10186
J. Phys. Chem. 1!J93,97, 10186-10191
Aggregation Behavior of Sodium Cholate in Aqueous Solution A. CoeUo, F. Meijide, E. Rodriguez Nuez, and J. Vdzquez Tato' Uniuersidade de Santiago, Campus de Lugo, Facultade de Ciencias, Deparramentos de Quimica Fisica e Fisica Aplicada, 27002 Lugo, Spain Received: December 21, 1992; In Final Form: June 24, 1993'
We analyze the freezing point depression, A T / k , and pNa measurements for aqueous solutions of sodium cholate (NaC). The results show the existence of a break point at 0.0197 m in the plot of AT/k us molality. Below this point (more exactly 0.0160 m) N a C behaves as a 1:l strong electrolyte. Above it, in the range of concentrations studied, N a C forms aggregates with an aggregation number equal to 3.09 f 0.06. The fraction of bound counterions is also deduced, ranging from 0.03 to 0.16 (average value 0.077). These results indicate that less than one Na+ counterion is bound per aggregate. The value for the equilibrium constant for the formation of the trimer is K(3) = (3.4 f 0.9) X lo3, from which a value of WO = - 2 . 6 k ~ Tis obtained for the reversible transfer of a free surfactant ion, together with the associated counterions, from the bulk solution to the aggregate. The effect of the addition of NaCl is also studied. The results show that the aggregation number increases slowly with NaCl concentration. This result is corroborated by viscosity measurements.
Introduction Sodium cholate, a trihydroxy bile salt, has been extensively studied by using different experimentaltechniques. Among them, we can mention conductivity,l-s density,3q7-lo diffusion,lI-l4 equilibriumultracentrifugation,lSESR,lG1*fluorescence,1G22 light ~cattering,2~5~'~.23-28 m i c r ~ c a l o r i m e t r y ,NMR,30-34 ~~ osmometry,ItM X-ray,34,41,42partition method?' potenti~metry,".~~ refractive index,'.& solubilization,6."14ss surface and interfacial tension,1,2.20.23~24.28149,5"1 spectral ultrasonic absorption,8 and ~iscosity.7.10,31,63,64 Apart from problems concerningthe use of unpurified samples (which have been commented on by Kratohvil et ~ 1 1 . 6 and ~3~~ Zana67), it is obvious that there are opposite conclusions even when the results have been obtained by the same author. For instance, from vapor pressure osmometry,37Fontell concludes that the aggregation number should be as low as 3-4, but from X-ray spectrometry42 he deduces a value of 16. The dispersion in values also concernsthe degree of bound counterions. Although most of the authors deduce values lower than 0.3, Sugihara et al.52 deduce a value of 0.7. The existence of a cmc is also subject of controversy, and the values at which association appears range from 0.00318.59,62 to 0.018 M.20 Ekwal145and Norman4 affirm that NaC behaves as a normal 1:1electrolyteup to concentrations of 0.014-0.015 M, but Lindenbaum et al.39showthat "very large deviations from Debye-Hiickel theory are observed even at the lowest concentrations used, typically 0.0025 M". The structure of the aggregates is also under discussion. Some authors accept the scheme proposed by SmaP,9,10.'5.2'.29,668 while others do not . 3 3 Chang and Cardinal,?6 Oakenfull and Fisher,' and Vadnere and Lindcnbaum43 have given values for the equilibrium formation constant of aggregates. In these papers two major simplifications are assumed: (a) solutes behave ideally, Le., activity coefficients are unity, and (b) bound counterion is not considered, although it is required according to mass action models. Furthermore, it is important to remark that some of the values for the counterion binding have been obtained from the Corrins expression. Since several authors8~~'~26~30.43~5~.60 accept that bile salt aggregates are highly polydisperse in size, it is interesting to notice here that, as Kratohvil et aI.'j9 have pointed out, 'there is no physical justification for using the quasi-cmc values observed for surfactants which show continuous growth of aggregates in solution, as a basis of evaluating the thermodynamic quantities for micelle *Abstract published in Aduance ACS Absrracrs, September 1, 1993.
0022-3654/93/2097- 10186$04.00/0
formation or the fraction of bound counterions". The values obtained by Lindman and c o - w o r k e r ~ ~are ~ - accompanied ~~*~~ by abnormally high values for the aggregation number compared with those obtained by other authors, and the one given by Sugihara et al.52seems to be too high. On the basis of thermodynamic principles developed for mixed electrolyte solutions,7° Burchfield and W o o l l e ~have ~ ~ proposed a model for the interpretation of osmotic coefficients for longalkyl chain surfactants in aqueous solution. However, this or similar studies have not been applied to the analysisof osmometry measurements for bile salts although such experiments have been carried out by different authors in studying bile salt ~ystems.'.2~,3M,~2,73 The only quantitative attempt is due to Allende et aL73although in this case some inconsistenciesbetween the thermodynamic model and the average aggregation number used for calculations can be detected. A simple inspection of a micellar system shows that it is not possible to obtain simultaneously the aggregation number and the fraction of bound counterions from freezing point depression measurements since both parameters are related to the number of solubilized particles; i.e., the same value for the freezing point depression may be obtained with different pairs of values of those parameters. To solve the problem, it is necessary to use another experimental technique or introduce a reasonable hypothesis. To clarify some of the aspects mentioned above, we analyze the freezing point depression and pNa for aqueous solutions of NaC. Cation (Na+ and K+)activity in aqueous cholate solutions have been measured previously by Moore and Dietschy36 and Sugihara et al.,52 and freezing point depression measurements have been carried out by Moore and Dietschy,36 Roepke and Mason,35 and Rains and Crawford.' Conclusions reached by these authors either are qualitative36 or are not in good agreement with other results.52
Thermodynamic Analysis Several authors*.l1.26,30,43,51.60 accept that NaC micelles are polydisperse in size. According to M~kerjee,'~ it implies that the average aggregation number increases with surfactant concentration. In their model for the interpretation of osmotic coefficients, Burchfield and W o o l l e ~take ~ ~ the aggregation number and the fraction of bound counterions from other sources. (Alternatively, they are estimated from values for similar compounds.) Furthermore, they assume that micelles are monodisperseand that the average aggregation number is constant 0 1993 American Chemical Society
Aggregation Behavior of Sodium Cholate
The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 10187
over the range of concentrations studied. As a consequence, such a model is not useful for the purposes of the present paper. We will accept here that a polydisperse system is well modeled by a monodisperseone. In the latter, all micelles have the average aggregation number which could be determined from the former. The averageaggregationnumber can increase with concentration. Knowing the dependence of the aggregation number with concentration and determining the equilibrium constant for the formation of the correspondingaggregates, we could approximate the distribution function of the aggregates at a single concentration. Furthermore, knowing thedependence of the equilibrium constant values with the average aggregation number, the equilibrium constant for a given aggregate (as for instance a dimer) could be estimated. According to mass action models, the formation of a micelle, M, having n monomers, A, and /3n bound counterions, B, is given by eq 1, while the equilibrium constant is given by eq 2.
TABLE I: Freezing Point Depression (Ex r d 88 Osmolality) and pNa Measurements at d e r e n t Total NaC Concentrations and Deduced Values for the A egation Number, a, and the Fraction of Bound Couote%s, 8, Following the Method Described in Text m,mol kg' A T / k , mol k g l pNa n B 0.0460 0.0529 0.0640 0.0775 0.0939 0.114 0.139 0.169 0.205
0.0608 0.0668 0.0758 0.0872 0.101 0.118 0.137 0.161 0.192
1.45 1.40 1.34 1.26 1.19 1.12 1.05 0.99 0.93
3.01 3.03 3.08 3.17 3.15 3.11 3.14 3.11 3.00
0.028
0.040 0.083 0.051 0.059 0.069 0.081 0.12 0.16
is the charge of the ith ion, and u b ) is a function given by
a(y)=3(1+y-l/(l+y)-2In(l The summation c c i is given by z c i
= CA
+ cg + CM
+ y ) ) / y 3 (11) (12)
Furthermore, Here, ut = cnt means the activity of the ith solute and yi the corresponding activity coefficient on the molality scale. The total surfactant concentration is given by
= CA + ncM
(3) where CA is the molal concentration of free monomers and CM the molal concentration of micelles. Similarly, the mass balance for counterions gives CL
;c
= CB
+
(4) where CB is the concentration of free counterions and /3 is the fraction of bound counterions. From previous equations CB
BncM
= cL(1 -0)
+ OCA
(5)
The ionic strength of solution is ~c
= O.~(C,
+ CB + a2n2(1 - @)'cM)
(6) Here 6 is a "screeningm or 'shielding" factor introduced by Burchfield and Woolley71to reduce the effect of micellar charge on the ionic strength of the solution. 6 = 0 indicates that micelles are omitted in the calculation of p, while 6 = 1 means that the full charge of the micelle must be considered. As we show below for the present system 6 = 1. From eqs 3 and 6 n = (2p - CA - cB)/[(cL
- cA)a2( 1 - /312]
(7)
According to G ~ g g e n h e i mthe , ~ ~freezing point lowering obeys the equation
AT/k = c p z c , (8) where is the osmotic coefficient defined by Bjerrum and ct is the molal concentration of the ith ion. The summation extends toall ions present in solution. Accepting Guggenheim's expression for the calculation of the activity coefficient for ions with the ion-ion interaction parameters being negligible (eq 9) and the Gibbs-Duhem equation, eq 10 may be deduced
where A, = 0.4918 kg112
for water at 0 OC, y = p1I2,zi
pNa = -log(aB) (13) Here two physical quantities, A T / k and pNa, are measured experimentally,and the hypothesis CA = constant is accepted (see Results and Discussion). The problem is solved as follows. With a given /3 value, CB is calculated from eq 5 , which is substituted in eq 13, allowing the calculation of YB. Now, we can calculate the ionic strength from eq 9, the aggregation number from eq 7 and micelle concentration from eq 3. Finally, zc, (eq 12), the osmotic coefficient I#I (eq lo), and the freezing point depression (eq 8) are obtained. This last value is compared with the experimental one. The process is repeated, and the value for /3 which makes minimum the relative error between the experimental and theoretical freezing point depression is accepted for every NaC concentration.
Experimental Section Freezing point determinationswere carried out in a Fiske Model OS osmometer equipped with 0.25-mL sample tubes. The osmometer was calibrated with a solution of 3.089 g of NaCl/kg of water. The measurements were repeated at least five times. The average reproducibility was better than 1%. Densities were measured at 0 "C in an Anton Paar Model DMA-45 densimeter. Viscosities were measured in a falling ball viscometer with a reproducibility better than 0.5%. pNa measurements werecarried out with a Radiometer pHM82 potentiometer by using a Na+ ion selective electrode model Radiometer G502Na. A calomel Ingold 303-K19 was used as reference electrode. Calibration was carried out with NaCl solutions. Since the minimum temperature available with the ion selective electrode is 10 "C, measurements in the range 10.525 OCwerecarriedout. It wasobserved that, withinexperimental error, pNa does not change with temperature. pNa values shown in Table I and plotted in Figure 3 were obtained at 25 OC. In the range of NaC concentrations, 0.034.205 m, the pH of solutions was constant (7.72 0.03) and pH diminishes at lower NaC concentrations,7.00 being the minimum value for the lowest concentration used in this work. The use of unpurified samples of bile salts leads to erroneous conclusions as Kratohvil and co-workers have repeatedly commented.65.66969 Therefore, we have dedicated particular attention to purify NaC (purchased from Sigma). We have followed the method used by Kratohvil et al.69 NaC was dissolved in hot ethanol until saturation, treated with charcoal for half an hour, filtered, and concentrated in a rotavapor. Ethyl acetate was added to the ethanolic solution (1:l) and kept in a refrigerator at
*
Coello et al.
10188 The Journal of Physical Chemistry, Vol. 97, No. 39, 1993
0
0
0.00 0.00
5
15
10
25
20
1Osc:@/( 0.05
0.10
0.15
0.20
0.25
ci/(mol kg-') Figure 1. Experimental osmolality us total NaC concentration.
mol kg-')
Figure 2. Experimental osmolality us total NaC concentration
