Presents the Math Original article © by A. Daniel Klarmann, 2007
How much Electricity does it take to make build the colorful oxide layer?
To start with, I'll assume you've read my pages on anodizing titanium
and the basic physics of the color.
So now you want to know if your power supply is up to the task.
Basic Numbers Needed to Find the Answer | |
---|---|
Titanium dioxide numbers
TiO_{2} specific gravity = 4.2 g/cc TiO_{2} refractive index = 2.7 TiO_{2} mole-weight = 47.87 + (2 x 16) = 79.87g/mole |
Relevant Physical Constants
Avogadro's Number = 6.02 x 10^{23} Electrons per Coulomb = 6.24 x 10^{18} 1 Coulomb = 1 amp-second Approximate Wavelength of titanium colors - bronze = 3 x 10^{-5} cm - blue = 4 x 10^{-5} cm - yellow = 6 x 10^{-5} cm - purple = 7 x 10^{-5} cm - green (second order) = 10 to 12 x 10^{-5} cm |
Assumptions for this approximate number:
- Optically smooth metal surface
- Incident and reflected light both normal to surface
- Color produced purely by reflective/refractive reinforcement
- Oxide thickness is uniformly half of wavelength in oxide at refractive index
- Pure titanium and pure titanium dioxide, no inclusions
- Benign electrolyte, nothing to bond directly to titanium (like nitrogen, chlorine, etc)
- No metal migration to cathode
- No sharp corners or burs to bleed current
- One oxygen atom/ion (O^{--}) migrates per 2 electrons injected (100% efficient = each electron strips off one hydrogen)
Steps to calculate the volume of oxide under these ideal conditions and the amount of electricity needed:
- OxideVolume = area * (wavelength / 2) / refractiveIndex
The refractive index is needed because light is slower in solids than in a vacuum
Divide by 2 because the distance traveled inand out is twice the thickness therefore the wavelength is that much shorter at the same photon energy. - OxideMass = oxideVolume * specGrav
- OxideMolecules = oxideMass / moleWt * Avagadro
Units: g / (g/mole) = moles, * molecules/mole = molecules - Total amp-seconds = 4 * oxideMolecules / ElectronsPerCoulomb
Times 4 because have 2 oxygen atoms per molecule and 2 free electrons per oxygen atom.
Example:
Let's turn a 1 square centimeter flat piece of titanium yellow:
Ignoring edges, that gives us 2 cm^{2}
OxideVolume = area * (wavelength / 2) / refractiveIndex
= 2 * (6 x 10^{-5} / 2) / 2.7
= 2.22 x 10^{-5} = 0.0000222 cc
= 2 * (6 x 10^{-5} / 2) / 2.7
= 2.22 x 10^{-5} = 0.0000222 cc
OxideMass = oxideVolume * specGrav
= 2.22 x 10^{-5} cc * (4.2 g/cc)
= 9.32 x 10^{-5} = 0.0000932 g
= 2.22 x 10^{-5} cc * (4.2 g/cc)
= 9.32 x 10^{-5} = 0.0000932 g
OxideMolecules = oxideMass / moleWt * Avagadro
= 9.32 x 10^{-5} g) / (79.87 g/mole) * 6.02 x 10^{23}
= 7.0 x 10^{17} = 700,000,000,000,000,000 molecules
= 9.32 x 10^{-5} g) / (79.87 g/mole) * 6.02 x 10^{23}
= 7.0 x 10^{17} = 700,000,000,000,000,000 molecules
Amp-seconds = 4 * oxideMolecules / ElectronsPerCoulomb
= 4 * (4.5 x 10^{21}) / (6.24 x 10^{18})
= .45 amp-seconds or 100 ma for 4.5 seconds
= 4 * (4.5 x 10^{21}) / (6.24 x 10^{18})
= .45 amp-seconds or 100 ma for 4.5 seconds
BUT in the real world, you probably need several times that much for reasons alluded to in the Assumptions section above.
Also, the current starts high, and drops exponentially fast as the oxide builds up.