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

In this example, we use buff-neq to reproduce the results of [this example from the scuff-neq documentation][scuffSIO2Spheres]. We will compute (1) the power radiated by a single SiO2 sphere, and (2) the heat transfer and non-equilibrium Casimir force between two SiO2 spheres.

The files for this example may be found in the share/buff-em/examples/SiO2Spheres subdirectory of your buff-em installation.

---

gmsh geometry file and volume mesh for a single sphere

The gmsh geometry file Sphere.geo describes a sphere of radius 1 micron. (This is the same file used in this example; as noted there, it is not the same .geo file that was used in the [scuff-neq version of this calculation][scuffSIO2Spheres], because for buff-em we need volume meshes instead of surface meshes. This file may be meshed to create coarse and fine volume meshes as follows:

% gmsh -3 -clscale 1 Sphere.geo
% RenameMesh3D Sphere.msh
% gmsh -3 -clscale 0.75 Sphere.geo
% RenameMesh3D Sphere.msh

(Here RenameMesh3D is a simple bash script that uses buff-analyze. to count the number of interior faces in a volume mesh and rename the mesh file accordingly; note that it also changes the file extension from .msh to .vmsh, which I find convenient for distinguishing volume mesh files from surface mesh files.

This produces the files Sphere_637.msh and Sphere_1727.msh, which you can visualize by opening in gmsh::

% gmsh Sphere_637.msh

% gmsh Sphere_1727.msh

Note: As you can see from the first image here, by default gmsh

---

buff-em geometry files

The buff-em geometry file SiO2Sphere_637.buffgeo describes a single SiO2 sphere.

The buff-em geometry files SiO2Spheres_637.buffgeo and SiO2Spheres_1727.buffgeo 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.

% buff-analyze --geometry SiO2Spheres_1727.buffgeo --WriteGMSHFiles
% gmsh SiO2Spheres_1727.pp

---

Spectral density of radiated power

As described in the buff-neq documentation, buff-neq computes the total power radiated by finite-temperature objects as an integral over angular frequencies $\omega,$ in which the integrand involves a temperature-dependent Bose-Einstein factor and a temperature-independent dimensionless flux $\Phi.$ To calculate this radiated-power flux at a given set of frequencies, we say

 % buff-neq --geometry SiO2Sphere_637.buffgeo --OmegaFile --PRad

where OmegaFile is a list of angular frequencies. (Here --PRad says that we are interested in the radiated power). This produces the file SiO2Sphere_637.SiFlux, which looks something like this:

# buff-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 $\omega_0=3\cdot 10^{14}$ 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 buff-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 buff-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_637.SIFlux and SiO2Spheres_1727.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).

---