Self-Diffusion of Constituent Cationic Surfactants in Threadlike

The self-diffusion coefficient (Dself) for the cetyltrimethylammonium (CTA+) cation in the micelle was calculated on the basis of a phantom network mo...
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Langmuir 1998, 14, 2020-2026

Self-Diffusion of Constituent Cationic Surfactants in Threadlike Micelles Toshiyuki Shikata,* Shin-ichiro Imai, and Yotaro Morishima Department of Macromolecular Science, Osaka University, Toyonaka, Osaka 560, Japan Received May 21, 1997. In Final Form: January 26, 1998 Self-diffusion mechanisms of a surfactant molecule in entangled threadlike micelles formed from a mixture of cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal) in aqueous solution (CTAB:NaSal/W) are discussed. The self-diffusion coefficient (Dself) for the cetyltrimethylammonium (CTA+) cation in the micelle was calculated on the basis of a phantom network model. Two formulations for the self-diffusion coefficient are derived for two separate regimes with respect to time and space. When the relaxation time (τm) for the micellar system is short, Dself is represented as 〈ξ2〉{6τmP(Le,τm)}, where 〈ξ2〉 is the mean square of the entanglement spacing which depends on the CTA+ concentration and P(Le,τm) is a probability for the CTA+ cation to remain in the initial part of the threadlike micelle with the average contour length of Le between two entanglements. On the other hand, when τm is long, Dself is expressed as (2Q/3)(Dlat/2τm)1/2, where Q and Dlat are the persistence length of the threadlike micelle and the lateral diffusion coefficient of the CTA+ cation along the micellar contour, respectively. Dself values and activation energies determined experimentally by use of a forced Rayleigh scattering technique agree with those calculated based on the model.

Some kinds of surfactant molecules form long and stable rodlike or threadlike micelles in aqueous solution, the micelles resembling polymer chains in dynamic behavior. For example, threadlike micelles form concentrated entanglement networks and exhibit profound viscoelastic behavior in a manner similar to a concentrated polymer system.1-5 However, in the longest relaxation process, threadlike micellar systems show a unique Maxwell type relaxation with a single relaxation time,3-5 which is completely different from polymer systems with broad relaxation spectra.6 These similarities and differences between the threadlike micelles and polymer systems are attributable to structural features of the micelle constructed by intermolecular interactions including hydrophobic bonding. We can classify the dynamics of the threadlike micelle into three regimes depending on the time and dimension for the motions of the micelle. The smallest and fastest regime, on the order of 1 nm and of 1-100 ns, is related to molecular motions of surfactants and additives in the micelle, which has been investigated by use of fluorescence probe techniques7,8 and nuclear magnetic resonance (NMR) spectroscopy.9 In this regime, diffusions of the micelle-forming species within micellar interior are important. Consequently, the diffusion of surfactant molecules corresponds to their lateral diffusion along the micellar surface because the headgroups of the surfactants are located on the micellar surface. The second regime, on the order of 10 nm to 1 µm and of 1 µs to 100 ms, is related to bending motions of the part

of the threadlike micelle between two entanglement points. This regime can be examined with high-frequency viscoelastic measurements10 and also electric birefringence measurements.11-13 It is well-known that the origin for the elasticity in the threadlike micellar system is a conformational entropy loss due to deformation of a coillike conformation.14,15 Thus, dynamics and statistics of the threadlike micelle in this regime can be described by the Gaussian statistics with good accuracy as in the case of polymer chain systems.15 The third regime is governed by the dynamics of the largest scale and slowest motion in the longest relaxation process with a single relaxation time as mentioned above. In this regime, both macroscopic flow behavior and the self-diffusion of the micelle-forming species are obviously controlled by the longest relaxation process with a single relaxation time. Because the self-diffusion of the micelleforming species is directly related to the dynamics in the first regime, these two, the first and the third regimes, are strongly interrelated. In the present study, we will propose a new theoretical model to predict the self-diffusion coefficient (Dself) of a surfactant molecule in the threadlike micelle based on our previous model3 for the longest relaxation process. Applicability of our present model will be discussed by comparing the prediction from the model with experimental data obtained by a forced Rayleigh scattering technique (FRS) for a typical aqueous threadlike micellar system consisting of cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal), CTAB:NaSal/ W.16,17

(1) Gravsholt, S. J. Colloid Interface Sci. 1976, 57, 575. (2) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1987, 3, 1081. (3) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1988, 4, 354. (4) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1989, 5, 398. (5) Shikata, T.; Hirata, H.; Takatori, E.; Osaki, K. Non-Newtonian Fluid Mech. 1988, 28, 171. (6) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; John Wiley & Sons: New York, 1980. (7) Shikata, T.; Morishima, Y. Langmuir 1996, 12, 5307. (8) Shikata, T.; Imai, S.; Morishima, Y. Langmuir 1997, 13, 5229. (9) Shikata, T.; Morishima, Y. Langmuir 1997, 13, 1931.

(10) Shikata, T.; Morishima, Y. In preparation. (11) Shikata, T.; Morishima, Y. In preparation. (12) Hoffmann, H.; Kra¨mer, U.; Thurn, H. J. Phys. Chem. 1990, 94, 2027. (13) Kra¨mer, U.; Hoffmann, H. Macromolecules 1991, 24, 256. (14) Shikata, T.; Dahman, S. J.; Pearson, D. Langmuir 1994, 10, 3470. (15) Shikata, T.; Pearson, D. Langmuir 1994, 10, 4027. (16) Nemoto, N.; Yamanura, T.; Osaki, K.; Shikata, T. Langmuir 1991, 7, 2607. (17) Nemoto, N.; Kuwahara, M. Colloid Polym. Sci. 1994, 272, 846.

Introduction

S0743-7463(97)00524-6 CCC: $15.00 © 1998 American Chemical Society Published on Web 03/26/1998

Self-Diffusion of Surfactants

Langmuir, Vol. 14, No. 8, 1998 2021

The macroscopic self-diffusion of the surfactant molecule in the threadlike micelle results from its lateral diffusion along the micellar surface or contour, because the surfactants scarcely escape from the threadlike micelle once they are incorporated in the micelle. Thus, the activation energy for the self-diffusion and that for the lateral diffusion of the surfactant molecule should be closely related to each other. On the other hand, the activation energy in the longest relaxation process is usually different from those for the lateral and self-diffusional processes, because the longest relaxation process is, in general, independent of the lateral diffusional and the selfdiffusional processes. However, in a special case where the self-diffusional process is due to the longest relaxation mechanism, activation energies for both the longest relaxation process and the self-diffusional process are identical to each other, as will be described in the next section. Theory Phantom Network Model. To interpret the unique longest relaxation process having a single relaxation time in the threadlike micellar system, especially in the CTAB: NaSal/W system, we proposed a models3,4 in which some very essential characters of the threadlike micelle were taken into consideration as follows. We called this model the phantom network model. In this model, we assumed that micelles could fuse at an entanglement point to form a temporary cross-link, which was followed by passing through each other, like a phantom, with a time constant equal to the rheological longest relaxation time (τm), as schematically shown in Figure 1a. Because the threadlike micelle is formed only via intermolecular hydrophobic interaction between the alkyl tails of the CTA+ cation, the formation of the temporary cross-link, followed by the reformation of two separate micelles after crossing each other, should be possible. This is an essential character of the threadlike micelle. According to the phantom network model, the selfdiffusion constant (Dself) of individual surfactant molecules within the threadlike micelle can be formulated under two distinctive conditions as follows (see Figure 1b). (i) The time scale for the lateral diffusional motion of the CTA+ cation is much longer than that of the phantom crossing of the micelle. In this case, the CTA+ cation is not able to escape by the lateral diffusion from a threadlike micelle part between two entanglements, where the CTA+ cation existed initially at t ) 0, within time shorter than the lifetime of the entanglement which is identical to τm: The dot in Figure 1b, t ) 0, schematically represents the location of a marked surfactant, and the dot in Figure 1b, t ) τm, mechanism i, represents the location of the marked surfactant at t ) τm. (ii) The lateral diffusional motion of the CTA+ cation is so fast that within time τm, the CTA+ cation can migrate into another micelle part different from the micelle part where the CTA+ cation existed at t ) 0: The dot in Figure 1b, t ) τm, mechanism ii, represents the location of the marked surfactant at t ) τm. Regime of Slow Lateral Diffusion. In condition i, τm is a step time for a random walk process of the CTA+ cation in the threadlike micellar system, which approaches a diffusional process at a large number of the steps. On the other hand, a step distance for the random process should be identical to the average entanglement spacing (ξ), as schematically shown in Figure 1b, because a micelle part between two entanglement points, which includes a marked CTA+ cation, can migrate by a distance of ξ at every step to produce a new entanglement in a different

Figure 1. (a) Schematic representation of a phantom crossing process for threadlike micelles. (b) Schematic representation of difference in contribution of the lateral diffusional process in conditions i and ii. Dots in threadlike micelles (solid lines) represent location of a marked surfactant molecule. An entanglement point (*) disappears at t ) τm by the phantom crossing (type II in this case). (c) Schematic diagrams representing existing probability (φ(L,τm)) of a marked surfactant molecule in the threadlike micelle as a function of a micellar contour (L). It was assumed that the surfactant existed at the origin at t ) 0. The ratio of the area of hatched portion to the whole one in the probability function means a fraction of existing probability (P(Le,τm)) of the surfactant at t ) τm in the original entangled micelle portion between two entanglement points.

place. Then, one can evaluate the self-diffusion coefficient (Dselfi) for condition i from eq 1 assuming three-dimensional diffusion of the micelle portion between entanglement.

Dselfi ) 〈ξ2〉/6τm

(1)

Here, 〈ξ2〉 is the mean square of entanglement spacings dependent on the concentration of the surfactant (CD). In condition i, the essential mechanism for the selfdiffusional process is the phantom crossing. Thus, the activation energy (Ei*) in condition i is close to the activation energy for the phantom crossing in the longest relaxation process, because ξ is weakly temperature dependent. Regime of Fast Lateral Diffusion. In condition ii, we consider contribution of the lateral diffusion of individual CTA+ cations. We assume that the lateral diffusion of the CTA+ cation is so fast that the CTA+ cation

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can migrate to another entangling micellar part within a time τ, which is identical to τm in the simplest case. The CTA+ cation can choose either of crossing directions, the type I and the type II phantom crossings, at the entanglement point, as depicted schematically in Figure 1a. If the choice of the crossing direction between the type I and the type II has the same probability at the time τ, the direction is random for every crossing and the time τ may be a step time of the random walk. Considering a very long period of time t ()nτ, n . 1), one can estimate the self-diffusion coefficient as follows. The mean square distance (〈R(t)2〉) that the CTA+ cation can migrate spatially within t may be given by eq 2, because the diffusive mode of the CTA+ cation must be a random process. Here, 〈R(τ)2〉 is the mean square distance at each step. Because the Gaussian chain statistics15 hold for the conformation of the threadlike micelle, 〈R(τ)2〉 is described with the persistence length (Q) and the average contour length 〈L(τ)〉 of the threadlike micelle along which the CTA+ cation can migrate within τ by the lateral diffusion (eq 3). Moreover, eq 4 connects 〈L(τ)2〉 and τ with Dlat because the diffusion of the CTA+ cation can be approximately regarded as a one-dimensional process along a contour of the micelle. Finally, we obtain Dselfii as eq 5 for condition ii with Q, Dlat, and τ. Here, we approximate 〈L(τ)2〉 ≈ 〈L(τ)〉2.

〈R(t)2〉 ) n〈R(τ)2〉

(n . 1)

(2)

〈R(τ)2〉 ) 2Q〈L(τ)〉

(3)

〈L(τ)2〉 ) 2Dlatτ

(4)

Dselfii )

( )

〈R(t)2〉 2Q Dlat ≈ 6t 3 2τ

1/2

(5)

τ must be equal to or longer than τm. Each CTA+ cation encounters the phantom crossing of entanglement points every τm in the average, wherever it exists in the long micelle, because every entanglement has the lifetime of τm. When the type I and the type II crossings occur randomly at every crossing, the simplest case, τ must be identical to τm. Because Q and Dlat should be independent of CD and CS*, eq 5 predicts the self-diffusion coefficient independent of both CD and CS* in the simplest case. The activation energy for the self-diffusional process (Eii*) in the case of condition ii is related to the activation energies of the lateral diffusional process (EL*), the longest relaxation process (Eτm* is identical to Ei*) and Q. In a special case in which τ is not identical to τm, estimation of Dself is much more complicated. If one of type I or type II has higher probability at the crossing process for some reason, time τ for the CTA+ cation to reach random choice at the crossing process is longer than τm. However, at the present time, we do not know how to estimate τ for this case. Hybrid Regime. Here, we consider the simplest case of condition ii. One can estimate a probability (P(Le,τm)) for a CTA+ cation to remain in the initial part of a micelle with the average contour length of Le between two entanglement points by use of eq 6 assuming that the CTA+ cation exists at the origin (x ) 0) at t ) 0 and that its probability (φ(L,t) dL) of existence between L and L + dL at t obeys a simple one-dimensional diffusional equation with the diffusion constant of Dlat along the threadlike micellar contour, as schematically represented in Figure 1c.

P(Le,τm) ) )

∫-LL /2/2 φ(L,τm) dL e

e

2 (4πDlat τm)

(

2

)

∫0L /2 exp - 4Dxlat τm e

1/2

dx

(6)

Because CTA+ cations with the concentration of P(Le,τm)CD remain in the initial micelle part at t ) τm, there is a possibility for them to migrate according to condition i. When we consider the self-diffusion coefficient of the CTA+ cation exactly by taking account of diffusing distance of one CTA+ cation, each of a cation performs a sequence of steps of conditions i or ii with probability P(Le,τm) or 1 P(Le,τm), respectively. Ultimately, the CTA+ shows a single diffusion at long times and possesses the average (true) self-diffusion coefficient of Dself ) DselfiP(Le,τm) + Dselfii(1 - P(Le,τm)). However, conventional experimental techniques such as FRS16,17 and PFG-NMR18 to determine the self-diffusion coefficient record concentration decay of molecules marked in some methods as a function of time at the same position. Because these ordinary techniques do not chase diffusional motions of the same molecule, the true self-diffusion coefficient cannot be determined in some conditions like this hybrid regime. If there are two kinds of diffusion processes with distinctively different self-diffusion coefficients for marked molecules as conditions i and ii, it would be possible that the concentration of the marked molecules shows two kinds decays dependent on both the diffusion processes. Since the amount of marked CTA+ cations always remains in the initial micellar part with probability P(Le,τm) and performs the diffusion owing to condition i, the concentration decay ascribed to the condition i can be detected separately. When P(Le,τm) is low, the contribution of condition i is not important. However, when P(Le,τm) is not low owing to a small Dlat value, two distinctive concentration decay modes ascribed to both conditions i and ii will be detectable. An equation of the concentration change of (marked) CTA+ cations for condition i is given as eq 7 with the time and position dependent concentration (CD(x,y,z,t)) of the marked CTA+ cation

∂CD(x,y,z,t) ) ∂t ∂2CD(x,y,z,t) ∂2CD(x,y,z,t) ∂2CD(x,y,z,t) + + ∂x2 ∂y2 ∂z2

P(Le,τm)

{

Dselfi

}

(7)

where we consider a much longer time scale than the step time τm. CD(x,y,z,t) is common to both conditions i and ii. Then, a new apparent self-diffusion coefficient (DselfiP(Le,τm)-1) can be obtained for condition i under the hybrid regime. The other apparent self-diffusion coefficient (Dselfii{1 - P(Le,τm)}-1) can be also obtained for condition ii. When the P(Le,τm) value is small, the DselfiP(Le,τm)-1 value is much larger than the Dselfi value, whereas an amplitude of concentration decays for condition i is small. Then, it becomes difficult to detect condition i. Most of experimental techniques to measure the selfdiffusion coefficients such as FRS and PFG-NMR have appropriate time regimes within which the obtained Dself values are reliable. If a true or apparent Dself value in question is out of the time regime for a technique employed, the Dself value will not be detected accurately. (18) Singer, J. R. J. Phys. E: Instrum. 1978, 11, 281.

Self-Diffusion of Surfactants

Kato et al.19 also consider the self-diffusion coefficient of a micelle-forming surfactant in aqueous threadlike micellar system formed with nonionic surfactants, polyethylene glycol n-alkyl ether (for example, C16E7; 16 and 7 are the numbers of the carbon atoms in the alkyl chain and ethylene oxide units, respectively). They assumed that the surfactant molecule resided in the threadlike micelle for a while and stepped out from the original position in the micelle and quickly reentered into a different position in the micelle. Consequently, they obtained an expression for Dself similar to eq 1. According to their model, τm and ξ are interpreted as the residence time of the surfactant in the micelle and the average spacing between centers of gyration of formed threadlike micelles, respectively. Cates and co-workers20 developed a living polymer model to understand dynamic features of threadlike micellar systems, and they proposed a theoretical Dself formulation. Their models are likely to hold only in a particular condition of aqueous CTAB systems with added simple salts such as sodium bromide21 and are not applicable to the CTAB:NaSal/W system and to the nonionic threadlike micellar systems. Another living polymer model proposed by Turner22 considering the bond interchange is quite similar to the phantom crossing model. Turner’s model cannot say anything about dependence of τm and Dself on CS, but τm and Dself in the model have only CD dependence in manners of τm ∝ CD1.7 and Dself ∝ CD-1.4 for the reptative regime. However, it is well-known that in the CTAB: NaSal/W system τm is independent of CD and is strongly dependent on CS*.3 Thus, Turner’s model is not applicable to the CTAB:NaSal/W system due to basic characters of τ m. Experimental Section Materials. CTAB was purchased from Wako Pure Chemical Industries, Ltd. (Osaka, Japan), and was purified by recrystallization from a mixture of ethanol and acetone. Extra pure grade NaSal was also purchased from the same company and used without further purification. Water was deionized with a Milli Q SP system (Millipore) and had a specific resistance higher than 14 MΩ cm-1. A fluorescence probe, cetylacridinium orange bromide (CAOB), was synthesized according to the method by Miethke and Zanker.8,23 Acridine orange hydrochloride was purchased from the same company, and HCl was removed by neutralization with equimolar aqueous sodium hydroxide. Acridine orange was treated with cetyl bromide in toluene under reflux for several hours. The resulting CAOB was recrystallized in methanol. CTAB concentration (CD) was kept at 10.0 mM, while NaSal concentrations (CS) were 20, 200, and 410 mM. The concentration of CAOB was kept constant at lower than 0.1 µM. Glycerine was used as a solvent to measure the instantaneous fluorescence anisotropy (r0) of a CAO+ cation, the fluorescence anisotropy at time zero, because it is slightly soluble in glycerine, which has a high viscosity and is glassy below -10 °C. The concentrations of CAOB in glycerine for the r0 determination were lower than 0.1 µM. Special care was not taken to remove oxygen, which can be a fluorescence quenching species in some cases, because the fluorescence lifetime of CAO+ was short enough ( 500 nm light to pass. The same quartz cell was employed for the fluorescence lifetime measurements, and temperature was varied from 25 to 50 °C.

Results Self-Diffusion Coefficient: In Figure 2, the Dself data for the CTAB:NaSal/W system with CD ) 10 mM at 25 and 40 °C and with CD ) 100 mM at 25 °C are quoted from refs 16 and 17, which were evaluated by FRS. The abscissa of this figure is the concentration of free Sal- anions (CS*) which can be evaluated from CS* ) CS - CD, because the threadlike micelle is constructed by a 1:1 complex between CTA+ and Sal-.3-5 It is known that the τm value is a function of only CS* but not of CD in the CTAB:NaSal/W system.5 In the high CS* region possessing short τm values, the Dself values at 25 °C for solutions with CD ) 10 and 100

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mM do not agree with each other. However, the Dself values at 25 °C with τm longer than a few seconds in the low CS* region become independent of CD. According to the phantom crossing model in condition i, Dselfi can be calculated from eq 1, and it depends on CD, because the ξ value is a function of CD in a manner14 〈ξ2〉1/2 ≈ 〈ξ〉 ∝ CD-0.7. Then, the relationship is obtained as

Dselfi ∝ CD-1.4τm-1

(8)

From rheological data, one can estimate the ξ values by use of the simple relationship24 ξ ) (kBT/GN0)1/3, where GN0, kB, and T are the plateau modulus, Boltzmann’s constant, and the absolute temperature, respectively. The Dselfi values predicted from eq 1 (or eq 8) based on condition i with the GN0 data2,3 were much smaller than the obtained Dself values even in high CS* range. Therefore, we do not conclude that condition i purely holds in the CTAB: NaSal/W system. However, the CS* dependence of the Dself values for both the solutions with the different CD shows qualitatively the same tendency as that predicted by the model in condition i. On the other hand, the agreement in the Dself values for both the solutions with different CD in the low CS* region suggests that contribution of the lateral diffusional process of the CTA+ cation along the micellar surface and/or contour to the Dself becomes more important than that of the phantom crossing process, as we discussed above with the model in condition ii. Dotted and dot-dash lines in Figure 2 represent the Dselfii values predicted from eq 5 based on condition ii in the simplest case (τ ) τm). For a dotted line, we used values of Q ) 26 nm and Dlat ) 10-5 cm2 s-1, which were previously estimated at 25 °C.15 As will be discussed later, the Dlat value is weakly temperature dependent. The Q value of the threadlike micelle should be directly related to the lateral diffusion of the micelleforming species, therefore, the Q value may also be weakly temperature dependent. The Dself values at 25 °C show good agreement to the predicted Dselfii values only if CS* < 150 mM. However, in the entire CS* range examined the Dself values at 40 °C are in fair agreement with the Dselfii values predicted from eq 5 with values of Q ) 17 nm and Dlat ) 1.5 × 10-5 cm2 s-1 estimated at 40 °C (a dotdash line in Figure 2). In the intermediate to high CS* region at 25 °C, the Dself values obtained by FRS are between the Dselfi and Dselfii values predicted by conditions i and ii. Then, we can estimate the P(Le,τm) value and draw DselfiP(Le,τm)-1 curves for solutions with CD ) 10 and 100 mM at 25 °C as functions of CS* in Figure 2 with solid and broken lines, respectively. Both the DselfiP(Le,τm)-1 values for CD ) 10 and 100 mM fairly agree to the experimentally obtained Dself values in the high CS* range. It is likely that the model for the hybrid regime holds in the intermediate to high CS* region at 25 °C, and the Dself data obtained by FRS are the apparent self-diffusion coefficient. The FRS technique is a useful method to detect very slow diffusion processes but not so sensitive in a short time range in the case of this threadlike micellar system because of a time lag to make clear optical fringes.16,17 If the system consists of two kinds of diffusional processes with markedly different self-diffusion coefficients according to the model for the (24) The basic form to obtain the equation is GN0 ) ckBT/Me, where c and Me are weight concentration of polymer (or surfactant) species and the average molecular weight between two entanglements, respectively. This is well described in ref 6. Because one can suppose the relationship of c ) Me/ξ3, the equation in the text is obtained. The equation was used by Granek and Cates (J. Chem. Phys. 1992, 96, 4758) and also by Shikata et al. in ref 14.

hybrid regime, the FRS technique will accurately provide only a slower apparent Dself value. Therefore, the FRS technique is likely to fail to detect another faster diffusion process. The self-diffusion coefficient, which is attributed to condition i in the solution with CD ) 10 mM and CS* ) 400 mM at 40 °C, is estimated to be Dselfi ≈ 1 × 10-8 cm2 s-1 and is not so different from that attributed to condition ii; Dselfii ) 3 × 10-8 cm2 s-1, DselfiP(Le,τm)-1 ) 2.9 × 10-8 cm2 s-1, Dselfii{1 - P(Le,τm)}-1 ) 4.6 × 10-8 cm2 s-1. These may imply that the obtained Dself value is close to the DselfiP(Le,τm)-1 value. We recently measured Dself values for the CTA+ cation in the same system by use of a pulsed field gradient nuclear magnetic resonance (PFG-NMR) technique at 25 °C in a preliminary experiment.25 Because the spin-spin relaxation times of all the protons in the CTA+ cation are short, the Dself values cannot accurately be determined. We obtained the Dself values of 7 and 8 × 10-8 cm2 s-1 for both the solutions with CD ) 10 and 100 mM, respectively at CS* ) 400 mM. Although these Dself values are not accurate, however, the fact that the values are weakly dependent on CD and are not far from the Dselfii values (the dotted line) predicted from eq 5 based on condition ii or Dselfii{1 - P(Le,τm)}-1, as shown in Figure 2, suggests that contribution of condition ii is detected by the PFG-NMR technique in the form of the faster apparent diffusion coefficient. Lateral Diffusion Coefficient. Lateral diffusion coefficient of the CAT+ cation in the threadlike micelle can be estimated by employing a fluorescence probe, cetylacridinium orange bromide (CAOB), incorporated into the micelle in substitution for the CAT+ cation.8 As we reported in our previous paper,8 the CAO+ cation in the threadlike micelle of the CTAB:NaSal/W system showed relatively high fluorescence anisotropies which were weakly dependent on CS*. Because the transition moment of the CAO+ cation is located in the direction from the 9-carbon to the 10-nitrogen in the acridinium headgroup and the axis of fluorescence emission is essentially parallel to the transition moment, the fluorescence anisotropy (r) directly reflects migrational motions of the CAO+ cation along the surface and/or contour of the threadlike micelle. A circular migrational motion of the CAO+ cation along the cross section of the threadlike micelle should be most effective to relax the r value. From these considerations, Dlat can be roughly estimated to be Dlat ≈ a2/τφ, where a and τφ represent the radius of the threadlike micelle and the rotational relaxation time of the fluorescence anisotropy, respectively. The τφ values can be evaluated by the Perrin equation26 (eq 9) with the instantaneous fluorescence anisotropy (r0 ) 0.37)8 and the fluorescence lifetime (τlife) of the CAO+ cation in the micelle.

(

)

3τlife 1 1 ) 1+ r r0 τφ

(9)

Temperature dependencies of r, τlife, and τφ values for the CAO+ cation in the CTAB:NaSal/W system at CD ) 10 mM and CS* ) 190 and 400 mM are shown in Figure 3. It is clear that the τφ-1 value is essentially independent of CS* and is weakly temperature dependent, the activation energy Eφ* being 32 kJ mol-1. Thus, the Dlat values can be estimated to be 1 × 10-5 cm2 s-1 at 25 °C by use of an approximate equation of Dlat ≈ a2τφ with a ) 2.5 nm.15 The Dlat value has the same activation energy (EL*) as the Eφ* (25) Nose, T. Private communication. (26) Perrin, F. J. Phys. Radium 1926, 7, 39.

Self-Diffusion of Surfactants

Figure 3. Dependence of the lateral diffusion constant (Dlat ≈ a2τφ-1) on the reciprocal of temperature for the CTAB:NaSal/W systems with CD of 10 mM and with different CS* of 190 and 400 mM.

value. Since the persistence length (Q) of the threadlike micelle is governed by fluidity of an assembly of micelleforming species such as the CTA+ cation, the weakly temperature dependent Dlat value suggests that the Q-1 value is also weakly temperature dependent, the activation energy EQ* ≈ Eφ* ()32 kJ mol-1). The Q value of 17 nm at 40 °C used to evaluate the Dself in Figure 2 is estimated with the activation energy of EQ* ) 32 kJ mol-1. Discussion Activation Energy of Dself. Temperature dependence of the Dself value for solutions17 with CD ) 10 mM and with CS* ) 10, 190, and 400 mM are plotted in Figure 4. The solution at CS* ) 10 mM only shows an Arrhenius type temperature dependence with a constant activation energy. However, the other solutions show complicated temperature dependencies as shown in Figure 4. These plots may suggest two distinctive activation energies with apparent breaks around T-1 ) 3.25 × 10-3 K-1. The lower activation energy in the high-temperature range for the solutions at CS* ) 190 and 400 mM is essentially identical to that for the solution at CS* ) 10 mM. Because the Dself value of the solution with CS* ) 10 mM at 25 and 40 °C is well described by the phantom crossing model in condition ii, as shown in Figure 2, the resulting activation energy of Dself for the solution, ca. 45 kJ mol-1, implies the Eii* value. Then, the activation energy of Dself in the higher temperature range (T-1 < 3.25 × 10-3 K-1) for the solutions at CS* ) 190 and 400 mM should also be the Eii* value. Since the contribution of the phantom crossing in condition i in the solution with CS* ) 190 and 400 mM at 25 °C cannot be neglected, as we discussed in the previous section, the higher activation energies in the low-temperature range in Figure 4 may be related to the Ei* value for condition i. Open symbols in Figure 4 represent the

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Figure 4. Dependence of the self-diffusion constant (Dself) on the reciprocal of temperature for the CTAB:NaSal/W systems with CD of 10 mM and with the several CS* values.17 Open symbols represent the DselfP(Le,τm) ()Dselfi) values which will express pure contribution of condition i for the solution with CD ) 10 mM and CS* ) 190 mM.

DselfP(Le,τm) ()Dselfi) values which will express pure contribution of condition i for the solution with CD ) 10 mM and CS* ) 190 mM. The slope of the DselfP(Le,τm) values does not look so different from that of the Dself values. Activation Energy of τm-1. Temperature dependence of the τm value for the CTAB:NaSal/W system17 with CD ) 10 and 100 mM and with CS* ) 190 mM are plotted in Figure 5. It is likely that all the τm-1 values are temperature dependent in the Arrhenius type and their activation energies are almost independent of both CD. The activation energy (Eτm *) estimated from this figure is ca. 140 kJ mol-1 and is essentially identical to that estimated for the DselfP(Le,τm) ()Dselfi) value at CS* ) 190 mM in Figure 4. Because the Ei* value should be identical to the Eτm * value on the basis of the phantom network model, the self-diffusional process in the high CS* range at 25 °C in Figure 4 is governed by the phantom crossing in condition i as discussed above. As we considered in the previous section, the Q and Dlat values are weakly temperature dependent with low activation energies (EQ* ≈ EL* ) Eφ* ) 32 kJ mol-1). Thus, the entire activation energy for the phantom crossing in condition ii can be approximately expressed as Eii* ≈ (Eτm* + Eφ*)/2 - EQ* (≈54 kJ mol-1) based on eq 5, supposing τm-1, Dself, and Q have the Arrhenius type temperature dependence with activation energies; τm-1 ∝ exp(-Eτm*/ kBT), Dself ∝ exp(-Eφ*/kBT), and Q ∝ exp(-EQ*/kBT). This value of 54 kJ mol-1 is fairly close to the Eii* value ()48 kJ mol-1) obtained in Figure 4 for the low CS* range at high temperature. Equation 5 is simple but essential to elucidate the self-diffusion behavior of the micelle-forming CTA+ cation in the threadlike micellar system. Ultimately, from considerations based on the activation energies, the self-diffusion process observed in the low

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corresponding to our EL* (or Eφ*) and Ei*, are 30 and 160 kJ mol-1, respectively. It is interesting to note that these values fairly agree with those obtained in this study although species of surfactants used in their studies are nonionic and of some specific characters different from our cationic surfactant.

Figure 5. Dependence of the reciprocal of the longest relaxation time (τm-1) on the reciprocal of temperature for the CTAB: NaSal/W systems with CD of 10 and 100 mM and with CS* ) 190 mM.17

CS* side is essentially governed by the phantom crossing condition ii, while that in the high CS* side is governed by condition i. Kato et al.19 also estimated the activation energy for the self-diffusion process of the nonionic surfactant in the threadlike micellar system. The values they obtained,

Conclusions Self-diffusion coefficient (Dself) of the CTA+ cation in the threadlike micelle formed in the CTAB:NaSal/W system was evaluated theoretically on the basis of the phantom network model. The self-diffusional processes are classified into two regimes. When the system has high CS* and the short relaxation time, τm, Dself is basically represented as 〈ξ2〉/6τm, where 〈ξ2〉 is the CD dependent square of the entanglement spacing. However, the probability, P(Le,τm), for the CAT+ cation to remain in the initial threadlike micellar part between two entanglements after the phantom crossing with τm is necessary for the precise expression of the Dself data obtained with FRS in this high CS* regime; Dself ) 〈ξ2〉/{6τmP(Le,τm)}. On the other hand, when the system possesses low CS* and long τm, Dself is expressed as (2Q/3)(Dlat/2τm)1/2, where Q and Dlat are the persistence length of the threadlike micelle and the lateral diffusion coefficient of the CTA+ cation along the micellar contour, respectively. Moreover, predicted activation energies for both the conditions based on the model well agree with the values experimentally determined. Acknowledgment. The authors thank Professor T. Nose of Tokyo Institute of Technology for PFG-NMR measurements with a JEOL GSX-270 spectrometer equipped with a magnetic field gradient attachment. We also thank him for helpful discussions. This work was partially supported by Grant-in-Aid (08651079) from Ministry of Education, Science, Culture, and Sports, Japan. LA970524L