3476
Langmuir 1999, 15, 3476-3482
Temperature Dependence of the Chemical Potential of Tetradecyldimethylaminoxide Micelles in D2OsA SANS Study N. Gorski,†,‡ J. Kalus,*,‡ G. Meier,§ and D. Schwahn| Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research JINR, 141980 Dubna, Moscow Region, Russia; Experimentalphysik I, Universita¨ t Bayreuth, D-95440 Bayreuth, Germany; Institut fu¨ r Festko¨ rperforschung, Forschungszentrum Ju¨ lich, D-52425 Ju¨ lich, Germany; and Max Plank Institut fu¨ r Polymerforschung, D-55021 Mainz, Germany Received November 13, 1998
By means of small-angle neutron-scattering (SANS) experiments, the dependence on concentration and temperature of the mean aggregation number 〈N〉 of rodlike tetradecyldimethylaminoxide (TDMAO) micelles in D2O was determined for 278 K < T < 370 K and for 2.69 mM/L < c < 118 mM/L. By an analysis via the so-called ladder model, the difference of the chemical potentials for a monomer located in the end cap and one in the cylindrical part of the spherocylindrical micelles, respectively, was extracted. This difference depends on the concentration c of TDMAO and strongly on temperature T.
1. Introduction
2. Experimental Details
The mechanism and the thermodynamic parameters influencing the self-organization of surfactants in aqueous solutions are of current interest.1 The aggregates are thermodynamically stable structures, which change their size and shape in response to changes in concentration, composition, temperature, pressure, and other conditions. In dilute solutions of zwitterionic tetradecyldimethylaminoxides (C14H29N+O-(CH3)2), abbreviated TDMAO, micellar aggregates exist. Small-angle neutron-scattering (SANS) experiments have shown that at 297 ( 0.5 K and for TDMAO concentrations of 0.61 mM/L < c < 53.4 mM/L they are of rodlike shape with a radius R of 1.85 ( 0.02 nm. The mean lengths increase steadily with concentration c.2 Previous work has shown that zwitterionic surfactants exhibit properties which could be described as being intermediate between those of a typical nonionic and a typical cationic surfactant of equal chain length:3 For example, no cloud point is observed, as is commonly found for nonionics.4 The mean aggregation number 〈N〉, which is related to the mean length 〈L〉 of the spherocylindrical micelle, shows a complex dependence on concentration and temperature. As previously found by light-scattering methods,3 〈N〉 first increases with temperature and then decreases above 315 K. This unusual behavior is discussed in this paper on the basis of SANS experiments and their detailed analysis.
TDMAO was obtained from Hoechst AG, Gendorf, and recrystallized twice from acetone. D2O of 99.8 % isotopic purity was obtained from ISOTOP in Moscow. The SANS experiments were performed at the time-of-flight spectrometer MURN5 of the pulsed reactor IBR-2 in Dubna, Russia, and at the KWS1 and KWS2 of the Dido-reactor in Ju¨lich, Germany.6 In Dubna, neutrons were used in the wavelength range 0.07-1 nm, and the accessible range of momentum transfer Q was 0.08-2.5 nm-1. Typically the neutron intensity at the sample position is 107 cm-2 s-1. In Ju¨lich, neutrons were used with the wavelength λ ) 0.7 nm and with the monochromatization ∆λ/λ ) 0.2. The samples were contained in 1 mm path length quartz glass Hellma cells with an electrical heater to control the temperature of the samples to (0.5 K. The data treatment followed standard procedures.7 The raw data were corrected for the cell background and detector variation and put on an absolute scale. Background scattering was subtracted by comparison with a corresponding D2O sample. Conversion of the scattered intensities into absolute differential cross sections was performed in Dubna by using an internal calibration standard (vanadium). In Ju¨lich the calibration standard was a vanadium-calibrated polyethylene specimen. Samples of TDMAO concentrations c of 2.69(D), 5.22(D), 12.5(D), 15.6(J), 25.0(D), 31.0(J), 50.0(D), 56.0(J), 101(D), 104(J), and 118(J) mM/L in D2O were studied. The measurements done at Ju¨lich and Dubna are marked by D and J, respectively. As can be seen from an inspection of Figures 2-4 (see below), the data from KWS1, KWS2, and MURN are entirely self-consistent.
* To whom correspondence should be addressed. † Frank Laboratory of Neutron Physics, Dubna, Russia. ‡ University of Bayreuth, Germany. § Max Plank Institut Mainz, Germany. | Forschungszentrum Ju ¨ lich, Germany. (1) Gelbart, W. M.; Ben-Shaul, A. J. Phys. Chem. 1996, 100, 13169. (2) Gorski, N.; Kalus, J. J. Phys. Chem. B 1997, 101, 4390. (3) Hoffmann, H.; Oetter, G.; Schwandner, B. Prog. Colloid Polym. Sci. 1987, 73, 95. (4) Nakagawa, T.; Shinoda, K. In Colloidal Surfactants; Shinoda, K., Nakagawa, T., Tamamuski, B., Isemara, T., Eds.; Academic: New York, 1963; pp 129 ff.
3. Theory 3.1. Neutron Scattering. The scattering cross section per particle dσ/dΩ is given by
dσ/dΩ ) 〈F2〉(F - Fs)2V2S(Q)
(1)
(5) Ostanevich, Yu. M. Makromol. Chem. Macromol. Symp. 1988, 15, 91. (6) Schwahn, D.; Meier, G.; Springer, T. J. Appl. Crystallogr. 1991, 24, 568. (7) Vagov, V.; Kunchenko, A. B.; Ostanevich, Yu. M.; Salamatin, I. M. JINR Report, Dubna, Russia, 1983; P14-83/898.
10.1021/la981592q CCC: $18.00 © 1999 American Chemical Society Published on Web 04/24/1999
Chemical Potential of TDMAO Micelles in D2O
Langmuir, Vol. 15, No. 10, 1999 3477
provided the particles are homogeneous and monodisperse. V is the volume of the particle. F and Fs are the scattering length densities of the micelles and the solvent, respectively. For homogeneous cylinder-like micelles, the mean squared form factor is given by8
〈F2〉 )
∫0π/2(sin(Q(L/2) cos φ)/
(Q(L/2) cos φ)2J1(QR sin φ)/(QR sin φ))2 sin φ dφ (2) φ is the angle between the axis of symmetry of the micelle and the vector of the momentum transfer Q B . J1 is the Bessel function of the first kind and of order 1. L and R are the length and the radius of the micelle, respectively. The value of |Q B | is given by Q ) 4π sin(φ/2)/λ, where λ is the incident neutron wavelength and φ is the scattering angle. S(Q) is the structure factor and is related to the interaction between the micelles. For a sufficiently high value of Q or for non-interacting micelles, S(Q) goes to 1. In our case, we did not observe a so-called correlation peak in the intensity distribution of the scattered neutrons, and so we could assume for our data evaluations at low TDMAO concentration that S(Q) is equal to 1 in the studied Q range. This assumption is strengthened by the results of static light-scattering experiments, where a correlation between the micelles was found above a concentration of about 100 mM/L.3 3.2. Theoretical Concepts for Micellar Aggregation. It is well-known that rodlike micelles show a wide distribution of lengths L. We use the so-called ladder model to analyze the measured intensity distribution curves, neglecting for a moment any possible influence of intermicellar interactions on the size distribution. According to this model,9,10 the size distribution (mole fraction of an N-mer) XN is given by
XN ) βN/K
(3)
where N ) N0, N0 + 1, N0 + 2, ... N0 and N are the aggregation numbers of the spherical and rodlike micelle, respectively. The rodlike micelle has spherical end caps on each side. The radius of the cylindrical section of the micelles is assumed to be equal to the radius of the spherical micelle independent of the lengths L of the micelles. Assuming the value V0 ) 0.480 nm3 for the volume of a monomer, as quoted by Gradzielski,11 and a radius of R ) 1.85 ( 0.02 nm, as quoted in ref 2, N0 turns out to be 4/3πR3/V0 ) 55 ( 2. β ) X1 exp(-δ/(kBT)) determines the width of the length distribution function, the number-averaged aggregation number 〈N〉, and the number-averaged squared aggregation number 〈N2〉:
〈N〉 ) N0 + β(1 - β)
(
〈N〉 ) 〈N〉 N0 +
(
β β 1+ 1-β N0(1 - β) + β
(4a)
))
(4b)
K ) exp((∆ - N0δ)/(kBT)) is a Boltzmann factor corresponding to a chemical potential gain of a surfactant (8) Guinier, A.; Fournet, G. Small-Angle Scattering of X-Rays; Wiley: New York, 1955. (9) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 84, 1044. (10) Chen, S.-H.; Sheu, E. Y.; Kalus, J.; Hoffmann, H. J. Appl. Crystallogr. 1988, 21, 751. (11) Gradzielski, M. Diplomarbeit, Universita¨t Bayreuth, 1989.
molecule to be situated in the cylindrical section of the micelle, δ, compared to the chemical potential of a molecule in the spherical end caps, ∆/N0. Both δ and ∆ have their origin in a substantial lowering of the chemical potential of the aggregates compared to having all the monomers dispersed in water. In order to get rodlike micelles, ∆ and δ must have negative values and the condition |δ| > |∆|/n0 must be fulfilled. Within the framework of the ladder model, one finds12
(
)
〈N2〉 - N0 /(X - X1)1/2 ) exp((∆ - N0δ)/(2kBT)) 〈N〉 β1-N0/2(1 + 1/(β + N0(1 - β)))/ (β + N0(1 - β))1/2 ≈ 2xK for β f 1 (5)
X is the total concentration of TDMAO molecules given in mole fraction. For high X, one can approximate (X X1) by (X - Xcmc). (The critical micellar concentration cmc is 0.122 ( 0.002 mM at 298 K.13) For high concentrations, β f 1 and therefore 〈N2〉/〈N〉 f 2〈N〉 - N0. If ∆ and δ do not depend or only weakly depend on concentration X and X . Xcmc, we see that 〈N〉 - Ν0 becomes proportional to X1/2 or c1/2, which was indeed observed.2 Intermicellar interaction Wint can influence the size distribution {XN}. Wint can be taken into account by means of a mean-field theory for that purpose,14 which is given by Wint ) -0.5∑i NiUi, where Ui ) U∑j fij‚Fj. Each i-mer is interacting with an average local potential Ui produced by other j-mers. This potential is proportional to the j-mer concentration Fj. The parameter U(T) measures the magnitude of the potential, and the coupling constant fij describes how the average interaction between an i-mer and a j-mer depends on the number of amphiphiles i and j in each of them. For three specific forms for the coupling constants fij, the size distribution {XN} can be evaluated:
XN ) Iν(β/Iν)n/K (ν ) 1, 2, 3)
(6)
with ν ) 1: fij ) ij, giving I1 ) 1 with ν ) 2: fij ) 1, giving I2 ) exp(γCM0/(kBT(1 + (γ - 1)X))) with ν ) 3: fij ) (i + j), giving I3 ) exp(γCX/(kBT(1 + (γ - 1)X))) where C(T) ) U(T)/V0 and γ ) V0/Vw. Vw ) 0.03 nm3 and V0 ) 0.48 nm3 are the effective volumes of a water and a solute molecule, respectively. M0 is the moment of order zero of the distribution {XN} and is given by M0 ) ∑N XN. The value of M0 depends for a given concentration X on the actual value of the cmc. Any interaction according to the model ν ) 1 gives no change in the size distribution {XN} at all. In this case, no information about U can be extracted. 4. Results of the SANS Experiments and their Discussion In Figure 1 we present as an example the SANS intensity distribution I at different temperatures for a (12) Sheu, E. Y.; Chen, S.-H. J. Phys. Chem. 1988, 92, 4466. (13) Po¨ssnecker, G. Dissertation, Universita¨t Bayreuth, 1991. (14) Blankschtein, D.; Thurston, G. M.; Benedek, G. B. J. Chem. Phys. 1986, 85, 7268.
3478 Langmuir, Vol. 15, No. 10, 1999
Figure 1. SANS scattering intensity for a 25 mM/L TDMAO solution in D2O. Notice the I versus Q plot in part a and the ln(I‚Q) versus Q2 plot in part b. The solid lines are due to fits as described in the text. The temperatures (283, 293, and 328 K) are indicated. For convenience the curves in part b are shifted by 1.5 and 3.0 units, respectively. The position of the maximum is related to the mean lengths of spherocylinders in this representation. To avoid confusion, in part a measured points and fitted lines for the 323 and 283 K curves are only shown for Q < 0.5 nm-1.
TDMAO concentration of 25 mM/L. Notice that in Figure 1b ln(IQ) versus Q2 is shown, since, for L f ∞, eq 2 yields 〈F2〉 ≈ π exp(-Q2R2/4)/(Q‚L) for Q‚L g 1 and Q‚R e 1.15 This latter representation of the data has the advantage that for infinitely long rods the scattering intensity distribution is characterized by a straight line, whereas for rods of a finite length a maximum occurs. The Qm value of this maximum can be used to get a first rough idea about the length of long rods according to L ≈ 2π/Qm. The solid lines in Figure 1 are due to fits as described now. Knowing the expression for the size distribution of the N-mers (see eq (3), we can get the length distribution of the micelles. By use of the form factors squared 〈F2〉 (see eq (2), which depend on the lengths, we can calculate the SANS intensity distribution I(Q). (I(Q) was convoluted with the known resolution of the SANS instruments.) Therefore, I(Q) depends on one unknown parameter, β, and will be compared with the measured intensity distribution. (15) Kostorz, G. In Treatise on Material Science and Technology, Neutron Scattering; Kostorz, G. Ed.; Academic Press: New York, 1979.
Gorski et al.
Having obtained β by fitting, we can calculate 〈N〉, 〈N2〉, and K, using eq 4a, 4b, and 5 and finally ∆µ ) (∆ - N0δ) itself. The values of 〈N〉 and (∆ - N0δ) depend on both temperature and concentration and are given in Table 1. We observed that at a given temperature the values of ∆µ ) (∆ - N0δ) stay constant or decrease with increasing concentration for all temperatures between 278 and 370 K. (Notice that in the original framework of the ladder model K and ∆µ do not depend on concentration.) At a given concentration we observe that ∆µ ) (∆ - N0δ) increases continuously with increasing temperature, eventually showing a tendency to become constant above 350 K (see Figures 2-4). (Notice that if ∆ and δ are constant, 〈N〉 decreases with increasing temperature according to our model.) We mention that for concentrations above 50 mM/L the mean length of the micelles is not very different from the mean distance between the micelles. Therefore, for these TDMAO concentrations we have a semidilute solution, where interactions between micelles may become important. An evaluation of ∆µ, based on a theory that does not take into account any interaction between micelles, may be questionable for concentrations above 50 mM/L (see below for a further discussion of this point). We mentioned that under certain restrictions the exponential size distribution as expressed by equ 3 is not influenced by intermicellar interactions (see eq 7 and ν ) 1).14 By an inspection of Figures 2-4, we observe that for 2.69 mM/L < c < 31 mM/L the values of ∆µ do not depend on c for the range of temperatures 295 K < T < 330 K. In this temperature range ∆µ can be approximated by a linear function of temperature. Below 295 K a tendency is seen that at a given temperature ∆µ decreases with increasing c. This behavior is found for 2.69 mM/L < c < 118 mM/L. Above 340 K, ∆µ levels off around ∆µ ≈ 24.5 with the exception of c ) 5.22 mM, where ∆µ seems to be approximately ∆µ ∼ 26. At the moment we have no explanation for this exception. For c g 50 mM/L, the limiting high-temperature values of ∆µ decrease with increasing concentration c. It is conceivable that for these concentrations a mutual interaction between the micelles becomes important, giving a structure factor S(Q) * 1. We tested this by use of a structure factor as proposed by Hayter and Penfold.16 S(Q) requires as input the volume fraction of the micelles, the charge Z0, the Debye screening length 1/κ, and the radus Reff of the micelles, which are assumed to be monodisperse and have spherical shape. It is not possible to calculate a structure factor for nonspherical particles in a simple closed form. As long as the micellar shape does not differ too much from spheres, we can assume that calculations according to the model of Hayter and Penfold can fit the experimental data quite well, because the center-to-center interaction potential for large distances between the micelles depends only weakly on the actual orientation of the nonspherical particles. Of course in the case of rather long particles an agreement with this theory cannot be expected. Nevertheless, we adopted this theory for our nonspherical, polydisperse micelles, assuming that the radius of the effective spheres is given by 4πReff3/3 ) 〈N〉‚V0. Taking the input data Z0 ) 0 and κ ) ∞ (which is a hard core model, neglecting any other attractive or repulsive forces), ∆µ changes. S(Q) itself shows practically no correlation peak for c ) 101 mM/L but has of course a value cmc. Knowing the temperature dependence of the cmc between 283 and 313 K,13 we can deduce the temperature dependence of δ and ∆, as shown in Table 2. We observe a steady and nearly linear increase of δ and ∆ with increasing temperature. The quoted errors for δ and ∆ are mainly due to the quoted errors for the cmc. But we have to be aware of the fact that for this evaluation one has to assume that both δ and ∆ do not depend on the concentration X. At least for temperatures below 288 K this seems to be not in accordance with the experimental results, as mentioned above. Unfortunately, no experimental data for the cmc are available for temperatures larger than 313 K where ∆µ reaches the limiting high-temperature value mentioned above. To test the temperature dependence of ∆µ further, we can express ∆µ ) (∆ - N0δ) in terms of enthalpic (HN0 N0H0) and entropic (SN0 - N0S0) contributions, respectively, giving an expression where the temperature appears as a prefactor:
∆µ ) (∆ - N0δ) ) (HN0 - N0H0) - T(SN0 - N0S0) (7) H0(S0) is the gain in enthalpy (entropy) of a molecule if it is transfered from the solvent to the cylindrical section of the micelle compared to the enthalpy (entropy) of a molecule transferred to the spherical end caps, HN0/N0‚ (SN0/N0). Again by comparing eq 6 with the measured values (see for example Figure 2), one cannot extract unambiguously the individual temperature dependences of the enthalpic or entropic contributions. For a moment
Gorski et al.
we can speculate that (HN0 - N0H0) and (SN0 - N0S0) are constant in a restricted region, where ∆µ ) ∆µ(T) is independent of c and where ∆µ can be approximated by a linear function. By following this procedure, we get, for 295 K < T < 330 K and for c < 31 mM, (HN0 - N0H0) ) -17 ( 2 kcal/mol and (SN0 - N0S0) ) -0.092 ( 0.003 kcal/(mol K). A negative entropic contribution of the same order of magnitude (-0.040 ( 0.002 kcal/(mol K)) was indeed found by Missel et al.9 for dodecyl sulfate micelles. But at the same time he found a positive value of 24.4 ( 0.5 kcal/mol for (HN0 - N0H0). For both quoted values he took into account solely hydrophobic interactions of the studied charged micelles. By way of contrast nonionic micelles, which exhibit a lower consolute point, we show qualitatively the same temperature dependence of ∆µ14,19 as that found for TDMAO at lower temperatures: a linear increase of ∆µ with increasing temperature. For nonionics this linear law holds up to the cloud point. Both (SN0 N0S0) and (HN0 - N0H0) are negative and have, for example for a micellar solution of C12E6 in water, as analyzed in refs 14 and 19, values exceeding the TDMAO values for submultiples of 3.2 and 4.7, respectively. In contrast to the normal nonionics, the aminoxides have no cloud points, even in the presence of excess electrolyte. Because no cloud point is observed, critical phenomena like concentration fluctuations,20 giving eventual rise to scattering close to the cloud point, are probably of minor importance. The experimental data of Hoffmann et al.3 show that TDMAO behaves like a nonionic system for low temperatures and like an ionic system for high temperatures. For increasing temperatures, the zwitterionic molecules may show a tendency for slight dissociation, giving rise to electrostatic effects. The aminoxides are coupled to an acid-base equilibrium, and the possibility exists that the size of the micelles is affected by this equilibrium (C14H29(CH3)2N+O- + H2O T C14H29(CH3)2N+OH + O-H). The basicity of the oxide group is, however, rather weak, and only a few tenths of a percent of the compound are present in the protonated form.3 It was reported21 that the degree of dissociation might be around 0.1%. Eventual effects related to such a dissociation are discussed now. Missel et al.22 calculated how counterions influence the micellar growth of spherocylinders. If we adopt their formula 13 for ∆Fel, which is the difference in electrostatic free energy per molecule in a spherical end cap versus a cylindrical micellar region, we end up with negative electrostatic contributions ∆µel, where |∆µel| would be large (small) for c ) 2.69 mM/L (c ) 101 mM/L). Unfortunately, we cannot use the results of this theory, because the Debye length, which is purely determined by the dissociated ions, exceeds by far the mean distance between the micelles. Such a behavior would reduce ∆µel but unfortunately is not covered by this theory. We try now to get some further hints about the order of magnitude of terms influencing ∆µ. According to Israelachvili et al.,23 one can split ∆µ into at least three terms: ∆µ ) ∆µ1 + ∆µ2 + ∆µrest. ∆µ1 is related to the zwitterionic charge group separation D, ∆µ2 is related to the interfacial free energy per unit area, and ∆µrest incorporates all other contributions. With the values (19) Thurston, G. M.; Blankschtein, D.; Fisch, M. R.; Benedek, G. B. J. Chem. Phys. 1986, 84, 4558. (20) Corti, M.; Minero, C.; Degiorgio, V. J. Phys. Chem. 1984, 88, 309. (21) Gradzielski, M.; Hoffmann, H. J. Phys. Chem. 1994, 98, 2613. (22) Missel, P. J.; Mazer, N. A.; Carey, M. C.; Benedek, G. B. J. Phys. Chem. 1989, 93, 8354. (23) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525.
Chemical Potential of TDMAO Micelles in D2O
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Table 2. Temperature Dependence of the Thermodynamic Parameters ∆µ ) (∆ - N0δ), δ, and ∆ (in units of kBT0, T0 ) 298 K) and the cmc T (K)
283
288
293
298
303
308
313
cmc (µM) ∆µ δ ∆/N0
149 ( 2
130 ( 2 15.9 ( 0.2 -12.44 ( 0.02 -12.15 ( 0.02
124 ( 2 16.9 ( 0.2 -12.72 ( 0.02 -12.41 ( 0.02
122 ( 2 18.2 ( 0.1 -12.95 ( 0.02 -12.61 (0.02
120 ( 2
118 ( 2 20.0 ( 0.2 -13.45 ( 0.01 -13.09 ( 0.01
115 ( 1 21.0 ( 0.2 -13.70 ( 0.01 -13.31 (0.01
representative for our system (R ) 1.85 nm, 0 ) permittivity of the free space, ) 81 ) permittivity of water, e ) unit charge, N0 ) aggregation number of a spherical micelle ) 55, V0 ) 0.480 nm-3, as ) surface per amphiphile in the spherical region of the micelle ) 0.78 nm2, ac ) surface per amphiphile in the cylindrical region of the micelle ) 2as/3 ) 0.52 nm2, D ) 0.2 nm, and the surface tension σ of TDMAO at 298 K, measured on a surfactant film at a plane water/air interface24 at 298 K from σ ) 3.09 × 10-2 N m-1), we get ∆µ1 ) N0‚e2‚D(1/as - 1/ac)/(20) ≈ -31 and ∆µ2 ) N0σ(as - ac) ≈ 107. The value for σ in principle must be corrected with respect to a surface curvature,25 giving a reduction of σ. We can estimate that we have to correct σ by a factor of the order of 0.68. (We used the value Z ) 0.4 nm for eq 2 from ref 25 for this estimation, neglecting that the principal local curvature radii are different for the spherical and cylindrical parts of the spherocylinders, respectively.) We mention that various authors use quite different values for actual σ-values.9,26,27 The sum of ∆µ1 + ∆µ2 is positive, but as soon as we assume that σ decreases with increasing temperature (we could not find measured values for the temperature dependence of σ), ∆µ1 + ∆µ2 decreases with increasing temperature, which is in contradiction to our experimental result for ∆µ. The considerations related to the magnitude of the terms ∆µ1 and ∆µ2 are not sufficient and seem to be too simplifying. Therefore, we have to conclude that another mechanism is important. It might be that the growth of ∆µ with increasing temperature is a consequence of a typical hydrophobic interaction, resulting from a change of order in the water structure.1,14,28-31 Especially near the interface between the zwitterionic head groups of the surfactant and solution, the water molecules have a preferred orientation while at the same time the rate of molecular motion is restricted.32 Orientation, motion, and hydration of water molecules are assumed to depend strongly on temperature33,34 and give rise to entropic and enthalpic temperature dependencies. The water in the shell around the micellar surface has a lower enthalpy and entropy than bulk water, and the former is more structured than the latter.18 An overlap of hydration shells of neighboring micelles gives a net reduction of the amount of structured water and finally an increased effective attraction between micelles.35 (24) Oetter, G.; Hoffmann, H. J. Dispersion Sci. Technol. 1988-1989, 9, 459. (25) Choi, D. S.; Jhon, M. S.; Eyring, H. J. Chem. Phys. 1970, 53, 2608. (26) Tanford, C. J. Phys. Chem. 1974, 78, 2469. (27) Hermann, R. B. J. Phys. Chem. 1972, 76, 2754. (28) Lang, J. C.; Morgan, R. D. J. Chem. Phys. 1980, 73, 5849. (29) Geetha, B.; Mandal, A. B. J. Chem. Phys. 1996, 105, 9649. (30) Jonstro¨mer, M.; Jo¨nsson, B.; Lindmann, B. J. Phys. Chem. 1991, 95, 3293. (31) Mitchell, D. J.; Tiddy, G. J. T.; Waring, L.; Bostock, T.; McDonald, M. P. M.; J. Chem. Soc., Faraday Trans. 1 1983, 79, 975. (32) Marcelja, S.; Radic, N. Chem. Phys. Lett. 1976, 42, 129. (33) Funari, S. S.; Holmes, M. C.; Tiddy, G. J. T. J. Phys. Chem. 1994, 98, 3015. (34) Nilsson, P.-G.; Wennerstro¨m, H.; Lindmann, B. J. Phys. Chem. 1983, 87, 1377. (35) Kjellander, R. J. Chem. Soc., Faraday Trans. 1 1981, 77, 2053.
It is conceivable that the entropy of the hydrated water molecules near the spherical end caps is different from their entropy near the cylindrical part. This is due to the differing mean curvatures and to the more open structure of the surface around the end caps. This open structure allows more water molecules to enter the surface region where they can interact more efficiently with the strong electric dipole field of the zwitterions, giving rise to a strong polarization of the water molecules. In addition this mechanism also influences the enthalpy of the water molecules, which becomes different for the two regions, spherical end caps and cylindrical region, respectively. Qualitatively we can try to explain the experimental results, if we assume that the reductions of entropy and enthalpy of the water molecules which are related to one TDMAO molecule (compared to bulk water) are both larger for the end cap region than for the cylindrical region. Then the contribution of only the hydrated water to ∆µrest results in a negative enthalpic (HN0 - N0H0)rest term and a positive entropic term -T(SN0 - N0S0)rest. The temperature dependence of ∆µrest can be explained by assuming that (HN0 - N0H0)rest (which adds to the other enthalpic contributions of ∆µ1 and ∆µ2) is small compared to -T(SN0 - N0S0)rest and by further assuming that predominantly both S0 and H0 adjust more rapidly to the bulk water values by an increase of temperature than HN0 and SN0. This seems to be conceivable, if the interaction energy of hydrated water in the cylindrical part of the micelles is lower than that in the spherical part and if the temperature dependences of the entropic and enthalpic contributions of the TDMAO molecule itself to ∆µrest are assumed to be of less importance. 6. Conclusions We observed an increase of the length 〈N〉 of the rodlike micelles respectively with increasing TDMAO concentration c and with temperature up to ∼340 K. For higher temperatures, 〈N〉 decreases again. We have shown that the micelles have a pronounced length distribution which was analyzed within the framework of the so-called ladder model. Basically two thermodynamic parameters are required to characterize the system: δ, the energy advantage of inserting an additional monomer into a micelle of size equal to or greater than that of a micelle having the minimum aggregation number N0 ) 55 ( 2, and ∆/N0, where ∆ is the chemical potential of micellization of the spherical micelle with aggregation number N0. The temperature dependence of ∆µ ) ∆ - N0δ was evaluated for each concentration and shows a steady increase with increasing temperature. For low concentrations (c e 31 mM/L) and for temperatures exceeding 290 K, ∆µ does not depend on concentration. Using published results of the cmc, we were able to get information about the temperature dependences of ∆ and δ, provided the so-called ladder model is applicable at all concentrations. Interaction between the micelles was introduced to analyze the results for the concentrated solutions (c g 50 mM/L), where the solution is semidilute. A first correction was applied via the structure factor S. By this correction the ∆µ ) ∆µ(T) curves of the 50 and 56 mM/L solutions coincide
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with the curves for lower concentrations, for T g 290 K. The ∆µ curves for the solutions with concentrations 101, 104, and 118 mM/L still show a deviation from the lowconcentration curves. For these concentrations we incorporated a second correction based on a theory developed by Blankschtein et al.14 In the framework of this theory a change of the length distribution of the spherocylinders was related to the mutual micellar interaction. Relevant interaction parameters were evaluated, showing that for these concentrations the interaction is attractive and increases in magnitude with increasing temperature. This sheds some light on the importance of hydration forces or
Gorski et al.
energies, which can be separated in enthalpic and entropic contributions. Acknowledgment. We thank the Joint Institute for Nuclear Research JINR, Dubna, and the Forschungszentrum Ju¨lich for getting access to the SANS spectrometers, and we thank Professor Dr. Heinz Hoffmann for the gift of TDMAO. This research was supported by the BMBF of the Bundesrepublik Deutschland under Grant No. 03-DU4BAY-6. LA981592Q
