A. Dilute suspensions: Recovery of Einstein intrinsic viscosity

For very dilute to semidilute suspensions, the relative viscosity is described by

where η_r is the relative viscosity, η is the viscosity of the suspension, η_s is the viscosity of the fluid solvent (or embedding fluid), η_o is the intrinsic viscosity (equal to 2.5 for suspensions composed of spheres), φ is the volume fraction of rigid bodies, and K_H is the Huggins coefficient. As a simple test, a single sphere with radius a = 5.511 was introduced into a well characterized fluid system where the viscosity was known to about one part in a thousand. The simulation cell was 45³ so that adding a single sphere made φ = 7.692×10^–3. At this small solid fraction, only the lowest order term in Eq. 14 is important. Here η_r ≈ 1 + 2.5φ = 1.0192. After shearing this system over 40 times the system size, the DPD simulation obtained η_r = 1.019±0.002 implying the intrinsic viscosity is 2.46±0.26 which is in good agreement with theory. The uncertainty is based on a standard deviation analysis of simulation data.