Fluctuations and slow dynamics in an ageing polymer glass D. Bagchi , S. Ciliberto, A. Naert, L. Bellon June 2 - 6, 2008 UPoN 2008 Mechanical design: Frédéric Arnoul Electronics: Marius Tanase
Fluctuations and slow dynamics in an ageing polymer glass D. Bagchi, S. Ciliberto, A. Naert, L. Bellon June 2 - 6, 2008 UPoN 2008 Mechanical design: Frédéric.
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Fluctuations and slow dynamics in an ageing polymer glass
D. Bagchi, S. Ciliberto, A. Naert, L. Bellon
June 2 - 6, 2008
UPoN 2008
Mechanical design: Frédéric ArnouldElectronics: Marius Tanase
Thermal fluctuations, non-equilibrium thermodynamics
Fluctuations of a polymer glass and its response to thermal stress
More refined experiments
Results
Conclusions and unsolved aspects
UPoN 2008
Outline
Fluctuations and Dissipation(Dynamics in Equilibrium)
F
FluctuationVoltage V(t), Velocity v(t)
DissipationResistance R, Viscosity η
Examples: Colloidal particles in a fluid / Electrons in a resistor (Nyquist noise)
Fluctuation Dissipation Theorem
TrkD B6fTRkV B 42
)](~Im[2
4)( f
f
TkfS B
V
In general, if Sv(f) is the spectral density ofFluctuations and ϰ(t) the response, then
How does a system respond when driven away from equilibrium?
How does one develop a thermodynamic description for these systems?
Questions:
Answer:
Derive useful information from fluctuations of a relevantvariable.
A typical off-equilibrium system
1. Increase in viscosity by many orders of magnitude2. Physical properties depend on thermal history3. Slow dynamics4. Dynamic heterogeneities
E.g., a fluid coupled to a heat bath is quenched very fast
System: A glassy system, e.g. a polymer after a quench
Study: How the Nyquist noise and response (dielectric losses) change as a polymer evolves after a quench
RC
o
o
o
C
C
RC
iCiCiZ
CiRZ
'
''
'''
1
][)()(
1
)(
1
Dissipation
Response
T
tw
Tg
System thermally driven away from equilibrium
101
102
103
104
105
100
101
102
tw
(secs.)
R(t
w ,
f)
/ R
o (t w
= 0
)
0.5 Hz
0.1 Hz
1 Hz11 Hz
91 Hz
Time evolution of the response(dielectric losses)
ϰ(t,tw) separated intoshort time part (obeys equilibrium FDT)Ageing part (obeys the following relation:
Away from equilibriumGeneralized FDT, Fluctuation Dissipation Ratio
Bw
wweff kft
fftSftT
2)],(Im[
),(),(
Ref.: L. F. Cugliandolo, J. Kurchan, L. Peliti, Phys. Rev. E 55 (1997) 3898.
)],(Im[),(2
),( ftf
ftTkftS w
weffBw
The Effective Temperature
RC
CiRZ
1
)(
1
FDT for a dielectric
)],(Re[),(4),( wweffBwv tZtTktS
Power spectral density of fluctuations
Response
2)(1
),(4),(
RC
RtTktS
weffBwv
Recent experiments
L. Buisson, S. Ciliberto, Physica D 204 (2005)
Intermittent bursts in the noise voltage of polycarbonate
Optimization of the geometry of the sample
Buisson’s Sample14 parallel capacitors with polycarbonate as dielectric
Present geometry:10 μm PVAc betweentwo aluminium electrodes
Advantages:1. Higher mechanical rigidity2. Higher dissipation3. Less bulky (aids in efficient
thermal design of the setup).4. Good electrical contact
NI-PXI 4472
6GΩPeltiers
Faraday Cage
Thermalinsulation Amplifier
1st Stage: Differential amplifierwithLow noise JFET 2N6453
Experimental Setup
Polymer: Polyvinyl Acetate, Tg=45°C
Minimisation of the influence of external sources of noise on the noise spectrum of the sample
The first stage of the amplifier is a differential amplifier made ofJFET 2N6453, which has a very low input current noise (1 fA/Hz1/2) and input voltage noise (5 nV/Hz1/2) above 2 Hz.
The entire experimental setup was housed in a Faraday cage.
The current through the peltier was kept constant during the waiting time after a quench, so as to prevent the influence of magnetic fields due to changing currents.
15 20 25 30 35 40 45 50 55 6010
6
107
108
109
1010
1011
Temperature (C)
R (
ohm
s)
1 Hz
0.125 Hz15 Hz
0 200 400 600 800 1000 120015
20
25
30
35
40
45
50
55
60
Time (mins.)
Tem
pera
ture
(C
)
Re[Z(f)]
Tg
15 20 25 30 35 40 45 50 55 600.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
-8
Temperature (C)
C (
Far
ads)
Im[Z(f)]
Thermal Cycles
Ageing
101
102
103
104
105
100
101
102
tw
(secs.)
R(t
w ,
f)
/ R
o (t w
= 0
)
0.5 Hz
0.1 Hz
1 Hz11 Hz
91 Hz
102
103
104
100
101
102
tw
(secs.)
R(t
w,f
) /
Ro(t
w,f
)
0.1 Hz
1 Hz11 Hz
91 Hz
tw =0 when the system just crosses the glass transition temperature 45°C
Tstop=22 °C, quench rate=6.8 °C/min. Tstop=23.5 °C, quench rate=3.25 °C/min.
As a function of Frequency and speed of the quench
103
104
105
100
tw
(secs.)
R(t
w,f
) /
Ro(t
w=
0)
1 Hz
0.1 Hz11 Hz
91 Hz
102
103
104
100
101
102
tw
(secs.)
R(t
w,f
) /
Ro(t
w,f
)
0.1 Hz
1 Hz11 Hz
91 Hz
Tstop=23.5 °C. Tstop=35 °C.
As a function of Frequency and depth of quench
ZZ
Ze vS
vZ
G
)//( eZ
e
eS ZZv
ZZ
ZGv
ξ η
Nyquist noise measurements
Current noise of Amplifier
Voltage noise of Amplifier
In Fourier space
)](Re[4)( fZTkfS Bv )](Re[2)](Im[ fZff
S
ZZ
ZSZZTk
Z
ZZGfS
eB
ev
22
22
//)Re(4
//)(
A typical noise voltage signal and its power spectrum when thesystem is near equilibrium
10-1
100
101
102
103
104
10-9
10-8
10-7
10-6
Frequency (Hz)
Sv(f
) (v
olts
/ H
z1/2 )
noise spectrum
FDT(uncorrected)FDT (corrected)
amplifier noise
0 2 4 6 8 10 12 14
x 104
-1.5
-1
-0.5
0
0.5
1
1.5x 10
-6
number of points
Vo
lta
ge
(V
)
10-1
100
101
102
103
104
10-9
10-8
10-7
10-6
Frequency (Hz)
Sv(t
w,f)
(vo
lts /
Hz1/
2 )
10-1
100
101
102
102
103
104
105
Frequency (Hz)
T eff(t
w,f
)
57 mins.
65 mins.
69 mins.195 mins.
335 mins.
Evolution of the power spectrum ofvoltage noise for a very slow quench(rate = 0.15°C/min), Tstop= 32°C, for tw=57 mins., 65 mins., 200 mins., and 335 mins.
Effective temperature (in Kelvins) for the same quench
Effect of very slow quenches crossing the glass transitionEffect of very slow quenches crossing the glass transition
Development of a DC polarisation voltage across the polymer film
• Follows temperature change• Insensitive to the presence of thermal gradients• Quite stable with time• Highest at high temperatures when the polymer
molecules are more rubbery• Direction of polarization is constant Vx
RxC
Equivalent Circuit
600 800 1000 1200 1400 1600 1800 2000 2200
0
1
2
3
4
5
6
7
8
9
time (mins.)
DC
volta
ge
( X
70
) (
volts
)
600 800 1000 1200 1400 1600 1800 2000 2200
20
25
30
35
40
45
50
55
time (mins.)
T (
C)
10-1
100
101
102
10-8
10-7
10-6
Frequency (Hz)
Sv(f
,tw
) (v
olts
/ H
z1/2 )
-500 0 500 1000 1500 2000 2500 300020
25
30
35
40
Waiting time,tw
(secs.)
Te
mp
era
ture
(C
)Effect of fast quenches crossing the glass transition
Quench rate: 7 °C/min.; Tstop= 21°C
Effective temperature as a function of the waiting time
Bw
wvweff kft
fftSftT
2)],(Im[
),(),(
Time evolution of Teff
Tf
Buisson’s slow quench
100
101
100
200
300
400
500
600
700
800
900
1000
f (Hz)
Te
ff(f, t
w)
720 secs.
20000 secs.10000 secs.
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x 10-6
100
101
102
103
104
105
V (volts)
P(V
)
20 mins6 mins.
2 mins.
30 secs.
3.6 Hrs.9 hrs.
Statistical analysis of the evolution of noise voltage after a quench
Thermal voltage fluctuations in an ordinary impedance has a Gaussian distribution.What happens when the impedance ages with time?
100 200 300 400 500 600-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
waiting time (mins.)
ske
wn
ess
0 100 200 300 400 500 600
2
2.5
3
3.5
4
waiting time (mins.)
kurt
osi
s
23
2
3
)()(
)()(
dxxPdxxPx
dxxPdxxPxskewness 22
4
)()(
)()(
dxxPdxxPx
dxxPdxxPxKurtosis
Deviations from the Gaussian shape
Conclusions and Future Perspectives
The relaxation dynamics of the polymer depends on the quench rate.
There is a small violation of the FDT at low frequencies for the fastest quench studied.
The PDFs of the noise voltage after a quench are Gaussian.
No intermittent bursts are observed in the noise voltage. The understanding of intermittency still remains an open question.
It is crucial to minimise the 1/f noise of the amplifier in order to do a thorough study of the important low frequency regime.
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