Langmuir 1994,10, 3955-3958
3955
Articles Determination of Aggregation Numbers and Ionization Degree of Micelles Using Surface Tension Isotherms of Maleic Acid Mono[2-(4-alkylpiperaziny1)ethylesters] R. Wiistneck,*lt P. Enders,S and H. Fiedler5 Department of Solid State Physics-Interfacial Dynamics, University of Potsdam, Rudower Chaussee 5, 12489 Berlin, Germany, Max-Born-Institut, Rudower Chaussee 6, 12489 Berlin, Germany, and Max-Planck-Institute of Colloid and Interface Science, Rudower Chaussee 5, 12489 Berlin, Germany Received December 27, 1993. In Final Form: May 17, 1994@ By use of the slopes of surface tension isotherms at concentrationsabove and below the critical micelle concentration,the aggregationnumbers of homogous maleic acid mono[2-(4-alkylpiperazinyl)ethylesters] micelles are determined at pH = 6.2, where these surfactants exist in the betain-like form. For pH values at which the adsorptionof almost only one surface active ionic component of these surfactants is guaranteed, the degree of counterion binding is determined for two samples of the homologous series. The results obtained show that the method applied to determine these values from surface tension measurements opens a new possibility to interpret surface tension isotherms. The method needs, however, very accurate measurements because even small deviations may alter the results significantly.
Introduction
checked using this method? despite its simplicity. Therefore, systematic data are not available. The goal of this paper is, thus, to fill this gap. A surfactant system is investigated which has been extensively characterized's8 using surface tension measurements. These experimental data are examined in the light of the thermodynamicmassaction theory of cmc.
Measurements of surface tension, u, are widely applied for characterizing the adsorption behavior of surface active materials. These measurements yield material constants, such as the minimum area demand of a molecule adsorbed or spread. Approximating adsorption models, further parameters can be determined, i.e., adsorption energy Experimental Details parameters and even interaction parameters. Usually, As surfactants maleic acid 2-(4-alkylpiperazinyl)ethyl moreliable measurements of surface tensions are limited to noesters with alkyl chain length from 9 to 13 have been used concentrations up to the critical micelle concentration (denoted below as PIP 9-PIP 13). In aqueous solution, these (cmc, c,) or to 1 order of magnitude higher. The surfactants exist in different ionic forms, depending on the pH dependence of the surface tension on the concentration value. Their adsorption behavior has been characterized over for the case c > cmhas been discussed as well. The results Surface , ~ tension measurements a wide range of pH v a l ~ e s . ~ however are offen inconsistent. Generally accepted is the have been carried out using the ring method version of du NoUy.' experimentally confirmed change of the slope of the u-logThese experimental data7j8 were sufficient t o describe the adsorption behavior at concentrationsc < c,,,. At concentrations ( c ) isotherm when the cmc is passed. According to phase c > cm,in some cases, additional measurements were necessary. models,l the parameters of state remain constant at c > At such high concentrations,the surface tension became constant cm. Despite a minimum of u that is caused by impurities within a few minutes. The measurements have been carried out having higher surface activity than the main component 30 min afier aspirating the surface. to be characterized, u is found to be more or less constant, Estimation of the Aggregation Number of if the surfactant is sufficiently clean. Nevertheless, there Micelles are experimental results showing a slight decrease of u . ~ - ~ Recently, Rusanov5 has argued for a decrease ofthe surface By use of the Gibbs adsorption equation and the mass tension after passing the cmc. Knowing the ratio of the action law for concentrations below and above the critical micelle concentration, cm,the number of molecules in a slopes of the a-log(c) curve a t c < cmand at c > c,, such micelle can be determined from the a-log(c) plots for interesting values such as the aggregation number of nonionic surfactant^.^ The number of aggregate monomolecules in a micelle or the degree of counterion binding mers in micelles equals are accessible. Up to now, only few surfactants have been University of Potsdam.
n=
Max-Born-Institut. 8
Max-Planck-Institute of Colloid and Interface Science.
Abstract published in Advance A C S Abstracts, July 1, 1994. (1)Shinoda, K.; Hutchinson, E. J . Phys. Chem. 1962,66,577. (2)Aratono, M.;Okamoto, T.; Ikeda, N.; Motomura, K. Bull. Chem. SOC.Jpn. 1988,61,2773. (3)Elworthy, P. H.; Mysels, K. J. J . Colloid Interface Sci. 1966,21, 331. (4)Wiistneck, R.; Miller, R.; Czichocki, G. Tenside, Surfactants, Deterg. 1992,29,265. ( 5 ) Rusanov, A. I. Micelloobrazovanie v rastvorach poverchnostnoaktivnych uescestv; Chimija: Saint Petersburg, 1992. @
0743-7463/94/2410-3955$04.50IQ
d d d In CI,=~ dold In CI,=~
c - c1 a=C
(1)
a is the degree of micelle formation, c the total surfactant concentration, and c1 the concentration of surfactant (6)Prokhorov, V.A.;Rusanov, A. I. Colloid J.USSR 1990,52,1109. (7)Fiedler, H.; Wiistneck, R.; Weiland, B.; Miller, R.; Haage, K. Langmuir in press. (8)Wiistneck, R.;Fiedler, H.; Miller,R.; Haage, Khngmuirinpress. (9)Rusanov, A.I.;Fainerman, V. B. Dokl. &ad. Nauk SSSR 1989, 308,651.
0 1994 American Chemical Society
Wiistneck et al.
3956 Langmuir, Vol. 10, No. 11, 1994
-
monomers. a = 0 a t c c,, while a 1in the limit c >> c1. The physically meaningful case n > 0 requires duldc 0 for both a = 0 and a = 1. duld ln(c)l,=o is the slope is of the adsorption isotherm for rzELm1,where ri(rl-) the (saturation) adsorption of the component i. Even in real systems, the value a = 0 is assumed for arbitrary c c,. The value a = 1, however, is reached for c >> cmonly. These concentrations may be out of reach when the surfactant concentration exceeds the limit of solubility. Another critical issue is the fact that the selfassembly process is coupled to the thermodynamics of the system and the micells vary their size and shape in response to any change in macroscopic state.1° Even in ideal solutions and at concentrations not much above the cmc, small spherical aggregates are known to transform themselves into extended, cylinder-like micelles upon increase in concentration.ll Therefore, eq 1has to be used with caution. Since adsorption rl and monomer concentration c1 increase only negligibly for c exceedingc,, one can assume c1 = cm and a = 1 - c,lc for c 2 c,. Then5J2
dold In c I a=O -I+--- n - 1 -cIn--n do/d In c, c, - c 1--
bm.7
30
I
4
PIP11 PIP10 PIP9
. , . . , , .., . PIP13 . . . , , .,PIP12 . , , . .. . a 3 -2 I.,
. . . . .. ,
. .. -1 log c [Wdd] ,
.0
Figure 1. 0-log(c) plots of homologous maleic acid mono[2(4-alkylpiperaziny1)ethylesters] (alkyl chain length Cg-C!13) at pH = 6.2: points, experimental values; solid lines, regression lines for c > c , and c < cm, respectively.
acid mono[2-(4-alkylpiperazinyl)ethylesters] at pH = 6.2, where, in the bulk, these surfactants are in a betain-like form7
(2)
c
R-NH
This relation can be used to determine n from experimental data for concentrations only slightly exceeding the cmc. The slope duld ln(c) should be almost constant for concentrations well above c,. There, n can be determined “locally”,for each corresponding a-value. However, the experimental values of u for a > 0 exhibit small scatters which lead actually to large deviations of the slope. For this, the resulting duld ln(c)l,>o values have been replaced with the slope resulting from a linear regression analysis of the isotherm branch c c,. Correspondingly, eq 2 has been solved for duld ln(c)l,.o and then averaged. This yields (i = 1, 2, ..., N, N being the number of points of measurement above c,)
However, these ni values depend too strongly on a. For this, a mean value of n is calculated such that eq 3a is fulfilledin the average. Thus, introducing the mean slopes
\
/
0
0
Figure 1 shows surface tension values of different homologous maleic acid mono[2-(4-alkylpiperazinyl)ethyl esters] which have been used for the calculations. Table 1 gives the cmc determined by the regression straight lines at c < c, and c > c, respectively, and the aggregation numbers n determined through eq 3c. These results are quite reasonable. They are in good agreement with results reported by h a c k e r et al.13J4for similar surfactants, where the aggregation numbers have been calculated from light scattering (e.g., n = 95 for decylpiperazinium bromide in 0.5 Wdm3NaBr). n does not depend on the alkyl chain length within the limits of measurement accuracy. It should be noted that the present method depends very sensitively on deviations of u at c > cm. Even inaccuracies of the measuring method as small as 0.1 mN/m may change the results. Therefore, extreme accuracy of measurement is necessary.
Estimation of the Degree of Counterion Binding For ionic surfactants the ratio (1)reads
n is computed from the equation
1
+ vl/v, -- (do/d In c ) , = ~
1- p
(du/d In c),=~
(4)
p being the degree of counterion binding by micelles and The error of n is determined mainly by that of 7122, since those of ml and ai are much smaller. It has been calculated by inserting the confidential intervals (95%) of m2 in formula 3. Strictly speaking, this approach is applicable only to nonionics. In this investigation, we will use it for maleic
vi the number of ions i, whereas index 1deotes the surface active ion and index 2 the counterion; a = 1 - c1/(cv1).12 Equation 4 can be used only for c >> c, because the same restrictions apply as for eq 1. A more realistic value for p can be determined using the equation (v v2/v1)5
(10)Bagdassarian, C.; Gelbart, W. M.; Ben-Shaul, A. J. Statistical Phvs. 1988.52. 1307. ?ll)Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976,72,1525. (12)Rusanov, A. I. Adv. Colloid Interface Sci. 1993,45, 1.
(13)h a c k e r , E. W.; Geer, R. D. J. Colloid Interface Sci. 1971,35, 441. (14) Jacobs, P. T.; h a c k e r , E. W. J. Colloid Interface Sci. 1973,43, 1105.
Langmuir, Vol.10,No. 11, 1994 3957
Aggregation Number Determination
Table 1. Critical Micelle Concentration, cm, and Aggregation Number, n,Determined for Different Homologous Maleic Acid Mono[2-(4-alkylpiperazinyl)ethylesters] at pH = 6.2
PIP 9
PIP 10
PIP 11
PIP 12
PIP 13
4.50 10-3 120 f 2
7.73 10-4 121 f 13
2.65 10-4 122 f 16
~
cm
[M/dm3]
7.67 x 104 f 3
n
do I
1 -vn@1-a
1.40 x 105 f 6
1-P 1-aB
*]oo l
+ n a + v n 4 18- 24 8L ‘ -a
X ) 1-aB
+1-
This equation, however, contains three unknown parameters. Assuming a value for n, it can be solved as a quadratic equation in@or, vice versa, as a linear equation inn, provided, d d d ln(c)la>ocan be determined. But even in that case, p depends on a , whereas the eq 5 has been derived under the assumption /3 = const. Figure 2 shows that the condition p = const is best fulfilled for unusual small aggregation numbers. Therefore, results forp determined in a domain, where p strongly depends on a , should be averaged. This has been done in recent work,6where these authors replaced d d d ln(c)la=l with the slope of a regression straight line fitted to the a-log(c) plot at c > Cm. Supposing the condition
n >> v(1 - @)/@(1
OS
0.4
0.7
0.9
0.8
a!
Figure 2. Dependence of p on n and a as calculated by eq 5 for results given in ref 6.
- P)
Rusanov12 has derived the simplified relation
-2
3
4
log c [Mldt~?]
It contains only one unknown value, p. But, again, because ddd ln(c)la>ohas to be set equal to the regression straight line, p has been chosen to solve the average (denoted by brackets ( )) of (7) over appropriate experimental values
where m1 = (ddd ln(c)),=o and m2 = (ddd ln(c)>,,o, the mean slope values for the regions c < c m and c > c m , respectively. Figure 3 displays the a/log(c) plots used for maleic acid mono[2-(4-alkylpiperazinyl)ethylesters] (C12 and C13) a t pH values where the adsorption of almost only 1 surface-active component is guaranteed.8 Consider first the validity of condition 6. As an example, Table 2 lists data calculated for maleic acid mono[2-(4dodecylpiperaziny1)ethylester] a t pH = 10 for different a-values. The ratios Y( 1- @)/@(1- p) were calculated by means of eq 6 using the p-values from eq 8. Assuming a realistic value of n 100, the condition 6 is observed. On the other hand, Table 2 shows that /3 does depend on a. For our experimental results this dependence is weaker than in the case of assuming n and using eq 5, but it is, again, still remarkable and shows the limitation of the theory used. Using these p-values, mean value and standard deviation were determined. The error propagated by m2 was found to be less, when compared with the one obtained from the dependence ofp on a. Table 3 shows the degrees .of counterion binding and the calculated confidential intervals of a 95%confidence level.
Figure 3. a-log(c) plots of homologous maleic acid mono[2(4-alkylpiperaziny1)ethylester] of alkyl chainlength C13 (curves 1-3) and C12 (curves4-7): pH = 10.5(curve4),pH = 10(curves 1and 5), pH = 3 (curves and 7), and pH = 0.5 (curves 2 and 6);points, experimental values; solid lines, regression lines for c > cm, and c < cm, respectively. Table 2. Degree of Counterion Binding, /3, Calculated by Equation 8 for Different Values of a,and the Ratio Y ( 1 - @)/@(1 - /3) of Maleic Acid Mono[2-(4-dodecylpiperazinyl)ethylester] at pH = 10 a 0.30 0.45 0.54 0.65 0.77 0.83
P
~ ( -1 O$)/O$(l- P )
0.88 22.7
0.89 13.8
0.90 10.5
0.91 7.7
0.93 5.5
0.93 4.5
In cases where the a-log(c) plot at c > c m is sufficiently well described by a regression line, p can be determined directly by solving eq 7. It results in a quadratic equation, the physical solution to which (negative sign of square root) leads to the following results:
The results determined by means of eqs 9 agree well with those calculated through eq 8. The p-values found are plausible. They are in good agreement with those reported for dialkylpyridinium iodides15(e.g. , p = 0.79 for 1-ethyl-4-dodecylpyridinium (15) Nusselder, J. H.; Engberts, J. B. F. N. J.Am. Chem. SOC.1989, 111,5000.
3958 Langmuir, Vol. 10,No. 11, 1994
Wiistneck et al.
Table 3. Degree of Counterion Binding,B, for Different pH Values and Ionic Forms of Maleic Acid Mono[2-(4-alkylpiperazinyl)ethylesters1 DH 0.5 3 10 R R R ionic form I I I
cj V2
PIP 12 PIP 13
2 0.88f 0.02 0.89 f 0.03
1
0.77f 0.03 0.81f 0.03
:$
1 0.91f 0.02 0.93 f 0.02
iodide). Generally, the degree of counterion binding is higher in the case of PIP 13. Assuming a similar relation between the aggregation numbers of PIP 12 and PIP 13 (Table 1)for the ionic species, too, the degree of counterion binding is higher in those cases where larger micelles are formed. The variation ofp with the pH value, however, is similar for PIP 12 and PIP 13. /3 varies significantly, depending on the position of the charges within the molecule. Due to the excellent accessibility of the ionic group, p becomes maximum in the case of the diaminocarboxylate group (pH = 10). Protonation of one tertiary amino group leads to the ionic form at pH = 3. The charged ammonium group is screened by the ring system. This restricts the accessibility of the ammonium group and, hence, decreases the degree of counterion binding. When
the second amino group is protonated, the double charged diammonium carboxylate is formed. The charge density increases, whereas the accessibility of the charged groups becomes better, when compared with the case pH = 3. As a result,3!, increases. Nevertheless, the accessibility is worse than in the case of the diaminocarboxylate group.
Conclusions The method to determine aggregation numbers of nonionic micelles and the degree of counterion binding of ionic micelles developed by Rusanov5J2have been applied to the case of concentrations exceeding the cmc by only 1order of magnitude for maleic acid mono[2-(4-alkylpiperazinyllethyl esters]. This method leads to reasonable values, where the results for the aggregation numbers agree with relevant data reported by other authors13J4 having used alternative measuring methods. This determination of agreegation numbers leads to values which only negligibly depend on the concentration of monomer surfactants and on alkyl chain length. The determination of the degree of counterion binding is different, however. The thermodynamictheory5.12yields one equation for p and n. Moreover, this theory supposes ,8 to be independent of a,in contrast to experimental findings. Assuming n to be large enough (see eq 6))n can be eliminated from the determination of p. The dependence of on a remains, but it is weaker in the present experimental case. From this point of view, this method to determine of the degree of counterion binding is preferable. Unfortunately, the analysis cannot be extended to concentrations exceeding the cmc by more than 1order of magnitude, since the possibility of changes in size of micelles increases and the parameters determined become more and more doubtful. Acknowledgment. The project was sponsored generously by the Senate of Berlin (ERP-2659).
