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Kinetics of Colloid Particle Adsorption at Heterogeneous Surfaces Zbigniew Adamczyk,* Barbara Siwek, and Elizeusz Musiał Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, 30-239 Krako´ w, Niezapominajek 8, Poland Received February 8, 2001. In Final Form: May 8, 2001 Adsorption kinetics of negatively charged polystyrene latex particles (average size 1.38 µm) over a heterogeneous surface was studied experimentally. The substrate of a controlled heterogeneity was produced by covering natural mica sheets by smaller (average size 0.55 µm) positively charged latex particles. The direct microscope observation method combined with the impinging jet technique was used to determine particle adsorption kinetics. The initial flux of particles (the slope of the kinetic curves in the limit of short adsorption time) was determined quantitatively as a function of the coverage of smaller particles. It was demonstrated that the initial flux attains the limiting values for the smaller particle (heterogeneity) coverage as small as a few percent. The experimental results were interpreted in terms of the generalized random sequential adsorption model by considering the coupling between the bulk and surface transfer steps. A good agreement was found which seems to confirm the validity of the theoretical model for predicting particle adsorption at heterogeneous surfaces.
Introduction Adsorption and deposition (irreversible adsorption) of colloids and bioparticles on solid/liquid interfaces is of large significance for many practical and natural processes such as filtration, water treatment, paper making, thrombosis, protein, bacteria, enzyme immobilization and separation, immunological assays, biofouling of transplants and artificial organs, and so forth. The effectiveness of these processes is often enhanced by the use of coupling agents preadsorbed at the interfaces, which promote attachment of particles. Thus, cationic polyelectrolytes are used to increase retention of filler particles (e.g., titania) in paper-making processes1 or to promote a selective self-assembly of colloid particles at patterned surfaces.2,3 In biological applications, special immobilized ligands (antibodies) are widely applied for a selective binding of a desired solute from protein mixtures, for example, in affinity chromatography4,5 or immunological assays.6,7 A characteristic feature of these processes is that particle or protein adsorption occurs at surfaces, which are inherently heterogeneous. This raises an important question: what is the correspondence between the number, size, and distribution of the ligands (which can be treated in the classical adsorption terminology as active centers) and particle adsorption kinetics and mechanism? Despite a considerable theoretical significance of these processes, little systematic work has been reported in the literature with the exception of the paper of Jin et al.4 who studied theoretically irreversible adsorption of disks * Corresponding author. E-mail:
[email protected]. (1) Baluk, M. Y.; van de Ven, T. G. M. Colloids Surf., A. 1990, 46, 157. (2) Clark, S. L.; Hammond, P. T. Adv. Mater. 1998, 10, 1515. (3) Chen, K. M.; Jiang, X.; Kimerling, L. C.; Hammond, P. T. Langmuir 2000, 16, 7825. (4) Jin, X.; Wang, N. H. L.; Tarjus, G.; Talbot, J. J. Phys. Chem. 1993, 97, 4256. (5) Chase, H. A. Chem. Eng. Sci. 1984, 39, 1099. (6) Peula, J. M.; Hidalgo-Alvarez, R.; Nieves, F. J. D. J. Colloid Interface Sci. 1998, 201, 139. (7) Miksa, B.; Wilczynska, M.; Cierniewski, C.; Basinska, T.; Slomkowski, S. J. Biomater. Sci., Polym. Ed. 1995, 7, 503.
on nonuniform surfaces covered by pointlike adsorption sites. A correspondence (mapping function) between the adsorption process at heterogeneous surfaces and the classical random sequential adsorption (RSA) model8-10 was found. However, no site geometry effects have been determined in these calculations. Adsorption at heterogeneous (precovered) surfaces was studied theoretically11 and experimentally12 by using monodisperse latex suspensions. However, these results were concerned with the situation when the interactions between the sites attached to the surface and the adsorbing particles were of a repulsive character, so adsorption occurred at uncovered areas only. In this case, a significant decrease in particle flux was observed when the number of sites (smaller sized spheres) present at the surface was increased. The goal of this work is to determine experimentally particle flux for the opposite situation, that is, when particle adsorption occurs at active sites only as a result of attractive interactions with adsorbing particles. Our experiments, performed for a model latex suspension, enable one to find the links with the former case of particle adsorption at precovered surfaces. Experimental Section Particle adsorption experiments were performed by exploiting the direct microscope observation method in the radial impinging jet (RIJ) cell used extensively before for similar studies.10,12-17 (8) Schaaf, P.; Talbot, J. J. Chem. Phys. 1989, 91, 4401. (9) Senger, B.; Voegel, J. C.; Schaaf, P. Colloids Surf., A. 2000, 165, 255. (10) Adamczyk, Z.; Siwek, B.; Zembala, M.; Belouschek, P. Adv. Colloid Interface Sci. 1994, 48, 151. (11) Adamczyk, Z.; Weron´ski, P. J. Chem. Phys. 1998, 108, 9851. (12) Adamczyk, Z.; Siwek, B.; Weron´ski, P. J. Colloid Interface Sci. 1997, 195, 261. (13) Da¸ bros´, T.; van de Ven, T. G. M. J. Colloid Interface Sci. 1982, 89, 232. (14) Da¸ bros´, T.; van de Ven, T. G. M. Colloid Polym. Sci. 1983, 261, 694. (15) Da¸ bros´, T.; van de Ven, T. G. M. PCH, PhysicoChem. Hydrodyn. 1987, 8, 161. (16) Adamczyk, Z.; Zembala, M.; Siwek, B.; Czarnecki, J. J. Colloid Interface Sci. 1986, 110, 188.
10.1021/la010208d CCC: $20.00 © 2001 American Chemical Society Published on Web 06/26/2001
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Because of the hydrostatic pressure difference, the particle suspension was driven through a circular capillary with an aperture having the radius of R ) 0.0735 cm, impinged against a perpendicularly oriented mica plate, and left the cell through the external tubing. After use, the particle suspension was discarded. The distance h between the top of the capillary and the mica plate was equal to 0.22 cm which gives h/R ) 3. The volumetric flow rate Q in the cell was regulated by the change of the diameter of the outlet capillary tube and its level. This allowed one to regulate the flow Reynolds number Re ) Q/πRν within broad limits (where ν is the kinematic viscosity of the suspension). Because of the underpressure prevailing in the cell, the mica plate was held fixed to the external tube without using any adhesive which reduced the possibility of cell contamination during the measurement. Adsorbed particles were observed in situ under an optical microscope (Leitz, Germany, dark-field illumination) coupled with a CCD TV camera (Hamamatsu type C-3077), an image processor (Argus 10 type C-3930), and a video recorder.
Adamczyk et al. between 2 × 108 cm-3 and 8 × 108 cm-3. Then, the adsorption kinetics was followed by determining the average surface concentration of larger particles as a function of time. To obtain a single point on the kinetic curve, 100-500 particles were counted over statistically chosen areas having typical dimensions of 100 per 100 µm. All these areas were located not further than 200 µm from the stagnation point (center of the cell) where the interface remains uniformly accessible for particle transport.10,21 The dimensionless surface coverage of larger particles was calculated as Θl ) πal2. The kinetic runs were fitted by the linear or parabolic regression lines (in the case of higher coverages) whose slope at t f 0 gave the initial flux values for a given coverage of smaller particles. It was proven in separate experiments that up to the maximum Re number studied particle adsorption was perfectly irreversible and localized. No lateral motion or particle desorption was observed for experimental time reaching 24 h.
Results and Discussion Materials and Methods In the experiments, a model suspension of polystyrene latex particles was used. The negatively charged polystyrene latex suspensions were synthesized according to the polymerization procedure using a persulfate initiator.18 The positively charged latex particles were obtained using a similar procedure with the azonitrile initiator. The concentrated stock samples obtained from the polymerization were purified by steam distillation and washed on a Sartorius membrane filter until the filtrate conductivity stabilized around the value of pure water (about 1-2 µS). The pH of the samples measured before and after each experiment was usually 5.6-6. Particle size and number concentration in suspensions of these latices was determined by the Coulter-Counter method with an accuracy of a few percent. The averaged size of the negative latex (L58) used in our experiments was 1.38 µm with the standard deviation of 8%. The smaller sized, positive latex had an average diameter of 0.55 (standard deviation of 8%). The specific density of the latex was 1.05 g/cm3. The ζ potential of the negative latex (measured by the Brookhaven Zeta PALS Apparatus) was -55 mV for ionic strength of 10-4 M, and for the positive latex the value was +45 mV. The ionic strength of the suspensions was fixed by high purity KCl recrystallized many times from quadruple-distilled water. As the adsorbing surface, mica sheets provided by Mica & Micanite Supplies Ltd., U.K., were used. The sheets were freshly cleaved before each experiment and used without any pretreatment. The ζ potential of mica was determined by the streaming potential method in the plane-parallel channel cell.19 For the ionic strength of 10-4 M, the potential was found to be -90 mV. The experimental procedure was the following: first, the mica surface was covered by positive latex in separate runs carried out for very low Reynolds number and ionic strength of 10-4 M. By changing the adsorption time and bulk suspension concentration, a desired surface concentration of particles was attained determined by direct microscope counting over statistically chosen areas. The total number of particles was about 1000-2000, which ensured a relative precision better than 3%. For the sake of convenience, the surface concentration of particles was expressed as the dimensionless coverage Θs ) πas2 (where Ns is the average surface concentration of adsorbed smaller particles). In our experiments, Θs varied between 0 and 30%. As discussed extensively in refs 16 and 20, particle distributions produced under such conditions are statistically uniform with the variance decreasing significantly with Θs. After the heterogeneous mica surface was prepared, the positive latex suspension was replaced in situ by the negative latex suspension of a given bulk concentration nb varied usually (17) Adamczyk, Z.; Siwek, B.; Zembala, M.; Warszyn´ski, P. J. Colloid Interface Sci. 1989, 130, 578. (18) Goodwin, J. W.; Hearn, J.; Ho, C. C.; Ottewill, R. H. Colloid Polym. Sci. 1974, 252, 464. (19) Zembala, M.; Adamczyk, Z. Langmuir 2000, 16, 1593. (20) Adamczyk, Z.; Szyk-Warszyn´ska, L.; Siwek, B.; Weron´ski, P. J. Chem. Phys. 2000, 113, 11336.
In a series of preliminary experiments, it was proven that no measurable deposition of larger particles was observed in the case of bare mica, when there were no preadsorbed smaller particles. Hence, larger particle adsorption occurred if Θs > 0 only. Micrographs of particle monolayers obtained in these experiments (Re ) 2) are shown in Figure 1 (part a corresponds to Θs ) 0.04 and Θl ) 0.04, and part b corresponds to Θs ) 0.04 and Θl ) 0.07). Both the smaller and larger particles can be directly seen at the interface, which enables a quantitative analysis of adsorption kinetics. Also, larger particle distribution remains uniform with no tendency to clustering as is often the case for monolayers dried before microscope observation. The uniformity of small and larger particle distribution has been confirmed quantitatively by the variance analysis. Some typical kinetic runs determined for larger particles adsorbing at heterogeneous surfaces characterized by Θs ) 0.018, 0.026, and 0.21 are plotted in Figure 2. The dependency of larger particle coverage Θl on adsorption time t remains linear in all cases with the slope monotonically increasing with the coverage of active centers (smaller particles). For comparison, the limiting theoretical results predicted for a uniform surface are also plotted in Figure 2 (the broken straight lines). These results have been obtained by a numerical solution of the governing mass transfer equation according to the procedure discussed extensively elsewhere.16,21,22 As can be observed in Figure 2, larger particle adsorption kinetics approaches abruptly with the increase in Θs the limiting results characteristic to a uniform surface. Thus, for Θs as low as a few percent, the experimental and theoretical runs coincide within the error bounds. These results, apparently the first of this kind in the literature, seem rather unexpected. However, they can be interpreted theoretically in a quantitative manner when expressed in a more suitable way as the initial flux versus Θs dependence. As mentioned, the initial flux of larger particles was calculated from the equation
|jl| )
1 ∆Θl πal2 ∆t
(1)
where ∆Θl/∆t is the slope of the experimental kinetic run fitted by linear regression lines. (21) Adamczyk, Z.; Siwek, B.; Warszyn´ski, P.; Musiał, E. J. Colloid Interface Sci., accepted. (22) Adamczyk, Z.; Warszyn´ski, P.; Szyk-Warszyn´ska, L.; Weron´ski, P. Colloids Surf., A. 2000, 165, 157.
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Figure 1. Micrographs of polystyrene latex particles (average diameter 1.38 µm) adsorbed at mica precovered with smaller particles (average diameter 0.55 µm), I ) 10-4 M, Re ) 2: (a) Θs ) 0.04, Θl )0.04; (b) Θs ) 0.04, Θl ) 0.07.
It was also shown in separate runs that the initial flux was proportional to the bulk suspension concentration nb which indicates that larger particle adsorption was indeed a linear process for the range of times studied (reaching 45 min). This fact enabled one to increase the accuracy of experimental determination of the initial flux expressed in the reduced form jjl ) |jl|/nb since averages from many kinetic runs performed for various nb could be taken. The dependence of the reduced flux determined in this way on the coverage of smaller particles Θs is plotted in Figure 3 (Re ) 2, I ) 10-4 M). After a dramatic initial rise, the reduced flux approaches the asymptotic value of 2.04 × 10-6 cm/s. As previously discussed, this is close to the theoretical value pertinent to uniform surfaces, equal to 1.92 × 10-6 cm/s, calculated numerically. Similar results as shown in Figure 3 were obtained for larger Re numbers (reaching 8) as well. These experimental evidences suggest, therefore, that the kinetics of particle adsorption at heterogeneous surfaces (at least for initial stages when
Figure 2. Kinetics of polystyrene latex adsorption at mica precovered with smaller particles expressed as the dependence of Θl on the adsorption time t, I ) 10-4 M, Re ) 2: (a) Θs ) 0.018, (b) Θs ) 0.026, and (c) Θs ) 0.21. The solid lines denote the regression fits, and the dashed line shows the limiting kinetics for a homogeneous surface.
the surface coverage remains low) can be well predicted by the convective theory elaborated for uniform surfaces.10,13-16,22 This has a considerable significance since many of the known theoretical and experimental data for
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Adamczyk et al.
jl(Θs) ) j0pl(Θs)
(4)
where j0 ) -kbnb is the flux of particles reaching the adsorption layer from the bulk and kb is the bulk transfer rate constant. This constant can be calculated for various transport conditions from the numerical solutions of the convective diffusion equation10 or determined empirically for the barrierless transport conditions. However, eq 4 is valid if the bulk resistance characterized by kb is comparable with the surface resistance. Otherwise, eq 4 should be generalized by considering the coupling between the bulk and surface transport steps. In this case, the appropriate expression for jl takes the form24,25
Ka〈pl(Θs)〉 jl ) j0 1 + (Ka - 1)〈pl(Θs)〉 Figure 3. The dependence of the reduced flux of larger particles jjl ) jl/nb on surface coverage of smaller particles Θs (I ) 10-4 M, Re ) 2), the points denote experimental results and the dotted line represents the theoretical prediction derived from the convective diffusion theory.
uniform surfaces can directly be transferred to heterogeneous system adsorption. A quantitative interpretation of the abrupt increase of jjl with the active center coverage Θs can be attained by exploiting the theoretical results developed previously for precovered surfaces.11,12The probability of larger particle adsorption over a surface covered by smaller sized particles (often referred to as the available surface function or less accurately as the surface blocking parameter) can be approximated by the analytical expression
{
Bl(Θs) ) (1 - Θs) exp -
(4λ - 1)Θs 1 - Θs
[
]}
(2xλ - 1)Θs 1 - Θs
2
(2)
where λ ) al/as is the larger to smaller particle size ratio. This formula has been derived by generalizing the scaled particle theory to bimodal sphere mixtures. Equation 2 describes the situation when the preadsorbed particles (heterogeneities) and adsorbing particles do not interact, so adsorption occurs at the interface only. Thus, eq 2 gives the probability of finding an empty surface element over the heterogeneous surface, large enough to accommodate a larger sphere without overlapping with smaller particles. The probability of particle adsorption in the opposite case pertinent to our experimental results, that is, when adsorption occurs upon physical contact between larger and smaller spheres, but not at the interface, is just
pl(Θs) ) 1 - Bl(Θs)
(3)
The validity of this expression has been confirmed by numerical simulations performed recently23 for heterogeneous surfaces covered by active centers. Knowing pl(Θs), one can calculate the effective particle flux to heterogeneous surfaces from the classical RSA model9,10 which postulates that (23) Adamczyk, Z.; Weron´ski, P. J. Chem. Phys., submitted.
(5)
where Ka ) ka/kb and 〈pl(Θs)〉 is the integrated probability (blocking function). The adsorption rate constant ka can be calculated as
ka )
∫δ
δa m
1 eφ/kT dh D(h)
where δm is the minimum distance between the particle and the interface, δa is the adsorption layer thickness, φ is the interaction energy of the particle with the interface, k is the Boltzmann constant, T is the absolute temperature, and D is the particle diffusion coefficient depending on the distance from the interface h. Because of mathematical difficulties, 〈pl(Θs)〉 has been formulated explicitly only for a limited range of coverage in the case of adsorption over uniform interfaces.25 However, for practical applications, when Ka . 1, the integrated probability can be well approximated by the classical function pl(Θs). Assuming this, one can apply eq 5 with pl(Θs) given by eq 2 to describe particle flux to heterogeneous surfaces. A useful analytical expression can be derived in this case if λΘs , 1. Then, eq 5 becomes
4λKaΘs jl ) j0 1 + 4λ(Ka - 1)Θs
(6)
The theoretical results stemming from the above approaches are compared with experimental results in Figure 4. The relevant parameters for our experimental conditions were λ ) 2.5 (particle size ratio) and Ka ) 7. The latter value was calculated by assuming the thickness of the adsorption layer δa to be equal to the diameter of smaller particles (heterogeneities) and taking kb ) 2.04 × 10-6 cm/s, that is, equal to the experimentally determined value for large Θs (see Figure 3). In Figure 4, the experimental results are quantitatively reflected by eq 5 for the entire range of smaller particle coverage Θs. As expected, eq 6 gives also a satisfactory agreement with the experimental data for Θs < 0.1. These facts suggest that the basic features of particle adsorption at heterogeneous surfaces are well reflected by eqs 5 and 6, which has a considerable practical significance in view of the simplicity of these analytical expressions. In particular, one can deduce from eq 6 that the flux for (24) Adamczyk, Z.; Senger, B.; Voegel, J. C.; Schaaf, P. J. Chem. Phys. 1999, 110, 3118. (25) Adamczyk, Z.; Weron´ski, P. Adv. Colloid Interface Sci. 1999, 83, 137.
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Figure 4. The dependence of the normalized flux jl/j0 on Θs. The points denote the transformed experimental results shown in Figure 3, the solid line represents the theoretical results derived from eq 5 for Ka ) 7 and λ ) 2.5, the dashed-dotted line denotes the results calculated from eq 6, and the dashed line shows the theoretical results calculated from eq 5 by neglecting the bulk transport (Ka ) 1).
heterogeneous surfaces attains the limiting value for uniform surfaces for the critical value of Θs given by
Θs >
askb 1 ) 4λKa 4alka
Figure 5. The dependence of jl/j0 on Θs; I ) 10-4 M, Re ) 8. The solid line represents the theoretical results calculated from eq 5 for Ka ) 5 and λ ) 2.5, the dashed-dotted line denotes the results calculated from eq 6, and the dashed line shows the results calculated from eq 5 by neglecting the bulk transport (Ka ) 1).
Because of relatively high ionic strength in our experiments, this condition was approximately fulfilled because the range of electrostatic interactions was much smaller than particle dimensions.
(7)
As can be deduced, this limiting value is proportional to as and kb, which means that the increase in the flux should be the most dramatic for smaller heterogeneity size and low flow rate (Reynolds number) when the thickness of the diffusion boundary layer becomes significantly larger than particle dimensions. This suggests that the effect of the bulk transport plays a decisive role in adsorption at heterogeneous surfaces. This conclusion is further supported by the fact that the theoretical results derived from eq 3 neglecting the bulk transport deviate significantly from the experimental results (see Figure 4). From eqs 6 and 7, one can deduce that the dependence of the reduced flux on Θs should be influenced by the bulk transfer rate constant kb which increases with the Reynolds number as discussed at length in refs 10 and 20. As a consequence, the increase in jl/j0 with Θs should be less steep for higher Re numbers. The experimental results shown in Figure 5 seem to support this hypothesis although the effect is less pronounced than one may expect theoretically. A good agreement of eq 6 with the experimental data is attained for Ka ) 5 instead of for Ka ) 1.2 as predicted theoretically. A plausible explanation of this deviation is that for higher Re numbers, particle transport in the vicinity of the interface is affected by the geometrical interception.10 This effect, due to the tangential velocity component, is expected to increase the value of kb (and consequently Ka) because a large particle population will be attached at the top of the heterogeneities. However, these many body problems can be analyzed theoretically only in terms of the time-consuming Brownian dynamic simulation method.22 Equations 2-6 are strictly valid for hard particles, which do not interact except upon forming a physical contact.
Concluding Remarks The experiments performed for model colloid systems revealed that adsorption flux at heterogeneous surfaces attains the limiting value j0 (predicted by the convective diffusion theory for uniform surfaces) for active center coverage Θs as low as a few percent. This can be attributed to the combined effect of geometrical interception, proportional to 4λΘs and the coupling with the bulk transport, described by the Ka parameter. In the general case, this effect can be well described by eq 5. For Ka . 1, the dependence of particle flux at heterogeneous surfaces can be well approximated by the Langmuir-like equation
4λKaΘs jl ) j0 1 + 4λ(Ka - 1)Θs As can be deduced from this equation, the flux increases abruptly with the size ratio particle/heterogeneity and with the thickness of the diffusion boundary layer, which is proportional to 1/kb. This means that after the critical value of Θs ) 1/4λKa is exceeded, the kinetics of particle adsorption at a heterogeneous surface (at initial stages) can be well predicted by the convective-diffusion theory derived for a uniform surface. Because of analogous transport conditions,10,22 our experimental results obtained for the impinging jet cell can be used as useful reference measurements for more complicated flows, for example, the spherical or cylindrical collectors placed in an uniform flow when the direct microscope observations seem considerably more tedious. Acknowledgment. This work was supported by the KBN Grant 3T09A 105 18. LA010208D
