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Determination of the Translational and Rotational Diffusion Coefficients of Rodlike Particles Using Depolarized Dynamic Light Scattering Dieter Lehner, Helmut Lindner, and Otto Glatter* Institute of Chemistry, University of Graz, Heinrichstrasse 28, A-8010 Graz, Austria Received July 29, 1999. In Final Form: October 18, 1999 Depolarized dynamic light scattering (DDLS) experiments are reported on three different systems: on tobacco mosaic virus as a well investigated sample; on PEO-PPO-PEO triblock copolymer micelles, which show sphere-to-rod transition with increasing temperature; and on a microemulsion of the type water/ octane/CiEj. DDLS measurements were performed using highly discriminating polarizers and single-mode fiber detection at different scattering angles, obtaining decay rates ΓVH versus scattering angle. The rotational and translational diffusion coefficients available from these plots were taken to evaluate size parameters of the systems using Broersma’s expressions for a stiff rod. Good agreement with theory and literature was found in the case of tobacco mosaic virus and at concentrations below the overlap concentration c* for the other two systems.
1. Introduction Dynamic light scattering (DLS) has been extensively applied to dynamic studies of macromolecules in solution. Most of these studies measure time correlation functions of polarized scattered light. However, depolarized scattered light can provide dynamic and structural information that is often not obtainable by other techniques.1,2 For particles that are small compared to the wave vector, depolarized dynamic light scattering (DDLS) provides the rotational and the translational diffusion coefficient. Furthermore, these parameters can be used together with hydrodynamic theories to access information about the size and shape of the particles in solution.1-3 Despite these advantages, DDLS has not become a very common technique. The major reasons for this low acceptance are the weakness of the depolarized signal compared to the polarized signal, which leads to obscurity by interfering signals due to optical imperfection in lenses; polarizer leakage or even multiple scattering especially in the forward scattering direction; and the need for a goniometer and for high power lasers. DDLS experiments can have very short decay times, and the first channels may be biased by afterpulsing of the photomultiplier. Depending on the time scale of the motion of the particles in solution, two different techniques are used; for small particles, interferometric studies using Fabry-Perot interferometers are favorable. The application range of this technique includes, typically, time scales of 50 ns and faster, which is usually valid for molecules with a molecular weight of less than 50 000 Dalton.2 One of the first interferometric studies was performed with lysozyme.4 Studies on several biological systems such as DNA and other biopolymers can be found in the literature5-7 as well * To whom correspondence should be addressed. Fax: (+43) 316 380 9850. E-mail:
[email protected]. (1) Berne, B. J.; Pecora, R. Dynamic Light Scattering: WileyInterscience: New York, 1976. (2) Zero, K.; Pecora, R. In Dynamic Light Scattering; Pecora, R., Ed.; Plenum Press: New York, 1985; p 59. (3) Russo, P. S., In Dynamic Light Scattering; Brown, W., Ed.; Clarendon Press: Oxford, 1993; chapter 12. (4) Dublin, S. B.; Clark, N. A.; Benedek, G. B. J. Chem. Phys. 1971, 54, 5158.
as studies on the reorientation times of small molecules in mostly organic solvents.8-10 For larger particles and therefore longer correlation times, the usual photon correlation technique is applied using the homodyne mode, where the scattered light is allowed to impinge directly on the photocathode by itself. The lower correlation time limit of this technique is determined by the photomultiplier tube and the correlator, leading to reliably detectable decay times of one microsecond or more. The depolarized homodyne time correlation function provides information about the rotational and the translational diffusion coefficients.1,2 However, in the forward scattering direction only the rotational diffusion term remains.11-13 Several studies on numerous systems have been applied: on biological sources such as tobacco mosaic virus (TMV) with DDLS13,14 and correlated techniques;15-17 on block copolymers with DDLS18-20 and other techniques;20-22 and also on other different systems23,24 as well as on spherical particles with internal anisotropy.25,26 (5) Eimer, W.; Williamson J. R.; Boxer, S. G.; Pecora, R. Biochemistry 1990, 29, 799. (6) Patkowski, A.; Eimer, W.; Dorfmu¨ller, T. Biopolymers 1990, 30, 93. (7) Haber-Pohlmeier, S.; Eimer, W. J. Phys. Chem. 1993, 97, 3095. (8) Rizos, A.; Fytas, G.; Lodge, T. P.; Ngai, K. L. J. Chem. Phys. 1991, 95, 2980. (9) Vogt, S.; Gerharz, B.; Fischer, E. W.; Fytas, G. Macromolecules 1992, 25, 5986. (10) Takaeda, Y.; Yoshizaki, T.; Yamakawa, H. Macromolecules 1995, 28, 682. (11) Degiorgio, V.; Bellini, T.; Piazza, R.; and Mantegazza, F. Prog. Colloid Polym. Sci. 1997, 104, 17. (12) Aragon, S. R. Macromolecules 1987, 20, 370. (13) Saiz, J. M.; Gonzalez, F. Spectrosc. Lett. 1991, 24, 1247. (14) Schurr, J. M.; Schmitz, K. S. Biopolymers 1973, 12, 1021. (15) Wilcoxon, J.; Schurr, J. M. Biopolymers 1983, 22, 849. (16) Maeda, T.; Fujime, S. Macromolecules 1984, 17, 1157. (17) Kubota, K.; Urabe, H.; Tominaga, Y.; Fujime, S. Macromolecules 1984, 17, 2096. (18) Hoffmann, A.; Koch, T.; Stu¨hn, B. Macromolecules 1993, 26, 7288. (19) Jian, T.; Anastasiadis, S. H.; Fytas, G.; Adachi, K.; Kotaka, T. Macromolecules 1993, 26, 4706. (20) Schillen, K.; Brown, W.; Johnsen, R. M. Macromolecules 1994, 24, 4825. (21) Mortensen, K.; Brown, W. Macromolecules 1993, 26, 4128. (22) Pan, C.; Maurer, W.; Liu, Z.; Lodge, T. P.; Stepanek, P.; von Meerwall, E. D.; Watanabe, H. Macromolecules 1995, 28, 1643.
10.1021/la9910273 CCC: $19.00 © 2000 American Chemical Society Published on Web 01/08/2000
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The aim of this work was to investigate the applicability of DDLS to colloidal systems such as block copolymer micelles and microemulsions. These systems show a variety of structures in the range of several to some hundred nanometers dependent on the concentration, temperature, or ionic strength.27,28 2. Theory The normalized electric field correlation function g1(t) of optically isotropic rods can be expanded as a weighted sum of exponential decays:29
g1(t) ) S(q,t) ) -q2DTt
S0(qL)e
-(q2DT + 6DR)t
+ S1(qL)e
+ ... (1)
where t is time, q is the length of the scattering vector, S(q,t) is the dynamic form factor, S0(qL) and S1(qL) are the scattering amplitudes, L is the rod length, DT is the translational diffusion coefficient, and DR is the rotational diffusion coefficient of the particle. Here q is given by
q)
θ 4πn′ sin λ0 2
()
(2)
where n′ is the real part of the refractive index of the solvent, λ0 is the free space wavelength of radiation, and θ is the scattering angle. The dynamic form factor S(q,t) shows the q- and t-dependence of the total scattered light intensity for an isotropic rodlike particle, and the scattering amplitudes can be expressed by
S0(qL) )
(qL2 ∫
qL/2
0
sin z dz z
)
2
(3a)
and
[qL1 (-3j (qL/2) + ∫
S1(qL) ) 5
1
qL/2
0
sin z z
)]
2
(3b)
where j1(qL/2) is the spherical Bessel function of first order. The angular dependence of the above-described relationship leads to the following observation: in the angular regime of qL < 3, only translational motion contributes significantly to the signal. As qL becomes greater, the rotational motion of the rod increasingly contributes to the signal. Higher terms than those shown in eq 1 can be neglected as long as qL < 8.3 For DDLS, g1(t) consists only of one term, SVH(qL)‚e-Γt, with the decay constant Γ defined by
Γ ) q2DT + 6DR
(4)
where in a plot Γ vs q2, the value of DR can be obtained directly by an extrapolation of q f 0 and DT from the slope of the curve. For rigid noninteracting rods at infinite dilution with an aspect ratio (L/d) greater than 5, DR and DT can be expressed using Broersma’s relationships:30,31
DR
2L kT ln -ξ 3πη0L d
[ ( ) ] 2L 3kT ) ln( ) - 0.5(γ + γ )] [ d πη L DT )
3
|
⊥
(5) (6)
0
(23) Camins, R.; Russo, P. S. Langmuir 1994, 10, 4053. (24) Sierra, A. Q.; Delgado Mora, A. V. Appl. Opt. 1995, 34, 6256.
where k is the Boltzmann constant, T the temperature in Kelvin, η0 the viscosity of the solvent, and d the rod diameter. The parameters ξ, γ|, and γ⊥ are the end-effect corrections which differ slightly in several approaches,30-34 and the validity of these expressions holds for different ranges of the aspect ratio. We use the expressions of Tirado et al.,35 which are valid for aspect ratios in the range of 2 < L/d < 30. 3. Materials and Methods 3.1. Materials. The tobacco mosaic virus sample was kindly provided by the group of Prof. Weber in Konstanz, Germany. The preparation and the characterization of the sample can be found in ref 36. The original concentration was 50.7 g/L, and all samples were stored at -5 °C. The triblock copolymer P94 is an ICI product obtained from Erbslo¨h KG, Krefeld (Germany), and has the formal formula for the average composition of HO-(C2H4O)21-(C3H6O)47-(C2H4O)21OH with an average molecular weight of 4700.37 The material was dissolved in water by constant rotation on a mixing wheel at a temperature of 20 °C. At this temperature, small spherical micelles with a radius of approximately 8 nm are formed.38 The microemulsion was of the type C10E4/n-octane/water with the volume fraction 0.01370:0.0166:0.9697. The tetraoxyethylene glycol mono-n-decyl ether (C10E4) was purchased from Nikko (Tokyo, Japan) with a quoted purity of >98%. The critical demixing temperature or cloud point of this system was determined as 18.5 ( 0.1 °C. The microemulsion was obtained by treatment described in the literature.39 All materials were used as received without further purification. 3.2. Instrumentation. 3.2.1. Dynamic Light Scattering Measurements. The experiments were performed on a laboratory built goniometer equipped with an argon+ laser (Spectra Physics, model 2060-55, Pmax ) 5 W, λ ) 514.5 nm). The scattering cells (10 mm cylindrical cuvettes, Hellma) were immersed in a thermostated index matching bath (decaline). The temperature for all measurements was adjusted to an absolute accuracy of 0.1 °C. The typical laser power for the polarized (VV) DLS measurements was 150 mW. In the case of depolarized (VH) measurements, the laser power was set to 800 mW. The thermostatization of the index matching bath was sufficient to exclude heating effects. For depolarized dynamic light scattering measurements, the primary beam and the scattered light passed through GlanThomson polarizers with an extinction coefficient better than
(25) Piazza, R.; Stavans, J.; Bellini, T.; Lenti, D.; Visca, M.; Degiorgio, V. Prog. Colloid Polym. Sci. 1990, 91, 89. (26) Degiorgio, V.; Piazza, R.; Jones, R. B. Phys. Rev. E 1995, 22, 2707. (27) Glatter, O.; Scherf, G.; Schillen, K.; Brown, W. Macromolecules 1994, 27, 6046. (28) Glatter, O.; Strey, R.; Schubert, K.-V.; Kaler, E. W. Ber. BunsenGes. Phys. Chem. 1996, 100, 323. (29) Pecora, R. J. Chem. Phys. 1968, 48, 4126. (30) Broersma, S. J. Chem. Phys. 1960, 32, 1626. (31) Broersma, S. J. Chem. Phys. 1981, 74, 6989. (32) Newman, J.; Swinney, H. L.; Day, L. A. J. Mol. Biol. 1977, 116, 593. (33) Tirado, M. M.; Martinez, C. M.; de la Torre, J. G. J. Chem. Phys. 1979, 71, 2581. (34) Tirado, M. M.; Martinez, C. M.; de la Torre, J. G. J. Chem. Phys. 1984, 73, 1986. (35) Tirado, M. M.; Martinez, C. M.; de la Torre, J. G. J. Chem. Phys. 1984, 81, 2047. (36) Hagenbu¨chle, M. Ph.D. Thesis, Konstanz, 1993. (37) Nace V. M. Nonionic Surfactants; Dekker: New York, 1996. (38) Bergmann, A., Fritz, G., Scherf, G., and Glatter, O., in preparation. (39) Strey, R.; Glatter, O.; Schubert, K.-V.; Kaler, E. W. J. Chem. Phys. 1996, 105, 1175.
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10-6. The first polarizer guaranteed that only vertically polarized light meets the sample; the orientation of the second polarizer (analyzer) was carefully adjusted to a crossed position with minimum scattering intensity. The measuring time was varied depending on the count rate; at least 106 counts/scan were measured, and consecutive scans guaranteed a total count number of above 107.40 Detection was performed via a single mode fiber with grin lens coupled to a Thorn-Emi photomultiplier (Type B2FBK/RFI), the output of which was analyzed by an ALV-5000 digital multiple-τ correlator (ALV, Langen, Germany) with 256 quasiexponentially spaced channels. The intensity autocorrelation function g2(t) was measured at different angles between 20° and 150° (q-range of 5.65 × 10-3 to 3.14 × 10-2 nm-1). 3.2.2 Viscosity Measurements. The measurements were performed with a DMA 5000 prototype (Anton Paar GmbH, Graz, Austria).41,42 This method of viscosity determination using ultralow shear is based on the measurement of the eigenfrequency of a U-shaped glass tube filled with sample (sample volume 1 mL). An external excitation force compensates the damping of the oscillator. The viscosity η can be derived from the ratio of the damping force to the elastic forces of the glass tube. For details, the reader is referred to the literature.41-43 3.3. Data Evaluation. The measured intensity autocorrelation function g2(t) in the homodyne mode is related to the normalized field correlation function g1(t) as
g2(t) ) 1 + β|g1(t)|2
(7)
where β is the coherence factor, which has a theoretical maximum of 1 and is for our experimental setup with single-mode fiber detection very close to this limit. In the case of DDLS, only scattering due to the optical anisotropy of the rod is collected. This leads to a single-exponential decay in the autocorrelation function, and the field correlation function in the VH geometry g1VH(t) is given by
g1VH (t) ) xβ e-Γt + baseline
(8)
where the decay rate Γ is given by q2DT + 6 DR (see eqn 4). The determination of the decay rates was performed by cumulant expansion,44 where the first cumulant was taken from secondorder cumulant analysis. The resulting decay rates Γ are graphically represented versus q2, and the diffusion coefficients are obtained by weighted linear regression. The size parameters correlated to the diffusion coefficients were estimated using the Tirado expressions35 (See eqs 5 and 6). The DLS data were analyzed as described above using the expressions for the first cumulant of Wilcoxon and Schurr15 and by the expressions of Maeda and Fujime16 where the angular h /q2) of dependence of the effective diffusion coefficient Deff () Γ a rodlike particle is given by15
(31 - F(qL)) + 2L D G(qL)
Deff(q) ) DT + 2∆
2
R
(9)
where F(qL) and G(qL) are universal analytical functions and ∆ is the anisotropy of the translational diffusion coefficient (∆ ) D| - D⊥).
4. Results 4.1. Tobacco Mosaic Virus. The most well-known cylindrical particle investigated by DLS and DDLS is the tobacco mosaic virus (TMV).13,15-17,45 TMV represents a stiff cylindrical particle with a length of 300 nm, a diameter of 18 nm, and a molecular weight of 4 × 107. The entanglement threshold or overlap concentration c* is (40) Degiorgio, V.; Lastovka, J. B. Phys. Rev. A 1971, 4, 2033. (41) Stabinger, H. Proceedings of the Sheffield Meeting on Calibration, Calibration & Quality Standards in the 1990s; Sheffield, 1994. (42) Stabinger, H.; Sommer, K. D.; Fehlauer H. ITG-Fachbericht 1995, 126, 549. (43) Glatter, O. J. Phys. IV 1993, 3, 27. (44) Koppel, D. E. J. Chem. Phys. 1972, 57, 4814.
Figure 1. Decay rate Γ vs q2 from DDLS data (O) of a TMV solution in pure water with a concentration of 0.245 g/L (0.1c*) at 20 °C, including linear regression (dashed line) and theoretical prediction using literature data (full line, see also text).
given by 1/L3, which is 2.45 g/L for TMV.36 To measure TMV in dilute solution and to exclude entanglement of the rods, we chose a concentration of 0.1c* for the DDLS experiments; the data were obtained at several scattering angles (20-150°) at a temperature of 20.0 ( 0.1 °C. The normalized polydispersity ratio (variance) was in all cases much less than 0.1. Figure 1 shows the plot of the decay rate Γ versus q2 of a TMV solution with a concentration of 0.1c* in pure double-distilled water: the data show a linear behavior, and the linear regression (dashed line) results in values of 1798 ( 60 s-1 for 6DR and (3.61 ( 0.10) × 10-12 m2/s for DT. Although the DR value with 300 ( 10 s-1 is in good agreement with the literature,12-13,15,46 the value of DT is approximately 20% lower when compared to the literature, where DT values between 3.4 and 4.3 × 10-12 m2/s 17,47 can be found. However, a good overview of DR and DT values can be found in ref 17, and a reasonable value of DT at 20 °C is (4.2 ( 0.1) × 10-12 m2/s. This value corresponds to the full line in Figure 1, where a mean literature value of 315 s-1 for DR was chosen. This discrepancy of literature and experimental data in pure water should be reduced by screening of the surface charges of TMV. To counter the polyelectrolytic nature of TMV, a 2 × 10-3 M NaCl solution was chosen in the next step for the dilution of the TMV stock solution. Figure 2 shows a comparison of the DDLS results with and without addition of salt in a 0.1c* solution. While the DR value remains almost unchanged after addition of salt, a significant increase of the slope and hence of the DT value is observed: in 2 × 10-3 M NaCl solution, a DR value of 299 ( 9 s-1 and a DT value of (4.05 ( 0.09) × 10-12 m2/s were determined. The hydrodynamic expressions of Broersma and others30,32,35 give for L ) 300 nm and d ) 18 nm quite similar values for DT and DR, (4.0 ( 0.1) × 10-12 m2/s and 295 ( 10 s-1, respectively. In addition to the DDLS data, DLS of a 0.1c* solution of TMV in 2 × 10-3 M NaCl was also investigated at 20 °C. Figure 3 shows the results of this experiment together with fits to the data15 using eq 9. The best fit was achieved using the following parameters for DT, DR and ∆: 4.3 × 10-12 m2/s, 300 s-1, and 1.76 × 10-12 m2/s () 0.41 × DT), respectively, (full line) and using the particle length of 300 nm. The fit was done using DR from the DDLS results, (45) Loh, E.; Ralston, E.; Schumaker, V. N. Biopolymers 1979, 18, 2549. (46) Newman, J.; Swinney, L. L. Biopolymers 1976, 15, 301. (47) Schaefer, W. D.; Benedek, G. B.; Schofield, P.; Breadford, E. J. Chem. Phys. 1971, 55, 3884.
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Figure 2. Comparison of the decay rates rate Γ vs q2 from DDLS data of a TMV solution of 0.1c* in pure water (O) and in 2 × 10-3 M NaCl solution (∇) at 20 °C, including a linear fit to the data.
Figure 3. Polarized DLS data (Deff vs q2) of a TMV solution of 0.1c* in 2 ×10-3 M NaCl solution at 20 °C (O), including fits to the data according to Wilcoxon et al.:15 (a) DT ) 4.3 × 10-12 m2/s, DR ) 300 s-1, ∆ ) 0.41DT (full line); and (b) DT ) 4.1 × 10-12 m2/s, DR ) 300 s-1, ∆ ) 0.25DT (dashed line).
and DT and ∆ were free fit parameters. In this fit DT was found to be app. 5% higher than from the DDLS analysis, and ∆ is rather close to the value given by Wilcoxon et al. (1.79 × 10-12 m2/s, 0.42 × DT).15 However, taking the DT and DR values from the DDLS results would also lead to a reasonable fit to the data but with a significantly lower ∆ of 1.03 × 10-12 m2/s () 0.25 × DT, see dashed line in Figure 3). Comparison with hydrodynamic theories35 leads to higher ∆ values in the range of 1.8 × 10-12 m2/s, which corresponds nicely to the first fit. It should also be mentioned that the first data point is taken at a scattering angle of 20°. At very low scattering angles, there is an increasing danger of stray light from dust particles and of heterodyning from imperfections of the experimental setup. All data shown until now imply the ideal case of infinite dilution. For finite concentration, the entanglement threshold or overlap concentration c* marks the end point of the dilute regime, whereas one should always be aware that the picture is gradually changing with increasing concentration. Figure 4 depicts the behavior of the DDLS data of TMV in 2 × 10-3 M NaCl solution at 20 °C at three different concentrations: 0.1-, 1.0-, and 3.0c* (0.245, 2.45, and 7.35 g/L, respectively). The data show that with increasing concentration in DDLS, the linear behavior is still obtained, whereas the slope and the offset of the linear
Lehner et al.
Figure 4. Decay rates Γ vs q2 from DDLS data of three solutions of TMV in 2 × 10-3 M NaCl at 20 °C: 0.1c* (O), 1.0c* (∇), and 3.0c* (0); the lines are linear fits to the data.
regression (see lines in Figure 4) are significantly changing: in this concentration series, decreasing DT values of 4.05, 3.81, and 2.84 × 10-12 m2/s are found with increasing concentration, whereas DR values of 299, 436, and 638 s-1 are received. Transforming these values into particle dimensions using hydrodynamic relations leads to an underestimation of the length of the particles and an overestimation of the diameter (e.g., at 1c* the DT and DR values can be achieved by a particle length of 230 nm and a diameter of 30 nm). For fixed values of the diameter d (often known from independent methods such as smallangle X-ray or neutron scattering, SAXS and SANS, respectively) the variation of L leads to inconsistent values for DT and DR. Fitting of DR as the most length-sensitive value leads to overestimated DT and underestimated L values (app. 10-15% at c* and 20-30% at 3c*). 4.2. Block Copolymers. Triblock copolymers of the poly(ethylene oxide) and poly(propylene oxide) (PEOPPO-PEO) type are interesting nonionic amphiphilic substances as they show a tendency to form micellar aggregates in aqueous solution.20,27 The investigated triblock copolymer has a formal composition of PEO21PPO47-PEO21 and is called P94. Triblock copolymers in aqueous solution form micelles with increasing temperatures due to the increasing hydrophobicity of the PPO block. These micellar systems with an outer water-swollen PEO-surface and an hydrophobic PPO-core with low water content are observed over a broad range in temperature and concentration. At low temperatures just above the critical micellization temperature (CMT), the micelles show spherical symmetry, and they grow with temperature. Evidence is given by SAXS data38 that spherical micelles are formed between 20 and 40 °C. At 40 °C, they have a radius of 8 nm. Figure 5 shows the development of the micellar structure of a 1% (w/w) solution of P94 in pure water followed by light scattering and viscosimetry: the polarized scattering intensity IVV at a 90° scattering angle increases almost linearly between 20 and 40 °C, indicating formation of micelles. Between 40 and 55 °C, the scattering intensity is almost constant, showing only a very slight increase due to the growth of the micelles. Between 55 and 70 °C, a strong increase is observed, indicating a strong growth of the micelles. Above 70 °C, the solution is becoming turbid and a soft gel is formed. The depolarized scattering intensity IVH at a 90° scattering angle shows essentially no signal below 55 °C except the leakage of the polarizers (app. 120 counts/s) indicating spherical symmetry of the micelles. Above 55 °C, the evolution of the scattering
Diffusion Coefficients of Rodlike Particles
Langmuir, Vol. 16, No. 4, 2000 1693 Table 1. Parameters Determined from DDLS and DLS Data of a 1% P94 Solution in Water DDLS T (°C)
DT (m2/s) × 10-11
DR (s-1)
L h (nm)
estimated c/c*
57.5 60.5 65.5
2.00 ( 0.56 1.24 ( 0.21 1.12 ( 0.10
3400 ( 290 2520 ( 150 1600 ( 100
190 220 270
2.1 2.7 4.1
T (°C)
DT (m2/s) × 10-11
DR (s-1)
L h (nm)
∆ (×DT)
57.5 60.5 65.5
1.90 1.20 0.85
3300 2500 1700
190 220 300
0.45 0.35 0.35
DLS
Figure 5. Indication of a temperature-dependent sphere-torod transition of a 1% P94 solution in water: (∇) depolarized scattering intensity IVH at a scattering angle of 90°, (O) polarized scattering intensity IVV at a scattering angle of 90°, and (0) square root of the viscosity η1/2.
Figure 7. Polarized DLS data (Deff vs q2) of a 1% P94 solution in water at different temperatures: (O) T ) 45 °C, shifted by a factor f ) 0.38 for better graphical presentation; (∇) T ) 57.5 °C, f ) 0.8; (0) T ) 60.5 °C, f ) 1.0; and (]) T ) 65.5 °C, f ) 1.0; the lines are fits to the data (see text for details). Figure 6. Decay rate Γ vs q2 from DDLS data of a 1% P94 solution in water at three different temperatures: (O) T ) 57.5 °C; (∇) T ) 60.5 °C; and (0) T ) 65.5 °C; the lines are linear fits to the data.
intensity scales with polarized scattering intensity. This increase can be understood as formation of nonspherical particles. Such a sphere-to-rod transition can also be found from SAXS data.38 In addition, ultralow shear viscosimetry was performed where the viscosity decreases with increasing temperature to 53 °C. Above 55 °C, the viscosity scales both with the polarized and depolarized scattering intensity, confirming a structural change from spheres to rods. Figure 6 shows the decay rates Γ vs q2 of a DDLS experiment using P94 in a 1% solution in water at three different temperatures in the regime of a strong increase of IVH, IVV, and viscosity: 57.5, 60.5, and 65.5 °C. At all temperatures, linear behavior of the decay rate Γ versus q2 is found. From SAXS data, the cross-sectional diameter of the elongated particles was determined as 15 nm.38 Table 1 shows the parameters of the linear regression and the estimated average particle length L h using Tirado’s expressions (see eqs 5 and 6).35 The average particle lengths L h were obtained by using a constant diameter d of 15 nm and were fitted to reach h can be estimated as approximately DR. The error of L 10%. This leads to an underestimation of the average particle length as described in Section 4.1. The estimated concentration range c/c* was calculated using an average h and the c*-value was particle volume V ) (d/2)2π L obtained by 1/ L h 3. The estimated concentration range
represents also a minimum value due to the underestimation of the particle length. Nevertheless, the obtained length scales are in good agreement with the increase of the forward scattering intensity of polarized scattering intensity. Polarized DLS of a 1% P94 solution in water at different temperatures are shown in Figure 7: at 45 °Cswhere spherical micelles are formedsno angular dependence of Deff () Γ/q2) is observed. The linear fit gives a diffusion coefficient of 5.2 × 10-11 m2/s, which is evaluated via D ) kT/6πηRH with T ) 318.15 K, η ) 0.596 mPa to a hydrodynamic radius of RH ) 7.5 nm. At higher temperatures, the curves were fitted using eq 9;15 here, the values from Table 1 were used to fit the data. Due to the fact that the sphere-to-rod transition is very sensitive to small changes in temperature and the DLS and DDLS experiments were performed in different series, small adaptations of these values were necessary, as can be seen in Table 1; the resulting parameters are close to the ones found from DDLS. The differences of the two sets of measurements are within the possible variation of temperature in the two series, and all obtained values differ less than 10%; which gives a rough estimation for the accuracy of the methods applied. 4.3. Microemulsions. Another challenging task is the investigation of microemulsions as self-organizing systems in the nanometer regime. The microemulsion was of the type CiEj/alcane/water and its composition is described in section 3.1. The total concentration of this system was 3.033% (vol %), and the demixing temperature was found at 18.5 °C. SANS investigations48 indicated a one(48) Strey, R.; Glatter, O. Unpublished results.
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Lehner et al. Table 2. Parameters Determined from DDLS Data of the Microemulsion of the Type CiEj/Alcane/Water T (°C)
DT (m2/s) × 10-12
DR (s-1)
L h (nm)
estimated c/c*
16.2 17.2 18.2
6.0 ( 1.0 3.7 ( 0.6 3.2 ( 0.4
1550 ( 70 1220 ( 60 260 ( 40
160 180 330
4.7 6.0 20.0
For the determination of L h and c/c*, see Section 4.2. The strong decrease of DR from 17.2 to 18.2 °C is remarkable. Also, the estimated concentration of 20c* or more indicates that at this temperature the solution properties are that of a semidilute solution.49 5. Discussion Figure 8. Temperature-dependent development of the depolarized scattering intensity IVH (O) at a 90° scattering angle and the hydrodynamic radius RH (0) obtained from DDLS data of the microemulsion C10E4/n-octane/water with an overall concentration 3.033% (composition: see text) in logarithmic presentation. The critical point was found to be 18.5 °C.
Figure 9. Decay rate Γ vs q2 from DDLS data of the microemulsion depicted in Figure 8: (O) T ) 16.2 °C; (0) T ) 17.2 °C; and (4) T ) 18.2 °C; the lines are linear fits to the data.
dimensional growth of the swollen micelles at temperatures close to the demixing temperature. The overall dimension was too large to be investigated by SANS, however, the cross section dimension was obtained by indirect Fourier transformation as 14.5 ( 0.5 nm. Figure 8 shows the temperature development of the depolarized scattering intensity IVH (circles) together with the increase of the determined RH (squares): The intensity IVH increases almost linearly up to 16.8 °C and exponentially thereafter. The hydrodynamic radius scales together with the IVH and also the polarized scattering intensity shows the same feature (data not shown). This behavior can be described by scaling laws of the critical phenomenon, but it is noteworthy that close to this demixing temperature the depolarized signal also increases exponentially. Therefore, this sample was also investigated using the DDLS method, and the result is depicted in Figure 9; although the measurements were performed rather close to the critical temperature, the plots Γ vs q2 show linear behavior, as already observed for TMV and P94. The results of the evaluation of these curves are given in Table 2. Since the turbidity also increases close to the critical point, we determined the transmission values at all three temperatures. The transmission values at 16.2, 17.2, and 18.2 °C were 0.99, 0.99, and 0.90, respectively. Multiple scattering could therefore be excluded in the first two cases, and even the highest temperature is only slightly affected.
DDLS experiments were carried out at different angles between 20° and 150° scattering angle. The equipment containing a high power Ar+ laser, highly discriminating polarizers, and single-mode fiber detection was used in order to overcome the conventional drawbacks of DDLS. TMV was chosen as a test case for DDLS because it is a well investigated model system for a rigid rod with uniform length and diameter. In dilute solution, TMV showed the behavior depicted in Figures 1 and 2; at a concentration of 0.1c*, almost no diffusion limitations and hindrances were expected. Nevertheless, there is an electrostatic interaction due to the polyelectrolytic nature of TMV; while DR was not affected by this interaction, DT was lowered by approximately 20% without screening with salt. However, by using a 2 × 10-3 M NaCl solution as solvent for TMV, all the interaction effects were sufficiently screened, and the obtained diffusion coefficients were in excellent agreement with the literature. In addition to the DDLS measurements, much less timeconsuming DLS measurements were also carried out with the TMV sample (see Figure 3): Even in the ideal test case it was almost impossible to estimate the diffusion coefficients from DLS data because of nonlinear behavior of Deff vs q2. In principle, DT could be found at q ) 0 (see eq 9), but this value was affected by lower data quality at smaller scattering angles. While DDLS experiments can provide information about DT and DR, DLS measurements were an appropriate tool to ascertain the obtained diffusion coefficients from DDLS data. TMV samples at and above the overlap concentration c* were also investigated to enlighten the behavior at concentrations between the regime of an ideal dilution and a semidilute solution (see Figure 4). The concentrations were, however, too low to be recognized as semidilute solutions where the tube model by Doi and Edwards49 would be valid. In the investigated regime at and just above c*, the diffusion coefficients were strongly affected by the increase of the concentration. Compared to the results in dilute solution, we found at 3 c* a doubled DR value, while DT was reduced by more than 30%. Relating the diffusion coefficients to particle dimensions using eqs 5 and 6 from data at concentrations above c* led to inconsistent values. The particle diameter of systems such as TMV is often obtainable by other methods such as SANS and SAXS. Therefore, we were mainly interested in the determination of the particle length and therefore DR, with its 1/L3 dependence as the essential parameter of interest. Since DR is strongly affected by the interaction effects with increasing concentration, one has to be aware of uncertainty of the resulting lengths, which are usually underestimated. The increase of DR with increasing concentration was not expected and is also not preferred (49) Doi, M.; Edwards, S. F. J. Chem. Soc., Faraday Trans. 2 1978, 74, 560.
Diffusion Coefficients of Rodlike Particles
by literature50 where a moderate decrease of DR until approximately 10c* is observed. The reason for this behavior could be incomplete screening of the polyelectrolyte TMV at higher concentrations. The sphere-to-rod transition of a triblock copolymer of the PEO-PPO-PEO type was investigated with different methods (see Figure 5); the polarized scattering intensity IVV, the depolarized scattering intensity IVH, and the square root of the viscosity η1/2 showed the same qualitative behavior with increasing temperature. At temperatures below 53 °C, only the leakage of the polarizers was detected for IVH while IVH was increasing strongly from 55 up to 70 °C. In this temperature regime, three sets of DDLS experiments were carried out, at 57.5, 60.5, and at 65.5 °C (see Figure 6). All three series showed linear behavior in a Γ versus q2 plot. In all cases, the estimated c/c* value (see Table 1) was above 1, indicating that the solution did not correspond to an ideal dilution. The values given for L h in Table 1 as well as the calculated c/c* ratio were certainly underestimated. In addition to the length polydispersity and the fact that the concentration was no longer that of an ideally diluted solution, the question arises whether the assumption of a stiff rod was applicable to this system. Taking the results from DDLS and applying Wilcoxon’s equation for stiff cylinders on DLS data (see Figure 7) led to reasonable fits to the data. Undoubtedly, the P94 cylinders will have a certain flexibility; however, this flexibility would not be very pronounced. Determination of the persistence length by combined light scattering and SAXS or SANS measurements51 could confirm this argument. In a recent publication,20 DLS and DDLS data were used for the determination of the structure of a similar triblock copolymer system named P85. Compared to P94, P85 has an equivalent molecular weight but a relatively higher part of PEO blocks (PEO27-PPO39-PEO27). P85 in 1% solution also showed a sphere-to-rod transition, but this transition started at 70 °C as a result of the higher ratio of PEO/PPO blocks. The authors investigated the possibilities of DLS and DDLS to determine DT and DR. They determined DR from the small, fast term in the decay time distribution obtained from DLS data by a regularized inverse Laplace transformation. P85 did not show a linearbehavior Γ vs q2 plot from DDLS data like we found for P94. This nonlinear behavior was not discussed by the authors; it could probably be explained by a much higher flexibility of the P85 cylinders. DDLS was also applied to a microemulsion of the type CiEj/alcane/water, which demixes at a temperature of 18.5 °C. Close to the critical phenomenon, not only the polarized scattering intensity but also the depolarized scattering intensity increased exponentially (see Figure 8). Also, in this case a linear dependence of Γ on q2 was found at temperatures between 16.2 and 18.2 °C where the depolarized signal was strong enough to allow for DDLS measurements (see Figure 9). In all cases, the overlap concentration c* was exceeded several times. The existence of rodlike particles is also confirmed by SANS experiments48 where a cross-sectional dimension of 14.5 ( 0.5 nm was determined. At 18.2 °C, the length estimation certainly failed. This might be due to the fact that this temperature is very close to the critical temperature of 18.5 °C. Our results indicated a sphere-to-rod transition which started smoothly below 14 °C until about 17 °C. Above this temperature, an enormous increase of polarized (50) Teraoka, I.; Hayakawa, R. J. Chem. Phys. 1989, 89, 6989. (51) Jerke, G.; Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. Phys. Rev. E 1997, 56, 5772.
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and depolarized scattering intensity was observed, indicating that the interaction forces between the particles changed dramatically toward attractive interaction potentials and a possible formation of locally ordered regimes of elongated particles. Such systems could certainly not be described within the theories existing for independent stiff rods. In addition to the measurement of angular-dependent DDLS signals, there exists the possibility to evaluate DR by zero-angle DDLS;11-13 the correlation function depends at q ) 0 only on rotation according to eq 4, and the translational component is completely eliminated. The drawbacks of this single-point measurements are polarizer leakage, stray light depolarized by the optics, and multiple scattering. On the other hand, rotational diffusion coefficients are available with one experiment. We prefer to measure angular-dependent DDLS signals because in addition to DR, DT is also received, and all the problems occurring at low scattering angle are diminished. So depolarized measurements at q > 0 may be somewhat easier for complex fluids, even though the experiments for a full ΓVH versus q2 plot may require several hours to days, depending on the intensity of the depolarized signal. We have shown that it is possible with careful experiments to determine the diffusion coefficients DT and DR with reasonable accuracy from DDLS. However, the interpretation of the results is complicated by two different facts: (1) Many amphiphilic systems show a sphere-to-rod transition where the rods easily exceed the overlap concentration even at rather low concentration. There exist two theories which handle different concentration regimes of stiff rods: the ideally diluted solution and the semidilute solution. While the first region is valid for concentrations of c/c* < 1, the semidilute region starts at much higher concentrations of c/c* > 50. Between these two limits, there exists no sufficient theory, and more theoretical work in this area is certainly needed. (2) Amphiphilic rodlike systems are normally not uniform in length and have a certain flexibility. Very flexible systems such as polymer coils show low depolarized scattering intensity. Their special conformation leads to an additional error in the determination of L h , on the other hand the value of c* is higher for such systems. A combination of DDLS and DLS experiments can be used for the study of relatively rigid rods, and generally it is very important to also use other complementary techniques in parallel: SAXS/SANS, electron microscopy techniques, viscosimetry, NMR, transient electric birefringence, and others. Nevertheless, DDLS is a rich source of information about dynamics and structural properties of rodlike systems. Acknowledgment. This work was supported by the O ¨ sterreichischer Fonds zur Fo¨rderung der wissenschaftlichen Forschung under Grants P-10682-CHE and P-12611CHE. We also thank Prof. Weber and Prof. Klein from Konstanz for providing the TMV sample and helpful discussions. LA9910273
