7628
J . Phys. Chem. 1990, 94, 7628-7634
Evaluation of Viscosity Measurements of Dilute Solutlons of Ionic Surfactants Forming Robshaped MlceHes H.-H. Kohler* and J. Stmad Institute of Physical and Macromolecular Chemistry, University of Regensburg, 0-8400 Regensburg, FRG (Received: February 13, 1990)
A theoretical model that describes the association of ionic surfactants into rodlike micelles under equilibrium conditions is constructed. The model is based on the concept of counterion binding. Moreover, a simplified formula for the concentration-dependent viscosity increase of a solution of rodlike colloids is developed, that provides a good approximation of the exact equation of Kuhn and Eisenschitz. In combination with the association model it is possible to evaluate all thermodynamic model parameters from viscosity and surface tension measurements. This approach is applied to micellar solutions of hexadecylpyridinium nitrate and bromide at various electrolyte concentrations. It is shown that the pronounced effect of counterion concentration on the growth of rod-shaped micelles reflects a small excess of counterion association in the cylindrical part of the micelle. The degree of counterion association is only by about 0.025 larger in this part of the micelle than in the hemispherical caps.
I. Introduction Two types of models are needed for the interpretation of viscosity data obtained from dilute solutions of rod-forming surfactants: one to describe viscosity as a function of the size distribution of the rods, and one to describe the actual size distribution of the micelles as a function of the composition of the solution. The former model will be called a viscosity model, the latter one a micelle association model. Numerous contributions have been, made in the past to both types of models.l-I* In this paper we will use such models with the aim of providing a more reliable basis for the evaluation of viscosity data. An effort will be made to simplify the theoretical relations as much as possible without risking substantial loss of accuracy. Our treatment will be restricted to solutions with high counterion concentrations where the ionic cloud virtually does not affect the hydrodynamic dimensions of the micelle. Geometrically, a rod-shaped micelle consists of a central cylindrical part and of more or less hemispherical caps at the ends. We assume these caps to be strictly hemispherical and to have the same radius as the cylindrical part (see Figure 1). This geometry seems to be reasonable at high counterion concentrations where the optimal head-group area of the surfactant ion will be relatively small.' The viscosity increase caused by short rod-shaped micelles-with an axial ratio of long axis over short axis smaller than about 4-is not very different from that due to spherical micelles. Therefore, these short rods can hardly be detected in viscosity measurements. Our interest therefore will be focused on micelles of an axial ratio greater than 4. In the last part of the paper we will apply the theoretical ( I ) Jeffery, G. B. Proc. R. Soc. London, A 1923, 102, 163. (2) Eisenschitz, R. Z. Phys. Chem. A 1933, 163, 133. (3) Simha, R. J . Phys. Chem. 1940, 44, 25. (4) Kuhn. W.: Kuhn. H. Helv. Chim. Acta 1945. 28. 97. (Sj Mukerjee,'P. J . Phys. Chem. 1972, 76, 525. (6) Tanford, C. J . Phys. Chem. 1974, 78, 2469. (7) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J . Chem. Soc., Faraday Trans. 2 1976, 72, 1525. (8) Tausk, R. J. M.; Overbeek, J. Th. G . J . Colloid Interface Sci. 1976, 2 119 -,
(9) Missel, P. J.; Mazer, N. A.; Benedek, C. Y.; Young, C. G . ;Carey, M. C . J . Phys. Chem. 1980, 84, 1044. (10) Ozeki, SI;Ikeda, S.J . Colloid Interface Sci. 1980, 77, 219. (11) Nagarajan, R. Colloids Surf. 1982, 4, 147. (12) Porte, G.; Appell, J. J . Phys. Chem. 1981, 85, 251 1 . (13) Nagarajan, R. J . Colloid Interface Sei. 1982, 30, 477. G.; Carey, M. C. J . Phys. (14) Missel, P. J.; Mazer, N. A.; Benedek, C. Y.; Chem. 1983,87, 1264. (15) Porte, G.; Poggi, Y.; Appell, J.; Maret, G. J . Phys. Chem. 1984,88, 5713.
(16) Eriksson, J . C.; Ljunggren, S.J. J . Chem. Soc., Faraday Trans. 2
1985, 81. 1209.
(17) Gamboa, C.; Sepulveda, L. J . Colloid Interface Sci. 1986, 113, 566. (18) Imae, T.; Abe, A,; Ikeda, S. J . fhys. Chem. 1988, 92, 1548.
relations to viscosity data of hexadecylpyridinium (Cp+) solutions containing the anions NO3- and Br- a t different concentrations. 11. Viscosity Model
The "intrinsic viscosity" or "shape factor" of colloid particles in solution is defined by s = A%eI/@
with
A h 1 = (7 - r l 0 ) / 8 0
(1)
where q is the viscosity of the solution containing colloid particles at a volume fraction a, 90 the viscosity of the corresponding colloid-free solution (which may be the pure solvent), and vrelthe relative viscosity increase. We equate the intrinsic viscosity of rods with that of prolate ellipsoids having the same axial ratio and assume weak shearing forces so that the thermal motion of the particles is only slightly disturbed. A corresponding expression for the intrinsic viscosity of an ideal solution of prolate ellipsoids of a given axial ratio f = b / a ( b is the long axis, a the short axis) has been derived by Simha3and by Kuhn and Kuhn4 in connection with earlier papers by Jeffery' and Eisenschitz.2 Unfortunately, the crucial equations of these latter papers, also referred to by Simha and Kuhn and Kuhn, are distorted by misprints. Therefore, the formula resulting from the corrected expression is given in the Appendix. The corresponding curve is shown as curve a in Figure 2. F o r f = 1 the value of s is 2.5, in agreement with the Einstein viscosity law. For large f the exact result can be approximated by the "Simha equationn3 s = 1.6
+ f2( 15 In (21) - 1.5 + In (21) - 0.5
As is seen from curve b in Figure 2, the approximation is poor for f < 8. For very large J eq 2 can be further simplified to
f2
s=- 4
(3)
15 In (0.941)
Taking advantage of the structure of this limiting expression, we have determined an approximate equation for s, which is simpler than eq 2 and is of reasonable accuracy forf> 3 or s > 3.5. This equation reads s = 0.25f2/ln (0.631)
f > 3 or s > 3.5
(4)
This relation is shown as curve c in Figure 2. We will use eq 4 throughout this paper. For polydisperse rod-shaped particles the intrinsic viscosity s, as defined by eq 1, equals the volume average of the right-hand side of eq 4. Denoting the volume average of a quantity x by (x)", we obtain s = 0.25(
f2
In (0.631)
0022-3654/90/2094-7628%02.50/0 0 1990 American Chemical Society
)
f>3
The Journal of Physical Chemistry, Vol. 94,No. 19, 1990 7629
Ionic Surfactants Forming Rod-Shaped Micelles nS -
f = -2 -n+ - 1
"S -
"C
2
2
3 ns
3
Under equilibrium conditions, formation of a micelle of aggregation number n, M,, can be regarded to be the result of the chemical reaction nA * M, b
Figure 1. Schematic drawing a rod-shaped micelle of length b and diameter u. The cylindrical part is built from nc, each of the two hemispherical caps from ns/2 surfactants ions.
(94
The role of counterions, whether directly bound to the micellar aggregate or forming part of the ionic cloud, will show up implicitly in the equilibrium constant and the activity coefficients. In effect, use of this equation has led to a number of theoretical contributions in which the possibility of counterion association in the sense of direct binding to the micellar aggregate is neglected.6~9.11~16~'9.20 As a consequence, the role of the diffuse double layer is liable to be overestimated. Alternatively, counterions can be taken into account explicitly by a reaction equation of the form nA
+ nBB
M,
(9b)
Here a micelle is assumed to consist of n surfactant ions and nB counterions. We write nB in the form
n~ = w
s + Yc(n - ns)
where ys and yc are the fractional counterion association coefficients of the hemispherical and the cylindrical parts. Reaction 9b then can be written as nA 4
2
6
8
10 -f
12
14
16
Figure 2. Intrinsic viscosity s of a solution of rod-shaped particles as a function of the axial ratio/. a: Exact relation (see Appendix). b Simha relation (eq 2). c: Relation according to eq 4.
As we assume incompressible particles, the volume average equals the weight average. The numerator,?, changes more rapidly with f than the denominator, In (0.63f). Therefore, it is justified to replace the denominator by the following linear approximation around a reference value fo: 1
In (0.63f)
In (0.63f0)
(1+
fo In (0.63f0)
Substituting this expression into eq 5 , we obtain
Choosingfo = cf3),/cf2),, the second term becomes zero, and the result simplifies to I
c2\
The lower validity bound, (f),O.s > 6, has been intuitively assumed to be twice that off in eq 4. Then s is necessarily greater than 4.5. 111. Association Model for Rod-Sbaped Micelles We consider rod-shaped micelles formed from a monovalent surfactant ion A in the presence of a monovalent counterion B. A micelle of aggregation number n consists of nc monomers of A, forming the cylindrical part of length 1, and diameter a, and of ns monomers forming the hemispherical caps of radius a / 2 (see Figure 1): n = nc ns (7)
+
Neglecting the volume contribution of counterions, the volume of a micelle is V = nu, where v is the volume of a monomer of A. We assume v, a, and ns to be independent of n. The axial a ) / a . From simple geometry ratio is/ = b / a = (Ic
+
+ [ w s + ~ c ( -n n d l B + Mn
(9c)
This equation gives special emphasis to direct counterion binding, while contributions of the ionic cloud are only implicitly taken into account. Since the surfactant ion plays the role of a co-ion, the influence of its concentration on ys and yc will be small. The influence of the counterion concentration on the y's is more difficult to assess, but there is experimental evidence that its influence, too, is rather small.12-21For the following theoretical treatment, we therefore postulate concentration independence of the coefficients ys and yc. This assumption will be reexamined in the discussion section. In principle, our treatment will be similar to that presented by Porte and Appell in their work on ionic rod-forming surfactants.12 However, there are two important differences: first, the aggregation number will be treated as a continuous variable throughout, and second, a special parameter representation of the size distribution function will be introduced. These features will lead to rather simple general expressions for the size distribution function and its moments. Let cA and cg denote the molar concentrations of free A and B, cM(n) the molar concentration of the micelle M,, and p,,, pg, and p M ( n )the respective chemical potentials. Chemical potentials will always be taken with respect to a single particle. The concentration of surfactant ions incorporated into micelles is given by
The total surfactant concentration is
A corresponding equation holds for cBm. Application of the law of mass action to reaction 9c yields
where the equilibrium constant K ( n ) can be written as (19) Stigter, D. J . Phys. Chem. 1974, 78, 2480. (20) Evans, D. F.; Ninham, B. W. J . Phys. Chem. 1983,87, 5025. (21) Heckmann, K.; Schwarz, R.; Stmad, J. J . Colloid fnrerfuce Sci. 1987, 120, 114.
7630 The Journal of Physical Chemistry, Vol. 94, No. 19, 1990
1
1
- npoA - (YCn + (7s - YC)nS)P0B1
K ( n ) = exp --[poM(n) kBT
(1lb) Here kBis the Boltzmann constant and co a standard concentration, which we identify with the surfactant concentration in the condensed state (co = (NAu)-],N A = Avogadro constant). The PO'S are the corresponding standard potentials. As mentioned above, we assume the y's to be independent of the composition of the solution. We further assume that poM(n) is a linear function of n (cf. refs 9 and 14): PoM(n) =
w 0 s + ( n - ns)poc
(12a)
where pos and poc are independent of n. The standard works, wos and woc, for the reversible transfer of a surfactant ion from the free solution to the hemispherical and to the cylindrical part of the micelle are
was=
p 0S
- 0A - YSPoB - poA - yCpoB
(1 2b)
= p°C ( 12c) Due to the above choice of co, w0s and w0c become zero if there is no specific interaction energy. (This means that the interaction energy of a particle within the aggregate equals that in the free aqueous solution.) The standard work for the reversible transfer of a surfactant ion from the hemispherical to the cylindrical part thus is w°C
AwO = WOc - wOS
(124 If the effect of a variable counterion concentration cBis included, the standard work becomes
Aw* = Awo - kBTAy In (cB/co)
( 12e)
where A y = ye - ys. Using the standard works gS and gc, we now define the characteristic concentrations c * and ~ c * by ~
Kohler and Strnad having n appreciably larger than % will be virtually absent unless cA equals cmc. Under these conditions we have (1 5a)
6
cM(nS)/cO
In an ideal solution the concentration ratio on the left-hand side, and thus 6, must be much smaller than unity. We conclude that rod formation can occur only if the condition 6 4.5, the corresponding value of is read from curve 11 in Figure 5 (or numericalb calculated from eq 24). According to eq 18a, the values of obtained at varying values of cA, but constant CB should give a straight line with slope l/(co6). This allows one to determine 6. According to eq 14b, the values 6,and Sj obtained for different counterion concentrations cB,and cB, should satisfy the relation
where Ay = yc - ys. If ns is known, this allows one to determine Ay, Approximate values of c * and ~ yc can be obtained from an experimental curve of the critical micelle concentration vs cB;see eq 13c. From yc and AT, y s can be determined. Then any pair of 6 and CB can be used to determine /3 and c*s from eq 14b. Finally, the standard work terms wos and w", can be calculated from eqs 13a,b. In summary, we see that apart from the monomer volume u and the aggregation number ns, all thermodynamic parameters used in the theory can be determined from viscosity and cmc measurements.
V. Experimental Section Reagents. The surfactants used were prepared by crystallization from solutions of CpCl (Merck AG, West Germany) with a surplus of the corresponding acid. They were recrystallized twice from absolute ethanol and dried under vacuum. The potassium bromide and nitrate (also from Merck AG) were of p.a. grade. Methods. The cmc's were determined from surface tension measurements by use of a digital tensiometer (K 10 T, Kruess, West Germany). The viscosities of the surfactant solutions were measured in two Ubbelohde viscometers thermostated at 25 and 36 f 0.05 "C, respectively. The viscosities of all solutions were found to be independent of capillary diameter and thus of flow rate. VI. Experimental Results and Discussion ( a ) Solutions Containing CpNO,. In Figure 6a experimental data for the relative viscosity increase Avd of hexadecylpyridinium nitrate (CpNO,) solutions are shown as a function of eA a t different KNO, concentrations CE (electrolyte concentration). The monomer volume u of a Cp+ ion is about 0.5 nm3, giving a molar
7632 The Journal of Physical Chemistry, Vol. 94, No. 19, 1990 W
4.0r 3.5 -
Kohler and Strnad 14 12
3.0 -
10
8 6 4
2
0. 0
I
I
I
I
2
4
6
8
0
1
10
0
5
10
15
20
25
FA
FA
exact
L
1 .Ooo cA/mc
30
0.995 approx.
0.990-
/
/ - - -
/--
I
0.985 -
/
/
0.980- /
I
0
5
10
15
20
25
30 PA
0
I
I
5
10
I
15
I
I
I
20
25
30 1' A
Figure 4. Comparison between exact and approximateexpressions related to micelle size distribution. All quantities plotted as functions of the reduced
concentration FA. a: w (from eqs 18b and 23b). b: cA/cmc(from eqs 16 and 23a). c-e;: ( n ) " , (n2),1/2,and (n3)01/3(from eqs 22b,d and 23d. For calculation of the exact curves in b e , w is used as an auxiliary parameter determining Fa according to eq 18b. volume UNA = I/co = 0.30 L mol-' (where NA is Avogadro's constant) and a volume fraction
CP = EA/CO = 0.30EA/(mol Le')
(26)
From this relation and eq 1, the experimental value of Aqrel obtained at cs = mol L-' and cA = 9.85 X lo-' mol L-' yields an intrinsic viscosity s = 3.0. Within experimental error, this compares favorably with the value of 2.5 in the Einstein viscosity law and shows first that at cs = IO-* mol L-' the micelles are practically of a purely spherical form, and second that at this and higher values of cs the hydrodynamic volume of a surfactant monomer contained in a micelle can be equated with the condensed-state monomer volume u. This also means that the thickness of the ionic cloud can be neglected. Note, on the other hand, that Avd, and thus the hydrodynamic volume, increase when CE is lowered from 1W2 to 0 mol L-I, which reflects the expansion of the ionic cloud at reduced electrolyte concentrations. As is seen from Figure 6a, there is a strong increase of viscosity with both EA and CE at electrolyte concentration above mol L-'which can be attributed to the formation of rod-shaped micelles. From the experimenjal data in Figure 6a and from curve 11 of Figure 5 , a value of r A can be determined for every s = Aqwl/CP (where Q, is obtained from eq 26). The results are plotted as FAvs f Ain Figure 6b. (Only data with Avml> 0.01 have been
taken into account. At lower values of Aqrel the scatter in the experimental data increases, and even negative values of Avml sometimes occur. We have no explanation for this strange effect.) At the high values of cE specified in Figure 6b, cE may be identified with the concentration of the free counterions, cB. Following the numbering in Figure 6b, we thus have cB, = 0.05 cB2 = 0.07 cB3 = 0.1 cB, = 0.2 (27) (a! c values in mol L-I). According to eqs 18a and 14b the points of ra vs EA belonging to the same ce, should lie on a straight line with slope l/(cob). From the fits we obtain 6, = 2.16 x 10-5 d2 = 9.34 x 10-6 J3 = 4.98 X lo4 b4 = 1.54 X 10" (28) Note that these values are in agreement with condition 15b. Using ns = 80 (ref 22) and applying eq 25, we obtain A712 = 0.031 A723 = 0.022 Ay34 = 0.022 (29a) where the subscripts correspond to the values of i and j used in eq 25. The mean value of the Ayijs is Ay = 0.025 ( 2 9 ~ (22) This value for ns is obtained if it is supposed that a Cp micelle has a radius of 2.2 nm and a density of 0.9 kg dm- .
The Journal of Physical Chemistry, Vol. 94, No. 19, 1990 7633
Ionic Surfactants Forming Rod-Shaped Micelles S
1.7 -
14r 1 7 d
12 10 8-
1.6
-
1.5
-
1.4
642-
01 0
I
'
1
.
'
'
1
20
10
1
'
11.2 .3
1
40
30
e
i I
0.01 M
+ 0.07 M
. x
0.10M 0.20M
1
/
/ x '
fA
2.5 -
log s 2.0 -
2
4
6
8
10
EA
1.5 -
10-3 mol 1-1
fn 1 600
-
0 0
1
2
3
log f
A Figure 5. Intrinsic viscosity s vs reduced concentration FA. a: Comparison between exact and approximate relation. b: Logarithmic plot (decadic) of approximate relation. 600
Surface tension measurements reported in ref 21 yield the estimates yc = 0.76 and c * = ~ 6.0 X lo4 mol L-]; with eq 29b one thus obtains ys = 0.735. Following the procedure outlined in section IV (using 6 = b4 and A y = A?,,), one finds j3 = 0.02 and c * =~ 1.O3clc. By comparison with eq 28 we conclude that the small 6,'s in eq 28 are not so much due to a small j3 but rather to small q ' s . Because of A y > 0, the ails decrease with increasing cB. This leads to an enhanced formation of long micelles and thus gives rise to the experimentally observed increase of viscosity. Finally, we calculate from eqs 13a,b @s = -14.91kBTand @, = -15.17kBT. The standard work A@ (eq 12d) for the reversible transfer of a single surfactant ion from the hemispherical to the cylindrical part thus is A@ = -0.26kBT. Because of the positive value of Ay found in eq 29b, the transfer energy Aw*, defined in eq 12e, will decrease with increasing cB. According to eq 14d, this entrains decreasing values of 6 and thus favors the formation of long rods. In our experiments rod-shaped micelles actually form only in the range cB > mol L-' (Figure 6b). With cB= lo-* mol L-I, eq 12e gives Aw* = -0.1 1kBT. Although this value is still negative, the low surfactant concentrations used in our experiments make rod formation entropically too unfavorable to occur. In combination with the square-root dependencies in eqs 23c,d, the regression lines in Figure 6b provide complete information about the aggregation number 2verages as functions of = 500 with ns = 80 EA. For example, eqs 23c,d yield for
( n ) = 1790
( n ) , = 3580
(n2):,5 = 4380
By virtue of eq 8, the axial ratio averages are
# = 15.2
0, = 30.2
(f2)fi5= 36.1
For the counterion concentrations cB, through CB, (eq-27), the following values of the surfactant concentrations giving = 500 are read from Figure 6b: CIA, = 37, EA, = 16, EA, = 8.4, and ?AI = 2.5 mmol L-]. The values of CIA, and ?At lie outside the experimental range. ( b ) Solutions Containing CpBr. Figure 7a shows viscosity data for CpBr solutions. As is well-known, rod formation is not so pronounced with Br- as with NO3-. Again, the r A vs EA data
1
cE= 0.10 MA
0
".".J
2
4
6
8
111
10
?A
10-3 mol. 1-'
Figure 6. Viscosity of CpN03solutions at different KN03concentrations cB. a: Relative viscosity qrrl = 7 / q o vs surfactant concentration tA. b R e d u d concentration r A vs 2,. Below the dashed line (following from the equality sign of eq 31) rotation overlap can be neglected.
shown in Figure 7b for cE = cB = 0.2 and 0.4 mol L-' can be reasonably fitted by straight lines. In the same way as described for CpN03, we find A y 1 2 = 0.035 (again u = 0.5 nm3 and ns = 80 are used). Porte and AppellI2 have deduced A y = 0.05 f 0.01 from light scattering experiments, which reasonably agrees with our value. In comparison with CpNO,, no fundamentally new features appear in the CpBr system. Therefore, we omit further discussion. Discussion ofthe Model. In the theoretical treatment of section 111 the counterion association coefficients ys and y c were assumed to be independent of cB. However, according to eq 29a, these parameters vary slightly with counterion concentration. It can be shown that such small variations will not affect the applicability of the model as long as the difference In (cB/c*~)- Iln (c*c/c*s)I is sufficiently large (some tenths of a unit may be sufficient). In the exposition of the model we have further assumed that the micelles are (1) nonineracting, ( 2 ) rigid, and (3) only weakly oriented by the shearing forces competing with thermal agitation. Carrying out viscosity experiments with capillaries of varying diameter, which implies varying shearing intensities, no changes of viscosity could be detected in the Cp+ solutions discussed under subsections a and b (see under Experimental Section). Thus assumption 3 can be taken for granted. From theoretical considerations concerning the bending vibrations of rod-shaped micelles we conclude that rigidity can be assumed in a good approximation if (fz)0.5is smaller than about
7634 The Journal of Physical Chemistry, Vol. 94, No. 19, 1990 1.30'lrel
1.25 -
cE= 0.0
1
i
1.20 -
~
~
I
1.15 -
I
I I
0.01 M 0.05M + 0.20M I
e
0.40M
x
l
l
2
1 4
i
I
I
l
I
a
6
I 10
EA
10-3 mol
1-1
Kohler and Strnad dashed curve in Figure 6b (it is outside the range of Figure 7b). Except the outermost point for CpN03, all experimental points satisfy condition 3 1 and thus assumption 1. (4 ConcludingRemarks. In summary, it has been shown that viscosity measurements of diluted solutions of rod-forming ionic surfactants can be interpreted on the basis of a counterion binding model of micellization in combination with a viscosity model. A considerable amount of quantitative information can be obtained about the structure and the thermodynamics properties of the micelles. Direct comparison with the exact formulas shows that, within the experimental range, relatively simple approximations can be used without substantial loss of accuracy. In this context it should be noticed that viscosity measurements require only relatively simple experimental setups and are of high reproducibility. In the experimental case of hexadecylpyridinium salts, rod formation is driven by a preferential adsorption of counterions to the central cylindrical part of the micelle. This is probably a general feature of rod-forming ionic surfactants. It is surprising that a micelle association model based on the assumption of concentration-independent counterion association coefficients provides a fairly quantitative description of rod formation. This point has already been discussed by Porte and Appell,Iz but a physical explanation of the phenomenon is still missing. Perhaps the answer will arise from a detailed comparison of the present counterion association model with a microscopic description based on a Gouy-Chapman-Stern approach (as an extension of the Gouy-Chapman approach used, e.g., in refs 9 and 20).
Acknowledgment. We thank Mrs. H. Brunner for experimental help and Mr. A. Heindl for valuable computer assistance.
cB= 0.20M
2
4
8
6
10
?A
10-3 mol I-'
-re 7. Viscasity of CpBr solutions at different KBr concentrations CE. a: Relative viscosity vs surfactant concentration,,?, b: Reduced concentration F A vs ?A.
Appendix Shape Factor of Prolate Ellipsoids in Ideal Solution under the Condition of Complete Brownian Motion. As has been indicated in section 11, some errors have crept into relations derived by Jeffery' and Eisenschitz2 that were later used in the theories of the shape factor s presented in refs 3 and 4. (Actually, the errors in the Jeffery paper were corrected in the Eisenschitz paper (footnote, page 138) but then new errors appear in eq 3 of the Eisenschitz paper.) Therefore, we give here the complete formula for s as a function of the axial ratiof: 14
- IO+
100. This is the case in our experiments, so that assumption 2 is also satisfied. Interaction of micelles will be small if they do not overlap in their rotary movement. This requires that the volume covered by random rotation (a sphere of radius b/2) be smaller than the solution volume per micelle. Taking linear number averages, this translates into
4f2 f2(4f2 - 10
3a)
+
(f2
+
6 1)(2fZ + 4 - 3f2.)
X
+ 2 - (4f2 - 1)a
+ 3a)(-6+ (2f2 + 1)a) + 38
f2(p
+ 1)(-2 + (2f2 - I).)
where
-
8 =fZ-
where-(N
ns(r,)6
is the solution volume per monomer. With ( n ) (eq 23c) and ( b / a ) = 2(n)/(3ns)(eq S), we obtain ?A
f,
5 11.3
mol L-' The relation following from the equality sign is indicated by the
a=- 1
fj6
In
1
f+fi
fr
1
f-fi
This formula corresponds to the general relations of eq 6 in ref 3 and eq 76 in ref 4.
