Experimental and modeled deposition kinetics of large colloidal particles

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Abstract

The kinetics of the deposition of polymeric microspheres (diameter 3.31–8.74μm, density 1.5gcm−3; diameter 4μm, density 1.055gcm−3), as well as of polymorphonuclear leukocytes (diameter 7μm, density 1.11gcm−3) on smooth, horizontal silica slides was measured using a vertical cylindrical sedimentation cell. The adsorbing surface was observed with an inverted optical microscope connected to a camera and a micro-computer for the image analysis. The experimental deposition kinetics were first compared to the prediction of the ballistic deposition (BD) model aimed at the description of the sedimentation and adhesion under the influence of a strong gravitational field. This model proves effectively to be appropriate in the case of the largest particles used. However, once the gravitational force ceases to strongly dominate the random thermal force, the diffusion of the particles in the liquid becomes an important feature of the process as far as its kinetic aspect is concerned. To take the diffusion into account accurately, especially in the vicinity of the interface, a non-sequential model is developed. Such a model is needed to avoid the problem raised by the treatment of particles trapped by those near the surface, which cannot be solved in a rigorous way in the framework of a sequential model, even though the bulk concentration is small.

Introduction

The concept of irreversible adhesion (or deposition) of particles on solid surfaces has been introduced in order to account for the adhesion kinetics of proteins, cells or colloidal particles on solid/liquid interfaces. “Irreversible” means that, once a particle has touched the surface, it neither desorbs from the surface, nor diffuses along it. Due to this irreversibility, the equilibrium statistical mechanics concepts cannot be used in general for predicting the properties of the assemblies of deposited particles.

In order to describe the irreversible adhesion kinetics and the structure of particle monolayers formed in this way, various models have been developed. The most widely used are those based on the random sequential adsorption (RSA) concept, which is expected to describe irreversible adhesion processes of colloidal particles or proteins in the absence of gravitational effects [1], [2], [3], [4]. On the other hand, the ballistic deposition (BD) model is aimed to describe the irreversible deposition of particles for which gravity dominates strongly over thermal motion [5], [6], [7]. In the intermediate cases, or in the presence of interaction forces, a realistic model should, however, account for the diffusion of the particles in the solvent in contact with the adsorbent. Only when the trajectory of the particles is followed from the bulk suspension to the interface, can in principle all the forces be taken into account. In spite of the difficulties that may arise from the clear identification of the most important of them, a diffusional model at least provides a potentially appropriate framework. In the special case of electrostatic mutual repulsion of the particles, an alternative way was proposed [8], [9], [10], [11], based on the concept of RSA for soft particles having an effective radius Reff (see also Ref. [12]), but it is not evident how to incorporate the effect of gravity in the evaluation of Reff [13].

The explicit description of the diffusion was discussed in former papers [14], [15], [16], [17]. However, even with a model incorporating the diffusion under the combined action of the random forces from the surrounding liquid and all of the deterministic forces that may act on the particle, a new, major difficulty arises, especially when the model is aimed at the determination of the time evolution of the coverage [18]. This difficulty is directly related to (i) the assumption that the particles can be treated sequentially (as in the RSA and BD models), and (ii) the fact that diffusional models are not purely geometrical models. The geometrical RSA and BD models are in fact “yes/no” models in some sense: either an adhesion trial succeeds or the particle is immediately rejected. It was shown earlier [14] that in the absence of gravity, the jamming limit coverage, θ(∞), is not significantly influenced by the bulk diffusion when compared to the RSA prediction (θ(∞)≈0.547). Nevertheless, this does not mean that the RSA model is also relevant for the description of the kinetics when gravity vanishes as was demonstrated recently [19], [20], in accordance with experimental results [21]. On the other hand, the rejection rule captures certainly the main features of the deposition process when gravity is the dominant driving force. In this case, when a particle hits another one, it rolls to the interface and adsorbs on it if enough room is available. If not, the particle is trapped by already fixed particles. Since the gravitational force is assumed to be virtually infinitely large, the probability for the trapped particle to escape is strictly zero. In the BD model, the particle is therefore removed. An alternative would have been to leave it in its trap for ever. In this case, it would have contributed indirectly to the adhesion kinetics by preventing another particle to be trapped at the same location. In contrast, if the gravity is not very large, one has to face the tricky question of how long a (pseudo-)trapped particle should try to jump out of the trap before being discarded or left definitively in its position.

In many experimental instances, the gravity, though important, may not be so strong that a particle can be considered as permanently trapped over the time scale of the experiment. It is then necessary, in a sequential model, to decide how long one has to wait before rejecting it or leaving it definitively in its trap [22]. This waiting time, tw, constitutes an additional arbitrary parameter in the model, which has no experimental counterpart. This is obviously a serious flaw because the estimated kinetics depends directly on the value of tw. In a sequential model, this parameter is unavoidable, or, in other words, the arbitrariness introduced by this parameter cannot be wiped out, even though, when the gravity is strong enough, the kinetics seems to converge to a limiting curve when tw is progressively increased [18]. In any case, to build up a general model, free of non-experimental parameters, the unique route consists in imitating as closely as possible the actual behavior of a suspension of colloidal particles. This means that all the particles must be treated at the same time. This is by far the prime feature of a model aimed at a realistic determination of the deposition kinetics. Such a model is called a non-sequential model [20], [23], [24], [25] and might also be called a parallel model.

In a parallel model, the notion of trapped particle becomes irrelevant. As long as a particle has not touched the adsorbing plane, it continues its diffusion whatever be its location in the system. This is notably the case for a particle which is (or seems to be) trapped by three or more adhering particles. However, even though one recognized that a particle moves within a trap, one would not take a special decision for it. Only at the end of the deposition process one can observe which particles have not reached the collector before the latter attained its jammed state. This inspection could reveal a posteriori whether a particle stayed trapped or not. This information would, however, be useless.

The irreversible deposition of hard spheres under the influence of gravity is characterized by the dimensionless radius R given by [15]R=RΔρg3kT1/4,where Δρ is the specific density difference between the particles and the liquid, g the acceleration due to gravity, k the Boltzmann constant, T the absolute temperature, and R the geometrical radius of the particles. Note that (R)4, which defines the Péclet number (Pe), represents the energy in kT units required to transfer a particle over a distance equal to its radius R against gravity. It was shown through the comparison of experiments and simulations that one can assume the ballistic model to be valid for R>3 as far as the radial distribution function and the local coverage fluctuation are concerned [21], [26], [27]. This, however, does not mean that the contribution of the gravitational force to the movement of the particles should be necessarily negligible when compared to the thermal, random displacement, even when R<3.

Most of the experimental results aimed at checking the validity of the BD model concerned the structure of the adsorbed layers, characterized in terms of the radial distribution function g(r) [21], [26], [27]. It is only relatively recent since the first results concerning the deposition kinetics had been published [28]. It was found that the adhesion kinetics deviated slightly from those predicted by the BD model in the intermediate coverage range, i.e., between 0.35 and 0.50, for Pe=2.53 and, more surprisingly, for Pe=98.9. Similarly, the jamming coverage θ(∞) was found equal to 0.58, i.e., smaller than the value of about 0.61 predicted by the BD model. This discrepancy was attributed to the electrostatic interactions among adsorbing particles.

The aim of the present work was to perform experimental kinetics measurements using polymeric particles of various sizes in order to scan a relatively wide domain of R. As will be seen in Section 2, we started with R=6.52(Pe≈1800), where the BD model was expected to be valid, and decreased R down to 1.72(Pe≈9), where the diffusion and deterministic forces other than the gravity may have a significant influence on the deposition process.

In Section 2, we briefly describe the experimental setup, the image processing and the determination of the experimental kinetics. Section 3 deals first with the BD model; more particularly, we derive there an equation intended to account for the kinetics simulated in the framework of this model. Then we describe the non-sequential model developed to reproduce the experimental findings over the whole R range investigated in this study. The comparison between the predictions of the BD model and of the non-sequential model with the experimental kinetics is exposed in Section 4. It appears that at high values of R the BD model is appropriate as expected, while as R gradually decreases, deviations become more and more striking and justify a more realistic model (for these R values) as well as the introduction of electrostatic repulsion. Concluding remarks are given in Section 5.

Section snippets

Experimental setup

The experimental system consists of a cylindrical vertical glass cell of internal diameter 5.3mm and of height comprised between 38.5 and 72.5mm following the particles used, connected near its top end to a reservoir containing the particle suspension [28], a microscope, a CCD video camera, and a computer devoted to the image analysis [18], [29]. The free, bottom end of the tube was closed with a 18×18mm2 silica slide (Micro Cover Glass Esco 100, Erie Scientific, Portsmouth, NH, USA),

Modeling

Since all the particles we used in the experiments are characterized by R>1.7, it seems reasonable not to neglect a priori the influence of gravity and to try first to reproduce the kinetics with the widely used BD model [5], [6], [7], [29]. This model will serve as a reference throughout this study. We shall, in Section 3.1, give the main features of this model, and especially derive an equation for the deposition kinetics as an alternative to that proposed earlier by Choi et al. [7]. We

Preliminary tests

Two parameters involved in the model, difficult to set a priori, may be essential in the liability of the results. They are the initial concentration of the particle suspension and the maximum number of repositionings a particle is allowed to attempt in case of collision. The following tests were performed with the melamine particles of diameter d=8.742μm.

The side length c of the adsorbing square surface was always set equal to 14d. The area A=c2 then corresponds approximately to that of the

Concluding remarks

The experiments performed for dense polymeric microspheres characterized by R=6.52 have shown that both the deposition kinetics and the jamming coverage are in a quantitative agreement with the BD model. Our experimental data constitute, therefore, a direct experimental proof of the validity of this model for characterizing particle deposition processes under a strong gravitational force. Although the experiments were performed for large particles, it can be postulated that the BD model is

Acknowledgements

The authors are grateful to J. Bafaluy for interesting discussions on the interfacial resistance. Ph.C. benefited from a grant of the Ministère de l'Enseignement, de la Recherche et de la Technologie (France). M.S. thanks the Faculté de Chirurgie Dentaire (Strasbourg, France) for financial support. The authors are indebted to V. Latger-Cannard, S. Viry and S. Balat (Service d'Hématologie Biologique, Centre Hospitalier Universitaire, Nancy, France) for the preparation of the PMN.

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