Vibrational pseudo-modes of a multilayer sphere embedded in a medium

General help

Using this javascript calculator, you can calculate the frequencies and width of the mechanical resonances of a (multilayer) sphere in a medium assuming isotropic viscoelasticity. In the presence of viscosity, the frequencies and FWHM do not scale as the reciprocal size of the sphere. Note that this calculator works for systems without viscosity too. For details, see the following paper which you may want to cite if you use this calculator: L. Saviot, C. H. Netting & D. B. Murray, J. Phys. Chem. B. 111, 7457 (2007).

This calculator also returns so-called "matrix modes" which have a very large FWHM. These modes are probably not the ones you are most interested in. The maximum FWHM considered by the root finding algorithm is printed when the calculation is over. It is chosen so that the calculator works as expected in most cases.

Input box help

The input box contains one line for the core and one for each shell. The last line describes the surrounding viscoelastic medium. Therefore there should be at least two lines. The content of the lines is organized by column as follow:

• if the first column does not contain the character v:
  1. ρ (g/cm³): mass density
  2. v_L (m/s): longitudinal sound velocity
  3. v_T (m/s): transverse sound velocity
  4. R_ext (nm): external radius of the shell or of the core for the first line. This value should increase for every shell.
  5. The rest of the line is ignored.
• if the first column contains the character v, the following ones are:
  2. ρ (g/cm³): mass density
  3. v_L (m/s): longitudinal sound velocity
  4. v_T (m/s): transverse sound velocity
  5. μ (Pa.s): shear viscosity
  6. ζ=λ+2/3μ (Pa.s): bulk viscosity
  7. R_ext (nm): external radius of the shell or of the core for the first line. This value should increase for every shell.
  8. The rest of the line is ignored.

The last line is for the medium around the nanoparticle. It is formatted as above except that the last value (radius) is not required.

Here are some possible entries for water at different temperatures. They are taken from the following article: M. J. Holmes, N. G. Parker & M. J. Povey, J. Phys.: Conf. Ser. 269 012011 (2011) ( arXiv ).

v 0.99981 1434.92 0 0.00143  0.00450 water at 07°C
v 0.99970 1447.29 0 0.00131  0.00403 water at 10°C
v 0.99897 1465.96 0 0.00114  0.00338 water at 15°C
v 0.99700 1496.73 0 0.000888 0.00247 water at 25°C
v 0.99222 1528.89 0 0.000653 0.00184 water at 40°C
v 0.98803 1542.57 0 0.000547 0.00148 water at 50°C

The number of steps for the root finding algorithm is proportional to the inverse of the step value. So increase this number if you are wondering if some roots were missed. Of course, it also slows down the calculation.

References

The implementation uses the ComplexNumber class.