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Comparison of Particle Deposition in a Parallel Plate and a Stagnation Point Flow Chamber Junlin Yang,† Rolf Bos,† Albert Poortinga,† Poppo J. Wit,† Gerald F. Belder,‡ and Henk J. Busscher*,† Department of Biomedical Engineering, University of Groningen, Bloemsingel 10, 9712 KZ Groningen, The Netherlands, and Philips Research Laboratories, Professor Holstlaan, 4 5656 AA Eindhoven, The Netherlands Received November 17, 1998. In Final Form: April 7, 1999 Deposition of negatively charged polystyrene particles on quartz in a parallel (PP) and stagnation point (SP) flow chamber is compared at similar Peclet numbers for different ionic strengths and pH values. Initial deposition rates have been derived from the experiments and numerically calculated from the convective-diffusion equation. An upper limit for deposition, neglecting interaction forces, is obtained from the Smoluchowski-Levich solution of the convective-diffusion equation. Initial deposition rates were on average 5 times higher in the SP than in the PP flow chamber, in accordance with theoretical results. Effects of ionic strength on the initial deposition rates and blocked areas were observed more strongly in the SP flow chamber than in the PP flow chamber, while the dependence of deposition on pH was absent in both flow chambers. Desorption probabilities of the polystyrene particles on the quartz surfaces were close to zero. The experimental reproducibilities obtained in the SP flow chamber were superior compared with those found in the PP flow chamber. Analysis of error propagation for the SP data could account for the final reproducibility observed, but for the PP data the final reproducibility could not be accounted for on the basis of system variabilities and enumeration errors, leading to the conclusion that mechanistically deposition in the PP flow chamber depends more on statistical processes, like diffusion and collisions between flowing and adhering particles, while deposition in the SP flow chamber is more interaction controlled.
Introduction Particle deposition on collector surfaces plays an important role in many technological and natural systems.1 By consequence, particle deposition is receiving attention from both an applied2,3 and a fundamental point of view.4-6 Different flow chamber devices have been developed to study particle deposition under controlled mass transport conditions, such as the rotating disk system,7-9 the stagnation point flow chamber,1,6 and the parallel plate flow chamber.5,10,11 Since it has been demonstrated theoretically12 and experimentally13 that the passage of a collector surface with adhering colloidal particles through a liquid-air interface can remove nearly all adhering particles, in situ observation techniques and image analysis have become indispensable in particle deposition studies. * Corresponding author. Fax: (+31)-50-3633159. E-mail:
[email protected]. † University of Groningen. ‡ Philips Research Laboratories. (1) Adamczyk, Z.; Zembala, M.; Siwek, B.; Czarnecki, J. J. Colloid Interface Sci. 1986, 110, 188. (2) Boluk, M. Y.; Van de Ven, T. G. M. Colloids Surf. 1990, 46, 157. (3) McCarthy, J. F.; Zachara, J. M. Environ. Sci. Technol. 1989, 23, 496. (4) Adamcyk, Z. Colloids Surf. 1989, 35, 283. (5) Adamczyk, Z.; Van de Ven, T. G. M. J. Colloid Interface Sci. 1981, 80, 340. (6) Charis, K.; Rajagopalan, R. J. Chem. Soc., Faraday Trans. 2 1985, 81, 1345. (7) Dabros, T.; Adamczyk, Z. Chem. Eng. Sci. 1979, 14, 1041. (8) Prieve, D. C.; Lin, M. M. J. Colloid Interface Sci. 1980, 76, 1178. (9) Rajagopalan, R.; Kim, J. S. J. Colloid Interface Sci. 1981, 83, 428. (10) Sjollema, J.; Busscher, H. J. Colloids Surf. 1989, 47, 382. (11) Sjollema, J.; Busscher, H. J.; Weerkamp, A. H. J. Microbiol. Meth. 1989, 9, 73. (12) Leenaars, A. F. M. In Particles on Surfaces 1; Mittal, K. L., Ed.; Plenum Press: New York, 1988; p 361. (13) Noordmans, J.; Wit, P. J.; Van der Mei, H. C.; Busscher, H. J. J. Adhesion Sci. Technol. 1997, 11, 957.
Mass transport in the parallel plate flow chamber is slow, and the convective-diffusion equation for the parallel plate flow chamber is difficult to solve as compared to for example that for the stagnation point flow chamber. On the other hand, the parallel plate flow chamber is conceptually simpler than the stagnation point flow chamber. Although several theoretical and experimental studies of particle deposition in parallel plate and stagnation point flow chambers are available in the literature,14,15 to out knowledge, no comparison of colloidal particle deposition in both systems has been made on an experimental basis. The aim of this paper is to present a comparison of polystyrene particle deposition in a parallel and stagnation point flow chamber under conditions of identical Peclet numbers (i.e. the ratio between convective and diffusional mass transport is the same). Mass Transport Equations for the Parallel Plate and Stagnation Point Flow Chambers Mass transport in a parallel plate and in a stagnation point flow chamber is governed by convection, diffusion, and migration under the influence of substratum-particle interaction and described by the so-called convectivediffusion (CD) equation5,16,17
c )0 (D‚∇φ kT )
∇‚(uc) - ∇‚(D‚∇c) - ∇‚
(1)
(14) Adamcyk, Z.; Van de Ven, T. G. M. J. Colloid Interface Sci. 1981, 84, 497. (15) Dabros, T.; Van de Ven, T. G. M. Physicochem. Hydrodyn. 1987, 8, 161. (16) Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1962. (17) Peters, M. H. J. Colloid Interface Sci. 1990, 76, 32.
10.1021/la981607k CCC: $18.00 © 1999 American Chemical Society Published on Web 05/27/1999
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where u is the particle velocity vector, c is the particle concentration, D is the particle diffusion tensor, kT is the product of the Boltzmann constant and the absolute temperature, and φ denotes the potential function of an external force working on the particles. The particlecollector surface interaction potential is assumed to be the sum of a Lifshitz-van der Waals interaction potential (φLW), an electrostatic interaction potential (φel), and a potential due to gravity and buoyancy (φgr).18 The van der Waals contribution is given by
φLW )
-A132 6H
(2)
where A132 is the Hamaker constant for the interaction of particle 1 in medium 3 with planar surface 2 and H is the distance between the surface of the particle and the collector surface normalized with respect to the particle radius ap
H)
y -1 ap
(3)
where y is the distance between the center of the particle and the substratum. Assuming constant surface potentials, the electrostatic interaction potential reads
φel ) π0ap(ζ21 + ζ22)
[
2ζ1ζ2
ζ21
+
ζ22
ln
(
)
1 + exp(-κapH)
1 - exp(-κapH)
+
]
ln(1 - exp(-2κapH)) (4) where is the relative dielectric constant of the liquid phase, 0 is the permittivity of a vacuum, ζ denotes the zeta potential, and κ is the inverse double-layer thickness. Finally, the combined effect of gravity and buoyancy follows from
4 φgr ) πa4p(Fp - Fl)gH 3
(5)
where g is the acceleration of gravity and Fp and Fl are the densities of the particle and the liquid, respectively. It is impossible to obtain analytical solutions of the convective-diffusion equation although several numerical solutions exist for both types of flow chambers.5,16 Apart from exact, numerical solutions, approximate solutions have been forwarded, like the Smoluchowski-Levich solution.19 Approximate Solutions. In the SmoluchowskiLevich approximate solution, it is assumed that the hydrodynamic interaction between a colloidal particle and a collector surface is counterbalanced by attractive Lifshitz-van der Waals forces, while all other external forces are neglected. Furthermore, it is assumed that all particles move with the undisturbed fluid velocity and that interception plays no role in deposition.19 Therewith, the Smoluchowski-Levich approximate solution to the convective-diffusion equation usually provides an upper limit for mass transport in the flow chamber system under consideration. Table 1 summarizes some expressions for mass transport in parallel plate (PP) and stagnation point (SP) flow chambers, including the Peclet number Pe, denoting the (18) Visser, J. Surf. Colloid Sci. 1976, 8, 3. (19) Elimelech, M.; Gregory, J.; Jia, X.; Williams, R. Particle Deposition & Aggregation, Measurement, Modelling and Simulation; Butterworth-Heinemann Ltd.: Woburn, MA, 1995; pp 68-156.
Table 1. Mass-Transport Equations for the Parallel Plate (PP) and the Stagnation Point (SP) Flow Chambers, Including the Peclet Number Pe, the Initial Deposition Rate j0SL According to Smoluchowski-Levich, and the Wall Shear Rate σ20,21 a PP Pe
3
3Vma
2b2D∞
)
SP
3Qppa
4wb3D∞
j0SL
D∞c 2 b Pe ‚ 0.89a 9 x
σ
3 Qpp 2 b2‚w
(
3
)
1/3
2asa 1.1Qsp3/2ν-1/2R-9/2a3 ≈ D∞ D∞ 3
cD∞ 1/3 ‚Pe a
0.616‚ Rsr
a The approximate equation for the Pe number is only valid for the specific SP chamber and here. Nomenclature: Vm, mean particle velocity in cell; a, particle radius; b, half depth of a parallel plate flow chamber; Qpp and Qsp, flow rate; D∞ , particle diffusion coefficient; w, the width of the flow chamber; ν, kinetic viscosity; r, the longitudinal distance from the stagnation point; Pe, Peclet number; x, the longitudinal distance from the flow chamber entrance; c, particle concentration; Rs, a constant that can be calculated numerically, which depends on the Reynolds number and the geometry of the system.
ratio between convective and diffusional mass transport, the theoretical initial deposition rates j0SL, according to Smoluchowski-Levich, and the wall shear rate σ.20,21 Exact Numerical Solutions. If the fluid flow field (v) over a collector surface is known, the particle velocity components can be determined from
uy ) f1(H) f2(H)vy
(6)
ux ) f3(H)vx
(7)
where the subscripts x and y denote components parallel and perpendicular to the collector surface, respectively. In addition, the perpendicular and parallel particle diffusion components depend on the particle-collector distance
Dx ) f1(H)D∞
(8)
Dy ) f4(H)D∞
(9)
where D∞ is the bulk diffusion coefficient of the particles and f1, f2, f3, and f4 are universal hydrodynamic correction functions accounting for the effect of hydrodynamic interactions between the particle and the collector surface.19,22 Parallel Plate Flow Chamber. Since the width of a parallel plate flow chamber is much larger than its depth, the fluid velocity can be expressed as
y 3 y vx ) Vm 2 2 b b
(
)
(10)
where Vm is the mean fluid velocity and b is half the distance between the plates of the parallel plate flow chamber. Under the assumption that particles diffuse only in the y direction, the convective-diffusion equation reads23 (20) Elimelech, M. Sep. Technol. 1994, 4, 186. (21) Kamiti, M.; Van de Ven, T. G. M. Colloids Surf. 1995, 100, 117. (22) Sanders, R. S.; Chow, R. S.; Masliyah, J. H. J. Colloid Interface Sci. 1995, 174, 230. (23) Sjollema, J.; Busscher, H. J. J. Colloid Interface Sci. 1989, 132, 382.
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∂C* ∂x* ∂ ∂φ* ∂C* + C* f (H) ) 0 (11) ∂H 1 ∂H ∂H
Pe f3(H)(H + 1)[2 - (H + 1)A]
[ [
]]
in which A ) ap/b, x* ) x/b, φ* ) φ/kT, C* ) C/C0, and Pe is the Peclet number. Both the hydrodynamic interaction functions and the interaction potential vary strongly in the region close to the collector surface, which makes it convenient to neglect convection in this region. Consequently in the region 0 < H < δ23 at H ) δ
-f1(δ)
[∂C* ∂H |
H)δ
| ]
∂φ* ) ∂H H)δ C*(H)δ) (12) δ exp(φ*(H)) dH exp(-φ(δ)) 0 f1(H)
+ C*(H)δ)
∫
where δ is a parameter denoting the thickness of a layer adjacent to the collector surface over which convection is neglected. Equation 12 implies a “perfect sink” boundary condition, that is
C* ) 0 for H ) 0
(13)
Assuming gravitational forces can be neglected, another boundary condition can be formulated for the center of the flow chamber
b ∂C* ) 0 for H ) -1 ∂H ap
(14)
Equation 11 has been solved numerically using the boundary conditions given by eqs 12 and 14, applying a central difference scheme with a nonuniform mesh for the y coordinate and explicit discretization with a constant step size for the x coordinate. In the y direction the step size decreases for decreasing H. The step size in the x direction is decreased by the computer program until a stable solution is obtained. Stagnation Point Flow Chamber. For the stagnation point flow chamber, close to the stagnation point, the convective-diffusion equation reads20
∂ (f (H) f2(H) Pe (H + 1)2C*) ∂H 1 ∂φ* ∂C* ∂ + C* f (H) )0 2f3(H) Pe (H + 1)C* ∂H 1 ∂H ∂H (15)
[ [
]]
Discretization of eq 15 was also done using a central difference scheme on a nonuniform step size. As boundary conditions, the perfect sink condition (eq 13) was used together with
C* ) 1 for H f ∞
(16)
Near the collector surface, large particle concentration gradients can occur, depending on the shape of the interaction potential. The computer program employed allocates the mesh points in such a way that the changes in the interaction potential and in the first derivative of the interaction potential between mesh points (∆φ* and ∆(∂φ*/∂H), respectively) obey
|∆φ*| < 1
| ( )| ∆
∂φ* ∂φ* ∂2φ* < 10-2 max for >0 ∂H ∂H i ∂H2
[( ) ]
(17) (18)
Table 2. Zeta Potentials (mV) of Polystyrene Particles, Determined by Particulate Microelectrophoresis and of Quartz Collector Surfaces from Streaming Potential Measurements for Different Ionic Strength KNO3 Solutions and pH Values, as Used in the Deposition Experiments (( Denotes the Standard Deviation over Triplicate Experiments) ζ pH
10 mM
3.0 5.6
Polystyrene Particles -52 ( 3 -41 ( 3 nd -51 ( 3
-25 ( 5 nd
Quartz -12 ( 4 -17 ( 5
-7 ( 1 nd
3.0 5.6
-2 ( 1 nd
60 mM
100 mM
where i numbers the mesh points. Each time new mesh points are inserted, the complete mesh is smoothened to eliminate abrupt changes in the step size. Using this procedure, stable, accurate solutions of the convectivediffusion equation can be obtained. Materials and Methods Polystyrene Particles and Collector Materials. Polystyrene particles (VS2A, AKZO Research, Arhem, The Netherlands) with a diameter of 783 nm were prepared as described by Brouwer and Zsom24,25 and were kindly provided by Dr. R. Zsom, AKZO Research, The Netherlands. The particles were washed twice by centrifugation in demineralized water and subsequently suspended in potassium nitrate (Merck, Darmstadt, Germany) solutions with ionic strengths of 10, 60, and 100 mM and a pH of 3.0 or 5.6. Particles were suspended to the concentration 1 × 108 cm-3, as determined in a Bu¨rker-Turk counting chamber. Table 2 shows the zeta potentials of the polystyrene particles in the different solutions as measured by particulate microelectrophoresis26 (a Laser Zee Meter 501 (PenKem, Bedford Hills, NY)). Quartz collector surfaces were cleaned thoroughly by sonication in a 2% surfactant RBS 35 (Fluka Chemie AG, Buchs, Switzerland), followed by alternately rinsing with methanol and demineralized water. Advancing type water contact angles on thus prepared quartz surfaces were 26° in accordance with the literature,27 while the zeta potentials of the quartz, obtained from streaming potential measurements,28 are listed in Table 2. X-ray photoelectron spectroscopy29 showed a minor carbon contamination of the quartz (11.2%), while 60.0% oxygen and 28.9% silicon were measured. Parallel Plate (PP) and Stagnation Point (SP) Flow Chambers and Image Analysis System. The PP11 and SP1 flow chambers used are schematically presented in Figure 1, together with their essential dimensions. For the PP flow chamber, deposition was observed on the center of the bottom plate with a CCD-MXR camera (High Technology, Eindhoven, The Netherlands) mounted on a phase contrast microscope (Olympus BH-2), while for the SP flow chamber deposition was observed in the stagnation point with a dark-field microscope (Leitz) equipped with a CCD-LDH camera (Philips, Eindhoven, The Netherlands). Live images were grabbed with a PC-Vision+ frame grabber and Gauss and Sharp filtered after subtraction of an out of focus image. Thereafter, deposited particles were discriminated from the background by single-gray-value thresholding. This yielded binary black and white images which were subsequently stored on disk for later analysis. (24) Brouwer, W. M.; Zsom, R. L. J. Colloids Surf. 1987, 24, 195. (25) Goodwin, J. W.; Hearn, J.; Ho, C. C.; Ottewil, R. W. Colloid Polym. Sci. 1974, 252, 464. (26) Wiersema, P. H.; Loeb, A. L. J. Colloid Interface Sci. 1966, 22, 78. (27) Janczuk, B.; Zdziennicka, A. J. Mater. Sci. 1994, 299, 3559. (28) Van Wagenen, R. A.; Andrade, J. D. J. Colloid Interface Sci. 1980, 76, 305. (29) Amory, D. E.; Genet, M. J.; Rouxhet, P. G. Surf. Interface Anal. 1988, 11, 478.
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Figure 1. Schematics of the PP and SP flow chambers used in this study, together with their essential dimensions. Experiments were carried out three times at the flow rate 0.05 cm3 s-1 for the PP and 0.00883 cm3 s-1 for the SP flow chamber, corresponding with wall shear rates of 22.5 and 17.3 s-1 at the jet radius, respectively, and a Peclet number of 3.95 × 10-3. Data Analysis. The total number of adhering particles per unit area n(t) was recorded as a function of time by image sequence analysis.30 The initial deposition rate j0 of the depositing particles was determined from the initial increase of n(t) with time, while the later stages of the deposition process were assumed to be described by
n(t) ) j0τ(1 - e-t/τ)
(19)
β ) 1/τ - j0A1
(20)
where A1 is the area blocked by an adhering particle, as can be calculated from the radial pair distribution function g(r), and β is the desorption rate coefficient, assumed to be independent of the residence time in the present study. The expression 1/τ can be calculated directly from the measured initial deposition rate (j0) and the number of particles adhering in a stationary end point (n∞), while subsequently combination with the initial deposition rate and blocked area yields the possibility to determine β. Blocked areas A1 can be derived from the radial pair distribution function g(r), which describes the relative density of adhering particles around a given center particle as a function of interparticle distance
g(r) )
F(r,dr) F0
(21)
where F(r,dr) is the density of adhering particles in a shell with thickness dr at a distance r from a center particle. Each adhering particle is taken once as a center particle. The region for which g(r) < 1 is taken as the blocked area A1,31 as illustrated schematically in Figure 2. (30) Wit, P. J.; Busscher, H. J. Colloids Surf., A 1995, 125, 85. (31) Sjollema, J.; Busscher, H. J. Colloids Surf. 1990, 47, 337.
Figure 2. Analysis of the spatial arrangement of all adhering particles, in which each adhering particle is taken once as a center particle, yielding the relative density of adhering particles as a function of the interparticle distance r. The region where g(r) < 1 is associated with the blocked area A1.
Results As an example, Figure 3 shows the deposition kinetics of the polystyrene particles on quartz from a 60 mM KNO3 solution (pH 3.0) both for the PP and the SP flow chamber. As can be seen, initial deposition in the SP flow chamber is faster than that in the PP flow chamber, while furthermore the curves seem to indicate that, at least within the time scale of the experiments, final surface coverage in the SP flow chamber will be higher than that in the PP flow chamber. Tables 3 and 4 summarize the quantitative data derived from the experimental deposition kinetics and spatial distributions as well as from the approximate and exact numerical solutions of the CD equation. From both tables it can be seen that the experimental initial deposition rates in the SP flow chamber are 5 times higher on average than those in the PP flow chamber. In line, also the Smoluchowski-Levich initial deposition rates and those calculated by numerically solving the convective-diffusion equation demonstrate 4.4 and 4.7 times higher deposition rates in the SP flow chamber, respectively. The ap-
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Langmuir, Vol. 15, No. 13, 1999 4675
initial deposition rates clearly increase with increasing ionic strength, especially for the stagnation point flow chamber. Experimental initial deposition rates are nevertheless always smaller than the initial deposition rates from the Smoluchowski-Levich approximation, which is independent of ionic strength. Stationary end-point numbers of adhering polystyrene particles do not show a consistent trend with ionic strength. The blocked areas in the PP flow chamber decrease 2-fold upon increasing the ionic strength of the solution from 10 to 60 mM, likely due to decreased electrostatic repulsion between adhering and flowing particles, but are independent of ionic strength when measured in a SP flow chamber. Desorption increases with ionic strength in both flow chambers. Discussion Figure 3. Example of the deposition kinetics for polystyrene particles on a quartz collector in a 60 mM KNO3 solution (pH 3.0) as studied in a PP and SP flow chamber. Table 3. Experimental and Theoretical Initial Deposition Rates j0, Numbers of Particles Adhering in a Stationary End Point n∞, Blocked Areas A1 Calculated from the Radial Pair Distribution Functions, and Desorption Rate Probabilities β from a 60 mM KNO3 Solution (pH 3.0 and 5.6) Measured in a Parallel Plate Flow Chamber (PP) and a Stagnation Point (SP) Flow Chambera pH 3.0 PP j0 (cm-2 s-1) n∞ (106 cm-2) 1/τj0 (µm2) 1/τ (10-5 s-1) A1 (µm2) β (10-5 s-1)
pH 5.6 SP
PP
Experimental Results 313 (114 1299 ( 92 387 ( 132 3.4 ( 0.7 9.8 ( 0.3 4.7 ( 1.2 20.9 ( 7.3 8.1 ( 0.8 15.8 ( 5.5 6.5 ( 4.7 10.5 ( 0.8 6.1 ( 4.2 2.9 ( 0.8 1.9 ( 0.4 3.4 ( 0.5 5.6 ( 4.1 8.0 ( 1.1 4.8 (3.6
j0SL (cm-2 s-1) 400 j0NUM (cm-2 s-1) 247
Theoretical Results 1770 400 1155 247
SP 1348 ( 104 14.6 ( 1.2 4.7 ( 1.1 6.3 ( 2.0 2.0 ( 0.3 3.6 ( 1.4 1770 1155
a All experiments were done at the Pe number 3.95 × 10-3 for both systems in triplicate (( denotes standard deviation).
proximate Smoluchowski-Levich initial deposition rates are generally higher than the experimentally observed ones, but the numerically calculated initial deposition rates are similar to the experimentally observed ones within the experimental accuracy. As noted above, stationary end-point numbers of adhering particles appear higher in the SP flow chamber than in the PP flow chamber within the time scale of the experiment. Blocked areas calculated for the SP flow chamber are smaller than those for the PP flow chamber, especially at low ionic strength, presumably because convective flow is directed toward the surface in the SP flow chamber. Desorption is somewhat higher in the SP flow chamber than in the PP flow chamber for the high ionic strength solutions, although it is emphasized that the desorption rate coefficients calculated are very small, on the average ∼×10-5 s-1. Finally, with regard to the comparison of PP and SP flow chambers, Tables 3 and 4 show that, particularly on a percentage basis, the standard deviations over triplicate experiments are smaller for the SP data than for the PP data. For the initial deposition rates, for instance, the percentage standard deviations over triplicate runs are 9% for the SP and 36% for the PP chamber. pH is of little or no influence on the deposition data measured (see Table 3). However ionic strength has a substantial influence (see Table 4). The experimental
This paper compares experimental results and theoretical predictions of particle deposition in PP and SP flow chambers. With regard to the effects of pH and ionic strength on deposition, both systems reveal the same trends. However, the effects appear greater in the SP chamber than in the PP chamber, which indicates that deposition in the SP chamber is more controlled by interaction forces than that in the PP chamber. Moreover, the reproducibility in the SP chamber appears superior to the one that could be achieved in the PP chamber. The reproducibility in initial deposition rates has, in our experiments, never been better than 20-30% over three separate experiments in a PP chamber,32 as also found here. A formal analysis of error propagation in initial deposition rates, based on the parameters occurring in the Smoluchowski-Levich solution, shows the extent to which deposition rates and their accuracy depend on Qpp, a, w, b, x, η, T, R, and c (see Table 5). On the basis of the estimated errors, it is calculated, however, that the accuracy for the PP chamber is similar to the one that is achieved in the SP chamber, the main source of variability being the particle concentration c. All other sources of variability occurring in the Smoluchowski-Levich solution are nearly negligible. An additional source of variability not accounted for in the above is the enumeration error, which can be taken as the square root of the number of particles counted per image. For the PP chamber, this yields an additional variability of around 5%, similar to what has been reported before on the basis of enumerations taken by the human observer as a gold standard.32 For the SP chamber, the enumeration error is around 2-3%. Although these calculations fully account for the reproducibility observed in SP data, they leave a major portion of the variability in the PP data unaccounted for. Therefore, it is concluded that, mechanistically, statistical processes, such as diffusion and collisions between flowing and deposited particles,33 have a stronger influence on particle deposition in a parallel plate flow chamber than on deposition in a stagnation point flow chamber (see Figure 4 for schematics) or that, as concluded before, interaction forces dominate the deposition process more strongly in the SP flow chamber than in the PP flow chamber. In addition, depositing particles in the PP chamber arrive at a grazing angle, which may influence their chance upon rejection. Experimental initial deposition rates j0 and theoretical values calculated numerically from the convective-diffusion equation compare well for both the PP and SP flow chambers, while the initial deposition rates from the Smoluchowski-Levich approach are only slightly higher (32) Wit, P. J.; Busscher, H. J. J. Microbiol. Meth. 1998, 32, 281. (33) Meinders, J. M.; Busscher, H. J. Langmuir 1995, 11, 327.
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Table 4. Experimental and Theoretical Initial Deposition Rates j0, Numbers of Particles Adhering in a Stationary End Point n∞, Blocked Areas A1 Calculated from the Local Radial Distribution Functions, and Desorption Rate Probabilities β from Different KNO3 Solutions at pH 3.0 Measured in a Parallel Plate (PP) and a Stagnation Point (SP) Flow Chambera 10 mM
60 mM
PP
SP
j0 (cm-2 s-1) n∞ (106 cm-2) 1/τj0 (µm2) 1/τ (10-5 s-1) A1 (µm2) β (10-5 s-1)
195 ( 74 2.5 (1.3 22.4 ( 6.7 4.4 ( 3.0 3.4 ( 0.5 3.7 ( 2.7
1182 ( 155 14.7 ( 0.6 4.7 ( 0.8 5.6 ( 1.7 1.5 ( 0.1 3.8 ( 1.4
j0SL (cm-2 s-1) j0NUM (cm-2 s-1)
400 247
1770 1155
a
PP Experimental Results 313 ( 114 3.4 ( 0.7 20.9 ( 7.3 6.5 ( 4.7 2.9 ( 0.8 5.6 ( 4.1 Theoretical Results 400 247
100 mM SP
PP
SP
1299 ( 92 9.8 ( 0.3 8.1 ( 0.8 10.5 ( 1.8 1.9 ( 0.4 8.0 ( 1.1
220 ( 80 2.1 ( 1.2 32.8 ( 16.9 7.2 ( 6.3 1.5 ( 0.3 6.9 ( 6.1
1535 ( 138 8.1 ( 0.5 11.0 ( 1.4 16.9 ( 3.6 1.5 ( 0.1 14.6 ( 1.9
1770 1155
400 247
1770 1155
All experiments were done at the Pe number 3.95 × 10-3 for both systems in triplicate (( denotes standard deviation).
Table 5. Parameters Involved in the Error Propagation of the Initial Deposition Rates According to Smoluchowski-Levich, Together with the Estimated Errors in the Individual Parameters and the Final Accumulated Error for the PP and the SP Flow Chambers PP parameter c Q T x w a b R η accumulated error a
estimated error (%) 9.0 4.0 1.7 16.7 5.3 3.1 7.7 naa 1.0 12.2
SP factor 1 1/3 2/3 1/3 1/3 2/3 2/3 2/3
estimated error (%) 9.0 4.0 1.7 na na 3.1 na 2.0 1.0 9.6
factor 1 1/2 2/3 2/3 1/2 2/3
na is not applicable.
than those experimentally found. Therefore, it can be concluded that the deposition efficiency of polystyrene particles on quartz is relatively high. Also Varennes and van de Ven34 observed good agreement at low shear rate between the experimental and theoretical deposition in a stagnation point flow collector. Under the present experimental conditions, the polystyrene particles are highly negatively charged, irrespective of pH and ionic strength, while the quartz collector surface is far less charged. Consequently, the initial deposition rates obtained by solving numerically the convective-diffusion equation are the same for all pH values and ionic strengths. This indicates that, under the measured conditions, electrostatic interactions are too small to influence deposition and deposition is above a critical threshold. Blocked areas are smaller in the SP flow chamber than in the PP chamber, most likely as a result of the greater convection toward the collector surface. The desorption probabilities found here for quartz in the PP chamber are about 100-fold smaller than those observed by Meinders et al.35 on glass using individual particle tracking. Individual particle tracking for the present experiments on quartz showed essentially zero desorption, likely because it is too small to be detected with the number of particles present within one microscopic field of view. The small desorption probabilities reported here are based on (34) Varennes, S.; Van de Ven, T. G. M. Colloids Surf. 1988, 4, 63. (35) Meinders, J. M.; Busscher, H. J. Colloids Surf., A 1993, 80, 279.
Figure 4. Collisions between flowing and adhering particles may yield a particle velocity component away from the collector surface, and the particle subsequently has to bridge a relatively great distance by diffusion in order to reach the collector surface (parallel plate flow chamber). In the stagnation point flow chamber, there is always a convective mass-transport component toward the collector surface.
combining the deposition kinetics and the blocked area from the radial pair distribution functions. Although it has been argued not only that radial pair distribution functions reflect blocking but also that there are influences of surface chemical heterogeneity on the spatial arrangement of the adhering particles, quartz is likely to be more homogeneous than glass used before. Hence, we expect that it will be reasonable to assume that on quartz the radial pair distribution functions reflect blocking. Summarizing, this study demonstrates that similar conclusions regarding particle deposition at different pH values and ionic strengths can be obtained from experiments in a PP and in a SP flow chamber. However, the
Particle Deposition
influence of experimental conditions such as pH and ionic strength is greater in the SP chamber than in the PP chamber, while SP chamber experiments also appear more reproducible. This is probably related to mechanistic differences in deposition between both flow chambers, with deposition in the SP flow chamber more strongly controlled
Langmuir, Vol. 15, No. 13, 1999 4677
by interaction forces and less by statistical processes, such as diffusion and collisions between flowing and adhering particles, than that in the PP flow chamber. LA981607K
