Transport of aerosol particles is a fundamental phenomenon in many environmental and industrial applications. Among the several computational fluid dynamical schemes used to study this problem, the lattice Boltzmann methods (LBMs) have shown great promise. Using a 2-D LBM model coupled with a Lagrangian formalism, this study investigates an early stage of particle-surface collisions in a free-stream flow over a semi-infinite array of staggered obstructions at operating conditions of woven-wire screens. After an initial validation of the model, the particle-surface collision efficiency with different diameters is then examined as functions of the number of staggered obstructions, obstruction morphology, and separation distance between two center points of obstructions. Particle motion mechanisms include drag, lift, and Brownian forces. Enhanced collision efficiency of particles to obstructions due to the presence of multiple staggered cylindrical obstructions is identified and highlighted. Based on these insights, our work also explores the possibility that collision efficiency of particles to obstructions can be enhanced by a change of obstruction morphology. Finally, the results highlight the range where particle-surface collision efficiency is sensitive to the longitudinal spacing between two cylindrical obstructions.