Resonant Enhancement and Dissipation in Nonequilibrium
van der Waals Forces
Adam E. Cohen (Stanford)Shaul Mukamel (UC Irvine)
Intermolecular forces and optical response
van der Waals: RTbVVap
)(2
a ~ index of refraction (empirically)
McLachlan:
0
2 )()()(2
)(
diirJrU ba
0
2 )()()()(n
nbnaB iirTJkrF
T = 0:
T > 0: ‘ =polarizability
J = coupling
/2 TnkBn
Gauss: if 23
13
21
1~ rdrdrr
f n
and f is insensible between macroscopic objects, then 6n
3 + 3 = 6
Relating Nonlinear Optical Response to Intermolecular ForcesStart with perturbations to the individual molecules:
)()( 0 tEpHtH aaaa )()( 0 tEpHtH bbbb
121a2a
t t t
12)2(
a11a1)1(
a0aa dtdt)t(E)t(E)t,t,t(Rdt)t(E)t,t(Rp)t(p
Can calculate or measure R(1), R(2), … for any initial state.
Response given by Volterra series:
To calculate R(n): L
i
)t()1()0( LLL Liouville superoperator
01n
n
1n)n( )t(p̂)t(p̂)t(p̂i
!n1)t,,t,t(R
T
Time-ordering superoperator X,AXA
Commutator
X,AXA 21
Anticommutator
titi
AeetA 00)(ˆ LL
Liou. Space interaction picture
)A(A 00Tr
Expectation value
coordinate operator classical source
Do the nonlinear response functions completely describe a molecule?
01n
n
1n )t(p̂)t(p̂)t(p̂i!n
1)t,,t,t(Rn
T terms
In a quantum system or a classical ensemble, fluctuations have a life of their own
Calculate response of fluctuations to a perturbation:
01nma
n
1nma )t(p̂)t(p̂)t(p̂)t(p̂i!n
1)t,,t,t,,t(Rnm
T
terms terms
R++ and R+- are related by the Fluctuation-Dissipation Theorem (FDT)
Call R+...+-...- Generalized Response Functions (GRFs)
Causality Generalized K-K relations
Thermal equilibrium Generalized FDT
(Compare with )
Two coupled moleculesCoupled molecules:
baba pptJHHtH )()( 00
1212
t t t
12)2(
111)1(
0baba dtdt)t(J)t(J)t,t,t(R~dt)t(J)t,t(R~pp)t(p)t(p
Want to evaluate:
L
i
)t()1()0(b
)0(a LLLL
Liouville superoperatorAgain
01b1anbnaba
n
1n)n( )t(p̂)t(p̂)t(p̂)t(p̂)t(p̂)t(p̂i
!n1)t,,t,t(R~
TUsing superoperator algebra, we can factor the joint response function:
n
mnbnan
n tttRtttRtttR mn mmn - m
0111
)( ),,,(),,,(),,,(~ terms terms terms terms
The joint response of the coupled molecules depends on all GRFsof the individual molecules.
Joint response function:
Example: Coupled Harmonic Oscillators
Phys. Rev. Lett. 91, 233202 (2003)
),(),(),(),(),(~11111
)1( ttRttRttRttRttR baba
1st order response to coupling J(t)papb:
Frequency domain: )()()1( Jpp ba
Steady state coupling: 2)1( )0(21 JF
Reproduces McLachlan formula for Ta = Tb
Time domain: 111)1( )(),(~ dttJttRpp
t
ba
0.51
1.52
0.5
1
1.5
2
- 20
- 10
0
10
0.51
1.52
0.5
1
1.5
2
b/a
b/a
(1)(0)
Dissipation between coupled SHOs
For time-varying J, need: )('')(')()1( i
Force Dissipation
Possibility of negative friction
-20
2
0.5
1
1.5
2
-10
0
10
-20
2
0.5
1
1.5
2
b/a
Re[(1)] Im[(1)]
-20
2
0.5
1
1.5
2
-5
0
5
-20
2
0.5
1
1.5
2
b/a
Example: FRET forceFluorescence Resonance Energy Transfer (FRET) is mediated by thesame dipole-dipole interaction that mediates the vdW force.
Do not fret for it leads only to evil.--Psalm 37
J. Phys. Chem. A 107 (19) 3633 (2003)
Forster rate of FRET:
d
nf
rck ad
dFRET
0
35
''
60
23
)()()(
83
orientational factordonor emission spectrum
lifetime of donor
Interaction energy from FRET:
d
nf
rcU ad
dFRET
0
35
'
60
23
)()()(
163
Kramers-Kronig relation between kFRET and UFRET
UFRET can also be thought of as optical trappingof acceptor in near-field of excited donor.
Sample calculationChlorophyll b in diethyl ether
-20
0
20
40
60
80
100
550 600 650 700 750 (nm)
Im(a) fD
-20
0
20
40
60
80
100
550 600 650 700 750
(nm)
Re(a) fD
FRET FRET force
FRET force may be either attractive or repulsive
FRET force may be much stronger than vdW force
Possibilities for experimental verification
• NLO effects in critical systems (gasses, binary mixtures, polymers)
• Conformational changes in tethered bichromophores• Concentration quenching• Solid state measurements (Casimir-type)
Conclusions
• Quantum ensemble described by Generalized Response Functions (GRFs)
• Response functions of two coupled systems may be expressed in terms of the GRFs of the constituents
• vdW forces between objects at different temperatures or in relative motion show resonant enhancement and (possibly negative) dissipation
• A mechanical force accompanies FRET
Acknowledgments
Professor Shaul Mukamel (UCI)
$$ Hertz Foundation $$
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