Electric potential difference → Electrical work

Matrix context

D = disequilibrium row · C = coupling family column. A cell reads the state of the atlas for that pair.

D.06 × C.01 · Electric potential difference × Thermoelectricfamily: ThermoelectricD.06 × C.40 · Electric potential difference × Thermopolarizationfamily: Thermopolarization

Typed bounds on this route

Every limit the atlas holds for this route, as constraint records: reached through bounded_by claims on the route's own entities. Hard bounds (an upper bound or a formula) decide the thermodynamic-bound check above; benchmarks, constitutive relations and resource limits are listed and never decisive. No prose limit exists anywhere (pass 42).

• Carnot limit · formula-bound · hard, applies to this route
  η ≤ 1 − T_c/T_h for any heat engine between reservoirs at T_h and T_c

Steps

1. 01
   Electric potential difference —drives→ Ettingshausen effectestablishedclaim record

   electrical → thermal

   • a current in a conductor with a perpendicular magnetic field

   solid · conductor required · magnetic field required

   refs [1] [2]
2. 02
   Ettingshausen effect —produces→ Temperature gradientestablishedclaim record

   thermal → thermal

   refs [2] [3]
3. 03
   Temperature gradient —drives→ Thermopolarization effectreportedclaim record

   thermal → electrical

   P_i = γ_ij ∇_j T · thermopolarization coefficient γ [C/(m K)]

   phenomenological tensor form; in Onishi et al.'s linear-response notation γ corresponds to −L̃₁₂/T for the relevant component; Kholkin 1982 report B₁₁ ≈ 3 × 10⁻¹² C/(cm·K) (≈ 3 × 10⁻¹⁰ C/(m·K)) for PMN at 310 K

   • a spatial temperature gradient is applied across an insulating solid dielectric
   • this claim represents the direct thermopolarization contribution; a signal established to arise through thermal expansion → strain gradient → flexoelectricity belongs to that longer pathway instead
   • the solid spelling; the polar-liquid spelling is claim:temperature-drives-liquid-thermopolarization, with its own (simulation) evidence

   refs [4] [5] [6] [7] [8]
4. 04
   Thermopolarization effect —produces→ Separated surface chargereportedclaim record

   electrical → electrical

   • the induced bulk polarization terminates at a surface or electrode as bound / screening charge; modulating the gradient produces a displacement / polarization current
   • the solid, electroded dielectric of the 1982–1994 measurements; the polar-liquid spelling has no charge-collection evidence (the simulations report bulk polarization and electrostatic potential) and does not inherit this sink

   refs [8] [6] [9]
5. 05
   Separated surface charge —converts into→ Electrical workestablishedclaim record

   electrical → electrical · loss to thermal

   • electrodes on the charged surfaces connected through a load

   refs [10]

Evidence for the complete composition · 0

No source on record for this exact composition. The sources below support the individual steps, not the route as a whole.

Evidence for the constituent relations · 10

1. [1]
   A. von Ettingshausen, W. Nernst. Ueber das Auftreten electromotorischer Kräfte in Metallplatten, welche von einem Wärmestrome durchflossen werden und sich im magnetischen Felde befinden (1886)

   Annalen der Physik 265, 343–347 · doi:10.1002/andp.18862651010 · crossref ✓ 2026-09-19 · record
2. [2]
   H. B. Callen. The application of Onsager's reciprocal relations to thermoelectric, thermomagnetic, and galvanomagnetic effects (1948)

   Physical Review 73, 1349 · doi:10.1103/PhysRev.73.1349 · crossref ✓ 2026-09-19 · record
3. [3]
   H. J. Goldsmid. Introduction to Thermoelectricity (2nd ed.) (2016)

   Springer Series in Materials Science 121 · doi:10.1007/978-3-662-49256-7 · crossref ✓ 2026-09-19 · record
4. [4]
   M. Marvan. The electric polarization induced by temperature gradient and associated thermoelectric effects (1969)

   Czechoslovak Journal of Physics B 19, 1240–1245 · doi:10.1007/bf01690309 · record
5. [5]
   A. K. Tagantsev. Pyroelectric, piezoelectric, flexoelectric, and thermal polarization effects in ionic crystals (1987)

   Soviet Physics Uspekhi 30, 588–603 · doi:10.1070/PU1987v030n07ABEH002926 · record
6. [6]
   V. A. Trepakov, K. M. Nurieva, A. K. Tagantsev. Recent developments of the thermopolarization effect investigation (1989)

   Ferroelectrics 94, 377–381 · doi:10.1080/00150198908014283 · record
7. [7]
   Y. Onishi et al.. Theory of Thermopolarization Effect (2025)

   Nano Letters 25, 2763–2768 · doi:10.1021/acs.nanolett.4c05870 · record
8. [8]
   A. L. Kholkin, V. A. Trepakov, G. A. Smolenskii. Thermopolarization currents in dielectrics (1982)

   JETP Letters 35 (3), 124–127 (Pis'ma Zh. Eksp. Teor. Fiz. 35, 103–106) · www.jetpletters.ru/ps/1313/article_19837.shtml · record
9. [9]
   E. Rafikov et al.. Frequency and temperature dependence of the thermopolarization response in dynamic investigations (1994)

   physica status solidi (a) 144, 471–477 · doi:10.1002/pssa.2211440228 · record
10. [10]
    H. B. Radousky, H. Liang. Energy harvesting: an integrated view of materials, devices and applications (2012)

    Nanotechnology 23, 502001 · doi:10.1088/0957-4484/23/50/502001 · crossref ✓ 2026-09-19 · record