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Pulse-Injection Chromatographic Determination of the Deposition and Release Rate Constants of Colloidal Particles in Porous Media Y. D. Yan† Institute of Terrestrial Ecology, Federal Institute of Technology (ETH), Grabenstrasse 3, CH-8952 Schlieren, Switzerland Received July 28, 1995. In Final Form: April 22, 1996X The packed column technique with injection of very short sample pulses was demonstrated in this study to be well suited for studying the mass transfer coefficients of colloidal particles flowing through porous media. It has been used here to examine the deposition and release kinetics of model colloidal latex spheres with sulfate surface functional groups in packed soda-lime glass beads. Apart from the commonly observed process of particle removal by the beads, strong evidence of spontaneous detachment of the deposited particles from the bead surfaces was found without even changing solution chemistry or flow conditions. In particular, the rate constants of the two processes were estimated from the measured particle breakthrough curves on the basis of the convection-dispersion equation coupled with a two-site first-order kinetics model; the reversible site was used to account for the observed particle release, while the irreversible site was assumed to be responsible for the apparent loss of particles in the column. Solution ionic strength was found to have a significant effect on the deposition process. Upon increasing ionic strength, the particle deposition rate constant showed transition from slow to fast regime, and this can be explained qualitatively using the DLVO theory. The particle release rate constant was, however, seen to be almost independent of ionic strength. One of the main advantages of this pulse-injection technique over the traditionally used continuous step input method is that many repeat experiments can be performed on the same column without causing significant blocking or filter ripening effects owing to the very small amount of particles injected into the column.
1. Introduction Colloidal particles flowing through porous media may be removed as a result of their interaction with the fixed solid (collector) surfaces. Depending on the magnitude of the potential energy barrier for detachment, an immobilized particle may also, due to its thermal motion and the hydrodynamic forces rendered by the flowing medium, detach from its collector surface and be carried away by the mobile fluid. Particle deposition and release processes have important applications in various industrial and engineering fields, e.g., water filtration and wastewater treatment1,2 and colloid and associated pollutant transport in soil and groundwater flows.3,4 They are also of significant importance in basic colloid science. A large body of research has been carried out, both experimentally5-9 and theoretically,10-12 on particle removal by collectors. It is understood that three underlying † Present address: Department of Chemical Engineering, The University of Newcastle, University Drive, Callaghan, NSW 2308, Australia. X Abstract published in Advance ACS Abstracts, June 15, 1996.
(1) Tobiason, J. E.; O’Melia, C. R. J. Am. Water Works Assoc. 1988, 80, 54. (2) Yao, K.-M.; Habibian, M. T.; O’Melia, C. R. Environ. Sci. Technol. 1971, 5, 1105. (3) Buddemeier, R. W.; Hunt, J. R. Appl. Geochem. 1988, 3, 535. (4) McDowell-Boyer, L. M.; Hunt, J. R.; Sitar, N. Water Resour. Res. 1986, 22, 1901. (5) Ryde, N.; Matijevic´, E. J. Colloid Interface Sci. 1995, 169, 468. (6) Litton, G. M.; Olson, T. M. Colloids Surf., A 1994, 87, 39; J. Colloid Interface Sci. 1994, 165, 522; Environ. Sci. Technol. 1993, 27, 185. (7) Song, L.; Johnson, P. R.; Elimelech, M. Environ. Sci. Technol. 1994, 28, 1164. (8) Rodier, E.; Dodds, J. Colloids Surf., A 1993, 73, 77. (9) Elimelech, M.; O’Melia, C. R. Langmuir 1990, 6, 1153; Environ. Sci. Technol. 1990, 24, 1528. (10) Dahneke, B. Can. J. Chem. Eng. 1976, 54, 26; J. Colloid Interface Sci. 1974, 48, 520. (11) Spielman, L. A.; Friedlander, S. K. J. Colloid Interface Sci. 1974, 46, 22. (12) Ruckenstein, E.; Prieve, D. C. J. Chem. Soc., Faraday Trans. 2 1973, 69, 1522.
S0743-7463(95)00629-9 CCC: $12.00
particle transport mechanisms exist which lead to particle-collector collisions:13 Brownian diffusion, interception due to fluid stream lines, and gravitational settling (sedimentation). When there are no net repulsive forces between a mobile particle and its collector, every collision will result in particle attachment to the collector (i.e., fast deposition). Under usual conditions net repulsive forces exist between the two surfaces which come into play at close particle-collector approach distances. These forces create an energy barrier for particle attachment, resulting in the so-called slow deposition. Study of release of immobilized particles is, by contrast, only a recent practice and particle mobilization was observed mostly when solution chemistry such as ionic strength and pH was changed or fluid flow conditions were altered subsequent to the deposition stage.14-16 It is generally believed that particles attached in the primary energy well cannot escape from their collector surfaces without introducing changes in chemical or flow conditions. Recent studies on colloidal particle interactions with glass plates17 and particle18 and bacterial19-24 transport in packed columns have, however, indicated the (13) O’Melia, C. R. Environ. Sci. Technol. 1980, 14, 1052. (14) Ryan, J. N.; Gschwend, P. M. J. Colloid Interface Sci. 1994, 164, 21; Environ. Sci. Technol. 1994, 28, 1717. (15) McDowell-Boyer, L. M. Environ. Sci. Technol. 1992, 26, 586. (16) Kallay, N.; Barouch, E.; Matijevic´, E. Adv. Colloid Interface Sci. 1987, 27, 1. (17) Meinders, J. M.; Busscher, H. J. Colloid Polym. Sci. 1994, 272, 478. (18) Saiers, J. E.; Hornberger, G. M.; Liang, L. Water Resour. Res. 1994, 30, 2499. (19) McCaulou, D. R.; Bales, R. C.; Arnold, R. G. Water Resour. Res. 1995, 31, 271. (20) McCaulou, D. R.; Bales, R. C.; McCarthy, J. F. J. Contam. Hydrol. 1994, 15, 1. (21) Tan, Y.; Gannon, J. T.; Baveye, P.; Alexander, M. Water Resour. Res. 1994, 30, 3243. (22) Lindqvist, R.; Cho, J. S.; Enfield, C. G. Water Resour. Res. 1994, 30, 3291. (23) Hornberger, G. M.; Mills, A. L.; Herman, J. S. Water Resour. Res. 1992, 28, 915.
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reversible nature of the corresponding deposition processes. Nevertheless, it is still unclear as to whether those mobilized species were from the primary or secondary energy minimum. The reversible deposition phenomena have also been studied theoretically by Ruckenstein and Prieve.25,26 On the experimental side, packed column techniques are very useful for studying particle deposition and release kinetics in porous media. Most of the earlier studies have used the traditional step-injection method, where a background electrolyte solution used for conditioning the column is switched to a continuous input of a suspension with constant particle concentration. While such column experiments mimic closely real industrial processes such as deep-bed water filtration, they are not ideal for many repeat runs, since collector surfaces can soon be contaminated due to accumulation of the incoming particles on the collectors and therefore the characteristics of the packed bed may be altered. Particle detachment in such experiments can be explored practically only by switching the influent containing particles to a blank solution after the deposition stage, i.e., via a negative step. In comparison, experiments with injection of only a very short sample pulse (Dirac delta function input) into the column, referred to here as pulse injection, circumvent the collector contamination problem as the total amount of injected particles is very small. In this technique small quantities of colloids are injected into a carrier solution which has the same background chemical composition as the colloidal suspension. Particle deposition and release processes can now be monitored simultaneously without having to perform the negative step. In addition, in an ideally packed column, particle release can be identified simply by occurrence of tailing behavior in the particle breakthrough curve (BTC), as will be shown in this work. Unfortunately, until now the pulse-injection technique has not been widely exploited for the study of particle deposition and release kinetics in porous media. It is worth noting that a similar technique involving injection of short square-wave sample pulses has been employed by McCaulou et al.20 and Hornberger et al.23 to study bacterial sorption and transport in packed columns. By comparing the convection-dispersion equation, incorporating assumed first-order kinetics for bacterial adsorption and desorption, to their measured bacterial BTCs, the rate constants of both processes were obtained. There was, however, significant mismatch between the fitted and measured BTC tails in these studies. As will be shown in this paper, such tails contain essential information about particle release kinetics, and hence care should be taken in the fitting procedure. In this study the pulse-injection technique was employed to explore the transport kinetics of model sulfate colloidal latex spheres through a packed bed of glass beads. The objective of this work is to use such a model system to show the methodology, reliability, and usefulness of this technique for determining particle deposition and release rate constants in porous media. 2. Theoretical Basis To describe the concentration profile C(x,t) of colloidal particles in a column of homogeneously packed porous media in response to a Dirac delta function input, the one-dimensional convection-dispersion equation is adopted, together with assumed two-site first-order kinetics (the (24) Fontes, D. E.; Mills, A. L.; Hornberger, G. M.; Herman, J. S. Appl. Environ. Microbiol. 1991, 57, 2473. (25) Ruckenstein, E. J. Colloid Interface Sci. 1978, 66, 531. (26) Ruckenstein, E.; Prieve, D. C. AIChE J. 1976, 22, 276; 22, 1145.
Yan
first site being reversible and the second site being irreversible, as prompted by the experimental observation of this study) to account for particle attachment and detachment processes:
∂C ∂C ∂Ca1 ∂Ca2 ∂2C + + )D 2 -U ∂t ∂t ∂t ∂x ∂x
(1a)
∂Ca1/∂t ) kdC - krCa1
(1b)
∂Ca2/∂t ) kiC
(1c)
with boundary conditions C(x)∞,t) ) 0, C(x,t)0) ) 0, and C(x)0,t) ) C0tpδ(t), where C0 is the particle number concentration which would be observed if the amount of particles injected into the column were uniformly dispersed in the column pore volume and tp is the mean travel time of nonretarded particles. In eq 1, t is the time, x is the travel distance, C (number of particles per unit pore volume) is the particle concentration in the mobile phase, Ca1 and Ca2 (number of particles per unit pore volume) are the respective concentrations of particles adsorbed on the collectors through the reversible and irreversible processes, kd(T-1) and kr(T-1) are, respectively, the particle deposition and release rate constants of the reversible site, ki(T-1) is the particle deposition rate constant of the irreversible site, D(L2T-1) is the dispersion coefficient, and U(LT-1) is the mean travel velocity of nonretarded particles. When size exclusion effects can be neglected, as in this study, U can be taken as being equal to the mean travel velocity of the mobile fluid. In this case, U can be related to the Darcy velocity q(LT-1) by U ) q/θ, where θ is the packing porosity and q is given by the background fluid flow rate Q (volume per unit time) divided by the cross-sectional area of the column. Equation 1 can be solved analytically in the following two cases: Case I: In the absence of kinetic effects (or in the case of nonadsorbing/ideal tracers), the particle breakthrough concentration at the column outlet, C(X,t), X being the packing height of the column, is given by
C(X,t) ) C0tp
(
)
2
(X - Ut) X exp 3 1/2 4Dt 2(πDt )
(2)
Case II: In the case of no dispersion effects it can be shown, after rewriting eq 1 in reduced time and travel distance, that the particle breakthrough concentration at the column outlet in the Laplace domain, C(s), can be written as
(
C(s) ) C0 exp(-s) exp -
{
)
skdtp - kitp s + krtp
(3)
Inversion of eq 3 to the reduced real time space, T (T ) t/tp) yields
C(T) ) C0 exp[-(kd + ki)tp] ×
(
δ(T - 1) + kdkrtp2 ×
exp[-krtp(T - 1)] ×
[
1+
0
])
kdkrtp2 (T - 1) + ... 2
for T g 1
(4a)
for 0 < T < 1
(4b)
If particle deposition and release of the reversible site
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Table 1. Characteristics of the Glass Beads and Latex Particles Used in This Work Soda-Lime Glass Beads 377.5 µm SiO2, 72.00-73.00%; Na2O, 13.30-14.30%; K2O, 0.20-0.60%; CaO, 7.20-9.20%; MgO, 3.50-4.00%; Fe2O3, 0.08-0.11%; Al2O3, 0.80-2.00%; SO3, 0.20-0.30%
mean diametera compositionb
Polystyrene Latex Particles with Sulfate Surface Functional Groups diameterb 0.200 µm size polydispersityb 4.6% density at 20 °Cb 1.055 g/cm3 surface charge densityb 1.1 µC/cm2
Figure 1. Schematic of the pulse-injection chromatographic setup used in this work.
a Average of the sieve mesh sizes. b As specified by the manufacturer.
are very slow processes, i.e., if kdkrtp2 , 1, then eq 4a may be approximated as
C(T) ) C0 exp[-(kd + ki)tp](δ(T - 1) + kdkrtp2 exp[-krtp(T - 1)]) (5) Equation 5 reveals that the particle breakthrough curve (BTC), in the case of slow particle deposition and release kinetics and also without dispersion effects, is composed of a nonretarded delta peak at T ) 1 and an exponentiallydecaying tail at longer times due respectively to the breakthrough of nonadsorbing and detached particles from the column. The overall particle deposition rate constant, kd + ki, can be estimated from the nonretarded peak area, whereas the slope of the release tail, when plotted on a natural semilogarithmic scale, yields the particle release rate constant. Having discussed the specific cases, it should be mentioned that no analytical solution can at present be found for eq 1 in general cases. Nevertheless, eq 1 can still be Laplace transformed. If eq 1 is first rewritten in reduced time and travel distance, then the particle concentration at the column outlet is given in the Laplace domain by
{
C(s) ) C0 exp -
}
1 [(1 + 4D′s(1 + M(s)))1/2 - 1] 2D′ (6a)
with
M(s) )
kitp kdtp + s + krtp s
(6b)
and D′ ) D/XU. Inversion of eq 6 to the reduced real time space can be done numerically and was accomplished in this work by implementing inverse Laplace transform routines (modified Weeks’ method) from the NAG FORTRAN Library.27 The general features of a BTC as obtained from the numerical inversion of eq 6 in the presence of dispersion effects resemble closely those without dispersion (eq 5), i.e., the former BTC also consists of a peak at T ) 1 and an almost exponentially decaying tail at longer times. 3. Experimental Section Glass Beads. Soda-lime glass beads (Sovitec, Florange, France) passing a 400 µm sieve and retained on a 355 µm sieve were used in this work. As shown in Table 1, they were mainly (27) NAG Fortran Library, Mark 15, NAG Ltd., Wilkinson House, Jordan Hill Road, Oxford, OX2 8DR, U.K.
composed of SiO2. They were washed before use with HCl (1 M)/chromic acid (2.5% (w/v)) following the same procedure as that adopted by Olson and Litton28 to eliminate surface contamination. The washed beads were dried in an oven at 85 °C for 24 h and then cooled down to room temperature in a desiccator. They were then wet packed to a height of 32 cm into a cylindrical glass chromatography column (OMNI) with an internal diameter of 2.42 cm. Prior to each measurement the column was equilibrated with the influent solution of interest. Colloidal Particles. Surfactant-free polystyrene latex particles carrying sulfate surface functional groups (Batch No. 1-FY200,1, IDC, Portland, OR) were used. These particles were fluorescently-labeled; their respective excitation and emission wavelengths were 490 and 515 nm. According to the manufacturer, these particles were negatively charged at pH > 2 and have an average diameter of 0.2 µm (cf. Table 1). Before use, the particles were deionized using a mixed-bed ion exchange resin (AG 501-X8 (D), BIO-RAD). Immediately prior to injection into the column the particles were dispersed in the medium of same chemical composition as the preequilibrating influent solution. Particle breakthrough concentrations were detected on-line using a flow-through spectrofluorometer (Jasco 821-FP) equipped with a 16 µL flow-through cell. Influent Solutions. The pH of all influent solutions was fixed at 9.5. Their ionic strength was varied using NaCl (Fluka, p.a.). Except for two solutions at the low ionic strength end, all solutions were buffered with NaHCO3 (Merck, p.a.) and in equilibrium with atmospheric CO2. These buffered solutions were prepared by first dissolving appropriate amounts of NaHCO3 and NaCl in deionized water (Barnstead NANOpure system) and then bringing the pH of the resulting solution to 9.5 with 2 M NaOH. For these buffered solutions the amount of NaHCO3 needed at each NaCl concentration was calculated from chemical equilibria for open systems, with ionic strength corrections taken into account for the activity coefficients of the relevant chemical species in solution. For the two unbuffered solutions at the low ionic strength end, no NaCl was added, and the NaHCO3 solutions were prepared according to chemical equilibria for closed systems. The actual ionic strength of each influent used in this work was calculated from the weighted sum of all ionic species in solution. Chromatographic Setup. Figure 1 illustrates the pulseinjection chromatographic setup used in this study. In operation, the influent was passed first through a degasser (Erma ERC3511) and then pumped downward through the packed column by an HPLC pump (Sykam S 1000). Prior to entering the column the influent was thermostated via a tubing loop (FIAstar 5101 Thermostat) to 25.0 ( 0.5 °C. The column was embraced in a water jacket and thermostated to 25.0 ( 0.1 °C. Both the influent and effluent pH values could be monitored with a flow-through electrode (Sensorex), and the effluent pH was found to reach that of the influent after 2-3 pore volumes of flushing. Tracers and particle suspensions having the same background chemical composition as the preequilibrating influent solution were manually injected into the column inlet through a 60 µL (equivalent to 0.001 pore volume in this study) valve injection loop. Their concentrations were measured at the column outlet with flow-through spectrophotometers, from which the output (28) Olson, T. M.; Litton, G. M. In Transport and remediation of subsurface contaminants: colloidal, interfacial, and surfactant phenomena; ACS Symp. Series No. 491; Sabatini, D. A., Knox, R. C., Eds.; American Chemical Society: Washington, DC, 1992; p 14.
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Figure 2. NaNO3 tracer breakthrough behavior (open circles) together with the best fit using eq 2 (solid line), showing the quality of the column packing. signals were digitized with an analog I/O board (DT2805/5716, Data Translation, Inc., Marlboro, MA) and then stored on a PC for later processing. The length of the connection tubings (0.8 mm inner diameter) from injector to detector was kept at a minimum in this work in order to reduce convolution effects from the tubings on the measured breakthrough curves (BTCs); the total internal volume (about 0.4 mL) of the connection tubings was very small compared to the pore volume (ca. 60 mL) of the column packing used here. Since a good column packing is essential for extracting particle deposition and release rate coefficients from the measured BTCs, UV absorption tests were carried out using molecular tracers NaBr (Fluka, MicroSelect) and NaNO3 (Merck, p.a.) to check each packed column. The tracer breakthrough concentration was monitored using a UV-vis detector (UV-vis, Linear Model 204) with a flow-through cell capacity of 18 µL. The selected wavelength was 215 nm for NaBr and 220 nm for NaNO3. These two tracers showed identical ideal breakthrough behavior. The quality of the packing in the column used in this work is illustrated in Figure 2, where raw NaNO3 breakthrough data are exemplified, along with the nonlinear least-squares fit using eq 2. The packing porosity, i.e., all pore spaces accessible to the tracer relative to the apparent packing volume, is 0.39, as calculated from the mean tracer travel time (about 20 min, as estimated from the first moment of the tracer BTC). The dispersivity, D/U, obtained from the above fit is 0.28 mm, which is about 70% of the size of the glass beads used. In this study an influent flow rate of 2.82 mL min-1 was used, which corresponds to a true (interstitial) fluid velocity of 2.61 × 10-4 m s-1.
4. Results and Discussion All particle breakthrough curves (BTCs) shown in this section in response to pulse injections are normalized by their respective particle concentration C0, where C0 is equal to the total amount of particles injected into the column divided by the column pore volume. Therefore, the integrated area under a whole particle BTC on a linear scale gives the total percentage of particles recovered at the column outlet.29 The deposition and release kinetics of the latex particles in the packed glass beads were examined as functions of ionic strength. Experimental particle BTCs are illustrated (29) In an actual experiment the percentage of particle recovery can be calculated using (Q/N0) ∫0∞C′(t) dt, where Q is the volumetric fluid flow rate, N0 is the total amount of particles injected, i.e., N0 ) Q × ∫0∞Cb′(t) dt. In the above C′(t) denotes the particle concentration detected at the column outlet, and Cb′(t) is the particle concentration measured from a bypass experiment by short-circuiting the column with a tubing loop. The experimental normalized concentration C(T)/ C0, as used in the pulse-injection breakthrough curves, was equated with (Qtp/N0)C′(t) when plotted against t/tp or (Qt0/N0)C′(t) when plotted against t/t0.
Figure 3. (a) Sample experimental breakthrough curves on a semilogarithmic scale as a function of the reduced time, t/t0 (namely, the number of effluent pore volumes), t0 being the mean travel time of NaNO3: dotted line, ideal NaNO3 tracer; solid line, latex particles (ionic strength I ) 0.033 M). (b) Latex particles (solid line) at I ) 0.092 M with other conditions the same as in (a). As a reference, the same NaNO3 tracer as in (a) is also plotted.
in Figure 3, together with that of NaNO3 for comparison. These curves are plotted on a semilogarithmic scale as a function of the reduced time, t/t0 (namely, the number of effluent pore volumes), where t0 is the mean travel time of the NaNO3 tracer through the column. In this work, t0 is very close to the mean travel time of the nonretarded particles, tp, as can be seen in Figure 3. No significant particle size exclusion effect was observed, probably due to the lack of micropores in the column packing from which the particles were excluded. There are two distinctive differences between the particle and tracer breakthrough behavior (see Figure 3). Firstly, the height of the breakthrough peak for particles is much lower than that for the ideal tracer. Secondly, there is significant tailing in the particle BTC. Since no tailing was present for the NaNO3 tracer, microscopic dead ends or stagnant pores must be negligible in the column. Therefore, as can be understood from the discussion of section 2, the tailing on the particle BTC manifests clearly the spontaneous detachment of the deposited particles from the bead surfaces. It can also be seen from Figure 3 that increase in ionic strength leads to a decrease in the peak height of the particle BTC. This must be due to increased deposition of particles onto the beads as ionic strength was raised. By contrast, change of ionic strength has a rather limited effect on the particle release tail. In theory the rate constants of both particle deposition and release can be extracted by fitting eq 6 to the experimentally measured BTCs, but in practice if no proper weighting is applied, the release tail in the BTCs will be neglected by least-squares fitting procedures since the former has only a minor effect on the sum of the fitted residuals. A weighting factor suitable for the release tail as well as for the nonretarded peak is, nevertheless, difficult to find. Consequently, the particle deposition and release rate constants reported here were estimated on the basis of eq 5 as follows: The overall deposition rate, kd + ki, was calculated from the total fraction of
Colloidal Particles in Porous Media
Figure 4. Observed particle breakthrough behavior (dotted line) as compared with the prediction (solid line) from the convection-dispersion equation (eq 6) using preestimated rate parameters (see text) at an ionic strength of 0.033 M.
nonretarded particles in the column effluent. The latter was calculated by taking twice the area under a BTC as integrated on a linear scale from time zero to the peak position. This effectively avoids influence of the released particles at longer times. On the other hand, when a BTC is plotted on a natural semilogarithmic scale, the absolute value of the slope of the release tail was simply taken as the particle release rate constant multiplied by tp. This simple intuitive method of estimating the rate constants proves to be very reliable as can be seen in Figure 4, where an example is shown of the numerically predicted particle BTC from eq 6 using the rate constants so obtained. Such a method is also very useful since it would help avoid extensive numerical calculations for future applications of the pulse-injection technique. The best visual fit in Figure 4 was achieved by keeping the sum of kd and ki constant while adjusting their individual values (and also the dispersion coefficient). It is worth mentioning that the particle dispersion coefficient at each ionic strength had to be obtained by repeating this fitting procedure. The typical value of D/U was 0.80 mm in this work. Following the tradition in colloid filtration studies, Figure 5 reports the particle collision efficiency30 as calculated from the estimated overall particle deposition rate constant as a function of ionic strength. As can be seen from this figure, below a critical point the particle deposition rate increased steadily with increasing ionic strength. The overall deposition process eventually moved from the slow to the fast regime upon further increasing the ionic strength. The measured fast deposition plateau value is about 0.3 (on the linear scale), compared to unity by theoretical definition. Effects of hydrodynamic forces on the particle collision efficiency have been estimated using the semiempirical expression given by Rajagopalan and Tien (eq 27 in ref 32) and were found to be insignificant for small particles such as those used in this study. The above-observed transition from the slow to the fast particle deposition can be qualitatively explained with (30) The particle collision efficiency, R, was calculated using the colloid-filtration theory31 which takes into account the effects of Brownian diffusion, interception, and gravitational settling, but neglects hydrodynamic effects. Under the experimental conditions used in this study the following relation between R and the overall deposition rate constant, kd + ki, exists: R ) 132.28(kd + ki). (31) O’Melia, C. R.; Tiller, C. L. In Environmental Particles; Buffle, J., van Leeuwen, H. P., Eds.; Lewis Publishers: Pearl River, NY, 1993; Vol. 2, p 353. (32) Rajagopalan, R.; Tien, C. AIChE J. 1976, 22, 523.
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Figure 5. Particle collision efficiency as a function of ionic strength on a double logarithmic scale as obtained from pulseand step-injection experiments.
the DLVO theory. Since both the sulfate latex particle and glass bead surfaces are negatively charged at pH 9.5,9 repulsive forces due to overlapping of the electric double layers surrounding both surfaces will create a potential energy barrier for particle deposition. This barrier will, however, decrease upon increasing ionic strength as a result of the compression of the double layers. Thus, more particles can overcome this barrier and be deposited on to the bead surfaces. At the critical ionic strength the energy barrier disappears, thus leading to the observed fast deposition. The particle release rate constant was found to be almost independent of ionic strength, averaging 6 × 10-4 s-1. This contrasts with an earlier finding on carboxylated latex particles in packed glass beads,33 where the release rate constant was found to remain constant at low ionic strengths but decrease markedly at high ionic strengths. Since both the particle and bead surfaces were different in these two studies, one would need an accurate estimate of the shape of the total potential energy curve for each system in order to understand the observed difference in the release behavior. Although particle release was observed here without even changing solution chemistry or fluid flow conditions, injected particles were not always fully recovered at the column outlet. The percentage of particle recovery depended strongly on ionic strength and ranged from almost 100% at the lowest ionic strength used to about 6% at the highest ionic strength. As a first approximation, it has been assumed here that there were only two different affinity sites on the glass bead surfaces; one of these sites was strongly adsorbing, while the other was less affinitive. This difference in affinity was seen to be most pronounced when ionic strength was increased. In view of the apparent loss of particles, the strongly adsorbing site was, in this work, simply treated as being irreversible. The two-site model describes the experimental data reasonably well (cf. Figure 4). Nevertheless, this may be an oversimplistic picture. It is likely that more than one reversible site should be used in order to describe adequately the particle breakthrough behavior, especially the observed smooth transition between the nonretarded particle breakthrough peak and the release tail. It should be mentioned that nonlinear processes were unlikely to be the source of the apparent loss of particles (33) Yan, Y. D.; Borkovec, M.; Sticher, H. Prog. Colloid Polym. Sci. 1995, 98, 132.
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Figure 6. Examples of particle and tracer breakthrough behavior from step-injection experiments.
Figure 7. Overlap of column responses to different input particle concentrations at an ionic strength of 0.033 M, indicating first-order rate processes in these systems.
found in this work. As will be shown later in this section, it has been confirmed that the particle deposition-release process involved here was linear, i.e., the response of the column was independent of particle concentration over the range used (typically 30 mg L-1). To assess the reliability of the above-described pulseinjection technique, the particle deposition rate constants obtained thereof were compared with those from the more traditionally used step-injection method. Conditions used in the latter case were the same as in the pulse measurements, except that in the step runs, influents containing the sulfate latex particles were pumped through the column by a peristaltic pump and were not degassed. In addition, the column was repacked with fresh glass beads for each step run in an effort to avoid possible bead surface contamination by leftover particles from previous runs. Figure 6 shows the observed BTCs from the step measurements. As can be seen from this figure, the particle breakthrough concentration is highly dependent on the ionic strength of the background solution. Also noticeable from Figure 6 is that the particle breakthrough concentration rises, although very slowly, and no plateau value is reached within the first three pore volumes of observation. Since particle suspensions used in these experiments were very dilute (about 0.5 mg L-1), no area blocking effect34 by the already deposited particles on the bead surfaces was expected. This small rise should, therefore, be attributed to the slow release of particles. As a result, strictly speaking, the commonly used method in step experiment for estimating particle deposition rate constant (or collision efficiency) from a plateau value9 cannot be applied in this case. For these step runs the overall particle deposition rate constant, kd + ki, was estimated also on the basis of eq 5. The percentage of nonretarded particles was obtained as follows: First, the first-order derivative was taken over a normalized BTC from the step injection. The x value corresponding to the maximum in the first-order derivatives could therefore be located. Then, twice the value of the breakthrough concentration at this x value on the original normalized step BTC was taken as the percentage of nonretarded particles. This approach is strictly valid only when linear dynamics were involved in the deposition and release processes. As will be shown, this condition
was indeed met for the systems studied here. The particle collision efficiency as calculated from the estimated overall deposition rate constant from these step experiments is also plotted in Figure 5. Excellent agreement between the results from the two methods demonstrates the reliability of the pulse-injection technique. Finally, it should be pointed out that the methods used here for obtaining the deposition and release rate constants of particles transporting through porous media rely on the assumption that the kinetic processes involved follow the first-order rate law. Therefore, it is necessary to check if this condition was actually satisfied. Figure 7 shows the pulse-injection BTCs for three particle concentrations. Overlap between the different concentrations is a good indication that in this study the particle-bead interaction was a first-order rate process and linear chromatography was observed.
(34) Johnson, P. R.; Elimelech, M. Langmuir 1995, 11, 801.
5. Concluding Remarks This work has demonstrated the use of the pulseinjection technique in studying the transport kinetics of model colloidal latex particles in a packed bed of glass beads. Spontaneous detachment of the attached particles from the bead surfaces has been observed. By employing this model system, it has been shown that the pulseinjection method is a very reliable tool for studying particle deposition and release kinetics in porous media. And it also has many advantages over its step-injection counterpart: The former is cleaner to use and many repeat experiments are possible without having to change the packing media. Particle release can also be more easily identified in the pulse technique. The very small amount of particles required for a pulse-injection measurement warrants the possibility to also study the transport kinetics of colloidal particles present in scarce quantities. One should, therefore, envisage increasing use of this neat method in future studies. Acknowledgment. The author wishes to thank Michal Borkovec and Jaro Ricˇka for many stimulating discussions. Thanks are also due to Ruben Kretzschmar and Markus Schudel for a critical reading of the manuscript and to Hans Sticher for his interest in this work. This research was supported by the Swiss part of the COST D5 program. LA950629T
