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Thermodynamic and Kinetic Study of the Sphere-to-Rod Transition in Nonionic Micelles: Aggregation and Stress Relaxation in C14E8 and C16E8/H2O Systems Georg Ilgenfritz,*,† Ralph Schneider,† Ernst Grell,‡ Erwin Lewitzki,‡ and Horst Ruf‡ Institut fu¨ r Physikalische Chemie der Universita¨ t zu Ko¨ ln, Luxemburger Strasse 116, D-50939 Ko¨ ln, Germany, and Max-Planck-Institut fu¨ r Biophysik, Marie-Curie-Strasse 15, D-60439 Frankfurt a.M., Germany Received October 16, 2003. In Final Form: December 16, 2003 The energetics and dynamics of the growth of nonionic n-alkyl poly(ethylene glycol) ether surfactant micelles in the isotropic L1 phase have been analyzed on the basis of a recently described, concentration dependent thermal transition.1 This highly cooperative endothermic transition is assigned as a sphereto-rod transition. We analyze the thermodynamic data in terms of aggregation-fusion of complete micelles and also discuss the kinetics in the frame of this “random micelle aggregation” model. The analysis of the calorimetric data leads to a cooperative unit of about 160 detergent molecules undergoing simultaneously the transition from spheroidal to cylindrical structure. We investigate the aggregation kinetics in the microsecond to millisecond range using the temperature-jump (TJ) method with observation of scattered and transmitted light and the stress relaxation dynamics monitoring transient electric birefringence (TEB). As expected from the model, the aggregation dynamics gets faster with increasing temperature, while the stress relaxation dynamics slows down with temperature, leading to apparent negative activation energies. At the transition temperatures, the dynamics of the structural transition for C14E8 is characterized by an association and dissociation rate constant of 4.4 × 106 M-1 s-1 and 1.8 × 103 s-1, respectively, compared to the much slower dynamics of C16E8 with the corresponding rate constants 8.5 × 104 M-1 s-1 and 29 s-1, respectively. The rate constants of micelle fusion increase, while those of micelle scission decrease with increasing temperature. The TEB results are discussed using the relations derived in the literature for the competition between rotational motion and chain scission. The fast dynamics of C14E8 compared to the much slower dynamics of C16E8 provides an understanding of the differences in stress compliance and viscosity for these systems. As a major result of the study, we present a consistent mechanism of micellar growth which involves the aggregation-fusion of two micelles in a single, rate-limiting reaction step and not the stepwise incorporation of single surfactant molecules into pre-existing micelles. The speeding up of the TJ kinetics in C16E8 at higher temperatures indicates a further structural transition, possibly network formation.
Introduction Amphiphilic molecules exhibit the phenomenon of selforganization and show a characteristic variety of aggregated forms and mesophases.2 In the isotropic L1 phase, above its critical micelle concentration (cmc), aggregated surfactants adopt the nanoparticulate structure of spheroidal micelles. At higher concentrations and/or above certain critical temperatures, further aggregation can occur, leading to larger structures, that is, rod- or wormlike aggregates up to networks. The existence of an aggregation process in micellar systems at a well-defined temperature and concentration has been known for a long time; its concentration dependence has led to the terminus of a “second critical micelle concentration”.3 The early investigations on the so-called sphere-to-rod-transition were all concerned with ionic surfactants which exhibit aggregation at higher salt concentration (sodium dodecyl sulfate (SDS),4 cetylpyri* Corresponding author. Phone: 0049 221 470 4557. Fax: 0049 221 470 5104. E-mail:
[email protected]. † Institut fu ¨ r Physikalische Chemie der Universita¨t zu Ko¨ln. ‡ Max-Planck-Institut fu ¨ r Biophysik. (1) Grell, E.; Lewitzki, E.; von Raumer, M.; Hormann, A. J. Therm. Anal. 1999, 57, 371. (2) Mitchell, D. J.; Tiddy, G. J. T.; Waring, L.; Bostock, T.; McDonald, M. P. J. Chem. Soc., Faraday Trans. 1 1983, 79, 975. (3) Porte, G.; Poggi, Y.; Appell, J.; Maret, G. J. Phys. Chem. 1984, 88, 5713.
dinium bromide (CpyBr),5 tetradecyltrimethylammonium chloride (TTAC)6). For the nonionic n-alkyl(Ci)-poly(ethylene glycol)(Ej)ethers, which are the subject in the present investigation, the increase in molecular weight has been deduced from osmotic measurements (C12E6, C16E97) and from light scattering investigations (C12E6,8,9 C14E8,10 C12E511). Recent investigations concentrate on small-angle neutron scattering (SANS)12 and electron microscopy13 providing detailed information on the structures involved. In contrast to these structural studies, dynamic measurements have only scarcely been reported.9,14-17 Transient electric birefringence (TEB, Kerr effect) is, apart from viscoelastic methods, a powerful (4) Young, C. Y.; Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1978, 82, 1375. (5) Porte, G.; Appell, J. J. Phys. Chem. 1981, 85, 2511. (6) Imae, T.; Ikeda, S. J. Phys. Chem. 1986, 90, 5216. (7) Attwood, D.; Elworthy, P. H.; Kayne, S. B. J. Phys. Chem. 1970, 74, 3529. (8) Brown, W.; Johnsen, R.; Stilbs, P.; Lindman, B. J. Phys. Chem. 1983, 87, 4548. (9) Strey, R.; Pakusch, A. In Surfactants in Solution; Mittal, K. L., Bothorel, P., Eds.; Plenum Press: New York, 1987; Vol. 4, p 465. (10) Richtering, W. H.; Burchard, W.; Jahns, E.; Finkelmann, H. J. Phys. Chem. 1988, 92, 6032. (11) Menge, U.; Lang, P.; Findenegg, G. H. J. Phys. Chem. B 1999, 103, 5768. (12) Glatter, O.; Fritz, G.; Lindner, H.; Brunner-Popela, J.; Mittelbach, R.; Strey, R.; Egelhaaf, S. U. Langmuir 2000, 16, 8692. (13) Bernheim-Groswasser, A.; Wachtel, E.; Talmon, Y. Langmuir 2000, 16, 4131.
10.1021/la0359326 CCC: $27.50 © 2004 American Chemical Society Published on Web 01/23/2004
Sphere-to-Rod Transition in Nonionic Micelles
method for investigating stress relaxation and has been applied for several ionic surfactant and microemulsion systems.18-22 There is extensive theoretical work to understand thermodynamics and kinetics of surfactant self-organization. Regarding thermodynamics, the ideas comprise simple mass action behavior,23 statistical mechanics formulations with different degrees of complexity,24-26 and concepts of critical fluctuation phenomena.27 Stress relaxation has been discussed in terms of orientation of rods18 and of “wormlike micelles” forming an entangled network,28 where chain reptation competes with chain breakage. Concerning the sphere-to-rod transition of micelles, it has been argued long before by Porte et al.3 that there must be an energy barrier involved in the elongation process. In recent investigations,1,29 it was possible to provide direct evidence for the existence of such a barrier and to measure its energy as well as to determine the transition temperature (TU) by differential scanning calorimetry (DSC). Using various CiEj surfactants, it was shown that the change in specific heat exhibits a rather narrow endothermic transition, a few kelvins wide. The calorimetric results allow further important characterization of structural changes, in particular giving information on the cooperativity of the transition. This can be achieved by comparing calorimetric and van’t Hoff reaction enthalpies. In the present paper, we perform an analysis of the calorimetric data for the nonionic C16E8 and C14E8 surfactants1,29 and determine estimates of the cooperative unit, that is, the number of surfactant molecules transformed simultaneously. We also present kinetic data for these systems obtained from temperature-jump (TJ) experiments, which characterize the dynamics of aggregation, and time-resolved electric birefringence measurements (TEB, Kerr effect), which probe the stress relaxation of the micellar system. We discuss thermodynamic and kinetic results in the common frame of the random micelle aggregation model,23 which is based on the assumption that growth of the micelles is brought about by the association/fusion of complete micelles. Materials and Methods Chemicals. The surfactants C14E8 (MW 566.8) and C16E8 (MW 594.9) were purchased from Sigma and Fluka and were used without further treatment (phase diagrams can be found in refs (14) Hoffmann, H.; Kielman, H. S.; Pavlovic, D.; Platz, G.; Ulbricht, W. J. Colloid Interface Sci. 1981, 80, 237. (15) Lessner, E.; Frahm, J. J. Phys. Chem. 1982, 86, 3032. (16) Kahlweit, M. J. Colloid Interface Sci. 1982, 90, 92. (17) Kositza, M. J.; Rees, G. D.; Holzwarth, A.; Holzwarth, J. F. Langmuir 2000, 16, 9035. (18) Schorr, W.; Hoffmann, H. In Physics of Amphiphiles; Degiorgio, V., Ed.; Elsevier North-Holland: New York, 1985; p 160. (19) Hoffmann, H.; Kramer, U.; Thurn, H. J. Phys. Chem. 1990, 94, 2027. (20) Mantegazza, F.; Degiorgio, V.; Giardini, M. E.; Price, A. L.; Steytler, D. C.; Robinson, B. H. Langmuir 1998, 14, 1. (21) Vanderlinden, E.; Bedeaux, D.; Hilfiker, R.; Eicke, H. F. Ber. Bunsen-Ges. Phys. Chem. Chem. Phys. 1991, 95, 876. (22) Schelly, Z. A. Curr. Opin. Colloid Interface Sci. 1997, 2, 37. (23) Mukerjee, P. In Physical Chemistry: Enriching Topics from Colloid and Surface Science; vanOlphen, H., Mysels, K., Eds.; Iupac Theorex: La Jolla, CA, 1975; Chapter 9. (24) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525. (25) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 84, 1044. (26) Ikeda, S. J. Phys. Chem. 1984, 88, 2144. (27) Eriksson, J. C.; Ljunggren, S. J. Chem. Soc., Faraday Trans. 2 1985, 81, 1209. (28) Cates, M. E. Macromolecules 1987, 20, 2289. (29) Grell, E.; Lewitzki, E.; Schneider, R.; Ilgenfritz, G.; Grillo, I.; von Raumer, M. J. Therm. Anal. 2002, 68, 469.
Langmuir, Vol. 20, No. 5, 2004 1621 2 and 10). All studies were performed within the isotropic, micellar L1 phase, above the cmc but below the cloud point temperatures. The Sigma substances gave a somewhat lower conductivity and were used for the Kerr effect measurements. All solutions were prepared with bidistilled or Millipore treated water. Only TJ studies were performed in the presence of 0.1 M KCl to achieve Joule heating within a few microseconds. Before use, all solutions were allowed to equilibrate for at least 3 h at elevated temperature (about 323 K) and were utilized for a maximum of 2 weeks. Differential Scanning Calorimetry. DSC studies were performed with the MCS DSC instrument of Microcal by applying a temperature gradient of 30 K per hour. To apply fit procedures for a detailed analysis, the DSC curve of 60 mM C14E8 has been remeasured, leading to parameters that differ slightly from the published mean values.1 The solutions were degassed under reduced pressure prior to every measurement. Temperature-Jump Method. TJ experiments were performed using two different apparatus. The Ko¨ln equipment uses a condenser discharge (20 nF, 20 kV, effective cell volume of 0.45 cm3), and the transmitted light was monitored at 350 nm. In the Frankfurt equipment, similar to that described by Rigler et al.,30 a voltage pulse (HV pulse generator Hilo Test, Karlsruhe) is applied (1 µF, 11-13 kV, pulse duration of 5-8 µs, effective cell volume of 0.65 cm3); 90° scattered light was observed at 478 nm. In general, temperature jumps of 2 K were generated with both setups. The transient signals were stored in transient recorders and analyzed with the program DISCRETE.31,32 All temperatures given in the text and the figures refer to the final temperature after the jump. Transient Electric Birefringence. A rectangular pulse of up to 20 kV and 10 ms duration was applied to the sample contained between platinum electrodes, separated by 2 cm. Switching on the electric field is done with electron tubes; switching off occurs via a triggered spark gap and thus is very fast. All relaxation data refer to the field-off condition. The pulsegenerating circuit has been described before.33 The current through the solution is measured with a current viewing resistor in series with the sample cell. The optics for Kerr effect measurement consists of a helium-neon laser, two polarizers (Halle), a quarter wave plate (Spindler & Hoyer), and an RCA 1P28 photomultiplier in the standard optical arrangement as described before.34 Stopped-Flow (SF) Kinetics. The experiments were performed with a SMX-18 setup of Applied Photophysics using the fluorescence detection to observe 90° scattered light at 350 nm. Viscosimetry. The viscosities were determined with a falling ball viscosimeter (HAAKE) in the range from 0.3 to 200 mPa‚s (four different balls applied). The temperature program and the time measurements for the ball passing the two light barriers were computer controlled. The system was calibrated with degassed bidistilled water and viscosity standard oils (Physikal. Techn. Bundesanstalt).
The Random Micelle Aggregation Model We will discuss the thermodynamic and kinetic results within the frame of the random micelle aggregation model. This is in fact the simplest model which accounts for the high cooperativity of the transition and was among the first models which have been applied to describe the sphere-to-rod transition.23 The mathematics involved can be found in textbooks;35 calculation of specific heats however has, as far as we know, not been done before. In the following, we review the model giving the basic equations for the description of the thermal transition curve and the aggregation relaxation kinetics. The model (30) Rigler, R.; Rabl, C. R.; Jovin, T. M. Rev. Sci. Instrum. 1974, 45, 580. (31) Provencher, S. W. J. Chem. Phys. 1976, 64, 2773. (32) Provencher, S. W. Biophys. J. 1976, 16, 27. (33) Runge, F.; Ro¨hl, W.; Ilgenfritz, G. Ber. Bunsen-Ges. Phys. Chem. Chem. Phys. 1991, 95, 485. (34) Schlicht, L.; Spilgies, J. H.; Runge, F.; Lipgens, S.; Boye, S.; Schu¨bel, D.; Ilgenfritz, G. Biophys. Chem. 1996, 58, 39. (35) Hunter, R. J. In Foundations of Colloid Science; Clarendon Press: Oxford, 1987; Vol. 1, Chapter 10.
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Ilgenfritz et al.
is characterized by the chemical reactions
Mi + Mj h Mi+j
i,j ) 1,2,...
(1)
where Mi and Mj denote rodlike micelles (concentrations Ci and Cj) consisting of a fused assembly of i and j small spheroidal micelles, respectively, acting as basic units which we will denote further on as “micellar units”. M1 represents the intact small spheroidal micelle (concentration C1). Each reaction in scheme 1 is in principle characterized by an equilibrium constant Kij and the corresponding rate constants. The model in its simplest form assumes equal equilibrium constants for each step. This assumption can easily be understood when considering rodlike micelles as cylinders with hemispherical end caps. Reaction of two micelles Mi and Mj to form Mi+j then involves the transfer of two end caps, that is, a complete spheroidal micelle M1, into the cylindrical structure. This should be independent of the length of the reacting rods. It has been pointed out by Mukerjee23 that this might however not hold for the formation of the dimer from two monomers. The necessity for such a modification is also borne out by the findings from SANS and electron microscopy in various systems that the cylinder radius is smaller than that of an end cap or a spheroidal micelle. As a consequence, fewer surfactants as corresponding to M1 are incorporated into the energetically more favorable cylindrical structure of a dimer M2. Thermodynamically, the model is equivalent to the model of stepwise incorporation of only small spheroidal micelles, which can be characterized by the equilibrium constant K and the reaction enthalpy ∆H for the reactions
Mm-1 + M1 h Mm for m ) 3,4, ...; K ) Cm/(Cm-1C1) (2) For the dimerization step 2M1 f M2, we set according to Mukerjee23 K2 ) RK with the factor R e 1, the value R ) 1 corresponding to the ideal limiting case. The temperature dependence of K obeys the van’t Hoff relation
d ln K ∆H ) dT RT2
(3)
R is assumed to be temperature independent. Thermodynamics. With the notation q ) KC1, the micelle concentrations are given by
C1 )
q K
Cm )
Rqm K
mg2
(4)
q can be determined as a function of the total concentration of micellar units, C0, from
Z(q) ≡
Rq + (1 - R)q ) KC0 (1 - q)2
(5)
The temperature dependence of q is determined by the temperature dependence of the equilibrium constant K. Using the van’t Hoff relation, eq 3, we have
( (
K(T) ) K(T0) exp
))
1 ∆H 1 R T0 T
(6)
The reference temperature T0 will be chosen as the temperature at the maximum of the transition curve at a surfactant concentration of 60 mM.
The fraction of micelles aggregated, p, corresponding to a mean aggregation number 〈m〉 ) 1/(1 - p) is given by
p)
Rq R + (1 - R)(1 - q)2
(7)
Equation 5 arises from mass conservation ∑1mCm ) C0; eq 7 results from the definition 〈m〉 ) ∑1mCm/∑1Cm. This can be verified by using the sum expressions ∑2qm ) q2/(1 - q) and ∑2mqm ) [q/(1 - q)2] - q. For the limiting case R ) 1, we have p ) q and expressions in closed form for all relevant thermodynamic and kinetic parameters can be obtained. In this case, one arrives at a practically exponential distribution of aggregated species with the mean aggregation number
〈m〉 ) 1/(1 - q) ) 1/2(1 + x4KC0 + 1)
(8)
Introducing this expression into eq 4 gives direct evidence of the exponential distribution for larger values of m: Cm ) (1 - 1/〈m〉)m/K ≈ exp(-m/〈m〉)/K. For the calculation of the heat capacity, we set H1 for the partial molar enthalpy of the spheroidal micelle and Hm ) mH0 (m g 2) for the cylindrical aggregate containing m micellar units. The enthalpy of the system is then given by
1
H)
C0
1 C0
∑1 HmCm
(9)
resulting in
C1 1 H ) H0 + (H1 - H0) C0 C0 ) H0 -
q ∆H Z(q)
(10)
where we have introduced the reaction enthalpy ∆H ) H0 - H1 and used eqs 4 and 5. Differentiation with respect to T results in the excess heat capacity per mole of micellar units,
∆CP ≡
1 1 dH ∆H2 1 ) C0 dT RT2 Z/q dZ/dq
[
]
(11)
Here we have used ∂ ln Z/∂T ) ∂ ln K/∂T together with eq 3. The polynomial Z(q) is given by eq 5; its solution for each temperature requires numerical calculations. In the limiting case R ) 1, eq 11 results in
∆CP )
2q(1 - q)2 (∆H)2 1+q RT2
(12)
Experimental values of the DSC study are the total surfactant concentration, C0S, and the enthalpy change per mole of surfactant, ∆HU, from the area under the transition curve. For modeling the transition curve according to eq 11 or 12, a value for the number NS of surfactant molecules contained in a spheroidal micelle is chosen. Setting C0 ) C0S/NS and ∆H ) NS∆HU and adjusting the value of K(T0) leads to curves which can be compared to the experimental data. We have used here the chemical notation in terms of equilibrium constants which was put forward by Mukerjee23 and used further by others.5,26 This is in fact formally equivalent to the statistical mechanics expres-
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Langmuir, Vol. 20, No. 5, 2004 1623
sions for the free energy as given in the literature36 or to the linear Ising model. It may be stressed that the analysis given is a pure thermodynamic one which allows a priori no direct conclusions about the structures involved. Interpretation of the aggregates in terms of fractal clusters, linearly aggregated spheres (pearl chains), stiff rods, or wormlike structures requires the application of size and shape sensitive methods. Kinetics. Regarding kinetics, one has to distinguish between the two mechanisms according to eq 1 and 2. We certainly have to allow for the elementary steps of random chain association and chain fission, as has also been assumed by Cates et al. in discussing wormlike micelles.28,37 The simple random aggregation model has been applied long before by Smoluchowski for the (irreversible) aggregation of lyophobic colloids and by Flory38 for the description of polycondensation and has also been used to describe aggregation processes of proteins.39,40 In the latter work, the relaxation kinetics has been discussed for both mechanisms. For the case R ) 1, the random aggregation model (eq 1) gives a single relaxation time, while the stepwise mechanism (eq 2) leads to a distribution of relaxation times. Denoting (2kass) and (2kdiss) the rate constants of association (Mi + Mj f Mi+j, the factor 2 is a statistical factor for i * j) and dissociation (chain fission Mk f ...), whereby K ) kass/kdiss, the relaxation time for the random micelle aggregation model is given by 1/τ ) kdiss + 2kass∑Cm. This relaxation time refers to small changes in the sum of micellar species, ∑Cm. It applies also to light scattering changes in the case of small perturbations, that is, in a T-jump relaxation experiment, as has already been shown in ref 39. Using eq 8, we obtain (R ) 1)
1/τ ) kdiss(1 + q)/(1 - q) ) kdiss(2〈m〉 - 1)
(13a)
At larger aggregation numbers, we thus expect
1/τ ≈ 2kdissxKC0
(13b)
The reciprocal relaxation time should therefore be proportional to the average aggregation number and scale with the square root of concentration. The relaxation time according to eq 13 can also be interpreted as the mean lifetime of the chain before breaking at any point. The random association model discussed above implicitly assumes discrete numbers of surfactants in the rodlike micelles, Mk, containing the amount of k micellar units. On the other hand, we have to allow for random scission of larger micelles at any point, leading to a continuous distribution of lengths. The use of discrete variables is however no real restriction. A formulation in terms of continuous lengths has been given by Marques et al.37 leading to equivalent expressions. Equation 13b can be brought into the form discussed by Cates28 by introducing the fission rate constant per unit length k∼ ) kdiss/l0, where l0 denotes the length of the cylindrical unit, corresponding to a spheroidal micelle or a pair of end caps; L ) 〈m〉l0 is the average length of the chain. Equation 13b then reads τ ≈ 1/(2k∼L), the mean lifetime of the chain being inversely proportional to the chain length. (36) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. (37) Marques, C. M.; Turner, M. S.; Cates, M. E. J. Non-Cryst. Solids 1994, 172, 1168. (38) Flory, P. Principles of Polymer Chemistry; Cornell University Press: New York, 1953; Chapter 8. (39) Thusius, D. J. Mol. Biol. 1975, 94, 367. (40) Thusius, D.; Dessen, P.; Jallon, J. M. J. Mol. Biol. 1975, 92, 413.
Figure 1. Heat capacity changes as a function of temperature for 60 mM C16E8/H2O. Squares: experimental values. Solid curves: random micelle aggregation model. (a) With R ) 1 for different numbers NS of surfactant molecules contained in a small micelle (from top to bottom: NT ) 320, 260, 200); ∆HU ) 2.1 kJ/mol surfactant. (b) With R ) 0.1, NS ) 160, and K(TU) ) 2.95 × 103 M-1 (the fit curve is hardly to be distinguished from the experimental course). The lower figure contains in addition the average number of aggregated micelles according to the model (right ordinate).
Results Calorimetric Studies. DSC data of the binary systems C14E8/water and C16E8/water have already been presented in previous papers.1,29 As a characteristic example, we reproduce here one of the DSC curves of 60 mM C16E8, which we use for the fit with the micelle aggregation model (eq 1). The experimental curve, characterized by squares in Figure 1a,b, indicates a rather narrow endothermic transition with a halfwidth of about 5 K. The maximum of the fit-curve defines the transition temperature TU ) 305.1 K. Integration of the DSC curve gives an enthalpy change of ∆HU ) 2.1 kJ/mol surfactant. From these data, it is clear that the process involved is highly cooperative, that is, a large number of surfactant molecules must be simultaneously transformed from the spheroidal to the new cylindric structure. The simplest model which accounts for this behavior is the aggregation of intact micelles above TU. To demonstrate that this concept leads to meaningful results, heat capacities have been modeled by eq 12 for the case R ) 1 or by eq 11 for the case R < 1 using different numbers NS of surfactant molecules forming a spheroidal micelle or a pair of end caps, respectively. For each value of NS chosen, the total concentration of spheroidal micellar units C0 and the reaction enthalpy ∆H are calculated as outlined above. The equilibrium constant K0 ≡ K(T0) is then adjusted such that the transition temperature for a surfactant concentration of 60 mM has the value TU ) 305.1 K. The solid
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Ilgenfritz et al. Table 1. Rate and Equilibrium Parametersa for the Formation of Rodlike Micelles (Surfactant Concentration C0S ) 60 mM)b TU NS ∆H τ(T)TU) Eτ kdiss(T)TU) kass(T)TU) Ediss Eass Eτ,TEB Eη
K kJ mol-1 ms kJ mol-1 s-1 M-1s-1 kJ mol-1 kJ mol-1 kJ mol-1 kJ mol-1
C14E8
C16E8
313.8 (40.7 °C) 160 290 0.29 110 1.8 × 103 4.4 × 106 -60 ((30) 230
305.1 (32.0 °C) 160 340 18 130 2.9 × 101 8.5 × 104 -70 ((30) 270 -200 -135
-55
a
Figure 2. Concentration dependence of the transition temperatures TU relative to TU0 at 60 mM surfactant concentration, ∆TU, for the system C16E8/H2O, as determined from the maximum of the heat capacities. Solid curves: dependence expected from the random micelle aggregation model with R ) 0.05-1.
lines in Figure 1a, calculated for three values of NS with R ) 1, indicate the high sensitivity of the model for this parameter, which thus can be determined quite accurately. As expected, smaller values of NS lead to broader, less cooperative transition curves. With R ) 1, the experimental curve cannot completely be described. But this is achieved with NS ) 160 and K(T0) ) 2.95 × 103 M-1 when R is reduced to 0.1 (Figure 1b). This value for R also follows from the dependence of the transition temperature on surfactant concentration shown in Figure 2. With increasing surfactant concentration, the transition temperature is shifted toward lower values, as expected from the model, and the experimental data are best described by R ) 0.1. The DSC curve for the 60 mM C14E8 system (not shown) is similar; TU is increased to 313.8 K with a halfwidth of 8.5 K and ∆HU ) 1.8 kJ/mol. A corresponding fit to the DSC curve and to the concentration dependence of TU leads to similar parameters (see Table 1). Actually, Figure 2 represents part of the phase diagram of the binary system C16E8/H2O describing the isotropic L1 phase. The curve transition temperature TU versus surfactant concentration divides this isotropic phase into two parts. The region below the curve can be considered as that of spheroidal micelles, and the region above as that of rodlike micelles. In the present study, we carry out kinetic experiments in pure water and in the presence of 100 mM KCl. Therefore, the influence of the salt on the thermal transition was investigated by DSC. For C16E8, the transition temperatures TU for the 60 mM and 20 mM aqueous solutions are lowered by 1 K in the presence of the salt. The halfwidth and the enthalpy change remain unchanged. For C14E8, the TU values for the 60 mM and 20 mM solutions are lowered by 1.5 and 2.0 K, respectively, in the presence of the salt. Again here, the halfwidth and the enthalpy change are not significantly altered. Although we observe a slight decrease of the transition temperature upon the addition of the salt, it is obvious that the endothermic transition is not markedly altered under the experimental conditions of our study. Temperature-Jump Relaxation Kinetics. To study the dynamics of the transition, extensive TJ relaxation
TU: transition temperature as obtained from the maximum of an individual DSC curve (Figure 1 for C16E8/H2O). NS: number of surfactant molecules in a small spheroidal micelle or two end caps (cooperative unit) from fit of the DSC curve according to the micelle aggregation model (see text). ∆H ) NS∆HU: reaction enthalpy per micellar unit (pair of end caps) of micelle aggregation according to the model; the reaction enthalpy ∆HU (per mole of surfactant) is obtained from the area under the DSC curve. τ: T-jump relaxation time at the transition temperature (from Figures 4 and 5). Eτ ) RT2 ∂ ln(τ-1)/∂T: apparent activation energy from TJ kinetics (Figures 4 and 5). kdiss: chain dissociation rate constant, determined from eq 13a with 〈m〉 ) 1.44; 〈m〉 at T ) TU is calculated from eq 7 using the parameters from the fit of the DSC curve (see Figure 1). kass: chain association rate constant. Ediss: activation energy for chain dissociation, calculated from eq 14 with y ) 0.6 (the errors given correspond to y ) 0.6 ( 0.1). Eass: activation energy for chain association. Eτ,TEB: apparent activation energy from TEB kinetics (Figure 9). Eη ) RT2 ∂ ln(η-1)/∂T: activation energy for viscosity (Figures 4 and 5). b Entries are rounded to two significant digits.
Figure 3. Temperature-jump relaxation curves: (a) for 60 mM C16E8/H2O at 306.4 K measured by transmitted light (τ ) 13.4 ms) and (b) for 60 mM C14E8/H2O at 314.4 K measured by scattered light (τ ) 0.276 ms). The curves demonstrate strict monoexponential behavior near the transition temperature. The residuals of the fits are in the insets. Ordinates give the apparent change in transmittance and light scattering, respectively.
experiments have been carried out. Typical curves obtained with both applied TJ apparatus are shown in Figure 3. Temperature increase leads to an increase in scattered light and a decrease in transmitted light. In the transition range, that is, from TU to about (TU + 15 K), both methods
Sphere-to-Rod Transition in Nonionic Micelles
Figure 4. Aggregation kinetics of 60 mM C16E8/H2O (0.1 M KCl). Temperature dependence of TJ relaxation times (large squares, left ordinate), TJ amplitudes (circles, arbitrary units), and viscosity (small squares, right ordinate). Relaxation times: Filled squares are for turbidity, and open squares are for 90°scattering observation. Solid line: fit to Arrhenius law, giving Eτ ) 130 kJ/mol. Relaxation amplitudes (connecting curve drawn by eye): Turbidity is observed at 350 nm, the maximum corresponding to an extinction per degree temperature change of 4.2 × 10-3 K-1. The cloud point is marked at 338 K.
give the same result, which has extensively been checked, especially with C16E8. In particular, with both setups the time course in this transition range is strictly monoexponential. We take this as clear evidence that there is no dependence of the observed dynamic mode on the observation angle of the scattered light. We are thus observing an isotropic chemical process. Temperature Dependence. Figure 4 shows for C16E8 the relaxation times, the amplitudes of the turbidity relaxation signal, and the viscosity as a function of temperature. It is seen that upon approaching the transition temperature, there is a steep increase in the TJ amplitude which reaches a maximum at the calorimetric transition temperature TU and then falls off with increasing temperature. In the temperature range between 302 and 318 K, the relaxation time course is characterized by a single time constant and the logarithm of its reciprocal value scales linearly with temperature resulting in an apparent activation energy of 130 kJ/mol. The amplitude behavior with its characteristic maximum is consistent with the static scattering reported in ref 29, which exhibits a sigmoidal temperature dependence in the transition temperature region. The relaxation amplitude corresponds to the derivative of this curve with respect to temperature. The monoexponential relaxation behavior and the Arrhenius type temperature dependence definitely break down at temperatures much above TU. In addition to a slower turbidity increase, also faster processes become observable. Since the signals are weak in this hightemperature range, a rigorous evaluation is not feasible. The entries in Figure 4 for temperatures greater than 320 K correspond to the slow time constant, obtained from a fit to two exponentials. The changed behavior correlates with the inflection of the viscosity curve as well as with the temperature at which the TJ and TEB relaxation times cross (see Figure 9). There is also a resolvable signal of low amplitude at temperatures below the thermal transition, and its relaxation time seems to level off (see the first points in Figure 4). The associated process probably involves temperature-induced changes in the structure or narrow size distribution of the small spheroidal micelles. Figure 5 shows the corresponding data for the C14E8
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Figure 5. Aggregation kinetics of 60 mM C14E8/H2O (0.1 M KCl). Temperature dependence of TJ relaxation times (large squares, left ordinate), TJ amplitudes (circles, arbitrary units; connecting curve drawn by eye), and viscosity (small squares, right ordinate). Relaxation times: entries are for 90°-scattered light. Solid line: fit to Arrhenius law, giving 110 kJ/mol; light scattering is observed at 478 nm.
Figure 6. Double logarithmic plot of the concentration dependences of reciprocal TJ and TEB relaxation times for C16E8 at the temperatures indicated. The scaling exponents are 0.76 ( 0.02 (TJ 313.5 and 309.2 K), 1.02 ( 0.04 (TJ 305.2 K), and 1.23 ( 0.04 (TEB 313.5 K). Only the curve at 305.2 K, where the concentration of the transition point is marked by an arrow, contains an experimental point below the thermal transition. The stretched exponent β is indicated. Solid lines represent linear fits to the data.
system in the transition range. It is seen that the general behavior is the same as for the C16E8 system; however, the dynamics is strikingly faster. While for C16E8 the relaxation time at the transition temperature τ(T)TU) ) 18 ms, it is reduced by almost 2 orders of magnitude for C14E8, τ(T)TU) ) 0.29 ms, with an apparent activation energy of 110 kJ/mol, somewhat smaller than in the case of C16E8. We have not tried with this system to go to still higher temperatures and to look for deviations from the linear behavior as described above. It is also apparent from Figure 5 that the viscosity does by far not reach values as high as in the C16E8 system. Concentration Dependence. TJ relaxation times as a function of C16E8 surfactant concentration are given in Figure 6. Each point in the figure is an average over 4-6 transients. At 313 K, that is, 8 K above the transition temperature, the sample with the lowest concentration is
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Figure 7. Concentration dependence of reciprocal TJ relaxation times for C14E8 at the temperatures indicated. Solid lines represent linear fits to the data. Scaling exponents: 0.70 ( 0.05 (TJ 315.1 and 321.7 K).
just above its transition temperature while the sample with the highest concentration is at the upper temperature limit of the transition range. It is demonstrated that the log-log plot exhibits strict linear behavior, yielding scaling exponents of n ) 0.76 ( 0.02 at 313 and 309 K, respectively. A corresponding data set at the transition temperature 305 K gives somewhat larger values, the scaling exponent approaching the value of 1. However, the data cover a smaller concentration range here, since aggregation occurs only from 60 mM on. The results for the C14E8 system are shown in Figure 7. It turns out that the scaling exponents are similar to those in C16E8 at larger temperatures, resulting in n ) 0.70 ( 0.06 at 322 K (about 8 K above the transition temperature) and 315 K, close to the transition temperature. Stopped-Flow Kinetics. The TJ experiments do not provide an independent determination of association and dissociation rate constants. It should however be possible to characterize preferentially the dissociation kinetics of rodlike micelles by carrying out a fast dilution experiment at constant temperature leading to mainly spheroidal micelles. This would allow determination of an estimate of the value of the dissociation rate constant kdiss. Such experiments have been performed for rodlike C16E8 micelles by carrying out a 3.5-time dilution of a 30 mM solution with water at 303 K in the stopped-flow instrument. An intensity decrease is observed, characterized by a reciprocal decay time of 24 s-1 on the basis of a single kinetic phase. This kinetic phase is assigned to the dissociation process. Electric Birefringence Dynamics. Although CiEj surfactants are not charged and can adopt symmetric structures with no resulting dipole moment in small micelles, their aggregated forms give rise to a Kerr effect due to the anisotropy of polarizability. The Kerr effect could well be measured for the C16E8/water system. Only small Kerr signals in the microsecond range could however be detected in the C14E8 system. In Figure 8, typical recordings of the Kerr effect for C16E8 are given, showing the transient curves of voltage, current, and birefringence phase shift φ as a function of time. Six recordings with varying voltage are superimposed. The upper figure monitors the applied voltage; the decrease in voltage during the pulse is due to the finite conductivity of the sample. The middle figure shows the current through the
Figure 8. Time-resolved electric field pulse experiments for 60 mM C16E8/H2O at 315.1 K. Summary of six experiments with varying field strength. From top to bottom: Voltage pulse, current through sample, birefringence phase shift φ; τ1 ) 260 µs ((10%), τ2 ) 50 µs; τKWW ) 86 µs, β ) 0.63; B ) 10.8 × 10-12 m/V2.
sample. The curves demonstrate that the current follows the voltage without delay, while the birefringence, shown in the bottom figure, exhibits a relaxation in the microto millisecond range. The birefringence has positive values (n| > n⊥), the Kerr law φ prop. E2 (E electric field strength) being fulfilled. The Kerr constant is given by B ) φ/(2πlE2); light path l ) 0.65 cm. As is seen, there is no current relaxation and the high field conductivity equals the low field value. In the present system, there is thus no evidence for fieldinduced structure changes altering the conductivity of the solution. This is in contrast to Igepal stabilized waterin-oil microemulsions investigated earlier where a strong current relaxation is observed due to field-induced aggregation of water droplets.33 The birefringence shows a clear relaxation in the accessible time range. The field-off decay is definitely not monoexponential, and we have applied several ways of characterizing the dynamics quantitatively. The area under the curve gives a welldefined average relaxation time, τav; analysis of the curves in terms of two relaxation times characterizes a slow (τ1) and a fast (τ2) phase. The analysis in terms of stretched exponentials, exp[-(t/τKWW)β] (Kohlrausch-WilliamsWatts), leading to τKWW and β, characterizes a distribution of relaxation times which is proposed on theoretical grounds. We have evaluated our curves with all three methods. Generally we find τ1 > τav > τKWW > τ2. There is no significant dependence of the relaxation times on the field strength; scattering of the data however is up to (10%. Also the exponent β is independent of the electric field strength.
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activation energy of -135 kJ/mol for C16E8 and -55 kJ/ mol for C14E8. The viscosity for the C16E8 system exhibits a maximum before the cloud point is reached at 338 K. The increase in viscosity with temperature is a clear indication that increasingly larger structures are forming with temperature. Discussion
Figure 9. Temperature dependence of reciprocal TEB relaxation times (left ordinate) and Kerr constants B (right ordinate) for 60 mM C16E8; TEB analysis is (i) in terms of two relaxation times, the slow one τ1 (full circles) being plotted, and (ii) in terms of stretched exponentials, τKWW (open circles). The stretch exponent β is indicated on the top scale. The full squares describing the dependence of the Kerr constant are connected by broken lines. TJ relaxation times (cf. Figure 4) are shown for comparison.
The concentration dependence of the KohlrauschWilliams-Watts decay characteristics is compared with the TJ dynamics at the same temperature in Figure 6. It is seen that over the whole concentration range the birefringence relaxation is much faster than the aggregation kinetics but slows down with increasing concentration. The TEB log-log plot is also linear, leading to a scaling exponent of n ) 1.2. The temperature dependence of τKWW and of the slow relaxation time τ1 from the biexponential fit are compared with the TJ relaxation times in Figure 9. The stretch exponents β become smaller with increasing concentration and temperature. In contrast to the TJ relaxation times, the TEB decay becomes slower both with increasing concentration and temperature. No TEB signal is measurable at the transition temperature where the amplitude of the TJ exhibits a maximum. This might be due to the fact that the rods which start to be formed around this temperature are still too short and do not yet lead to a detectable contribution. With increasing temperature, there is a strong increase in the Kerr constant up to a maximum of B ) 20 × 10-12 m/V2 (Figure 9) and the kinetics show Arrhenius behavior. Both sets of data, τ1 and τKWW, give nearly the same formally negative, apparent activation energy of -200 kJ/mol. No further slowing down is observed, and even a slight increase in rate is seen from about the temperature where the TJand TEB-Arrhenius lines seem to cross. This point also correlates with the inflection point of viscosity (see Figure 4). Viscosity. Viscosity data are shown together with the TJ results in Figures 4 and 5 for the C16E8 and the C14E8 aqueous system. It is seen that there is a direct correlation with the calorimetric transition. Below the transition temperature TU (the maximum in the DSC curve and also the maximum in the TJ amplitudes), the relative viscosity is constant, independent of temperature. The value at 293 K is η/η0 ) 1.18 for both systems at a concentration of 60 mM. From TU on, there is a strong increase in viscosity for C16E8 but only a moderate increase for C14E8. In the range TU to (TU + 17 K), log(η/η0) scales linearly with reciprocal temperature yielding also a negative, apparent
We have presented thermodynamic and kinetic data on the sphere-to-rod transition of the nonionic surfactants C16E8 and C14E8 in aqueous solution. The combined results from thermoanalytics, aggregation kinetics, dynamics of stress relaxation and viscosity allow now a comprehensive characterization of the self-aggregation process. Calorimetry. The calorimetric data show that the transition from spherical to cylindrical structures is an endothermic process, 2.1 kJ/mol of surfactant in the case of C16E8 and a similar value for C14E8. Since rodlike structures are favored with increasing temperature, the positive enthalpy change has to be overcompensated by a positive entropy contribution. On the molecular level, the cause of the transition may lie in a loss of hydration of the polar ethoxy chain, leading to a change of its conformation. The dehydration process requires the breaking of hydrogen bonds, that is, a positive enthalpy; the subsequent release of the water molecules into the bulk water gives the positive entropy change. The loss of hydration would also lead to a reduced effective headgoup area. In the simple picture of packing constraints put forward by Israelachvili,41 a larger packing parameter S ) v/(al) favors rodlike structures (v ) volume, l ) length of hydrophobic chain, a ) headgroup area of surfactant molecule; for spherical micelles, S < 1/3, and for rods, 1/3 < S < 1/2). For a more quantitative treatment, the curvature of the surfactant film and the bending moduli would have to be introduced.42 The transition occurs in a rather narrow temperature range; thus the process must be highly cooperative. A considerable number of surfactant molecules must simultaneously change their conformation. This is consistent with the view that spheroidal surfactant aggregates such as globular micelles or end caps, and not individual surfactant molecules, are involved in the elongation process. If two rodlike micelles combine and fuse, all the surfactant molecules in both end caps are tranformed in a single step. The degree of cooperativity should thus depend essentially on this number of involved surfactant molecules. It has been argued in the preceding section that reasonable estimates of this number can be deduced from the analysis of the thermal transition curve in terms of the micelle aggregation model. From the analysis of the DSC curve and the concentration dependence of TU, we arrive at values of NS ) 160 for both surfactants. Estimation of the reaction enthalpy per mole of micellar units ∆H ) NS∆HU then yields ∆H(C16E8) ) 160 × 2.1 kJ/mol ) 336 kJ/mol and ∆H(C14E8) ) 160 × 1.8 kJ/mol ) 288 kJ/mol. It is this large positive reaction enthalpy which is responsible for the strong temperature dependence of the equilibrium constant and thus for the relatively narrow thermal transition, and it is the fine balance between enthalpy and entropy which determines the transition temperature. The number of surfactant molecules per spheroidal micelle or per two end caps of a rodlike micelle estimated here from the thermal analysis turns out to be quite large. On the basis of the geometrical (41) Israelachvili, J. N. In Aggregation of Amphiphilic Molecules into Micelles; Academic Press: London, 1985; Chapter 16. (42) Helfrich, W. Z. Naturforsch. 1973, C28, 693.
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parameters for the hydrophobic part of a surfactant molecule (all-trans length lmax/nm ) 0.15 + 0.1265nC, volume v/(nm)3 ) 0.0274 + 0.0269nC; nC is the number of C atoms43), such a large number of hydrocarbon chains cannot be accommodated in a spherical core. Even the small values estimated for C12E8 from sedimentation data (NS ) 120)44 or for C14E8 from light scattering (NS ) 119)10 would not fit. There is thus evidence that the small micelles have a nonspherical, possibly oblate, core. For C16E8, Zulauf et al.45 have determined from SANS measurements an aggregation number of 150 ( 15, which is very close to our value. Application of the random micelle aggregation model (R ) 1) leads to an exponential distribution of aggregates with a mean aggregation number 〈m〉 ∼ (KC0)1/2 (cf. eq 8). The temperature dependence of 〈m〉 is determined by the temperature dependence of K. Thus for the concentration dependence at constant temperature we expect a square root dependence. Menge et al.11 have indeed found from detailed light scattering studies a square root dependence of the molar mass as a function of concentration for C12E5/ water and C12E5/water/decane systems. The same conclusion is reached for C12E646 and for ionic systems.19,47,48 There are however also contradictory reports. For the C16E6/water system, an exponent in the range 1.1-1.2 proved necessary.49 Model calculations for R ) 0.1 (C16E8) and 0.3 (C14E8) predict a mean length 〈m〉 ∼ (C0)y with y ) 0.65 and 0.55, respectively. The deviation from the mean field value of y ) 0.5 can also be rationalized by the scaling theory of semidilute chains, leading to y ∼ 0.6.36 Dynamics. It is seen from Figures 6 and 9 that the dynamics resulting from TJ studies employing transmitted light as well as light scattering detection and the dynamics as monitored by electric birefringence show opposite behavior. We have a strong increase in the TJ rate both with temperature and concentration but a comparably strong decrease in the rate of the Kerr effect. This is indeed to be expected. Light scattering is sensitive to the volume of the particles; the TJ kinetics gives the rate of growth. Electric fields impose a stress to the system. In dilute and semidilute systems, rodlike or threadlike molecules will be oriented; an entangled network, for example, will be distorted. In the Kerr effect measurements, under conditions where the field is switched off, we observe the relaxation of the system back to stress-free conditions. It is to be expected that the orientational relaxation is slowed down with increasing persistence length of rodlike/ threadlike molecules. Aggregation Kinetics. The random micelle aggregation model predicts a single time constant in a relaxation experiment when applying a small perturbation. Experimentally, this is fulfilled with high accuracy for both surfactants in the temperature range from TU to about (TU + 15 K). It is seen from Figures 6 and 7 that the expected dependence 〈m〉 ∼ (C0)y with y ) 0.65 and 0.55 is not strictly fulfilled, the scaling exponent giving a value of 0.77 (C16E8) and 0.70 (C14E8) at about 8 K above the (43) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; Wiley: New York, 1980; p 52. (44) Tanford, C.; Nozaki, Y.; Rohde, M. F. J. Phys. Chem. 1977, 81, 1555. (45) Zulauf, M.; Weckstrom, K.; Hayter, J. B.; Degiorgio, V.; Corti, M. J. Phys. Chem. 1985, 89, 3411. (46) Kato, T.; Kanada, M.; Seimiya, T. Langmuir 1995, 11, 1867. (47) vonBerlepsch, H.; Dautzenberg, H.; Rother, G.; Jager, J. Langmuir 1996, 12, 3613. (48) Buhler, E.; Munch, J. P.; Candau, S. J. J. Phys. II 1995, 5, 765. (49) Schurtenberger, P.; Cavaco, C.; Tiberg, F.; Regev, O. Langmuir 1996, 12, 2894.
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transition temperature. At TU, the exponent does not change for C14E8 but increases to a value of about 1.0 for C16E8. We thus find for both systems clear deviations from the expected mean field value. Nevertheless, we consider eq 13a as a reasonable approximation for estimating the dissociation rate constant (see Table 1). The only example where aggregation kinetics has been investigated in the past is the C12E6/H2O system.9 There, a square root dependence seems to be fulfilled. Equation 13b predicts that the apparent activation energy of the reciprocal relaxation time is given by
Eτ ) Ediss + y∆H
(14)
with y ) 0.5. Incorporating the results for the case R
