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Langmuir 1999, 15, 1896-1898
First in Situ Determination of Confined Brownian Tracer Motion in Dense Random Sphere Packings Sebastiaan G. J. M. Kluijtmans† 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 September 25, 1998. In Final Form: January 4, 1999 The long-time self-diffusion of fluorescent molecular and colloidal tracers in a dense random packing of large (nondiffusing) spheres has been studied for the first time in situ, using fluorescense recovery after photobleaching. The long-time tracer diffusion coefficient depends uniquely on the ratio β of the tracer size to the packing sphere size. This dependence is much stronger than that predicted by theories which neglect hydrodynamic interactions, emphasizing the importance of hydrodynamics on confined diffusion of finitesized tracers. Hydrodynamic friction can be described qualitatively by a mapping model of the pore space, which reproduces experimental diffusivities fairly well.
Introduction Random sphere packings are introduced here as model porous media to investigate hindered Brownian motion of colloids or molecules. The packings mimic porous structures such as sand beds, porous ceramics, and chromatographic columns, in which hindered diffusion is difficult to measure and to quantify. Theory on transport in sphere arrays dates back to Maxwell’s description of the electrical conductivity of such arrays1 and has only recently been extended to Brownian motion of finite-sized tracers in random packings of fixed spheres.2 One evident drawback of present theory is that confined Brownian motion is still treated as a purely geometrical problem, neglecting any hydrodynamic effect due to liquid in the pore space. This leaves one important issue unresolved, namely, the relative contribution of geometrical and hydrodynamic effects on diffusion in random sphere packings. However, no experimental data are available at present for comparison. Theory and experiments on other confining geometries such as cylinders,3 bicontinuous porous glasses,4,5 and fibrous porous media6-9 suggest that the effect of hydrodynamics must be substantial. In this Letter we report on the first quantitative, in situ determination of (colloidal) tracer diffusion in well-defined random sphere packings. Our experiments are directly comparable with present theory.2 We prepared various random sphere packings of volume fraction φ ∼ 0.6 containing equisized silica spheres with diameters in the range 0.7-5 µm (see Figure 1). Sphere packings were made optically transparent by immersing them in an isorefractive solvent (mixture). This allows the use of optical probing techniques to monitor in situ the Brownian motion of tracers in the liquid-filled voids between the spheres (see Figure 1, bottom). The tracer particles are nonad* Corresponding author. † Current address: Fuji Photo Film, Tilburg Research Laboratory, Oudenstaart 1, PO box 90156, 5000 LJ, Tilburg, The Netherlands. (1) Maxwell, J. C. Electricity and Magnetism; Clarendon: London, 1873. (2) Kim, I. C.; Torquato, S. J. Chem. Phys. 1992, 96, 1498. (3) Brenner, H.; Gaydos, L. J. J. Colloid Interface Sci. 1977, 58, 312. (4) Bishop, M. T.; Langley, K. H.; Karasz, F. E. Macromolecules 1989, 22, 122. (5) Kluijtmans, S. G. J. M.; Philipse, A. P.; Dhont, J. K. G. Langmuir 1997, 13, 4982. (6) Kluijtmans, S. G. J. M.; Koenderink, G. H.; Philipse, A. P. In preparation. (7) Cukier, R. I. Macromolecules 1984, 17, 252. (8) Altenberger, A. R.; Tirrell, M. J. Chem. Phys. 1984, 80, 220. (9) Phillies, G. D. J. J. Phys. Chem. 1989, 93, 5029.
Figure 1. Scanning electron micrographs (XLFEG30, 10 kV, 8 nm Pd/Pt coating) of a cross section of a dried packing of micrometer-sized silica spheres (volume fraction φ ∼ 0.6). The 20 µm scale bar (top) indicates the distance over which diffusion is monitored with FRAP (see Figure 2).
sorbing fluorescent dye molecules fluorescein isothiocyanate, and small spherical silica colloids of 40 nm radius in which the dye is incorporated in the cores of the particles.10 With the combination of dye molecules, silica tracers, and packing spheres, the ratio β of the tracer to packing sphere size can be varied from virtually zero to (10) van Blaaderen, A.; Vrij, A. Langmuir 1992, 8, 2921.
10.1021/la9813275 CCC: $18.00 © 1999 American Chemical Society Published on Web 02/24/1999
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Langmuir, Vol. 15, No. 6, 1999 1897
Table 1. Characteristics of Packing Spheres and Tracer Particles packing sphere SIP(1) SIP(2) BB10 BB14 BB19 BB25 AS30 BB50
RTEM (nm) 347 395 526 735 1075 1275 1675 2808
FS40 tracer 40.1 (DLS)
size refractive β polydispersity σ index ()rFS40/RTEM) 0.04 0.04 0.06 0.09 0.07 0.06 0.13 0.05
1.45/46 1.45/46 1.36 1.36 1.36 1.36 1.45/46 1.36
0.12
1.45
0.115 0.101 0.076 0.055 0.037 0.031 0.024 0.014
β ≈ 0.1. Using fluorescence recovery after photobleaching (FRAP),11-13 we measure the self-diffusion coefficient of the fluorescent tracers. Briefly, a fluorescence pattern is created in the pore space of the sphere packing by a short intense laser pulse, which bleaches fluorescent groups of the tracer. This bleaching pattern decays at a rate governed by the tracer self-diffusion coefficient D. Because FRAP measures diffusion over distances much larger than the pore spaces in the packing (typically the length of the scale bar in Figure 1, top), pore structure details are averaged and a long-time self-diffusion coefficient is obtained. Methods In this section we give an overview of particle synthesis and analysis, sample preparation, and measuring methods. Details are described in refs 11-13. Size, polydispersity, and refractive index of packing spheres and fluorescent tracer spheres FS40 are summarized in Table 1. Monodisperse silica packing spheres SIP(1) and SIP(2) of respectively 379 and 424 nm radius were synthesized by hydrolysis of tetraethoxysilane and subsequent polymerization in an alkaline ethanol-water mixture.14 BB silica spheres of 500 nm radius and larger were purchased as powders from Bangs Laboratories (Fishers, IN). Because these powders contain dumbbells and larger aggregates, they were purified by repeated sedimentation in N,N-dimethylformamide (DMF) retaining only the upper part of the sedimenting dispersion. Silica column packing material AS30 (Alltech, Deerfield, USA) needed no further purification Optically transparent sphere packings were prepared by centrifugal sedimentation of a suspension of silica spheres in an isorefractive solvent. For SIP(1), SIP(2), and AS30 with a refractive index of 1.46 the isorefractive solvent is a mixture of N,N-dimethylformamide (DMF) and dimethyl sulfoxide (DMSO). To optically match the BB spheres, which have a surprisingly low refractive index of 1.36, we used ethanol. All solvents had an electrolyte concentration of 5 × 10-3 M lithium chloride. The cross-section area of the resulting sphere packings in Figure 1 (top) shows a homogeneous, random arrangement of spheres without ordered regions. Fluorescent molecules or silica tracers are added to the liquid above the sediment and enter the sphere packing by diffusion. This diffusion process, which can be followed by measuring the fluorescence across the sample, takes several weeks to months. Fluorescent tracer spheres FS40 are silica spheres with a fluorescent core of silica-coupled fluorescein isothiocyanate (FITC) prepared following van Blaaderen.10 In the close-up of the packing (Figure 1, bottom) the silica tracer spheres, with a radius of 40 nm (obtained by dynamic light scattering), are visible as small dots. Due to capillary forces during drying of the electron microscopy sample, tracer particles accumulate in the narrow inclusions between touching spheres. (11) van Blaaderen, A.; Peetermans, J.; Maret, G.; Dhont, J. K. G. J. Chem. Phys. 1992, 96, 4591. (12) Imhof, A.; Dhont, J. K. G. Phys. Rev. E 1995, 52, 111. (13) Kluijtmans, S. G. J. M. Dynamics of colloids in porous media; Thesis, Utrecht University, Utrecht, The Netherlands, 1998. (14) Sto¨ber, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26, 62.
Figure 2. Long-time self-diffusion coefficient DS,L of fluorescent tracer spheres with radius βR in optically transparent, dense random packings (volume fractions ca. 0.6) of static micrometersized spheres with radius R. DS,L is normalized on the free particle diffusion coefficient D0 as obtained by FRAP measurements on dilute bulk solutions of tracer colloids. Purely geometric calculations2 (dotted line) severely overestimate the experimentally determined mobility for any finite tracer size (0). Note that for β > (2/3x3 - 1) ≈ 0.155 the tracers are completely immobilized; the tracers cannot escape from interstitial spaces in the sphere packing. The drawn lines are the product of the purely geometrical hindrance in the packing simulated by Hagen (dashed line) and the purely hydrodynamic hindrance of a tracer in a cylinder or a slit pore, with a diameter equal to the average pore size of the available pore space in the packing. The tracer volume fraction is only 0.01, so the tracers diffuse independently of each other. The solid volume fraction of the sediments was determined by preparing a packing with a dispersion of known volume fraction and measuring the height of sediment relative to the total height of the fluid in the vial. The volume fraction of the dispersion was measured by drying a certain volume of the dispersion and weighing the solid residue. For FRAP an argon laser with a wavelength of 488 nm (Spectra Physics series 2000) was used. The bleach time typically was 1 s with a laser power of about 100 mW. In the reading mode the fluorescence was monitored with a low-intensity laser beam (typically 100 times less intensity). FRAP measurements were performed in the long-time self-diffusion limit corresponding to scattering vectors smaller than 3 × 105 m-1. Ten FRAP curves were measured at different positions in the sediment. After baseline subtraction these FRAP curves were scaled and averaged. The long-time self-diffusion coefficient DS,L was obtained by a second-order cumulant fit to the resulting average curve. DS,L was normalized on D0, which was obtained in the same sample by measuring ten FRAP curves in the liquid above the sediment. Details of the FRAP setup are described in refs 11 and 12. Examples of FRAP curves and their evaluation are given in ref 13.
Results and Discussion Figure 2 summarizes the results of our diffusion measurements. We find that diffusion of confined tracers, irrespective of their size, is always retarded considerably
1898 Langmuir, Vol. 15, No. 6, 1999
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with respect to their free particle diffusion D0 in a bulk liquid. Even the mobility of the small molecular probe (β ≈ 0) is already reduced with 30% in the packing. Since hydrodynamic friction with surrounding pore walls is virtually absent for such a “point” particle, the reduction at β ≈ 0 must be entirely due to geometrical effects, like the volume occupied by the packing spheres and their mutual arrangement. For diffusion of point particles in a static arrangement of medium spheres with volume fraction φ, Torquato and co-workers15-17 derived
1 1 - φ - (1 - φ)ξ DS,L 2 -1 ) (1 - φ) 1 D0 1 1 + φ - (1 - φ)ξ 2 2
(1)
where ξ is a structural parameter accounting for the microstructure of the porous medium. Miller and Torquato16 showed that the first-order approximation ξ ) 0.21φ provides a reasonable estimate for ξ. Using the volume fraction of experimental sphere packings φ ) 0.6 in eq 1, we obtain DS,L/D0 ) 0.71, which agrees well with the experimental value DS,L/D0 ) 0.7 at β ≈ 0. These values are also close to the value of 0.67 found in recent computer simulations of Hagen18 for point-tracer diffusion in a random sphere packing of volume fraction φ ) 0.636. Kim and Torquato2,15 adapted eq 1 to finite-sized tracers using their isomorphism theorem. This theorem states that diffusion of a finite-sized tracer with size r in an array of medium spheres with size R and volume fraction φ is equal to diffusion of a point tracer in a medium of overlapping spheres with size r + R and volume fraction φ*. As a first approximation Kim and Torquato rescaled the volume fraction using φ* ) φ(1 + β)3, thereby neglecting sphere overlap. (Comparison with expressions from scaled particle theory19 indicates that this approximation is reasonable for r/R < 0.1.) As shown in Figure 2, however, experimental diffusivities of finite-sized tracers strongly deviate from this geometrical theory.2,15 Although Hagen’s computer simulations of hard sphere diffusion in the absence of hydrodynamics show a somewhat stronger dependency on the tracer size, they are also clearly not sufficient to account for the retarded Brownian motion in our experiments. This discrepancy between experiments on one hand and theory and simulations on the other is not due to any attractive (van der Waals) forces between tracers and pores walls. For the salt concentration used in our experiments (5 × 10-3 M LiCl), classical DLVO theory20 predicts a net repulsive double layer interaction between the negatively charged tracers and pore walls. Indeed we find no indication of flocculation in the samples for a time span of at least a year. Absence of permanently trapped tracer particles is also corroborated by the full decay of the fluorescence bleach pattern in the FRAP measurements. Besides attractive forces also long-ranged repulsions between tracer and pore walls could affect their mobility. Longrange double layer repulsions have been invoked to explain the very low mobility of charged tracers in porous media with charged pore walls at low ionic strength.21 However at the fairly high salt concentrations in this study electrical double layer repulsions only can have a minor effect. So in the absence of significant attractions or long-range repulsions, the strong reduction of the tracer mobility in (15) Torquato, S. J. Appl. Phys. 1985, 58, 3790. (16) Miller, C. A.; Torquato, S. J. Appl. Phys. 1990, 68, 5486. (17) Torquato, S.; Lado, F. Phys. Rev. B 1986, 33, 6428. (18) Hagen, M. H. J. Diffusion of colloidal particles in confined media; Thesis, Utrecht University, Utrecht, The Netherlands, 1997.
fluid-filled porous sphere packings as in Figure 2 must be due to hydrodynamic interaction between tracers and pore walls. In lieu of an appropriate theory we have estimated the hydrodynamic friction in a random sphere packing by a mapping model. This mapping model is based on calculating an average pore size of the packing, which is then used in theoretical expressions for hydrodynamic friction of spheres in model confining geometries, like a cylinder or a slit pore. First the pore size distribution of a computer-generated random packing is probed by inflating spherical volumes at random positions in the voids of the packing until a packing sphere is hit. This so-called first passage sphere technique was also employed by Torquato.22 For every tracer size inaccessible pores are excluded from the pore size distribution when calculating the average pore size. The average pore size is then used as input in models for diffusion in a slit pore and a cylinder, for which the hydrodynamic friction is a known function of the tracer to pore size ratio λ.23,24 For a slit pore, for example, we have24
D/D0 ) (1 - λ)-1[1 + (9/16)λ ln λ 1.19358λ + 0.159317λ3] λ e 0.5
(2)
Finally the resulting hydrodynamic friction factor is multiplied with a factor accounting for the geometrical hindrance of the packing. Herefore one can either choose eq 1 of Kim and Torquato or the simulation results of Hagen. We have selected the latter since eq 1 is known to fail at high effective volume fractions. It is seen in Figure 2 that the cylinder mapping agrees fairly well with the experiments, particularly at small size ratios. At larger size ratios, cylinder mapping overestimates hydrodynamic hindrance. The slit pore mapping seems to be somewhat better at large size ratios. The calculations with both model geometries merely illustrate that the hydrodynamic friction can be estimated with our mapping method. We conclude that Brownian tracer motion in sphere packings is much slower than that predicted by the present geometrical theory as a result of strong hydrodynamic friction. Thus to obtain reliable estimates of Brownian particle displacements of finite tracers in porous structures, one cannot simply account for the pore liquid by its viscosity as for diffusion in a bulk fluid. It is essential to take hydrodynamic forces between tracers and pore walls into account, for example, by mapping the available pore space for a tracer in a sphere packing on a liquid-filled cylinder or slit pore. Acknowledgment. Fluorescent particles were kindly provided by Dr. A. Imhof. Dr. M. Hagen, Professor D. Frenkel, and Professor J. L. Anderson are thanked for valuable discussions on the subject of this Letter. P. van Maurik is thanked for his contribution to Figure 1. 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). LA9813275 (19) Reiss, H.; Frisch, H. L.; Lebowitz, J. L. J. Chem. Phys. 1959, 31, 369. (20) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the stability of lyophobic colloids; Elsevier: Amsterdam, 1948. (21) Kluijtmans, S. G. J. M.; de Hoog, E. H. A.; Philipse, A. P. J. Chem. Phys. 1998, 108, 7469. (22) Torquato, S.; Yeong, C. L. Y. J. Chem. Phys. 1997, 106, 8814. (23) Bungay, P. M.; Brenner, H. Int. J. Multiphase Flow 1973, 1, 25. (24) Pawar, Y.; Anderson, J. L. Ind. Eng. Chem. Res. 1993, 32, 743.
