|MadSci Network: Physics|
This is, unfortunately, a difficult question. Thermal conductivity in solids is dictated by how thermal energy is stored at the atomic level. You didn't indicate a grade level with your question, but your insight regarding phonons and electrical conductivity tempts me to tailor this response for an advanced undergraduate or graduate student. I will instead, however, try to simply summarize my answer and point you to additional references.
You specifically stated copper in your question, so I'll address that metal, but the following explanation can be generalized to other conductors.
Below around 20 K, thermal conductivity is limited by surface scattering. Therefore the temperature dependences comes entirely from the T**3 dependence of the specific heat. As the temperature rises, a processes known as Umklapp scattering becomes more prevalent. Thermal conductivity becomes reaches a maximum when the mean free path due to phonon-phonon scattering is roughly equal to the surface scattering. Where this maximum occurs is dependent on the dimensions of the sample. As temperature rises even more, the thermal conductivity falls because the phonon-phonon scattering rate rapidly increases while the phonon specific heat levels off.
I hope this has helped
James R Holliday
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