# Thermal radiation, heat transfer, and non-equilibrium Casimir forces between silicon dioxide spheres

In this example, we use scuff-neq to compute **(1)** the
power radiated by a single SiO2 sphere, and **(2)** the heat
transfer and non-equilibrium Casimir force between two SiO2
spheres. We compare the results of scuff-neq to
the predictions of the
T-matrix formalism of Krueger et al.

The files for this example may be found in the
`share/scuff-em/examples/SiO2Spheres`

subdirectory
of your scuff-em installation.

## gmsh geometry file and surface mesh for a single sphere

The gmsh geometry file `Sphere.geo`

describes a sphere of radius 1 micron; it may
be meshed to generate coarse and fine surface meshes as follows:

% gmsh -2 -clscale 1 Sphere.geo % RenameMesh Sphere.msh % gmsh -2 -clscale 0.5 Sphere.geo % RenameMesh Sphere.msh

`RenameMesh`

is a simple `bash`

script
that uses scuff-analyze to count the number of interior
edges in a surface mesh and rename the mesh file accordingly.)
This produces the files `Sphere_501.msh`

and `Sphere_1479.msh,`

which you can visualize by opening in gmsh::
% gmsh Sphere_501.msh

% gmsh Sphere_1479.msh

## scuff-em geometry files

The scuff-em geometry file
`SiO2Sphere_501.scuffgeo`

describes a single SiO2 sphere.

The scuff-em geometry files
`SiO2Spheres_501.scuffgeo`

`SiO2Spheres_1479.scuffgeo`

each describe the same configuration: two SiO2 spheres
separated by a center--center distance of 10 microns.
You can visualize this configuration by typing e.g.

% scuff-analyze --geometry SiO2Spheres_1479.scuffgeo --WriteGMSHFiles % gmsh SiO2Spheres_1479.pp

## Spectral density of radiated power

As described in the
scuff-neq documentation,
scuff-neq computes temperature-independent quantities
known as *generalized fluxes,* commonly denoted ,
which describe the ability
of systems to exchange energy and momentum via radiation
at specific frequencies
The *total* power radiated by a finite-temperature object is
obtained as an integral over angular frequencies
in which the integrand involves a
temperature-dependent Bose-Einstein factor
and a temperature-independent dimensionless flux
The distribution of labor in the scuff-em
suite is that scuff-neq computes
frequency-dependent generalized flux data and writes these
in the form of data files, while a separate tool named
scuff-integrate
post-processes the flux data to yield temperature-dependent
heat-transfer rates, powers, and torques.

To use scuff-neq to calculate radiated-power flux at a given set of frequencies, we say simply

% scuff-neq --geometry SiO2Sphere_501.scuffgeo --OmegaFile MyOmegaFile

where `MyOmegaFile`

is a list of
angular frequencies.

This produces the file
`SiO2Sphere_501.SiFlux`

, which looks something
like this:

# scuff-neq run on superhr2 (07/11/15::00:31:36) # data file columns: # 1 transform tag # 2 omega # 3 (sourceObject,destObject) # 4 PRad flux spectral density DEFAULT 1.000000e-01 11 4.18911788e-06 DEFAULT 1.300000e-01 11 1.38869207e-05 DEFAULT 1.600000e-01 11 3.93335327e-05 DEFAULT 1.900000e-01 11 1.05263974e-04

As the file header says, the second column here
is the angular frequency
in units of rad/sec
and the fourth column is the dimensionless power
flux. (The first column lists the
geometrical transformation; since
we didn't specify the `--transfile`

option to
scuff-neq, we have just a single geometric
configuration, labeled `DEFAULT`

. The third
column identifies the source and destination objects;
since this geometry only has a single object,
the source and destination object are both
always object 1 and this column always reads
`11`

.)

Here's a plot of the data:

In this plot, the solid line is the prediction of
the Krueger formalism, as computed
by a julia code called `KruegerFormulas.jl`

.

The plot is produced by gnuplot using this script.

## Spectral density of power transfer and non-equilibrium force

Here's a bash script that runs scuff-neq
for both the coarsely-meshed and finely-meshed two-sphere
geometry to compute the fluxes of power transfer
and nonequilibrium force between the spheres.
Running the script produces files `SiO2Spheres_501.SIFlux`

and `SiO2Spheres_1479.SIFlux.`

Here are plots (produced
by the same gnuplot script referenced above)
of the heat-transfer flux from sphere 1 to sphere 2,
and the force fluxes from sphere 1 to sphere 2 and
from sphere 2 to sphere 2, compared to the Krueger
T-matrix results (again computed using the julia
code referenced above).