(713f) Application of Surface Element Integration to Diffuse Double Layer Pair Interactions between Non-Simple, Non-Smooth, and Oriented Particles

Electric double layer interactions are critical in governing colloidal behavior including aggregation, deposition, and dynamics in solution. Rigorous calculation of the double layer force requires solving the electric potential distribution around interacting objects and integrating the stress tensor around a surface enclosing one of the objects. This rigorous calculation is numerically intensive and typically requires finite element methods to solve the potential distribution, especially for oriented, non-simple, and rough particles. As a result, approximate methods are commonly used to calculate double layer interactions, the most prominent of which is Derjaguin’s approximation.

Derjaguin’s approximation calculates diffuse double layer interactions for simple geometries and approaches the rigorous calculation when the Debye length is small relative to the particle size. In practice, however, particles and surfaces often have morphological and charge heterogeneity that precludes application of Derjaguin’s approach. The surface element integration (SEI) method provides an alternative, commonly used method that can be applied to arbitrary particle and surface morphologies. Originally, the SEI method was developed to assess the interaction between a particle (of any morphology) and an infinite flat surface¹ and was thought to be exact in the constant potential limit (but not the constant charge limit¹). Later the SEI method was extended to approximate interactions between two particles,² and a particle and rough surface.^3,4 Given the versatility of the SEI method, it has been applied to characterize the effects of charge inhomogeneity,⁵ roughness,^3,4 and particle orientation⁶ on diffuse double layer interactions.

Despite its widespread use, the accuracy of the SEI method for double layer interactions has not been demonstrated by comparison to the rigorous calculation. Further, the SEI method was originally derived for general DLVO interactions so the limits of rigorous applicability to double layer forces have not been elucidated.

Here we derive the SEI method from the rigorous calculation and show that the SEI method neglects radial potential gradients. It is thus accurate only in the limit of large κa, the ratio of particle size to Debye length. Additionally, we find that the SEI method is aphysical for double layer interactions with the particle backside. We derive a physically consistent, simpler SEI methodology. We use finite element methods to assess rigorously double layer interactions and evaluate the accuracy of the SEI method towards accounting for surface roughness, particle orientation, and charge inhomogeneity. Figure 1 shows the dimensionless force calculated from finite element methods and the SEI method between a spherical particle and a sinusoidal rough surface. We find that the SEI method is accurate for both the constant potential or charge limits when κa is large (>5) but deviates significantly from the rigorous calculation for small κa, even for the particle-flat plate interaction which was previously thought to be exact.

References:

1. Bhattacharjee, S. & Elimelech, M. Surface element integration: A novel technique for evaluation of DLVO interaction between a particle and a flat plate. J. Colloid Interface Sci. 193, 273–285 (1997).
2. Bhattacharjee, S., Elimelech, M. & Borkovec, M. DLVO Interaction between Colloidal Particles: Beyond Derjaguin’s Approximation. Croat. Chem. Acta 71, 883–903 (1998).
3. Bhattacharjee, S., Ko, C. H. & Elimelech, M. DLVO interaction between rough surfaces. Langmuir 14, 3365–3375 (1998).
4. Rajupet, S. DLVO Interactions between Particles and Rough Surfaces: An Extended Surface Element Integration Method. Langmuir 37, 13208–13217 (2021).
5. Bendersky, M. & Davis, J. M. DLVO interaction of colloidal particles with topographically and chemically heterogeneous surfaces. J. Colloid Interface Sci. 353, 87–97 (2011).
6. Bhattacharjee, S., Chen, J. Y. & Elimelech, M. DLVO interaction energy between spheroidal particles and a flat surface. Colloids Surfaces A Physicochem. Eng. Asp. 165, 143–156 (2000).