RAYLEIGH LIMIT CALCULATOR

A droplet tends to have a spherical shape because of the surface tension of the liquid. If it is electrically charged, the electrostatic repulsion between ions might overcome the surface tension, leading to its breakup. Upper estimates for the charge in spherical and cylindrical systems are given by the Rayleigh limits:

Sphere
q2 = 64π2 ε σ r3

Cylinder
q2 = 6π2 ε σ r l2

where:

  • ε is the vacuum permittivity.
  • σ is the surface tension of the liquid.
  • r is the radius of the droplet (jet).
  • l is the length of the jet.

The calculator on the right uses these formulas to calculate the Rayleigh limit for a droplet of radius r or for a jet of length l and radius r. Input data:

  • Permittivity of the vacuum, ε.
  • Surface tension of the liquid, σ (the default value is for water).
  • Radius of the droplet (jet), r.
  • Length of the jet (must be zero for droplets), l.

If the radius of the droplet (jet) is not known, it can be estimated using the following quantities:

  • Molecular mass of the liquid molecules (the default value is for water).
  • The density of the liquid, ρ (the default value is for water).
  • The number of molecules in the system.
  • The length of the jet. The system is considered spherical if the length is zero and cylindrical otherwise.

Version 2.0 04/18/2005: added the cylinder case.

Surface tension, σ (N m-1)

Permittivity, ε (C2 N-1 m-2)
Molecular mass, M (au)

Density, ρ (kg m-3)

Number of molecules, N
Constant volume
Radius, r (nm)
Length, l (nm)
Rayleigh limit (# charges)
Concentration (mol L-1)
Reference: Lord Rayleigh Phil. Mag. 1882, 184-186.

How to use the calculator: Change any of the fields (except for the concentration), the rest of them will be automatically updated.

Checking the box Constant volume prevents the number of molecules from changing in case there is another parameter that can be changed instead. It works only for cylindrical systems, when changing the Radius (for example) could impact the Length or the Number of molecules. If the box is not checked, after changing the Length or Radius the calculator will recalculate the volume and the Number of molecules. When changing the Rayleigh limit the checkbox is ignored. For a spherical system, a change in the Rayleigh limit will have effect on Radius and Number of molecules while for a cylindrical system, the Radius and the Length will be recalculated keeping the volume (Number of molecules) constant.

We used this calculator when we simulated the behavior of charged water nanodroplets in vacuum, click to read how solvated ions evaporate from charged water nanodroplets.

References: