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Interactions between spherical colloids and nano-structured membranes have been investigated using atomic force microscopy and numerical modeling of colloid-substrate interaction energies. In the past, polymeric surfaces with nano-scale roughness features were simulated by re-constructing randomly sized, located, and oriented hemispherical asperities on a flat plane based on AFM derived roughness statistics for polymeric membranes. Lifshitz-van der Waals, Lewis acid-base, and electrostatic double layer interactions between spherical particles and rough surfaces were determined via numerical simulations using the surface element integration (SEI) technique.

These previous simulations demonstrated that nano-scale surface roughness features reduce the magnitude of colloid-substrate interaction energies and produce distributions of interaction energy profiles across a rough substrate surface. Further, these simulations suggested that nano-scale surface roughness can create a small number of locally favorable sites for particle deposition onto membranes, which may act to initiate surface fouling phenomena, even for highly repulsive colloid-substrate pairs. A modified Derjaguin's integration (DI) model was developed, which considered the average effects of nano-scale surface roughness on colloid-substrate interactions. The modified DI model uses a weighted average of sphere-plate and sphere-sphere interactions where the weighting factor is the fraction of interactions involving an average sized roughness asperity (ç). The modified DI (MDI) model agreed well with numerical SEI model prediction of average interaction energy profile, but the analytical method lacks generality because ç could only be determined from SEI simulation results due to the complex, irregular nature of rough surfaces simulated.

In this paper, our objective is to develop a simple analytical model of colloid-rough surface interactions by exploring rough surfaces with regular nano-structured patterns. A series of simulation were performed systematically by modeling surfaces comprised of regularly patterned asperities and the analytical (modified Derjaguin approximation) is applied by determining the weighting factor from the geometry of the nano-structured patterns. Hence, ç may be predicted from membrane morphological statistics that could be derived from AFM roughness analysis.

Polymeric membrane and aquatic colloidal surface properties are used as characteristic input parameters for interaction energy calculations using SEI and the modified DI methods. We systematically vary the asperity size, density, and distribution, as well as the interacting particle size and chemical properties. SEI, standard DI, and MDI simulations (with predicted weighting factors) are compared. SEI simulation results typically fall between the two extreme scenarios using the standard DI method, i.e., the results of sphere-plate and sphere-sphere interaction energy computations and the average trend over a larger surface is well-described by the modified Derjaguin integration method. The ability to analytically predict interactions between colloidal particles and substrates with nano-scale surface roughness feactures has important practical implications for many aggregation and deposition processes. In addition, these results provide valuable insight into potential surface nano-structures that may yield fouling-resistant surfaces.