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Fines Deposition on Pulp Fibers and Fines Flocculation in a Turbulent-Flow Loop T. G. M. van de Ven,*,† M. Abdallah Qasaimeh,‡ and J. Paris§ Departments of Chemistry and Chemical Engineering, Pulp & Paper Research Centre, McGill University, Montreal, Canada H3A 2A7, and De´ partement de Ge´ nie Chimique, E Ä cole Polytechnique, Montreal, Canada H3T 3A7
Fines retention is a combination of fines deposition on fibers and fines flocculation, followed by entrapment of fines flocs in a forming sheet. In the laboratory, fines flocculation is often studied in a dynamic drainage jar (DDJ) to mimic the hydrodynamic shear on a paper machine. However, the shear in a DDJ is very different from the shear on a machine. A flow geometry that might approximate shear in a headbox of a paper machine better is high Reynolds number flow through a tube because many headboxes contain a series of parallel pipes. We studied the deposition of fines and the flocculation of fines in a flow loop, with flow velocities on the order of a few meters per sceond, using a poly(ethylene oxide)-cofactor retention aid system. We found that fines deposition and flocculation follow the predictions of kinetic theories of Langmuir and Smoluchowski rather well despite the fact that fines are highly polydisperse. Fines were found to be flocculated even in the absence of a retention aid probably because of mechanical entanglements of fibrillar fines. Adding retention aids resulted in further aggregation. The detachment and floc breakup rates were found to be rather high, and extrapolation to papermaking conditions leads to the conclusion that fines deposition and flocculation are negligible in a headbox, at least for the retention aid system considered. This contradicts findings from DDJ experiments, which usually show appreciable fines retention. Perhaps a flow loop better represents flow conditions in a headbox, and a DDJ better represents flow conditions during drainage and formation. Introduction To understand and control fines retention by retention aids on a paper machine, one needs to know how fines deposit on pulp fibers, how fines flocculate, and how flocs of fines are captured in a forming sheet. Although several studies of fines deposition and flocculation are available,1-5 including studies in which poly(ethylene oxide) (PEO) was used as a retention aid6-8 (similar to this study), several important questions remain unanswered. For instance, what is the state of fines aggregation prior to the addition of a retention aid? How do retention aids affect flocculated fines? How much fines deposition on fibers occurs in a headbox and how much in the drainage section of a machine? How much fines flocculation occurs in the short circulation loop? To answer some of these questions, one needs to know the flocculation behavior of fines and fibers under realistic hydrodynamic conditions. Most fines retention studies are performed in a dynamic drainage jar (DDJ)1-8 or simply in a stirred beaker,9,10 but hydrodynamic conditions in such a jar are quite different from those on a paper machine. Another technique to study flocculation is using a flow loop, through which a pulp suspension is passed at industrially relevant velocities. So far, such loops have been used to study fiber flocculation,11,12 but not to study fines deposition on fibers or fines floccula* To whom correspondence should be addressed. Tel.: (514) 398-6177. Fax: (514) 398-6256. E-mail: theo.vandeven@ mcgill.ca. Paprican-NSERC Industrial Research Chair. † Department of Chemistry, McGill University. ‡ Department of Chemical Engineering, McGill University. § E Ä cole Polytechnique.
tion. In this study, we followed the kinetics of fines deposition and flocculation in such a loop, induced by a PEO-cofactor retention aid system, widely used for mechanical-type furnishes with high fines content. To facilitate observations, we somewhat slowed the kinetics of deposition and flocculation by performing experiments with lower pulp consistencies and lower velocities than those used in industry but, nevertheless, for velocities high enough for the flow to be turbulent. Materials and Methods 1. Fines and Fibers. To simulate realistic conditions, a pulp suspension was taken from an actual mill and separated in the long fiber fraction and fines fraction, using a float-wash unit with a 200-mesh screen. The pulp was a mixture of mechanical pulp, sulfite and kraft pulps, and deinked pulps. An analysis of the fractions showed that the concentration of the long fiber fraction in the suspension was 4.44 g/L and that of the fines fraction was 5.05 g/L. The supernatant further contained dissolved and colloidal substances. The fraction of colloids that did not pass a 300-nm filter was 0.08 g/L. The fines and fiber fractions were washed prior to use. 2. Retention Aid System. As a retention aid system, we used PEO with a cofactor. PEO (Floc 999) has a mass-average molecular weight of 6 × 106, and the cofactor (Interac 1323) was a mixture of modified phenolic resin and sulfonated kraft lignin.9 Both PEO and the cofactor were provided by EQUIP Inc., Montreal, Canada. PEO-cofactor complexes are known to be good flocculating agents and, at the low concentra-
10.1021/ie0499456 CCC: $30.25 © 2005 American Chemical Society Published on Web 12/30/2004
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Figure 1. Schematic of the flow loop.
tions used, are expected to adsorb on fibers and fines, with negligible concentrations in the supernatant.9 3. Flow Loop. The flow loop in which the flocculation experiments were performed is shown schematically in Figure 1. The length of the tube is 5.2 m, and the inner diameter of the tubing is 4.0 cm. The suspension was circulated by a specially designed centrifugal pump, which is open at the top. A cylindrical tank was mounted (height of 8 cm and width of 20 cm) on top of the pump to allow for easy filling of the loop. Velocities in the loop were varied from 1.5 to 2.5 m/s, corresponding to Reynolds numbers in the range of (0.6-1.0) × 105, implying that the flow was fully turbulent. A bypass stream (sampling loop) was passed through a photometric dispersion analyzer (PDA), via 3-mm-innerdiameter tubing. The velocity in the bypass loop was about 30 cm/s, and it took about 0.5 s for the suspension to reach the PDA detector after entering the bypass. The entrance of the tubing contained a screen (200-mesh), through which only the fines could pass, but not the fibers. For fines flocculation experiments, the screen was omitted in most experiments. Fines aggregates can grow to sizes exceeding the mesh size, but under our experimental conditions, fines deposition on fibers is faster than fines flocculation (see below), and fines flocculation is negligible during fines deposition. An injection point in the loop allowed for injection of the retention aids. 4. PDA and Effects of Fines Flocculation on PDA Readings. The PDA measures fluctuations in turbidity, which provide insight into the flocculation process. It measures the root-mean-square (rms) fluctuations in transmittance, Trms, the average transmittance, T (dc signal in Figure 1), and their ratio, R.13 The fines concentration can be obtained from T, using a calibration curve, which is not very sensitive to the state of fines flocculation. Changes in R are mainly due to fines flocculation, and the slope of the initial increase of R with time is taken as an apparent flocculation rate. Repeat experiments of fines flocculation performed with the PDA show a reproducibility of the initial slope of R readings (after addition of the flocculant) better than 5%.14 Before performing experiments in the loop, we wanted to know how the state of fines flocculation (in the absence of retention aids) affects the ratio reading R. To elucidate this, we suspended fines in a beaker with a paddle stirrer and circulated some of the suspension through the PDA. The results for a 0.11% fines suspen-
Figure 2. PDA ratio of a fines suspension (0.11%) in the absence of a retention aid as a function of shear in the beaker (stirring) or in the bypass stream (tube shear).
sion are shown in Figure 2. The shear can be modified in two ways: by the rotational speed of the paddle stirrer (rpm) and by the velocity with which the suspension flows through the tubing (tube shear), which passes through the PDA. Increasing the shear in either way causes a decrease in the ratio R. A plateau was reached at R = 0.3. These results imply that at low shear fines are in a flocculated state. With increasing shear, these flocs break down, and at high shear, fines are present as individual particles. The flocculation at low shear, which occurs in the absence of any flocculating agent, is likely due to mechanical entanglements of fibrillar fines, similar to fiber flocculation. This entanglement of fibrillar fines was proposed earlier,15 to explain the anomalous flocculation of fines fully coated by poly(ethylenimine). Because fines are in a different state of flocculation at different shear rates, this prevents a direct comparison of flocculation results at different shear conditions, based on the ratio R (as is customary). Instead, we calculate the apparent number, n, of fines particles (or flocs) (a measure of flocculation) from13
[
ln Vo/V n ) np ln V/Vp
]( ) 2
Rp R
2
(1)
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that system the fines detachment was ascribed to polymer transfer from fibers to fines.17 Except for the decrease in fines deposition at long times, the deposition follows the trend expected from Langmuir kinetics:18
dθ/dt ) Rdepkfast(n0 - θ)(1 - θ) - kdetθ
Figure 3. Fines deposition on fibers detected by a PDA cell and displayed as a dc reading for various fiber/fines ratios indicated in the figure. In run 1, no retention aid was used. Table 1. Amount of Fines Deposited on Fibers
run
[fibers] (g/L)
[fines] (g/L)
[cofactor] (mg/g solids)
1 2 3 4
1.1 0.55 1.1 1.1
0.979 0.475 0.880 0.852
0 0.18 0.18 0.5
[PEO] (mg/g solids)
t (s)
Γm (mg of fines/ g of fiber)
0 0.04 0.04 0.1
22 9 8
0 36 23 18
Here np is the total number of fines (in a well-dispersed state). V’s are the voltages associated with the transmittance: Vo for pure water, V for a given fines suspension, and Vp for a fully dispersed suspension. R and Rp are the ratio readings for a given suspension and one that is fully dispersed. Results and Discussion 1. Fines Deposition on Fibers. The loop was filled with a mixture of fines and fibers at various consistencies, and the suspension was circulated with various speeds. dc readings (V), corresponding to the average transmittance, were taken with the PDA. At steadystate conditions, i.e., when the PDA readings were steady, a cofactor was added to the loop, followed several seconds later by the addition of PEO. PEO injection was done gradually and took about 3 s (closely corresponding to one-fluid circulation in the loop). Adding the cofactor did not affect the PDA readings, but PEO injection resulted in near-instantaneous changes. The results are shown in Figure 3. The concentration of fibers, fines, cofactors, and PEO are shown in Table 1, together with the amounts of fines deposited on the fibers Γm (at the maximum), calculated from the dc signal and the calibration curve. The times τ to reach the maximum are shown as well. The maximum amounts of deposited fines are rather small but significant. Even for the smallest increase (run 2), considering that experimental errors in the slope are about 5%, the initial slope is (1.5 ( 0.1) × 10-3 s-1. Theory (see below) predicts that the deposition of fines reaches a plateau. Instead, Figure 3 shows that the fines detach after reaching a maximum. The same has been observed for the deposition of clay on fibers by PEO and was ascribed to the flattening of adsorbed PEO, resulting in polymer layers too thin to bridge.16 A similar detachment from fibers was observed for fines, using cationic poly(acrylamide) (CPAM) as a retention aid. For
(2)
Here θ is the fractional coverage of fibers by fines, defined as θ ) Γ/Γmax, with Γ being the amount of fines adsorbed on fibers and Γmax the maximum amount of fines that can adsorb on fibers, which equals about 250 mg of fines/g of fiber.10 The deposition rate constant is Rdepkfast, which is the product of the deposition efficiency Rdep and the fast rate constant kfast; kdet is the detachment rate constant, and n0 is the initial concentration of fines, relative to the concentration corresponding to a full coverage of fibers by fines. For the fines concentrations in Figure 3, n0 is in the range 3.1-3.6. From Table 1, we can see that Γ ) Γmax and thus θ ) n0 and 1. Equation 2 can then be simplified to
dθ/dt ) Rdepkfastn0 - kdetθ
(3)
which has the solution
θ)
Rdepkfastn0 (1 - e-kdett) kdet
(4)
Fitting the data from run 2 to the above equation and neglecting the fines detachment at long times, one obtains Rdepkfast = 2.0 × 10-3 s-1 and kdet = 0.05 s-1. The fact that more fines detachment occurs at long times implies that kdet increases with time but on a time scale much longer than the initial deposition. The deposition efficiency, Rdep, can be estimated from the PEO dosage and the surface area of fines. Assuming a maximum adsorption capacity of PEO of 0.6 mg/m2,19 assuming a surface area of fibers of 0.3 m2/g,19 and that of fines of 10 m2/g,15 and assuming that the adsorption of PEO on fibers and fines scales with their volume fraction,18 one obtains that the coverage by PEO on fibers is θ1 = 0.1 and that on fines is θ2 = 0.003 (for runs 2 and 3). According to standard polymer bridging theory, the deposition efficiency can be estimated from
Rdep ) θ1(1 - θ2) + θ2(1 - θ1)
(5)
and thus Rdep = 0.1 for run 2. Hence, kfast = 0.02 s-1. This result can be compared with the theoretical fast rate at which spheres deposit on slender spheroids:20
kfast ) βd
cF GVF GφF N ) βd ) βd G π F π πFF
(6)
Here VF is the volume of a fiber, NF is the number of fibers per unit volume, φF and cF are the fiber volume and weight fraction, and FF is the apparent density of the fiber (including the water in the lumen and fiber wall). G is the effective shear rate, and βd is the collision efficiency for fines-fiber collisions. Assuming FF = 1.3 kg/L, one obtains that βdG = 150 s-1 for a flow velocity of 1.5 m/s. A similar calculation for run 3 yields kfast = 0.03 s-1 and βdG = 125 s-1. The ratio of velocity to radius yields a macroscopic shear rate of Gm ) 1.5/0.02 ) 75 s-1 and a turbulent shear rate (defined as the velocity of a critical eddy divided by its length, as defined in the isotropic turbulent theory) of G ) Gm
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xRe = 2 × 104 s-1, implying that βd is somewhat less than 1%. This value is similar to values calculated for small spheres interacting with slender spheroids in simple shear.20 Thus, the results of fines deposition on fibers are consistent with Smoluchowski and Langmuir kinetics, with a collision efficiency βd of somewhat less than 1% and with a deposition efficiency Rdep of about 10%, determined by the fractional coverage of fibers by PEO. Runs 3 and 4 were performed with similar fines and fiber concentrations, but 2.5 times more PEO and cofactor were used in run 4. Surprisingly, there is little effect on the fines deposition. Theory would predict that θ1 and thus also Rdep should be 2.5 times as high, and thus the initial slope should be about 2.5 times as high as well. Instead (cf. run 4 in Figure 3), the slopes are almost the same. The reason for this is not clear. It has been found that the efficiency of PEO depends on its dissolution history,21 which determines whether PEO is present in the form of single coils or as entanglements. Possibly PEO at higher concentrations is more entangled, which might lead to a lower coverage than that for well-dissolved PEO. Another effect that might contribute is how uniform PEO is distributed over the fibers. Adding more concentrated PEO might lead to a less uniform distribution. Because fines (which are larger than 75 µm) are relatively large (compared to the sizes of PEO molecules, typically of size around 100 nm), the fiber-fines contact area (of a fines particle adsorbed on a fiber) is large and can contain hundreds of molecules if the polymer coverage on fibers is dense. However, the fines particle may deposit also if the polymer density on the surface is far less. Thus, the rates for fines deposition on densely and less densely covered areas could be similar, implying that the overall deposition rate is highest when polymer adsorption is uniform. If this explanation is true, eq 5 does not apply to fines deposition. It could be that Rdep ) 1, for both θ ) 0.1 and 0.25. This would imply that βd is 10 times smaller than that estimated above. For runs 2 and 3, the detachment rate kdet = 0.050.10 s-1, implying that fines stay on the fiber on average 10-20 s before they detach. This may seem like a long time, but the detachment rate is 2-3 times as fast as the deposition rate. This is the reason the maximum fines adsorption is rather small. The deposition rate is expected to increase with shear nearly linearly, whereas the detachment rate is expected to increase exponentially with shear.18 Extrapolating to shear conditions on a paper machine, one predicts that the detachment rate will be much larger than the deposition rate. This suggests that in the headbox of a paper machine, where the shear is much larger than that in our flow loop (in which fines deposition is already small), the deposition of fines on fibers is expected to be negligible. 2. Fines Flocculation in a Flow Loop. The results for fines flocculation in a flow loop are presented in Figure 4, which shows how the ratio reading R of the PDA varies with time after the addition of PEO. These runs were done without a screen in the bypass stream. An additional run was done with a screen (not plotted), which showed a lower apparent flocculation rate and a longer time to reach steady state. This shows that fines flocculate to sizes that exceed 75 µm. The different ratio readings at t ) 0 (t ) t1) show that fines are flocculated to various extents prior to the addition of a retention aid. As explained above, this is
Figure 4. Flocculation of fines in the flow loop at various flow rates. For all runs, the cofactor (0.25 mg/g fines) was added first to the fines (0.1%), followed by PEO (0.1 mg/g fines). The initial flocculation rates were determined from the initial slopes. Table 2. Fines Flocculation Rates v V1 V2 (m/s) (mV) (mV) R1 1.5 2.0 2.5
7.1 6.9 6.9
R2
6.95 1.89 2.25 6.7 0.65 1.61 6.7 0.65 1.61
t2 - t 1 kfl ) (n1 - n2)/ n1∆t (s-1) (s) n1/np n2/np 48 24 24
0.025 0.020 0.075 0.035 0.25 0.034
0.0043 0.022 0.034
likely due to mechanical entanglements of fibrillar fines. The initial slopes in Figure 4 were converted to slopes in the apparent number of fines using eq 1. From the slope, one can obtain the flocculation rate constant, kfl, because the Smoluchowski theory22 predicts that for short times
n ) no(1 - t/τfl)
(7)
with τfl being the half-time of the flocculation process, given by τfl ) 1/kfl. The estimated rate constants are shown in Table 2 and vary from 0.4 × 10-2 to 3.4 × 10-2 s-1. These rate constants can be compared with those of the Smoluchowski theory, valid for the coagulation of spherical particles, modified by a hydrodynamic collision efficiency βfl and a flocculation efficiency Rfl given by
Rfl ) 2θf(1 - θf)
(8)
Here θfl is the fractional coverage of fines by PEO, which for the conditions in Figure 4 is estimated as 0.03 and thus Rfl = 0.03. With these modifications, the rate constant equals
4 kfl ) RflβflGφf π
(9)
Assuming that βflG = βdG = 125 s-1 (at v ) 1.5 m/s), it follows that kfl = 0.4 × 10-2 s-1. This is close to the value in Table 2, indicating that the Smoluchowski theory, modified by a polymer bridging efficiency, reasonably accounts for the flocculation rates of fines, despite the fact that fines have very complicated shapes and are polydisperse in size. The time period t2 - t1 in Table 2 is the time of the initial linear increase in R (t1 ) 0). Table 2 shows that at higher flow velocities the flocculation rate increases, in agreement with theory. Even at v ) 2.5 m/s, not all fines are initially dispersed, but fines flocs contain only about 4 fines particles on average (n1/np = 0.25). By extrapolation, it
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can be concluded that all fines will be well-dispersed in the headbox of a paper machine in the absence of retention aids. It is not clear from the data in Figure 4 whether fines aggregates break up or not. For intermediate velocities, a maximum is observed, but at other velocities a plateau appears to be reached. If this is due to a dynamic equilibrium between floc formation and breakup, then the breakup times are comparable to the times a plateau is reached, i.e., kbr = 0.03 s-1 for v ) 2.5 m/s (with kbr being the breakup rate constant). By the time steady state is reached, the number of fines particles per aggregate has changed from 4 to about 30 (cf. Table 2). At high shear rates on a paper machine, one would expect the number of fines per floc to be much less. Trying to extrapolate to high shear is dicey, but the observation that an increase from 1.5 to 2.5 m/s results in a decrease in the number of fines particle per floc from about 50 to 30 suggests that the number of fines particles per floc in a headbox will be rather small. Also, floc breakup rates are of a magnitude similar to that of the rates for fines detachment from fibers, for which we also concluded that fines deposition is expected to be negligible. Conclusion We have shown that the rates of fines deposition on fibers and of fines flocculation in a flow loop, induced by a PEO-cofactor retention aid system, closely follow the theory of shear-induced collisions. Fines deposition is followed by a detachment of fines, implying that the detachment rate “constant” is not constant but increases with time, most likely because of PEO flattening on the surface. Extrapolating to the much larger shear conditions in the headbox of a paper machine, one reaches the conclusion that both fines deposition and flocculation are likely to be very small. This suggests that most of the flocculation and deposition occurs after the papermaking suspension leaves the slice of the headbox. These conclusions are for PEO-cofactor retention aid systems; other retention aids are expected to have similar adsorption and attachment kinetics but might have different desorption and detachment kinetics. Experiments in a DDJ usually show appreciable fines retention, implying that shear in a flow loop is rather different from shear in a DDJ. Perhaps a flow loop better represents flow conditions in a headbox (including the headbox approach section, fan pump, etc.), and a DDJ better represents flow conditions during drainage and formation. Acknowledgment This work was financed by the NCE on “Wood Pulps” and by a NSERC/CRD grant. The industrial partners were Hydro-Que´bec, Paprican, and three Quebec newsprint mills. The authors thank EQUIP Inc. (Baie d’Urfe´, Quebec, Canada) for providing PEO and cofactor.
Literature Cited (1) Britt, K. W. Paper Sheet Formation: Observations Concerning Retention. Tappi J. 1981, 64 (5), 53. (2) Pelton, R. H.; Allen, L. H.; Nugent, H. M. Measuring Fines Retention of Newsprint Pulps (with the Dynamic Drainage Jar). Pulp Pap. Can. 1979, 80 (12), 125. (3) Klungness, J. H.; Fahey, D. J. Pulp Floc Stability of Hardwood-Softwood Mixtures. South. Pulp Pap. 1980, 43 (12), 37. (4) Gerischer, G.; van Wyk, W. J. Effectiveness of Polyelectrolytes as Retention and Drainage Aids in the Presence of Kraft Lignin. Papier 1988, 42 (3), 129. (5) Xu, J.; Bousfield, D. W. Effect and Modeling of Shear Rate and Drainage Rate on Retention of Fines. Tappi Papermaking Conf. 1993, 1, 95. (6) Pelton, R. H.; Allen, L. H.; Nugent, H. M. Survey of Potential Retention Aids for Newsprint Manufacture. Prepr. CPPA Annu. Meeting (Montreal) 1979, 65B, 163. (7) Pelton, R. H.; Allen, L. H.; Nugent, H. M. Factors Affecting the Effectiveness of Some Retention Aids in Newsprint. Sven. Papperstidn. 1980, 9, 251. (8) Braun, D. B.; Ehms, D. A. Filler and Fiber Retention in Newsprint and Groundwood Specialty Using Poly(Ethylene Oxide). Tappi J. 1984, 67 (9), 110. (9) Carignan, A.; Garnier, G.; van de Ven, T. G. M. The Flocculation of Fines by PEO/Cofactor Retention Aid Systems. J. Pulp Pap. Sci. 1998, 24 (3), 94. (10) Asselman, T.; Garnier, G. Dynamics of Polymer-Induced Hetero-Flocculation of Wood Fibres and Fines. Colloids Surf. A 2000, 174, 297. (11) Swerin, A.; Sjo¨din, U.; O ¨ dberg, L. Flocculation of Cellulosic Fiber Suspensions by Model Microparticulate Retention Aid Systems. Nord. Pulp Pap. Res. J. 1993, 8 (4), 389. (12) Kerekes, R. J.; Schell, C. J. Effects of Fiber Length and Coarseness on Pulp Flocculation. Tappi J. 1995, 78 (2), 133. (13) Abdallah Qasaimeh, M. Role of polymer entanglements in poly(ethylene oxide) induced fines flocculation. Ph.D. Thesis, Department of Chemical Engineering, McGill University, Montreal, Canada, 2001. (14) Gregory, J. Turbidity Fluctuations in Flowing Suspensions. J. Colloid Interface Sci. 1985, 105, 357. (15) Porubska´, J.; Alince, B.; van de Ven, T. G. M. Homo- and Heteroflocculation of Papermaking Fines and Fillers. Colloids Surf. A 2002, A210, 223-230. (16) van de Ven, T. G. M.; Alince, B. Heteroflocculation by Asymmetric Polymer Bridging. J. Colloid Interface Sci. 1996, 181, 73. (17) Asselman, T.; Garnier, G. Mechanism of Polyelectrolyte Transfer during Heteroflocculation. Langmuir 2000, 16 (11), 4871. (18) van de Ven, T. G. M. Particle Deposition on Pulp Fibers: The Influence of Added Chemicals. Nord. Pulp Pap. Res. J. 1993, 1 (8), 130. (19) van de Ven, T. G. M. A Model for the Adsorption of Polyelectrolytes on Pulp Fibers: Relation between Fiber Structure and Polyelectrolyte Properties. Nord. Pulp Pap. Res. J. 2001, 15 (5), 494. (20) Petlicki, J.; van de Ven, T. G. M. Shear-Induced Deposition of Colloidal Particles on Spheroids. J. Colloid Interface Sci. 1992, 148 (1), 14. (21) van de Ven, T. G. M.; Abdallah Qasaimeh, M.; Paris, J. PEO-Induced Flocculation of Fines: Effects of PEO Dissolution Conditions and Shear History. Colloids Surf. A 2004, submitted for publication. (22) Smoluchowski, M. Towards a Mathematical Theory of the Kinetics of Coagulation of Colloidal Dispersions. Z. Phys. Chem. 1917, 92 (29), 129.
Received for review January 16, 2004 Revised manuscript received October 12, 2004 Accepted October 18, 2004 IE0499456
