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Permittivity of a metal-loaded fluid

How much does suspending metal particles in the water raise its dielectric constant?

The formula
ε=e⁢1+2⁢φ1−φ
ε
Effective permittivity
e
Host water permittivity at T
φ
Metal volume fraction, %
LaTeX 
ε = e \cdot \frac{1 + 2 \cdot φ}{1 - φ}

Work it out

Fill from a part the library

The host fluid. Sets the permittivity at 20 °C.

°C

The water's temperature. Its permittivity falls about 0.4 % per degree of warming.

%

The metal's share of the volume.

Compare with a variation

Result

ε Effective permittivity 89.76

The mixture's relative permittivity — multiply a cell's vacuum capacitance by this instead of by the water's.

k Enhancement 1.158

The mixture's permittivity over the water's. Capacitance rises by this factor; resonant frequency falls by its square root.

With your numbers
89.76=77.52⁢1+2⁢5 %1−5 %
LaTeX 
89.76 = 77.52 \cdot \frac{1 + 2 \cdot 5\,\mathrm{%}}{1 - 5\,\mathrm{%}}

This result is a link — the address bar holds your numbers, so it can be pasted into a post and opened to the same answer.

What this looks like

Effective permittivity against metal loading Metal loading swept from 2.5 % to 7.5 % with everything else held at your numbers. The dashed lines cross where you are.
Effective permittivity against metal loadingEffective permittivity rises from 83.5 to 96.4 as metal loading rises from 2.5 % to 7.5 %. At your metal loading of 5 % it is 89.8.8085909510024685 %89.8Metal loading (%)Effective permittivity
The formula behind the curve
ε=e⁢1+2⁢φ1−φ
ε
Effective permittivity
e
Host water permittivity at T
φ
Metal volume fraction, %
LaTeX 
ε = e \cdot \frac{1 + 2 \cdot φ}{1 - φ}
What moves the answer Each input moved 10% either way, with the others held still, and the effect on effective permittivity.
What moves the answerEffective permittivity is most sensitive to Metal loading, which moves it by about 1.44% for a 10% change. It is least sensitive to Temperature, at about 0.588%.Change in the answer when each input moves by 10%-2%-1%0%1%2%Metal loading±1.44Temperature±0.588
The formula behind the curve
ε=e⁢1+2⁢φ1−φ
ε
Effective permittivity
e
Host water permittivity at T
φ
Metal volume fraction, %
LaTeX 
ε = e \cdot \frac{1 + 2 \cdot φ}{1 - φ}

Method

  1. Take the water's permittivity at 20 °C for the chosen profile and adjust it for temperature, exactly as the water permittivity calculation does.
  2. Convert the loading to a fraction by dividing by 100.
  3. The Maxwell Garnett rule relates the mixture's permittivity to the host's and the inclusions' through their volume fraction. For inclusions that are conductors the inclusion permittivity is effectively infinite, and the rule collapses to the host's permittivity times (1 + 2φ) ÷ (1 − φ).
  4. The enhancement factor is that fraction on its own. A cell's capacitance rises by it; its resonant frequency falls by its square root.

Assumptions

  • Spherical, well-separated, conducting inclusions in a dielectric host — the dilute Maxwell Garnett picture. It is good to a few per cent loading and increasingly poor above about 20 %; near 30 % by volume randomly placed spheres begin to touch, the mixture percolates, and it becomes a conductor rather than a better dielectric. This formula does not know that and will keep returning a permittivity.
  • Static, or low-frequency, permittivity. Metal particles in a conducting host also make the mixture lossy — the interfacial (Maxwell–Wagner) polarisation — which is a large imaginary part this real-valued figure leaves out. The Cole-Cole calculation covers the water's own loss; the particles add to it.
  • The particles are not magnetised and not chained. Under a field a ferromagnetic particulate lines up into strings along it, and a mixture of aligned chains is anisotropic — a different permittivity along the field than across it.
  • The host is water at the chosen profile. Dissolved solids change its conductivity by orders of magnitude and its permittivity by a few per cent; the profile picks both.