Reply to “Comments on ‘Effect of Electric Charge on Collisions between Cloud Droplets’”

Abstract

Tinsley and Zhou (2014) identify a very significant disagreement between the simple calculations presented in my recent paper (Fletcher 2013) and the earlier detailed calculations by Khain et al. (2004). As expressed in their comments, there is a difference of about a factor of 4 between these two papers when predicting the charge ratio necessary to achieve an attractive force between two equal sized droplets, and a difference of sign between attraction and repulsion at small distances. It is the purpose of this brief response to identify the source of the disagreement and to resolve it, at least partially. Fletcher (2013) aimed to explain the interaction between charged droplets in a simple way by considering the forces between droplet charges and induced dipoles without examining higher-order forces involving quadrupoles. A crucial element of the theory and, as we shall see later, the cause of the major disagreement is the magnitude of the induced dipole. Fletcher (2013) assumed, without any discussion, that the magnitude of this induced dipole m on a droplet of radius r situated at a distance R from a point chargeQ is given by Eq. (2) of that paper asm’Qr/4R. This differs significantly from the more detailed expression used by Khain et al. (2004) and cited as Eq. (A2) in the appendix to their paper. In the limit where R r, this expression is equivalent to m ’ Qr/R and so differs from my assumed value by a factor of 4. It is here that the major discrepancy between the two papers occurs, and my assumed dipole magnitude is probably incorrect. Even when this change is made, however, my approximate calculations indicate a repulsive force at small distances between droplets of equal size bearing charges of the same sign and with charge ratio less than about 5, instead of the original ratio of 20. To correct any discrepancy inmy simple theorywould require the inclusion of quadrupoles and perhaps even higher terms—changes equivalent to moving the positions of the dipoles away from the droplet center and equivalent to the ‘‘imaginary charges’’ of Khain et al. (2004). While it is tempting to accept the calculations of Khain et al. (2004) as being practically correct, they cannot be considered to be based upon an exact theory that is applicable to droplets. The original work of Davis (1964), which does appear to be exact, involves the interaction between two rigid charged conducting spheres on which the surface potential is uniform. The material of a water droplet is, however, almost electrically insulating and has a large dielectric constant, and therefore the situation is rather different. There is also another, and perhaps more important, deviation from spherical symmetry. Because the electrostatic force varies across the volume of the droplet and because the surface tension of water is finite, it results in a distortion of the original spherical droplet shape. This refinement, which could be very significant at small separations, does not appear in any of the published treatments of which I am aware. I thank the authors for bringing this apparent error inmy paper to the attention of readers, and I hope that my further remarks above prove helpful to those concerned with the subject. There appears to be more work to be done.