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Nucleation process of micelle formation in apolar solvents
Nucleation Process of Micelle Formation in Apolar Solvents HAxNS F R I E D R I C H E I C K E axi) HEINZ CHRISTEN Institute for Physical Chemistry of the University, CH-4056 Basel, Switzerland Received August 31, 1973; accepted December 17, 1973 Vapor pressure osmometric, dipole moment and dielectric dispersion measurements of aerosol OT, AY, di-2-ethylhexyl sodium phosphate, and lecithin in apolar solvents give strong indications as to the existence of nuclei below the cmc. Based on these observations, a model has been proposed to interpret the micellization process as a phase change. According to this model, explicit calculations of the enthalpy and entropy changes for the micelle formation lead to an estimation of the size of the miceUe, of the micellar nucleus, and in addition, to a determination of the free energy of formation of the nucleus. The dependence of these quantifies on the selfenergy of the micelle, the temperature, and the cmc is discussed. INTRODUCTION
The formation of micelles has been treated mostly with the help of the so-called "pseudochemical equilibria" (see, e.g., Refs. 1-3) where only two types of particles, monomers and n-mers, were considered. This description corresponds in some respects to the initial treatment of Kreutzer (4). In many cases another possible approximate interpretation of micelle formation is to view the micellization process as a phase change, since its thermodynamic behavior resembles in many respects that of systems with two macroscopic homogeneous phases (5). One indication for the phase concept is the jumplike change of the measurable quantity which follows the transformation in many aqueous and nonaqueous colloidal solutions (6). Furthermore, vap0r-pressure-osmometric and light-scattering measurements in nonaqueous solutions of some surfactants (7) indicate a considerable monodispersion of the micellar phase. This finding is important with respect to the phase model. The experimental observations on nucleation in a series of nonaqueous
colloidal solutions below the critical micelle concentration provide additional evidence. In view of the fact that this phase transformation proceeds in the solute within a binary system (solute and solvent), the classification of such phase change is difficult. The Ehrenlest scheme (8) does not apply, but the change might belong to the anomalous transformations of the first kind as described by Mayer (9). In relation to the description of micellization as a phase transition, the nucleation process is of primary importance and will be described by means of a model related to the present experimental observations and to a recent theoretical paper on the stability of micelles in apolar media. EXPERIMENTAL
Substances
The ionic surfactant di-2-ethylhexyl sodium sulfonate (aerosol OT) molecular weight 444.57, was supplied by Fluka AG, Buchs, SG, in analytical quality; the di-2-pentyl sodium sulfosuccinate (aerosol AY), molecular weight 360, was supplied by the Cyanamid Company
281 Copyright ~ 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 48, No. 2, August 1974
282
E I C K E AND CHRISTEN
in the same quality. Both products were dissolved in benzene for further purification and stirred for several hours in the presence of active charcoal. After separating the charcoal, the solution was evaporated under vacuum. Then the aerosols were dissolved again in ethyl ether of highest purity and dried under high vacuum. The purified products were stored in a desicator over granulated phosphorous pentoxide in a warming cabinet at 50°C. This procedure proved to be most successful in drying the substances. Di-2-ethylhexyl sodium phosphate was prepared according to the description in an earlier paper (6). Chromatographically homogeneous lecithin was prepared from egg phospholipid by column chromatography over aluminium oxide. The method has been described elsewhere (10). The solvents used--cyclohexane, benzene, 1,4-dioxane, 2,2-dimethyl-butane--were of high reagent grade quality from FLUKA or M E R C K AG. The solvents were dried over calcium chloride, phosphorous pentoxide, or sodium/potassium alloy. Measurements
Vapor-pressure-osmometric, and dielectric dispersion measurements were used for the direct determination of the mean number of monomers per aggregate and its concentration dependence. The measurements were made at 20 4- 0.1°C with a Hitachi-Perkin-Elmer vapor-pressure-osmometer type 115, and an apparatus for the measurement for the dielectric absorption based on a sensitive resonance method (11). The equipment permits loss measurements at room temperature (20 40.05°C) in the frequency range 0.5-120 MHz with an accuracy of Atg 8 4- 1 X 10-6. Constructional details are given in Ref. 12. From dipole moment and conductivity measurements additional information on the nucleation could be derived. Dipole moment determination was carried out with the WTW-dipole meter type D M 01 (Weilheim, Germany). The dielectric constants of the solvent (el) and
solutions (e12) have been determined. Measuring frequency was about 2 MHz; the sensitivity of the measurement Ae/e-= 5 X 10-5 • The measuring temperature was adjusted to 20 40.05°C. The evaluation of the results and the computation of dipole moments followed the method of Guggenheim (13) and Smith (14). It is apparent that the plots of Ae = e12 -- el against weight-fraction of solute (w2) deviate from a straight line at low concentrations, depending on the cmc, i.e., the solvent used. Whether or not information on aggregates smaller than micelles can be obtained from the curved part of the plot below the cmc depends on the sensitivity (Ae/e) of the dipole meter. In this part of the curve a parabolic approximation was used (15) to fit the experimental plot, which resulted in extrapolated values of the dipole moments of the monomers. Conductivity measurements were made at 204-0.05°C with a Keithley electrometer type 602, or, if the specific conductivity was above 10-11 mho cm -1, a Wayne-Kerr-bridge type B221 at a constant frequency of 1592 (4-1%) I-Iz. The cell consisted of a thermostatable cylindrical metal condenser with gold plated electrodes (the same as used with the WTW dipole meter). Additional details are given elsewhere (6). RESULTS AND DISCUSSION
Experimental Evidence on Nucleation If the phase model approximation is accepted, the question arises as to the possible existence of nuclei. The first hint was given by conductivity measurements where two different particles below the cmc had to be hypothetically assumed in order to interpret theoretically the two experimentally observed distinct regions of the conductivity-weighedin-concentration plot. The first one is thought to correspond to the conductivity of monomers ; the second range should be due to colloidal aggregates of a few monomeric particles. The marked break in the initial slope of the conductivity curve (see Ref. 6), which separates these two regions, could be tacitly assumed
Journal of Colloid and Interface Science, Vol. 48, No. 2, August 1974
MICELLE FORMATION
/
283
/* /
-13.
/ zX /
Fro. 1. Specific conductivity (~2-1 cm-l) of aerosol OT in benzene at 20 4- 0.05°C as a function of weighed-in-concentration Emole 2-11 (see Ref. 6). to be a second critical concentration, as was done by other authors (16) for aqueous colloidal systems. The conductivity of the small colloidal aggregates, which is attributed to the almost horizontal second region of the conductivity plot, is now thought to be that of particles which may play the role of nuclei (see Fig. 1). This working hypothesis is strongly supported by a number of dipole moment measurements on aerosol OT, AY, di-2-ethylhexyl sodium phosphate, and lecithin in different apolar solvents. The dipole moment (~) divided by the m e a n aggregation number of the monomeric molecules ((n)) i.e., the contribution of one monomer to the dipole nloment of the aggregate, has been plotted against (n). From Fig. 2 it is seen that ~/
i.e., to an increase in the dielectric increment while here a strong decrease with the aggregation number is observed. These remarkable observations indicate that subunits of only three or four molecules ha~-e reached a stable (electrostatic) equilibrium configuration which is retained in the ultimately formed micelle. The stability of the trimeric subunits is demonstrated by the present results of the dielectric dispersion and dipole moment measurements in 1,4-dioxane. Both methods demonstrate, up to concentrations of 5.10 .2 m, that only trimers have been formed which show a normal Debye dispersion behavior. In view of the fact that dioxane is only a "pseudo" apolar solvent, since two dipoles are compensating for each other, the solvent will interact with the dipoles of the surfactant molecules as in all polar solvents. This is the reason that, in dioxane and in polar media, miceltes of the inverted type, as usually found in apolar solvents, are not formed. The occurrence of trimers in dioxane may indicate that this configuration has a preferred stability. In this connection, the findings and considerations of Ekwall et al. (17) on X-ray investigations of aerosol OT should be mentioned. According to molecular model studies, these authors assume that there is probably no room
Journal of Colloid and Interface Science, Vol. 48, No, 2, A u g u s t 1974
284
EICKE AND CHRISTEN
765-
321-
i X x
~
~x
X
{{
I0
I;
i
D
35
5-
~_ 'x A3-
21-
X ~ x - - x
FIG. 2. Dipole moment ~ divided by the mean aggregation number of monomers (n), i.e., the contribution of one monomer to the dipole moment of the aggregate (~ in [D]), as a function of (n} (Solvents: benzene, cyclohexane, carbontetrachloride, 1,4-dioxane, heptane, and 2,2-dimethylbutane:) (a) aerosol OT; (b) aerosol AY.
in the center of the cylindrically proposed micelle for six sulfonate groups in the same plane as is the case in the solid state. The molecules will presumably be arranged in two subgroups, each containing three sulfonate groups and located one above the other. These observations strongly support the idea of nucleation as a prerequisite for the phase approximation model.
will be discussed as a function of the number of interacting monomers (n). I t is possible to obtain AH(n) from a previous calculation (7) of the "selfenergy" AW(n) of micelles as a function of their aggregation number n. Referring to N1 monomers which form N2 = N1/no micelles where no equals the number of monomers of the equilibrium micelle (neglecting entropy contributions) -AH(n) = (Xl/no) ~W(n).
A Model for _PhaseChange and Nucleation
[11
To investigate the phenomenon of nucleation and the phase change, it seems suitable first to consider the course of AG, the free energy change for the entire process: monom e r - + micelle. Thus the expression
To determine the entropy difference AS(n) when a solution containing N1 monomers and No solvent molecules changes into a solution with N~ micelles, one could imagine the following processes :
A~(n) = ~H(~) - T/,S(n)
1. Nx monomeric molecules are mixed with
Journal of Colloid and Interface Science, Vol. 48, No. 2, August 1974
285
MICELLE FORMATION
sl /4-
\
x
2-
x ~ x
I0
8765-
A ~" V
32-
\ X
~
X
" X
10 '
15 ,
2=0
FIo. 2 (Cont'd.). (c) Di-2-ethylhexyl sodium phosphate; (d) lecithin.
No solvent molecules. The mixing entropy
of this process is AS1. 2. N2,= N { n micelles are mixed with No solvent molecules to give the entropy of mixing AS2. The entropy of mixing for the process: monomer ~ micelle is therefore AS = AS2 -AS~. The explicit calculation of the mixing entropy m a y follow the model of Hildebrand (18). This gives
AS1 = k N o l n
No(vo - bo) + X~(v~ - b~) No(vo - bo)
No(Vo
-
-
volumes, bo and bl their intrinsic van der Waals volumes, and k the Boltzmann constant. In Eq. [2] not only the number of arrangements of the particles but also their differing sizes are taken into account with the introduction of the free volume ( v - b). A corresponding expression can be written for AS2: No(Vo - bo) + N2(v2 - b2) AS2 = k No in No(vo - bo)
+N21.
No(Vo -- bo) + N2(v2 - 52)',,
N2( 2 b2-
)" E3]
bo) + N~(v~ -- b~)\
where vo and vt are the respective molal
With the assumption that % -- b2 = n(vl -- b~) and substituting N { n for N2, the entropy of mixing for the transfer: monomer--+ micelle
Journal of Colloid and Interface Science, Vol. 48, No. 2, August 1974
286
E I C K E AND C R H I S T E N
Finally, AG(n), the free energy change per mole, is given by
AGCn)
T AS
N Lho
&G(n) _~ no
_
_
_
_
_
(n ~ -- 2nno + 2no -- 1) no(no -- 1) ~
_
+RT ho
1-
In
[-6] N1
-3
.........
,b . . . .
n ....
io .........
FIG. 3. Schematic representation of the enthalpy AH, entropy AS and the free energy change &G of the nucleation and micellation processes with the number of interacting monomers, n.
reads AS=k
1--
Nlln n/
,
[-4-]
Xl
where a = (% -- bo)/(vl -- bl). The value of a may be estimated neglecting the contribution of the intrinsic van der Waals volumes bo and 51- Thus a ~ vo/vl = ( M o / p o ) / ( M 1 / m ) , which gives, for example, for the aerosol OT/benzene system a ~ 0.22, if m (aerosol OT) = 1.1 g cm -a according to Ekwall el al. (17). The detailed calculation of AH (n) is possible with the help of the results on the selfenergy calculations of micelles in apolar media (7), The gain of potential energy (zXW) per micelle follows approximately the square of the number of monomers, i.e., the variation of AW with n m a y be described by a parabola that passes through the point (n = 1, AG = 0) and has a minimum value ho at n = no (Fig. 3). The parabola ho
In general, this function (Eq. E6]) will show two extreme values (see Fig, 4a, b). The parameters ho, no, and a depend on the selected systems. The variables are the number of interacting monomers n, the respective mole numbers of solvent (No) and solute (NI) (referring to monomeric units), and the temperature, T. At constant temperature there will be a certain concentration of monomers N1 where AG(n) is zero. This is the critical micelle concentration (cmc). The height of the maximum of the AG(n) curve at n = nN can be interpreted in terms of a potential barrier and corresponds to the energy required to form a micellar nucleus. From the existence of such a potential barrier, it can be inferred, with respect to the micellization process, that it is energetically more favorable to attach monomers to a nucleus than to assume an aggregation of colloidal subunits. I t is now possible with the help of Eq. [-6] to derive quantitatively the following quantities which characterize the micellar system, especially with respect to the formation of nuclei. Determination of the micelle size f r o m the critical micelle concentration. The cmc occurs under the condition A G ( n ) = 0, i.e., at the
point where the curve touches the n-axis. From the physical situation it is evident that a + 2 n o - 1) [5-] second intersection of the curve with the fulfills these conditions. Hence from Eq. E1l n-axis has to exist at the point (n = 1, AG = 0) since growth and decomposition of the colloidal NlhoNL AH(n) _~ (n 2 -- 2nno aggregates start and end respectively with n o ( n o - 1) 2 +2no -- 1). [-5a-] monomers. When A G ( n ) = 0, Eq. [-6] gives
zXW (n)
~
(n ~ -
(no -
2nno
1) 2
[ournal of Colloid and Interface Science, Vol. 48, No. 2, August 1974
MICELLE FORMATION
287
giving
the following cubic equation for n:
2no -- 1
no(.o-
1) ~
~7~1,2
2
n 3 -- 2non 2 + 2non -- n + hoN L
R T n o ( n o _ l ) 2 1 n _ _a N o + N 1
[ (2n°-1)2 a N o -J- N1 × (n - i ) R r i.
IV1
- 0.
[7]
r7b]
Ni
The remaining two roots must be equal for the existence of a critical micelle concentration where the condition AG(n) = 0 is fulfilled. Therefore n l , 2 = n o - - ½ = r i m and consequently
One solution from the above argument is n = 1, and this reduces to no(no n 2 -- (2no -- 1 ) n +
-
1) 2
-
hoNz aNo + N i
(2no -- 1) 2
RT
4
hoYL
-m(no -
o, ETa]
XRTln
xln
1) 2
aNo + cmc
N1
[7c]
cmc
2.000
T
b c d
u
,x, i.OOO
o ~3 0.000
-i
.OOO
' -
.OOO
'
'
~
i
'
S.O00
'
~ fl
'
I i0.00
'
,
,
~
I
'
IS.O0
'
'
,
I
,
,
,
, ~"
20.00
FIG. 4a. Variation of the free energy change AG (kcaI mole -1) as a function of the aggregating monomers n (see Eq. [-6]). P a r a m e t e r s : T e m p e r a t u r e T = 293°K; micelle size n M = 17; N1 = 5"10 -4 mole-l.
~a) 2.9
|b) 3.0 3c) 3.1
ho (selfenergy of micelle) (eV) ] d ) 3.2
]e) U)
328 3.4
Journal of Colloid and Interface Science, Vol. 48, No. 2, August 1974
288
EICKE AND CHRISTEN
O .OOO
-t
.000
.
.
.
.
~
0.000
.
.
.
S.O00
.
~
rl
10.00
.
.
.
.
~
'
1~.00
,
I
.
.
.
.
~"
20.00
FIG. 4b. Variation of the free energy change AG (kcal mole -i) as a function of the aggregating monomers n (see Eq. I-6]). P a r a m e t e r s : T e m p e r a t u r e T = 293°K; micelle size nM = 17; selfenergy of micelle ho = 3.28 eV (see text a n d Ref. 7)
Ia) b) Ni.10 4mole-i c) d) e) Since it is known from a previous paper (7) that a relation between no and ho exists on physical grounds, no may be expressed with the help of Eq. [-7c-] and combined with the above equation for n~ to give h oNL n~u =
aN o + cmc
1 2'
however, that the apparent strong temperature dependence of n~, which is not experimentally observed, is considerably reduced by the fact that the cmc is also a function of temperature (see Fig. 5), namely cmc ~ aNoe-h°lvLl'~°~T,
[-7d~
R T in cmc if 1 is neglected compared with no. Eq. [-Td-] describes the size of the micelle under the explicit consideration of the entropy. This equation demonstrates that the size of the micelle is almost exclusively determined by the selfenergy ho at constant temperature (Table I). Hence the influence of the entropy is negligible with respect to the aggregation number of monomers. It should be noted,
1.0 2.5 -5.0 7.5 10.0
[-8-]
where the same condition applies as in Eq. [-7d~. Size of the micellar nucleus. With respect to Fig. 3, the extreme values of &G(n) (Eq. [6-]) have to be investigated to obtain information about the size and energy of formation of the nucleus. Hence &G is differentiated and equated to zero. The region of the formation of the nucleus is of special interest, i.e., the maximum value of the function f~G(n) at the
Journal of Colloid and Interface Science, Vol. 48, No. 2, ALxgt~st1974
289
MICELLE FORMATION TABLE I DEPENDENCE OF TItE SEL~ENERGY
]¢oj THE
SIZE OF THE
/
10,0
N U C L E U S nN~ A N D TIIE E N E R G Y OF FOR3~AT10N
AGN O N
0~" TI-IEN U C L E U S
T K E SIZE 0P TIIE
C M C = 5-10 -~ M AND T = 293 K
.E.
TEMPERATURE nM
14
15
16
17
18
ho (eV) nN zXGN (kcal/mole)
2.85 2.90 2.01
3.06 3.00 2.09
3.28 3.10 2.15
3.49 3.20 2.21
3.70 3.25 2.27
point n
=
×
/
M I C E L S E n M A T CONS:rAN~r
/
5,0
x
x/x
nN. Here
(o%
2'73 283 T 293[°K]303
.=o 2hoNL n
-
--
/go)
1) 2 RT
+ --
a2¢o + N1
in
,
[-9-]
IV I
152
where # N refers to the chemical potential of monomers in the nuclei,/~ to that in the monomeric phase. Eq. [9~ can be rewritten to give
which is the condition for determining the number of monomers in the nucleus (n~;). One root of the equation (n = no -- ½) is known from Eq. [-7c] which applies at the cmc. Hence, replacing the third term by (2no -- 1)2/8 (see Eq. [7b]) and dividing b y n - no + ½, 152 - - -
n a -- non 2 +
no(no
--
-- ½(no
- - ½) =
0
[-9b]
2
1) 2
RT
X
3;~3
Fro. 5. Critical miceIle concentration (cmc) (mole-1) as a function of temperature T(°K). Parameters: Micelle size n M = t 6; selfenergy of micelle ho = 3.28 eV.
On I n=~v
no(no
3i3
is obtained with the solution:
a d o q- N1
in
- 0
[-9a-]
N1
2hoNL
nl,2 = ~ 4- 4 ~
+ ½0~o - ½),
",× 2,3'
\ X
2,2"
2,1.
~ × ~ 2,0"
~,~c.lO"
slo
'[mill
1<0
"
FIG. 6. Free energy of nucleation as a function of the critical micelle concentration. T h e cmc is varied by changing the micelle size. P a r a m e t e r s : T e m p e r a t u r e T = 293°K; selfenergy of micelle ho = 3.28 eV. Journal of Colloid and Interface Science, Vol. 48, No. 2, August 1974
290
EICKE AND CHRISTEN
where only the plus sign is of physical significance. Substitution for no from Eq. [-7c] yields
1 =
-
/
hoNL
3
aNo q- cmc
16'
[io]
+
4
R T In
cmc giving the size of the micellar nucleus as a function of the critical micelle concentration and temperature for a specified system (a, ho). It should be noted that nN is--in contrast to nM--dependent on the square root of ho, from which it can be inferred that the size of the micellar nucleus is relatively insensitive to the stabilizing energy ho per micelle and to variations of the critical micelle concentration. This is consistent with the present experiments.
Calculation of the free energy of nucleation. With the help of Eq. [-10] the necessary energy for the formation of a micellar nucleus may be calculated. Substitution of n~; from Eq. [-10] in Eq. [6-] neglecting the first term (~), generally small compared with the value of the square root, yields:
AGz¢ = no(no -- 1) 2
aN o + cmc T in cmc
--2no R T in
aNo + cmc
[-2no---16
cmc
aNo -}- cmc
16
T in cmc
-}
l R T In
hoNL
3
aNo -}- cmc
16
cmc
X R T in
aN o -]- cmc
[n]
cmc
This equation [-11] describes the dependence of the energy of nucleation on the critical micelle concentration (Fig. 6) and the selfenergy ho (see Table I) at constant temperature. Figure 6 demonstrates that AG~vincreases
with decreasing critical micelle concentration at constant temperature, as is to be expected. The energy necessary to overcome the potential barrier, represented by AGN, is supplied by thermal fluctuations. [A detailed discussion on the origin of the nuclei, based on these fluctuation phenomena with the help of a Monte-Carlo model, is in print (19).-] With respect to the experimental results and the theoretical considerations presented above, it seems indeed possible to discuss the formation of micelles in the investigated systems as phase changes with the expected preformation of nuclei. REFERENCES 1. JONES,E. R., AND BURY, C. R., Phil. Mag. 4, 841 (1927). 2. Scmctq M. J. (Ed.), "Nonionic Surfactants," Marcel Dekker, New York, 1967. 3. ELIAS, i . G., "The Study of Association and Aggregation via Lightscattering," in "LightScattering from Polymer Solutions," (M. B. Huglin, Ed.), New York, 1971. 4. KREUTZER,J., Z. Phys. Chem. B53, 213 (1943). 5. HALL, D. G., Trans. Faraday Soc. 66, 1351 (1970). 6. EICKE, H. F., AND ARNOLD,V., J. Colloid Interface Sci. 46, 101 (1974). 7. EICKE, tI. F., AND CI~RISTEN, H., J. Colloid Interface Sd. 46, 417 (1974). 8. M[/NSTER, A., "Statistische Thermodynamik," Springer, Berlin, 1956. 9. MAYER,J. E., AND GOEPPERT-MAYER,M., "Statistical Mechanics," Wiley, New York, 1959. 10. SINGLETON, W. S., GRAY, M. S., BROWN, M. L., A~D W~IITE, J. L., J. Amer. Oil Chem. Soc. 42, 53 (1965). 11. BERGMANN,K., EIGEN, M., A N D :DE MAEYER, L., Ber. Bunsenges. Phys. Chem. 67, 819 (1963). 12. BERGMANN,K., Ber. Bunsenges. Phys. Chem. 67, 826 (1963). 13. GUGGENKEI~, E. A., Trans. Faraday Soc. 45, 714 (1949). 14. S~ITI~, J. W., Trans. Faraday Soc. 46, 394 (1950). 15. OE~M~, F., "Bestimmung des molekularen Dipolmoments," WTW, Weilheim Obb. (1960). 16. MR~RA, M., AND KODAMA, M., Bull. Chem. Soc. Jap. 45, 428, 2265 (1972). 17. EKWALL, P., MANDELL, L.~ AND FONTELL, K., J. Colloid Interface Sci. 33, 215 (1970). 18. HILDEBRAND,J. H., P~AUSNITZ,J. M., AI~I)ScoTT, R. L., "Regular and Related Solutions/' Van Nostrand Reinhold, New York, 1970. 19. CIrRISTEN H., AND EICKE, H. F., J. Phys. Chem.
Journal of Colloid and Interface Science. Vol. 48. No. 2, August 1974
(i974).
The formation of micelles has been treated mostly with the help of the so-called "pseudochemical equilibria" (see, e.g., Refs. 1-3) where only two types of particles, monomers and n-mers, were considered. This description corresponds in some respects to the initial treatment of Kreutzer (4). In many cases another possible approximate interpretation of micelle formation is to view the micellization process as a phase change, since its thermodynamic behavior resembles in many respects that of systems with two macroscopic homogeneous phases (5). One indication for the phase concept is the jumplike change of the measurable quantity which follows the transformation in many aqueous and nonaqueous colloidal solutions (6). Furthermore, vap0r-pressure-osmometric and light-scattering measurements in nonaqueous solutions of some surfactants (7) indicate a considerable monodispersion of the micellar phase. This finding is important with respect to the phase model. The experimental observations on nucleation in a series of nonaqueous
colloidal solutions below the critical micelle concentration provide additional evidence. In view of the fact that this phase transformation proceeds in the solute within a binary system (solute and solvent), the classification of such phase change is difficult. The Ehrenlest scheme (8) does not apply, but the change might belong to the anomalous transformations of the first kind as described by Mayer (9). In relation to the description of micellization as a phase transition, the nucleation process is of primary importance and will be described by means of a model related to the present experimental observations and to a recent theoretical paper on the stability of micelles in apolar media. EXPERIMENTAL
Substances
The ionic surfactant di-2-ethylhexyl sodium sulfonate (aerosol OT) molecular weight 444.57, was supplied by Fluka AG, Buchs, SG, in analytical quality; the di-2-pentyl sodium sulfosuccinate (aerosol AY), molecular weight 360, was supplied by the Cyanamid Company
281 Copyright ~ 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 48, No. 2, August 1974
282
E I C K E AND CHRISTEN
in the same quality. Both products were dissolved in benzene for further purification and stirred for several hours in the presence of active charcoal. After separating the charcoal, the solution was evaporated under vacuum. Then the aerosols were dissolved again in ethyl ether of highest purity and dried under high vacuum. The purified products were stored in a desicator over granulated phosphorous pentoxide in a warming cabinet at 50°C. This procedure proved to be most successful in drying the substances. Di-2-ethylhexyl sodium phosphate was prepared according to the description in an earlier paper (6). Chromatographically homogeneous lecithin was prepared from egg phospholipid by column chromatography over aluminium oxide. The method has been described elsewhere (10). The solvents used--cyclohexane, benzene, 1,4-dioxane, 2,2-dimethyl-butane--were of high reagent grade quality from FLUKA or M E R C K AG. The solvents were dried over calcium chloride, phosphorous pentoxide, or sodium/potassium alloy. Measurements
Vapor-pressure-osmometric, and dielectric dispersion measurements were used for the direct determination of the mean number of monomers per aggregate and its concentration dependence. The measurements were made at 20 4- 0.1°C with a Hitachi-Perkin-Elmer vapor-pressure-osmometer type 115, and an apparatus for the measurement for the dielectric absorption based on a sensitive resonance method (11). The equipment permits loss measurements at room temperature (20 40.05°C) in the frequency range 0.5-120 MHz with an accuracy of Atg 8 4- 1 X 10-6. Constructional details are given in Ref. 12. From dipole moment and conductivity measurements additional information on the nucleation could be derived. Dipole moment determination was carried out with the WTW-dipole meter type D M 01 (Weilheim, Germany). The dielectric constants of the solvent (el) and
solutions (e12) have been determined. Measuring frequency was about 2 MHz; the sensitivity of the measurement Ae/e-= 5 X 10-5 • The measuring temperature was adjusted to 20 40.05°C. The evaluation of the results and the computation of dipole moments followed the method of Guggenheim (13) and Smith (14). It is apparent that the plots of Ae = e12 -- el against weight-fraction of solute (w2) deviate from a straight line at low concentrations, depending on the cmc, i.e., the solvent used. Whether or not information on aggregates smaller than micelles can be obtained from the curved part of the plot below the cmc depends on the sensitivity (Ae/e) of the dipole meter. In this part of the curve a parabolic approximation was used (15) to fit the experimental plot, which resulted in extrapolated values of the dipole moments of the monomers. Conductivity measurements were made at 204-0.05°C with a Keithley electrometer type 602, or, if the specific conductivity was above 10-11 mho cm -1, a Wayne-Kerr-bridge type B221 at a constant frequency of 1592 (4-1%) I-Iz. The cell consisted of a thermostatable cylindrical metal condenser with gold plated electrodes (the same as used with the WTW dipole meter). Additional details are given elsewhere (6). RESULTS AND DISCUSSION
Experimental Evidence on Nucleation If the phase model approximation is accepted, the question arises as to the possible existence of nuclei. The first hint was given by conductivity measurements where two different particles below the cmc had to be hypothetically assumed in order to interpret theoretically the two experimentally observed distinct regions of the conductivity-weighedin-concentration plot. The first one is thought to correspond to the conductivity of monomers ; the second range should be due to colloidal aggregates of a few monomeric particles. The marked break in the initial slope of the conductivity curve (see Ref. 6), which separates these two regions, could be tacitly assumed
Journal of Colloid and Interface Science, Vol. 48, No. 2, August 1974
MICELLE FORMATION
/
283
/* /
-13.
/ zX /
Fro. 1. Specific conductivity (~2-1 cm-l) of aerosol OT in benzene at 20 4- 0.05°C as a function of weighed-in-concentration Emole 2-11 (see Ref. 6). to be a second critical concentration, as was done by other authors (16) for aqueous colloidal systems. The conductivity of the small colloidal aggregates, which is attributed to the almost horizontal second region of the conductivity plot, is now thought to be that of particles which may play the role of nuclei (see Fig. 1). This working hypothesis is strongly supported by a number of dipole moment measurements on aerosol OT, AY, di-2-ethylhexyl sodium phosphate, and lecithin in different apolar solvents. The dipole moment (~) divided by the m e a n aggregation number of the monomeric molecules ((n)) i.e., the contribution of one monomer to the dipole nloment of the aggregate, has been plotted against (n). From Fig. 2 it is seen that ~/
i.e., to an increase in the dielectric increment while here a strong decrease with the aggregation number is observed. These remarkable observations indicate that subunits of only three or four molecules ha~-e reached a stable (electrostatic) equilibrium configuration which is retained in the ultimately formed micelle. The stability of the trimeric subunits is demonstrated by the present results of the dielectric dispersion and dipole moment measurements in 1,4-dioxane. Both methods demonstrate, up to concentrations of 5.10 .2 m, that only trimers have been formed which show a normal Debye dispersion behavior. In view of the fact that dioxane is only a "pseudo" apolar solvent, since two dipoles are compensating for each other, the solvent will interact with the dipoles of the surfactant molecules as in all polar solvents. This is the reason that, in dioxane and in polar media, miceltes of the inverted type, as usually found in apolar solvents, are not formed. The occurrence of trimers in dioxane may indicate that this configuration has a preferred stability. In this connection, the findings and considerations of Ekwall et al. (17) on X-ray investigations of aerosol OT should be mentioned. According to molecular model studies, these authors assume that there is probably no room
Journal of Colloid and Interface Science, Vol. 48, No, 2, A u g u s t 1974
284
EICKE AND CHRISTEN
765-
321-
i X x
~
~x
X
{{
I0
I;
i
D
35
5-
~_ 'x A3-
21-
X ~ x - - x
FIG. 2. Dipole moment ~ divided by the mean aggregation number of monomers (n), i.e., the contribution of one monomer to the dipole moment of the aggregate (~ in [D]), as a function of (n} (Solvents: benzene, cyclohexane, carbontetrachloride, 1,4-dioxane, heptane, and 2,2-dimethylbutane:) (a) aerosol OT; (b) aerosol AY.
in the center of the cylindrically proposed micelle for six sulfonate groups in the same plane as is the case in the solid state. The molecules will presumably be arranged in two subgroups, each containing three sulfonate groups and located one above the other. These observations strongly support the idea of nucleation as a prerequisite for the phase approximation model.
will be discussed as a function of the number of interacting monomers (n). I t is possible to obtain AH(n) from a previous calculation (7) of the "selfenergy" AW(n) of micelles as a function of their aggregation number n. Referring to N1 monomers which form N2 = N1/no micelles where no equals the number of monomers of the equilibrium micelle (neglecting entropy contributions) -AH(n) = (Xl/no) ~W(n).
A Model for _PhaseChange and Nucleation
[11
To investigate the phenomenon of nucleation and the phase change, it seems suitable first to consider the course of AG, the free energy change for the entire process: monom e r - + micelle. Thus the expression
To determine the entropy difference AS(n) when a solution containing N1 monomers and No solvent molecules changes into a solution with N~ micelles, one could imagine the following processes :
A~(n) = ~H(~) - T/,S(n)
1. Nx monomeric molecules are mixed with
Journal of Colloid and Interface Science, Vol. 48, No. 2, August 1974
285
MICELLE FORMATION
sl /4-
\
x
2-
x ~ x
I0
8765-
A ~" V
32-
\ X
~
X
" X
10 '
15 ,
2=0
FIo. 2 (Cont'd.). (c) Di-2-ethylhexyl sodium phosphate; (d) lecithin.
No solvent molecules. The mixing entropy
of this process is AS1. 2. N2,= N { n micelles are mixed with No solvent molecules to give the entropy of mixing AS2. The entropy of mixing for the process: monomer ~ micelle is therefore AS = AS2 -AS~. The explicit calculation of the mixing entropy m a y follow the model of Hildebrand (18). This gives
AS1 = k N o l n
No(vo - bo) + X~(v~ - b~) No(vo - bo)
No(Vo
-
-
volumes, bo and bl their intrinsic van der Waals volumes, and k the Boltzmann constant. In Eq. [2] not only the number of arrangements of the particles but also their differing sizes are taken into account with the introduction of the free volume ( v - b). A corresponding expression can be written for AS2: No(Vo - bo) + N2(v2 - b2) AS2 = k No in No(vo - bo)
+N21.
No(Vo -- bo) + N2(v2 - 52)',,
N2( 2 b2-
)" E3]
bo) + N~(v~ -- b~)\
where vo and vt are the respective molal
With the assumption that % -- b2 = n(vl -- b~) and substituting N { n for N2, the entropy of mixing for the transfer: monomer--+ micelle
Journal of Colloid and Interface Science, Vol. 48, No. 2, August 1974
286
E I C K E AND C R H I S T E N
Finally, AG(n), the free energy change per mole, is given by
AGCn)
T AS
N Lho
&G(n) _~ no
_
_
_
_
_
(n ~ -- 2nno + 2no -- 1) no(no -- 1) ~
_
+RT ho
1-
In
[-6] N1
-3
.........
,b . . . .
n ....
io .........
FIG. 3. Schematic representation of the enthalpy AH, entropy AS and the free energy change &G of the nucleation and micellation processes with the number of interacting monomers, n.
reads AS=k
1--
Nlln n/
,
[-4-]
Xl
where a = (% -- bo)/(vl -- bl). The value of a may be estimated neglecting the contribution of the intrinsic van der Waals volumes bo and 51- Thus a ~ vo/vl = ( M o / p o ) / ( M 1 / m ) , which gives, for example, for the aerosol OT/benzene system a ~ 0.22, if m (aerosol OT) = 1.1 g cm -a according to Ekwall el al. (17). The detailed calculation of AH (n) is possible with the help of the results on the selfenergy calculations of micelles in apolar media (7), The gain of potential energy (zXW) per micelle follows approximately the square of the number of monomers, i.e., the variation of AW with n m a y be described by a parabola that passes through the point (n = 1, AG = 0) and has a minimum value ho at n = no (Fig. 3). The parabola ho
In general, this function (Eq. E6]) will show two extreme values (see Fig, 4a, b). The parameters ho, no, and a depend on the selected systems. The variables are the number of interacting monomers n, the respective mole numbers of solvent (No) and solute (NI) (referring to monomeric units), and the temperature, T. At constant temperature there will be a certain concentration of monomers N1 where AG(n) is zero. This is the critical micelle concentration (cmc). The height of the maximum of the AG(n) curve at n = nN can be interpreted in terms of a potential barrier and corresponds to the energy required to form a micellar nucleus. From the existence of such a potential barrier, it can be inferred, with respect to the micellization process, that it is energetically more favorable to attach monomers to a nucleus than to assume an aggregation of colloidal subunits. I t is now possible with the help of Eq. [-6] to derive quantitatively the following quantities which characterize the micellar system, especially with respect to the formation of nuclei. Determination of the micelle size f r o m the critical micelle concentration. The cmc occurs under the condition A G ( n ) = 0, i.e., at the
point where the curve touches the n-axis. From the physical situation it is evident that a + 2 n o - 1) [5-] second intersection of the curve with the fulfills these conditions. Hence from Eq. E1l n-axis has to exist at the point (n = 1, AG = 0) since growth and decomposition of the colloidal NlhoNL AH(n) _~ (n 2 -- 2nno aggregates start and end respectively with n o ( n o - 1) 2 +2no -- 1). [-5a-] monomers. When A G ( n ) = 0, Eq. [-6] gives
zXW (n)
~
(n ~ -
(no -
2nno
1) 2
[ournal of Colloid and Interface Science, Vol. 48, No. 2, August 1974
MICELLE FORMATION
287
giving
the following cubic equation for n:
2no -- 1
no(.o-
1) ~
~7~1,2
2
n 3 -- 2non 2 + 2non -- n + hoN L
R T n o ( n o _ l ) 2 1 n _ _a N o + N 1
[ (2n°-1)2 a N o -J- N1 × (n - i ) R r i.
IV1
- 0.
[7]
r7b]
Ni
The remaining two roots must be equal for the existence of a critical micelle concentration where the condition AG(n) = 0 is fulfilled. Therefore n l , 2 = n o - - ½ = r i m and consequently
One solution from the above argument is n = 1, and this reduces to no(no n 2 -- (2no -- 1 ) n +
-
1) 2
-
hoNz aNo + N i
(2no -- 1) 2
RT
4
hoYL
-m(no -
o, ETa]
XRTln
xln
1) 2
aNo + cmc
N1
[7c]
cmc
2.000
T
b c d
u
,x, i.OOO
o ~3 0.000
-i
.OOO
' -
.OOO
'
'
~
i
'
S.O00
'
~ fl
'
I i0.00
'
,
,
~
I
'
IS.O0
'
'
,
I
,
,
,
, ~"
20.00
FIG. 4a. Variation of the free energy change AG (kcaI mole -1) as a function of the aggregating monomers n (see Eq. [-6]). P a r a m e t e r s : T e m p e r a t u r e T = 293°K; micelle size n M = 17; N1 = 5"10 -4 mole-l.
~a) 2.9
|b) 3.0 3c) 3.1
ho (selfenergy of micelle) (eV) ] d ) 3.2
]e) U)
328 3.4
Journal of Colloid and Interface Science, Vol. 48, No. 2, August 1974
288
EICKE AND CHRISTEN
O .OOO
-t
.000
.
.
.
.
~
0.000
.
.
.
S.O00
.
~
rl
10.00
.
.
.
.
~
'
1~.00
,
I
.
.
.
.
~"
20.00
FIG. 4b. Variation of the free energy change AG (kcal mole -i) as a function of the aggregating monomers n (see Eq. I-6]). P a r a m e t e r s : T e m p e r a t u r e T = 293°K; micelle size nM = 17; selfenergy of micelle ho = 3.28 eV (see text a n d Ref. 7)
Ia) b) Ni.10 4mole-i c) d) e) Since it is known from a previous paper (7) that a relation between no and ho exists on physical grounds, no may be expressed with the help of Eq. [-7c-] and combined with the above equation for n~ to give h oNL n~u =
aN o + cmc
1 2'
however, that the apparent strong temperature dependence of n~, which is not experimentally observed, is considerably reduced by the fact that the cmc is also a function of temperature (see Fig. 5), namely cmc ~ aNoe-h°lvLl'~°~T,
[-7d~
R T in cmc if 1 is neglected compared with no. Eq. [-Td-] describes the size of the micelle under the explicit consideration of the entropy. This equation demonstrates that the size of the micelle is almost exclusively determined by the selfenergy ho at constant temperature (Table I). Hence the influence of the entropy is negligible with respect to the aggregation number of monomers. It should be noted,
1.0 2.5 -5.0 7.5 10.0
[-8-]
where the same condition applies as in Eq. [-7d~. Size of the micellar nucleus. With respect to Fig. 3, the extreme values of &G(n) (Eq. [6-]) have to be investigated to obtain information about the size and energy of formation of the nucleus. Hence &G is differentiated and equated to zero. The region of the formation of the nucleus is of special interest, i.e., the maximum value of the function f~G(n) at the
Journal of Colloid and Interface Science, Vol. 48, No. 2, ALxgt~st1974
289
MICELLE FORMATION TABLE I DEPENDENCE OF TItE SEL~ENERGY
]¢oj THE
SIZE OF THE
/
10,0
N U C L E U S nN~ A N D TIIE E N E R G Y OF FOR3~AT10N
AGN O N
0~" TI-IEN U C L E U S
T K E SIZE 0P TIIE
C M C = 5-10 -~ M AND T = 293 K
.E.
TEMPERATURE nM
14
15
16
17
18
ho (eV) nN zXGN (kcal/mole)
2.85 2.90 2.01
3.06 3.00 2.09
3.28 3.10 2.15
3.49 3.20 2.21
3.70 3.25 2.27
point n
=
×
/
M I C E L S E n M A T CONS:rAN~r
/
5,0
x
x/x
nN. Here
(o%
2'73 283 T 293[°K]303
.=o 2hoNL n
-
--
/go)
1) 2 RT
+ --
a2¢o + N1
in
,
[-9-]
IV I
152
where # N refers to the chemical potential of monomers in the nuclei,/~ to that in the monomeric phase. Eq. [9~ can be rewritten to give
which is the condition for determining the number of monomers in the nucleus (n~;). One root of the equation (n = no -- ½) is known from Eq. [-7c] which applies at the cmc. Hence, replacing the third term by (2no -- 1)2/8 (see Eq. [7b]) and dividing b y n - no + ½, 152 - - -
n a -- non 2 +
no(no
--
-- ½(no
- - ½) =
0
[-9b]
2
1) 2
RT
X
3;~3
Fro. 5. Critical miceIle concentration (cmc) (mole-1) as a function of temperature T(°K). Parameters: Micelle size n M = t 6; selfenergy of micelle ho = 3.28 eV.
On I n=~v
no(no
3i3
is obtained with the solution:
a d o q- N1
in
- 0
[-9a-]
N1
2hoNL
nl,2 = ~ 4- 4 ~
+ ½0~o - ½),
",× 2,3'
\ X
2,2"
2,1.
~ × ~ 2,0"
~,~c.lO"
slo
'[mill
1<0
"
FIG. 6. Free energy of nucleation as a function of the critical micelle concentration. T h e cmc is varied by changing the micelle size. P a r a m e t e r s : T e m p e r a t u r e T = 293°K; selfenergy of micelle ho = 3.28 eV. Journal of Colloid and Interface Science, Vol. 48, No. 2, August 1974
290
EICKE AND CHRISTEN
where only the plus sign is of physical significance. Substitution for no from Eq. [-7c] yields
1 =
-
/
hoNL
3
aNo q- cmc
16'
[io]
+
4
R T In
cmc giving the size of the micellar nucleus as a function of the critical micelle concentration and temperature for a specified system (a, ho). It should be noted that nN is--in contrast to nM--dependent on the square root of ho, from which it can be inferred that the size of the micellar nucleus is relatively insensitive to the stabilizing energy ho per micelle and to variations of the critical micelle concentration. This is consistent with the present experiments.
Calculation of the free energy of nucleation. With the help of Eq. [-10] the necessary energy for the formation of a micellar nucleus may be calculated. Substitution of n~; from Eq. [-10] in Eq. [6-] neglecting the first term (~), generally small compared with the value of the square root, yields:
AGz¢ = no(no -- 1) 2
aN o + cmc T in cmc
--2no R T in
aNo + cmc
[-2no---16
cmc
aNo -}- cmc
16
T in cmc
-}
l R T In
hoNL
3
aNo -}- cmc
16
cmc
X R T in
aN o -]- cmc
[n]
cmc
This equation [-11] describes the dependence of the energy of nucleation on the critical micelle concentration (Fig. 6) and the selfenergy ho (see Table I) at constant temperature. Figure 6 demonstrates that AG~vincreases
with decreasing critical micelle concentration at constant temperature, as is to be expected. The energy necessary to overcome the potential barrier, represented by AGN, is supplied by thermal fluctuations. [A detailed discussion on the origin of the nuclei, based on these fluctuation phenomena with the help of a Monte-Carlo model, is in print (19).-] With respect to the experimental results and the theoretical considerations presented above, it seems indeed possible to discuss the formation of micelles in the investigated systems as phase changes with the expected preformation of nuclei. REFERENCES 1. JONES,E. R., AND BURY, C. R., Phil. Mag. 4, 841 (1927). 2. Scmctq M. J. (Ed.), "Nonionic Surfactants," Marcel Dekker, New York, 1967. 3. ELIAS, i . G., "The Study of Association and Aggregation via Lightscattering," in "LightScattering from Polymer Solutions," (M. B. Huglin, Ed.), New York, 1971. 4. KREUTZER,J., Z. Phys. Chem. B53, 213 (1943). 5. HALL, D. G., Trans. Faraday Soc. 66, 1351 (1970). 6. EICKE, H. F., AND ARNOLD,V., J. Colloid Interface Sci. 46, 101 (1974). 7. EICKE, tI. F., AND CI~RISTEN, H., J. Colloid Interface Sd. 46, 417 (1974). 8. M[/NSTER, A., "Statistische Thermodynamik," Springer, Berlin, 1956. 9. MAYER,J. E., AND GOEPPERT-MAYER,M., "Statistical Mechanics," Wiley, New York, 1959. 10. SINGLETON, W. S., GRAY, M. S., BROWN, M. L., A~D W~IITE, J. L., J. Amer. Oil Chem. Soc. 42, 53 (1965). 11. BERGMANN,K., EIGEN, M., A N D :DE MAEYER, L., Ber. Bunsenges. Phys. Chem. 67, 819 (1963). 12. BERGMANN,K., Ber. Bunsenges. Phys. Chem. 67, 826 (1963). 13. GUGGENKEI~, E. A., Trans. Faraday Soc. 45, 714 (1949). 14. S~ITI~, J. W., Trans. Faraday Soc. 46, 394 (1950). 15. OE~M~, F., "Bestimmung des molekularen Dipolmoments," WTW, Weilheim Obb. (1960). 16. MR~RA, M., AND KODAMA, M., Bull. Chem. Soc. Jap. 45, 428, 2265 (1972). 17. EKWALL, P., MANDELL, L.~ AND FONTELL, K., J. Colloid Interface Sci. 33, 215 (1970). 18. HILDEBRAND,J. H., P~AUSNITZ,J. M., AI~I)ScoTT, R. L., "Regular and Related Solutions/' Van Nostrand Reinhold, New York, 1970. 19. CIrRISTEN H., AND EICKE, H. F., J. Phys. Chem.
Journal of Colloid and Interface Science. Vol. 48. No. 2, August 1974
(i974).