Diffusional Coating of Nanoparticles
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Diffusional Coating of Nanoparticles James P. Lavine Digital and Applied Imaging, Image Sensor Solutions, Eastman Kodak Company, Rochester, NY 14650-2008, U.S.A. ABSTRACT The time-dependent diffusion of particles to an absorbing sphere is investigated with three random walk models. The first uses consecutive independent particles and finds the capture time distributions are exponential for a range of values of the surface absorption probability. The next two models are of the ensemble variety and assume that only a finite number of particles may be absorbed by the sphere. These models investigate depletion effects and concentration dependence. The latter is probed by varying the initial number of diffusing particles. It is found that the capture time distributions now resemble asymmetric Gaussians. INTRODUCTION The diffusion of particles to an absorbing sphere is a model system for aspects of nanoparticle fabrication such as capping layer formation [1,2]. The capping layers are used to improve the properties of nanoparticles. In addition, metal gettering at cavities or silicon oxide precipitates in silicon device wafers is often analyzed in terms of solutions of the diffusion equation in three spatial dimensions [3]. Results for time-independent diffusion between spheres are found in Berg [4], while Redner presents solutions for time-dependent situations [5]. The present report simulates the time dependence of the coverage of a sphere through a set of random walk models. The first assigns a fixed absorption probability to each encounter of the diffusing particle with a sphere, which represents the nanoparticle. The second is an ensemble calculation with all the unabsorbed diffusers moving at each time step. This model is aimed at studying the effects of depletion, because the number of diffusers decreases with time. The final ensemble model changes the numbers of initial free diffusers in order to investigate concentration dependencies. The latter two models vary the absorption probability when diffusing particles are successfully absorbed. The distribution of the absorption or capture times is obtained for each model. Surface diffusion is not included at present. The second section describes the numerical computations. The third section contains the numerical results for the three models and the final section has the conclusions. NUMERICAL MODEL The model space consists of two concentric spheres with radii of 1 nm and 10 nm, respectively. The inner sphere is absorbing. Each diffusing particle starts with randomly generated spherical coordinates. The radial coordinate is uniformly distributed between 1.5 and 9.5 nm, while the angles are uniformly distributed over 4π steradians. The particle undergoes a random walk with step sizes in the spatial coordinates x, y, and z that are Gaussian-distributed [6] with a standard deviation of the square root of (2D ∆t). Here, the diffusion coefficient D is 5 × 10–6 cm2/s and ∆t is the time step of 0.18 ns. The Gaussian is used [7], as it is the Green’s functions for the
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