11742
J. Phys. Chem. 1995, 99, 11742-11751
Thermodynamics of Micelle Formation as a Function of Temperature: A High Sensitivity Titration Calorimetry Study Stefan Paula,? Willy Siis, Jiirgen Tuchtenhagen,”and Alfred Blume* Department of Chemistry, University of Kaiserslautem, Erwin-Schrodinger-Strasse, 0-67653 Kaiserslautem, Germany. Received: February 22, 1995; In Final Form: May 8, 1995@
Titration calorimetry was employed to measure the critical micelle concentration (cmc) and the heat of demicellization A H d e m i c of the four surfactants octyl glucoside, sodium dodecyl sulfate (SDS), sodium cholate, and sodium deoxycholate at temperatures between 10 and 70-80 “C. From these data, the thermodynamic parameters AGdemic, ASdemic, and ACp.demic associated with the demicellization process were calculated. Titration calorimetry has the advantage that the cmc and the thermodynamic parameter m d e m i c can be directly measured, whereas with other methods f%emic has to be calculated from the temperature dependence of the cmc, which requires high precision for the cmc data. Changes in temperature caused large variations of A H d e m i c and ASdemic, whereas AGdemic remained virtually constant. Therefore, the changes in enthalpy and entropy almost completely compensate each other. At room temperature, the entropy was found to be the dominant factor responsible for micellization, whereas at elevated temperatures contributions from enthalpy dominate. These observations are in agreement with data of other processes where hydrophobic effects play a major role and were used to discuss the nature of the driving forces that rule micelle formation at various temperatures. Furthermore, predictions regarding the degree of hydration of the micelle interior were made. It is shown that titration calorimetry is an easy and fast method to determine the cmc and the demicellization enthalpy from a single experiment. For surfactants with low aggregation numbers the titration curves could be simulated using a mass action model.
Introduction The self-assembly of amphiphiles in water to form micelles has been subject of many investigations. In particular, the critical micelle concentration (cmc) has been determined by numerous researchers using different techniques.’-I2 Thermodynamic quantities of micellization like the Gibbs energy AGmIc, the enthalpy AHmlc, or the entropy AS,,, can be derived from either the temperature dependence of the critical micelle concentration or from direct measurement of the enthalpy by microcalorimetry.1.3.13-17 The availability of these parameters at various temperatures can give valuable insight into the principles which govern the formation of micelles. The formation of micelles was always found to be connected with a large, negative change in Gibbs energy AGmIc; Le., the aggregation process is thermodynamically favored and sponAlthough for most amphiphiles AG,,, exhibits a slight minimum somewhere between 90 and 140 “C, it seems to be only slightly dependent on temperature in the observed range between 10 and 160 “C. The major driving forces for micelle formation are hydrophobic interactions. At room temperature these are due to a large gain in entropy when water molecules in hydration shells around the hydrophobic parts of the monomeric amphiphiles are released during the micellization process. At elevated temperatures, however, the increase in entropy cannot account for the large negative AGm1c value.’ 1-12-18,22 For many hydrophobic compounds AS,,, ap* T o whom correspondence should be addressed at Department of Chemistry, University of Kaiserslautern, Postfach 3049, D-67653 Kaiserslautem, Germany. ’ Present address: Department of Chemistry and Biochemistry, University of California, Santa Cruz, CA 95064. Present address: Max-Planck-Institut fur Kolloid- und Grenzflachenforschung, D-12489 Berlin, Germany. @Abstractpublished in Advance ACS Abstracts, July 1, 1995.
0022-3654/95/2099-11742$09.00/0
proaches zero at higher temperatures and becomes negative at temperatures above 130 “C. In this case, the major contribution for AGmichas to come from AHmic,which should become large and negative in order to compensate for the changes in ASmic. In other words, micelle formation at high temperatures is thought to be “enthalpy-driven”, whereas at room temperature the gain in entropy is the major factor leading to the negative change in Gibbs energy.7,’1-16-22 There are unfortunately not much data available for micellization at elevated temperatures, as most measurements carried out in the past were restricted to room temperature. More recent studies covering a broader temperature range confirm indeed the idea of enthalpy-entropy compensation.20-26The implied temperature dependence of A H m i c manifest itself in large negative values of the change in heat capacity (ACP,,ic), which is the unique feature of all processes related to the hydrophobic effe~t.~~-~~ ACP.,ic has been found to be a linear function of the hydrophobic surface area of an amphiphile that gets excluded from water throughout micelli~ation.~~ Therefore, predictions regarding the water content of the micelle core are possible. Micelle structure and the extent of water exposure of the micelle interior can be described in terms of various models, such as the “fjord” and “reef” model or the “lattice” model, for e~ample.~~-~~ The application of calorimetry to studies of the micellization process is quite ~ l d . ~ It. ’was ~ used, however, only for a few systems because the early calorimeters were not very sensitive and mostly of the heat flow type so that the experiments were very time consuming. With the availability of power compensated titration calorimeters running in the quasi-isothermal mode, titration calorimetry has become very attractive again.I6 As the sensitivity has become higher, the time for determining a complete titration curve at one particular temperature has been reduced to approximately 2 h. 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 30, 1995 11743
Thermodynamics of Micelle Formation
b-
OH
CH,OH
octyl glucoside
0'
-0-s-0
NE
+
sodlum dodecylsulfate
experiments. The pH of the solutions containing ionic surfactants was measured with a pH meter equipped with a glass electrode. Titration Calorimetry. Heats of dilution and demicellization were measured using an OMEGA titration microcalorimeter (MicroCal, Northampton, MA). The sample cell had a volume of 1.34 mL and was filled with water prior to each experiment. Micellar surfactant solutions were placed in a 250-pL continuously stirred syringe. Injected into the sample cell were 25 10pL aliquots in intervals of 6 min. The concentration of the surfactant in the syringe was chosen in such a way that, with increasing surfactant concentration in the sample cell, the cmc was reached during the experiment. Each measurement was repeated at different temperatures in the range between 10 and 70-80 "C. Data analysis was carried out using the MicroCal ORIGIN software. Results
sodium cholate
sodium deoxycholate
Figure 1. Chemical structure of the surfactants sodium dodecyl sulfate, octyl glucoside, sodium cholate, and sodium deoxycholate.
In this study, we employ high sensitivity titration calorimetry to determine the cmc and the enthalpy of demicellization of four surfactants within a temperature range from 10 to 7080 "C. The surfactants we chose for the measurements were the anionic sodium dodecyl sulfate (SDS) as a reference, because a large amount of data is available for this compound, the bile salts sodium cholate and sodium deoxycholate, because of their importance in protein isolation and their physiological relevance, and, finally, octyl glucoside in order to have a representative from the group of non-ionic surfactants. The Gibbs energy change AGdemic,the entropy change ASdemic, and the heat capacity change ACp,demicare calculated, and the contributions of ASdemic and A H d e m i c to AGdemicat various temperatures are discussed. The titration curves of deoxycholate were simulated using a mass action model for the aggregation process, as this surfactant shows only low micellar aggregation numbers. From the temperature dependence of the demicellization enthalpy, M d e m i c , we calculated ACp,demic,which is positive as expected for processes involving hydrophobic affects. ACp,demicis a measure for changes in exposed hydrophobic surface area and can be used to make suggestions for the structure of micelles, i.e. for the fraction of the hydrophobic surface area of the micelle that is exposed to water. Materials and Methods Chemicals and Sample Preparation. The surfactants SDS, sodium cholate, sodium deoxycholate, and octyl glucoside were obtained from Aldrich (Steinheim, Germany) and were used without further purification. Surfactant solutions of a definite concentration were always freshly prepared by weighing out a certain amount of surfactant and diluting it up to the required volume with water. Twice deionized water was used for all
CmC and M d e m i c . The chemical structure of the four surfactants studied by titration calorimetry are shown in Figure 1. We will first discuss the calorimetric results obtained for the non-ionic surfactant octyl glucoside. A typical experimental titration curve obtained from dilution of a micellar octyl glucoside solution into water at 70 "C is shown in Figure 2a. The enthalpogram can be subdivided into two concentration ranges where the reaction enthalpies are almost constant. For the first injections the final concentrations in the sample cell are below the cmc. Here, the large enthalpic effects observed are due to dilution of micelles, the demicellization process, and dilution of the resultant monomers. The sharp decrease in the reaction enthalpy in the curve at a concentration of 23-24 mM indicates that the cmc in the sample cell has been reached. If more micellar solution is added, the micelles are no longer dissolved and the only heat that is measured is caused by dilution of micelles. This is the second concentration range. The cmc corresponds therefore to the concentration where the first derivative of the curve in Figure 2b displays a minimum (Figure 2c). The heat of demicellization AHdemic is equal to the enthalpy difference between the two extrapolated lines in Figure 2b. Thus, cmc and m d e m i c can be measured in one and the same experiment. We report here all thermodynamic quantities for the demicellization process using A H d e m i c as the directly measured quantity. AHdemic is, of course, equal in magnitude but opposite in sign to the heat of micellization A H m i c . When the titration experiment is performed at lower temperatures, the sudden change in reaction enthalpy gradually becomes smaller and finally changes sign. This is shown in Figure 3 for five different temperatures. It is also evident from the experimental curves that the cmc changes and passes through a minimum at a temperature between 40 and 50 "C. The same types of titration experiments were performed with SDS in water and in 0.1 M NaCl solution, and qualitatively very similar titration curves were observed. The major difference compared to octyl glucoside is the observation that the cmc minimum, i.e. the temperature at which m d e m l c is zero, occurs at a lower temperature, namely at 20-25 "C. Measurements at even lower temperature are below the Krafft point for SDS, which is 8-12 "C in water and increases to -18 "C in In many cases the micellar SDS solutions 0.1 M NaC1.34%35 could be cooled to a temperature below the Krafft point without precipitating the surfactant, allowing for experiments to be performed. The experimental data at these temperatures therefore represent values for a metastable micellar solution being diluted to concentrations below the cmc and the saturation concentration.
11744 J. Phys. Chem., Vol. 99,No. 30, 1995 0
25
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Paula et al. 125 m 5 1 'OoC
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Figure 2. Titration of 10-pL aliquots of octyl glucoside micelles (270 mM) into 1.34 mL of water at 70 "C: (a) calorimetric traces (heat flow against time); (b) reaction enthalpy (obtained by integrating the peaks of the upper curve) vs the total concentration in the sample cell (the heat of demicellization is represented by the length of the arrow); (c) first derivative of curve b calculated numerically from interpolated values. The cmc is defined as the concentration where this curve has a minimum.
Compared to SDS and octyl glucoside, the bile salts exhibit a significantly broader transition region at the cmc; Le., there is no abrupt change in enthalpy. Consequently, the cmc and m d e m i c Can be measured less accurately for these two compounds. The broad transition region is due to the much smaller aggregation number of these surfactants, which can be attributed to the absence of long alkyl chains (see Discussion). Figure 4 shows as an example the titration curve for sodium deoxycholate at 50 "C, together with its first derivative. The enthalpy of demicellization is difficult to determine as the transition range is very broad and the final concentration range, in which addition of more surfactant leads only to a dilution of micelles, cannot be reached in the same experiment. The lower line in Figure 4a used to determine M d e m i c represents this dilution enthalpy which was determined in a separate experiment (not shown). Despite the broad transition range, the cmc can be determined fairly accurately from the first derivative of the titration curve. For all surfactants a difficulty for the cmc determination arises in the temperature range where AHdemic becomes small or even zero. Then, no break in the enthalpy versus concentration curve is observed, and the determination of cmc and AHdemic becomes inaccurate or even impossible. However, the cmc for this specific temperature range can be extrapolated from measure-
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2
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8
ments performed at other temperatures. One possibility for achieving this is to use the van't Hoff equation:
where cmc' is now the critical micelle concentration in mole fraction units and .&$dem,c represents the enthalpy of demicellization as a function of temperature, which is an experimentally
J. Phys. Chem., Vol. 99,No. 30, 1995 11745
Thermodynamics of Micelle Formation -3.4 :SDS in water 0 :SDSin0.1MNaCl
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The relation between K and AG is therefore
AG = -RT In K = RT In([Sl“/[M,I) = nRTln([S]/[M,]””) = nAG, (14)
Tem perature/K Figure 9. Thermodynamic parameters for demicellization of sodium cholate as a function of temperature. cholate. We have therefore attempted a different approach, namely the direct simulation of the calorimetric titration curves using a mass action model without taking into account the effects of counterion: 1 1
For a given value of K , i.e. an aggregation number n and a value of AGO,the Gibbs energy of micellization, the concentrations of the surfactant monomer [SI and of the micelle [M,] can be calculated as a function of Ctotalusing eq 13. This implicit equation can be programmed and solved by iterating procedures. An example calculated for our specific experimental conditions of a dilution experiment of 60 mM sodium deoxycholate is shown in Figure 12 with C,,, = [SI and Cmic = n[M,] and the parameters as indicated. The arrows point at the particular concentrations of monomer and micelle in the syringe (arrow 3 and arrow 1, respectively) and in the cell after injection of the first 25 p L of 60 mM sodium deoxycholate (arrow 4 and arrow 2, respectively). Because the micelles do not completely disappear upon dilution to a concentration below the cmc, as would be the case for the pseudo-phase separation model, the differences in concentrations of monomers and
Paula et al.
11748 J. Phys. Chem., Vol. 99, No. 30,1995
techniques or to the strong temperature dependence of this quantity. Although the experimental data of the present study are somewhat larger than the ones reported in the past, they are still similar. Data for octyl glucoside are in good agreement with already published results at limited temperatures. However, in contrast to previous results,s we show that A H d e m l c increases with temperature and changes sign at -50 "C, so that at this temperature a cmc minimum is predicted and also observed. Figures 6- 10 show the temperature dependence of the thermodynamic demicellization parameters of all four surfactants. At room temperature, the contribution of the enthalpy to the Gibbs energy is very small compared to the entropy term, with the exception of octyl glucoside for which a slightly higher negative demicellization enthalpy is observed. This can be understood in terms of the hydrophobic effect.'o-'2.2' The large loss in entropy is caused by an increase in hydrophobic surface area that is exposed to water when micelles are dissolved. At the temperature where m d e m l c equals, zero the cmc should have its minimal value. This is shown as follows:
SDSIwater SDSlO.1 M NaCl
0.15
Octyl glucoside Cholate
0.10
0.05
...- ...Deoxycholate 1
280
.
I
,
300
I
320
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1
340
.
I
360
.
380
Temperature / OC Figure 11. Heat capacity changes observed for the demicellization of surfactants as a function of temperature. The lines were obtained from a second order polynomial fit for the demicellization enthalpies (see text).
micelles before and after injection have to be taken into account. A simulation of the experimental titration curve thus requires, in addition to the equilibrium constant K and the aggregation number n, three reaction enthalpies. These are the enthalpy of dilution of monomers AHdil(mon), the enthalpy of dilution of micelles mdi](mic), and the demicellization enthalpy AHdemic. Three examples for these simulations are shown in Figure 13 with AGdemic= -AGO. There is a fairly good agreement between experimental and calculated curves. This proves that the previously determined low aggregation numbers for cholates and deoxycholates are reflected in the shape of the titration curves. On the other hand, the deviations at lower concentrations show that the assumed model is obviously still too simple. An increase in the number of fitting parameters, such as required by a more complicated stepwise aggregation model and the inclusion of effects of counterion binding, is not reasonable, however. It would only increase the number of fitting parameters without supporting information from the experimental data. With this example we only wanted to show that a simulation of these titration curves is possible in principle. The problems are the large number of adjustable parameters and the time consuming fitting procedure because for each value of n and AGOthe complete set of curves, as shown in Figure 12, has to be calculated.
This is indeed the case as comparison of the data of Figure 5 with those of Figure 6-10 show. If the cmc is the directly measured quantity, this minimum may not be easily detected. The temperature dependence of the m d e m i c values provide a more precise determination for the temperature of the cmc minimum. Experiments performed with SDS at temperatures between 340 and 350 K demonstrate that most of the contributions to AGdemic are provided by the demicellization enthalpy and that the entropy term gets steadily smaller when the temperature is increased. Extrapolation to temperatures beyond the experimental range predict a AGdemicmaximum at temperatures between 360 and 390 K for SDS in water and 0.1 M NaC1, respectively, depending on the type of extrapolation used. As
this means that the entropy approaches zero at the point where the Gibbs free energy has a maximum. This has been confirmed by studies of other amphiphiles and hydrophobic comp o u n d ~ . ' ~ This . ~ ~ observation .~~ has also been explained by a molecular model. The water structure at room temperature, characterized by extensive hydrogen bonding, breaks down at elevated temperatures, and water becomes a more "normal" solvent. The entropic effects that dominate the hydrophobic effect at room temperature and are responsible for the association of amphiphiles in water are reduced at higher temperature. The association is now driven by an exothermic association enthalpy. For the demicellization process we therefore observe an endothermic AHdemic. We will not go into a discussion of the origin of the hydrophobic effect, which has been discussed recently in much detai1,'2,20,23-25 but will only state that our findings are in complete agreement with the results of studies of the solubility of hydrocarbons and the demicellization of other surfactants.l.16-i8.20,23
Discussion Table 2 presents a summary of the values of the cmc and m d e m i c as reported by other workers and compares them to the results presented in this study. Our values for the cmc determined by titration calorimetry are in excellent agreement with the reported data. Literature values of A H d e m i c for SDS diverge considerably, which might in part be attributed to the application of different
Enthalpies from head group interaction can be considered, but they are probably small and have been predicted to be essentially temperature independent. For octyl glucoside, the temperature at which m d e m i c is zero is approximately 25 degrees higher than that for SDS. The whole m d e m i c curve is shifted to the right. Thus at room temperature A H d e m i c for octyl glucoside is more negative than that for SDS. This effect can be explained assuming changes in head group hydration between
J. Phys. Chem., Vol. 99, No. 30, 1995 11749
Thermodynamics of Micelle Formation
0.08
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0.06
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Ctotal I M Figure 12. Calculated concentration curves for a mass action model for aggregation of sodium deoxycholate with parameters as indicated. Arrows 1 and 3 point at the concentrations of deoxycholate in the syringe, and arrows 2 and 4, at the concentrations in the cell after the first injection of 10 p L
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the monomeric and the micellar state of the surfactant, adding a constant enthalpic term. The differences between octyl glucoside and SDS are obviously due to their different head groups, uncharged for octyl glucoside vs negatively charged in the case of SDS. However, it is difficult to assign these differences in A H d e m i c to particular changes in head group interactions in the charged or uncharged surfactant when demicellization occurs. The change in heat capacity ACp,demlcat room temperature has been shown to be a linear function of the hydrophobic surface that gets exposed to water during the demicellization p r o c e ~ s . ~In~ the ~ * case ~ of unbranched saturated hydrocarbon chains, like the dodecyl group of SDS, this hydrophobic surface
can be expressed by the number of hydrogen atom nH, such as
ACp,demic = 33n, [J/(mol*K)]
(17)
For SDS, a ACp,demicvalue of -450 J/(mol-K) at 298 K for demicellization indicates that only -14 additional hydrogen atoms, which corresponds to one terminal methyl group and -5-6 methylene groups, are exposed to water upon demicellization. Consequently, the remaining 5-6 methylene groups must be in contact with water even in the micellar state. This effect has been observed before and has been accounted for in various models for micellar s t r u c t ~ r e . ~ However, ~-~~ the approximation presented above is probably too simple because
11750 J. Phys. Chem., Vol. 99, No. 30,1995
Paula et al.
TABLE 2: Comparison of Literature Data for Demicellization of Surfactants with Data of This Study literature
surfactant
T (K)
cmc (mM)
AHdemic
(kT/mOl)
cmc‘ (mM)
this study A&mIca
(kT/mOl)
SDS
water 0.1 M NaCl sodium cholate
sodium deoxycholate octyl glucoside
a
298 313 323 298 298 298 288 298 323
8.1,398.3; 8.46 9.66 1.93 13,* 1636 2,39 102.36 28.8’ 20.9: 257,8 23.9’
-0.678,3 0: 0.209,26-0.42713 7.340 7. 1066 2.15,3 1.88313 -13.08: -13.38’ - 14.0709 - 1.268
8.0 8.6 9.4 1.54 15.3 5.5 27.5 23.0 22.6
-.024 6.50 10.90 1.9 -1.59 0.6 - 12.55 -8.24
1.05
Values are interpolated values from experimental curves determined from polynomial fits of the experimental data as shown in the figures.
electrostatic effects have to be considered also. The degree of counterion binding to SDS micelles is -0.7-0.8.” Counterion binding reduces the number of water molecules in the solvation shell of the sodium ion and the negatively charged SDS head groups as the ions can share waters of hydration. Upon demicellization additional water molecules are now required for the complete hydration of the sodium ion and the sulfate head group. This solvation process is associated with a negative AC, and can be quite large.41 For Na2S04 a AC, value of -270 J/(mol.K) is e~timated.~’ A AC, contribution of approximately -100 J/(mol*K) caused by increased solvation of ions upon demicellization could therefore be quite reasonable. This would reduce the number of methylene groups in contact with water in a SDS micelle to a value of 3-4. In 0.1 M NaCl solution the ACp,demlc values are smaller, particularly at higher temperature. Increasing ionic strength leads to higher aggregation numbers because of better shielding of the negatively charged head groups.” One would expect a higher ACp,demlc value as hydrophobic contacts in the micelles should be reduced. The observed opposite behavior is possibly a consequence of the overcompensating effect of increased “hydrophilic hydration” of the previously condensed counterions, when the micelles are dissolved. Octyl glucoside differs from SDS in one important point. As mentioned above, the curves describing the thermodynamic parameters are shifted toward higher temperatures (Figure 6 ) . The cmc minimum where A H d e m l c equals zero occurs at 314318 K. Such behavior has been observed for other non-ionic surfactants as well and is regarded as a characteristic property for this class of compound^.^ Similar to the cmc of SDS, the cmc of octyl glucoside is well defined and the transition from micelle to monomers in the enthalpogram is sharp. Again, we can use ACp,demlc to estimate the number of methylene groups that are in contact with water in the micellar state. A ACpdemIc value of 450 J/(mol-K) at 298 K means that 2-3 methylene groups of the octyl residue are hydrated in the micelle. The ACp,demlc values decrease with increasing temperature as A H d e m l c is not a linear function of temperature. Our linear decrease of ACp,demlc is in agreement with data for the solubility of hydrocarbons in water. These data also show an almost linear decrease of AC, within the temperature range of fluid water.28 The variation of A H d e m l c and ASdemlc of the bile salts with temperature is less pronounced compared to the other surfactants, leading to lower heat capacity changes. The overall feature, entropy dominance in the low temperature range and increasing enthalpy contribution at higher temperatures, is similar to SDS and octyl glucoside. It has been mentioned above that the cmc of the bile salts compared to those of SDS and octyl glucoside is not well defined. This has to do with the molecular structure of these compounds that makes the formation of “common” micelles of
spherical or disk- or rodlike shape impossible. Bile salts do not possess a polar head group or an apolar hydrocarbon tail. Instead, they are better described in terms of a hydrophobic and a hydrophilic molecule s u r f a ~ e .These ~ ~ ~structural ~ ~ ~ ~ properties are believed to be the reason for a micelle aggregation number which is strongly dependent on the total surfactant concentration. This makes a clear definition of a cmc difficult, and it might be more accurate to talk about a critical concentration range instead of a definite concentration. At low concentrations, like the ones in the titration experiments, bile salts are able to form so-called “primary micelles”. These are small aggregates which are composed of 2-9 monomer^.^'-^^ If the concentration is increased, “secondary micelles” with an aggregation number between 9 and 60 are created.39 We have tried to simulate some titration curves of deoxycholate using a mass action model with an aggregation number of 5 (see Figure 13). This model is able to reproduce the experimental data sufficiently well. A more elaborate model would only increase the number of adjustable parameters without giving more insight. It should be noted that the cmc of sodium deoxycholate is smaller than the cmc of sodium cholate. This is expected, as sodium deoxycholate has one less apolar hydroxyl group and therefore the hydrophobic molecule area is increased. There is a slight difference in the temperature at which AHdemicis zero and the ACp,demic values. The latter ones are probably not significant as the temperature range and number of data points are limited. Summary and Conclusions
Titration calorimetry can be used as a routine method for the determination of all thermodynamic quantities for the micellization of surfactant^.'.^^^^-'^ Particularly, the availability of power compensated microtitration calorimeters with high sensitivity has improved the speed and precision of the method.I6 The cmc and the micellization enthalpy can be determined from a single experiment. The temperature dependence of these quantities is easily accessible in the temperature range between 0 and 80 “C. From the temperature dependence of the demicellization enthalpy, udemic, the heat capacity change, ACp,demic, can be directly determined and thus information on the change in hydrophobic contacts upon demicellization can be estimated. The temperature at which the cmc minimum occurs can be determined with great precision from the temperature where AHdemic is zero. The shape of the titration curves also contains information on the aggregation number. A simulation of the titration curves for deoxycholate was attempted using a simple mass action model with a low aggregation number and the association constant and three enthalpic terms as adjustable parameters. The calculations support previous findings that cholates and deoxycholates form micelles with low aggregation numbers.
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