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205, 290 –304 (1998)

CS985644

Computer Simulation of Flocculation Processes: The Roles of Chain Conformation and Chain/Colloid Concentration Ratio in the Aggregate Structures Serge Stoll1 and Jacques Buffle Analytical and Biophysical Environmental Chemistry (CABE), Department of Inorganic, Analytical and Applied Chemistry, University of Geneva, Sciences II, 30 quai E. Ansermet, CH - 1211 Geneva 4, Switzerland Received September 3, 1997; accepted May 8, 1998

The flocculation of colloidal particles in the presence of adsorbing polymers is a key process in colloid science, as well as in the chemical and biological regulation of aquatic systems. Polymers can influence important physical properties of colloidal aggregates such as their densities and settling velocities, as well as their chemical properties, affecting the probability that two colloidal particles will stick together when they collide. The presence of polymers usually makes more difficult the application of a coagulation theory to colloidal suspensions and the interpretation of experimental observations. Knowledge of floc structures is a key factor in the understanding of flocculation processes, and simulation may provide useful insights required to interpret the results of experimental studies and elaborate new theoretical models. Although modeling leaves much room for more progress, researchers now find it indispensible from a fundamental point of view and for environmental applications. In this paper, we report a computer simulation study of a two- and three- dimensional model for bridging flocculation between large linear polymer chains and comparatively small colloidal particles. The floc structures are investigated as a function of chain/particle concentration ratio, chain conformation, and space dimension. The values of the sticking probabilities are chosen to emphasize colloid– chain interactions compared to colloid– colloid or chain– chain interactions. The results suggest that the floc morphology is strongly dependent on the chain conformation and to a slight extent on the chain/ particle concentration ratio. In particular, colloid interactions with linear rods result in a network characterized by fractal dimensions significantly higher than those obtained on the basis of the Cluster–Cluster Aggregation models of colloids only, or by flocculation of colloids with coiled chains. © 1998 Academic Press Key Words: flocculation; bridging; modeling; chains; colloids; fractal dimensions.

I. INTRODUCTION

Understanding colloidal aggregate structures and formation is one of the central issues of colloid science and an important topic in many industrial, biological, and environmental studies. Due to the development of computational chemistry (1, 2), of new math1

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0021-9797/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.

ematical concepts such as scaling laws (3, 4), of fractals (5, 6), and of models such as the Cluster–Cluster Aggregation model (7, 8), computer simulations have been successfully applied to the investigation of the factors influencing the morphology and formation kinetics of aggregate structures (9, 10). The majority of computer simulations have been carried out under the assumption that all the elementary particles are identical in shape and size. In most practical cases, however, aggregation occurs between particles of different sizes or chemical nature (heterocoagulation, 11, 12, 13) or between particles and macromolecules (flocculation, 14, 15, 16). In natural waters containing mixtures of colloids and macromolecules originating from many sources (plants, microorganisms, minerals), aggregation is a key physicochemical regulation process of the concentration of colloidal matter and of the associated vital or toxic trace metals (17, 18, 19). A key process is the floc formation of submicrometer particles with extracellular polymers (18, 20, 21) released from phytoplankton cells, in particular the polysaccharides which form 80 –90% of the total extracellular material (22). Transmission electron microscopic observations of flocs in natural waters (Fig.1) as well as in the biological step of water treatment plants support the existence of large, rigid, filamentous polysaccharides interacting with comparatively small, spheroidal inorganic particles (23, 24, 25). Owing to the fact that high molecular weight organic matter seems to induce and facilitate particle aggregation and the subsequent removal and elimination of contaminants from the water column, the investigation of aggregate characteristics is therefore essential for the understanding of trace compounds’ circulation in environmental systems. Computer simulation studies of aggregation processes can provide the critical insights required to interpret the results of these experimental observations. In a previous paper (26), we have presented in a complete manner a flocculation model involving large linear chains and small particles, where aggregate structures and formation kinetics were qualitatively and quantitatively described for different chain/particle concentration ratios and chain conformations. It has been shown that the formation of large aggregates

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FIG. 1. TEM image of a centrifuged natural water sample showing associations between large organic fibrils and fulvic compounds present as small spheroidal colloids (size of individual colloid ; 1–2 nm). Note that fulvic colloids also form aggregates together.

is a slow reaction-limited process, even when the reaction between particles and chains is rapid, and that aggregation kinetics are strongly dependent on the chain/particle concentration ratio and chain conformations. These results have also demonstrated that bridging flocculation can be described in terms of scaling concepts and that dynamic scaling appears then to be quite general and applicable to a wide range of aggregation processes. To gain more insights into flocculation mechanisms, we have extended the previous approach to study the morphologies of aggregates composed both of colloidal particles and of comparatively large linear polymer chains. The goal of this paper is to understand how the presence of polymers in suspensions containing interacting particles affects the structure of the aggregates to be formed (Fig. 2). For that purpose, the aggregate fractal dimensions were computed as a function of (i) the space dimensions, (ii) the polymer/ particle concentration ratio, and (iii) the chain conformations (comparison of coiled and rod-like structures). Threeas well as two-dimensional simulations have been performed. Although floc formation is usually a three-dimensional process, two-dimensional processes may play important roles, e.g., at the air–water or solid–water interface of various environmental processes. Furthermore, apart from

their lesser CPU time consuming aspect, they are useful from a theoretical point of view. Indeed, bidimensional simulations often point out space dimensionality aspects related to the formation mechanism and aggregate structures, and comparison of bidimensional and three-dimensional simulations often allows one to better understand the role of key aggregation factors. The organization of the paper is as follows. Section II contains a short description of the simulation model (a detailed description is given in Ref. (26)). Section III describes the mathematical methods for the measurement of aggregate fractal dimensions: a direct application is made on a Reaction Limited Aggregation process containing particles only. Section III then presents a detailed analysis of the floc fractal dimensions, as a function of chain conformation and space dimension. Section IV contains our conclusions. II. SIMULATION MODEL

Off-lattice simulations are carried out in 2d and 3d, with the colloidal particles and polymer chains confined in a square plane or cubic cell, respectively, using normal periodic boundary conditions (27) to avoid distorsions due to the presence of boundaries in a finite size system. When the initial starting

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y5

vu np N , v c n c 1 v u Nn p

[1]

where v c represents the elementary colloidal particle volume (or surface), n c is the initial number of particles, v u is the volume (or surface) occupied by a monomer unit, N the number of monomers per chain, and n p the initial chain number. Electrostatic and solvent effects (30, 31, 32, 33) related to temperature, number and location of functional groups, counter-ion concentration, presence of binding species, hydrogen bond formation, moisture sorption, etc. give rise to macromolecules of diverse structural conformations in solution, influencing thus the particle–polymer interactions and subsequently the kinetics of aggregate formation, as well as the aggregate structures. Although many synthetic polymers are flexible and exhibit a more or less random coil structure, others, the natural ones such as polysaccharides or DNA, are less or not flexible in particular when they associate into double or triple helices forming rigid rod-like structures (34, 35). Hereafter, the word “chain” will be used to cover these various situations. To take into account a wide range of possible structural configurations from random coils to rods, the model permits us to adjust the chain conformation by imposing angular constraints between the unconnected monomers using an unbiased chain sampling method. Below, both random coils and rods are used and characterized using either an effective fractal dimension D f, because of the finite size of the chains, or a Kuhn statistical segment length b k defined as b k 5 ^R 2 &/R max ,

FIG. 2. Schematic illustration of the influence of polymer chains on aggregation processes.

configuration is set up, the individual particles and chains are placed at random with the condition that they do not overlap with each other. Polymer chains are modeled by a succession of jointed spherical monomer units according to the pearl necklace model (28). The monomer unit size is not necessarily that of a chemical monomer, but rather that of a group of successive chemical monomers. Its size is on the order of the persistance length (29) along the chain (size over which the directional correlation between the “chemical” monomers disappears). Particles and chains are both monodisperse in size and total length, respectively; their relative sizes are controlled by adjusting the particle diameters, on the one hand, and the length of the chains through the number and size of the monomers, on the other hand. A chain/particle concentration ratio y is based on the total volume (3d) or surface (2d) of both the colloidal particles and the chains and is given by the equation

[2]

where ^R 2 & represents the mean square end-to-end length of the chain and R max its maximum length. The evolution of aggregate formation is computed using the Cluster–Cluster Aggregation scheme where particles, chains, and aggregates are allowed to move in random directions (Brownian diffusional motion) according to their respective diffusion coefficients. In our calculations, these latter values are assumed to vary inversely to the mass of the diffusive object. No structural parameters have been included in the diffusion coefficient calculations for mainly two reasons. First, owing to the collision efficiency values used in this paper, aggregate mobility is expected to play a second-order role in the aggregate static properties (36). Second, mathematically describing diffusion coefficients of complex structures, i.e., the hydrodynamic properties of the fractal flocs to be formed, for all but the simplest shapes of objects has proven to be a very complicated process. Some theoretical questions of importance such as the floc porosity and the influence of the underlying polymeric network and number of adsorbed particles are not totally clear yet and are beyond the limits of this work. In a single calculation loop, also used as the reference step to compute the relative time, all the particles, chains, and aggregates are randomly chosen and moved (the displacement length is equal

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to the radius of a particle). At the end of this step, the smallest diffusion coefficient of the moving entities in the cell is determined and taken as a reference diffusion coefficient for the next loop. The number of random walk steps is then set equal to one for the corresponding entity. Then particles, chains, and aggregates are chosen successively at random. Their diffusion coefficient is calculated and normalized with respect to the reference diffusion coefficient. A link between the relative and physical times may be established by relating the mean square displacement of an individual particle, taken as a reference particle, to its diffusion coefficient and to its number of elementary displacement during one unit of relative time. If only Van der Waals attraction forces were operating on suspended particles or chains in water, we might expect them to coagulate together immediately. However, they also undergo electrostatic repulsions due to ionizable chemical groups (-OH, -COOH, -NH2, -SH) or surface complexation reactions which may result in both attraction or repulsion forces. Particles or chains then may react quickly, slowly, or not at all, depending on the relative strength of each of the contributions (37). In the following, we do not consider all the details of possible interaction energies that describe aggregation; they are all included in the sticking probability a, which represents the net energy barrier between particles or chains. Probability a may be one, zero, or comprised between 0 and 1. The particle sticking probability is close to 0 when there is a large repulsive energy barrier between particles and/or chains and they have to collide on each other many times before sticking together. A random number x is generated with a uniform distribution in the range 0 , x , 1 after a move which brings two or more aggregates into contact via nearest neighbor occupancies (see Ref. (26) for a detailed description). If the random number x is smaller than the sticking probability a, the contacting clusters (aggregates) are joined to form a large cluster. The representation of the interaction potentials by sticking probability parameters allows us to model a material class rather than a particular type of chain or particle and is an obvious way to limit the number of degrees of freedom. In each particular case, a can be computed based on molecular level interaction energies according to the DLVO theory (38, 39). An additional assumption of the model is that after the energy barrier between particles or between particles and chains has been overcome, the attraction energies are large enough for the formation of irreversible bonds. The values of the sticking probabilitities used in this paper are: for particle–particle interactions, a c– c 5 0.01; for particule–monomer unit interactions, a c– u 5 1; and for monomer unit–monomer unit interactions, a u– u 5 0. They have been arbitrarily chosen to impose slow interactions of particles and no interactions of chains with each other thus enabling an important proportion of interactions to occur between particles and chains. From an experimental point of view, this set of a values is representative of a low ionic strength suspension containing charged particles and large linear chains with either opposite charges and/or hydrophobic moieties, enabling chain– colloid interactions to play a predominant role. Presently, no

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experimental a values have been reported for the interactions between natural colloids and monomer units of organic chains. Nonetheless, a values of 1024 to 1022 have been reported for interactions of inorganic particles in fresh water (17). When chains are introduced in a cell containing free particles, a large “variety” of aggregates is expected to be formed. In order to distinguish them, an arbitrary classification has been established (26). Aggregates composed of particles only are referred to as type I; those consisting of a single chain associated with one or several colloidal particles are referred to as type II. Type III aggregates are considered to be formed of several chains and particles (flocs). Chain dynamics or reorganization processes arising from rotational motions of chain sequences may be important in determining the floc structures when chain reorganization processes are fast compared to the aggregation kinetics and the chain are “flexible” enough to allow conformational changes. Nevertheless, it appears that in many cases natural biopolymer can associate in fibrils or double or triple helices (23, 24, 25) for which conformational changes are negligible in the time frame of aggregation. Conformational changes would greatly complicate the model and thus will not be considered here. The present model, in which no adjustments of the chain structures are performed during the simulations, is then applicable to chains whose conformations are time independent. They will be part of further refinements of the model. Nevertheless, a speculative interpretation of these effects is presented in paragraph g of the discussion. III. RESULTS AND DISCUSSION

Homocoagulation of Particles: Structures of Type I Although this study is not the main purpose of this paper, it has been performed to enable comparison with the results of heterocoagulation between chains and colloids. Two different modes of homocoagulation kinetics can occur depending on the particle sticking probability value. As a approaches unity, Diffusion Limited Aggregation (DLA) occurs and tenuous structures with low fractal dimensions are formed because particles or aggregates do not penetrate each other since they attach rapidly and preferentially on the aggregate surface exterior. On the other hand, as a approaches zero, Reaction Limited Aggregation (RLA) occurs; particles or aggregates have to collide many times before sticking, allowing them to interpenetrate their structure, thus increasing the density of the resulting aggregate. Thus RLA results in higher fractal dimensions. The fractal dimension can be directly measured by simulations since each particle position is stored during a simulation run, allowing the mathematical fractal dimension definition to be applied directly. Two methods are commonly used to calculate the aggregate fractal dimensions. The first method consists (40, 41) of choosing, for each aggregate, a given primary particle in the central part of the aggregate, and considering

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FIG. 3. Homocoagulation of colloids: Snapshots of 2d and 3d off-lattice Cluster–Cluster aggregation using 1000 particles. ac– c 5 0.01.

larger and larger circular areas centered on this particle. The primary particle number contained in this area is then plotted as function of the radius of the corresponding circle. A log–log

plot gives the fractal dimension D f. This method gives a good estimate of D f when only the center of the aggregate can be analyzed, far away from the edges (this is only possible for

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dimensions of aggregates formed by RLA ( a 5 0.01) homocoagulation processes (Figs. 3 and 4). Calculations were performed using 1000 primary particles. Based on gyration radius calculations and the second method, D f values were found to be, respectively, 01.54 6 0.03 (2d ) and 2.03 6 0.03 (3d), in agreement with the literature values, i.e., 1.55 6 0.02 and 2.09 6 0.02 in 2d and 3d, respectively (9, 39). The run-to-run reproducibility of the results has been verified. Flocculation: Aggregate Structures in the Presence of Polymer Chains a. Fractal dimension calculations. Simulations have been carried out with the assumption that there are strong interactions between monomer units and particles (ac– u 5 1) and weak interactions between particles or monomers together (ac– c 5 0.01, au– u 5 0), allowing both the concomitant occurence of homocoagulation and flocculation and the specific study of the influence of the chains (see introduction). The influence of chain/particle concentration ratio on the aggregate structures has been evaluated by increasing the chain

FIG. 4. Dependence of the mean square radius of gyration ^R2g& on the aggregate mass in 2d and 3d for Cluster–Cluster Homocoagulation. ac– c 5 0.01.

large aggregates), or when a cutoff function is used to interpret the data to take into account the finite size of most studied aggregates (42). It also permits us to quantify the short-range structure of simulated aggregates through the correlation function and the theoretical Fourier transform of the positions of the particles, allowing thus a direct comparison of computed structures with small-angle neutron or light-scattering experiments (5, 7, 10). In the second method, the mass of aggregates is plotted as a function of characteristic size, commonly the radius of gyration (1). The slope of the corresponding log–log plot gives the fractal dimension. This method gives a statistical value of Df for the whole of the aggregates which have been formed in the suspension, i.e., on a wide number of structures. It is based on the following geometric scaling relationship between the mean square aggregate radius of gyration ^R2g& and its particle number N, ^R 2g&}N 2y,

with ~2 y 5 1/D f ! ,

[3]

where y represents the scaling exponent and D f the aggregate mass fractal dimension or number if the elementary particles are identical in shape and size. Our simulation model and algorithm were checked in 2d and 3d by computing fractal

FIG. 5. (a) Number evolutions of free particle and type I aggregates as a function of the normalized time. A 3d homocoagulation simulation using 1000 particles. (b) Number evolutions of particle, chain, and aggregates (types I, II, III) as a function of the normalized time on a log–log plot. A 3d flocculation and concomitant aggregation using 1000 particles and 80 chains (number of monomers N 5 28, Kuhn length bk 5 25 monomer units, effective chain fractal dimension Df 5 1.02).

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FIG. 6. Effect of the chain/particle concentration ratio on the floc structures using rod-like chains (N 5 28, bk 5 26 monomer units, effective chain fractal dimension Df 5 1.02). Snapshot of 2d simulations using 500 particles and (a) 5 chains, (b) 10 chains, (c) 20 chains and, (d) 40 chains. In (a) the floc morphology is similar to that of Cluster–Cluster Aggregates (Fig. 3a). In (d) the floc morphology is similar to that of a network.

COMPUTER SIMULATION OF FLOC STRUCTURES

FIG. 6 —Continued

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by considering three limiting situations with respect to the sticking probability values and the chain/particle size ratio used in this paper. At low chain-to-particle concentration ratio (Figs. 6a and 9a), irrespective of the chain conformation, aggregates are essentially composed of particles. Thus, a simple calculation based on the particle positions is assumed to be representative of the aggregate morphologies. At high chain-to-particle concentration ratio, two situations have to be considered depending on the chain conformation. With rod-like chains, particles will be uniformely distributed onto the chains (Fig. 6d), reflecting thus their intrinsic linear conformation. Thus, the fractal dimension calculated on the particle positions will be similar to that of the subsequently formed underlying polymeric network. With coiled chains, the situation is more confused, because particles will be preferentially bound on the chain exterior, acting thus as a colloidal glue (25). In this condition, particle positions and thus D f values should be more representative of the aggregation mechanism, itself controlled by the number of particles adsorbed on the chain exterior. At intermediate particle concentration, the concomitant formation of both type I and type II clusters would require a multicomponent fractal approach including both particle and monomer

FIG. 7. The 2d dependence of the mean square radius of gyration on the aggregate mass (or particle number) for various particle/chain concentration ratios, as indicated in each frame. Polymeric rods (N 5 28, bk 5 26 monomer units, Df 5 1.02) are used. Open squares correspond to an aggregation process using particles only.

number from 0 to 80, and keeping the particle number constant, i.e., 1000 units. In the following, the fractal dimension computed from Eq. [3] is used to describe quantitatively the structure of flocs, i.e., aggregates composed of both colloids and polymers, and to investigate the role of the chain/particle concentration ratio and chain conformation on these structures. For that purpose, ^R 2g & values have been computed for each aggregate formed, based on the primary particle positions. This implies the important a priori assumption that the fractal dimension calculated on the particle position only is representative of that of the whole flocs. This choice has been used for several reasons. First, D f values calculated in this way can be directly compared with D f values obtained for the homocoagulation results of the previous section, thus allowing us to determine straightforwardly the influence of chains. Second, results can be directly compared with experimental values of D f determined by means of techniques such as light scattering, small-angle neutron scattering, image analysis of TEM pictures, or Coulter Counter analysis for which the signals are mainly attributed to the electrondense part of the flocs, i.e., the inorganic primary particles. However, the underlying influence of chains may be evaluated

FIG. 8. The 3d dependence of the mean square radius of gyration on the aggregate mass for various particle/chain concentration ratios as indicated in each frame. Rods (bk 5 25 monomer units, Df 5 1.02) are used. Open squares correspond to an aggregation process using particles only.

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TABLE 1 Flocs Fractal Dimension Df Involving Rods (Particle Number 5 1000) Chain number:

0

10

20

40

80

D f (2d ) D f (3d )

1.54 6 0.03 2.03 6 0.02

1.61 6 0.04 2.09 6 0.04

1.75 6 0.06 2.12 6 0.04

1.81 6 0.07 2.29 6 0.07

1.92 6 0.04 2.52 6 0.04

contributions. It is not considered here and will be part of another study. b. Flocculation kinetics. The mixing of chains and particles induces the formation of several types of clusters (types I, II, III), whose relative number is essentially controlled by the chain/particle concentration ratio and the set of sticking probabilities used. Due to the assumption that the sticking probability is zero for monomer–monomer contacts, no aggregate composed solely of chains can be formed in our case. To follow the aggregation kinetics, the number of clusters as well as free particles and naked chains have to be determined as a function of time. Typical evolution curves of the numbers of free particle and type I aggregates are presented in Fig. 5a for a three-dimensional homocoagulation process. The free particle number is continuously decreasing whereas the aggregate number follows a bell shaped curve. Since the system contains a finite number of particles, a single aggregate is obtained at the end of the simulation. Heterocoagulation curves of chains and particles are presented in Fig. 5b. Three successive modes of cluster formation are now operative. In the first mode, both type I and type II are rapidly formed owing to (i) the high specific surface area of the naked chains and (ii) the large Brownian diffusion coefficient of the free particles. In the second mode of cluster growth, type I and II aggregates coagulate with each other (or together) to form, via particle bridges, larger structures containing more than one chain (type III). In the third mode, these type III aggregates combine to form a single floc. These results suggest that both the absolute and the relative concentration of particles and chains are important variables on the time evolution of the number of each aggregate type and to some extent on the aggregates morphologies. For example, by increasing the number of polymeric rods at constant particle concentration, the number and size of the type I aggregates are expected to decrease because of the increase of the specific surface area of the adsorbing chains in the cell. Whereas, by decreasing the chain fractal dimension, the number of type I aggregates is expected to increase due to the decrease of the specific surface area of the adsorbing chains and the increase in the particle collision rate. These effects are studied quantitatively in Sections (c) to (e). c. Aggregate structures in the presence of rod-like chains. Typical structures composed both of particles and of rod-like chains are presented in Fig. 6 as a function of the chain/particle concentration ratio. Qualitatively, high chain concentrations lead to the formation of a polymeric network, a structure on

which particles are regularly distributed, a few of them acting as ligands or “reticulation” points. On the other hand, low chain concentrations result into the formation of structures whose morphology is similar to those of the “classical” RLA clusters (Fig. 3). To examine quantitatively the dependence of the aggregate fractal dimensions on the initial polymer concentration, ^R2g& has been determined as a function of the aggregate mass for different chain-to-particle concentration ratios (Figs. 7 and 8). The floc fractal dimensions (Table 1) were calculated using Eq. [3]. The results clearly demonstrate that there is a significant increase of D f with the increase of the chain/particle concentration ratio. This is due to the fact that at large ratio, in the first regime, particles more rapidly adsorb on the randomly positionned polymer rods to form type II aggregates than they aggregate to form type I aggregates. In the second regime, type II aggregates collide with each other to form a network structure. Particles are then distributed on a more or less regular network whose fractal dimension tends to be equal to the space dimensionality of the simulation run. When the chain number is decreased, homocoagulation of particles becomes the predominant aggregation process. Then, D f values are closer to those of type I aggregates whose fractal dimension is equal to that of RLA clusters, since a c– c 5 0.01. At intermediate chain concentrations, the concomitant formation of type I aggregates and flocs results in structures whose fractal dimensions are between those of RLA aggregates and a network structure. d. Aggregate structures in the presence of coiled chains. Typical floc structures are presented in Fig. 9 as a function of the chain/particle concentration ratio. As in the previous situation, at low chain number, the aggregate structures, which result essentially from the association of type I aggregates, are poorly affected by the presence of a few polymeric coils. But, contrary to the case with rods, at high chain/particle concentration ratio, the overall floc morphologies do not change significantly. They remain poorly interconnected without formation of a long-range network. These observations are consistent with the evolution of the radius of gyration with aggregate sizes (Figs.10 and 11), from which no significant differences in the D f values are found, irrespective of the chain/particle concentration ratio (Table 2). This result is explained by the fact that there is a non-negligible number of monomer units located inside the coils and inaccessible to the particles. Thus, in the first growth regime, particles

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FIG. 9. Effect of the particle/chain concentration ratio on the floc structures using coiled chains (N 5 28, bk 5 5 monomer units, Df 5 1.28 for chains). Snapshot of 2d simulations using 500 particles and (a) 5 chains, (b) 10 chains, (c) 20 chains, and (d) 40 chains. In all cases, the floc structure is similar to that of Cluster–Cluster Aggregates (Fig. 3a).

COMPUTER SIMULATION OF FLOC STRUCTURES

FIG. 9 —Continued

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are adsorbed at the surface of the polymeric coils to form type II aggregates. In a second regime, type II aggregates behave like new spherical “entities” interacting with each other via a classic CCA process for which the result of the inter-aggregate collisions depends on the geometric characteristics of the external surface of the coil, including the adsorbed particles. As in the La Mer model (42), where the covered area determined the aggregation probability, we expect the number of adsorbed particles on the chains to influence mainly the aggregation rate, and to a slight extent the aggregate structure. Since a c– u 5 1, the formation of “RLA” or “DLA” aggregates will depend on the chain surface coverage. e. Role of the chain/particle ratio in the aggregate fractal dimensions. Figure 12 sums up the change in D f as a function of the chain/particle concentration ratio y, the chain conformation, and space dimension. It clearly shows that the aggregate fractal dimension depends on both the chain conformation and the chain/particle concentration ratio. It is worth noting that when rods are used and y is increased, the floc fractal dimensions approach those of the space dimensionality in which simulations are performed. However, the situation where the floc fractal dimension would be strictly equal to the space

FIG. 11. The 3d dependence of the mean square radius of gyration on the aggregate particle number for various particle/chain concentration ratios as indicated in each frame. Coils are used (N 5 28, bk 5 3 monomer units, and Df 5 1.49).

FIG. 10. The 2d dependence of the mean square radius of gyration on the aggregate particle number for various particle/chain concentration ratios as indicated in each frame. Polymeric coils (N 5 28, bk 5 5 monomer units, and Df 5 1.28) are used.

dimension is expected to be rarely achieved because very well ordered and regular structures should be formed. Due to the finite number of chains and particles and a values, flocculation always results in the formation of one single large cluster (gelation). If our simulations were made with an excess of chains, the evolution of the aggregate size would be characterized by a steady-state regime including a number of small stable flocs (non-gelation). Note that nongelation and gelation region boundaries may be seen as a polymeric network where each particle would act as an efficient “reticulation” point for the chains. It is worth to note that smooth curves (Fig. 12) instead of step functions are observed for the change in Df, probably partly due to aggregate size effects resulting from the finite number of particles and chains in the simulations. According to the theories dealing with critical phenomena (43), if our fractal dimension measurements were performed in very large systems at infinite aggregation times, the resulting Df values would only change at a specific chain/particle concentration ratio. Because of the model complexity and the finite sizes of the simulations, a variety of crossovers occurs, such as the crossover from particle decorated chains (type II) on short length

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TABLE 2 Flocs Fractal Dimension Df Involving Coils (Particle Number 5 1000) Chain number:

0

10

20

40

80

Df (2d ) Df (3d )

1.54 6 0.03 2.03 6 0.02

1.59 6 0.04 1.97 6 0.03

1.56 6 0.04 2.01 6 0.04

1.61 6 0.03 2.08 6 0.03

1.57 6 0.05 2.03 6 0.03

scales (first regime) to CCA-like structures on long scales (type III, second regime) or the crossover from a mixture of type I and II aggregates to the type III formation on long scales. These crossovers explain the curvatures in Figs. 7 and 8 which are strongly dependent on the chain/particle concentration ratio y. For that reason the data corresponding to the formation of types I and II (structures on short scales) were not considered in the D f determination of the long scale CCA like structures. f. Influence of the sticking probability values. This paragraph discusses briefly and qualitatively some results obtained when sticking probability values are modified. When simulations are performed with large a c– c (0.1–1.0) compared to a c– u, large type I aggregates are rapidly formed, with mean masses (and consequently diffusion coefficient) similar to those of the chains. As opposed to the previous case, the

FIG. 12. The dependence of the aggregate fractal dimensions as a function of the chain/particle concentration ratio y. The floc fractal dimension is increasing when rods are used and tends to the space dimension in which they are formed.

elementary particles initially adsorbed on the chains do not participate significantly to the bridging mechanism of the chains since they are essentially bound to each other. Here D f values are also expected to be strongly dependent on the particle/chain concentration ratio and to vary in a range comprised between the fractal dimension of DLA aggregates and those of the networks. It is worth noting that the network formation is expected to occur only when type I aggregates of small size compared to that of the chain are formed, i.e., for dilute systems involving a large number of chains. g. Influence of chain reorganization processes on the floc structures. It is noteworthy that in real flocculation processes as well as in simulation models, a large number of factors may influence the aggregate structures and/or aggregation rate. One important parameter is chain dynamics. However, our simulation model assumes that both the shape of the chains and the flocs are fixed, i.e., no adjustments of the chain structures are performed during the calculations. This condition may be “a priori” considered valid only for stiff chains or when the rate of flocculation is fast compared to reorganization processes. For flexible chains and under the conditions chosen in the calculations, since the particle– particle coagulation efficiency is low, it might be suggested that chains could experience repulsive forces when the DLVO applies and thus would become more rigid and extended. A transition from flexible to rigid rods affecting the subsequent floc structures might be expected. This reconformation process will consequently be strongly dependent on the chain “flexibility”, solution chemistry affecting the chain and particle electrostatic properties, and number of adsorbed particles, i.e., the chain/particle concentration ratio. Nevertheless, it might be also suggested that the occurence of intra-molecular monomer–particle–monomer bridges could, on the contrary, freeze the coiled chain structures. In addition, floc reorganization processes such as collapse or even fragmentation might be crucial in determining the structures, provided chains and flocs are “flexible” enough to allow large conformational changes. However, owing to the amount of microscopic and macroscopic information needed to interpret such phenomena, a speculative discussion is beyond the limits of the present paper and model. They had to be part of other complementary studies using computational techniques such as Monte Carlo or Molecular Dynamics simulations. The use of finite systems and rigid chains of course have limitations. Nevertheless, they represent real systems which should also be understood.

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STOLL AND BUFFLE

IV. CONCLUSION

Based on the above results, one important conclusion can be made: the spatial disposition of particles in flocs results not only from the chain/particle concentration ratio but also from the chain conformations. In particular, flocculation processes with linear rods may be regarded as a network-forming process characterized by fractal dimensions higher than those obtained on the basis of the classic DLA or RLA models. Despite the highly loose structure of the aggregate formed, these fractal dimensions reflect the high order of particles in such networks. We believe that these results may be very helpful to studies of the morphologies of natural aggregates and may be a useful starting point for studying the circulation, role, and function of flocs in natural and industrial processes. Indeed physical properties such as densities, settling velocities of such aggregates, or diffusion of compounds inside aggregates depend much on their fractal dimensions which are still largely unknown. The results reported here have been obtained using a specific sticking probability set. The model, however, is not restricted to it, and a values may be changed or calculated rigorously based on experimental data or theoretical studies. In addition, it should be possible to extend the present results to the study of the influence of the length of polymeric rods on the aggregate morphologies. ACKNOWLEDGMENTS We wish to thank Dr. E. Pefferkorn, Dr. F. Rizzi, Dr. K.J. Wilkinson, and Dr. J. Zhang for helpful discussions. Financial support from the Fonds National Suisse is gratefully acknowledged (Projects 2000-037598.93/1 and 2000043568.95/1).

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

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