Dynamics of Uncharged Colloidal Silica Spheres Confined in

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Dynamics of Uncharged Colloidal Silica Spheres Confined in Bicontinuous Porous Glass Media Sebastiaan G. J. M. Kluijtmans, Jan K. G. Dhont, and Albert P. Philipse* Van’t Hoff Laboratory for Physical and Colloid Chemistry, Debye Research Institute, Utrecht University, Padualaan 8, 3584 CH Utrecht, the Netherlands Received February 20, 1997. In Final Form: June 17, 1997X The restricted diffusion of uncharged colloidal silica spheres in an optically matched, porous glass with uncharged internal surfaces is measured by means of heterodyne dynamic light scattering (DLS) and fluorescence recovery after photobleaching (FRAP). Glass and sphere surfaces have been chemically treated such that, in addition to hydrodynamic interactions, only hard particles-wall interactions are present. The wavevector-dependent effective diffusion coefficient inside the porous glasses, obtained from the initial decay of the correlation functions, is always smaller than the bulk diffusion coefficient. In particular the diffusion coefficient at small K, probing diffusion over several pore diameters, is strongly reduced due to increasing hydrodynamic drag at large particle to pore size ratios. The long-time self-diffusion coefficient DLS of fluorescent silica spheres measured by FRAP is in good agreement with the small angle DLS results. The reduction of the diffusion coefficient as a function of the size ratio qualitatively agrees with various models describing the restricted motion of a particle in an infinitely long cylinder.

1. Introduction Diffusion of molecular and colloidal species in porous media plays an important role in a large variety of topics such as diffusion-controlled catalysis, membrane and chromatographic separation procedures, and spreading of soil polution. Dynamics in confining geometries is an intricate subject because of the complex pore structure and hydrodynamic and potential interactions between a diffusion particle and the porous medium. In many cases experimental results are mapped on theories describing the motion of a hard spherical particle in a cylinder. This approach works fairly well for, e.g. polymer diffusion across track-etched membranes,1,2 but surprisingly also holds for polymer diffusion in more complicated media like controlled pore glasses3-6 (CPG). Diffusion in such complex porous systems has also been investigated using computer simulations. Kim and Torquato7 analyzed the conductivity of tracers in a random sphere packing, incorporating geometric effects only. Recently, computer simulations of Brownian particles in porous media including both hydrodynamic and geometric effects became available.8 In this work we present, to our knowledge, the first experiments on the diffusion of rigid, uncharged, monodisperse colloidal spheres, retarded by uncharged porous media. The aim is to determine the relation between the diffusion coefficient and the ratio of the particle size and pore size. For this purpose CPG with three different mean pore radii (70, 107, and 138 nm) are used. Since porous glasses can be made transparent by optical matching, light-scattering techniques can be applied3 to monitor directly particle diffusion in the porous medium. Here, * Corresponding author. X Abstract published in Advance ACS Abstracts, August 15, 1997. (1) Malone, D. M.; Anderson, J. L. Chem. Eng. Sci. 1978, 33, 1429. (2) Cannell, D. S.; Rondelez, F. Macromolecules 1980, 13, 1599. (3) Bishop, M. T.; Langley, K. H.; Karasz, F. E. Macromolecules 1989, 22, 122. (4) Zhou, Z. M.; Teraoka, I.; Langley, K. H.; Karasz, F. E. Macromolecules 1994, 27, 1759-1765. (5) Guo, Y.; Langley, K. H.; Karasz, F. E. Macromolecules 1992, 25, 4902. (6) Teraoka, Y.; Langley, K. H.; Karasz, F. E. Macromolecules 1993, 26, 287. (7) Kim, I. C.; Torquato, S. J. Chem. Phys. 1992, 96, 149. (8) Hagen, M. H. J.; Pagonabarraga, I.; Lowe, C. P.; Frenkel, D. To be published.

S0743-7463(97)00178-9 CCC: $14.00

we use heterodyne dynamic light scattering (DLS) and fluorescence recovery after photobleaching (FRAP). DLS experiments are performed with silica spheres with mean radii of 12, 23, 30, and 43 nm. In the FRAP experiments, fluorescently labeled silica particles of 40 nm are used. Combination of various porous glasses and silica tracers allows a particle to pore size ratio, ranging from 0.08 to 0.6. First we briefly describe some properties of our experimental model systems. Porous glasses or controlled pore glasses (CPG) are porous silica glasses with a bicontinuous structure. Optical and structural properties of these glasses are described in an accompanying paper.9 The surface of both pores and tracer particles is grafted with an alkane layer of octadecyl chains.9 Optical matching of the porous glasses is achieved in an isorefractive solvent mixture of cyclohexane and toluene. In this solvent mixture octadecyl-coated silica spheres are known to form stable dispersions at ambient temperature.10,11 Coating of the glass and tracer surface with an alkane layer has the following advantages: (a) particle and pore walls are uncharged, (b) particles and walls have a uniform and chemically equivalent surface, and (c) due to the steric repulsion of alkane layers in a good solvent the shortranged van der Waals attraction is screened. Absence of charge together with screening of van der Waals attraction gives rise to a steeply repulsive interaction between sphere and wall and between the spheres themselves. In this way we hope to monitor purely geometrical and hydrodynamic effects on the tracer diffusion. In a later study we investigate the additional influence of double layer effects in systems with charged surfaces.12 This paper is organised as follows. Section 2 reviews several theoretical approaches of the diffusion of particles in porous media, starting with the case of a hard sphere confined in a cylindrical geometry. In Section 3 principles of heterodyne dynamic light scattering and FRAP are (9) Kluijtmans, S. G. J. M.; Philipse, A. P.; Dhont, J. K. G. Langmuir 1997, 13, 4976. (10) Jansen, J. W.; de Kruif, C. G.; Vrij, A. J. Colloid Interface Sci. 1986, 114, 501-504. (11) Jansen, J. W.; de Kruif, C. G.; Vrij, A. J. Colloid Interface Sci. 1986, 114, 492-500. (12) Kluijtmans, S. G. J. M.; de Hoog, E. H. A.; Philipse, A. P. To be published.

© 1997 American Chemical Society

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shortly summarized, together with a description of the model systems and measuring procedures. Static light scattering of the porous media is analyzed in ref 9. In section 4 DLS and FRAP results are discussed. In Section 5 conclusions are summarized. 2. Diffusion in Porous Media In 1987 Deen13 has extensively reviewed transport phenomena in confining geometries. In this section we will briefly recapitulate the most important equations. For a sphere with radius a, the diffusivity in an unbounded fluid is given by the Stokes-Einstein relation

D0 )

kBT 6πηa

(1)

with kB the Boltzmann constant, T the temperature, and η the solvent viscosity. Near a wall, however, a particle experiences an extra hydrodynamic drag in addition to the friction in eq 1. A freely moving particle has an isotropic diffusion coefficient, but the Brownian motion of a particle near a wall is anisotropic. There are two diffusion coefficients: D⊥ for diffusion perpendicular to the wall, and D| parallel to the wall. Various authors13-18 have treated the Brownian motion of a spherical particle confined in a cylindrical pore of comparable size. In a cylindrical geometry, at long times, the particle is forced to move in the direction parallel to the cylinder axis and acquires an average diffusion coefficient Dpore along the cylinder axis. Averaging requires that the particle has probed all radial positions many times, which is equivalent to the condition that14

6D0t (Rp - a)2

.1

(2)

with t the observation time and RP the cylinder radius. The center-line (CL) approximation15,16 is based on a particle which can only move on the cylinder axis. Within this approximation the diffusion coefficient Dpore, which can be expressed as a function of the ratio λ ) a/Rp only, equals15,16

Dpore ) D0[1 - 2.1044λ + 2.089λ3 - 0.948λ5] for λ e 0.4 (3) For λ up to 1 the more elaborate centerline expression of Bungay and Brenner18 can be used, which extents eq 3 to higher order in λ. Recent computer simulations of Hagen et al.8 demonstrate that the CL results from Bungay and Brenner are reasonably accurate for all λ. The applicability of models for a single straight cylinder to the confined motion of colloids in porous media with a complex pore structure, like controlled pore glasses (CPG; see Figure 1 and ref 9) is not obvious. For a point particle (λ f 0) eq 3 reduces to the diffusivity D0 in eq 1. In a porous medium, however, the net displacement of a point particle still depends on the volume fraction and detailed geometry of the pore space. This dependence may be h , defined cast3-6 in the form of a conductivity factor X experimentally as (13) Deen, W. M. AIChE J. 1987, 33, 140. (14) Brenner, H.; Gaydos, L. J. J. Colloid Interface Sci. 1977, 58, 312. (15) Anderson, J. L.; Quinn, J. A. Biophys. J. 1974, 14, 130. (16) Pappenheimer, J. R.; Renkin, E. M.; Borrero, L. M. Am. J. Physiol. 1951, 167, 13. (17) Davidson, M. G.; Deen, W. M. Macromolecules 1988, 21, 34743481. (18) Bungay, P. M.; Brenner, H. Int. J. Multiphase Flow 1973, 1, 25.

Figure 1. Top left: transmission electron micrograph of coated silica spheres (radius 43 nm bar ) 200 nm). Top right: transmission electron micrograph of the fluorescent silica spheres with a radius of 40 nm (bar ) 200 nm). Bottom: scanning electron micrograph (bar ) 1 µm) of a grafted porous glass, with computer-drawn spherical particles, for which λ is about 0.2.

Dmacro λf0 D0

X h ) lim

Dmacro ) X h Dpore

(4)

Dmacro is the diffusion coefficient describing the particle displacement over a macroscopic distance in the porous medium, i.e. a distance much larger than the details of the pore space. For bicontinuous porous glasses X h is indeed a macroscopic parameter provided that 6Dporet/ Λ2>1, with Λ the characteristic (density) wavelength present in the porous glass.9 3. Experimental Section 3.1. Dynamic Light Scattering (DLS). 3.1.1. Background. Consider the light scattered by the rigid porous glass medium in which colloidal silica particles diffuse through the pores. The medium and the particles are sufficiently transpararent to eliminate multiple scattering. In this situation, which is referred to as a heterodyne dynamic light scattering experiment,19 the normalized intensity auto correlation function (ACF) gˆ het I is given by

gˆ het I (t) ) 1 +

2IB gˆ (t) It E,B

(5)

where It is the total scattered intensity, IB is the scattered intensity of the Brownian particles, and gˆ E,B is the normalized scattered field ACF of the Brownian particles. Equation 5 assumes that the scattering of the static background is at least 50 times larger than IB. In general for Gaussian fluctuations gˆ E ) exp(-Γt), in which Γ is the decay rate of the Kth Fourier mode of the intensity fluctuation. For intensity fluctuations due to Brownian (19) Chu, B. Laser Light Scattering; Academic Press: New York, 1974.

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Table 1. Size of the Silica Spheres and Average Pore Radius (Rp) of the CPG silica particles aTEMa (nm)

aDLSb (nm)

CPG Rpc (nm)

25.1 ( 1.3 36.1 ( 1.3

12.0 ( 0.4 23.2 ( 0.5 30.4 ( 1.1 43.4 ( 0.7

138 ( 6 107 ( 4 70.7 ( 3.8

motion we have

(6)

Here K ) 4πηλ0-1 sin(θ/2), with n the refractive index of the solvent, λ0 the wavelength in vacuo of the inciding light and θ the scattering angle. Deff is the effective diffusion coefficient which may depend on K. The StokesEinstein diffusivity D0 (eq 1) of the colloidal particles is determined in absence of scattering from the medium using the well-known Siegert relation.

(t) ) 1 + gˆ E,B2(t) gˆ hom I

system codea

λ

12SG138

0.087 0.168 0.220 0.314

30SG107

0.284 0.217 0.328

43SG138

Radius determined by transmission electron microscopy (TEM). Hydrodynamic radius determined by dynamic light scattering of very dilute samples in CHX. c Pore radius of the porous glasses as measured with mercury intrusion.

∂lngˆ E(K,t)) ) DeffK2 ∂t

λ

30SG138

b

tf0

system codea 23SG138

a

Γ ) -lim

Table 2. Code and Particle/Pore Size Ratio (λ) of the Combined Sphere (S)/Glass (G) Systems

(7)

3.1.2. Sample Preparation. To vary the pore/particle size ratio, silica spheres with various radii and porous glasses with various pore radii were used (see Table 1). Monodisperse silica with a TEM radius of 25 nm was prepared using the microemulsion method described by Osseo-Asare et al.20 These particles were grown stepwise to 36 nm following van Blaaderen.21 The polydispersity of these microemulsion silica particles was only 4%. The dispersion of 36 nm particles contained a small amount of dumbells, which have been formed during the growing step. Also Ludox HS40 (DuPont) and Sto¨bersilica22 of respectively 12 and 23 nm radius were used. These particles have a somewhat larger polydispersity of about 10%. Eventually, all particles were grafted with octadecyl alcohol following van Helden et al.23 and dispersed in cyclohexane. Controlled pore glasses (Controlled Pore Glass Inc., Lincoln Park, NJ; grain size 1-4mm) with three different pore diameters (see Table 1), were also grafted with octadecyl alcohol. The grafting procedure and characteristics of the porous glasses are described in the accompanying paper.9 Grafted porous glasses were stored in cyclohexane. Light-scattering samples were prepared as follows: 2 mL of a 2% v/v silica dispersion in cyclohexane was mixed with 2 mL of toluene. (Then the refractive index of the solvent mixture is approximately 1.46, which is similar to the refractive index of the porous silica glasses.) A piece of CPG was added to the suspension and the sample was allowed to equilibrate for at least 1 week. Table 2 summarizes the seven combinations of particles and glasses used. The solution was then poured into a lightscattering cuvette where the glass grain was firmly mounted. (20) Osseo-Asare, K.; Arriagada, F. J. Colloids Surf. 1990, 50, 321. (21) van Blaaderen, A.; Imhof, A.; Verhaegh, N. A. M.; Mason, N. Manuscript in preparation. (22) Sto¨ber, W., Fink, A., Bohn, E. J. Colloid Interface Sci. 1968, 26, 62. (23) Helden, A. K. v.; Janssen, J. W.; Vrij, A. J. Colloid Interface Sci. 1981, 81, 354.

23SG107 23SG71

a The left superscript stands for the hydrodynamic radius of the particle (aDLS) and the right superscript for the pore radius of the CPG, both taken from Table 1.

At ambient temperature octadecyl-coated silica spheres are known to be hard-sphere systems in various apolar organic solvents like cyclohexane and toluene.10,11 The DLS determination of the bulk diffusion coefficient in a solvent mixture of CHX and toluene shows no significant angular dependence, corroborating the small polydispersity and the nonaggregated state of the spheres. Measured diffusion coefficients were reproducible within a time span of at least 6 months. No visual sign of aggregation was observed, neither in the starting dispersions nor in the dispersions with CPG grains. 3.1.3. Measurements. Dynamic light scattering experiments were performed with a Krypton laser source (10-50 mW, Spectra Physics, Model 2020) at a wavelength of 647 nm in vacuo. The scattering angle was varied from 15 to 120°. Scattered intensities were detected using a photomultiplier tube with a pinhole in front. Correlation functions were measured with a Malvern 7032 CE, 128 channel correlator. Sample times were varied from a few milliseconds to microseconds. To study the long-time behavior of the correlation function occasionally a software correlator was used which measured the IACF up to 5 s. The cuvette was thermostated and matched in refractive index by a toluene bath. No special attempts were made to make the dispersions dustfree since dust particles are too large to intrude the pores of the porous glasses (no indications for the presence of dust or other contaminants were found when checking the DLS results for grains in a pure solvent). Restricted diffusion coefficients were measured by fixing the scattering volume precisely inside a CPG grain, eliminating the influence of freely moving particles on the correlation function. The amount of heterodyning was controlled in two ways. First, the temperature was adjusted to minimize the scattered intensity of the glasses to avoid multiple scattering. Even at minimal optical contrast (matching point) the scattering of the porous glass is fairly strong due to small refractive index inhomogeneities in the silica glass.9 At the porous glass matching point, the nearly matched spheres scatter just enough for detection. (the refractive index of the glasses and the silica particles is slightly different9). The second way to control the amount of heterodyning is based on the dependence of the static background on the orientation of the sample. Depending on the orientation of the sample the detector is fixed at a speckle or just in-between the speckles. Heterodyne correlation functions were taken at three to five different positions in the glass grain. This procedure averages any influences of inhomogeneities or defects inside the glass grain. Depending on the sample it took about 30 min to 7 h to obtain at least 1 × 107 counts. In most cases, this proved to be sufficient for a representative IACF. The IACFs were then scaled and averaged. Measurements of unrestricted diffusion coefficients are performed by fixing the scattering volume in the dispersion just above the grain. 3.2. Fluorescence Recovery after Photobleaching (FRAP). 3.2.1 Introduction. In a FRAP experiment, a fringe pattern is created in the sample by crossing two

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Table 3. Code and Particle/Pore Size Ratio (λ) of the Combined Fluorescent Sphere (FS)/Glass (G) Systems system codea

λ

system codea

λ

41FSG138

0.30 0.39

41FSG71

0.58

41FSG107 a

See footnote a in Table 2.

high intensity coherent laser beams at an angle θ for a short period of time. During this short pulse, typically 1 s, part of the fluorescent dye on the colloidal particles is irreversibly decomposed and a sinusoidal fluoresence profile is created. With a low intensity “reading” laser beam the decay of this profile, which is due to the Brownian motion of the particles, is measured. For a Brownian particle the fluorescence signal If is recovered according to

If ∝ exp(-DLSk2t)

Figure 2. Semilogarithmic plot of the normalized electric field ACFs of the samples 43SG138 and 23SG71 for confined silica spheres in porous glasses and for freely diffusing silica particles in bulk liquid.

(8)

with k ) 4πλ0-1 sin θ and t the time after the bleach pulse. Since scattering angles are small (∼0.5°) diffusion is probed over large distances, typically on the order of 30 µm, yielding the long-time self-diffusion coefficient DLS of the particles. For a detailed description of the FRAP technique and experimental setup we refer to Imhof et al.24 3.2.2. Sample Preparation and FRAP Measurements. We used fluorescently labeled silica tracer particles of 39 nm prepared by a modified Sto¨ber method in presence of fluoresceine isothiocyanate.25 The characteristics of these fluorescent particles are described elsewhere.26 The particles were then grafted with octadecyl alcohol and suspended in CHX/toluene. During the coating procedure at high temperatures part of the dye molecules is destroyed, making the particles somewhat less fluorescent. Grafting times should therefore be kept as short as possible. FRAP samples were prepared in a similar way as the DLS samples (see Table 3 for codes). It should be emphasized that the properties of fluorescent and nonfluorescent silica dispersions are very similar.27 The fluorescent core is shielded by a thin layer of silica (∼5 nm) and an octadecyl coating. So the fluorescent core does not influence the interparticle or particle-glass wall interactions. In a FRAP experiment the scattering volume cannot be focused inside a single grain. To avoid contribution of particles outside the glass, the grain was taken out of the suspension of colloidal particles and immersed in a “clean” solvent, i.e. a pure CHX/toluene mixture. During the measurements, which typically take 1 h, the release of fluorescent particles by the grain is small and their contribution to If can be neglected. At least 5 FRAP curves were measured at different positions in the grain. After scaling and averaging, the curves were fit to eq 8 to obtain DLS of the particles inside the grain. Details of the FRAP procedure and data evaluation will be given elsewhere.12 4. Results and Discussion 4.1 DLS Correlation Functions and FRAP Decay Curves. Figure 2 shows typical DLS correlation functions of silica spheres confined in Controlled pore glasses. ACFs of the 23SG71 and 43SG138 system (for codes see Table 2), (24) Imhof, A.; van Blaaderen, A.; Maret, G.; Mellema, J.; Dhont, J. K. G. J. Chem. Phys. 1994, 100, 2170-2181. (25) van Blaaderen, A.; Vrij, A. Langmuir 1992, 12, 2921-2931. (26) Imhof, A.; Dhont, J. K. G. Phys. Rev. E. 1995, 52, 111. (27) Imhof, A. Dynamics of concentrated colloidal dispersions. Thesis, Utrecht University, Utrecht, The Netherlands, 1996.

Figure 3. Semilogarithmic plot of a typical FRAP decay curve of silica particles confined in a porous glass.

both at the same scattering angle of 15°, clearly deviate from single exponentiality. The strongest deviation is found for the porous glass with the largest pore size where even two diffusion regimes are observed. It is conceivable that the nonexponentiality is (partly) due to local variations in the pore structure. One can imagine that a particle has to probe a fairly large region of porous glass for these variations to average out. Karasz et al.3 indeed showed that the deviation from singleexponential (SE) behavior, as indicated by the secondorder cumulant of the fits of their correlation functions, depends upon the distance over which diffusion is observed. This distance is proportional to 1/K. In order to compare glasses with a different pore size the wavevector is scaled on the pore radius of the porous glass as KRp. For distances larger than the pore radius, where KRp 1 large deviations were found. The fact that our FRAP curves are all single exponentials agrees with the findings of Karasz et al.3 In contrast, all DLS curves deviate from SE behavior and we do not find a strong K dependence of this deviation. The nonexponentiality rather depends on the pore size of the porous glass and the particle to pore size ratio. However, we cannot quantify the deviation from SE behavior using a second-order cumulant fit because this fit only describes the initial decay; for larger decay times a cumulant fit gives poor results. Since the ACFs are very well described by a double exponential, we quantify the deviation by the ratio of the amplitude of both exponents of this fit. Figure 4 shows this ratio for the various glass/tracer systems as a function of KRp. A1 is the fastest decay time, and A2 characterises the slow decay. All fits were performed with ACFs measured at the same scattering vector. For glasses with a small pore size A1/A2 > 1, which means that the amplitude of the fastest decay dominates the slow decay. For glasses with a large pore size both decay times have

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Figure 5. Deff/D0 vs KΛ for the systems 23SG71 (λ ) 0.33), 23SG138 (λ ) 0.17), and 43SG138 (λ ) 0.314). Lines are drawn to guide the eye. The scattering vector K is scaled on the spinodal decomposition wavelength Λ of the porous glasses in order to compare the different glasses with respect to the different scattering regimes. For KΛ < 2, the reduced diffusion coefficient, which is notably smaller than the free-particle value D0, becomes almost independent of the scattering vector. Figure 4. Ratio A1/A2 of amplitudes resulting from a double exponential fit of the intensity auto correlation functions as a function of KRp (Rp is the pore radius). A1 refers to the fastest decaying component. Effective diffusion coefficients are obtained at K ) 4 × 106 m-1.

about the same amplitude. Furthermore we find that for glasses with the same pore size the ratio A1/A2 increases with decreasing particle size. The influence of the pore size can again be explained in terms of structural averaging; the smaller the pore size the larger the relative diffusion distance probed at a certain scattering vector. The interpretation of the particle size effect is yet unclear. Note that also size polydispersity may account for the observed nonexponential decay. The FRAP fluorescence decay curves (see Figure 3), however, are single exponentials, and also in the DLS correlation functions of the “free” particles (see Figure 2) we do not find a significant influence of polydispersity on the ACF, as we can expect from the relative small spread in TEM diameters. One may argue that polydispersity in particle/pore size ratio σλ is the relevant quantity. Numerical calculations28 of polydisperse correlation functions for σλ up to 30%, however, show that also this polydispersity effect is much too small to account for the observed nonexponentiality. We obtain Deff, as defined in eq 6, by a second-order cumulant fit of the initial decay of the correlation function. The range of the initial region depends on the specific sample (see, e.g., Figure 2). For the majority of the samples this range is sufficiently large for a reliable determiniation of Deff. (The uncertainty in Deff is typically 10-30%). D0 is obtained from a second order cumulant fit of the ACF of the particles outside the glass. 4.2. Results for the Effective Diffusion Coefficient. For three systems the effective diffusion coefficient was measured with DLS as a function of the scattering vector over the whole range of experimentally accessible K vectors. To compare scattering regimes for the different systems the scattering vectors are scaled on the spinodal decomposition wavelength Λ in the porous glasses.9 Figure 5 shows the reduced diffusion coefficient as a function of KΛ. All reduced diffusion coefficients are smaller than unity, implying that the diffusion is hindered at all length scales. The reduced diffusion coefficients seem to reach a plateau value for KΛ < 2. For KΛ > 2, Deff/D0 does not show a clear trend as a function of K. As we discussed in the previous section, structural details will average out when a particle probes large enough distances, i.e. K f 0. In case of DLS, where scattering angles are relatively large compared to FRAP, we can find the long-time self-diffusion coefficient by (28) Kluijtmans, S. G. J. M. Dynamics of colloids in porous media; Thesis, Utrecht University: Utrecht, The Netherlands, to appear 1998.

Figure 6. Deff(K f 0)/D0 obtained by DLS measurements for 6 glass-colloid combinations (9) as a function of the particle to pore size ratio λ. Also included in this figure are the three FRAP results (0) and the calculation of Bungay and Brenner18 for spheres in a cylinder.

extrapolating the curves in Figure 5 to small K. This small-K limit of the reduced diffusion coefficients is plotted in Figure 6 (filled symbols) as a function of the ratio of the particle to pore radius λ. With increasing λ, the reduced diffusion coefficient decreases in reasonable agreement with the values predicted by the CL-model discussed in section 2. The open symbols in Figure 6 represent the reduced diffusion coefficients found by FRAP. The FRAP data are consistent with DLS results, and extend the trend in Deff/ D0 to higher λ. The consistency of FRAP results and small-K DLS data confirms that the extrapolation procedure to small K to obtain the long-time self-diffusion coefficient from DLS experiments is reliable. From Figure 6 we estimate that the “conductivity factor” (see eq 4) X h is in the range 0.8-1.0, which is consistent with the value X h ∼0.8 found by Karasz et al.3 for polystyrene spheres in these porous glasses. To determine the value for X h more precisely, systems with smaller λ are required. In case of our silica colloids, however, welldefined particles with a radius smaller than 10 nm are not available. Furthermore there are no porous glasses with larger pore radii than ca. 200 nm. With the combination of porous glasses and silica colloids it is not possible to obtain data in the regime where λ < 0.05. 5. Summary and Conclusions We have demonstrated the possibility to directly observe the Brownian motion of uncharged colloidal silica particles in an optically matched porous glass by means of DLS and FRAP, employing systematic variation of the pore

Bicontinuous Porous Glass Media

size and the size of the rigid, impenetrable diffusant. Charge interactions have been eliminated by grafting both particles and pore walls with a long chain alcohol. This steric stabilisation layer screens the van der Waals attraction, preventing the particles from sticking to the pore walls. In this way diffusion is only influenced by hydrodynamic and geometrical factors. Both DLS and FRAP results show a decrease of the reduced diffusion coefficient (Deff/D0) with increasing particle size. Despite some uncertainty with respect to the conductivity factor, we may conclude that the reduction of the diffusion coefficient is in reasonable agreement with the centerline model. In this model there is, in addition to hydrodynamic interactions, a steep repulsion when the hard sphere tracer contacts the hard wall of the cylinder. This complies with our silica spheres, which experience a steep short-range repulsion when they contact the walls of the porous glass; there is no experimental evidence of additional interactions due to surface charge or van der Waals attractions. So in this respect there is an analogy with the cylinder model. At the same time the agreement between our experiments and these models is surprising, because the pore geometry of the porous glass media (see Figure 1) clearly does not resemble a long straight cylinder.

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Nevertheless, both the absolute value and the λ dependence of Deff are reproduced fairly well. This agreement suggests that long-time self-diffusion is not very sensitive to the details of the pore structure. It would be interesting to measure or calculate, for example, tracer diffusion in a sphere packing and see if the size-dependent diffusion coefficient is very different from the results in Figure 6. In a follow-up study we will address the influence of charge interactions on the dynamics of colloidal particles in a porous glass. Charge interactions can be easily tuned, using different amounts of added electrolyte. Acknowledgment. Arnout Imhof is thanked gratefully for providing the fluorescent silica spheres and for his guidance with the FRAP experiments. Maarten Hagen is thanked for fruitfull discussions concerning this work. This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM) with financial support from the Netherlands Organization for Scientific Research (NWO). LA9701788