thermodynamics of colloidal particles
Thomas Speck
Valentin Blickle
Laurent Helden
Udo Seifert
Clemens Bechinger
contents
• mesoscopic thermodynamics
• TIRM experiment
• energy conservation
• work distribution / fluctuation theorems
thermodynamics
macroscopic:
W
ideal gas
p, V, n
external force
heat bath
Q
T
mesoscopic:
• suspended colloidal particle
W
• protein pulling
Q
F
• Brownian motion
• fluctuations
brownian motion
Brownian particle: random walk
Langevin equation
d 2z
dz
∂V
m 2 +γ
=−
+ ξ (t )
∂z
dt
dt
overdamped system
stochastic force:
friction coefficient
external potential
(Sekimoto 1998)
ξ (t ) = 0
ξ (t )iξ (t ') = 2γ T δ (t − t ')
energy balance: equilibrium
time-independent potential:
Langevin equation γ
dV
=0
dt
V only z dependent
∂V
dz
=−
+ ξ (t )
dt
∂z
∂V
dz
0 = −[−γ
+ ξ (t )]dz +
dz
∂z
dt
dz
dQ ≡ −[−γ
+ ξ (t )]dz heat
dt
∂V
potential difference
dV ≡
dz
∂z
0 = dQ + dU
energy balance
energy balance: time dep. potentials
V [ z , λ (t )]
λ (t ) control parameter
e.g.: λ (t ) = sin(ωt )
no total differential
add: ∂V ( z , λ (t ))
∂λ
dz
∂V ( z , λ (t ))
+ ξ (t )]dz +
0 = −[−γ
dz
∂z
dt
dλ
∂V ( z , λ (t ))
dz
∂V ( z , λ (t ))
∂V ( z , λ (t ))
d λ = −[−γ
+ ξ (t )]dz +
dz +
dλ
∂λ
dt
∂z
∂λ
dW
dQ
dV
work
heat
potential difference
microscopic energy balance
trajectory picture
τ
dz
∂V
dQ = −[−γ
+ ξ (t )]dz = −
dz
∂z
dt
z0 ; λ0
ze ; λe
parametric trajectory
Langevin equation
integration along trajectory
ze , λe
e
∂V
∂V •
Q = ∫ − dz = ∫ −
z dτ
∂z
∂z
z0 , λ0
t0
t
∂V •
W =∫
λ dτ
∂λ
t0
te
ΔV = V (te ) − V (t0 )
γ
dz
∂V
=−
+ξ
dt
∂z
trajectory
determination of Q,W, and V
experiment
• exact measurement of the trajectory
• externally controllable force to drive particle into non equilibrium
TIRM Technique
Z
3000
Streulicht:
I streu ~I ev
2500
Scattering signal
Probe particle
Streuintensität
Mikroskopobjektiv
2000
1500
1000
500
0
Iev
0
100
200
300
400
500
600
700
Zeit [s]
4000
r
3000
Histogram
Häufigkeit
Lase
2000
1000
0
0
500
1000
1500
2000
2500
3000
Streuintensität
• Creation of an evanescent wave
• Exponentially decaying scattering itensity
Inversion of the
Boltzman
distribution
• Determination of the particle wall distance z
Potential
Potential [kbT]
8
6
4
2
0
0,0
0,2
0,4
Abstand z [µm]
0,6
0,8
experiment
Photomultiplier
Intensität
sampling rate 2 kHz
polystyrene particles
Interference Filter
Zeit
Tubus
Lens
IR
Filter
d=4 µm
CCD
Camera
Beamsplitter
Objective
50X
upper tweezers 10mW
d
z
Polarizing
Beamsplitter
l/2
lower tweezers 300mW
lens 50 mm
Monitor Diode
Rotation Stage
EOM
Laserdiode:
652 nm
Intensity modulation
optical tweezers
Light
Pressure
Gradient
Force
Dipole interaction energy for
a particle in an electric field:
W ∝ - α ∫ E2 dV
α = εparticle / εoutside - 1
⇒
Particle moves towards
higher fields.
lower tweezers calibration
• increasing tweezers intensity
(3)
140
(1)
(2)
120
Force [fN]
Potential Energy [kBT]
8
6
4
100
80
60
40
20
22
24
26
28
30
32
Laser Power [mW]
2
0
0.0
0.1
0.2
0.3
0.4
z [µm]
0.5
0.6
0.7
• linear dependance of light pressure
• tweezers calibration
Vl = c i I l i z
interaction potential
V [ z (τ ), λ (τ )] = A ⋅ e −κ z + B0 ⋅ z + I lt [λ (τ )] ⋅ c ⋅ z
pulse protocol
interaction potential
1,2
Ilt
tp=700 ms
0,6
0,0
4,5
ts
120 ms
5,0
tp=700 ms
5,5
6,0
t [s]
30
tp
0.3
25
20
0.2
z [µm]
• strong coupling to
the bath
Laser Intensity [mW]
ts
15
10
0.1
5
0
0.0
493.6
493.8
494.0
time [s]
494.2
494.4
energy conservation
∂V •
1 ∂V •
Q = −∫
z dτ = − ∑
z (τ )
∂z
ν τ ∂z
t0
•
∂V •
c
W = −∫
λ (t )dτ = − ∑ I lt ⋅ z (τ )
∂λ
ν τ
t0
te
te
δ = W − Q + ΔV = 0
energy balance:
3
(a)
10
2
5
1
δ [kBT]
Energy [kBT]
15
0
-5
W
-Q
ΔV
δ
-10
-15
5950
6020
6090
Pulse Number
(b)
energy resolution
about 0.25 kT
0
-1
-2
-3
0.00 0.05 0.10 0.15
probability
work distribution
particle wall potential is not harmonic
harmonic potentials
symmetric gaussian work distribution
• demonstrated with colloidal particles in 3d traps
non harmonic potentials
theory predicts:
asymmetric non gaussian distribution
0.20
probability
0.15
theory:
•no fit parameters
0.10
0.05
0.00
-4
-2
0
2
4
6
8
W [kBT]
<W>=2.4 kT
10 12 14 16 18
fluctuation theorems
0.20
ΔF = 0
0.15
probability
symmetric protocol:
Jarzynski relation:
0.10
0.05
0.00
<e
exp.
− βW
-4
>= 1.03
8
ln[P(W)/P(-W)]
6
4
2
statistics!
0
-4
-6
-8
-8
-6
-4
-2
0
W
2
4
6
8
0
2
4
6
8
W [kBT]
P(−W )
= e − βW
detailed fluctuation theorem:
P (W )
-2
-2
10 12 14 16 18

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