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Langmuir 2007, 23, 12010-12015
Unexpected Slow Near Wall Dynamics of Spherical Colloids in a Suspension of Rods Peter Holmqvist,* Dzina Kleshchanok, and Peter R. Lang Forschugszentrum Ju¨lich, Institut fu¨r Festko¨rperforschung, Soft Matter DiVision, D-52425 Ju¨lich, Germany ReceiVed May 23, 2007. In Final Form: August 11, 2007 In this paper, we will show the influence of an additional rodlike component, that is, fd-virus, on the diffusion of spherical polystyrene colloids close to a wall. The sphere diffusivity normal to the wall, D⊥, is strongly affected by the presence of the rods, while the effect on the parallel diffusivity, D||, is less pronounced except in the immediate vicinity of the wall. We show that this observation cannot be explained by describing the effect of the rods as a simple mean field depletion potential alone.
Introduction The slowing down and the anisotropy of Brownian motion close to a wall due to hydrodynamic drag forces has been theoretically predicted1-3 and recently experimentally verified.4-6 So far, only the case of particles interacting by excluded volume with a wall has been considered. However, the introduction of a static interaction potential between the colloids and the wall is expected to alter the near wall dynamics of the spheres. There are different ways to introduce such a potential to a colloidal suspension, for example, steric repulsion, electrostatic interaction, or depletion. An elegant way to control the strength and the range of an attractive particle-wall potential is to add nonadsorbing ideal polymer chains to the colloidal suspension. In this case, the strength of the resulting depletion potential is determined by the polymer concentration, and its range is set by the polymer radius of gyration, RG. Several studies which report direct measurement of the depletion interaction have been published in the past decade. Good agreement between theory and experiment for different depletants (polymers,7,8 rods,9 and spheres10) has been shown. Effects of the nonideality of the depletant have been reported for polydisperse polymers11 and spheres,12 charged depletants,13 and semiflexible rods.14 The effect of depletion on the dynamics close to a wall has received very little or no attention, neither theoretical nor experimental. There are probably two main reasons for the lack of experimental results. First, the range of depletion interaction is limited to about 100 nm if typical polymers are used as depletants. This renders the identification of depletion effects on the particle dynamics very difficult with the techniques available, that is, video microscopy and dynamic light scattering with (1) Brenner, H. Chem. Eng. Sci. 1961, 16, 242. (2) Faxe´n, H. Ark. Mat., Astron. Fys. 1923, 17, 1. (3) Goldman, A. J.; Cox, R. G.; Brenner, H. Chem. Eng. Sci. 1967, 22, 637. (4) Lin, B. H.; Yu, J.; Rice, S. A. Phys. ReV. E 2000, 62, 3909. (5) Holmqvist, P.; Dhont, J. K. G.; Lang, P. R. Phys. ReV. E 2006, 74, 021402. (6) Kihm, K. D.; Banerjee, A.; Choi, C. K.; Tagaki, T. Exp. Fluids 2004, 37, 811. (7) Pagac, E. S.; Tilton, R. D.; Prieve, D. C. Langmuir 1998, 14, 5106. (8) Verma, R.; Crocker, J. C.; Lubensky, T. C.; Yodh, A. G. Macromolecules 2000, 33, 177. (9) Lin, K. H.; Crocker, J. C.; Zeri, A. C.; Yodh, A. G. Phys. ReV. Lett. 2001, 87, 088301. (10) Crocker, J. C.; Matteo, J. A.; Dinsmore, A. D.; Yodh, A. G. Phys. ReV. Lett. 1999, 82, 4352. (11) Kleshchanok, D.; Tuinier, R.; Lang, P. R. Langmuir 2006, 22, 9121. (12) Piech, M.; Walz, J. Y. J. Colloid Interface Sci. 2000, 225, 134-146. (13) Sharma, A.; Walz, J. Y. J. Chem. Soc., Faraday Trans. 1996, 92, 4997. (14) Lau, A. W. C.; Lin, K. H.; Yodh, A. G. Phys. ReV. E 2002, 66, 020401.
evanescent illumination (EWDLS). In video microscopy experiments, the determination of the particle’s distance from the wall becomes increasingly susceptible to systematic errors, due to spherical aberration,4 if the particle approaches the wall. Consequently, the measurement of particle mean square displacements becomes inaccurate at separation distances as small as the range of the potential caused by a polymeric depletant. EWDLS experiments result in diffusivity data which are averaged over the entire volume illuminated by the evanescent wave, which has a penetration depth of typically 100-1000 nm. Therefore, the effect of a potential which ranges only to approximately 100 nm will hardly be detected with this technique. Second, as one would intuitively expect, the effect of a static potential between the wall and the particles would have an influence on the particle mobility mainly in the direction normal to the wall, while the diffusivity parallel to the wall is expected to be less affected. Therefore, it is desirable to measure the diffusion parallel and normal to the surface independently to see any anisotropic effect on the dynamics. Several reports can be found on dynamic measurements near a wall both with EWDLS15-23 and total internal reflection microscopy (TIRM),24,25 but none of these techniques, so far, could distinguish between the parallel and the normal component of the particle diffusivity. On the other hand, microscopy techniques can resolve the parallel and normal component, but it is not possible to measure close enough to the wall to probe the effect of depletion interactions with a range of 100 nm.6 To circumvent these problems, we chose to apply fd-virus as a depletant. These rodlike particles are monodisperse and nonadsorbing neither on glass nor on polystyrene latex particles. They have a contour length of 880 nm which is comparable to the maximum penetration depth applicable in an EWDLS experiment. In other words, the depletion potential mediated by fd-virus is expected to be effective (15) Feitosa, M. I. M.; Mesquita, O. N. Phys. ReV. A 1991, 44, 6677. (16) Fytas, G.; Anastasiadis, S. H.; Seghrouchni, R.; Vlassopoulos, D.; Li, J.; Factor, B. J.; Theobald, W.; Toprakcioglu, C. Science 1996, 274, 2041. (17) Garnier, N.; Ostrowsky, N. J. Phys. II 1991, 1, 1221. (18) Hosoda, M.; Sakai, K.; Takagi, K. Phys. ReV. E 1998, 58, 6275. (19) Lan, K. H.; Ostrowsky, N.; Sornette, D. Phys. ReV. Lett. 1986, 57, 17. (20) Loppinet, B.; Petekidis, G.; Fytas, G. Langmuir 1998, 14, 4958. (21) Matsuoka, H.; Morikawa, H.; Tanimoto, S.; Kubota, A.; Naito, Y.; Yamaoka, H. Colloid Polym. Sci. 1998, 276, 349. (22) Yakubov, G. E.; Loppinet, B.; Zhang, H.; Ru¨he, J.; Sigel, R.; Fytas, G. Phys. ReV. Lett. 2004, 92, 115501. (23) Filippidi, E.; Michailidou, V.; Loppinet, B.; Ruhe, J.; Fytas, G. Langmuir 2007, 23, 5139. (24) Prieve, D. C. AdV. Colloid Interface Sci. 1999, 82, 93. (25) Oetama, R. J.; Walz, J. Y. Langmuir 2006, 22, 8318.
10.1021/la701516s CCC: $37.00 © 2007 American Chemical Society Published on Web 10/18/2007
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throughout the entire scattering volume. Further, we performed EWDLS measurements with our tipple axis setup, with which we can determine the parallel and normal component of the diffusivity independently.5 So far, no theories or simulation addressing the near wall dynamics of spherical colloids affected by an external potential are reported. Though, in a recent paper, we included a static potential into the expression of the initial decay rates, Γ1 and Γ2, of the scattered field autocorrelation function measured in EWDLS. From this, the mean diffusivities can be calculated with an external potential as a function of the distance from the wall. In the expressions for Γ1,2, an attractive static potential appears as a Boltzmann factor which increases the probability density with respect to the bulk density to find a particle at a distance from the surface smaller than the range of the potential. The depletion potential mediated by rodlike particles has been calculated by Mao et al. to third order in rod number density.26 In the first order approximation, there is a simple closed analytical form, which can be introduced in the expressions for Γ1,2. Our experimental TIRM data for the depletion potential mediated by fd-virus is in quantitative agreement with these predictions. Therefore, we used the first order approximation to estimate the lowest rod concentration at which the depletion potential would cause a notable effect on the near wall dynamics of spherical colloids with a radius of R ) 85 nm. We note that the Derjaguin approximation27 is not valid at this radius/rod length ratio, but it will give a lower boundary for the required rod number density, because the strength of the depletion potential decreases with decreasing R/L at constant density.28 From this, we chose as a starting point a rod concentration of twice the overlap concentration which would give a contact potential of approximately 0.5kBT. Further, at this rod concentration, the solvent viscosity will be changed only by a few percent and the fd-virus contribution to the scattering is essentially negligible, which keeps the data treatment on a tractable level. To our surprise, we found a much larger effect on the near wall particle dynamics than expected from these considerations. After a brief introduction to evanescent wave scattering techniques and a description of the instrumentation and sample in the Experimental Section, we will report on TIRM measurements of the depletion potential mediated by fd-virus and the influence of fd-virus on the diffusion of colloids close to a wall. We present and discuss EWDLS data for polystyrene spheres with R ) 85 nm in a solution of fd-viruses. The result will be compared with the theoretically calculated mean diffusivity assuming a mean field static potential showing that fd-viruses have to be treated as a second hydrodynamically active species. Experimental Section Total Internal Reflection Microscopy (TIRM). The interaction potentials between a single charge stabilized polystyrene particle and a glass wall were obtained using evanescent field scattering in TIRM.24 A single colloidal sphere, interacting with an evanescent wave, will scatter light depending on its position as24 Is(z) ) I(z ) 0) exp{-4z/κ}
(1)
where z is the shortest distance from the sphere to the wall and 2/κ is the penetration depth of the evanescent wave. Recording intensity fluctuations for a sufficiently long period of time provides the (26) Mao, Y.; Cates, M. E.; Lekkerkerker, H. N. W. J. Chem. Phys. 1997, 106, 3721. (27) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1991. (28) Yaman, K.; Jeppesen, C.; Marques, M. Europhys. Lett. 1998, 42, 221.
probability density of separation distances, which can be converted into a potential energy profile using Boltzmann’s law p(z) ) K exp
(
)
-φtot(z) kBT
(2)
where φtot(z) is the total interaction potential and K is a constant normalizing the integrated distribution to unity. The experimental TIRM setup was the same as that described by Kleshchanok et al.11,29 With this instrument, it is possible to exchange solvents while the observed particle is kept in place by an optical trap. For all experiments, we applied an angle of incidence of 62.9°, which corresponds to a penetration depth of 2/κ ) 224 nm as calculated from the optical path. The experimental protocol was as follows: First, a potential was obtained in the absence of fd-viruses. Afterward, the solvent was replaced by a solution with the same Debye length and containing the fd-virus. At the end of the experiment, a solution with a high electrolyte concentration (0.1 M NaCl) was pumped into the cell to make the particle stick to the surface to enable the measurement of I(z ) 0), which is required to convert relative separation distances to absolute values. It was possible to use the same particle to obtain a set of potential profiles for one particular Debye length with and without the fd-viruses in the solution. Thus, a direct comparison between potential profiles was possible. Evanescent Wave Dynamic Light Scattering (EWDLS). Here, we present a short description of the EWDLS setup; for a more detailed description, see ref 5. A triple axis diffractometer is used where the mechanical basis is a three axis goniometer custom-made by Huber Diffraktionstechnik, Rimsting, Germany. The instrument is equipped with a Diod-pumped solid-state laser (Excelsior from Spectra-Physics) with an output power of 150 mW operating at λ0 ) 532 nm mounted on the source goniometer arm. During experiments, the primary beam is polarized parallel to the plane of incidence by means of a λ/2 plate and a polarizer (Bernhard Halle Nachfl, Berlin, Germany). Varying the angle of incidence, Ri, the penetration depth of the evanescent wave, 2/κ ) λ0{2π[(n12 sin2 Ri - n22)1/2]}-1 (n1 is the refractive index of the glass, and n2 is the refractive index of the solution), can be changed in the range of 80 nm < 2/κ < 1 µm. The scattered light is collected with a monomode optical fiber, which is attached to a splitter with a splitting ratio of 55/45 (SuK Hamburg). From the splitter, two monomode fibers guide the scattered light to two Perkin-Elmer avalanche diodes. The transistor-transistor logic (TTL) output of the avalanche diodes is cross-correlated with a multiple tau correlator, ALV-6010 (ALV-Laservertiebsgesellschaft Langen, Germany), to obtain the time autocorrelation function of the scattered intensity (ITACF), g2(t). The detection unit can be moved by variation of the two angles θ (in the plane of the reflecting interface) and Rr (off plane) with two goniometers which are mounted normal to each other. From these two angles, the scattering vector components parallel, Q|| ) 2π(1 + cos2 Rr - 2 cos Rr cos θ)1/2/λ, and normal, Q⊥ ) 2π sin Rr/λ, to the surface can be determined. The total scattering vector is then given by Qtot ) (Q⊥2 + Q||2)1/2. Here, λ is the wavelength of the evanescent wave and the scattered light. The sample cell (custom-made by Hellma GmbH, Mu¨llheim, Germany) consists of a hemispherical lens as the bottom part, made of SF10 glass, which has an index of refraction of n1 ) 1.723 at λ0 ) 532 nm. The sample solution is contained in a hemispherical dome sitting on top of the lens. The ITACFs were analyzed following our standard routine reported previously. We use that the ITACF is related to the electric field time autocorrelation function (ETACF), g1(t), by g2(t) ) 1 + 2C1 g1(t) + (C2 g1(t))2
(3)
where C2 ) 1 - (1 - A)0.5 and C1 ) C2 - C22 with A being the intercept of g2(t). Using this expression and taking proper care of (29) Kleshchanok, D.; Wong, J. E.; von Klitzing, R.; Lang, P. R. Prog. Colloid Polym. Sci. 2006, 133, 52.
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the baseline (see ref 30), g1(t) can be calculated from the measured g2(t). The baseline is introduced to account for all relaxation processes which are not related to the colloidal dynamics i.a. a longtime tail observed in most EWDLS ITACFs. It also sums up all small contributions,that is, parasitic surface scattering and extra relaxation processes. The initial decay rate of the ETACF, Γ, is given by30 g1(t) ) exp(-Γ1t + O(t2))
(4)
where Γ1 can be written as
(
Γ1 ) Q||2〈D||〉 + Q⊥2 +
)
κ2 〈D⊥〉 4
(5)
Here, 〈D||〉 and 〈D⊥〉 are the mean diffusivity parallel and normal to the surface, respectively, which are a function of 2/κ. To determine 〈D||〉 and 〈D⊥〉 at a given penetration depth, Q|| is scanned at constant Q⊥ and vice versa. This is then done for a number of different penetration depths, 2/κ, which are varied by changing the incident angle Ri. Samples and Preparation. TIRM. Polystyrene (PS) sulfonate latex particles with a radius of 1.42 µm (σ ) 0.13 µm) were purchased from Polyscience Inc. The particles were diluted from the stock suspension down to a volume fraction of 10-9 for the experiments. The solutions were contained in a carbonized poly(tetrafluoroethylene) frame sandwiched between two microscope slides of BK-7 glass, which were received from Fischer Scientific Co. The glass slides were thoroughly cleaned in an ultrasonic bath for 30 min in ethanol before assembling the sample cell. All the measurements were performed in 2 mM TRIS (trihydroxymethylaminomethane) buffer, which corresponds to a solution Debye length of 9.6 nm. The concentration of fd-viruses was 0.17 g/L. EWDLS. Aqueous buffer solutions (20 mM TRIS) of PS latex spheres (Interfacial Dynamics Corp., Portland, OR) with a radius of R ) 85 nm were investigated with two different fd-virus concentrations, that is, 0.05 and 0.17 g/L, above and below the overlap concentration, c* ) 0.075 g/L. The PS latex spheres are charge stabilized by sulfonate surface groups, and they were diluted from their stock solutions to a volume fraction of 2 × 10-4. The fd-virus has a cross section of d ) 6 nm and a length of L ) 880 nm with a persistence length of 2.2 µm. At the ionic strength used for this investigation, the isotropic to nematic transition starts at 11 g/L, which is about 100 times higher than the concentrations used in our experiment.31 At these conditions, the particles may be regarded as hard spheres, since the Debye screening length is in the range of 3 nm, while the mean interparticle distance is of the order of several thousand nanometers. To make sure the cumulant analysis will not lead to a systematic overprediction of the relaxation rates, we checked the scattering amplitudes of the two species at their respective concentrations by independent bulk scattering experiments. A close examination by both model calculations and experimental measurements of the scattering profiles of fd-virus and latex spheres shows that in the light scattering q-range the virus contributes less than 3% to the total scattering. For EWDLS, this relation is even smaller due to the depletion of rods from the wall.32 We can therefore safely neglect any contribution of the fd-virus to g2(t) if there is no density enhancement of the fd-virus at the wall. No indication of ordering of the rods close to the wall was found which could give rise to an increase of the fd-virus scattering. Further, no experimental findings or theoretical predictions in the literature suggest any such phenomenon at the rod and sphere concentrations used in our experiment. For bulk solutions, several theoretical predictions and simulations show that one needs a much higher rod and/or sphere concentration to induce any alignment effect of the rods.33,34 In a pure rod system, one needs a high concentration (about 90% of the (30) Holmqvist, P.; Dhont, J. K. G.; Lang, P. R. J. Chem. Phys. 2007, 126, 044707. (31) Tang, J. X.; Fraden, S. Liq. Cryst. 1995, 19, 459. (32) Mao, Y.; Cates, M. E.; Lekkerkerker, H. N. W. Physica A 1995, 222, 10. (33) Lekkerkerker, H. N. W.; Stroobants, A. NuoVo Cimento Soc. Ital. Fis., D 1994, 16, 949.
Figure 1. Interaction potential with the gravitational contribution subtracted between a charge stabilized PS particle, R ) 1.42 µm, and a glass surface as measured with TIRM for cfd ) 0.0 g/L (O), cfd ) 0.17 g/L (0), and cfd)0.17 g/L with the electrostatic contribution subtracted (9). The line is the best fit of eq 6 with φ(z ) 0) ) 4.5kBT and Leff ) 690 nm. volume fraction needed to drive the isotropic nematic transition in bulk) to have any alignment at the wall.35,36 Finally, using only fd-virus for EWDLS, no g2(t) with a visible decay can be measured due to the low intensity from such a solution.
Experimental Findings The interaction potential between a charge stabilized PS particle with a radius of 1.42 µm and a glass surface as measured with TIRM is displayed in Figure 1. The open circles represent the potential in the absence of fd-virus which is dominated by electrostatic repulsion. For the sake of clarity, the gravitational contribution has been subtracted. The open squares are the total potential measured in the presence of 0.17 g/L fd-virus and represent the superposition of a repulsive electrostatic contribution and an attractive depletion potential. Assuming that the electrostatic part is not or only minimally influenced by the presence of the fd-virus, we can determine the depletion potential by subtracting the electrostatic repulsion measured in the absence of fd-virus (open circles) from the total potential obtained in the presence of fd-virus (open squares). The result is plotted as full squares in Figure 1 and matches quantitatively to the solid line in the figure, which was calculated using the first order density approximation by Mao et al.26,32 for the depletion interaction mediated by rodlike particles
crodNAπ 2 φdep,sphere-wall(z) z3 )RL 1 kBzT 3Mrod L
(
)
(6)
Here, NA is Avogadro’s number, crod ) 0.17 g/L is the rod concentration which is assumed to be constant throughout, R is the particle radius, L is the rod length, and Mrod ) 1.64 × 107 g/mol is the rod molecular mass.37 In our fit, we allowed for a finite flexibility of the fd-virus, giving an effective length of Leff ) 690 nm, which is in good agreement with the number given by Lau et al.14 With this, eq 6 yields a contact value of φ(z ) 0) ) 4.5kBT and the range in which it is significantly different from zero (i.e., φ(z) < 0.5kBT) extends to zmax ≈ 400 nm. In the context of the EWDLS experiments, which will be described in (34) Tuinier, R.; Taniguchi, T.; Wensink, H. H. Eur. Phys. J. E 2007, 23, 355-365. (35) Dijkstra, M.; van Roij, R.; Evans, R. Phys. ReV. E 2001, 63, 6305. (36) van Roij, R.; Dijkstra, M.; Evans, R. J. Chem. Phys. 2000, 113, 7689. (37) Newman, J.; Swinney, H. L.; Day, L. A. J. Mol. Biol. 1977, 116, 593.
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Figure 2. Initial decay ln(g1(t) - 1) for 2 × 10-4 volume fraction PS latex spheres (R ) 85 nm) in cfd ) 0.17 g/L at Qtot ) 0.0157 nm-1 and 2/κ ) 270 nm for bulk (0) and for two different combinations of Q|| and Q⊥ as follows: Q|| ) 0.0136 nm-1 and Q⊥ ) 0.00785 nm-1 (O), and Q|| ) 0.00785 nm-1 and Q⊥ ) 0.0136 nm-1 (4)). The lines are the expected ln(g1(t) - 1) for the same system without fd-viruses for bulk (s), for Q|| ) 0.0136 nm-1 and Q⊥ ) 0.00785 nm-1 (- - -), and for Q|| ) 0.00785 nm-1 and Q⊥ ) 0.0136 nm-1 (‚ ‚ ‚ ). The inset shows ITACFs for the same measurements with the same symbol representation.
the next section, this means that 2zmax/κ ≈ 3 for the smallest and 0.4 for the largest penetration depth applied. To study the influence of the presence of fd-virus on the near wall dynamics of colloidal spheres, we conducted EWLDS experiments on solutions containing fd-virus concentrations of 0.05 and 0.17 g/L. In bulk solutions with the same fd-virus content, the diffusion of the colloids is reduced by less than 20%.38 To illustrate the effect of the fd-viruses on the colloidal diffusion close to the wall, intensity time autocorrelation functions, g2(t) (ITACF), were measured at a constant penetration depth of 2/κ ) 270 nm and a total scattering vector, Qtot ) 0.0157 nm-1, for two different combinations of Q|| and Q⊥. These g2(t) values together with the bulk g2(t) values, recorded at the same scattering vector, are shown in the inset of Figure 2 for a fd-virus concentration of 0.17 g/L. There is an obvious difference between the EWDLS measurements, in that the relaxation time of the ITACF is significantly larger, if Q⊥ > Q||. This observation is much more evident if the ETACFs are plotted semilogarithmically (shown for the same data in Figure 2 as symbols). In this representation, the difference between the bulk and surface dynamics can be seen more clearly. The EWDLS g1(t) show a much smaller slope than that of the bulk curve. Further, the slope of the ETACF at Q⊥ > Q|| is significantly smaller than that at Q⊥< Q||, which indicates a strong anisotropy of the near wall diffusion. If these data are compared to the expected g1(t) for the same system without depletant5 (lines in Figure 2), it is clear that the sphere near wall dynamics is affected much more than the bulk dynamics by the presence of the rods. In bulk, the g1(t) values with and without the fd-virus are almost identical (the 20% reduction is hardly measurable), while a large difference is obvious for the EWDLS correlation functions. Not only do the near wall dynamics slow down significantly, but the anisotropy increases. This slowing down of the diffusion close to the surface cannot be explained solely by the small reduction of the bulk diffusion. To investigate this in detail, systematic measurements of the diffusivity parallel and normal to the surface at different penetration depths were performed. (38) Kang, K.; Gapinski, J.; Lettinga, M. P.; Buitenhuis, J.; Meier, G.; Ratajczyk, M.; Dhont, J. K. G.; Patkowski, A. J. Chem. Phys. 2005, 122, 044905.
Figure 3. Initial decay ln(g1(t) - 1) for 2 × 10-4 volume fraction PS latex spheres (R ) 85 nm) in cfd ) 0.17 g/L for two different penetration depths 2/κ ) 968 nm (solid symbols) and 2/κ ) 191 nm (open symbols) at (a) constant Q⊥ ) 0.0136 nm-1 at Q|| ) 0.00785 nm-1 (0), Q|| ) 0.01031 nm-1 (O), Q|| ) 0.01224 nm-1 (4), and Q|| ) 0.01407 nm-1 (3) and (b) constant Q|| ) 0.00785 nm-1 at Q⊥ ) 0.01360 nm-1 (0), Q⊥ ) 0.01345 nm-1 (O), Q⊥ ) 0.01003 nm-1 (4), and Q⊥ ) 0.00664 nm-1 (3).
The initial decay of g1(t) for two different penetration depths and four different Q|| values at constant Q⊥ ) 0.0136 nm-1 are shown in Figure 3a. For both penetration depths, there is a clear Q||-dependence of the initial relaxation rate. This Q||-dependence is less pronounced for the smaller penetration depth, indicating a smaller diffusivity (see eq 5) as compared to the larger penetration depth. The complementary ETACFs for constant Q|| ) 0.00785 nm-1 and varying Q⊥ are shown in Figure 3b. A similar behavior but with an overall weaker Q-dependence of the slope is observed. To emphasize this observation, we chose to use the same scale in both parts of Figure 3. A qualitative comparison of the ETACFs in both figures indicates a slower diffusivity normal to the surface at both penetration depths. To quantify this effect, we determined the mean diffusivities parallel, 〈D||〉, and normal, 〈D⊥〉, to the interface according to eq 5. As representative examples, we show the dependence of Γ on Q||2 (squares) at constant Q⊥ and the variation of Γ with (Q⊥2 + κ2/4) at constant Q|| (circles) for two penetration depths, that is, 2/κ ) 968 nm (solid symbols) and 2/κ ) 190 nm (open symbols) in Figure 4. The data obtained from a solution with a fd-virus concentration of 0.05 g/L are shown in Figure 4a. In Figure 4b, the corresponding data from a solution with a rod content of 0.17 g/L are presented. For all cases, a linear dependence of Γ on Q2||,⊥ is observed, the slope of which gives the corresponding mean diffusivity. Due to the scanning of the two Q-vectors by
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Figure 5. 〈D||〉 (squares) and 〈D⊥〉 (circles) normalized to the bulk diffusion, D0, are plotted against the reduced penetration depth, 2/κR, for cfd ) 0.05 g/L (solid symbols) and cfd ) 0.17 g/L (open symbols). The lines are the expected curves for the colloid solution with added depletion potential with contact potentials of 0kBT (solid lines), 2kBT (dashed lines), and 5kBT (dotted lines).
Figure 4. Dependence of Γ on Q||2 (squares) at constant Q⊥ and on (Q⊥2 + κ2/4) at constant Q|| (circles) at 2/κ ) 968 nm (solid symbols) and 2/κ ) 191 nm (open symbols) for (a) cfd ) 0.05 g/L and (b) cfd ) 0.17 g/L. The lines are the fits to eq 5. The bars indicate the general error determined as described in ref 30.
over more than 10 different points together with a consistent linear relation with Q||2 or (Q⊥2 + κ2/4) and a rigorous data analysis, as described earlier,30 the determined diffusivities have errors much smaller than those of individual Γ’s. The general trend which was observed for the pure colloidal system can also be seen here. The diffusivity normal is slower than the one parallel to the interface for both penetration depths. Further, the diffusivity slows down for both the parallel and normal components with decreasing penetration depth. It is immediately apparent that, for the higher fd-virus concentration of 0.17 g/L (Figure 4b), the slowing down with decreasing penetration depth is much more pronounced. To get a more complete picture of the effect of fd-virus on the colloidal dynamics close to the wall, the procedure described above was performed for a series of six penetration depths. In Figure 5, the extracted mean diffusivities normalized to the bulk diffusion, D0, are plotted against the reduced penetration depth, 2/κR (where R is the colloidal radius). The anisotropy of the diffusion can nicely be seen as the separation between 〈D||〉 (squares) and 〈D⊥〉 (circles) for both concentrations, 0.05 g/L (solid symbols) and 0.17 g/L (open symbols). For comparison, we show the expected curves for the colloid solution without fd-virus as solid lines. It is notable that the anisotropy is always larger for the high fd-virus concentration, while for the low fdvirus concentration the anisotropy is close to what can be expected for a system without depletant. If one compares the data with
the expected mean diffusivities without depletant (solid lines), no effect on 〈D||〉 can be seen at large penetration depths for both fd-virus concentrations. However, at penetration depths below about 450 nm, that is, 2/κR ) 5, a pronounced effect can be seen for both fd-virus concentrations. 〈D||〉 is decreasing much more with decreasing penetration depth than was observed for the system where no depletant was present. This effect is more pronounced in the solution with the high fd-virus concentration. If one looks at the normal diffusivity, 〈D⊥〉, a significant reduction can be seen as compared to the nondepletant system also at large penetration depths (distances) for both fd-virus concentrations. As for 〈D||〉, 〈D⊥〉 decreases with decreasing penetration depth, and for the high fd-virus concentration system 〈D⊥〉 drops more strongly at low penetration depths.
Discussion To summarize the experimental findings, when adding fdvirus to the suspension of spherical colloids, both 〈D||〉 and 〈D⊥〉 of the spheres decrease more strongly than in the absence of fd-virus when the penetration depth decreases. This effect is stronger with higher fd-virus concentrations. Intuitively, this is what one can expect together with the increase in anisotropy when considering the virus as a rodlike depletant. To quantify, this we start with a straightforward approach where we treat the effect of the rods as a static depletion potential. As we have verified with our TIRM experiments, this depletion potential can be quantitatively described by
(
)
φdepl(z) z )R 1kBT Leff
3
(7)
in the Derjaguin approximation, where R is the contact potential, which depends on the effective rod length, Leff, the particle radius and the rod concentration. It has been shown in a recent paper that the shape of the potential is almost unaffected by the R/L relation and only the amplitude changes with R/L < 1.39 This closed analytical expression for the potential can be easily introduced into the expression for Γ and the mean diffusivities close to a surface as shown in our recent work.30 It has to be pointed out that this potential does not affect the hydrodynamic interaction of the spherical colloids with the wall. This is assumed (39) Lang, P. R. J. Chem. Phys. 2007, 127, 124306.
Slow Near Wall Dynamics of Spherical Colloids
to be the same as in a system where no rods are present. The static potential influences only the particle dynamics by changing the probability density to find a particle at a given distance from the wall. Therefore, the depth and the range of the potential are important while its exact shape is less crucial. Using eq 7, we calculated the dependence of the mean diffusivities on the penetration depth for our system with contact potentials of 2kBT and 5kBT, which are shown as dashed lines and dotted lines, respectively, in Figure 5. It is immediately evident that this approach cannot even qualitatively capture the dependence of the experimental diffusivity data on the penetration depth. At low 2/κ, the experimental data are systematically much smaller than predicted using the depletion potential except for the diffusivity normal to the interface measured from the solution with the smaller fd-virus concentration. The nice agreement of these data with the calculated curve for R ) 2kBT is very likely a coincidence for the following reasons. Introducing the values for particle radius and effective rod length into eq 7, one obtains a contact potential of R ) 0.07kBT at crod ) 0.05 g/L. Further, as mentioned above, eqs 5 and 7 were derived using Derjaguin’s approximation, which is a simplification that for R/Leff ≈ 0.05, as in our experiments, can certainly not be justified. According to numerical calculations by Yaman et al.,28 the depletion interaction mediated by a rod between two spheres would drop by approximately 1 order of magnitude below the Derjaguin limit at R/Leff ≈ 0.05. For the depletion between a sphere and wall, a similar argument holds.39 This means in turn that the miniscule variation of the particle density close to the wall, caused by the depletion potential which is mediated by the fd-virus, is way too small to cause the observed effect on the near wall dynamics of the spherical colloids. In the light of these considerations, the observed slowing down of the particle dynamics and the increase of the diffusivity anisotropy are unexpectedly large. We conjecture that this effect is due to a
Langmuir, Vol. 23, No. 24, 2007 12015
hydrodynamic effect of the fd-rods, which couples to the hydrodynamic interaction of the spherical particles with the wall. These hydrodynamic interactions arise from the motion of the rods, creating a flow field which directly influences the motion of the spheres and additionally can be reflected at the wall and therefore affects the spheres in a more complicated way than in bulk. A coarse grained molecular dynamics study to test this hypothesis is currently in progress.
Conclusion We have performed EWDLS experiments on solutions of spherical latex particles with fd-virus as a second solute component to study the influence on the near wall dynamics. We started with a rod concentration, which was chosen as a lower estimate of the rod density, at which an effect of a static depletion potential might have an influence on the near wall dynamics of the spheres. However, we observed an unexpectedly pronounced slowing down of both the normal and the parallel component of the colloids’ diffusivity at penetration depths of the evanescent wave smaller than about half the rod length. In parallel, the anisotropy of the diffusivity increased. We observed the qualitatively similar behavior at an even lower fd-virus concentration. We therefore conclude that we found an effect of the rodlike cosolute on the colloids’ dynamics which cannot be explained by the effect of a static depletion potential. We conjecture that the rods rather have to be regarded as a second hydrodynamically active component on equal footing with the spherical particles. Since to our knowledge there is no theory available for this problem, we are investigating this question with coarse grained particle dynamics simulations in an ongoing project. LA701516S
