7753
J. Phys. Chem. 1989, 93,1153-1155
Reevaluation of Osmotic Pressure Measurements of Micellar Solutions Sudhakar Puvvada and Daniel Blankschtein* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received: July 25, 1989)
The peculiarities of osmotic pressure measurements of self-associating micellar solutions are examined. It is shown that these measurements can yield the weight-average micellar molecular weight, M,, contrary to the generally accepted notion that they yield the number-average micellar molecular weight, M,,. In view of this surprising new result, we have reevaluated osmotic pressure measurements of aqueous micellar solutions of the nonionic surfactant n-dodecyl hexaoxyethylene glycol monoether (C12E6).We have found that, contrary to previous interpretations indicating the presence of monodisperse micelles, the measurements are consistent with the presence of polydisperse micelles which grow with increasing surfactant concentration and temperature.
It is universally accepted' that measurements of solution osmotic pressure yield, in the limit of dilute solutions, the number-average solute molecular weight, A,. In this Letter, we show that due to the unique nature of micellar solutions2 and special features characterizing membrane osmometry in these self-associating system^,^ osmotic pressure measurements can yield the weightaverage micellar molecular weight, A,, instead of the generally accepted number-acerage micellar molecular weight, M,. We also examine the implications of this surprising new result for the interpretation of osmotic pressure measurements aimed at determining the extent of micellar size (molecular weight) polydispersity. In particular, we have reevaluated available osmotic pressure data4 of aqueous solutions of the nonionic surfactant n-dodecyl hexaoxyethylene glycol monoether (CI2E6)and have found that, contrary to the original interpretation indicating the presence of monodisperse micelle^,^ the data are consistent with the presence of large polydisperse micelles. In a conventional osmotic pressure measurement,' a test solution is contacted with pure solvent through a membrane that ideally is permeable only to the solvent. Under such conditions, the solvent will permeate through the membrane until thermodynamic equilibrium is attained, that is, until the solvent chemical potential is equal on both sides of the membrane. At fixed temperature, the solvcnt chemical potentials will be equal if the side containing the test solution is maintained at a pressure exceeding that of the side containing the pure solvent by the osmotic pressure, x . If x is expressed as a virial expansion in powers of the total solute concentration, c, one obtains, to quadratic order in concentration, the well-known result' x
= (RT/M,)c
+ B2c2
the mole fraction of micelles composed of m surfactant monomers (m-mer). It is essential to recognize2 that the entire micellar size distribution, {Xm),and any property derived from it, for example, A,, and A?, necessarily respond in a reversible manner to changes in total surfactant concentration, X,and other solution conditions. Utilizing the laws of multiple chemical equilibrium to model the reversible association of surfactant monomers into m-mers, one can obtain2 the following expression for {X,)
where /3 = l / k T with k the Boltzmann constant, X m and XI are the mole fractions of m-mers and free monomers, respectively, and gmic(m)is the free energy of micellization representing the free energy change associated with transferring a surfactant monomer from bulk water into an m-mer. Note that gmiC(m)is not a function of X but can depend on other solution conditions such as temperature, ionic strength, pH, e t c 2 Equation 2 has been widely used2 to predict various aspects of micellization and is recognized to be valid in dilute micellar solutions where intermicellar interactions are negligible. In addition, eq 2 was found to be valid for certain types of intermicellar interaction^,^,^ suggesting that its range of applicability may in fact be quite broad. It is possible to show6 that eq 2 leads to a simple mathematical relation between the weight-average micellar molecular weight
M,
I
(3)
(4)
that is,
l/A, = d(X/A,)/dX
(5)
where M I is the molecular weight of a surfactant monomer. Equation 5 is of central importance to the discussions that follow. The typical membranes used in osmotic pressure measurements are permeable to surfactant monomers, and as a result the measured pressure decreases with time as monomers permeate through the membrane into the solvent side. Under such nonequilibrium conditions the osmotic pressure is usually determined ( I ) For a comprehensive account of experimental and theoretical aspects of membrane osmometry, see: Tombs, M. P.; Peacocke, A. R. The Osmotic Pressure of Biological Macromolecules; Clarendon Press: Oxford, 1974. (2) For an introduction to the field of micellar solutions, see: Micellization, Solubilization and Microemulsions; Mittal, K . L., Ed.; Plenum: New York, 1977; Vols. 1 and 2. (3) Coil, H. J . Phys. Chem. 1970, 74, 520, and references therein. (4) Attwood, D.; Elworthy, P. H.; Kayne, S. B. J . Phys. Chem. 1970, 74, 3529, and references therein. (5) Ben-Shaul, A,; Gelbart, W. M . J . Phys. Chem. 1982, 86, 316. (6) Blankschtein, D.; Thurston, G. M.; Benedek, G . B. J . Chem. Phys. 1986, 85, 7268, and references therein.
* T o whom correspondence should be addressed. /
m
an = MI (CmXm) /(CXm) m m
where c is the total solute concentration in units of weight/volume, B2 is the second virial coefficient reflecting solution nonidealities, T is the absolute temperature, and R is the gas constant. Typically,' the measured osmotic pressure data are analyzed by plotting the reduced osmotic pressure, (x/RT)/c, against c and extrapolating the resulting straight line (see eq 1) to zero concentration. The intercept at c = 0 yields l/M,, and the slope yields B2. This extrapolation procedure is meaningful only when there is no association or dissociation reaction occurring in the solution, so that A,does not vary with concentration, c. However, in micellar solutions A, can vary with concentration, and therefore this extrapolation procedure must be implemented with great care (see below). When surfactant molecules are placed in water, they can self-associate to form micelles2 in order to avoid contact of their hydrophobic moieties with water. Typically, micellization occurs beyond a threshold surfactant concentration known as the critical miccllar concentration (cmc). A salient feature of micellar solutions is that micelles are often where X, denotes present in a broad distribution of sizes, {X,),
,
m
and the number-average micellar molecular weight
(1)
0022-3654/89/2093-1153$01.50/0
= MI(13m2Xm)/(CmXm)
Q 1989 -
American Chemical Societv
7754 The Journal of Physical Chemistry, Vol. 93, No. 23, 1989 by extrapolating the measured pressure to zero time.3 However, in micellar solutions the decline of pressure with time becomes rapid and concentration dependent, thus making the procedure ~ n r e l i a b l e .These ~ nonequilibrium complications can be reduced to tolerable levels by contacting3 the test solution with a reference solution having approximately the same monomer concentration as the test solution. In p r a ~ t i c ethis , ~ is achieved by choosing the reference solution to be a micellar solution having a total surfactant concentration, X', of about 3 times the cmc. Under this unique experimental configuration the measured pressure, AT, is given by A T = a ( X ) - 7r(X') (6) where a ( X ) and a ( X ' ) are the osmotic pressures (measured against pure solvent) of the test and reference solutions, respectively. For the sake of brevity, we will omit hereafter second and higher order virial corrections which are negligible in the limit of int e r e ~ t . Using ~ , ~ eq I , expressed for convenience in mole fraction units, in eq 6 one obtains A ~ V J M , R T = x/A,(x)- xr/A,(xr) (7) where Vsis the partial molar volume of the solvent. In micellar solution^,^.^ the reduced osmotic pressure, = (AaVs/MIRT)/ (X- X'), is usually plotted against ( X - X ' ) , and the resulting fitted line is extrapolated to X = X'to obtain the average micellar as a function of molecular weight. Indeed, if one measures ( X - X ' ) sufficientl-v close t o X = X', then using eq I and 5 at X = X', one obtains lim
X-X'
(S)= I / A , ( x ~ )
(8)
Equation 8 constitutes a surprising new result since it indicates that membrane osmometry in micellar solutions, conducted under the conditions stated above and analyzed by using the procedures described above,7 yields the weight-average micellar molecular weight. A , ( X ' ) , instead of the universally accepted numberacerage micellar molecular weight, A,. It is of interest to examine the implications of this result on the experimental determination of average micellar sizes and micellar size polydispersity using membrane o s m ~ m e t r y . ~ , A ~-* convenient measure of micellar size polydispersity is provided by the polydispersity index, N = A,/A,. For micellar solutions, a value of CY N 1 would indicate the presence of monodisperse micelles, whereas a value of 01 2 would indicate the presence of large polydisperse micelles that exhibit one-dimensional growth.2-6Typically, A, is obtained from measurements of the is obtained from absolute intensity of scattered light,4 and A,, colligative methods such as membrane osmometry.' However, if osmotic pressure measurements, conducted as described above, are correctly interpreted as indicated above, then it would follow that both experimental methods (light scattering and osmometry) should yield the same property, A,. Consequently, a deduced Ljalue of (Y N I should not be interpreted as indicating the presence of monodisperse micelles, but rather as a manifestation of experimental consistency. The determination of the extent of micellar growth in aqueous solutions of nonionic surfactants belonging to the polyoxyethylene glycol monoether family (C,Ej) constitutes a very important and still controversial p r ~ b l e m . Therefore, ~ it is of considerable interest to examine the implications of the observations made in the previous paragraph in the context of this problem. T o this end, we have reexamined available osmotic pressure data4 of aqueous solutions of C12E6. In ref 4, AT was measured
-
( 7 ) Note that if X' < cmc (including X ' = 0), then as X X'the test solution will only consist of monomers dispersed in water (recall that no micelles are present below the cmc). Therefore, in this case eq 8 will yield the monomer molecular weight, M , . (8) Birdi, K . S . Kolloid Z. Z. Polym. 1972, 250, 7 3 1 . (9) For a detailed exposition of this problem see: Degiorgio, V . In Proceedings of the International School of Physics Enrico Fermi-Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Degiorgio, V., Corti, M.. Eds.; horth-Holland Physics Publishing: Amsterdam, 1985; p 303.
Letters 10,
,
,
,
I
"
"
,
t
0 0 ' ' 0 0010
I
I
,
,
,
,
1
'
00015
0 0020
0 0025
X
Figure 1. Predicted (full line) dimensionless osmotic pressure, 7rrw RT, / as a function of surfactant mole fraction, X , and temperature for an aqueous micellar solution of the nonionic surfactant (C,*E6). The experimental points are from ref 4 and correspond to 25 O C (H), 30 OC (A), and 36 O C ( 0 ) .
in the concentration range 1.2 X I X I 2.5 X with a reference solution having a concentration X' = 3cmc N 6 X IO". versus ( X - X ' ) was obtained and Subsequently, a linear fit of extrapolated to X = X'. Following common practice, the intercept at X = X'was interpreted as corresponding to 1/fin.Note that the experimental measurements and the subsequent linear fit were conducted4 a t concentrations X much larger than X'. Consequently, the intercept at X = X ' cannot be obtained from eq 8 since, as explained above, this equation is only valid when the actual measurements are performed sufficiently close to X = X'. Indeed, in the context of a linear extrapolation, the intercept at X = X'does not, in general, yield l/Anor l/A,,,,but instead a more complicated quantity (see eq 9). A careful analysis of the linear extrapolation procedure, which accounts for the possible dependence of A, on surfactant concentration, indicates that for X = X * (where X * >> X' is an arbitrarily chosen value in the measured concentration range) this intercept is approximately given by I /A&(X*)
2 / A , ( X * ) - 1 / A W ( X * )2 ( X ' / X * ) { 1/ M , ( X * )
+ 1/ A " ( X ' ) j
(9)
Equation 9 clearly shows that Herr depends explicitly on X * and, in general, is different from A,. Consequently, any conclusions based on the assumption that , ~ the micelles the intercept at X'is equal to l / A n , for e ~ a m p l ethat are monodisperse, should be carefully reexamined. For this purpose, we have calculated the osmotic pressure as a function of X and T, allowing f o r micellar size polydispersity and growth, and have compared the predicted values with the original experimental data.4 As can be seen from Figure I , the agreement is very good, suggesting that the data are in fact consistent with micellar size polydispersity and growth. The theoretical predictions were made in the context of a recently developed6 theory of phase behavior and phase separation of micellar solutions which can exhibit micellar growth. This theory has been successfully used to self-consistently predict6 the experimentally observed coexistence curve, average micellar size, polydisperse micellar size distribution, and osmotic compressibility of aqueous solutions of C,,E6 as a function of surfactant concentration and temperature. By use of this theory, the dimensionless osmotic pressure is given by6
aV,,,/RT = -In ( 1 - X ) - X
+ M , X / A , ( X ) - (1/2)Ca2/y (10)
where V,,, is the partial molar volume of water, y is the ratio
J . Phys. Chem. 1989, 93, 7755-1756 between the molar volumes of C12E6and water (approximately equal to 25), is the total volume fraction of surfactant, and C is a phenomenological interaction parameter. Utilizing results from ref 6. we have computed A,(X.T) and deduced that y C / k = 14.3T - 4220. Using this information in eq IO,we have been able to predict aV,,,/RT as a function of surfactant concentration, X.at the three temperatures 25, 30, and 36 "C; see Figure 1. The favorable comparison with the original data of ref 4 adds further support to the central claim of this Letter, that, in general, osmotic prcssure measurements of micellar solutions, performed and analyzed a b described above, do not yield the number-aoerage micellar molecular weight.
+
7755
Acknowledgment. We are grateful to Professor J. Th. G. Overbeek for many illuminating discussions on membrane osmometry. This research was supported in part by the National Science Foundation under Grant DMR-87-19217 administered by the Center for Materials Science and Engineering a t MIT. Daniel Blankschtein acknowledges the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of his research and is grateful for the support by the Texaco-Mangelsdorf Career Development Professorship at MIT. Registry No. CI2E6.3055-96-7.
Oscillatory Coupling of Chemical Oscillators and Other Reactive Systems Mihily T. Beck* and Istvin P. Nagy Department of Physical Chemistry, Kossuth Lajos University, Debrecen IO, Hungary 401 0 (Receiued: June 13, 1989)
The effect of periodic mass transfer between two reactive systems can be achieved by applying the principle of the hydrodynamic oscillator. The oscillatory coupling of chemical oscillators and the periodic perturbation of a reactive system by a suitable agent results in novel oscillatory responses.
Introduction The coupling of oscillatory chemical systems is basically important both from a chemical point of view and as modeling of regulation in biochemical systems.' So far, all these studies dealt with the coupling of two or more CSTRs (continuous-flow stirred tank reactors). Martin's discovery* that rhythmic downward and upward flows occur when two aqueous salt solutions of different density are connected by a vertical capillary indicated to us that it is possible to make a periodic mass transfer between two reactive systems. From among the many obvious possibilities the following appeared to us the most interesting. First is the coupling of two different chemical oscillators, and second, the periodic perturbation of an oscillatory system by a substance which either initiates or inhibits the oscillatory kinetics. The periodic mixing of the ingredients of an oscillatory system by the hydrodynamic oscillator also offers a deeper insight to the nature of such systems. Experimental Section The experimental setup is shown in Figure 1. The frequency and the amplitude of the hydrodynamic oscillator depend on the density difference of the solutions in the two compartments and on the length and the diameter of the capillary and the heights of the liquid columns in the compartments. In our experiments the frequency was about 0.238 m i d , while the volume of the transferred solution was 7.68 X dm3 per period. This means that about 5% of the full volume of each compartment was exchanged in 2 h. The solutions in both compartments were stirred (80 rpm). (A higher stirring rate would interfere with the hydrodynamic oscillator.) The volumes of the solutions in compartments A and B were always 31.5 and 45 cm3, respectively. In all cases, the hydrodynamic oscillation started by a downward stream. The potentials in one or both compartments were measured by a Radelkis OP-208/ 1 precision potentiometer using bright Pt electrodes and S C E reference electrodes and KN03-KCI double salt bridges. Reagent grade chemical were used without further purification. ( I ) Rehmus, P.;Ross, J. In Oscillarions and Traveling Wawes in Chemical Systems; Field, R . J., Burger, M., Eds.; Wiley: New York, 1985; pp 287-332. (2) Martin, S. Geophys. Fluid Dyn. 1970, I , 143.
0022-3654/89/2093-7755~01S O / O
All the experiments were performed a t room temperature (22 f 2 "C).
Results and Discussion The mutual effects of two BZ systems containing two different catalysts depend to a large extent on the state of the two reactions at the moment of coupling, but the effect of the ferroin-catalyzed system on the manganese(I1)-catalyzed one is always much greater than vice versa. As appears from the curves of Figure 2 the length of the oscillatory stage of the Mn2+-catalyzed system can be increased or decreased by the coupling, while only rather small, but definite, changes in the amplitude and frequency of the oscillations in the potential of the ferroin-catalyzed system were found. Bromide ions play a crucial role in the BZ reactions3 A relatively small concentration of bromide inhibits the oscillatory character. When bromide ion is periodically introduced by the hydrodynamic oscillator into the BZ system, the response greatly depends on the concentration of the bromide solutions, as illustrated by curves of Figure 3. At smaller concentrations, first irregular changes occurred, and then a rather stahle oscillatory behavior is observed. At higher bromide concentrations mixedmode oscillation is observed for more than an hour. Later the period time increases and at even higher bromide concentrations the oscillatory character is totally suppressed. A most interesting phenomenon occurs when the reactants of the BZ system are mixed by the hydrodynamic oscillator (Figure 4). While the periodic mixing leads to an oscillatory change of the potential, the continuous addition of the same amount of bromate to the other reactants leads to a monotonous change of the potential. Note that the frequency of the potential change is much larger than that of the hydrodynamic oscillation! When the total amount of the bromate is added to the other reactants in a single dose, also oscillatory change of the potential occurs, but the shape of the curve and the characteristics of the oscillatory reaction are different. It has been found that while the oxidation of oxalic acid by bromate in the presence of a catalyst is not an oscillatory reaction, (3) Field, R. J.; Kijros, E.; Noyes, R. M. J. Am. Chem. Soc. 1972, 94, 8649.
0 1989 American Chemical Society
