Analysis of the drop weight method

The drop weight method is an accurate yet simple technique for determining surface tension sigma. It relies on dripping a liquid of density rho at a low flow rate (Q) over tilde from a capillary of radius R into air and measuring the combined volumes of the primary and satellite drops that are formed. The method's origin can be traced to Tate, who postulated that the volume (V) over tilde (ideal) of the drop that falls from the capillary should be given by rho g (V) over tilde (ideal)=2 pi R sigma, where g is the gravitational acceleration. Since Tate's law is only an approximation and the actual drop volume (V) over tilde (f)<(V) over tilde (ideal), in practice the surface tension of the liquid-air interface is determined from the experimental master curve due to Harkins and Brown (HB). The master curve is a plot of the fraction of the ideal drop volume, Psi equivalent to(V) over tilde (f)/(V) over tilde (ideal), as a function of the dimensionless tube radius, Phi equivalent to R/(V) over tilde (1/3)(f). Thus, once the actual drop volume (V) over tilde (f), and hence Phi, is known, sigma is readily calculated upon determining the value of Psi from the master curve and that Psi=rho g (V) over tilde (f)/2 pi R sigma. Although HB proposed their master curve more than 80 years ago, a sound theoretical foundation for the drop weight method has heretofore been lacking. This weakness is remedied here by determining the dynamics of formation of many drops and their satellites in sequence by solving numerically the recently popularized one-dimensional (1-d) slender-jet equations. Computed solutions of the 1-d equations are shown to be in excellent agreement with HB's master curve when (Q) over tilde is low. Moreover, a new theory of the drop weight method is developed using the computations and dimensional analysis. The latter reveals that there must exist a functional relationship between the parameter Phi, where Phi(-3) is the dimensionless drop volume, and the gravitational Bond number G equivalent to rho gR(2)/sigma, the Ohnesorge number Oh equivalent to mu/(rho R sigma)(1/2), where mu is the viscosity, and the Weber number We equivalent to rho Q(2)/pi R-2(3)sigma. When We -> 0, the computed results show that Phi depends solely on G. In this limit, a new correlation is deduced which has a simple functional form, G=3.60 Phi(2.81), and is more convenient to use than that of HB. The computed results are also used to show how the original drop weight method can be extended to situations where We is finite and resulting drop volumes are not independent of Oh. (c) 2005 American Institute of Physics.