CPO3D 65th example: enhancement factor for a 'spoilt' array of nanotubes all having the form of 'hemisphere on post'.
The 'posts' have a height h = 0.5 micron (ie 0.0005mm) and a radius a = 1nm.
They are placed here at a distance s = 160nm apart to create a field an anode is placed at an arbitrary 4 micron away, with a potential that gives a 'far field' of 1V/mm.
3 symmetry planes are used.
There are 25 tubes (an array 5 times 5). The central tube has more segments than the other tubes.
A 'boundary plate' surrounds the top ends of the tubes, in the z = 0 plane. The potential of this plate is V3 = 0.000123, which is aproximately the mean potential in the plane z = 0 in the region of the tubes. This potential is added to the anode potential to give the required far field. The value of V3 is determined iteratively to ensure that the potential above the central tube at (x,y,z) = (0,0,.0003) is approximately equal to that above the boundary plate at (0,0.001,0.0003).
The system is enclosed in a square shield of side 0.0016.
Results:
x y z ex ey ez etotal (V/mm)
0.00E+00 0.00E+00 5.00E-08 0.00000E+00 -4.54590E-03 -9.55504E+01 9.55504E+01
0.00E+00 0.00E+00 1.00E-07 0.00000E+00 -3.27893E-03 -8.69116E+01 8.69116E+01
0.00E+00 0.00E+00 1.50E-07 0.00000E+00 -2.74110E-03 -7.95675E+01 7.95675E+01
0.00E+00 0.00E+00 2.00E-07 0.00000E+00 -2.35123E-03 -7.31693E+01 7.31693E+01
0.00E+00 0.00E+00 2.50E-07 0.00000E+00 -2.03977E-03 -6.75677E+01 6.75677E+01
0.00E+00 0.00E+00 3.00E-07 0.00000E+00 -1.80338E-03 -6.26371E+01 6.26371E+01
0.00E+00 0.00E+00 3.50E-07 0.00000E+00 -1.61292E-03 -5.82723E+01 5.82723E+01
0.00E+00 0.00E+00 4.00E-07 0.00000E+00 -1.43547E-03 -5.43875E+01 5.43875E+01
0.00E+00 0.00E+00 4.50E-07 0.00000E+00 -1.30136E-03 -5.09133E+01 5.09133E+01
0.00E+00 0.00E+00 5.00E-07 0.00000E+00 -1.18581E-03 -4.77923E+01 4.77923E+01
Analysis of these results:
Extrapolate field to z=0 by extrapolation of E(z)*(z+a)**2. But this extrapolation is not smooth at small z. So take the lowest value of E(z)*(z+a)**2 (which is at z=1E-7) and divide it by a**2, which gives E(a) = 10516. Therefore enhancement factor, gamma = 105.16
This can be compared with the value 99.50 found in xmpl3d65.dat, with more tubes and more segments (because 3 reflection planes were used).
The boundary plate has V3 = 1.31E-4, to make potential approximately independent of y at z = 0.0005 (the value was 1.23E-4 in xmpl3d65).
Please see the footnotes to the data file for a considerable amount of further information.
Please also see papers 55 and 59 in the publications list.