An uncertainty budget analysis on the Hamaker constant determined when fitting force-distance curves for a sphere-plate system

Abstract

This work presents an analysis of factors that contribute to the uncertainty on the Hamaker constant A determined by curve fitting theoretical equations to forces measured between macroscopic bodies. The model geometry is a colloidal sphere of radius Rs approaching from a distance z above flat surface plate. The governing equation for non-contact force FNC(z) in a perfect system includes only van der Waals FvdW and Born FBr forces, perfect measurements include other forces FOT,j, and real systems consider experimental noise ΔFm and non-equilibrium forces FNE. The expression for the overall uncertainty budget Δr2A has three groups of factors. The first are relative uncertainties Δr2Xj on each terms Xj in the theoretical force equation. The second are weighting parameters ϕj2 or ωj2 that scale the relative contribution of Δr2Xj. The third are sensitivity factors Sj,k2 as partial derivatives of Xj with regard to xk that scale the contribution of relative measurement uncertainties δr2xk. The general expression Δr2A=∑ωj2∑Sj,k2δr2xk+∑ϕOT,j2Δr2FOT,j+ϕFNE2Δr2FNE+ϕFm2Δr2Fm includes relative contributions from measurement uncertainties δr2Rs,δr2z, measurement noise Δr2Fm, non-equilibrium effects Δr2FNE, and uncertainties in other force terms Δr2FOT,j. Conclusions are that uncertainty in separation distance has a greater contribution to Δr2A than uncertainty in sphere radius, that one point measurements should expect Δr2A≈16Pr2 where Pr is the average relative uncertainty over all measured inputs, and that Δr2A≈2Δr2Fm for multi-point curve-fitting results.