Langmuir 1992,8, 2605-2610
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Art ic 1es Kinetics of Localized Adsorption of Colloid Particles Z. Adamczyk,’ B. Siwek, M. Zembala, and P. Werodski Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, 30-239 Cracow, Niezapominajek 1 , Poland Received April 6,1992. In Final Form: June 22, 1992
Kineticsof localized adsorption of charged colloid particles on homogeneous solidlliquidinterfaceswas analyzed. Limitinganalyticalequationswere formulatedfor the maximum surface concentration,e, the blocking parameterB(O),and adsorptionkineticsin terms of the lateral interactionparameter h* depending on the ionic strength of the colloid suspension. In the general case particle adsorption kineticsproceeding according to the random sequential adsorption (RSA) mechanism was simulated numerically by using the Monte Carlo method. The theoretical predictions were experimentally tested by applying the direct observation method based on the stagnation point flow cell. Monodisperse suspensions of negatively charged latex particles of micrometer size range were used with freshlycleaved mica sheets as the adsorbing surfaces. The theoreticalpredictionswere quantitativelyconfirmedshowingthat the RSA model describes well the kinetics of localized adsorption of interacting colloid particles especially for higher flow intensity. The widely used Langmuir model was found inappropriate for colloid particle adsorption from liquid phases.
Introduction Interactions of macromolecules and colloid and larger particles with interfaces leading to adsorption, deposition, and adhesion are of great practical interest, e.g., for various filtration processes. A quantitative analysis of adsorption phenomena has also significance for polymer and colloid science, biology and biophysics, i.e., in studies of bioadhesion, biofouling, thrombosis formation, immobilization of enzymes, protein and bacteria deposition, etc. Numerous experimental works concerning protein adsorption,’+ and bacteria depositionlOJ1were mainly concerned with finding the appropriate “isotherm” equation (although the true thermodynamic equilibrium was rarely the case) rather than on kinetic aspects of these processes.12-16 Most of these equilibrium and kinetic studies of particle (1) Feder, J.; Giaever, I. J. Colloid Interface Sci. 1980, 78, 144. (2) Feder, J. J. Theor. Biol. 1980,87, 237. (3) Young, B. R.; Pitt, W. G.; Cooper, S. L. J. Colloid Interface Sci. 1988, 124, 28. (4) Schmitt, A.; Varoqui, R.; Uniyal, S.; Brash, J. L.; Pusineri, C. J. Colloid Interface Sci. 1983, 92, 25. (5) Beiesinger, R. L.; Leonard, E. F. J. Colloid Interface Sci. 1982,85, 521. (6) Aptel, J. D.; Voegel, J. C.; Schmitt, A. Colloids Surf. 1988,29,359.
(7) Vincent, B.; Young, C. A.; Tadros, Th. F., J.Chem. SOC.,Faraday
Trans. 1 1980. 76.665. (8) Vincent, B.; Jaffelici, M.; Luckham, P. F. J. Chem. SOC.,Faraday
Tram. - - . 1 1980. -.. 76. 674. (9) Kallay, N.; Tomic, M.;Biskup, B.;Kunjasic,I.;Matijevic,E. Colloids Surf. 1987, 28, 185. (10) Gibbons, R. J.; Moreno, E. C.; Etherden, I. Infect. Immun. 1983, ~
39,280. (11) Moncla, B. J.; Halfpap, L.; Birdsell, D. C. J . Gen. Microbiol. 1985, 131,2619. (12) Dgbrofi, T.; van de Ven, T. G. M. J . Colloid Interface Sci. 1982, 89, 232. (13) Dgbrd, T.; van de Ven, T. G. M. Colloid Polym. Sci. 1983,261, 694. (14) Adamczyk, 2.;Zembala, M.;Siwek, B.; Czamecki, J. J . Colloid Interface Sci. 1986,110, 188. (15) Adamczyk, 2.;Siwek, B.; Zembala, M.; Warszyhski, P. J . Colloid Interface Sci. 1988,130,578. (16) Sjollema, J.; Busscher, H. J. J . Colloid Interface Sci. 1982, 132, 382.
adsorption and d e p ~ s i t i o n ~ ~ ~ were - ’ ~ J ’interpreted by introducingthe geometricalblockingparameter analogous to the Langmuir model derived originally for a discrete distribution of “active centers”.l* However, simple geometrical considerations suggest that when the distance between active centers becomes much smaller than particle size (asis the case for aerosol,protein, and colloid particles) then the surface exclusion effects (geometrical blocking) should be more significant than the Langmuir model would predict especially for large surface concentrations. Similarly, due to surface charge present on colloid particles, their adsorption kinetics should be significantlyinfluenced by the electrostatic interactions, Le., by the ionic strength of the suspension. These aspects of colloid particle adsorption were rarely considered except for our previous Therefore, the goal of this paper is a quantitative description of adsorption kinetics of interacting colloid particles on homogeneous surfaces. In particular, we shall determine under which transport conditions (flow intensity) the particles adsorb accordingto the RSAmechanism. The theoretical predictions shall be verified by direct experimental measurements performed on monodisperse latex suspensions of micrometer-sized particles.
Theoretical Section The Random Sequential Adsorption Model. Gemera1 Considerations. Let us consider a dispersion of spherically-shaped colloid particles moving in the vicinity of a smooth, homogeneous solid-liquid interface (see Figure 1). Due to various transport mechanisms (convection, diffusion, migration) the particles may occasionally approach the interface close enough to establish a permanent contact as a result of the local field of forces (17) Ruckenstein, E.;Marmur, A,;Gill, W. N. J. Colloid Interface Sci. 1977, 61, 183. (18) Langmuir, I. J. Am. Chem. SOC.1918,40, 1361. (19) Adamczyk, 2.;Zembala, M.;Siwek, B.; Warszyfbki, P. J. Colloid Interface Sci. 1990, 140, 123. (20) Adamczyk, 2.;Siwek, B.; Zembala, M.Biofouling 1991,4, 89. (21) Adamczyk, 2.;Siwek, B.; Zembala, M. J. Colloid Interface Sci. 1992,151, 351.
Q143-1463/92/2~Q~-26Q5$Q3.O0/0 0 1992 American Chemical Society
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Vol. 8,No.11, 1992
Figure 1. Illustration of the RSA of interacting particles on homogeneous interfaces.
exerted by the interface. The particles accumulated at the interface influence the transport and adsorption of other particles moving in their vicinity not only due to geometrical volume excluding effect but also as a result of repulsive lateral interactions stemming from the presence of electricaldouble layers. As a result, the adsorption process becomes the classical many-body problem whose exact solutions is, at the present time, impractical. Therefore, in order to deal with this complex situation, we adopt the RSA model developed originally for hard (noninteracting) sphere a d s ~ r p t i o n ~and ~ p generalized ~~ later for interacting particles.lS21 The main assumptions of the model are as follows: (i) the particles are placed at random with a constant rate denoted as the initial fluxj , on a homogeneous interface; (ii) if the adsorbing (virtual) particle overlaps with any of those already adsorbed, it is immediately removed (see Figure 1); (iii) otherwise the probability density of placing the virtual particle is calculated from the Boltzmann law, i.e., Sp = exp(-$/kT), where $ is the repulsive energy of interaction of the particle with the environment, k is the Boltzmann constant, and T is the absolute temperature. The adsorption is assumed localized, Le., there is no lateral motion on the interface. By use of the RSA model, particle adsorption can be described by the simple kinetic equation21written in the dimensionless form --=de B(8) dT where 8 = ru2Nisthe dimensionless surface concentration of adsorbed particles (Nis the number of particles adsorbed per an unit area), T = (?ru2jbnb/8,h)tis the dimensionless adsorption time, OCh is the characteristic dimensionless surface concentration of adsorbed particles (we assumed hereafter Och = 0.546, i.e., the maximum surface coverage of RSA of hard spheres), nb denotes the bulk suspension concentration, and t is the time. B(8)in eq 1 is the surface blocking parameter which can be interpreted as the averaged probability of placing particles at the interface for a given 8 value. Using the statistical mechanical and generalized approaches developed for hard for interacting particles,lS21 one can formulate the following explicit expression for B(8) valid for low surface concentrations
B(e) = 1 - c,e + c2e2+ o(e7 (2) where the constants C1 and C2 stemming from the cluster (22) Hinricheen, E. L.; Feder,J.; Jossang, T. J. Stat. Phys. 1986,44, 793. (23) Schaaf, P.; Talbot,J. J. Chem. Phys. 1989,91,4401. (24) b i s e , H.; Frisch, H. L.; Lebowitz, J. L.J. Chem. Phys. 1959,31, 369.
0
'
a-'
2a [ m ]
Figure 2. Maximum surface concentration of interacting particles 6, adsorbed according to the RSA mechanism vs the particle radius a: T = 293 K,lfpl = 50 mV,7 = 0.5; (1)I = 10-2 M, (2) I = 1W3M, ( 3 ) 1 = l W M , ( 4 ) 1 = 1WM.
integrals involving two and three particles are complicate functions of the repulsive interaction potential 4. However, these constants can be approximated well for thin double-layers by2'
c, = 4 -= 4 (1 em
8,
e,
=
h*)2
8,
(1 + hq2
8, = Och = 0.546 is the "random monolayer" coverage for noninteractingparticles (hard spheres). These equations are valid for h* 1. Our MC simulations of the irreversible RSA of interacting particles on homogeneous surfaces can be most conveniently expressed in terms of the reduced quasitime variable r c defined as NatJNch, where Natt is the number of computer attempts at placing the particles on the simulation area and Nch = 0.546/m2 is the number of particles forming a random “monolayer”. By introducing this definition, the “computer” time rc becomes fully equivalent to the dimensionlesstime r defined previously. In this way the theoretical results plotted in the coordinate system 0 vs rc can be directly compared with experimental ones. Experimental Section The ExperimentalCell. The particleadsorptionexperiments were performed by using the direct microscope observation method described in detail elsewhere.14J6 The method is based on the confiied impinging-jet principle which ensures controlled and uniform transport conditions over large areas of the adsorbing surface. By changing the volumetric flow rate of the suspension in the cell (see Figure 3), one can easily change the Reynolds number within broad limits and consequently the initial particle flux jo. In the vicinity of the stagnation point (the center of the cell) particle transport toward the interface is dominanted by diffusion which ensures that the RSA mechanism is operating. It should also be mentioned that the direction of flow in our cell was opposite to the direction of gravity which proved to be advantageous because, due to sedimentation, all large particles and aggregates did not adsorb. As a result the size distribution of adsorbed particles was characterized by a smaller spread than in the bulk of the suspension. The adsorbed particles were directly observed under a microscope by using the dark-field illumination. In order to follow adsorption kinetics many micrographs of adsorbed particlea were taken at the appropriate time intervals. From these micrographs (totalmagnification ca. lo00times) coordinates of typically 12000 particles were recorded using a digitizer (KD 4600, Graphtec Corp., Tokyo, Japan). Then the coordinates were processed by a computer and the averaged particle surface concentration was determined together with the 2D pair correlation function of adsorbed particles. Determination of particle adsorption kinetics as well as the pair correlation function was carried out for the interface areas close to the stagnation point (maximum distance
The particle number concentration in stock suspensions of these latices was determined by the dry content method, whereas the concentrations of dilute suspensions used directly in experiments were determined by the Coulter-Counter method with an accuracy better than a few percent. The concentration derived from the dry weight method (by considering the dilution factor) agreed well with the Coulter-Counter values. The average particle size (determined by the Coulter-Counter method) of our negative latex was found to be 1.0 pm (standard derivation 6 % ). As the adsorbing surface we used mica sheets supplied by Mica & Micanite Supplies, Ltd.,England. In order to ensure localized and irreversible adsorption of particles, the natural negative charge of mica was converted to a positive one. This was accomplished by bringing freshly cleaved mica sheets in contact with a 1% solution of [3-(N-(2-aminoethyl)amino)propyl]trimethoxysilane for 15 min at 80 OC. Then the sheets were washed and heated for 24 h at 80 OC. Experimental Results and Discussion. The experimental procedure applied for determining particle adsorption kinetics is described in detail e l ~ e w h e r e . ~ ~Under J ~ J ~ our experimental conditions (low and moderate flow inteneity) particle adsorption was perfectly irreversible and localized; no lateral motion or particle desorption was observed in prolongated washing experiments. Using the particle suspensions of micrometer size range is advantageous because the adsorption kinetics can be determined directly by measuring the coordinates of individual particlee from the photographs taken under microscope. In this way the averaged surface concentration of adsorbed particles can be determined as well as the radial distribution function (pair correlation function) gla which is the measure of the averaged local surface concentration in the vicinity of an adsorbed particle. From the 2D correlation function one can draw important conclusions about the range and magnitude of the lateral interactions between adsorbed and adsorbing particlea in a direct way, something which is difficult by other experimentalmethode. In Figure 4 experimental resulta concerning distributions of adsorbed particles over silanized mica (pair correlation function) are compared with the theoretical MC simulations performed according to the irreversible RSAmechaniim. The ionic strength I in this experiment was l W b M (a = 0.5 pm, KO = 5.3) and 8 = 22.2 % As can be seen in Figure 4 the agreement between the experimental results (obtained from coordinates of more than 1500individual particlee) and the MC simulations is satisfactory. It can be o b s e ~ that d for the relatively low surface concentration (for comparison the hard-sphere model predicta 0, = 54.6%9 the 812 function deviates considerably from the Boltzmann distribution, i.e.,glZ(Zf) = exp(-.$(H)/kT) (depicted by the broken line in Figure 4) and exhibits an oscillatory character. This indicates that strong correlations between positions of adsorbed particles occur leading to a 2D short-range ordering quite analogous to that observed for 3D colloid and liquid One can draw, therefore, the conclusion that the resulta obtained
.
(26) Goodwin,J.W.;Hearn,J.;Ho,C.C.;Ottewill,R.H.ColloidPolym.
Sci. 1974, 252, 464. (26) Goodwin, J. W.;Ottewill, R. H.;Pelton, R. Colloid Polym. Sci. 1979, 257, 61. (27) Ottewill, R. M.Langmuir 1989,5,4.
Langmuir, Vol. 8, No.11, 1992 2609
Adsorption of Particles
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3 4 5 6 r12/a Figure 4. Experimentally determined 2 0 radial distribution function 812 for latex particles on silanized mica: a = 0.5 pm, Z = 1od M (Ka = 5.6),fp = -50 mV, y = 0.25,0 = 22.2 % ;full circles, Re = 2; empty circles, Re = 0.7. The continuous line denotes the Monte Carlo (MC) simulated radial distribution function obtained for the aame parameters, and the broken line denotes the Boltzmann distribution, i.e., glz(Zf) = exp(-$(Zf)/k!O. 0
1
.....................
20
I
2
!.......................
.................................
1
0 .i
........................
............
0
0.5
1.0
15
r
2o Figure 6. Measured adsorption kinetics expressed in the dimensionlesscoordinate system Ova T (whereT = (raz~~d0.546)t is the reduced adsorption time): negative latex particles (a = 0.5 pm); silaniied mica; Z = lo-' M; fp = -50 [mVl; y = 0.25. The points show experimental results obtained for various flow intensities (Re number): 0,Re = 16;&A, Re = 8; 0,Re = 2. The dotted line represents the theoretical MC simulations. The broken l i e showsresults obtained from the approximate equation (6),the continuous l i e shows the long-time asymptotic results derived from eq 10(with 0, = 43.6 % ) and the- -line represents the results calculated from the Langmuir model, i.e. 0 = o,[~~P(-(~~o,)T)I.
...........................
0 (00 100 t [MI Figure 5. Experimentally determined adsorption kinetics of negative polystyrene particles (a = 0.5 pm) on silanized mica: 0 [%] va t [min] dependence for I = lV M (KC1) ( K O = 16.6); fp = -50 mV; y = 0.25; nb = 3 x 108 cm"; (1) Re = 16,(2) Re = 8, (3)Re = 2. The lines denote the theoretical results derived from the MC simulations performed accordingto the irreversible RSA mechanism.
for relatively large colloid particles can be useful for predicting adsorption processes involving large molecular species, e.g., proteins and surfactants which can only be measured indirectly. One can also observe in Figure 4 that the effective range of the lateral double-layer interactions among adsorbed particles is about one particle radius, i.e., 5000 A,which exceeds the doublelayer thickness many times in accordance with the analytical eq 4. The results shown in Figure 4 and the other obtained for various ionic strength and surface concentration values demonstrate spectacularly the sisnificance of the repulsive doublelayer interactions in adsorption processes involving charged colloid particles. Note also that the experimentally found pair correlation function depends very little on the flow intensity charactarized by the flow Reynolds number Re = QlrvR (where Q is the volumetric flow rate of the suspension, v is the liquid kinematic viscosity, and R is the capillary radius), i.e., the results obtained for Re = 2 and Re = 0.7 are identical within the error bounds. As stated above, the advantage of our method is that it allows one to determine in a direct way not only the statistical featuree of adsorbed particle distributions but also the adsorption kinetics as a function of measurable physicochemical parameters. In Figure 5 such kinetic plots are presented in the form of the dependence of the dimensionless surface concentration 0 [ % ] on the adsorption time t for various Reynolds numbers. (28) Bowles, R.J.; Tildeskey,D.J.; Quirke, N.J. Chem. SOC.,Faraday Trans. 2, 1989,85,1505.
The results shown in this and next figures concernthe interface region in the immediate vicinity of the stagnation point where the flow shear rate is negligible (it should be mentioned that for the impinging-jet cell used in our experiments the flow shear rate depends linearly on the distance from the cell center). One can expect that the measured adsorption kinetics can be adequately described by the RSA mechanism discussed above. This conclusion is confirmed by the g o d agreement of the experimental results with the theoretical MC simulations especially for moderate Reynolds numbers. The small deviation of the experimental results from theoretical predictions observed for Re = 16 can be interpreted as discuaeed later on by the flowinduced increase of the blocking parameter B(O). The best agreement between theory and experiment was attained for f = -50 [mV] and y = 0.25. The small value of the y parameter indicates that the electric field generated by the mica interface exerted a considerable effect on the particleparticle interactions. As implied by the RSA mechanism (cf. eqs 1 4 ) all the kinetic results shown in Figure 5 should be reduced to one universal curve when using the coordinate system 0 va T , where T = (ra2nhjb/ 0,)t is the dimensionless time. Such a plot of experimental data is shown in Figure 6 for various Reynolds numbers Re and the bulk suspension concentrations nb. As can be seen in thief i e the theoretical and experimental results are in a quantitative agreement especially for moderate flow rates (Re = 8) when particle adsorption kinetics is adequately described by the proposed RSA model for the entire range of T . It should also be noted that the approximate analytical results derived from eq 6 (short time) and eq 10 (long time) reflect well the experimental results practically for the entire range of the surface concentrations studied. By contrast, the commonly used Langmuir mode11+13 expressed by eq 7 produces theoretical results which deviate considerably from the experimental ones, especially for larger T (see Figure 6). It should be also underlined that our RSA model enables one to calculate the maximum surface concentration values ,e, whereas in the Langmuir model they are empirical parameters whose interpretation is rather obscure. A closer inspection of experimental results shown in Figure 6 reveals that the increase of the flow intensity decreased particle adsorption rate especially for 7 > 1 and Re = 16 (higher surface concentrations). This indicates that the blocking parameter B(0) was dependent to some extent on the particle transport mechanism (flow rate of particle suspensions) in accordance with
2610 Langmuir, Vol. 8,No.11,1992 previous theoretical predictionsm based on simulations of individual particle trajectories (low surface concentration limit). At this moment, however, there are no theoretical approaches available which can account for the dependence of B(B)on Re for arbitrary eurface concentration. Development of such an approach and further experimental studies of flow-affected colloid adsorption phenomena should be the aim of our future works. Neverthelese, it seems that our results obtained for idealized syatems in which the flow-induced blocking effects were minimized can be used as useful reference states for more complex adsorption phenomena involving, e.g., bioparticles.
Concluding Remarks The theoretical RSA model presented in this paper can be applied for a quantitative analysis of adsorption kinetics of interacting particles on homogeneous surfaces. Direct experimental measurements performed using suspensions of monodisperse latex particles confirmed the validity of our theoretical predictions. In particular, it was found that the approximate analytical expressions (Le., eq 6 and eq 10)can be effectively used in place of the cumbersome numerical simulations for a broad range of particle sizes and ionic strength provided that h*
