJ/kn. The main reason is, of course, the composition fluctuation effect discussed above. We note that the gN and SN functions depend on the concentration of added salt. However, both
Langmuir, Vol. 8, No. 1, 1992 41
Formation of Long Rod-Shaped Micelles Table I. Weight-Average Aggregation Numbers, (N)and ( N e ) ,at Different Concentrations of Added Salt and Surfactant.
M
mM 0.3780 0.2501
I
I
I
(N) 543 1399
(Ne) 449 971
N,, 209 493 1.0 a ( N ) is an overall average aggregation number, whereas ( N e )is the weight-average aggregation number for micelles of equilibrium composition. N,, is the aggregation number where the peak of the size distribution is located. cn,
0.5
c2,
I
I
I
I
i
1 Y
'
I
01 k
I
I
2000
400
i
I
I
4000
6000
8000
10000
Aggregation number
Ir
P
Figure 7. Fluctuation entropy In SN as a function of the aggregation number N, at the salt concentrations cs = 0.5 M (upper curve) and c8 = 1.0 M (lower curve) a t the same monomer concentrations as in Figures 3 and 4, respectively.
e
I
L
200
0'
I
2000
I
4000
I
6000
I
8000
I
10000
Aggregation numbex
Figure 5. Degeneracy factor g N as a function of the aggregation number N, a t different salt concentrations. The solution states are the same as in Figures 3 (c, = 0.5 M) and 4 (c, = 1.0 M). I
1
I
1
I
i
"
2000
4000
6000
8000
10000
Aggregation number
Figure 8. Quotient between the fluctuation entropy contribution and the average excess free energy of a micelle, kT In SN/( N c N ) , plotted against the aggregation number N, at different concentrations cs of added salt. The monomer concentrations of the two components are the same as in Figures 3 and 4. ,
601 I
2000
I
I
I
I
4000
6000
SO00
10000
Aggregation number
Figure 6. Fluctuation entropy factor SN plotted against the aggregation number N, a t the same solution states as in Figures 3 and 4.
of these functions have a very weak dependence on the total volume fraction of rod-shaped micelles, 4. Figure 7 shows the fluctuation entropy contribution to the overall excess free energy as a function of the aggregation number N . The similarity with a square root function is obvious. The reason why the fluctuation entropy is larger at low than at high salt concentrations is related to the difference in composition for the equilibrium micelle at various salt concentrations c,. The equilibrium mole fraction of aggregated alcohol monomers is closer to 0.5 a t low salt concentrations (x& = 0.214 at 0.5 M) than at high salt concentrations (xiH = 0.184 at 1.0 M). This means that the number of numerically significant states is larger at low than at high salt concentrations. The quotient of the fluctuation entropy contribution to the free energy and the average excess free energy for a
T!
I
I
I
I
t
I
0'
'
I
I
2000
4000
I
6000
i
8000
10000
Aggregation number
Figure 9. Average excess free energy ( N c Nplotted ) against the aggregation number N, at the same solution states as in Figures 3 and 4.
rod-shaped micelle, k Tln SN/(N E N,is ) plotted as a function of the aggregation number in Figure 8 at two different concentrations of added salt. After reaching a weak maximum, these functions begin to decay, and at suffi-
42 Langmuir, Vol. 8,No. 1, 1992
Bergstr6m and Eriksson
ciently large aggregation numbers, the contribution of the fluctuation entropy to the overall excess free energy becomes negligible. This, in fact, illustrates the transition from a small to a large (i.e., macroscopic) system. The slopes of these functions are also strongly dependent on the salt concentration. The rate of decrease at large N is more rapid at lower salt concentration. This is due to the more rapid increase of the average excess free energy, ( N c N )with , the aggregation number at low salt concentrations, as seen in Figure 9. As a consequence, the shape of the size distribution depends on the salt concentration and it broadens a t high cs due to the less rapid increase with N of the average excess free energy while the fluctuation entropy is approximately uneffected (cf. Figures 7 and 9). The composition fluctuation effect on the size distribution considered here does not appear to be very significant for spherical micelles. This is related to the comparatively narrow and symmetric size distribution of spherical micelles in the pure state.
important fluctuation entropy contribution to the free energy. An implication of this fact is that a maximum of the volume fraction 4~ is obtained at aggregation numbers higher than would otherwise be expected when cB1 0.5 M and that this maximum occurs a t even higher N when the salt concentration is further raised. Due to the fact that the overall excess free energy is proportional to the aggregation number and the fluctuation entropy part is approximately proportional the importance of the fluctuation entropy term eventually becomes negligible at sufficiently high N . This reflects the transition from a small to a large thermodynamic system. The composition fluctuation effect promotes the formation of larger aggregates and is likely to be one main reason for the fact that long-chain alcohols promote the formation of long rod-shaped micelles, as has been observed in several experiments."-l3
Conclusions The larger number of accessible states for a mixed rodshaped SDS/DOH micelle of a certain length as compared with the corresponding pure micelle gives rise to an
(11) Ljosland, E.; Blokhus, A. M.; Veggeland, K.; Backlund, S.; H0iland, H. B o g . Colloid Polym. Sci. 1986, 70, 34. (12)Backlund, S.;Rundt, K.; Veggeland, K.; Hailand, H. Prog. Colloid Polym. Sci. 1987, 74, 93. Hoffmann, H.; Ulbricht, W. Surfactants in Solution; (13) Bayer, 0.; Mittal, K., Bothorel, P., Eds. Plenum: New York, 1987; Vol. IV, p 343.
Registry No. SDS,151-21-3; DOH,112-53-8.
