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Cake Collapse in Pressure Filtration† D. Antelmi,‡,§ B. Cabane,*,⊥ M. Meireles,# and P. Aimar# Equipe mixte CEA-RP, RHODIA, 93308 Aubervilliers Cedex, France; Research School of Chemistry, Institue of Advanced Studies, Australian National University, Acton, ACT 0200, Australia; Laboratoire PMMH, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France; and CNRS UMR 5503, Laboratoire de Ge´ nie Chimique, Universite´ Paul Sabatier, 31062 Toulouse Cedex 4, France Received March 26, 2001. In Final Form: July 31, 2001 Aqueous dispersions of latex particles have been aggregated by addition of Ca2+ ions and then filtered in a pressure filtration cell. The filtration cakes have been examined through small-angle neutron scattering, void volume fraction measurements, and permeate flux measurements. At all filtration pressures (20-400 kPa), the cakes were extensively collapsed (latex volume fractions φ ) 0.27-0.46), and the remaining porosity had a tenuous structure (fractal dimensions df ) 1.3-1). However, the local structures remained the same as in the aggregates of the original suspensions. The mechanisms that produce this collapse are made of very small relative motions of the particles, which leave the local coordination of the latex particles unchanged but allow large voids to be reduced. These motions could be inhibited by using particles with nonspherical shapes and by increasing the friction forces that act between particle surfaces, thereby reducing the extent of collapse and increasing the permeability of the cakes.
Introduction Dispersions of fine solid particles in a liquid are commonly made or encountered in manufacturing processes (e.g., ceramics, pigments, adhesives, paints, pharmaceuticals) as well as in biotechnology and biomedical applications (biofluids, including blood and whey, fermentation broths) and in oil recovery (drilling fluids). In many cases, it is at some point necessary to separate the particles from the liquid. This may be achieved through a variety of processes including sedimentation, centrifugation, membrane filtration, and drying. All of these processes have some inherent inefficiency, and none of them are totally satisfactory. Often, membrane filtration is one of the most practical options, particularly if the solid/liquid system cannot be heated. One of the main factors that determine the operating costs of membrane systems is the magnitude of permeate flux that can be achieved. When filtering dispersions of very fine particles, the magnitude of permeate flux can be severely limited by membrane fouling and cake formation, often rendering the technique economically unattractive.1 The formation of a cake results from accumulation of particles that are driven toward the membrane by the Stokes force, despite the interparticle forces and backdiffusion that tend to keep them apart. If the pressure applied to the fluid is high, the Stokes force will cause the particles to pack densely in the cake on top of the filter membrane. Such cakes have very small pores, and they reduce the flow rates to very low values.2-5 †
This work used the neutron beams of ILL. Equipe mixte CEA-RP, RHODIA. Australian National University. ⊥ Laboratoire PMMH, ESPCI. # Universite ´ Paul Sabatier. * To whom correspondence should be addressed. ‡ §
(1) Schweitzer, P. A., Ed. Handbook of Separation Techniques for Chemical Engineers, 3rd ed.; McGraw-Hill: New York, 1997. (2) Harmant, P.; Aimar, P. AIChE J. 1996, 42, 3523. (3) Bacchin, P.; Aimar, P.; Sanchez, V. AIChE J. 1995, 41, 368. (4) Benkhala, Y. K.; Ould-Ris, A.; Jaffrin, M. Y.; Si-Hassen, D. J. Membr. Sci. 1995, 98, 107. (5) McDonogh, R. M.; Fane, A. G.; Fell, C. J. D. J. Membr. Sci. 1989, 43, 69.
Higher flow rates may be achieved by aggregating the particles before filtration.6,7 Indeed, the packing of irregular aggregates in the cake leaves larger pores than a regular packing of particles that repel each other. The magnitude of the improvement may be estimated according to the expressions for the flow rates. Assume that a pressure difference ∆P is applied to the fluid that flows through the cake (not to the cake itself; otherwise, the experiment measures resistance to expression rather than filtration8). Then the flow rate dV/dt across the cake and membrane assembly is given by Darcy’s law:9
dV/dt ) A∆P/µ(Rm + Rc)
(1)
where A is the area of the membrane, µ the kinematic viscosity of the solvent, Rm the hydraulic resistance of the bare membrane, and Rc that of the filter cake. The cake resistance, Rc, can be used to define a specific resistance, R, according to the mass of deposited particles, M:
Rc ) RM/A
(2)
For a uniform cake, made of densely packed particles, the specific resistance R may be calculated from the Carman-Kozeny equation:10
R ) 180(1 - )/(FD23)
(3)
where is the pore volume fraction (on the order of 0.26 for densely packed particles), D the mean diameter of the particles, and F their density. If the particles are small (D in the nanometer range), the flow rates can be extremely slow, leaving little scope for enhancement. On the other hand, a cake made by the packing of fractal aggregates can have a larger effective particle size. Assume for (6) Waite, T. D.; Scha¨fer, A. I.; Fane, A. G.; Heuer, A. J. Colloid Interface Sci. 1999, 212, 264. (7) Waite, T. D. Colloids Surf. A 1999, 151, 27. (8) Sorensen, P. B.; Moldrup, P.; Hansen, J. A. A. Chem. Eng. Sci. 1996, 51, 967. (9) Kim, K. Y.; Chen, V.; Fane, A. G. J. Colloid Interface Sci. 1993, 155, 347. (10) Carman, P. C. Trans. Inst. Chem. Eng. 1938, 16, 168.
10.1021/la0104471 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/05/2001
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instance that each aggregate has a fractal dimension df and an overall diameter ξ. The packing of such aggregates in the cake yields an average volume fraction of particles, φ, which is7,11,12
φ ≈ (a/ξ)3-df
(4)
If the fractal dimension df is 2 or above, the corresponding aggregates would be essentially impermeable to the fluid, which will drain mostly between the aggregates. Hence, for a dense packing of the aggregates, the effective particle size put into the Carman-Kozeny equation should be ξ, and the porosity will correspond to the volume of voids between the aggregates, again of the order of ) 0.26. Taking for example df ) 2 and φ ) 0.1, the effective particle size ξ is 10 times the size of elementary particles, a. According to the Carman-Kozeny equation, the flux would then be 100 times higher than for a packing of nonaggregated particles. In some cases, the gains in flux obtained by preaggregating the particles are indeed spectacular. For instance, the filtration of mineral particles at pH values where they spontaneously aggregate (isoelectric point) may give flow rates that are 10 times faster than the filtration at pH values where they resist aggregation. Further gains may be obtained by producing aggregates that do not pack densely.6 In other cases, the gains obtained by preaggregating the particles are much less. The reason is that the cake collapses, despite the interparticle bonds that keep the particles aggregated.13-15 The collapse of filtration cakes is the most severe limitation to the efficiency of pressure filtration. To overcome this limitation, one must have detailed information on the processes by which the cakes collapse. Specific questions are as follows: (1) Does the whole cake collapse, or is there a thin collapsed layer next to the membrane that limits the flux and prevents densification of the rest of the cake? (2) What is the structure of the compressed cake, what are the pore sizes, and how are they connected? (3) Through what mechanisms did the cake collapse, e.g., rupture of interparticle bonds and interpenetration of aggregates, or continuous deformation of the aggregates leading to progressive collapse of the voids? (4) Can these mechanisms be controlled by changing the structures and the mechanical strengths of the aggregates? In the present work, we addressed these questions through a determination of the structures of filtration cakes at various stages of compression. We used smallangle neutron scattering (SANS) to measure the distribution of interparticle distances in the whole cake. A description of this technique, and of its use in the characterization of deposits, is found in our previous work on the filtration of clay suspensions.16 To facilitate the analysis of the results, we used a model system consisting of monodisperse, spherical polystyrene latex particles. These particles were preaggregated through addition of Ca2+ ions and filtered on “Anopore” alumina membranes. The solvent was a D2O/H2O mixture that canceled the scattering from the membrane, thereby making it possible (11) Cabane, B.; Dubois, M.; Lefaucheux, F.; Robert, M. C. J. NonCryst. Solids 1990, 119, 121. (12) Dubois, M.; Cabane, B. Macromolecules 1989, 22, 2526. (13) van de Ven, T. G. M. In Colloidal Hydrodynamics; Ottewill, R. H., Rowell, R. L., Eds.; Academic Press: London, 1989; p 351. (14) Tiller, F. M.; Kwon, J. H. AIChE J. 1998, 44, 2159. (15) Lee, D. J.; Ju, S. P.; Kwon, J. H.; Tiller, F. M. AIChE J. 2000, 46, 110. (16) Pignon, F.; Magnin, A.; Piau, J. M.; Cabane, B.; Aimar, P.; Meireles, M.; Lindner, P. J. Membr. Sci. 2000, 174, 189.
Figure 1. SEM micrographs showing the surface and cross section of a clean Anopore membrane with a nominal pore size of 20 nm.
to observe the undisturbed cake + membrane assembly. We obtained scattering curves for the original aggregated dispersion, for cakes made from this dispersion at different filtration pressures, and also for cakes made form dispersions of nonaggregated particles. We also measured the extent of cake collapse, through simple gravimetric methods, and changes in the hydraulic resistance of these cakes. The comparison of the microstructures determined through SANS with the overall characteristics of the cakes yields some of the structural information needed to answer the questions listed above. Experimental Section A. Materials. Latex Dispersions. The latex particles were made through emulsion polymerization of styrene (main component) with acrylic acid (1%), acrylamide (1%), and styrenesulfonate (0.5%); sodium dodecyl sulfate (SDS) was used as an emulsifier and persulfate as an initiator. The core was made of polystyrene (glass transition temperature Tg ) 100 °C, density 1050 kg/m3), and the surface contained the hydrophilic monomers, some sulfonate groups, and some SDS. The average size of the particles was 150 nm as measured from neutron scattering and electron microscopy. These dispersions are stabilized by electrostatic repulsions originating from the ionization of the particle surfaces. The dispersions of latex particles used in filtration experiments had a concentration of 0.5 wt %. Flocculated dispersions were prepared by diluting stock latex dispersion with 0.06 M CaCl2. At this salt concentration, the electrostatic interactions are effectively screened, and rapid coagulation takes place, forming large aggregates of the order of 10 µm as approximated from scanning electron microscopy. In these flocculated dispersions the overall latex concentration was also 0.5 wt %. Membranes. Whatman Anopore membranes with a nominal pore size of 20 nm and surface area of 1555 mm2 were used throughout this study. Figure 1 shows scanning electron microscopy (SEM) pictures of the Anopore membrane. The top view shows that the pores are regularly spaced, with a spacing
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of about 250 nm. This repetition shows up as a peak in the SANS scattering curves, when the membrane is not matched by the solvent. The side view of a broken piece of membrane shows that the pores are nearly straight. This feature is important, since it minimizes the internal fouling of the membrane during filtration by any particles that enter inside the pores. B. Methods. Filtration Experiments. Ultrafiltration experiments were performed in an unstirred ultrafiltration cell (Amicon 8050) of 80 mL capacity, consisting of a cylindrical section with two end pieces. The top piece was used to maintain the cell at a constant pressure, obtained from a regulated N2 supply. The bottom piece was used as a support for the membrane. Before each experiment, 200 mL of distilled water was passed through the membranes at 200 kPa to clean the membranes and ensure a complete wetting of the pores. For membranes to be used in the neutron scattering experiments, immediately prior to the experiment, two 50 mL volumes of D2O/H2O solvent (in the ratio 75:25, see below) were passed through the membrane to fill the pores with solvent at the contrast match composition. The experiments were conducted with cells completely filled with the dispersion, submitted to fluid pressures in the range 0-400 kPa at a temperature of 20 °C. During the filtration of the dispersion, the particles accumulated on the membrane to form a cake. The experiment was terminated, however, just before the level of the suspension reached the surface of the cake. The permeate flux was determined gravimetrically by weighting the permeate on an electronic balance at regular time intervals. The resistance of the membranes was determined by measuring the particle-free solvent flux before each experiment. It was observed that the membrane resistance to solvent containing 0.06 M CaCl2 was slightly lower (∼8%) than for Milli-Q water, presumably due to electroviscous effects.17 Latex Volume Fraction of the Cakes. The latex volume fraction of the cakes was determined in two ways. First, it was measured directly through gravimetric analysis at the end of filtration. Second, it was calculated using a simple mass balance between the latex content of the initial dispersion and that in the retentate, assuming complete retention of the particles. Agreement between the two methods was within 5-10%. In the direct method, the filter membranes were first weighed in their clean, dry state before being placed in the filtration cell. At the end of filtration, the wet filter membranes supporting the wet cake were blotted on absorbent paper to remove excess water. The wet membrane and deposits were then weighed before being placed in an oven at 120 °C overnight with dry N2 constantly circulating through the oven. The dry membranes with the dry deposit were then reweighed. The weights of the wet and dry cakes are then given by straightforward differences. (The amount of water trapped in the pores of the membrane was measured separately, for five separate membranes, and consistently found to be around 10% of the weight of the dry membrane. This was taken into account to calculate the mass of the wet cake on a wet membrane.) The weight of latex deposited on the membrane was also calculated from the known volume fraction in the initial dispersion and the weight of permeate that passed through the membrane during the filtration (measured gravimetrically). This analysis assumes that no latex passed through the membrane, which is well justified given that the pores (20 nm) were much smaller than the individual latex particles. The mass of latex in the cake determined in this manner was used to check against the gravimetric method described above. Finally, the measured weights of wet and dry cakes were used to calculate the latex volume fraction of the cake, Φ, using the densities known as 1.05 g/cm3 for latex and 1 g/cm3 for water. The volume fraction of the pores in the deposit is then
) 1 - Φ ) 1 - [Vlatex/(Vlatex + Vwater)]
(5)
Small-Angle Neutron Scattering. Small-angle neutron scattering is a technique that measures interference patterns produced by the spatial variations of a quantity called the “density of scattering length”, which is the equivalent, for neutrons, of an (17) Huismans, I. H.; Dutre´, B.; Persson, K. M.; Tra¨gårdh, G. Desalination 1997, 113, 95.
index of refraction.18,19 If the sample contains particles dispersed in a solvent, the scattered intensity is proportional to the square of the difference in density of scattering length between the particles and the solvent. The angular distribution of scattered neutrons is a Fourier transform of the distribution of distances between the particles. The resulting interference pattern consists of intensity values collected at various values of the scattering vector Q, which is the difference between the wave vectors of incident and scattered neutrons. For isotropic samples, this scattering pattern can be regrouped radially to give a scattering curve, where the intensity varies according to the magnitude Q of the scattering vector; Q is related to the scattering angle θ and to the wavelength λ by
Q ) (4π/λ) sin(θ/2)
(6)
The high Q part of the scattering curve corresponds to interferences between pairs of scattering centers that are quite close to each other, e.g., on the same particle. This part of the scattering curve is used to count how many particles contribute to the scattering. In practice, the scattering curves are scaled to give the same scattering intensity in this range, making it possible to compare the scattering from different structures with the same number of particles. The intermediate part of the scattering curves is controlled by interference between neighboring particles. This part of the scattering curves is used to determine what is the coordination shell of a particle; e.g., is it surrounded by a complete shell of neighbors at the contact distance, as in a crystal, or does it have a less dense environment? Finally, the low Q part of the scattering curves is controlled by interference between remote scattering centers. In this range, samples with a uniform particle density give no scattering because the contributions from individual particles cancel each other through destructive interference. Therefore, scattering in this range originates from large-scale fluctuations in the density of particles within the aggregates or within the cake. SANS was considered an ideal method to probe the structure of the filtration cakes since it allowed the cake to be left undisturbed on the membrane. Indeed, using the contrast match method, the solvent could be chosen to cancel the scattering from the membrane to obtain the scattering from the cake alone.19 The composition of the solvent (a D2O/H2O mixture) that matched, and thereby canceled, the scattering from Anopore membranes was determined in the following way. An Anopore membrane was loaded into the filtration unit and charged with solvent of specific D2O/H2O composition by passing about 80 mL of solvent through at 200 kPa two times, making sure that the surface of the membrane was always covered with solvent. A roughly rectangular piece of membrane was broken off and quickly placed in a quartz cell of 1 mm thickness filled with the appropriate solvent. The procedure was repeated for a range of solvents with varying D2O/H2O compositions. The scattered intensity produced by the neutron beam passing normally through the membrane was measured for each isotopic composition. The usual plot of the square root of intensity vs isotopic composition gave the contrast match point. This was found to be located at 75% D2O + 25% H2O, which is significantly lower than the contrast match point for pure alumina (92% D2O). The accuracy of the contrast match point determined in this way was assessed by examining the scattering from a membrane in the region of Q that corresponds to distances between neighboring pores of the membrane (250 nm). With solvents that are away from the match point, the membrane scattering has a very intense peak at the corresponding Q value. With solvents that are at the match point, this peak is completely extinct. Adsorption of a partial layer of latex particles on the membrane returns a weak peak at this location. For SANS experiments on the filtration cakes, the procedure was essentially the same. A latex dispersion made with a solvent of the same isotopic composition (75/25) was filtered on the Anopore membrane in the filtration cell while the amount of permeate was monitored as a function of time. At the completion (18) Champeney, D. C. Fourier Transforms and Their Physical Applications; Academic Press: New York, 1973. (19) Jacrot, B. Rep. Prog. Phys. 1976, 39, 911.
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this size was obtained from their sedimentation rate. Indeed, in H2O, these aggregates were seen to slowly sediment, due to their small density increment (∆F ) 50 kg/m3). For fractal aggregates of size R, the gravitational force is calculated from the number of particles per aggregate, N ) (R/a)df, and the volume per particle, V ) (4π/3)a3
Fg ) (R/a)df(4π/3)a3∆F
(7)
The frictional force is calculated from the aggregate radius, R, the aggregate velocity, u, and the viscosity of water, η ) 10-3 N m-2 s:
Fu ) 6πηRu Figure 2. Scattering curves of the aggregated dispersion (crosses) and of the cake made from this aggregated dispersion (circles) filtered at 20 kPa. of the filtration, the remaining retentate was gently swirled and tipped out of the filtration cell. A rectangular piece of membrane, with the cake on it, was broken off and placed in a quartz cell with a 1 mm path length. Some permeate was used to top up the quartz cell before closing it tightly. Great care was taken not to disturb the cake on top of the membrane surface. The quartz cells were then mounted on a holder for the scattering experiments.
Results Structures. In the following sections we present the small-angle neutron scattering curves obtained with the aggregated suspension, before filtration, and with the membrane + cake assembly, after filtration. Structures of the Aggregated Dispersions. The scattering curve of the original latex dispersion, aggregated by addition of Ca2+ ions, is shown in Figure 2. At high Q, the oscillations of the scattering curve reflect the size and shape of the latex particles. For monodisperse spheres these oscillations would be quite strong (the intensity would have a series of zeros), and for polydisperse particles they are damped. The observed oscillations reflect a weak polydispersity (standard deviation ) 10%). This is in agreement with SEM pictures of the latex dispersion on the membrane. At Q values that correspond to one or two particle diameters ((2-4) × 10-3 Å-1), the shape of the spectrum reflects correlations between neighboring particles. A structure where each particle was surrounded by a complete shell of neighbors (8-12 particles) would show a correlation peak at this location of the spectrum. The experimental scattering curve does not have a peak or an oscillation in this range of Q; hence, it may be concluded that each particle is in direct contact with a reduced number of particles. Thus, on a local scale, the aggregates are not dense. At small Q values, the scattering curve reflects largescale correlations in the distribution of matter within an aggregate. In this range of distances (150-1500 nm) the intensity follows a power law I(Q) ≈ Q-2.45. This power law (df ) 2.45) indicates that the aggregates have a “bushy” fractal structure, somewhat more dense than the structures of aggregates made by reaction limited cluster aggregation (df ) 2.2).7 The high value of the exponent indicates that some reordering has taken place after aggregation. The overall size of these aggregates may be estimated from the fact that they could barely be seem by eye; this indicates a size on the order of 10 µm. A confirmation of
(8)
In steady-state sedimentation, these forces are equal. For a sedimentation velocity u ) 10 mm/h, this yields sizes slightly in excess of 10 µm. Structures of Cakes Made at 20 kPa. The scattering curve of a wet filtration cake is presented in Figure 2. At high Q values, corresponding to intraparticle distances, the spectrum shows the usual oscillations and Q-4 decay due to intraparticle interferences, as in the original dispersion. Consequently, the intensity scales of the cake and of the dispersion were shifted so that the spectra superposed in this range of Q, corresponding to scattering by identical numbers of particles. At intermediate Q values, comparable to distances between neighboring particles, the spectra are also similar. Accordingly, the packing of particles in the cake does not modify the organization of particle-particle contacts. At small Q values, which correspond to large-scale fluctuations of the particle distribution, the intensity scattered by the cake is far below that scattered by the dispersion. Accordingly, the destructive interference is stronger in the cake than in the dispersion. This is caused, of course, by the denser packing of particles in the cake, which limits the magnitude of fluctuations in the particle density. This is a well-known effect for random packing of hard spheres20 or dense packing of fractal aggregates. At high volume fractions, the effect is so strong that the total scattered intensity is predicted to decrease when more particles are added (Figure 3). In this regime, it is easier (and equivalent, through Babinet’s theorem) to describe the scattering as produced by the voids that separate the particles. Accordingly, the power law observed at low Q for the intensity scattered by the cake reflects directly the fractal structure of the voids. (It also reflects the fractal structure of the aggregates, but then interference between neighboring aggregates must be taken into account.) Then, the value of the exponent, which is -1.33, implies that these pores are tenuous. Structures of Cakes Made at Pressures 120 and 400 kPa. Figure 4 shows the scattering curves of cakes made by filtering, at different pressures, identical amounts of aggregated latex dispersions. At high Q values, in the Q range where the scattering reflects interference within a particle, the raw spectra are identical, as expected for samples containing the same amount of latex particles. At intermediate Q values, in the range corresponding to distances between neighboring particles, the spectra are also similar. The absence of a scattering peak indicates that the particles still do not have complete coordination shells. Finally, at small Q values, all spectra are power laws of Q, with exponents that depend on the applied pressure. (20) Ashcroft, N. W.; Lekner, J. Phys. Rev. 1966, 145, 83.
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Figure 3. Intensity scattered by a hard sphere liquid, calculated through the Percus-Yevik equation.20 At volume fractions in excess of φ ) 0.12, interparticle interferences depress the scattered intensity so much that the total intensity decreases upon adding more particles into the liquid.
Figure 4. Scattering curves of cakes made from aggregated dispersions filtered at 20 (highest curve), 100, 200, and 400 kPa (lowest curve).
The evolution of the spectra is interesting: the application of higher pressures produced cakes that gave a weaker scattering, with an exponent of lower magnitude. Since the volume fraction of particles in the cake had increased, a weaker intensity must result from stronger destructive interference between particles. As indicated above, in this regime of strong interparticle interference, it is easier and equivalent to describe the intensity in terms of the voids that separate the particles. Accordingly, the variation of the exponent, from -1.33 to -1, indicates that the voids become still more tenuous as the pressure is increased. Structures of Cakes Made from Nonaggregated Dispersions. A remarkable feature of the cakes made from aggregated dispersions is that they retain a set of tenuous voids. Thus, they do not collapse to a fully dense, ordered structure, within the range of applied pressures studied here. To confirm that this set of tenuous voids is a memory of the structure of the original aggregates, we also made filtration cakes deposited from nonaggregated dispersions. The filtration was performed using a nonaggregated latex dispersion, with NaCl added rather than CaCl2. In
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Figure 5. Scattering curves of a cake made from an aggregated dispersion filtered at 400 kPa (triangles) and of a cake made from a nonaggregated dispersion, filtered at 200 kPa and washed by a CaCl2 solution in order to lock the structure (stars).
these conditions, the latex particles do not adhere upon mutual contact, and normally, the cake that forms on the membrane redisperses spontaneously as soon as the flow is stopped. To counteract the redispersion, a small amount of CaCl2 solution was fed into the filtration cell under the same pressure difference at the end of the filtration. The Ca2+ ions formed bonds between the particle surfaces in the cake and locked them into their positions, preventing redispersion. Although this technique does not give an ideal representation of the structure that a cake comprised of repelling particles would have, it does provide a useful contrast to the cakes formed from preaggregated particles. Once again, the intensity scales have been adjusted to give the same intensities at high Q, corresponding to the same numbers of particles. The spectrum of the cake made from nonaggregated particles shows a peak corresponding to the distance between neighboring particles, indicating that each particle is surrounded by a complete shell of neighbors (Figure 5). At lower Q values, the intensity given by cake is depressed. This depression is caused by strong destructive interference at scales beyond the distances between first neighbors, indicating that the cake is homogeneous at those scales. The spectrum of the cake made from an aggregated dispersion differs by the excess scattering at low Q, caused by heterogeneities (voids) with sizes ranging from 1 to 10 particle diameters. At this stage, the microstructures of the filter cakes may be described as follows. Filter cakes made by packing repelling particles have a homogeneous structure (few large-scale fluctuations of the particle density) with a strong short-range order, as expected. Filter cakes made by packing preaggregated particles show large-scale fluctuations of the particle density. In comparison with the density fluctuations of the original aggregates, the density fluctuations of the filter cakes are strongly attenuated. These density fluctuations are spatially selfsimilar, with fractal dimensions in the range 1-1.33, characteristic of tenuous objects such as pores with low connectivity. Overall Characteristics of Cakes. In the following sections we present the overall characteristics of cakes formed at different pressures in terms of void fraction (or latex volume fraction) and hydraulic resistance. These overall characteristics were evaluated independently. The
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Figure 6. Overall collapse of filter cakes. Vertical scale: average volume fraction of latex particles in the filter cake, from gravimetric measurements. Horizontal scale: pressure applied to the supernatant fluid.
cake void fraction was evaluated both from gravimetric experiments and from a mass balance, as explained in “Methods”. The cake hydraulic resistance was evaluated from filtration experiments, also in two independent ways. Void volume fraction and cake hydraulic resistance measurements indicate a strong evolution of the cake with the applied pressure. Latex Volume Fraction. The values of the latex volume fraction φ are presented in Figure 6. They show two successive stages for the packing of latex aggregates (flocs) in the cake under the effect of applied pressure: (i) Application of moderate pressures (0-20 kPa) is enough to pack loose flocs into a fairly dense cake (φ ) 0.27). This is achieved through collapse of the spaces between flocs and also of the largest voids within the flocs. In this regime, the dispersion is still highly compressible. (ii) Application of much higher pressures (up to 400 kPa) produces a (modest) further densification of the cake (up to φ ) 0.46). This is achieved through further collapse of voids in the cake. Since the increase in volume fraction achieved at this stage is comparatively small, the cake may be considered as practically consolidated. This is the point where the compressive yield stress of the material diverges with further increases in volume fraction.21,22 These volume fractions are averages over the whole volume of the cake. Of course, the local solid pressures and volume fractions must vary according to the depth z within the cake. However, it is obvious from the variation shown in Figure 6 that cakes made at high pressures must be collapsed over most of the cake height. Consider, for instance, a cake made at a total pressure difference ∆P ) 400 kPa. At each depth in this cake, the pressure exerted on the particles is the sum of hydrodynamic forces exerted by the permeating fluid on all particles located above this level. At the bottom of the cake, this pressure is the highest, and it equals the applied pressure, i.e., 400 kPa, leading to full collapse at the bottom of the cake. Strong collapse must also occur at other depths in the cake, as long as these layers support the forces transmitted by a substantial fraction of the cake mass. The only part of the cake that is submitted to pressures less than 20 kPa is the top layer, which, according to the pressure gradient, may contain no more than 5% of the cake mass. According to this pressure, this top layer may have a volume fraction on the order of 0.27. Therefore, the cake is collapsed over (21) Channel, G. M.; Zukoski, C. F. AIChE J. 1997, 43, 1700. (22) Channel, G. M.; Miller, K. T.; Zukoski, C. F. AIChE J. 2000, 46, 72.
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Figure 7. Specific resistance of cakes made at different pressures, calculated from the final permeation rate and final weight of the cake (filled diamonds), or from the variation of the permeation rate during the course of filtration (circles).
most of its height. This is in agreement with the results of cake dissection experiments, performed by Meeten:23 filtration cakes made from polystyrene latex had a uniform volume fraction profile at φ ) 0.55, whereas filtration cakes made from kaolinite had a continuous gradient of volume fraction through the cake. Hydraulic Resistance of the Cakes. The specific resistance of the cakes was determined from the flow rate measured at the end of the filtration (hydraulic resistance of the cake + membrane assembly), the flow rate measured before filtration (membrane resistance), and the mass of cake deposited on the membrane, according to eqs 1 and 2. These resistances are rather high (1013-1014 m/kg), compared with those of cakes made of other colloidal particles. For instance, the specific resistances measured for aggregated hematite suspensions submitted to similar pressures are in the range 1012-1013 m/kg.6 Since the individual particles have about the same sizes, the higher values found in the present case must reflect a stronger collapse of the cakes. The variation of the specific resistance with pressure (Figure 7) follows the law
R ) R0Pn
(9)
where the exponent n is 0.5 for both sets of data, and the prefactor R0 is 5 × 1013 m/kg. This increase in specific resistance must result from a corresponding increase in volume fraction throughout the cake. Here, a 4-fold increase in resistance results from a rise in average volume fraction from φ ) 0.27 to φ ) 0.46. Thus, the variation of resistance with volume fraction must be quite fast in this range. This is indeed expected for nearly dense packings of colloidal particles, as shown by eq 3. The flow rates measured during the formation of the cake can also be used to calculate the specific resistance of these intermediate states. A common observation in pressure filtration experiments where the membrane resistance is comparatively small is that the volume of permeate grows as a square root function of time.24 There is a pedestrian explanation for this behavior, which assumes that particles are deposited on the cake at a rate determined by the flow of permeate (i.e., there is no gravitational sedimentation and no diffusion) and that (23) Meeten, G. H. Chem. Eng. Sci. 1993, 13, 2391. (24) Langman, K. A.; White, L. R. AIChE J. 1995, 41, 1687.
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Figure 8. Unstirred filtration of 0.5 wt % dispersions of latex aggregates in a solvent of 75% D2O + 25% H2O at different applied pressures plotted according to the cake filtration model (eq 11). The linear behavior indicates that the cake grows at constant specific resistance.
the cake grows at a constant specific resistance which solely arises from Darcyan flow of the fluid relative to the particles in the immobilized bed. Indeed, the mass of deposited latex, M, used in eq 2, is related to the volume of permeate, V(t), and to the mass concentration W of latex in the dispersion:
M(t) ) V(t)W
(10)
Consequently, the filtration data can be expressed in a way that determines the average specific resistance R:
dt ) (µRm/A∆P) dV + (µRW/A2∆P)V(t) dV (11) If the specific resistance R remains constant during the growth of the cake, this can be integrated to give a filtration law with only two parameters, the initial flux, Q0, and the specific resistance, R:
t/V(t) ) (1/Q0) + (µRW/2A2∆P)V(t)
(12)
Thus, if the conditions stated above are met, then t/V(t) must be a linear function of V(t).9 Figure 8 presents the filtration results, plotted in this way, for suspensions of latex aggregates in the D2O/H2O mixtures. In this case, there was no sedimentation, and the usual linear behavior was observed. The specific resistance, calculated from the slope of these lines, is close to that determined from the permeation rate of the final cake. This suggests that the cake grows at a nearly constant specific resistance. At this stage, the macroscopic features of the filter cakes made with preaggregated latex particles may be summarized as follows. All cakes are fairly dense, even those made at rather low pressures. This dense packing produces a high resistance to permeation. Thus, the gain in permeability obtained from preaggregating the particles is almost totally lost through the mechanisms that lead to cake collapse. Discussion The aim of this work was to shed some light on the mechanisms by which filter cakes may collapse under applied pressure. The term “collapse”, in the present context, is defined as the processes by which a dispersion of bushy aggregates is compressed into a dense cake. The term “mechanisms” is defined as the set of particle motions
that occur under the effect of applied pressure and that produce this collapse. The link between the observed collapse and the mechanisms that make it possible will be examined in four steps: (a) What is the extent of collapse? (b) At what spatial scale does collapse take place? (c) Which type of particle motions make it possible? (d) Which properties of the particles may favor or inhibit these motions? What Is the Extent of Collapse? There is no doubt that collapse, or some type of densification, has occurred. The initial dispersion contained bushy aggregates that had a fractal exponent df ) 2.45 and overall dimensions ξ in excess of 10 µm. Simple packing of these aggregates, as in a sedimentation experiment, would give a volume fraction φ ≈ (a/ξ)3-df ) 0.1. In the filtration experiment, the applied pressure caused these aggregates to pack at a higher volume fraction. At the lowest pressure (20 kPa) the overall volume fraction of latex in the cake was already φ ≈ 0.27, and at the highest pressure (400 kPa) it went up to φ ) 0.46. Accordingly, these cakes are of the “supercompactible” type. The distributions of volume fraction of solids throughout such cakes have been measured by Meeten23 and discussed by Tiller et al. and Lee et al.25,26 At What Spatial Scales Does the Collapse Take Place? The macroscopic collapse of the cakes must originate from structural rearrangements that take place at shorter spatial scales. Changes in the local structure (i.e., the coordination shell of each latex particle) show up in the scattering curves at Q values that match the distances between nearest neighbors (3 × 10-3 < Q < 6 × 10-3 Å-1). In this range, cakes made from nonaggregated dispersions show a diffraction peak (Figure 5), indicating that each particle has a complete coordination shell and therefore that the local structure is dense and regular. However, cakes made from preaggregated suspensions do not produce this peak (Figure 5); instead, their scattering curves are identical, in this Q range, with that of the original aggregated suspension (Figure 2). Therefore, these cakes have retained the same local structures, with incomplete coordination shells, as the original aggregates. Thus, the processes that caused the collapse must have taken place at larger spatial scales. At large spatial scales, the collapse is evident. Indeed, at Q values corresponding to large groups of particles (Q < 3 × 10-3), the intensities scattered by the cakes are systematically lower than that of the original dispersion (Figure 2). Moreover, the application of higher pressures makes them lower still (Figure 4). This depression of the intensity reflects the suppression of fluctuations of the particle concentration, such as alternations of lumps and voids. The remaining voids are tenuous, as indicated by the fractal exponent df, which is 1.3 at lower pressures and 1 at the highest pressure. This interpretation states that the large-scale porosity is collapsed throughout the cake. At this point, it is instructive to examine the alternative interpretation, according to which the shape of the scattering curves would be related to a spatial gradient in the cake. Assume that the bottom part of the cake were fully collapsed, with a dense ordered structure similar to that of the cake made from a nonaggregated dispersion (Figure 5). Assume also that the top part of the cake had resisted the collapse and retained the same aggregate structure as the original suspension. Then the scattering curve of the whole cake (25) Tiller, F. M.; Kwon, J. H. AIChE J. 1999, 44, 2159. (26) Lee, D. J.; Ju, S. P.; Kwon, J. H.; Tiller, F. M. AIChE J. 2000, 46, 110.
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would be a linear combination of both scattering curves. This combination would show a steep decay at very low Q values (Q-2.45 power law in Figure 2) and a peak at the nearest-neighbor distance (scattering curve of the cake made from nonaggregated particles, shown in Figure 5). None of these features are observed in the experimental scattering curves of the cakes. Thus, the conclusion is that, throughout the cake, the local structure has remained the same as in the original suspension, while the largescale porosity has collapsed and left only tenuous voids. What Types of Particle Motions Make This Collapse Possible? The deformations that cause the collapse must result from relative motions of particles that are in contact. These motions must have small amplitudes, since the local structures remain essentially unchanged. Thus, particles that are in contact are displaced or rotated by small amounts, and these relative motions accumulate to produce large-amplitude continuous motions of structural elements such as clusters or voids. Why Does the Cake Collapse in This Way? A continuous deformation, such as that postulated in the mechanism described above, requires that the particles glide or rotate around each other. The main factor that favors this type of deformation is the fact that the latex particles are spherical. Indeed, simple rotation of one particle around another may be enough to bend a branch of an aggregate. Only friction forces that tend to maintain the established interparticle contacts oppose these rotations. In the latex aggregates that we have examined, the friction forces originate from Ca2+ ions that bind the particle surfaces together. Since the aggregates collapse under moderate pressures, the yield stress of these bonds must be low compared to the forces applied to particle surfaces. More specifically, the yield stress must be low compared to forces applied to large voids (these are collapsed efficiently), but not compared to forces applied to the smallest voids (since some porosity remains in the cakes).
Antelmi et al.
How Can This Collapse Be Controlled? In practice, two main factors should limit the collapse of filter cakes. First, aggregates of nonspherical particles, containing facets and edges, would resist collapse better, because the relative displacements of particles would require shear of interparticle contacts instead of pure rotations. Alternatively, collapse and densification will be limited if the friction forces between particles are high, as dictated by the strength of interparticle bonds. Enhancing either of these two effects should stabilize a cake to pressure increases. Conclusions Some aggregated suspensions, when filtered, form “supercompactible” cakes. The mechanism by which such cakes collapse and consolidate involves very small relative displacements of particles that are in contact with each other. These motions leave the local coordination of each particle unchanged; however, the continuous deformations that result from the accumulation of such motions cause all large voids to collapse. The remaining voids are tenuous, and the flux of permeate though the consolidated cakes is reduced to very low values. The suspensions that form these cakes are made of spherical particles, held together by physical forces. The collapse of these cakes is easy, because the friction forces that oppose relative motions are low. Control of this collapse may be achieved by increasing the friction forces that act on interparticle contacts or by using nonspherical particles. This would provide a way to maintain a low hydraulic resistance in the cakes. Acknowledgment. We thank O. Aguerre-Chariol for the scanning electron micrographs and S. Egelhaf and P. Lindner for help and useful suggestions with the neutron scattering experiments. LA0104471
