Albert Einstein’s 18-pages long doctoral thesis (submitted in July 1905 and published next year in the German journal Annalen der Physik) is his most cited paper. While the objective of Einstein’s work was to determine the size of atoms and molecules, an important result of the thesis was the viscosity of a suspension, or a dilute mixture consisting of a liquid phase and small solid particles with a volume concentration of ϕ.
Introduction of a solid phase into a liquid reduces the fluidity and, therefore, increases the viscosity k* of the composite two-phase system in comparison with that of the pure liquid (k). However, to determine a quantitative relationship between the dynamic viscosity of the pure liquid and that of the composite system may constitute a challenging problem.
Amazingly, Einstein’s original result, k*=(1+ϕ)k (see Annalen der Physik 19: 289 page 300), contained an error, which had not been discovered for several years, until experiments were conducted by Jean Perrin. These experimental findings troubled Einstein, because they contradicted his theoretical result. The calculation error was found only in 1911 with the help of Einstein’s collaborator, Ludwig Hopf, and an erratum was published in Annalen der Physik. The corrected formula is known today as the Einstein equation for the viscosity of a suspension:
The coefficient of 5/2 is hard to deduce from any intuitive considerations of similarity or dimensions. This is why neither Einstein himself nor his thesis reviewers found the error. A naïve analysis would suggest that due to the no-slip boundary conditions at the surfaces of the spheres, the effective velocity gradients would reduce by the factor of (1+ϕ), and the amount of dissipated energy per unit volume per unit time thus reduced by the factor of (1+ϕ)^2~(1+2ϕ) for small ϕ (Fig. 1). Furthermore, one could conclude that the volume of integration is reduced by the factor of (1-ϕ), which yields (1+2ϕ)(1-ϕ)~(1+ϕ). However, this naïve (and incorrect) speculation does not take into account the change of pressure/stresses (otherwise can be viewed as a contribution of the rotational motion), which, upon the proper integration, produces the factor of (1+0.5ϕ) and thus (1+2ϕ)(1+0.5ϕ)~(1+2.5ϕ).
The no-slip boundary condition on suspended particles decreases the velocity gradient (and hence increases the viscosity). However, one should take into account that pressure / stresses are also affected by the presence of the solid particles.
However, for most real liquids, particularly, for nanofluids containing submicron-sized particles, the dependency is governed by the generalized Einstein equation
k*=(1 + aϕ)k
where a is a parameter depending on various considerations including the particle shape, its Debye length, etc. Typically a>2.5, and in many cases it is significantly greater than this original value in the Einstein equation.
In tribology, nanolubrication is an important area, where nanoparticle additives are used for novel lubricants, and it is important to control the viscosity. An equation for the coefficient of friction, similar to the Einstein equation, can be formulated. Together with my colleague Dr. Alexander Breki from St. Petersburg (Russia) we are working on obtaining such formulation and experimentally verifying it.