Quantum Theory of Triboelectricity

Quantum Theory of
Triboelectricity
Alejandro Jenkins (ICTQT, Gdańsk & U. de Costa Rica)

work with Robert Alicki (ICTQT)

PRL 125, 186101 (2020)

Dissipation Mechanisms in Nano/Mesoscale Tribological Systems (smr 3713)

ICTP, Trieste

2 June 2022

Minimal tribo-generator
2
E =
I
C
E · ds =
d
dt
Z
B · da = 0
<latexit sha1_base64="1nn7cNlgf90I8a8Mqv1LDymWovc=">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</latexit>
Faraday-Maxwell:
Irreversibility: -
𝛺
does not undo +
𝛺

emf
• Introduced by Volta in 1801 for “nonelectrostatic
action on charges in conductors that causes
unlike charges to separate and to remain
separated”; see Varney & Fisher, Am. J. Phys.
48, 405 (1980)

• Active, non-conservative force; cannot be
obtained from any potential function

• analogous to hydraulic pumping

• Builds up a potential difference, but is not the
action of any potential

• Fundamentally out of equilibrium
• cf. Deryagin, Krotova & Smilga, Adhesion of
Solids (1978); Collins, Camara, Van Cleve &
Putterman, Rev. Sci. Instrum. 89, 013901 (2018)
Alessandro Volta

(1745–1827)

“What Zel’dovich knew”
Angular frequency of wave:
𝜔
= v k

Momentum conservation:

Kinetic energy of wind (non-relativistic):
4
Air
Water
x
<latexit sha1_base64="BBA+dhdA6yPRwzXfybWhtAcRQy4=">AAACEXicbVDLSgMxFM3UV62vUZdugkXoqsyUom6EogguK9gHtLVkMpk2NMkMSUYpw/yCG3/FjQtF3Lpz59+YtrOorQcuHM65l3vv8SJGlXacHyu3srq2vpHfLGxt7+zu2fsHTRXGEpMGDlko2x5ShFFBGppqRtqRJIh7jLS80dXEbz0QqWgo7vQ4Ij2OBoIGFCNtpL5duu4nXcnhIxV+Ci9gN5AIJ9GceF9JkwpP+3bRKTtTwGXiZqQIMtT79nfXD3HMidCYIaU6rhPpXoKkppiRtNCNFYkQHqEB6RgqECeql0w/SuGJUXwYhNKU0HCqzk8kiCs15p7p5EgP1aI3Ef/zOrEOznsJFVGsicCzRUHMoA7hJB7oU0mwZmNDEJbU3ArxEJlMtAmxYEJwF19eJs1K2T0tV2+rxdplFkceHIFjUAIuOAM1cAPqoAEweAIv4A28W8/Wq/Vhfc5ac1Y2cwj+wPr6BUm0nVU=</latexit>
Ewind =
p2
wind
2m
<latexit sha1_base64="VbX9WQCzylRhNEttpoH5c+rIhRw=">AAACPnicbVDLSgMxFM3UV62vqks3wSK4KjNS1I1QFMFlBfuAzlAyaaYNTTJDklHKMF/mxm9w59KNC0XcujTTzqK2vRA4nHPPzb3HjxhV2rbfrMLK6tr6RnGztLW9s7tX3j9oqTCWmDRxyELZ8ZEijArS1FQz0okkQdxnpO2PbjK9/UikoqF40OOIeBwNBA0oRtpQvXLT7Yca3vYSV3L4REU/hVfQDSTCSeJypId+AKN0Rk4TnkIXZ67MurypV67YVXtScBE4OaiAvBq98qsZhmNOhMYMKdV17Eh7CZKaYkbSkhsrEiE8QgPSNVAgTpSXTM5P4Ylh+jAIpXlCwwk760gQV2rMfdOZLavmtYxcpnVjHVx6CRVRrInA04+CmEEdwixL2KeSYM3GBiAsqdkV4iEy2WmTeMmE4MyfvAhaZ1XnvFq7r1Xq13kcRXAEjsEpcMAFqIM70ABNgMEzeAef4Mt6sT6sb+tn2lqwcs8h+FfW7x+YGbCh</latexit>
Ėwind =
pwind
m
· ṗwind
<latexit sha1_base64="hG+aN986YUnwLJQkxw47p8+mqhI=">AAACGXicbVBNS8MwGE7n15xfVY9egkPw4mhlqBdh6MXjBPcBaylplm5haVqSdDJK/4YX/4oXD4p41JP/xnQropsPJDw8z/sm7/v4MaNSWdaXUVpaXlldK69XNja3tnfM3b22jBKBSQtHLBJdH0nCKCctRRUj3VgQFPqMdPzRde53xkRIGvE7NYmJG6IBpwHFSGnJM61LeAJTJ0Rq6AewnUEH9yMFHX39qHHmpY4I4T0ak8wzq1bNmgIuErsgVVCg6Zkf+jGchIQrzJCUPduKlZsioShmJKs4iSQxwiM0ID1NOQqJdNPpZhk80kofBpHQhys4VX93pCiUchL6ujIfVs57ufif10tUcOGmlMeJIhzPPgoSBlUE85hgnwqCFZtogrCgelaIh0ggrHSYFR2CPb/yImmf1uyzWv22Xm1cFXGUwQE4BMfABuegAW5AE7QABg/gCbyAV+PReDbejPdZackoevbBHxif35g4oB4=</latexit>
= V · ṗwave
<latexit sha1_base64="6rY+PZa6GcT5uCfqozFe6q7pcs4=">AAACFHicbVDLSsNAFJ3UV62vqEs3g0WoiCWRom6EohuXFewDmlAmk0k7dDIJMxOhhHyEG3/FjQtF3Lpw5984aQNq9cCFwzn3cu89XsyoVJb1aZQWFpeWV8qrlbX1jc0tc3unI6NEYNLGEYtEz0OSMMpJW1HFSC8WBIUeI11vfJX73TsiJI34rZrExA3RkNOAYqS0NDCPLuAxTJ0QqZEXwE4GHexHCtYC6Iw8JL6tcXY4MKtW3ZoC/iV2QaqgQGtgfjh+hJOQcIUZkrJvW7FyUyQUxYxkFSeRJEZ4jIakrylHIZFuOn0qgwda8WEQCV1cwan6cyJFoZST0NOd+Yly3svF/7x+ooJzN6U8ThTheLYoSBhUEcwTgj4VBCs20QRhQfWtEI+QQFjpHCs6BHv+5b+kc1K3T+uNm0a1eVnEUQZ7YB/UgA3OQBNcgxZoAwzuwSN4Bi/Gg/FkvBpvs9aSUczsgl8w3r8AxfOczw==</latexit>
= V · (f~k)
<latexit sha1_base64="hYpnbDo253ardQyiKFIsFbaK7Gw=">AAACHnicbVBNSyNBEO1x/cy6Gtejl8Ig7CnMLH6BCGFF8KhgVMiEUNOpSZp0zwzdPZEwxD+yF/+KFw+KCJ7039iJOfj1oODxXhVV9aJMCmN9/8Wb+jE9Mzs3v1D6ufhrabm88vvMpLnmVOepTPVFhIakSKhuhZV0kWlCFUk6j3oHI/+8T9qINDm1g4yaCjuJiAVH66RWeStspxYOW0WoFVxin4awDzGE3Qg1hKmiDr4T+tCDqyvYa5UrftUfA76SYEIqbILjVvnJ7eG5osRyicY0Aj+zzQK1FVzSsBTmhjLkPexQw9EEFZlmMX5vCBtOaUOcaleJhbH6fqJAZcxARa5Toe2az95I/M5r5DbebRYiyXJLCX9bFOcSbAqjrKAtNHErB44g18LdCryLGrl1iZZcCMHnl7+Ss7/VYLu6ebJZqf2bxDHP1tg6+8MCtsNq7Igdszrj7D+7YXfs3rv2br0H7/GtdcqbzKyyD/CeXwGa9KBI</latexit>
Ėwave = f~! = f~vk ;
<latexit sha1_base64="QaFkVxStgFlAwTwM0tbs3pU4Yx4=">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</latexit>
V > v , Ėwind > Ėwave
Zel’dovich, JETP Lett. 14, 180 (1971);

Sov. Phys. JETP 35, 1085 (1971)
*
* Thorne, 2013
<latexit sha1_base64="8FzfSd3rkCoBg0x8QtFikcqg5Lg=">AAACJHicdVDLSsNAFJ34rPUVdelmsAhuLIkUFUQounFZwT6gCWEynbRDJ5MwM6mUkI9x46+4ceEDF278FidtFtrqgYHDOffeuff4MaNSWdansbC4tLyyWlorr29sbm2bO7stGSUCkyaOWCQ6PpKEUU6aiipGOrEgKPQZafvD69xvj4iQNOJ3ahwTN0R9TgOKkdKSZ144vUjB1AmRGvgBjDMvdUQI7ynvZfASHsN/fDQimWdWrKo1AZwndkEqoEDDM9/0NJyEhCvMkJRd24qVmyKhKGYkKzuJJDHCQ9QnXU05Col008mRGTzUSg8GkdCPKzhRf3akKJRyHPq6Ml9Wznq5+JfXTVRw7qaUx4kiHE8/ChIGVQTzxGCPCoIVG2uCsKB6V4gHSCCsdK5lHYI9e/I8aZ1U7dNq7bZWqV8VcZTAPjgAR8AGZ6AObkADNAEGD+AJvIBX49F4Nt6Nj2npglH07IFfML6+AYz3pMc=</latexit>
ṗwind = ṗwave

Superradiance
5
Ėwind Ėwave > 0 available for dissipation in the air
• Similar to Landau criterion for critical velocity of super
fl
uids
• Also sonic booms, shock waves, Cherenkov radiation,
electromagnetic “non-contact friction”, etc.

See: R. Alicki & AJ, Ann. Phys. (NY) 395, 69 (2018) [arXiv:1702.06231]

• Used by Zel’dovich in 1971 to predict radiation from
rotating black hole (BH); motivated Hawking’s discovery of
thermal radiation from static BHs in 1974

• Our original result: No superradiance of fermions, but
motion can still induce disequilibrium that pumps current
between two baths

Open quantum system
6
system: surfaces a + b

2 baths: bulks A & B
Ha
⌦ =
X
,m
[!a( , m) m⌦] c†
a( , m)ca( , m)
<latexit sha1_base64="uZKCFXIOZoL29DeI7AQuChhtFJI=">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</latexit>
HA
⌦ =
X
,m
[!A(, m) m⌦] c†
A(, m)cA(, m)
<latexit sha1_base64="nvfZrInQD0+SVrwNRRseaNFl0T0=">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</latexit>
In motion:
Surface-bulk tunneling: Hx
X =
X
, ,m
gx
X(, , m)c†
X(, m)cx( , m) + h.c.
<latexit sha1_base64="CXDR5QPgo0Hsemns6AJBxoz1xvw=">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</latexit>
(Doppler shift)
See also: R. Alicki & AJ, Ann. Phys. (NY) 395, 69 (2018)
Hx
0 =
X
,m
!x( , m)c†
x( , m)cx( , m) for x = a, b
<latexit sha1_base64="kQ/BlU63nFn4l1uuEmR5vpeW+58=">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</latexit>
HX
0 =
X
,m
!X(, m)c†
X(, m)cX(, m) for X = A, B
<latexit sha1_base64="q6kNbSwaOwk6Df9i4586EKW4+pk=">AAACV3icbVFNa9tAEF0pX67z5aTHXpaaQgLBSEkguQSS9JJjCnFisBwxWo/lxfshdlelRjg/MvSSv9JLu3JMSZ0OLLx5b4aZeZsVglsXRS9BuLK6tr7R+NDc3Nre2W3t7d9bXRqGXaaFNr0MLAqusOu4E9grDILMBD5kk6+1/vAdjeVa3blpgQMJueIjzsB5Km2pmzR67NELmthSplUygaKAIypnNNESc0h79OAveUjZYzKEPEezzC/lT080GWf6RzXSZlZn9Yiro+u01Y460TzoexAvQJss4jZtPSdDzUqJyjEB1vbjqHCDCozjTOCsmZQWC2ATyLHvoQKJdlDNfZnRL54ZUr+Cf8rROfu2owJp7VRmvlKCG9tlrSb/p/VLNzofVFwVpUPFXgeNSkGdprXJdMgNMiemHgAz3O9K2RgMMOe/oulNiJdPfg/ujzvxSef422n78nphR4N8Ip/JAYnJGbkkN+SWdAkjP8mvYCVYDV6C3+F62HgtDYNFz0fyT4R7fwBqdLAJ</latexit>
At rest:
( = 1)

Kinetic eqs.
7
KMS conditions: aA
" ( , m) = e (!a( ,m) µA) aA
# ( , m)
<latexit sha1_base64="3Va4J5CrTMMhOxaKX5ARlyEyejo=">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</latexit>
aB
" ( , m) = e (!a( ,m) m⌦ µB) aB
# ( , m)
<latexit sha1_base64="B09SoTq2+Kp9xeoyuikfYhDoMxo=">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</latexit>
Steady state: na = n̄a ⌘ aA
" + aB
" / a
= const.
<latexit sha1_base64="N2qT/SrSRfKhk5J5zjgCon1iodE=">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</latexit>
b
⌘ bA
" + bA
# + bB
" + bB
#
<latexit sha1_base64="BR0wAlzOTcOS1AauksB8T0HVA1E=">AAACVnicdVFNSwMxFMxu1db6terRS7AIglB2RdBjrQc9VrAqdNvyNk3bYJJdk6xSlv5JvehP8SKm7eJX9UFgMjPvJZlECWfa+P6r4xYWFpeKpeXyyura+oa3uXWt41QR2iQxj9VtBJpyJmnTMMPpbaIoiIjTm+jubKLfPFClWSyvzCihbQEDyfqMgLFU1xPhOQgBnQiH9D5lDzgcTPbdME1Aqfixk0WnY3zwSffiR/mn8OWv/+evj7texa/608LzIMhBBeXV6HpPdgJJBZWGcNC6FfiJaWegDCOcjsthqmkC5A4GtGWhBEF1O5vGMsZ7lunhfqzskgZP2e8dGQitRyKyTgFmqH9rE/IvrZWa/kk7YzJJDZVkdlA/5djEeJIx7jFFieEjC4AoZu+KyRAUEGN/omxDCH4/eR5cH1YDiy+PKrV6HkcJ7aBdtI8CdIxq6AI1UBMR9IzeHNcpOC/Ou7voFmdW18l7ttGPcr0P9pq1LA==</latexit>
a
⌘ aA
" + aA
# + aB
" + aB
#
<latexit sha1_base64="g8rqO/jrJRe3RIMGcNKVmAye1LU=">AAACVnicdVFNSwMxFMxu1db6terRS7AIglB2RdBjrQc9VrAqdNvyNk3bYJJdk6xSlv5JvehP8SKm7eJX9UFgMjPvJZlECWfa+P6r4xYWFpeKpeXyyura+oa3uXWt41QR2iQxj9VtBJpyJmnTMMPpbaIoiIjTm+jubKLfPFClWSyvzCihbQEDyfqMgLFU1xPhOQgBHcAhvU/ZAw4Hk303TBNQKn7sZHA6xgefdC9+lH8KX/76f/76uOtV/Ko/LTwPghxUUF6NrvdkJ5BUUGkIB61bgZ+YdgbKMMLpuBymmiZA7mBAWxZKEFS3s2ksY7xnmR7ux8ouafCU/d6RgdB6JCLrFGCG+rc2If/SWqnpn7QzJpPUUElmB/VTjk2MJxnjHlOUGD6yAIhi9q6YDEEBMfYnyjaE4PeT58H1YTWw+PKoUqvncZTQDtpF+yhAx6iGLlADNRFBz+jNcZ2C8+K8u4tucWZ1nbxnG/0o1/sA7ge1Jw==</latexit>
;
nb = n̄b ⌘ bA
" + bB
" / b
= const.
<latexit sha1_base64="Z2dbtewpcIQKWng+tyHxIAnVzlU=">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</latexit>
Kinetic eq. for each surface electronic mode.:
ṅx = xA
" + xB
"
xA
# + xB
# + xA
" + xB
" nx
<latexit sha1_base64="5Q+KceZNkJNfxtzMfcclaKs9Lfo=">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</latexit>
Htot = Ha
⌦ + Hb
0 + HA
⌦ + HB
0 + Ha
A + Ha
B + Hb
A + Hb
B
<latexit sha1_base64="AgcH3XpTbAgCTWGoWQDNsinDLJE=">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</latexit>
Total Hamiltonian:
(Doppler shift)
𝜇
A and
𝜇
B are respective electrochemical potentials

Tribocurrents
8
ja = aA
"
aA
# + aA
" n̄a =
aA
"
aB
#
⇥
1 e (m⌦+µB µA)
⇤
a
jb = bB
"
bB
# + bB
" n̄b =
bA
#
bB
"
⇥
1 e (m⌦+µB µA)
⇤
b
J = e
0
@
X
,m
ja( , m) +
X
0,m
jb( 0
, m)
1
A
<latexit sha1_base64="GuWLWot0gav1GQI+uw4Ju1Zn9n4=">AAACQnicbVBNaxsxFNS6aeO6X25z7OURU2pT1+yGQnspmORSenIgTlK8ZtHKb23V0u4ivS2Yxb+tl/6C3PIDcskhpeSaQ+SPlCbpgGDezDwkTZwracn3T73Kg42Hjzarj2tPnj57/qL+8tWhzQojsC8ylZnjmFtUMsU+SVJ4nBvkOlZ4FE/3Fv7RDzRWZukBzXIcaj5OZSIFJydF9W9f4TO8RwgVJtSE0BY6KkMrx5q3Qc/he8Sh+XduwbtbkbftZSS+ibi5BaGR4wm1onrD7/hLwH0SrEmDrdGL6ifhKBOFxpSE4tYOAj+nYckNSaFwXgsLizkXUz7GgaMp12iH5bKCObxxygiSzLiTEizVfzdKrq2d6dglNaeJvestxP95g4KST8NSpnlBmIrVRUmhgDJY9AkjaVCQmjnChZHurSAm3HBBrvWaKyG4++X75HCnEzi+/6HR3V3XUWWv2TZrsoB9ZF32hfVYnwn2k52xC/bb++Wde3+8y1W04q13ttgteFfXnK+tSA==</latexit>
Net current:
Only surface modes satisfying m 𝛺
<
𝜇
A -
𝜇
B contribute to ja ja > 0
Only surface modes satisfying m
𝛺
> 𝜇
A -
𝜇
B contribute to jb jb > 0
Self-limiting process: As charging increases |
𝜇
A -
𝜇
B|, fewer modes
contribute to corresponding jx, and more to opposing jx
Sign of J depends on relative magnitudes of and
aA
"
aB
# / a
<latexit sha1_base64="Vho8AaUxIdcrMeaiKy3cgE0v5b4=">AAACHnicbZDLSgMxFIYzXmu9VV26CRbBVZ2pii5rXeiygr1Ab5xJ0zY0yQxJxlKGPokbX8WNC0UEV/o2phdBWw8Efr7/nCTn90POtHHdL2dhcWl5ZTWxllzf2NzaTu3slnQQKUKLJOCBqvigKWeSFg0znFZCRUH4nJb93tXIL99TpVkg78wgpHUBHcnajICxqJk6q3VACGjWohCUCvqNGC6HP6wV9OUPzQ/xMa5dj4wGNFNpN+OOC88LbyrSaFqFZurDXkYiQaUhHLSuem5o6jEowwinw2Qt0jQE0oMOrVopQVBdj8frDfGhJS3cDpQ90uAx/T0Rg9B6IHzbKcB09aw3gv951ci0L+oxk2FkqCSTh9oRxybAo6xwiylKDB9YAUQx+1dMuqCAGJto0obgza48L0rZjHeSyd6epnP5aRwJtI8O0BHy0DnKoRtUQEVE0AN6Qi/o1Xl0np03533SuuBMZ/bQn3I+vwF5IaNG</latexit>
bA
#
bB
" / b
<latexit sha1_base64="7Yxy4yFNgR/NDTaLHyap8HEqHeg=">AAACHnicbZDLSgMxFIYzXmu9VV26CRbBVZ2pii5rXeiygr1Ab5xJ0zY0yQxJxlKGPokbX8WNC0UEV/o2phdBW38I/HznHE7O74ecaeO6X87C4tLyympiLbm+sbm1ndrZLekgUoQWScADVfFBU84kLRpmOK2EioLwOS37vatRvXxPlWaBvDODkNYFdCRrMwLGombqrNYBIaBZawV9CUoF/UbsXw5/aBT+sPwQH+Pa9Qg3/GYq7WbcsfC88aYmjaYqNFMfdgGJBJWGcNC66rmhqcegDCOcDpO1SNMQSA86tGqtBEF1PR6fN8SHlrRwO1D2SYPH9PdEDELrgfBtpwDT1bO1EfyvVo1M+6IeMxlGhkoyWdSOODYBHmWFW0xRYvjAGiCK2b9i0gUFxNhEkzYEb/bkeVPKZryTTPb2NJ3LT+NIoH10gI6Qh85RDt2gAioigh7QE3pBr86j8+y8Oe+T1gVnOrOH/sj5/AZ+E6NJ</latexit>

Tribocurrents II
• Sign of J depends on relative
magnitudes of and

• Compatible with triboelectric
series, correlated with materials’
work functions

See: Zou et al., Nat. Comm. 11, 2093
(2020)

• Charging shifts
𝜇
X’s; explains why
current
fl
ows in opposite
directions at Van de Graaf
generator’s two terminals
aA
"
aB
# / a
<latexit sha1_base64="Vho8AaUxIdcrMeaiKy3cgE0v5b4=">AAACHnicbZDLSgMxFIYzXmu9VV26CRbBVZ2pii5rXeiygr1Ab5xJ0zY0yQxJxlKGPokbX8WNC0UEV/o2phdBWw8Efr7/nCTn90POtHHdL2dhcWl5ZTWxllzf2NzaTu3slnQQKUKLJOCBqvigKWeSFg0znFZCRUH4nJb93tXIL99TpVkg78wgpHUBHcnajICxqJk6q3VACGjWohCUCvqNGC6HP6wV9OUPzQ/xMa5dj4wGNFNpN+OOC88LbyrSaFqFZurDXkYiQaUhHLSuem5o6jEowwinw2Qt0jQE0oMOrVopQVBdj8frDfGhJS3cDpQ90uAx/T0Rg9B6IHzbKcB09aw3gv951ci0L+oxk2FkqCSTh9oRxybAo6xwiylKDB9YAUQx+1dMuqCAGJto0obgza48L0rZjHeSyd6epnP5aRwJtI8O0BHy0DnKoRtUQEVE0AN6Qi/o1Xl0np03533SuuBMZ/bQn3I+vwF5IaNG</latexit>
bA
#
bB
" / b
<latexit sha1_base64="7Yxy4yFNgR/NDTaLHyap8HEqHeg=">AAACHnicbZDLSgMxFIYzXmu9VV26CRbBVZ2pii5rXeiygr1Ab5xJ0zY0yQxJxlKGPokbX8WNC0UEV/o2phdBW38I/HznHE7O74ecaeO6X87C4tLyympiLbm+sbm1ndrZLekgUoQWScADVfFBU84kLRpmOK2EioLwOS37vatRvXxPlWaBvDODkNYFdCRrMwLGombqrNYBIaBZawV9CUoF/UbsXw5/aBT+sPwQH+Pa9Qg3/GYq7WbcsfC88aYmjaYqNFMfdgGJBJWGcNC66rmhqcegDCOcDpO1SNMQSA86tGqtBEF1PR6fN8SHlrRwO1D2SYPH9PdEDELrgfBtpwDT1bO1EfyvVo1M+6IeMxlGhkoyWdSOODYBHmWFW0xRYvjAGiCK2b9i0gUFxNhEkzYEb/bkeVPKZryTTPb2NJ3LT+NIoH10gI6Qh85RDt2gAioigh7QE3pBr86j8+y8Oe+T1gVnOrOH/sj5/AZ+E6NJ</latexit>
9
I1
I2 I2
I3
I4
V1 V2
I6
Figure 4.18.
Currents and potentials at the nodes of a
network.
machine, too, there is a downward-directed electric field. Here charge
carriers can be moved against the field if they are stuck to a nonconduct-
ing belt. They are stuck so tightly that they can’t slide backward along the
belt in the generally downward electric field. (They can still be removed
from the belt by a much stronger field localized at the brush in the termi-
nal. We need not consider here the means for putting charge on and off
the belt near the pulleys.) The energy needed to pull the belt is supplied
from elsewhere – usually by an electric motor connected to a power line,
but it could be a gasoline engine, or even a person turning a crank. This
Van de Graaff generator is in effect a battery with an electromotive force,
under these conditions, of V0 volts.
In ordinary batteries it is chemical energy that makes the charge car-
riers move through a region where the electric field opposes their motion.
That is, a positive charge carrier may move to a place of higher electric
potential if by so doing it can engage in a chemical reaction that will
+
+
+
+
+
+
+ + + + +
+
+
+
+
+
+
+
+
+
–
–
–
–
–
–
–
–
–
–
+
+
+
+
+
+
+
+
+
+
V = V0
V = 0
R
I
E
E
E
E
Figure 4.19.
In the Van de Graaff generator, charge carriers
are mechanically transported in a direction
opposite to that in which the electric field would
move them.
yield more energy than it costs to climb the electrical hill.
To see how this works, let us examine one particular voltaic cell.
Voltaic cell is the generic name for a chemical source of electromotive
force. In the experiments of Galvani around 1790 the famous twitching
frogs’ legs had signaled the chemical production of electric current. It
was Volta who proved that the source was not “animal electricity,” as
Galvani maintained, but the contact of dissimilar metals in the circuit.
Volta went on to construct the first battery, a stack of elementary cells,
each of which consisted of a zinc disk and a silver disk separated by
cardboard moistened with brine. The battery that powers your flashlight
comes in a tidier package, but the principle of operation is the same.
Several kinds of voltaic cells are in use, differing in their chemistry but
having common features: two electrodes of different material immersed
in an ionized fluid, or electrolyte.
As an example, we’ll describe the lead–sulfuric acid cell which is
the basic element of the automobile battery. This cell has the important
property that its operation is readily reversible. With a storage battery
made of such cells, which can be charged and discharged repeatedly,
energy can be stored and recovered electrically.
A fully charged lead–sulfuric acid cell has positive plates that hold
lead dioxide, PbO2, as a porous powder, and negative plates that hold
pure lead of a spongy texture. The mechanical framework, or grid, is
made of a lead alloy. All the positive plates are connected together and
Source: Purcell, Electricity &
Magnetism, 3rd ed., (Cambridge
U. P., 2013 [1963]), sec. 4.9

Tribocurrents III
• Exponential dependence of
tunneling rates on energy
levels in complex surface
landscape

• can explain mosaic of
charging at ~1
𝜇
m roughness
scale

• also, dependence on pre-
strain of materials

Wang et al., Phys. Rev. Materials 1,
035605 (2017)
10
dating back to Thales of Miletus and his exper-
iments with amber charging against wool (4).
Although contact electrification has been success-
fully applied in several useful technologies [e.g.,
photocopying (5), laser printing (6), and electro-
text, it is commonly assumed that (i) contact
charging derives from spatially homogeneous
(on length-scales larger than molecular) surface
properties of contacting materials (1–3, 16–20)
and (ii) within a given pair of materials, one charges
uniformly positively and the other, uniformly
negatively (1, 7, 15, 21–24) (Fig. 1A, upper right).
These assumptions, however, make it difficult to
explain numerousexperimentalobservationswhere-
by different particles made of the same bulk mate-
rial (25) or even different macroscopic regions
electric series (13, 30, 31). Instead, contact charg-
ing phenomena derive from the chemical and
possibly micromechanical properties at and near
the surfaces of the contacting polymers.
Uniform versus nonuniform surface charging.
The starting point of the present study is a recent
observation that contact electrification can occur
between flat pieces of identical materials (32). Ac-
cording to the conventional view of contact elec-
trification, this should not happen because the
chemical potentials of the two surfaces/materialsare
Fig. 1. Possible scenarios of contact elec-
trification and experimental KFM surface
potentialmaps.Thetraditionalview(A,upper
right) assumes that upon contact and sep-
aration, one surface charges uniformly positively and
the other, uniformly negatively. In contrast, the mosaic
picture (A, lower right) assumes that each surface is a union of regions that
have different propensities to accept or donate charge. Upon contact charg-
ing, these regions are then expected to develop different charge polarities.
The mosaic picture of contact electrification is validated by the potential
maps shown in (B to D). The map in (B) corresponds to PDMS before contact
electrification. Such approximately uniform profiles were also observed for
other native materials tested. In contrast, contact-charged surfaces in (C) and
(D) feature a mosaic of (+) and (–) regions. (C) PDMS charged negatively
against another PDMS piece. (D) PC contact-charged positively against
PDMS. The left column has two-dimensional (2D) projections of the maps; on
the right, the corresponding 3D maps are displayed. Color scale is adjusted to
–1 V to 1 V in all images. Net surface charge densities on the contact-
charged pieces (measured by Faraday cup before KFM measurements) were
(B) 0.005, (C) –0.20, and (D) 0.16 nC/cm2
. All images have a scan area of
4.5 mm by 4.5 mm. Scale bars, 500 nm.
Department of Chemistry and Department of Chemical and
Biological Engineering, Northwestern University, 2145 Sheridan
Road, Evanston, IL 60208, USA.
*To whom correspondence should be addressed. E-mail:
grzybor@northwestern.edu
15 JULY 2011 VOL 333 SCIENCE www.sciencemag.org
308
on
www.sciencemag.org
Downloaded
from
Source: Baytekin et al.,
Science 333, 308 (2011)

Maximum tribovoltage
• In cylindrical coords. k = (km, kz)

• Fermi wave vector

• Upper bound on open-circuit tribovoltage (ja ≫ jb ,
or vice-versa):

• Not a ‘Volta potential’; rather, true emf from active
pumping of charges

• Macroscopic motion Vs prevents equilibration of
Fermi levels

• Can drive current along a closed circuit
11
|m⌦| = |kmVs|  kFVs
<latexit sha1_base64="wSaogi88czLMB1rWXAhuVQwOl4o=">AAACInicbVDLSgMxFM34rPU16tJNsAiuykwV1IVQFMSdFewDOmXIpLdtaDIzJhmhTPstbvwVNy4UdSX4MaYPUFsPXDg5515u7glizpR2nE9rbn5hcWk5s5JdXVvf2LS3tisqSiSFMo14JGsBUcBZCGXNNIdaLIGIgEM16F4M/eo9SMWi8Fb3YmgI0g5Zi1GijeTbp30ssHctoE1wH5+Z6voCV/zUkwKrgXl7HO6MOBIuBz+Wb+ecvDMCniXuhOTQBCXffveaEU0EhJpyolTddWLdSInUjHIYZL1EQUxol7ShbmhIBKhGOjpxgPeN0sStSJoKNR6pvydSIpTqicB0CqI7atobiv959US3ThopC+NEQ0jHi1oJxzrCw7xwk0mgmvcMIVQy81dMO0QSqk2qWROCO33yLKkU8u5hvnBzlCueT+LIoF20hw6Qi45REV2hEiojih7QE3pBr9aj9Wy9WR/j1jlrMrOD/sD6+gZNwKJW</latexit>
e oc = |µA µB|at zero current . ~kFVs
<latexit sha1_base64="Tx5ES3DUePbDm55FeUTPGiZxdus=">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</latexit>
kF
p
k2
m + k2
z
<latexit sha1_base64="P7p+saoUi88XIEGjxReG3ITxZSE=">AAACDHicbVBPS8MwHE39O+e/qUcvwSEIwminoMehIB4nuD+w1pJm6RaatF2SCrP0A3jxq3jxoIhXP4A3v43p1oNuPgh5vPd+JL/nxYxKZZrfxsLi0vLKammtvL6xubVd2dltyygRmLRwxCLR9ZAkjIakpahipBsLgrjHSMcLLnO/c0+EpFF4q8YxcTgahNSnGCktuZVq4Ka24PAqg/aAjKAtR0KlwV3d5fAY5vdDplNmzZwAzhOrIFVQoOlWvux+hBNOQoUZkrJnmbFyUiQUxYxkZTuRJEY4QAPS0zREnEgnnSyTwUOt9KEfCX1CBSfq74kUcSnH3NNJjtRQznq5+J/XS5R/7qQ0jBNFQjx9yE8YVBHMm4F9KghWbKwJwoLqv0I8RAJhpfsr6xKs2ZXnSbtes05q9ZvTauOiqKME9sEBOAIWOAMNcA2aoAUweATP4BW8GU/Gi/FufEyjC0Yxswf+wPj8ATACmm0=</latexit>

Phenomenology
• For kF ≃1010 m-1 (interatomic
scale) and VS ≃1 m/s,
𝜙
oc ≲ 10-5 V

• Voltage enhanced by mechanical
separation of electri
fi
ed
surfaces:

• increase from inter-atomic to
meter scale gives
𝜙
oc ≲ 105 V, as
in Van de Graaff generator

• Consistent with other qualitative
features of triboelectricity: e.g.,
𝜎
min ≃ -
𝜎
max for the charge
densities reported by Zou et al.
12
Source: Zou et al., Nat. Commun. 11,
2093 (2020)

Summary & outlook
• Single moving bath coupled to bosonic
fi
eld gives
Zel’dovich superradiance

• Fermionic
fi
eld coupled to two baths in relative motion can
describe triboelectrication as irreversible process,
generating an emf

• Precise control over Vs and surface separation needed for
further experimental testing of our theory

• Currents ja and jb dissipate kinetic energy into heat, even
without signi
fi
cant net charging

• May contribute directly to dry friction
13

Quantum Theory of Triboelectricity

Recommandé

Recommandé

Contenu connexe

Similaire à Quantum Theory of Triboelectricity

Similaire à Quantum Theory of Triboelectricity (20)

Plus de Alejandro Jenkins

Plus de Alejandro Jenkins (20)

Dernier

Dernier (20)

Quantum Theory of Triboelectricity