Langmuir 1996,11, 327-333
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Influence of Interparticle Interactions on Blocked Areas and Desorption during Particle Deposition to Glass in a Parallel Plate Flow Chamber J. M. Meinders and H. J. Busscher" Laboratory for Materia Technica, University of Groningen, Bloemsingel 10, 9712 Kz Groningen, The Netherlands Received May 16, 1994. In Final Form: September 13, 1994@ In this paper we measured the influence of interparticle interactions between flowing and adhering particles on initial deposition rates, blocked areas, and residence time dependent desorption duringdeposition of polystyrene latex particles to glass. The initial deposition rates increased linearly with particle concentrations in suspensions as expected from theoretical mass transport calculations. Deposition efficiencies of 0.78and 0.62were obtained for wall shear rates, G, of 15 and 50 s-l, respectively. Blocked areas, A1 expressed as the number of particle cross sections, y, decreased from initially high to low values = 10 anddlp=ao= 18,at infinite particle concentration) as a function of the particle concentration. This decrease in blocked area with particle concentration was explained as a consequence of the increased backscattering of flowing particles colliding with adhering particles. Backscattering due to collisions between flowing particles is obviously more frequent at higher concentrations. Blocked areas, calculated using pair distribution functions, corresponded well with those obtained from deposition kinetics. Initial desorption rate coefficients, POranging from 0.3 x to 2.0x s-l, increased linearly with particle concentration in suspension due to an increase in the number of collisions between adhering and flowin particles per unit time. The final desorption rate coefficients,P,, rangingfrom 0.002x to 0.01x 10s-l, and the relaxation time, 116, of the desorption rate coefficient, ranging from 50 to 1000 s, decreased as a function of the particle concentration because, as the effective interaction potential minimum is not the same for each adheringparticle, weakly adheringparticles will be removed first. The effective interaction potential minimum decreases as a function of shear rate due to increased hydrodynamic drag and lift forces resulting in higher desorption rate coefficients at higher shear rates.
f
Introduction Deposition of colloidal particles to solid surfaces is of importance in a broad field of applications. Depending on the situation, adhesion or, oppositely, lack of adhesion and desorption of colloidal particles is required. For example in industrial bioreactors irreversible adhesion of microorganisms or cells to carriers is required, whereas in biomedical applications microorganisms are prevented from adhering to surgical instruments or implants.lS2Also, pigment particles in paint and paper applications are supposed to adhere strongly to collector surface^.^ Therefore, numerous experimental setups, e.g. the impinging jet system4 or parallel plate flow chamber^,^'^ have been developed to investigate deposition of colloidal particles to collector surfaces. During deposition the number of adhering particles will reach a stationary state, predominantly due to blocking of particles residing on the collector surface.'j8 Analysis of the number of adhering particles as a function of time may yield the initial deposition rate while comparison of the deposition rates observed with calculations based on the convective-diffusion equation results in information about the interaction between collector surface and depositing/adhering p a r t i c l e ~ . ~Also J ~ the number of adhering particles in the stationary state can be analyzed Abstract published in Advance ACS Abstracts, December 1, 1994. (1)Meadows, P.S. Arch. Mikrobwl. 1971, 75,374. (2) Weiss, L.; Dimitrov, D. S. Cell Biophys. 1984,6,9. (3)Van de Ven, T. G. M. J . Colloid Interface Sci. 1988,124, 138. (4)Dabros, T.; Van de Ven, T. G. M. Colloid Polym. Sei. 1983,261, 694. (5)Bowen, B. D.; Levine, S.; Epstein, N. J . Colloid Interface Sci. 1976,54,375. (6)Absolom, D. R.;Strong, A. B.; Ledain, C.; Thompson, B. E.; Zingg, W. Trans. Am. SOC.Artif. Intern. Organs 1982,28,413. (7)Ruckenstein, E.; Marmur, A.; Gill, N. N. J . Theor.Biol. 1976,58, 439. (8)Schaaf, P.;Talbot, J. J . Chem. Phys. 1989,91,4401.
using pair distribution functions, providing direct information on the interaction between adhering particles.1l However, desorption can have a major influence on deposition kinetics as well, since the number of adhering particles in a stationary state will be lower than expected, due to desorption.12 In the first approach to determine interactions between colloidal particles and collector surfaces, the number of adhering particles can be directly obtained by using a microscope, photographs, or computer-aided image analysis,13whereas the desorption is usually measured after a deposition experiment by changing the suspension to a particle-free solution. A major drawback of this method is that adsorption and desorption rates derived cannot be combined to describe the entire kinetics involved in deposition processes, since in these procedures the particle concentration gradient between collector surface and suspension is changed and because collisions between flowing and adhering particles are absent. Therefore adsorption and desorption must be measured simultaneously. Meinders et all2 developed a method using computer-aided image analysis to determine not only adsorption and desorption simultaneously but also the residence time of desorbing particles.14J5 It is the aim of this paper to determine the influence of interparticle interactions on blocked areas and desorption (9)Adamczyk, 2.; Van de Ven, T.G. M. J . Colloid Interface Sei. 1981, 80,340. (10)Adamczyk, 2.; Siwek, B.; Zembala, M.; Belouwschek, P. Adu. Colloid Interface Sei. 1994,48,151. (11)Sjollema, J.; Busscher, H. J. Colloid Surf. 1990,47,337. (12)Meinders, J. M.; Noordmans, J.; Busscher, H. J. J . Colloid Interface Sei. 1992,152,265. (13)Meinders, J.M.;VanderMei, H. C.; Busscher, H. J.J . Microbiol. Meth. 1992,16,119. (14)Dabros, T.; Van de Ven, T. G. M. J . Colloid Interface Sei. 1982, 89,232. (15)Dabros, T.; Van de Ven, T. G. M. J. Colloid Interface Sci. 1983, 93,576.
0743-7463/95/2411-0327$09.00/00 1995 American Chemical Society
328 Langmuir, Vol. 11, No. 1, 1995
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of adhering particles using the method described by Meinders et a1.12 Measurements are performed in a parallel plate flow chamber using glass as a collector surface, while the (polystyrene) particle concentration in suspension and the shear rate are varied.
Theory In a first-order approach, depositing particles and particles already adhering to a collector surface are assumed not to interact with each other. Therefore the deposition rate of colloidal particles to the collector surface is a constant and the total number of adhering particles, n(t), is initially a linear function of time t according to
in which j o is the initial deposition rate. Theoretically, the number of particles arriving per unit time a t a collector surface can be calculated using the convective-diffusion equation. Assuming all arriving particles adhere to the collector surface, i.e. neglecting the influence of all interaction forces between particles and the surface upon the mass transport, the initial deposition rate becomesg
in which a is the particle radius, b the flow chamber half depth, D , the Stokes-Einstein diffusion coefficient, c the particle concentration in suspension a t the flow chamber entrance, x the longitudinal distance from the flow chamber entrance, and Pe the Peclet number defined as the ratio between convectiveand diffusion controlled mass transport toward the collector surface
(3) in which Q is the volume ofthe suspension passing through the flow chamber per unit time, w the width ofthe parallel plate flow chamber, and G the wall shear rate assuming a Poiseulle flow between the parallel plates. From eq 2 it can be seen that the initial deposition rate is a linear function of the particle concentration in suspension, c, and thus defining a mass transfer coefficient, K O , also referred to as the adsorption rate ons st ant,^ can be written as
j , = k,c
(4)
The initial deposition rate can be expressed in terms of a dimensionless mass transfer coefficients, the Shenvood numb er
repulsion between a collector surface and the depositing particles, Q is usually smaller than unity. Blocking of deposition by an adhering particle will decrease the adsorption rate, J&(t), according to14
(7) in whichAl is the blocked area per particle. The blocked area depends mainly on electrostatic and hydrodynamic interactions between a flowing and a n adhering particle and is often assumed to be independent of the particle concentration in suspension. The deposition rate also decreases due to desorption
in whichjd,,(t) is the desorption rate. Since desorption of a particle can be residence time dependent, i.e. the probability of desorption of a n adhering particle can change during the time a particle adheres to a surface (“bond aging“), the desorption rate can be described as14J5
in which /3(t - t) is the desorption rate coefficient, t the time a t which a particle was adsorbed and t the time a t which a particle desorbed. Though exact forms of /3(t z) are not known, it seems reasonable to propose that @(t - t) changes exponentially from an initial value 6 0 to a final desorption rate coefficient p, with a relaxation time 1/6,14e.g.
(10) Desorption of adhering particles is stimulated by collisions between flowing and adhering particles in a concentration-dependentfashion.16J7Neglecting all other interactions between particles and particles-surface and assuming that only the top half of an adhering particle suffers collisions, we can approximate the number of such collisions per second, f C l 7 fc
= 2cG
16 S,2a( ( 2 ~-)x 2~) dx = cGa3 3
(11)
Furthermore, each collision has a duration time t,; hence per unit time each particle suffers collisions during fd,. When we separate the desorption rate coefficient, p, in a component pCdue to collisions and a component Pncdue to other factors, then
3
with The ratio between the observed deposition and the theoretically calculated one according to eq 2 denotes the deposition efficiency ad. Hence
in which P, is the chance per unit time that a collision will yield desorption and k, a desorption rate constant. As can be seen from eq 13, the desorption rate coefficient, ,&, increases linearly with the particle concentration in suspension. ~~
in which the asterisk denotes the theoretically calculated values. Since in most practical cases there is electrostatic
(16) Varennes, S.; Van de Ven, T. G. M . Physicochem. Hydrodyn. 1987,9, 537. (17) Van de Ven, T. G. M . Colloid Su$. 1989,39,107.
Interparticle Interactions
Langmuir, Vol. 11, No. 1, 1995 329
Equation 14 can be further evaluated using eq 10 for p(t - r),to yield the number of adhering particles as a function of time12J4
Glass plates, used as collector material, were extensively washed with surfactant RBS in demineralized water, followed by a rinsing procedure with demineralized water. After this cleaning procedure, a zero water contact angle was observed on all glass plates, and the zeta potential (by streaming potentials) in the 50 mMpotassium nitrate solutionwas - 19f 1mV. Further surface characteristics of these prepared glass plates have been described in detail by Sjollema and Busscher.2l Experiments were carried out in 3-fold at flow rates of 0.034 and 0.11 cm3s-l, equivalent to wall shear rates of 15 and 50 s-l and Peclet numbers of 2.09 x and 6.96 x respectively. Data Analysis. To obtain j30,&, and 1/6,a least-squares fitting procedure of eq 10 to the data points of j3(t - t) was employed. Subsequently,BO,j3. and 1/6 were held fmed during a least-squares fitting procedure of eq 15 to the data points of n(t),yielding the initial adsorption rate,jo, and the blocked area,
where
A1.’2
Combining eqs 7-9, we can obtain the deposition rate as a function of time14
The spatial arrangements of adhering particles were analyzed according to their pair distribution functions. The radial pair distribution function, g(r), describes the relative density of particles in a circular shell around a central particle and is given by
1
and
A2 = (Ado
+ Bo +
+
- 46(Aj0 /3J
(17)
Materials and Methods The Parallel Plate Flow Chamber. The deposition experiments were carried out in a parallel plate flow chamber, which has been described by Sjollemaet aZ.18 h an important feature, the flow chamber, with dimensions 16 x 8 x 1.8 cm, has a gradually changing inlet and outlet region. This facilitates the establishment of laminar flow in the center of the chamber. The bottom and top plates, with dimensions of 5.5 x 3.8 cm, are made out of glass. A Teflon spacer between the two plates yields a separation distance of 0.06 cm. A pulse-free flow was created by hydrostatic pressure, and the suspension was recirculated by a roller pump. The entire flow chamber is placed on the stage of a phase contrast microscope (Olympus BH-2) equipped with a 40x objective with an ultralong working distance (Olympus ULWDCD Plan 40 PL). A CCD camera (CCD-MX High technology, Eindhoven, The Netherlands) is mounted on the phase contrast microscope and is coupled to an image analyzer (TEA, imagemanager, Difa, Breda, The Netherlands), installed in a 33-MHz IBM AT personal computer. With this setup, direct observation ofthe depositionprocessin situ is possible without any additional shear forcesacting on the deposited particles. Full details of the experimental setup are given in refs 13, 18, and 19. Enumeration of the total number of adhering particles as well as of the number of adsorbing and desorbingparticles was carried out on the bottom plate of the flow chamber according t o the method describedby Meinderset al .l2J3 Briefly,during deposition images are seized and after background subtraction, Laplace filtering, and thresholding, stored on disk. Further analysis consisted of comparison of successivelystored images, resulting in the total number of adhering particles, their times of arrival, and desorption and the desorption rate coefficient B(t - t)as a function of the residence time. Polystyrene Particles, Glass Collector, and Experimental Conditions. Monodispersepolystyrene particles (W-148, Akzo Research, Amhem, The Netherlands) with a diameter of 736 nm were kindly provided by Dr. R. Zsom, Akzo, The Netherlands. The particleswere washed twiceby centrifugation in demineralized water and suspended to concntrations ranging from 0.4 x lo8 to 4.0 x lo8 mL-’ in a 50 mM potassium nitrate (KNO3) solution. Zeta potentials of this suspension of the polystyrene particles (by particulate microelectrophoresis) at different concentrations averaged to -68 f 4 mV. Further surface characteristics as well as the preparation of the polystyrene particles have been described by Brouwer and Zsom.20 (18)Sjollema, J.; Busscher, H. J.; Weerkamp, A. H. J. Microbiol. Methods 1989, 9, 73. (19) Sjollema, J.; Busscher, H. J.; Weerkamp, A. H. J . Microbiol. Methods 1989, 9, 79. (20) Brouwer, W. M.; Zsom, R. L. J. Colloids Surf. 1987, 24, 195.
(18) in which r is the radius of the shell, dr the thickness of the shell, e(r,dr)the density in the shell, and eo the average density of the total field of view.’l Since,g(r)assumes a radial symmetry, we used the local pair distribution function,g(xy ) in this paper which can distinguish between upstream and downstream deposition because the times of arrival and desorption of each adhering particle are known, yielding an asymmetrical distribution function
(19) in which g ( x , h z y , A y )is the density in the area Ax6y at position (3iy). For statistical reasons, g(xy)has only been calculated for the higher particle concentrations in suspension,viz. 2 1.6 x lo8 mL-1 and at the end of the deposition experiment. Results Figure 1(top) shows the initial deposition rate,j o , as a function of the particle concentration in suspension for the two different shear rates employed. The solid lines represent least-squares fits between the data points and eq 4,resulting in experimental mass transfer coefficients cm k0,~=15 = 237 x cm s-l and k 0 , ~ = 5 0= 283 x s-l, respectively, which can be compared with the theocm retical mass transfer coefficients k * 0 , ~ = 1 5= 304 x s-l and k * o , ~ = 5 0= 454 x cm as calculated from eqs 2 and 4. Hence, using eq 6, we obtain deposition efficiencies, Q, of 0.78and 0.62for wall shear rates of 15 and 50 s-l, respectively. Figure 1 (bottom) presents the blocked areas AI, expressed in particle cross sections, y , as a function of the particle concentration in suspension. Note that at low concentrations A1 is largest and that it decreases to a constant value a t higher concentration. This constant value appears to be influenced by the wall shear rate applied (see also Table 1). The initial desorption rate coefficient,PO,in suspension is presented in Figure 2 (top) as a function of the particle concentrationfor both shear rates applied. It can be seen that P o increases linearly with particle concentration (see also Table 1). From a linear least-squares fit to the data points, we obtain an intercept value, which we associate with Pncand a slope from which we calculate (see eq 13) cm3 cm3s-’ and k c , ~ = 5= 0 321 x kc,G=15 = 149 x (21) Sjollema, J.; Busscher, H. J. Colloids Surf. 1990, 47, 323.
Meinders and Busscher
330 Langmuir, Vol. 11, No. 1, 1995 3 -2 -1 10 cm s ]
io[ 1.5
5
2.0
I
"
0
1
2
3
4
5
4
5
Concentration [ lo8 cm R m [ 1o%-'l
1501
\ 0 5OO
I
\
1
2
4
"
5
0
1
2
3
Concentration 1l o 8 cm
3 8
Concentration 110
1ia[1o3sl
Figure 1. Initial deposition rate, j o , and the blocked area, AI, of polystyrene particles adhering to a glass collector, as a function of the polystyrene particle concentration in 50 mM potassium nitrate solution (0,15s-l; M,50 s-l). The solid lines are least-squares fits between the data points and eq 4 (top) and an exponentially decaying function (bottom). Data points are averages of triplicate runs, with a standard deviation injo of f15% (top) and in AI of 20% (bottom). Table 1. Blocked Area, AI, Initial Desorption Rate Coefficients, BO,and Final Desorption Rate Coefficients, B-, Obtained from Figures 1 and 2, Respectively at Zero Particle Concentration, i.e. No Interparticle Interactions between Flowing and Adhering Particles, and at Infinite Particle Concentration
shear rate 15
concentration
120 10 1132 18
C - P
C-m
50
A1 (cross sections)
C--0"
c--
BO
s-l)
(x
287 -b)
533 -b)
(x
B9-l)
6.5 0.9 9.5 1.3
a AI,& may have a predominantly mathematical significance. Within the concentration range employed, BO increases linearly with concentration.
s-l. These values allow one to calculate the ratio between the desorption chances P, a t different wall shear rates from eqs 11 and 13 while assuming t, = G-' 'c,G=50 'c,G=15
- kc,G=50 - 2.2 kc,G=15
(20)
Thus at a shear rate of 50 s-l a collision is 2.2 times more effective in stimulating desorption than a t a shear rate of 15 s-l. The intercept values of P o at c = 0 in Figure 2 represent the initial desorption rate coefficients in the absence of collisions, and equal 287 x s-l and 533 x s-l for wall shear rates of 15 and 50 s-l, respectively. Figure 2 (middle) displays the final desorption rate coefficient, p-, as function of the particle concentration in suspension. Note that ,& decreases with increasing concentration in contrast to PO(see also Table 1). The relaxation time, 116,of the exponential decay of the desorption rate coefficient from an initial to a final value is also plotted against the particle concentration in Figure 2 (bottom) and shows an initial decrease a t low particle
1.00
0*75 0.50
I\
tt Concentration 1C8cm -31
Figure 2. Initial desorption rate coefficient,BO(top), the final desorption rate coefficient, /3- (middle),and their relaxation time, 1/6(bottom) of polystyrene particles adheringt o a glass collector as a function of particle concentration in 50 mM potassium nitrate solution (0,15s-l; M,50 s-l). The solid lines represent linear least-squares fits between the data points (top), and a least-squares fit between the data points and an exponentially decaying function (middle and bottom). Data points are averages of triplicate runs, with a standard deviation of f20%.
concentrations which levels off to a constant value at higher particle concentrations. The number of adhering particles in stationary state as a function of the particle concentration is presented in Figure 3. Note that the number of adhering particles in stationary state increases with increasing particle concentration reaching a final value which depends on the shear rates used. From eqs 4 and 15 it can be shown that (21)
Eq 21 has been used to draw a line through the data points in Figure 3, using the values of ko and A1 derived from Figure 1, and two different approaches to estimate &, Le., that it varies with concentration as measured in Figure 2 (solid lines) and that there is no desorption, Le., & = 0 (dashed lines). The pair distribution functionsg(x)andgb), being slices of g(xy) (see Figure 4) parallel and transverse to the direction of flow, respectively, are presented in Figure 5 for both shear rates applied. The examples presented are representative for all pair distribution functions measured a t the two shear rates employed. Note that g(x) is asymmetrically shaped, i.e. fewer particles adhere behind than in front of a n adhering particle, a feature which is more pronounced a t the high shear rate (Figures 4, bottom,
Langmuir, Vol. 11, No. 1, 1995 331
Interparticle Znteractions
20
t
,--.--r--i Concentration [ 10 8cm-3]
Figure 3. Number of adheringpolystyrene particles adhering on glass in the stationary state, n(-), as function of the particle concentration in a 50 mM potassium nitrate solution (0,15s-l; B, 50 s-l). The solid line has been calculated on the basis of eq 22 and the data obtained from Figures 1and 2, while the dashed lines have been drawn assuming zero desorption. Data points are averages of triplicate runs having a standard deviation in n(m)of 15%.
0-
'e V
and 5, right side). The blocking distance a t which g(x) attains its average valueg(x)= 1 amounts to 3.6 and 4.0 particle radii upstream and 5.2 and 7.2 particle radii downstream of a n adhering particle, for wall shear rates of 15 and 50 s-l, respectively. Similarly, a blocking distance of 3.5 particle radii can be derived from g(y), irrespective of the shear rate. The shape of the blocked area is highly irregular (Figure 4), but nevertheless blocked areasA1 of 16 and 20 particle cross sections can be estimated for G = 15 and G = 50 s-l, respectively, as averaged for all particle concentrations 21.6 x lo8 mL-l which correspond well with those obtained from deposition kinetics a t high particle concentration (Figure 1, bottom).
Discussion Using a parallel plate flow chamber and image analysis techniques, we were able to measure simultaneously the adsorption and desorption of colloidal particles to and from a collector surface. In particular, by varying the particle concentration in suspension, we could determine the influence of interparticle interactions, i.e. collisions between flowing and adhering particles, on the blocked areas and the desorption rate coefiicients. The number ofparticles adheringin the stationary state varies with concentration (see Figure 3). Since substitution of the values derived forjdc), Al(c), and /3&) into eq 21 yields lines that match the data points well within the accuracy possible for conditions of repulsive electrostatic interactions, it is concluded that the analysis method is consistent. Since in Figure 3 the lines calculated neglecting desorption also match the data points rather well, it can furthermore be concluded that desorption has a minor effect on the number of adhering particles in the stationary state. Also, due to the repulsive electrostatic conditions prevailing in the current set of experiments, the deposition efficiencies Q are smaller than unity and blocked areas amount to several times the particle cross section.
The Concentration and Shear Rate Dependence of the BlockedArea. The concentration dependence as well as the shear rate dependence of the blocked areas observed, shown in Figure 1 (bottom), can be explained by the following mechanisms. Fluid shear parallel to a collector surface stimulates collisions between flowingand adhering particles. As a result, flowing particles can be pushed away from the collector surface into a higher stream line, a process stimulated by the electrostatic repulsions between the particles (see also Figure 6). By
Figure 4. Local pair distributiong(xy) from an analysis ofthe spatial arrangementof polystyrene particles adheringto glass in a 50 mM potassium nitrate solution at a shear rate of 15 s-l (top) and 50 s-l (bottom). At the high shear rate an asymmetrical zone of lower density is clearly visible, a feature less pronounced at the low shear rate. diffusion, sedimentation, and interaction forces between collector surface and particles, particles are able to return to the collector surface and adhere behind a n already adhering particle. In addition, however, a particle pushed into a higher stream line may also collide with other flowingparticles and therewith gain an additional velocity component toward the collector surface. Obviously this is a concentration-dependent process which explains why in Figure 1 (bottom) the blocked area decreases with concentration. The latter model was proposed by Varennes and Van de Ven based on experiments with poly(ethy1ene oxide) coated latex particles in a stagnation point flow chamber.22 In the present analysis we have assumed that the mass transport toward the collector surface is constant and have derived the concentration dependence ofA1 based on this assumption. However, the effective mass transport may be affected by collisions of flowing and adhering particles and a distinction should be made between a so-called (22) Varennes, S.; Van de Ven, T. G. M. PhysicoChem. Hydrodyn. 1988, 10, 229.
332 Langmuir, Vol. 11, No. 1, 1995
Meinders and Busscher
g(x) 2
-1
Direction of Flow
1
Directiar a i Flow
1
1
010
-5
0
10
5
-5 0 5 Distance in Object Diameters
Distance in Object Diameters sty)
10
S(Y) 2
i
I 1
1
7 L
!lo
-5
0
10
5
0YO
0
-5
5
10
Distance in Object Diameters
Distance in Object Diameters Figure 5. Pair distribution functions for polystyrene particles adhering t o glass in a 50 mM potassium nitrate solution (particle concentration 1.6 x lo8 mL-l) at a shear rate of 15 s-l (left side) and 50 s-l (right side). g(x) (top) and gCy) (bottom)describe the relative density as function of the distance from a central particle, in the direction of flow and transverse t o the direction of flow,
respectively. Low shear rate
High shear rate
No collisions
with collisons
/
Figure 6. Schematic representation of a surface collision between a freely flowing suspended particle (particle2) and an adhering one (particle 1)at a low (top) and a high shear rate (bottom). Since collisions between particles with the same surface charge are repulsive, the distance between the flowing and an adhering particle will increase after collision and the flowing particle may be pushed into higher stream lines. Subsequently, due to collisions with other flowing particles (particle 3), particles may be pushed back toward the surface more rapidly than by diffusion, sedimentation, and interaction forces only, therewith increasing their probability to adhere. At higher wall shear rates, particles will be transported over a larger distance before they reach the surface. undisturbed blocked area AI,& and the experimentally observed blocked area which accounts for the influences of particles in suspension, particularly &-. Exact physical interpretation of AI,& is tedious, however, and it may not be ruled out that its interpretation is mathematical only. In order to obtain these blocked areas, we did a n exponential curve fitting on the data in Figure 1 yielding blocked areasAl,& andAl,, a t both shear rates employed (see Table 1). The variations of the blocked areas Al,-o and AI,- are completely in line with the collision model depicted in Figure 6, although the extrapolated values a t zero particle concentration may be irrealistically high due to the extrapolation involved and its predominantly mathematical significance. Note that
the influence of shear rate, i.e. the particle velocity, is more pronounced for Al,& than for AI,?-. Blocked Areas from Deposition Kinetics and Distribution Functions. Flow has a distinct influence on the spatial arrangements of adhering particles, resulting in an asymmetrical pair distribution function as can be seen best from g(x) (Figure 5, top). From g(x) it is obvious that the probability of deposition behind a n already adhering particle is small; i.e., the relative density is smaller than unity. Especially a t the high shear rate, the downstream blocking distance is large, which is completely in line with the previously presented model of interparticle collisions, predicting larger blocked areas a t higher shear rates. The blocked areas obtained from pair distribution functions are slightly larger than those from deposition kinetics (c -1, probably because within our definition of the blocking distance, there is still a reduced adhesion of particles possible. The sideward and upstream blocking distances are comparable and seem independent on the particle concentration and shear rate. Obviously, these distances represent the distance of closest approach between two adhering particles; i.e., interparticle electrostatic repulsion, immobilization forces, and hydrodynamic conditions prevent adhesion a t smaller distances. For the present system, the distance of closest approach amounts to a 3.5-particle radii equivalent to a surface to surface distance of 1125
-
A.
Desorption Due to InterparticleInteractionsand Flow. Another result of interparticle interactions is that collisions may stimulate desorption of adhering particles. Obviously, these collisions will be more frequent (eq 11) and more efficient (eq 20) in stimulating desorption a t higher shear rates and more frequent a t higher concentrations (see Figure 2, top). An additional effect of increased shear is that the interaction potential between particle and collector surface decreases. This creates an elastic force, pulling the particle back to its original interaction minimum, which is equal to the hydrodynamic
Langmuir, Vol. 11, No. 1, 1995 333
Znterparticle Interactions drag forces according toI6 I
(22) in which 7 is the viscosity and & is the change in the interaction minimum, #m, due to the hydrodynamicforce. Note that Ah, the potential distance a n adhering particle would have been dragged away from the interaction minimum, is assumed independent of shear rate. Since the effective interaction potential determines the desorption probability (23)
Using eq 23 and the measured values ko and calculate &,~=50 - &+15
= 0-44kT
we can (24a)
From eq 22 it can be seen that & varies quadratically with G; hence
- 50 -- (id &,G=15 &,G=~o
Thus from eq 24 we obtain & , G = ~ S = 0.04kT and &+SO = 0.48kT. Subsequently, by inserting these values in eq 22, a relation between & and Ah is found, which, combined with eq 23 yields Ah = 135 nm and q5m = -6.45kT. A similar analysis indicating similar mechanisms can be carried out on the pmdata, resulting in slightly deeper interaction minima between -lOkT and -12kT, indicative of strengthening of the bond during aging. A major underlying assumption in the above calculations is that all particles suffer the same interaction potential minimum. In reality, there will be a distribution of &, values present for all adhering particles, due to heterogeneities in surface characteristics of the particles and the collector surface. The distribution of values narrows down during the course of an experiment because the weakly adhering particles are removed due to collision with flowing particles. Hence, at higher shear rates and at higher concentration, the narrowing of the & distribution will occur more rapidly, which explains the decrease in the final desorption rate coefficient, Bm, and the relaxation time, l/d, as function of the concentration (see Figure 2): only adhering particles, whose bonds rapidly
grow strong, i.e. with a low relaxation time, can withstand collisions with flowing particles and remain adhering. In order to determine how the desorption rate coefficients in the presence of collisions relates to desorption rate coefficients in the absence of flowing particles, as mostly reported in literature,16 we changed the flow to a particle-free solution of the same ionic strength a t the end of each experiment and measured the number of desorbing particles as a function of time. From the decrease in the number of adheringparticles, we can obtain a desorption rate coefficient Best according to
The obtained desorption rate coefficients showed hardly any relation with concentration and averaged to 0.4 x lom6and 1.2 x s-l for wall shear rates of 15 and 50 s-l, respectively. These values are smaller than Bm as measured during deposition, i.e. including collisions, and even smaller than pnC,-,derived excluding effects of collisions, illustrating that desorption measured separately after a deposition experimentreflects the desorption of the most strongly adhering particles, as more weakly adhering particles are already removed by collisions with flowing particles.
Conclusions We can draw the following conclusions from the results of this study. Initial deposition rates increase linearly as the concentration of suspended particles increases. Deposition efficiencies decrease from 0.78 to 0.62 as the shear rate increases from 15 to 50 s-l. Blocked areas decrease nonlinearly as the concentration of suspended particles increases, due to an increasing probability of backscattering of flowing particles which have collided with adhering particles. Blocked areas are larger a t higher shear rates. Initial desorption rate coefficients increase linearly as the concentration of suspended particles increases due to higher numbers of collisions per unit time. At higher shear rates initial desorption rate coefficients are larger due to higher energy transfer collisions. The final desorption rate coefficient and the relaxation time of the residence time dependent desorption decrease nonlinearly as the concentration of suspended particles increases, since weakly adhering particles are removed quicker as the number of collisions increases, a process stimulated a t higher shear rates. Acknowledgment. The authors wish to thank Dr. Th.F. Tadrosfor critically commentingon this manuscript. LA9404043
