Thermocapillary piezoelectric micro heat engine

Matrix context

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

D.01 × C.34 · Temperature gradient × Marangonifamily: MarangoniD.01 × C.10 · Temperature gradient × Piezoelectricfamily: Piezoelectric

Measured or reported performance

1. predicted output voltage1.2–1.3 mVmodel2013
   simulated closed 60 × 60 × 15,000 μm channel on a PZT substrate, the thickest membrane; 0.5 mV predicted for a 300 nm membrane — a model result, never a laboratory measurement
   refs [4]
2. predicted system efficiency≈ 1.6 %model2013
   the same simulated geometry — a model result, never a laboratory measurement
   refs [4]

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
• Electromechanical coupling factor · formula-bound · hard, applies to this route
  converted fraction per cycle ≤ k² = d²/(s·ε)

Steps

1. 01
   Temperature gradient —drives→ Marangoni effectestablishedclaim record

   thermal → kinetic · loss to thermal

   • thermocapillary case: surface tension falls with temperature, so a temperature gradient along the interface drives flow toward the cold side

   liquid · temperature gradient required

   refs [1] [2]
2. 02
   Marangoni effect —produces→ Pressure gradienttheoretically predictedclaim record

   kinetic → mechanical · loss to thermal

   • a thermocapillary-driven liquid plug moves in a closed-ended microchannel and compresses the trapped gas; the pressure rise is the load presented to the compliant membrane

   Kept below demonstrated: the Odukoya device measured heater and droplet behaviour, while the pressure field feeding the membrane (about 0.1 kPa for a 13 μm deflection in the model) comes from the force balance, not a measurement.

   refs [3] [4]
3. 03
   Pressure gradient —drives→ Elastic deformationestablishedclaim record

   mechanical → mechanical

   • pressure acting on a diaphragm or shell

   solid

   refs [5]
4. 04
   Elastic deformation —produces→ Elastic stressestablishedclaim record

   mechanical → mechanical

   refs [5]
5. 05
   Elastic stress —drives→ Direct piezoelectric effectestablishedclaim record

   mechanical → electrical · loss to thermal

   • stress transmitted into a piezoelectric element

   solid · non-centrosymmetric crystal required

   refs [6]
6. 06
   Direct piezoelectric effect —produces→ Separated surface chargeestablishedclaim record

   electrical → electrical

   refs [5]
7. 07
   Separated surface charge —converts into→ Electrical workestablishedclaim record

   electrical → electrical · loss to thermal

   • electrodes on the charged surfaces connected through a load

   refs [7]

Evidence for the complete composition · 3

1. [3]
   P. S. Glockner, G. F. Naterer. Surface tension and frictional resistance of thermocapillary pumping in a closed microchannel (2006)

   International Journal of Heat and Mass Transfer 49(23–24), 4424–4436 · doi:10.1016/j.ijheatmasstransfer.2006.05.005 · record
2. [4]
   A. Odukoya, G. F. Naterer. Transient response of thermocapillary pumping of a droplet in a micro heat engine (2013)

   International Journal of Heat and Mass Transfer 58, 202–212 · doi:10.1016/j.ijheatmasstransfer.2012.11.009 · record
3. [8]
   P. S. Glockner, G. F. Naterer. Recent advances in nano-electromechanical and microfluidic power generation (2007)

   International Journal of Energy Research 31, 603–618 · doi:10.1002/er.1282 · record

Evidence for the constituent relations · 7

1. [1]
   L. E. Scriven, C. V. Sternling. The Marangoni Effects (1960)

   Nature 187, 186-188 · doi:10.1038/187186a0 · crossref ✓ 2026-09-20 · record
2. [2]
   Q. Dai et al.. Directional interfacial motion of liquids: Fundamentals, evaluations, and manipulation strategies (2021)

   Tribology International 154, 106749 · doi:10.1016/j.triboint.2020.106749 · crossref ✓ 2026-09-20 · record
3. [3]
   P. S. Glockner, G. F. Naterer. Surface tension and frictional resistance of thermocapillary pumping in a closed microchannel (2006)

   International Journal of Heat and Mass Transfer 49(23–24), 4424–4436 · doi:10.1016/j.ijheatmasstransfer.2006.05.005 · record
4. [4]
   A. Odukoya, G. F. Naterer. Transient response of thermocapillary pumping of a droplet in a micro heat engine (2013)

   International Journal of Heat and Mass Transfer 58, 202–212 · doi:10.1016/j.ijheatmasstransfer.2012.11.009 · record
5. [5]
   J. F. Nye. Physical Properties of Crystals: Their Representation by Tensors and Matrices (1985)

   Oxford University Press · book, no DOI · record
6. [6]
   A. Erturk, D. J. Inman. Piezoelectric Energy Harvesting (2011)

   Wiley · doi:10.1002/9781119991151 · crossref ✓ 2026-09-19 · record
7. [7]
   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