Multilayer Deposition of Stable Colloidal Particles during Flow within

Langmuir 1998, 14, 4435-4444

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Multilayer Deposition of Stable Colloidal Particles during Flow within Cylindrical Pores Venkatachalam Ramachandran and H. Scott Fogler* Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109 Received November 4, 1997. In Final Form: May 18, 1998 Multilayer deposition of stable colloidal particles during the low Reynolds number flow of aqueous suspensions of latex particles through cylindrical pores in track-etched membranes has been experimentally studied. Though multilayer deposition or the role of deposited particles as additional collectors has been recognized before this work, the reason multilayer deposition of particles occurs in the presence of strong interparticle repulsion and the interactions governing the transition from single layer to multilayer deposition have not been elucidated. Because sufficient repulsion exists between particles in a stable suspension to prevent their aggregation by Brownian flocculation, the deposition of particles on top of previously deposited particles to form multilayers is flow-induced. In this paper, we first show that only a single layer of particles will be deposited on the pore surface if strong repulsion exists between particles. We then demonstrate the transition from single layer to multilayer deposition at a sufficiently high flow rate when the interparticle repulsion is reduced by adding an electrolyte (but not exceeding the critical flocculation concentration). Under these conditions, the hydrodynamic force acting on a flowing particle in the vicinity of a deposited particle can overcome the net interparticle repulsion and multilayers of particles can accumulate within the pores. Approximate calculations of the trajectory of a flowing particle in the vicinity of a stationary particle that qualitatively demonstrate the flow-induced nature of the phenomenon are also presented.

Introduction The flow behavior of colloidal particles suspended in a liquid through a porous medium is of fundamental importance to a wide variety of natural and engineering processes. Examples include water and wastewater treatment using deep-bed filters, membrane microfiltration of biomaterials such as proteins and microorganisms, fines migration in oil reservoirs, and the transport of groundwater colloids. “Flow behavior” refers to how the suspended particles are retained or captured within the porous medium and how particle retention relates to macroscopic, measurable quantities such as the effluent concentration and the hydraulic conductivity of the porous medium. A clear understanding of the particle flow behavior is imperative for the optimal design and control of these processes. We will first discuss the three main mechanisms by which particles can be retained during suspension flow through porous media, namely, straining, deposition, and hydrodynamic bridging. Straining or size exclusion refers to the capture of a particle at a pore constriction that is smaller in size than the particle. If the particles are smaller in size than the pores, they can penetrate the porous medium. While flowing within the individual pores, particles can be deposited on the pore surface by different mechanisms such as inertial impaction, interception, Brownian diffusion, and sedimentation.1 In capture by inertial impaction, inertia causes particles to deviate from fluid streamlines and leads to deposition on the pore surface. Inertial impaction is an important mechanism of particle deposition from aerosols. When a particle travels along a streamline that is less than a particle radius away from the collector surface, it can be retained due to interception. For submicrometer particles, Brownian diffusion can play a significant role in particle (1) Tien, C. Granular Filtration of Aerosols and Hydrosols; Butterworth Publishers: Woburn, MA, 1989.

transport to the pore surface. Under conditions when gravity forces are significant (large and/or dense particles), particles can also be retained by sedimentation. The efficiency of particle deposition by these mechanisms is significantly affected by surface forces such as dispersion and electrostatic forces and solvation forces. Hydrodynamic bridging is the phenomenon of blocking of pores by stable particles, whose sizes are smaller than the pore size, arriving simultaneously at the pore entrance.2 The term “stable” refers to the particles in the suspension remaining dispersed and not aggregating as a result of Brownian motion. At a sufficiently high flow velocity in the low Reynolds number regime, hydrodynamic forces at the pore entrance can overcome interparticle and particlepore surface electrostatic repulsion resulting in the formation of a particle bridge across the pore constriction. Particle retention by bridging can therefore occur in the absence of straining and deposition. The situation of interest in this paper is the retention or filtration of stable colloidal particles from a dilute, liquid suspension during low Reynolds number flow through porous media. Under these conditions, if the particle size is much smaller than the average pore size in the porous medium, then deposition is the only mechanism by which significant retention of particles can occur. Many aspects of the deposition process have been intensively studied both theoretically and experimentally in the last three decades. Studies have been conducted using model deposition systems such as the stagnation point flow system, the rotating disk apparatus, and the parallel plate channel.3,4 With these systems, fundamental aspects of the deposition process such as the rate of deposition and the spatial distribution of the deposited particles have (2) Ramachandran, V. and Fogler, H. S. Submitted for publication. (3) van De Ven, T. G. M. Colloidal Hydrodynamics; Academic Press: San Diego, CA, 1989. (4) Elimelech, M.; Gregory, J.; Jia, X.; Williams, R. Particle Deposition and Aggregation. Measurement, Modeling and Simulation; ButterworthHienemann: Woburn, MA, 1995.

S0743-7463(97)01207-9 CCC: $15.00 © 1998 American Chemical Society Published on Web 07/17/1998

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been directly determined and compared with theory. Deposition has also been studied in packed bed systems where the focus includes macroscopic effects such as the change in effluent particle concentration and the bed permeability.1,4 The extent of deposition generally depends on factors such as the particle and pore surface properties and sizes, the suspension physicochemical conditions, the flow geometry, and the flow intensity. During deposition from a stable colloidal suspension, usually a single layer of particles is deposited on the wall. The strong interparticle electrostatic repulsion prevents multilayer deposition. Another feature of deposition from a stable suspension is that monolayer particle coverage (with particles touching each other) of the surface is not possible because of hydrodynamic and colloidal interactions between the flowing and deposited particles.5 During deposition from unstable suspensions, the formation of multilayers of particle deposits is possible because net attraction exists between particles. The presence of irregular surface aggregates during deposition from an unstable suspension on a rotating disk collector has been observed by Marshall and Kitchener6 and Ryde et al.7 have studied multilayer particle deposition from suspensions that are marginally stable or unstable during flow through packed columns. Physical and chemical heterogeneities on the particle and collector surfaces also can significantly affect the deposition process especially when unfavorable particle-wall interactions exist. Their effect on the initial deposition rate during suspension flow through packed beds of glass beads has been modeled by Vaidyanathan and Tien8 and Song and Elimelech.9 Elimelech et al.4 have summarized and reviewed the different deposition studies under conditions of net particle-wall repulsion. The role of deposited particles as additional collectors has been recognized prior to this work. During aerosol filtration, the formation of dendritic structures due to inertial impaction and interception of flowing particles on previously deposited particles has been observed10 and theoretically modeled.11 However, such dendrites cannot form during the low Reynolds number flow of stable hydrosols because of negligible particle inertia and strong interparticle electrostatic repulsion. In granular filtration of hydrosols, the term “filter ripening” refers to the filter performance improving with time and is attributed to deposition on previously deposited particles. Phenomenological models developed to describe such flow behavior are discussed by Tien1 and can also be found in works by Privman et al.12 and Song and Elimelech.13 However, these studies cannot explain why multilayer deposition (5) Dabros, T.; van de Ven, T. G. M. Surface Collisions in a Viscous Fluid. J. Colloid Interface Sci. 1992, 149 (2), 493. (6) Marshall, J. K.; Kitchener, J. A. The Deposition of Colloidal Particles on Smooth Solids. J. Colloid Interface Sci. 1966, 22, 342. (7) Ryde, N, N.; Kallay, Matijevic, E. Particle Adhesion in Model Systems Part 14.-Experimental Evaluation of Multilayer Deposition. J. Chem. Soc., Faraday Trans. 1991, 87 (9), 1371. (8) Vaidyanathan, R.; Tien, C. Hydrosol Deposition in Granular Media under Unfavorable Surface Conditions. Chem. Eng. Sci. 1991, 46 (4), 967. (9) Song, L.; Elimelech, M. Transient Deposition of Colloidal Particles in Heterogeneous Porous Media. J. Colloid Interface Sci. 1994, 167, 301. (10) Davis, C. N. Air Filtration; Academic Press: New York, 1973. (11) Payatakes, A. C.; Gradon, L. Dendritic Deposition of Aerosol Particles in Fibrous Media by Inertial Impaction and Interception. Chem. Eng. Sci. 1980, 35, 1083. (12) Privman, V.; Frisch, H. L.; Ryde, N.; Matijevic, E. Particle Adhesion in Model Systems Part 13.-Theory of Multilayer Deposition. J. Chem. Soc., Faraday Trans. 1991, 87 (9), 1371. (13) Song, L.; Elimelech, M. Dynamics of Colloid Deposition in Porous Media: Modeling the Role of Retained Particles. Colloids Surf. 1993, 73, 49.

Ramachandran and Fogler

of particles from a stable suspension occurs nor can they delineate the conditions for the transition from single layer to multilayer deposition. These issues are the focus of this study. A flowing particle can deposit on a previously deposited particle if the hydrodynamic force acting on it in the vicinity of the deposited particle can overcome the interparticle colloidal repulsion. The onset of multilayer deposition is therefore governed by the competition between the hydrodynamic force acting on the flowing particle and the interparticle colloidal repulsion. For a given particle and pore system, suspension ionic strength, and pH, there exists a critical velocity below which flowinduced multilayer deposition will not occur. The phenomenon of flow-induced deposition of particles on previously deposited particles is conceptually similar to orthokinetic flocculation of stable particles in the bulk. The build up of multilayers of particles in pore constrictions during suspension flow through a porous medium can lead to drastic reduction in the permeability of the porous medium.14-16 Such severe pore clogging is undesirable in applications such as water injection for oil recovery from reservoirs and water and wastewater treatment using deep bed filters. It is therefore crucial that the nature of deposition for a given system be well understood. We have used a model experimental system of latex colloidal particles and track-etched membrane with cylindrical pores to study multilayer deposition. The first set of experiments studies the effect of velocity on deposition during the flow of particles having a surface charge opposite in sign to that of the pore surface. In these experiments, only single layer deposition of particles was observed. In the next set of experiments, particles having like charge as the pore surface are flowed through the membrane. Upon a gradual increase of the suspension ionic strength while staying well below the critical flocculation concentration, the transition from single layer to multilayer deposition is observed at sufficiently high flow rates. In all the experiments, the Reynolds number for suspension flow within pores based on the particle diameter and the interstitial velocity is very small (Re < 0.01). Approximate theoretical calculations of the trajectory of a flowing particle in the vicinity of a stationary particle demonstrating the flow-induced nature of the phenomenon are presented in the last section. Experimental System Porous Medium. Nuclepore track-etched polycarbonate membranes having straight cylindrical pores were obtained from Corning Costar Corp. (Acton, MA). Figure 1 shows a scanning electron micrograph of the membrane. The porous structure in track-etched materials is obtained by preferentially etching tracks created by the passage of heavily ionizing, nuclear particles.17 These tracks are characterized by intense damage on an atomic scale and can be etched by a properly chosen chemical reagent. Circular membranes of size 25 mm and nominal pore sizes 1, 2, and 3 µm were used. The membrane thickness specified by the manufacturer is 10 µm. Examination of the membrane surface using the scanning electron microscope (SEM) revealed that the pore structure is not entirely regular. There is a distribution of (14) Muecke, T. W. Formation Fines and Factors Controlling their Movement in Porous Media. J. Pet. Technol. 1979, Feb, 144. (15) Vitthal, S.; Sharma, M. M. A Stokesian Dynamics Model for Particle Deposition and Bridging in Granular Media. J. Colloid Interface Sci. 1992, 153 (2), 314. (16) Choo, C.; Tien, C. Simulation of Hydrosol Deposition in Granular Media. AIChE J. 1995, 41 (6), 1426. (17) Fleischer, R. L.; Price, P. B.; Walker, R. M. Nuclear Tracks in Solids. Principles and Applications; University of California Press: Berkeley, CA, 1975.

Multilayer Deposition of Stable Colloidal Particles

Figure 1. SEM picture of 2 µm pores in a Nuclepore tracketched membrane (entrance face of the membrane shown at ×5K).

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Figure 2. TEM picture showing 0.25 µm latex particles (×50K).

Table 1. Pore Sizes in Nucleopore Membranes Used nominal pore size (µm)

measured pore size (µm)

% std dev

2 3

1.65 2.76

13.7 15.5

pore sizes together with the presence of a significant number of overlapping pores (∼14%). Table 1 lists the two nominal pore sizes used and the measured average pore size and standard deviation determined from SEM micrographs. Similar observations for Nuclepore membranes have been reported in the literature.18,19 It was also evident from the micrographs that the pores are not perfectly straight. The maximum deviation of the pore axis from the normal to the membrane surface is reported by the manufacturer to be 35°. For a membrane thickness of 10 µm, this deviation implies that interconnection between pores will be present in the membrane. Martinez-Villa et al.19 have reported that the pore number densities determined for the two sides of a membrane are not significantly different. This observation indicates that the number of pores that do not penetrate the entire thickness of the membrane is very small. Keesom et al.20 have measured the pore surface zeta potential using the streaming potential technique. They report that the pore surface is negatively charged due to the dissociation of carboxyl groups on the surface. The maximum value of the pore zeta potential is about -27 mV for pH values above 6 and an ionic strength of 10-3 M KCl. Particles. Charged polystyrene microspheres dispersed in water were used in the experiments. Strong interparticle electrostatic repulsion arising from the interaction of charged groups present on the surface of particles imparts stability to the suspension. Figure 2 shows a micrograph of particles used. Table 2 lists the latex particles used and their properties. Concentrated particle suspensions were purchased from Interfacial Dynamics Corp. (Portland, OR) and diluted as necessary for use in experiments. According to the manufacturer, the particles are prepared without the use of surfactants and are rigid since polystyrene is an amorphous polymer with a high glass transition temperature. Transmission electron microscopy (18) Liabastre, A. A.; Orr, C. An Evaluation of Pore Structure by Mercury Penetration. J. Colloid Interface Sci. 1978, 64, 1. (19) Martı´nez-Villa, F.; Arribas, J. I.; Tejerina, F. Quantitative Microscopic Study of Surface Characteristics of Track-etched Membranes. J. Membr. Sci. 1988, 36, 19. (20) Keesom, W. H.; Zelenka, R. L.; Radke, C. J. A Zeta-Potential Model for Surfactant Adsorption on an Ionogenic Hydrophobic Surface. J. Colloid Interface Sci. 1988, 125 (2), 575.

Figure 3. Apparatus setup for flow experiments. Table 2. Properties of Latex Particles in Experiments

particle type

surface properties

polystyrene sulfate (PSS) polystyrene amidine (PSA)

negatively charged, hydrophobic positively charged, hydrophobic

nominal diameter (nm)

surface charge density (µC/cm2)

241

-1.7

188

+8.3

(TEM) was used to determine the size distribution of the particles. TEM micrographs show that the particle cross section is perfectly circular and that the particle surface is smooth at the magnification used (×40K-80K). The particle zeta potentials were measured with a Pen Kem 501 Laser Zee Meter, which operates based on the technique of microelectrophoresis. Experimental Setup and Procedure. Particle retention within the membrane pores was monitored by measuring the pressure drop across the membrane housed in a filter holder (Figure 3). A sensitive differential transducer capable of measuring pressure drops as small as 10 Pa was used. The suspensions were injected through the membrane at constant volumetric flow rates using a ISCO-500D syringe pump. The pump is capable of pulse-free injection over a wide range of flow rates. To reduce the polydispersity in particle size, the suspensions were prefiltered prior to use in experiments. The suspensions were prefiltered four times through 1 µm filters at very low flow rates. To prevent bacterial growth, 50 ppm of the biocide sodium azide was added to the suspensions. Dissociation of sodium azide (NaN3) results in a background ionic strength of 8 × 10-4 M Na+ and N3- ions, respectively. Particle concentrations used in the experiments were 50-100 ppm (0.005-0.01 vol %). The filter holder with the membrane was vacuum saturated with deionized water to remove air. Prior to suspension injection, the pressure drop across the membrane was determined for three different flow rates of deionized water having the same ionic strength and pH as the suspension. The choice of flow rates

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Figure 5. Dependence of initial rate of deposition on Peclet number for runs in Figure 4. Figure 4. Effect of velocity on deposition (0.188 µm positively charged amidine latex spheres). depended on the flow rate at which the suspension was to be injected. This step is essential since the measured pressure drop during suspension flow is normalized with the pressure drop during the flow of water for meaningful representation of the experimental data. At the end of each experiment, the membranes were examined using the SEM to observe the nature of particle retention within pores. Membrane preparation for observation using the SEM involved cleaning by immersing the membrane in the particle-free dispersion medium (deionized water having the same ionic strength and pH as the suspension in the experiment). This procedure removed the thin film of suspension present on the membrane surface when retrieved from the filter holder while preserving the particles retained during flow. The gentle cleaning procedure does not disturb the retained particles because they are usually in primary minimum contact with the membrane surface. Visual examination aided in correctly interpreting the particle flow behavior from the pressure drop data.

Experimental Results and Discussion Single Layer Deposition of Particles. The focus of experiments in this section is to study the effect of velocity on the single layer deposition of particles within cylindrical pores. Experiments in this regime are characterized by the following conditions: strong particle-pore surface attraction; strong interparticle repulsion; low particle concentration; large aspect ratio. The aspect ratio is defined as the ratio of the pore size to the particle size. Positively charged 0.188 µm polystyrene amidine latex particles and 2 µm (nominal) pore membranes corresponding to an actual aspect ratio of 8.7 were used in the experiments. The suspension pH was adjusted to 4 to ensure suspension stability. The conditions chosen are conducive for particle deposition in the pores due to the strong attraction between particles and the pore surface. The choice of a large aspect ratio and low particle concentration ensured that retention by straining and hydrodynamic bridging was insignificant in the experiments.2 The effect of velocity on particle deposition is shown in Figure 4. Here, the ratio of pressure drop across the membrane during suspension flow to that during the flow of deionized water at the same flow rate, ∆p/∆pw, is plotted as a function of the total volume of suspension injected. The x axis is also proportional to the number of particles injected through the pores because the concentration is the same for all runs. During each run, the rate of particle retention is initially rapid but quickly decreases to virtually zero, leading to a final, steady pressure drop

across the membrane. The pressure drop reaching a nearly steady value in the runs indicates that the deposition of particles within pores is no longer occurring. The fact that only a single layer of deposited particles is present within the pores in these experiments can also be ascertained by considering the reduction in pore size due to particle deposition. On the basis of the measured aspect ratio of 8.7, if it is assumed that the pore radius is reduced by two particle diameters after deposition, the pressure drop across the membrane would increase by a factor of 2.84 for laminar flow within the pores. However, the assumption of a uniform reduction in the pore radius due to particle deposition is obviously incorrect because the particles are spheres. The factor of 2.84 is therefore an upper bound for the expected increase in pressure drop due to single layer deposition of particles and the actual ∆p/∆pw will be less than 2.84. The experimental observations in Figure 4 are consistent with this analysis. Both the initial rate of deposition and the number of particles retained depend on the flow velocity. The initial rate of deposition of particles within the pores increased with increasing velocity in the experiments in Figure 4. (This trend is not apparent in Figure 4 because the total number of particles is plotted on the x axis.) In Figure 5, the initial rate of increase in pressure drop is plotted as a function of the Peclet number (Pe) for the results shown in Figure 4. The Peclet number is defined by the following expression:

Pe ) vj a2/RD∞ where vj is the interstitial velocity, a is the particle radius, R is the measured average pore radius, and D∞ ) kT/6πµa is the Stokes-Einstein diffusion coefficient. From Figure 5, it can be seen that the initial rate of deposition is proportional to Pe0.53. (It is assumed that the initial rate of increase in pressure drop across the pores is a direct measure of the initial particle deposition rate.) The Smoluchowski-Levich approximation, which neglects both surface forces and particle-wall hydrodynamic interactions, predicts that the deposition rate varies as Pe1/3.21 This result is valid for collectors of different geometries. This result also holds at large Pe for deposition within parallel plate or cylindrical channels (modified by surface interactions). However, in flow geometries where there exists a component of the fluid motion normal to the collector surface (rotating disk, stagnation point flow, etc.) the deposition rate for large Pe varies as Pe.22 Deposition (21) Adamczyk, Z.; Dabros, T.; Czarnecki, J.; van de Ven, T. G. M. Particle Transfer to Solid Surfaces. Adv. Colloid Interface Sci. 1983, 19, 183.


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Figure 6. (a) Blocking by deposited particles and (b) asymmetric blocking due to hydrodynamic interactions.

during flow through a track-etched membrane such as the type used here is a combination of deposition from stagnation point flow (membrane face) and deposition during flow within cylindrical channels (membrane pores). Therefore, the observed Pe0.53 dependence in our experiments is consistent with theory. Given that only a single layer of particles is being deposited within the pores, the trend of decreasing plateau pressure drop with increasing velocity in Figure 4 indicates that the number of deposited particles decreases as the velocity increases. This trend can be understood by considering the effect of velocity on blocking by deposited particles. Note that the term blocking (in the context of particle deposition) refers to the loss of pore surface area surrounding a deposited particle for further deposition and not the plugging of pores by particles. Blocking arises due to two independent interactions between a flowing particle and a deposited particle:23 strong interparticle colloidal repulsion and interparticle hydrodynamic interaction (see Figure 6). Because deposited particles can block a surface area that is much larger than its geometric cross section, the final coverage of the pore surface by deposited particles will be less than the maximum possible monolayer coverage (with particles touching). The decrease in the extent of deposition with increasing velocity observed in the experiments can be explained by considering the contribution of the hydrodynamic interaction between a flowing and a deposited particle to blocking. For sufficiently high flow intensities, experimental observations of particle deposition24-26 as well as theoretical calculations5,27,28 show that the area blocked by a deposited particle is asymmetric with respect to the direction of flow (Figure 6b). Therefore, the blocked area downstream of the particle is larger than that on the upstream side. As a result of asymmetric blocking by deposited particles, the total area available for deposition is less at higher velocities. Hence, the total number of particles deposited within a pore decreases as velocity increases. Figure 7 compares SEM pictures of typical pores in membranes used in the high and low velocity runs in Figure 4. These pictures confirm that the particles are retained in a single layer within pores in the experiments. It can also be seen that fewer particles have been deposited within pores at the higher velocity, which is consistent (22) Adamczyk, Z.; van de Ven, T. G. M. Deposition of Particles under External Forces in Laminar Flow through Parallel Plate and Cylindrical Channels. J. Colloid Interface Sci. 1981, 80 (2), 340. (23) Dabros, T.; van de Ven, T. G. M. Kinetics of Coating by Colloidal Particles. J. Colloid Interface Sci. 1982, 89 (1), 232. (24) Meinders, J. M.; Noordmans, J.; Busscher, H. J. Simultaneous Monitoring of the Adsorption and Desorption of Colloidal Particles during Deposition in Parallel Plate Flow Chamber. J. Colloid Interface Sci. 1992, 152 (1), 265-280. (25) Adamczyk, Z.; Siwek, B.; Zembala, M.; Belouschek, P. Kinetics of Localized Adsorption of Colloidal Particles. Adv. Colloid Interface Sci. 1994, 48, 151. (26) Johnson, P. R.; Elimelech, M. Dynamics of Colloid Deposition in Porous Media: Blocking Based on Random Sequential Adsorption. Langmuir 1995, 11, 801. (27) Dabros, T. Colloids Surf. 1989, 39, 127. (28) Adamczyk, Z.; Siwek, B.; Szyk, L. Flow-Induced Surface Blocking Effects in Adsorption of Colloid Particles. J. Colloid Interface Sci. 1995, 174, 130.

Figure 7. SEM micrographs comparing particle deposition within pores at 0.03 cm/s (left) and 0.73 cm/s (right) for an aspect ratio of 8.7. The membranes are from experiments shown in Figure 4.

with the observed decrease in the final pressure drops upon increasing the velocities in the experiments. Multilayer Particle Deposition. The focus of the experiments reported here is to demonstrate flow-induced deposition of stable particles on previously deposited particles. Experiments in this section are characterized by moderately strong interparticle electrostatic repulsion, net attraction between particle and pore wall, large aspect ratio, and low particle concentration. These conditions are similar to that in the single layer deposition experiments but with the exception that the interparticle repulsion is not as strong as that in the single layer deposition experiments. In these experiments, negatively charged 0.241 µm PSS particles were flowed through 3 µm pore membranes. Particle deposition on the pore surface is induced by the addition of salt (NaCl) to the suspension. Because the particles have a much higher charge than the pore surface, it is possible to increase the ionic strength such that the particle-pore surface repulsion is suppressed without causing particle aggregation. The particle stability has been verified from size measurements using dynamic light scattering for all ionic strengths reported here. The aspect ratio based on the measured pore size is 11.6. A low particle concentration of 50 ppm was used in the experiments. Figure 8 shows the flow behavior of the particles at an ionic strength of 0.01 M NaCl. At the lowest velocity, discrete jumps in the pressure drop were observed because the magnitude of the pressure drop was near the lower end of the measurement range of the transducer. It can be seen that the effect of velocity on particle retention under these conditions is similar to that observed in Figure 4. Particles are being retained in a single layer within

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Figure 8. Effect of velocity on deposition at 0.01 M NaCl.

Figure 9. Effect of velocity on deposition at 0.05 M NaCl.

the pores and the number of deposited particles decreases with increasing velocity due to asymmetric blocking by deposited particles. However, the magnitude of the pressure drop in these experiments is small compared to that obtained in experiments shown in Figure 4 because the aspect ratio is larger. SEM examination of the membranes used in the experiments confirmed these conclusions. Figure 9 shows the effect of velocity for an ionic strength of 0.05 M NaCl. In the run at the lowest flow rate, plugging similar to that in regime II is observed indicating single layer deposition of particles. At the higher velocities of 0.15 and 0.25 cm/s retention is still characterized by an initial rapid deposition followed by a gradually decreasing deposition rate. However, the normalized pressure drop ∆p/∆pw in these experiments does not reach a constant value but increases continually as flow proceeds, indicating continuous retention of particles. SEM pictures of membranes used in experiments at 0.05 M NaCl are shown in Figure 10. At a low velocity of 0.051 cm/s, particles have been deposited in a single layer within pores similar to the experimental results reported in Figure 4. However, at a high velocity of 0.25 cm/s, multilayers of particles can be seen at the pore entrance. The presence of multilayers of deposited particles at the pore entrance implies that flowing particles have been deposited on previously deposited particles. The capture of flowing particles by deposited particles is flow-induced because the experimental conditions are such that sufficient electrostatic repulsion exists between particles preventing their flocculation by Brownian collisions. As will be seen in the next section, in the vicinity of a deposited particle, there exists a component of the hydrodynamic force acting on a flowing particle that tends to push it


toward the deposited particle. It is the competition between this hydrodynamic force component and the net electrostatic repulsion that determines whether flowinduced deposition can occur. At low ionic strengths when the interparticle electrostatic repulsion is strong or at low flow velocities when the magnitude of the hydrodynamic force is small, only a single layer of particles will be deposited within the pores. The experimental observations in Figure 9 are consistent with the above explanation of the phenomenon. In the 0.051 cm/s run, a single layer of particles is deposited because of the colloidal repulsion dominating over the weak hydrodynamic force. ∆p/∆pw for this run therefore levels out during the experiment. ∆p/∆pw during the 0.15 cm/s run in Figure 9 crosses over that for the 0.051 cm/s run because of the fact that multilayers of particles are being deposited indicating that at this flow velocity, the hydrodynamic force acting on the flowing particle near the pore entrance is able to overcome the interparticle net repulsion. Further, it can be seen that for velocities above the critical velocity, the rate of multilayer deposition, and therefore pore plugging, increases as the flow velocity increases for the range studied. The transition from single-layer deposition to multilayer deposition for the system being studied is clearly demonstrated in the experimental results shown in Figure 11. Here, runs were performed with different suspension ionic strengths but the same flow rate. In the absence of NaCl, no increase in pressure drop across the membrane is observed because the strong particle-pore wall repulsion prevents deposition. At low ionic strengths (ionic strength e0.01 M NaCl), by comparing the observations with experiments in Figure 4, it can be concluded that particles are deposited in a single layer within the pores. However, at higher ionic strengths and for the flow rate used, the interparticle electrostatic repulsion has been sufficiently reduced so that flow-induced deposition of particles on previously deposited particles can occur. Trajectory of a Flowing Particle in the Vicinity of a Stationary Particle The goal of these calculations is to show that the phenomenon of multilayer deposition of stable particles is governed by the interactions between a flowing particle and a deposited particle. The schematic shown in Figure 12 illustrates the flow problem that needs to be solved. The trajectory of the flowing particle can be determined from a knowledge of the hydrodynamic force and colloidal force acting on it (the particle is assumed to be neutraly buoyant). The hydrodynamic force acting on the flowing particle derives from its hydrodynamic interaction with both the solid plane wall and the deposited particle. In general, hydrodynamic interactions between bodies in Stokes flow can be divided into a far-field contribution and a near-field or lubrication interaction.29 The term “far-field” refers to interactions when the particles are widely separated. Typically, far-field interactions are out of range for colloidal forces to influence the flow behavior. When the interparticle separation (or particle-plane wall separation) is of the order of a particle radius, lubrication forces dominate. “Lubrication” interaction is caused by the slow drainage of liquid between two particles as they approach each other. Lubrication forces come into play at approximately the same separation distances at which van der Waals attraction and double-layer interactions become appreciable. It is therefore this near-field regime that determines whether flow-induced deposition can (29) Brady, J. F.; Bossis, G. Stokesian Dynamics. Annu. Rev. Fluid Mech. 1988, 20, 111.


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Figure 10. SEM micrographs comparing membranes from experiments shown in Figure 9 (PSS particles, aspect ratio 11.6, 0.05 M NaCl). Single layer deposition at 0.05 cm/s (left) and multilayer deposition at 0.73 cm/s (right).

Figure 13. Motion of a particle in the neighborhood of a deposited particle.

Figure 11. Transition from single-layer deposition of particles to flow-induced multilayer deposition.

Figure 12. Schematic of flow problem for studying flow-induced deposition. Our calculations include only the lubrication interaction between the flowing particle and the stationary particle.

occur. In our calculations, we have only considered the lubrication two-body interaction between the flowing and the deposited particle. While our calculations are approximate in nature, they clearly demonstrate flowinduced deposition and are qualitatively consistent with our experimental trends. Only the two-dimensional particle motion is being considered, as that is sufficient to elucidate the physics of the phenomenon. Dabros and van de Ven5 have studied surface collisions between a freely mobile particle and one deposited on a solid plane in a viscous medium. While they qualitatively demonstrated blocking by a deposited particle in the presence of strong interparticle repulsion, they did not consider the situation where flow can induce surface coagulation when the magnitude of the interparticle repulsion is reduced by the addition of an electrolyte. Kamiti et al.30 have determined the kinetics of surface coagulation from suspensions undergoing stagnation point

flow in a manner analogous to the Smoluchowski model for orthokinetic coagulation. They however did not include the case of net colloidal repulsion between flowing and deposited particles. Vitthal and Sharma15 have modeled the deposition of spherical colloidal particles during flow through an assembly of spheres using Stokesian dynamics. They calculated the trajectory of one particle at a time taking into account the hydrodynamic interactions and van der Waals attraction between the flowing particle and the sphere assembly including all previously deposited particles. In this manner, they demonstrated that previously deposited particles significantly affect the deposition process by both altering the flow field and acting as additional collectors. They did not consider the case of net repulsion between the depositing particles and the effect of velocity under such conditions. We start by considering the motion of a spherical particle of radius a near a stationary sphere of radius af (a much smaller than af). Goren and O’Neill31 have shown that the general motion of the moving particle can be taken to be the superposition of three independent motions owing to the linearity of the Stokes equations: (i) axisymmetrical stagnation point flow past a stationary sphere near a plane wall; (ii) linear shear flow past a stationary sphere parallel to a plane wall; (iii) motion of the sphere through stationary fluid toward the stationary sphere along the line of centers. The three independent motions of the flowing sphere are shown in Figure 13. From the solutions to the Stokes equation for these three different situations, the velocity of the flowing particle can be determined. Because the Peclet numbers in all our experiments are greater than 1, convective motion of particles dominates over their Brownian motion. We have therefore neglected particle Brownian motion in our analysis. (30) Kamiti, M.; Dabros, T.; van de Ven, T. G. M. Kinetics of Surface Coagulation. J. Colloid Interface Sci. 1995, 172, 459. (31) Goren, S. L.; O’Neill, M. E. On the Hydrodynamic Resistance to a Particle of a Dilute Suspension When in the Neighborhood of a Large Obstacle. Chem. Eng. Sci. 1971, 26, 325.

4442 Langmuir, Vol. 14, No. 16, 1998


Fcol vr )

Figure 14. Coordinate system for calculation of trajectory of a particle around a deposited particle.

Figure 14 shows the coordinate system used in the trajectory calculations; the origin is located at the center of the stationary sphere. The trajectory of the particle is given by

dr/dθ ) rvr/vθ

(1)

where vr is the radial velocity and vθ is the tangential velocity of the flowing particle. The expressions for the velocity components are obtained from the consideration that the particle has no net force or torque acting on it (for low Reynolds number motion). The tangential velocity of the particle near the deposited particle is given by31

vθ )

3U h ˜ sin θf* 8a a

()

(2)

where U is the fluid velocity far away from the stationary sphere (undisturbed flow), f * is the correction for hydrodynamic interaction between the flowing and stationary spheres, and h ˜ is the distance from the surface of the stationary sphere to the center of the moving sphere. The correction factor f * is obtained from Goldman et al.32 The radial velocity of the particle (along the line of centers) is determined from a radial force balance

(Fr)hyd + Fcol ) 0 (Fr)hyd )

()

()

(3)

()

9 a3 h ˜ h ˜ πµU 2 cos θf0 - 6πµavrf 4 a a af

(4)

The hydrodynamic force in the r direction is given by eq 4 and derives from two different contributions: one due to the axisymmetric stagnation point flow past a stationary sphere, and the other due to the motion of sphere toward an identical stationary sphere in a stationary fluid. In eq 4, f0 and f are hydrodynamic correction factors and are obtained from Goren and O’Neill31 and Jeffrey,33 respectively. In the absence of other external forces, these two contributions always oppose each other with the stagnation point flow contribution pushing the flowing particle toward the deposited particle on the upstream side (θ < π/2). In the presence of net interparticle repulsion, the net hydrodynamic force will act to push the flowing particle toward the deposited particle. Thus, the magnitude of the stagnation point flow contribution to the radial component of the hydrodynamic force determines whether flow-induced deposition can occur. Solving for vr from the radial force balance, we get (32) Goldman A. J.; Cox, R. G.; Brenner, H. Slow Viscous Motion of a Sphere Parallel to a Plane WallsII Couette Flow. Chem. Eng. Sci. 1967, 22, 653-660. (33) Jeffery, D. J. Low-Reynolds-Number Flow Between Converging Spheres. Mathematika 1982, 29, 58.

()

()

9 a3 h ˜ πµU 2 cos θf0 4 a af h ˜ 6πµaf a

()

(5)

Note that the hydrodynamic force arising from the stagnation point flow component is proportional to the cosine of the angle θ. Thus, as can be seen from eq 4, this force acts to oppose the colloidal repulsion in the upstream side of the deposited particle (θ < π/2) and its magnitude is greatest at the front stagnation point (θ ) 0). To relate U to the flow conditions in the vicinity of a deposited particle present at the pore entrance, U was estimated as the fluid velocity (parallel to the pore axis) at the pore entrance at a radial location one particle diameter away from the pore edge. The magnitude of the fluid velocity at that location was determined using the work of Manton34 who calculated the flow field approaching a Nuclepore filter in the low Reynolds number limit. For an aspect ratio of 11.6, U was determined to be 0.98U h, where U h is the initial interstitial velocity in the pore. Colloidal Forces. The colloidal interaction energy between two charged particles dispersed in a polar liquid is calculated in the framework of the DLVO theory and is a combination of van der Waals attraction (for particles of the same material) and electrostatic repulsion (like charge). A short-ranged repulsion attributed to solvation or structural forces35 is also included here. The expression for the retarded London-van der Waals attraction between two spheres of equal size given by Schenkel and Kitchener36 has been used here

VA(h) ) -

Aa 11.12h 12h 1 + λ

(

)

(6)

where VA is the attractive interaction energy, A is the Hamaker constant, h is the surface separation between the particles, λ is the characteristic wavelength of interaction (retardation length) often assumed to be 100 nm, and a is the particle radius. The range of validity of this expression is

0 < h < λ/π,

h,a

By comparing Schenkel and Kitchener’s expression with the exact results of Clayfield et al.,37 Gregory38 concluded that eq 5 is accurate up to about 5% of the particle radius. Gregory also points out that the departures at greater separations appeared to be due primarily to the geometric restriction h , R, rather than to exceeding the upper limit of h ) λ/π . However, for colloidal particles, the interaction energy usually becomes insignificant relative to the thermal energy of the particles at separations for which the condition h , R begins to cause inaccuracies. The use of eq 6 for describing van der Waals attraction is therefore justified. (34) Manton, M. J. The Impaction of Aerosols on a Nuclepore Filter. Atmos. Environ. 1978, 12, 1669. (35) Israelachvili, J. N. Intermolecular and Surface Forces-With Application to Colloidal and Biological Systems; Academic Press: New York, 1985. (36) Schenkel, J. H.; Kitchener, J. A. A Test of the Derjaguin-VerweyOverbeek Theory with a Colloidal Suspension. Trans. Faraday Soc. 1960, 56, 161. (37) Clayfield, E. J.; Lamb, E. C.; Mackey, P. H. Retarded Dispersion Forces in Colloidal Particles-Exact Integration of the Casimir and Polder Equation. J. Colloid Interface Sci. 1971, 37, 382. (38) Gregory J. Approximate Expressions for Retarded van der Waals Interaction. J. Colloid Interface Sci. 1981, 83 (1), 131.


Langmuir, Vol. 14, No. 16, 1998 4443

The expression for the electrostatic repulsion between two identical colloidal particles used here is based on the linearized Poisson-Boltzmann equation and the Derjaguin approximation for constant potential interaction39

VR(h) ) 2πa ψ02 ln(1 + e-κh)

(7)

where VR is the electrostatic interaction energy, is the permittivity of water, ψ0 is the surface potential of the particles, and κ is the Debye-Huckel parameter (κ-1 is the screening length). By exactly solving the linearized Poisson-Boltzmann equation for two equal, charged spheres via a multipole expansion, Glendinning and Russell40 have identified regions where this approximation is valid. For κR ≈ 10, which is the case in our experiments, the interparticle repulsive force calculated using eq 7 is within 10% of the exact force for h e 3κ when the potential is small. Feke et al.41 have derived an expression for the shortranged Born repulsion between two identical spherical particles. They obtained this expression using the repulsive part of the Lennard-Jones 6-12 molecular potential and assuming pairwise additivity similar to the integration procedure of Hamaker.

VB(h) )

() {

σ A 37800 a

6

1 R2 - 14R + 54 60 - 2R2 + + R (R - 2)7 R7 R2 + 14R + 54 (8) (R - 2)7

}

where VB is the Born interaction energy, σ is the collision diameter (4 Å), and R is the center to center particle separation scaled using the particle diameter (R ) h/2a + 1). The total interparticle colloidal interaction (VT) is the sum of van der Waals attraction (VA), electrostatic repulsion (VR), and the short-ranged Born repulsion (VB). The expression for the total colloidal force acting on the particles is then given by

Fcol(h) ) -

Figure 15. Total colloidal force between two spheres as a function of surface separation at three different ionic strengths.

d [V (h) + VR(h) + VB(h)] dh A

Figure 16. Layout for representing particle trajectories.

(9)

Figure 15 shows a plot of the calculated colloidal force between two spheres as a function of their surface separation at three different ionic strengths. In the calculations, particle zeta potentials of -65, -40, and -35 mV were used at the ionic strengths 0.001, 0.01, and 0.05 M of univalent electrolyte, respectively. These values were chosen based on measurements of the zeta potential of the PSS particles as a function of NaCl concentration. At an ionic strength of 0.001 M, it can be seen from Figure 15 that the interparticle interaction is strongly repulsive. The force barrier at this ionic strength that has to be overcome for bringing the particles into primary minimum contact (separation distance for which the total force is zero) is also apparent. As the ionic strength is increased, the magnitude of the repulsive force for a given separation and the force barrier, in particular, decreases. Trajectories of the Flowing Particle. The trajectory equation, eq 1, can be numerically solved for a given initial (39) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (40) Glendinning, A. B.; Russel, W. B. The Electrostatic repulsion between Charged Spheres from Exact Solutions to the Linearized Poisson-Boltzmann Equation. J. Colloid Interface Sci. 1983, 93, 95. (41) Feke, D. L.; Prabhu, N. D.; Amin, J. A., Jr.; Amin, J. A., II A Formulation of the Short-Range Repulsion between Spherical Colloidal Particles. J. Phys. Chem. 1984, 88, 5735.

Figure 17. Calculated particle trajectories under conditions of strong interparticle repulsion (0.001 M Na+).

position of the particle, particle surface potential, ionic strength, and flow velocity. Figure 16 shows the representation used for presenting the particle trajectories. The distances have been scaled using the particle diameter. In the schematic, the smaller semicircle represents the deposited particle. The horizontal axis passes through the center of the deposited particle. The larger semicircle (dashed line) represents the boundary that the center of the flowing particle cannot penetrate; that is, if the center of the flowing particle lies on this larger semicircle, the particles will touch each other. The particle trajectory shown in the figure is the locus of the center of the flowing particle in the vicinity of the deposited particle. If the trajectory intersects the boundary of closest approach, it means deposition will occur for the conditions under consideration. Figure 17 shows the trajectory of the flowing particle near the deposited particle for two different interstitial velocities under conditions of strong repulsion. In the calculations, a particle radius of 0.125 µm (a ) af) and a particle zeta potential of -65 mV have been used. For strong repulsion between particles, it can be seen from

4444 Langmuir, Vol. 14, No. 16, 1998

Figure 18. Calculated particle trajectories for different flow rates under conditions of moderate interparticle repulsion (0.05 M NaCl).

Figure 17 that deposition on a previously deposited particle will not take place for values of U h as large as 3.5 cm/s. The hydrodynamic force acting on the particles (in the Stokes regime) is insufficient to overcome the interparticle electrostatic repulsion. When the repulsion between the particles has been sufficiently reduced by increasing the ionic strength, Figure 18 shows that particle deposition is possible if the velocity is sufficiently high. At an ionic strength of 0.05 M 1:1 electrolyte, the particle trajectory for U h ) 3.5 cm/s intersects the boundary of closest approach indicating that particle deposition will occur. For the conditions specified, this velocity is the critical velocity required for flowinduced deposition. A zeta potential of -35 mV was used in the calculations. These calculations clearly show that the onset of multilayer deposition of stable particles is dictated by the colloidal and hydrodynamic interaction between a flowing particle and a deposited particle. For a given ionic strength, the existence of a critical velocity below which multilayer deposition will not occur also has been shown. The calculations are therefore qualitatively consistent with our experimental observations in Figure 9. In the runs involving the flow of positively charged PSA (Figure 4) and negatively charged PSS for ionic strengths up to 0.01 M NaCl (Figure 8), only single layer deposition was observed because the flow rates were not high enough to overcome the strong interparticle repulsion. Analogous to the existence of a critical velocity at moderate ionic strength for flow-induced deposition, there exists a critical ionic strength at a sufficiently high velocity below which flow-induced deposition will not occur. This behavior was experimentally demonstrated in the results shown in Figure 11. Figure 19 shows the trajectories of a particle in the vicinity of a deposited particle for different ionic strengths but for the same flow rate. It can be seen from the trajectories that, for the chosen flow rate, flowinduced deposition will not occur for ionic strengths up to 0.01 M. At an ionic strength of 0.05 M, the flow rate is sufficiently high for flow-induced deposition to occur. We point out that for a given ionic strength our calculated critical velocity for deposition of a flowing particle on to the first layer of deposited particles (i.e., formation of a second layer) is also the critical velocity required for multilayer deposition. In our experiments, the interstitial flow velocity in the pores increases as particles are deposited because the suspension volumetric flow rate is constant. If the conditions are suitable for the formation of a second layer of deposited particles, then


Figure 19. Calculated particle trajectories for different suspension ionic strengths at the same flow rate.

deposition of flowing particles on the second layer to form a third layer will inevitably follow because the interstitial velocity continues to exceed the critical velocity. Predicting the deposition of a flowing particle on the first layer of deposited particles is therefore sufficient to predict conditions for multilayer deposition of stable particles. Conclusions We have experimentally demonstrated flow-induced multilayer deposition of stable colloidal particles during the low Reynolds number flow of dilute suspensions within cylindrical pores. A model experimental system consisting of submicrometer latex particles and track-etched membranes having cylindrical pores was used in the study. Multilayer deposition of stable particles will occur if the hydrodynamic force acting on a flowing particle in the vicinity of a deposited particle overcomes the net colloidal repulsion, thereby enabling deposition on a previously deposited particle. By flowing suspensions of positively charge particles through membranes characterized by a negative surface charge, it was first shown that only single layer deposition is possible for the range of flow rates studied when strong interparticle repulsion exists. However, if the magnitude of the interparticle repulsion is reduced (say, by the addition of an electrolyte), multilayer deposition is possible for flow velocities greater than a critical velocity. Experiments involving the flow of negatively charged particles through the membrane in the presence of NaCl elucidated this important flow behavior. The magnitude of the critical velocity required is determined by the aspect ratio, particle and pore surface charge, suspension ionic strength and pH, and the flow geometry. Approximate calculations of the trajectory of a flowing particle in the neighborhood of a stationary sphere confirmed that flow-induced deposition is indeed possible for the system and experimental conditions chosen. In these calculations, the interparticle colloidal interaction and the hydrodynamic interaction in the lubrication limit (presence of wall not accounted for) were included. For a given particle-pore surface system and flow geometry, the critical conditions required for multilayer deposition can be identified in terms of the suspension ionic strength (and pH if H+ is a potential determining ion) and flow rate. This work thus provides the foundation for including multilayer particle deposition in predictive models describing particulate flow through porous media. LA971207H

Multilayer Deposition of Stable Colloidal Particles during Flow within

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