Abstract—Deposition of large particles such as colloidal or bio-particles on a solid surface is usually modeled by the random sequential adsorption (RSA). The model was previously described by the integral-equation theory whose validity was proved by Monte Carlo simulation. This research generalized the model to include the concentration effect of added particles on the surface. The fraction of particles inserted was varied by the number density of 0.05, 0.1, and 0.2. It was found that the modified integral-equation theory yielded the results in good accordance with the simulation. When the fraction of particles added was increased, the radial distribution function has higher peak, due to the cooperative and entropic effects. This work could bridge the gap between equilibrium adsorption, where all particles may be considered moving and RSA, where there is no moving particles. Index Terms—Colloid, Deposition, Integral Equation Theory, Monolayer Films. I. INTRODUCTION To understand the onset of irreversibility, it is very useful to generalize the adsorption model by considering the time scales that characterize its various steps. Thus, the generalized adsorption process will depend on at least three distinct characteristic times, assuming that the activation energies for desorption are much higher than for the other processes. Those times are a surface relaxation or diffusion time ( d τ ) measuring time for moving one step, a reaction time ( r τ ) measuring the time for irreversible attachment to the surface, and an adsorption time ( a τ ) measuring the separation between two consecutive adsorption events. The relative magnitude of the three times defined above lead to very different physical situations. In the equilibrium case there is no reaction with the surface and the adsorbed molecules equilibrate rapidly between successive additions such that r a d τ τ τ >> >> . In another limiting case, Random Sequential Adsorption (RSA), the adsorbed molecules react with the surface instantaneously but not diffuse, so that d a r τ τ τ >> >> . In the third limit, Manuscript received December 28, 2007. This work was financially supported by the office of National Research Council of Thailand in the fiscal year 2008. Panu Danwanichakul is with the Department of Chemical Engineering, Faculty of Engineering, Thammasat University, 99 Moo 18 Paholyothin Rd., Khlong Nung, Khlong Luang, Pathumthani, 12120 Thailand (e-mail: dpanu@engr.tu.ac.th). Sequential Quenching, the separation in the characteristic times such that a r d τ τ τ >> >> . A similar discussion about relative magnitude of the characteristic times was also given by Schaaf and Talbot [1]. Usually, the adsorption of large particles such as colloidal particles and proteins on a surface is irreversible. Feder and Giaever [2] reported that the monolayer structures of adsorbed ferritin on carbon surface could be explained by the model of random sequential adsorption (RSA). Onoda and Liniger [3] also found that the configurations of polystyrene spheres on a glass surface finally reached the jamming limit of RSA. However, there have been many indirect and direct evidences showing that among the immobile particles there exists mobile fraction of particles. Reports of surface coverages significantly greater than the RSA jamming limit provide indirect evidence for lateral diffusion [4]. An early study of surface diffusion of adsorbed proteins was carried out by Michaeli et al. in 1980 [5]. In their experiment the distribution of bovine serum albumin (BSA) adsorbed on glass was imaged by autoradiography. They saw no evidence of desorption and noticed that a protein front migrated over distances proportional to the square root of time, as expected for diffusion. Later, Burghardt and Axelrod [6] published a quantitative study of BSA surface diffusion on quartz. They used an elliptical spot fluorescence recovery after photobleaching (FRAP) in a total internal reflection configuration to measure simultaneous surface diffusion and exchange between adsorbed and dissolved proteins in solution. They found that adsorbed BSA exists in three distinct states: irreversible, slowly reversible and rapidly reversible. Tilton et al. [7], [8] used FRAPP based on interference of two coherent beams in total internal reflection at a solid-liquid interface. In their studies, BSA was adsorbed on polymer surfaces: spin-cast polymethyl-methacrylate (PMMA) films and cross-linked spin-cast films of polydimethylsiloxane (PDMS). They found coexistence of a mobile and an immobile population of BSA. Tilton et al. [8] also found that the mobile fraction does not depend on surface concentration. The coexistence of mobile and apparently immobile proteins appears to result not from aggregation of adsorbed BSA but from a change in conformation or orientation of the adsorbed protein. Other experiments on protein adsorption, such as that of DNA oligonucleotides on APTES glass [9] and of BSA on silica-titania surfaces [10], have also revealed surface diffusion in the adsorbed state. If the protein-protein interactions are favorable, surface diffusion will lead to clustering [11], [12]. Radial Distribution Functions of the Structures Built through Fractional Deposition of Hard Spherical Particles Panu Danwanichakul IAENG International Journal of Applied Mathematics, 38:4, IJAM_38_4_05 _______________________________________________________________________________ (Advance online publication: 20 November 2008)