Static electrification of the same material at different temperatures In the following books it is stated that when you have 2 samples of the same material at different temperatures and they are rubbed together, the colder one gets electrified positively, the hotter one negatively.
Electricité. Magnétisme. Courants électriques. Radiations (1840) by Gabriel Lamé
Electricity (1882) by Robert M. Ferguson & James Blyth
A course of lectures on electricity, delivered before the Society of Arts (1888) by George Forbes
Now, in principle I would expect the opposite to happen because of thermionic emission. The hotter one would lose more electrons and thus be electrified positively, yet the opposite happens.
Maybe the reduction of conductivity in metals at higher temperatures would account for this phenomenon? But that would not explain why this phenomenon also happens in non-metals.
I am quite puzzled, any help in the matter would be greatly appreciated.
Thanks
 A: The 19th century claim that the cold sample is always going to be electrified positively and the hot sample negatively is almost certainly wrong. It is also incorrect to assume that triboelectric charge is always transferred by electrons.
My guess is that the observations you cite were affected by water absorbed on their surfaces or were mostly insulators with positive charge carriers.
Before focusing on tribo-electrification, I think it is simpler to first just consider the thermoelectric effect where a Seebeck emf is generated by a temperature gradient across a single sample.
If a simple free carrier model can't explain this, then it is unlikely to work better if we split the sample into two pieces and start rubbing them against each other as in your scenario.
In a simple free electron/ion model we expect the electron/ions at the hot end to have higher energies and diffuse more rapidly towards the cold end. The number of charge carriers at the cold end will build up until a balancing electric field is established, so the cold end will have the same sign electric as the charge carriers, while the hot end will have the opposite charge.
This thermoelectric effect is often parameterized by the Seebeck coefficient
$$S=-\frac{\Delta V}{\Delta T}$$
In a simple free carrier model, the sign of $S$ corresponds to the sign of the dominant charge carriers.
For example, it is observed that $S<0$ in n-type semiconductors, and $S>0$ in p-type.
Unfortunately, this simple model fails when applied to metals, where the best conductors (Cu, Ag, Au) all have positive Seebeck coefficients.
Good theoretical first-principles explanations of this need to consider electron-phonon
scattering, electron band structures, densities of states, Fermi
surfaces, phonon dispersion curves, …. It is only the last few years that the signs of the Seebeck coefficients in lithium and other simple alkali (Li, Na, K) and noble metals (Cu, Ag, Au, and Pt) have been reasonably modelled.
Coming back to frictional electrification, it is hard to overemphasize how difficult it is to make controlled, reproducible, and well documented experiments.
As Castle noted in 1997,

*

*"reproducible experiments remain a challenge and a generally agreed upon theory of insulator-insulator charging remains elusive. … Many contradictory data for insulator charging exist in the literature."

*"impurities and oxides, surface roughness, geometry, type of pretreatment etc. … ensure that supposedly identical materials … display completely different microscopic surface properties."

*"prior sample history, contact pressure, temperature, etc., often differ between experimenters and even between … experiments in the same studies."

*"variations due to friction forces, mass transfer from one surface to another, local temperature effects, etc., may make tremendous differences …"

*"charge may flow back across the interface … due to either electron tunnelling or air breakdown."

*"For many years, it was believed that electrons were the mechanism for charge transfer.
However, evidence now exists that in some cases ions can act as the charge carrier and … material mass transfer may also contribute. In some cases, all three of these mechanisms may be present!"

Despite these difficulties, there has been a huge amount of research on tribo-electrification because it of ubiquitous importance in fields as diverse as meteorology, textiles and clothing, any industry (e.g. cement, grain) handling granular material, fire and explosion safety, electronics and many other areas of engineering, transportation, ….
Nominally identical insulators can electrify each other as long as there is some asymmetry, such as a large sample rubbing a smaller one, large grains impacting on small grains, rough surface against smooth, concave against convex. Colder against hotter also provides an asymmetry, but direct evidence on such electrification for identical samples is sparse :

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*The 19th century sources claim "cold-positive, hot-negative" is a general and apparently well-known rule, but the only specific example seems to be cork-on-cork (mentioned by Forbes).


*Shaw and Jex (1926) observed that a cold glass rod would charge positively when rubbed against a warm glass rod, but this was explained by water from the air being adsorbed on cooler glass surfaces, but driven off by heating.


*Lantham and Mason (1961) reported that when pieces of ice of different temperatures were brought into contact and separated, the warmer acquires a negative charge and the colder an equal positive charge, which was what was expected because of the much higher mobility of positive H$^+$ ions compared to negative OH$^-$ ions.


*Bowles (1961) reported evidence of contact electrification due to temperature differences for polyethylene spheres in air, and that the charge carriers involved seemed to have positive charge.


*Lowell (1986) studied identical polymer samples rubbing against each other in vacuum and concluded that the observed electrification "cannot be the result of a temperature difference" due to asymmetric frictional heating.
It would certainly be nice to have some similar "identical sample" measurements for metals which have (negative) electron charge carriers. I can't immediately find any, but
Lin (2019)  studied metal-semiconductor friction where electrons are also the dominant charge carrier. They observed "hot-positive, cold-negative", as expected for a simple electron thermionic-emission model, which confirms that "cold-positive, hot-negative" is not a universal rule.
Triboelectricity is easy to produce but fiendishly hard to study and understand, as indicated by the bottom right corner of this XKCD comic "Easy or Hard"

