Answer to the Question 12/02
CHARGED DROP
The question was:
The shape of a freely suspended liquid drop is kept spherical (with radius
R) by the surface tension g. [For simplicity we assume
weightlessness.] Assume that the liquid
is conducting and it is being gradually charged. What will happen as the charge
Q increases?
(2/06) A partial solution has been submitted (27/7/2005) by
J.I.I. de la Torre
(e-mail nacho@usal.es).
The answer:
Surface tension of a spherical drop produces positive pressure difference between
then inside and outside of the drop; this difference is 2g/R. Surface charge
produces negative pressure difference equal to 2π multiplied by the squared
charge density (in Gaussian units). When the charge is large enough, the total
pressure difference becomes negative, and, obviously the spherical configuration
is not stable. This has been shown (27/7/2005) by
J.I.I. de la Torre. His solution
can be found in the following PDF file.
Thus we can expect instability, when Q=QR,
where QR2=16 π R3g
in Gaussian units, or
QR2=64 π2 ε R3g
in MKSA units. We will denote QR Rayleigh charge, because
Lord Rayleigh was the first to analyze this problem in Philos. Mag.,
14, 184 (1882). Although we need to consider all possible distrotions
of a sphere, we will start with a specific example: We will examine what happens
when a charged sphere is distorted in ellipsoid of revolution. Deviation of the
ellipsoid from a speherical shape can be described by eccentricity e
which is defined by
e2=1-(b/a)2, where b and
a are the minor and major semiaxes. (e=0 is a sphere, e=1
is a needle.) The graphs below depict the ratio between the total energy
of an ellipsoid and the energy of the sphere as a function of squared eccentricity
for various values of charging parameter x=(Q/Q)R)2.
(The graphs, from top to bottom, correspond to x =0.88, 0.89, 0.0895,
0.898, 0.90, 0.91, 0.92.) [The graphs were taken from the paper by Y. Kantor and
M. Kardar, Phys. Rev. E51, 1299 (1995); many items discussed here
can be found in that paper. Here is a copy of the paper in
PDF format.] We note that for small charges (small x)
the spherical configuration has the minimal energy; a new local minimum appear at
x=0.887; it becomes a global minimum at x=0.899. Finally, at
x=1, the spherical configuration becomes locally unstable.
Thus, at Q=QR, the sphere is certainly unstable. However, one may
ask whether the "ellipsoidal deformation" is sufficiently general to represent
arbitrary deformations. The answer was given by Lord Rayleigh in 1882: he
studied all possible (infinitessimal) defromations of a sphere (using spherical
harmonics) and established that the lowest order deformation described by
Legendre polynomial
P2(cos θ) is the least stable and becomes unstable
exactly at the charge x=1. The remaining deformations require higher
charges to become unstable. Thus carefully charged drop will become unstable
at Q=QR.
However, we have seen that even at lower charges there are configurations that
have lower energy than the sphere, although the system will have to overcome
energetic bariers before getting to those minima. One option, is to divide
a sphere into several equal smaller spheres. (An example of division
into two parts was provided by J.I.I. de la Torre
(see the following PDF file.)) However,
even deeper minima can be found if the drop is divided into unequal
parts. In fact it can be shown, that if we divide a drop into one large drop
and many very small (and equal) drops, we can lower the energy to the level
of uncharged drop: One big drop (uncharged) will carry all the surface tension
energy, while a dust of infinitessimal drops will carry all the charge
with negligible electrostatic energy. [See this proven in the paper by Kantor
and Kardar mentioned above (in Appendix C).]
Pictures of exploding charged drops can be found in the early experimental
study by J. Zeleny, Phys. Rev. 10, 1 (1917).
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