![Page 1: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/1.jpg)
Experimental tests of the Fluctuation-Dissipation-Relation in aging glassy
systems
collaborators:
Hassan OukrisPhil Crider
Matt Majewski
Northeastern UniversityBoston
![Page 2: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/2.jpg)
Outline
• Nonequilibrium Fluctuation-Dissipation-Relation (FDR) Concept, Theory,
Simulations
• Experiments thus far: a mixed bag
• New results on a polymer glass.
– Try to “catch it in the act” of falling out of equilibrium
• Can we measure local correlation and response functions?
– Test local FDR violations
– Space-time correlation functions and dynamical heterogeneity
![Page 3: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/3.jpg)
Log ()
”()
Debye
glassy
Die
lect
r ic
susc
epti
bil i
tySignatures of glassy systems: Slow- nonexponential relaxation.
Rough energy landscape?
exp[-(t/ )]
Broadened response
Diverging relaxationtimes below Tg
(fragile glasses)
Aging after T-quench
Cooperative dynamics –jamming
![Page 4: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/4.jpg)
Fluctuation-Dissipation Relations (FDR)
Stokes-Einstein Relation D= kBT /60r
Nyquist RelationSV = 4kBTR
Violations expected in systems far from equilibrium
Brownian motion: Diffusion constant scales inversely with viscosity (1906)
Voltage noise scales with resistance (1928)
Aging glass: ideal system to study non-equilibrium FDR Cugliandolo and Kurchan, PRL 1993, PRE 1997, …
Configuration coordinate
•Universality in the violations?•Model dependent?•Effective temperature useful?
Teff =SV /4kBR
Ener
gy
![Page 5: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/5.jpg)
Time-dependent FDR violations and effective temperature
twait tobs
tobs
For tobs<< tw looks like equilibrium
FDR holds Teff = T
kB T
t= tw +tobs
C(t,tw)=<O(tw )O(t)> noise
(t,tw) =O(t)/h(tw) susceptibility
(t,tw) = [1/kB T][C(tw,tw )-C(t,tw)]
R(t,t
w)
C(t,tw)
Slope=-1/kBT
h(t)
(t,t
w)
![Page 6: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/6.jpg)
Time-dependent FDR violations and effective temperature
twait tobs
tobs
For tobs ≥ tw looks non-equilibrium
FDR fails Teff > T
kB T
t= tw +tobs
C(t,tw)=<O(tw )O(t)>
(t,tw) =O(t)/h(tw)(t,tw)
(t,tw) = [1/kB Teff][C(tw,tw )-C(t,tw)]
R(t,t
w)
C(t,tw)
Slope=-1/kBTeff
Slope=-1/kBT
h(t)
mean-field models(
t,tw)
![Page 7: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/7.jpg)
Frequency-dependent FDR violations and effective temperature
twait tobs
tobs
For tw < 1 looks non-equilibrium
FDR fails Teff > T
kB T
/tobs
)(2
COSo
)(")(')( i
)("
Bk
ST O
eff
h(t)
Difficult to access low ftw –
need rapid quench
0.1 1 10
ftw
T1
Mean-field T2
![Page 8: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/8.jpg)
Evidence from simulations
p-spin Ising modelCugliandolo, Kurchan, 1997
Lennard-JonesBarrat, Kob 1998
Domain growth- infinite Teff
Barrat, 1998
![Page 9: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/9.jpg)
Experiment on FDR in aging supercooled liquid
Oscillator as thermometer:Eosc = ½kBTeff Cugliandolo et. al. 1997
Resonant circuit driven by thermal fluctuations in dielectric sample
C<V2> = kBTeff <V2> is integrated noise power under resonance
Grigera and Israeloff, PRL 1999
![Page 10: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/10.jpg)
Small Long-Lived FDR Violations Observed
Violations persisted up to the average relaxation time of the material, suggested series or stringy kinetics
C’=C0’ C”=C0” tw ~ 105
![Page 11: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/11.jpg)
FDR violations in spin glasses
Herisson and Ocio PRL 2002
![Page 12: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/12.jpg)
FDR violations in Laponite and polymer glassElectrical: large FDR violations and non-Gaussian Teff ~106 K
Buisson, Bellon, Ciliberto, J. of Phys.: Cond Mat. 2003
But these samples are macroscopic:
Spikes require the coherent fluctuation of entire 10 cm3 sample!
In any case, these measurementsare tricky and extrinsic noise is challenging.
Large violations dueto non-Gaussian spikes.Attributed to intermittency Intermittency found in simulationsof mesoscopic glass models: Sibani, PRE 2006
![Page 13: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/13.jpg)
Summary of experimental resultsMaterial Property FDR violations? tw Ref.Glycerol electrical small short-moderate Grigera, 1999Spin glass magnetic large short Herisson, 2002Laponite electrical large short-moderate Buisson, 2003 Laponite rheological none Buisson, 2004
“ “ large long Abou, 2004 “ “ large long Strachan, 2006
“ “ large long Bartlett, 2006“ “ none Jabbari-Farouji, 2007
Poly-carbonate electrical large short-moderate Buisson, 2005
![Page 14: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/14.jpg)
Measure dielectric susceptibility and current noisepolymer glass: PVAc, Tg =308 K
’i” FDR: Si =4kbTC0”
![Page 15: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/15.jpg)
Aging of dielectric susceptibility
Rapid quench 330K to 300K
ftw scaling
![Page 16: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/16.jpg)
Current noise measurements
Ultra-low-noise current amplifier 0.5 fA/√Hz
"4 CTkS BI
FDR prediction:
![Page 17: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/17.jpg)
Equilibrium noise and Teff
1.E-32
1.E-31
1.E-30
1.E-29
1.E-28
0.1 1.0 10.0 100.0
S (
A/H
z)
Frequency (Hz)
0
100
200
300
400
500
0.1 1.0 10.0 100.0
Tem
per
atur
e (K
)
Frequency (Hz)
"4 CkST
B
Ieff
![Page 18: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/18.jpg)
Two temperature quench profiles
T(K)
time (s)time (s)
Initial dT/dt=0.15 K/s
“fast”“slow”
aging
300
305
310
315
320
325
330
0 2 4 6 8 10 12 14
T fictive 13.3 Hz
TInitial dT/dt=8 K/s
cooling
![Page 19: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/19.jpg)
Current noise during and after rapid quench
0
0.05
0.1
0.15
0.2
0.25
0.3
0 10 20 30 40 50
I (pA)
t(s)
cooling aging
Tg
Average of 840 quenches
![Page 20: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/20.jpg)
Dielectric response measurements
Conventional measurement Apply V=V0sin(t)
Measure I with Lock-in → Admittance Y=I/V
But fails for highly non-stationary early tw
V is white noise, measure I noise
FT- I, V and Admittance Y=I/V
![Page 21: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/21.jpg)
Slow quench: effective temperature
No clear FDR violations found for slow quench
![Page 22: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/22.jpg)
Effective temperature during fast quench
![Page 23: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/23.jpg)
Scaling of effective temperature in aging regime
100
200
300
400
500
600
700
800
0.01 0.1 1 10 100
Tef
f(K
)
ftw0.45
tw =tQ -5 from 1.5s to 400 s
Slower decay than ftw scaling expected Shape also disagrees with mean-field models
![Page 24: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/24.jpg)
1E-12
1E-11
1E-10
0.01 0.1 1 10
C''
Frequency (Hz)
0.470.60.91.522.75.1102060100200300450
Equilibrium 318 K tQ (s)
Spectrum of response, ”(f), is distorted during quench
”C0
![Page 25: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/25.jpg)
Time evolution of spectrum: noise and response
0
0.5
1
1.5
2
2.5
0.1 1 10
"
Frequency (Hz)
tQ (s)
0.9
1.5
2
2.75
Equilibrium 318 K
during quench
0.07
0.7
0.01 0.1 1 10
"
tQ = 5
10 20 80
200
during aging
responsenoise
responsenoise
One interpretation: for response is lower than for noise
![Page 26: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/26.jpg)
FDR violations in aging Lennard- Jones
Barrat and Kob 1998
Correlation
Response
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.01 0.1 1 10 100 1000 10000 100000
![Page 27: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/27.jpg)
0
0.5
1
1.5
2
2.5
0.1 1 10 100 1000
t(MCS)Co
rrel
ation
Correlation
1-kBT ·Response
Noise·/kBT
Susceptibility
”(a
rb. u
nits
)
ftw
Frequency domain susceptibility and noise for aging Lennard-Jones
Barrat and Kob 1998
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000 100000 1000000
tw =40000
![Page 28: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/28.jpg)
Noise is Gaussian even when FDR violated
Large extrinsic spikes (> 5 do occur, but very rarely, and are removed
![Page 29: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/29.jpg)
FDR violations during cooling and aging
fdt
dT
T
m
g
gTdT
dm
/
log
Hypotheses:
•Noise decorrelates faster during cooling and aging due to energy lowering transitions
•significant violations when quench rate, dT/dt, is high
•E.g. when fragility index
• Nonequilibrium noise saturates at ~ equilibrium -peak noise –this is reasonable since there are a finite number of dipoles.
• Practical upper limit on Teff ~ T ”(peak)/”(earliest tw) ~ 3T
![Page 30: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/30.jpg)
Caught polymer melt in the act of falling out of equilibrium
Moderate FDR violations observed: but only for high quench rates.
Violations are short-lived: but modified ftw scaling.
Noise is Gaussian
Interesting results:
Apparent response < corr noise much less stretched
Teff < T regime observed, disagrees with mean-field models but consistent with Lennard-Jones
Summary of FDR violation experiments
![Page 31: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/31.jpg)
Cr is correlation function (noise)
r is response function
Local aging is heterogeneous in a model spin glassCastillo, Chamon, Cugliandolo, Kennett PRL 2002
Castillo, Parsaeian, Nature Physics 2007
FDR violations heterogeneous
Non-Gaussian distributions and possibly intermittent noiseChamon et. al. PRE 2003 Crisanti and Ritort cond-mat/0307554.
![Page 32: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/32.jpg)
PVAc
Au film
V
Glass substrateDens ity P lot: |E |, V /m
2.145e-001 : > 2.200e-0012.090e-001 : 2.145e-0012.035e-001 : 2.090e-0011.980e-001 : 2.035e-0011.925e-001 : 1.980e-0011.870e-001 : 1.925e-0011.815e-001 : 1.870e-0011.760e-001 : 1.815e-0011.705e-001 : 1.760e-0011.650e-001 : 1.705e-0011.595e-001 : 1.650e-0011.540e-001 : 1.595e-0011.485e-001 : 1.540e-0011.430e-001 : 1.485e-0011.375e-001 : 1.430e-0011.320e-001 : 1.375e-0011.265e-001 : 1.320e-0011.210e-001 : 1.265e-0011.155e-001 : 1.210e-001< 1.100e-001 : 1.155e-001
Local dielectric spectroscopy
resresres fVz
C
kf
z
F
kf
k
kf 2
02
2
8
1
4
1
4
1
2
2
1VCU tip
F=dU/dz
UHV SPM
Electric Force Microscopy
Probed depth 20 nm
+
-
tVV sin0
20
202
20 tsin2
2
tcos21
4 PP VVVVdz
Cd
k
fdf
(susceptibility ) (polarization, charge)
Select 1 or 2 with lockin
![Page 33: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/33.jpg)
Time (s)
VP / VP(0)
Relaxation after a dc bias reduction
![Page 34: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/34.jpg)
Polarization images in PVAc near Tg
600 x 600 nm
t=0 t=17 min t= 48 min
303.5 K, we find rms spatial <VP > = 23±4 mV . 305.5 K <VP > =28±4 mV
![Page 35: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/35.jpg)
Tim
e (s
)
0
2500
0
2500
0 position (nm) 700
Imaging spatio-temporal dipolar fluctuations near Tg =308 K
Longer time correlations at lower temperatures seen.
Hints of dynamicalheterogeneity and web-like structures
Can study various correlationFunctions
e.g. global C(t)
301.5 K
305.5 K
![Page 36: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/36.jpg)
![Page 37: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/37.jpg)
Time (s)
C(t)
C(x)
X (nm)
![Page 38: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/38.jpg)
Local Response vs. Correlation
0
0
R(t
)
C (t)
Q=Ceff VP
Ceff = 7.2x10-18 F
R(t)=A-Q(t)/V
C(t)=<Q(t’)Q(t’+t)>
T (K) -1/kB slope
305.5 262 ± 15303.5 258 ± 30302.5 253 ± 40
305.5 K
303.5 K302.5 K
![Page 39: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/39.jpg)
Four-point space-time correlation functions
Various four-point space-time correlation functions have been studied in simulations. A recurring one is
g4(x,t) = <V(0,0)V(0,t)V(x,0)V(x,t)> - <V(0,0)V(0,t)><V(x,0)V(x,t)>
When integrated over all x, a generalized susceptibility, 4(t), is obtained.
4(t) is variance of C(t) Glotzer et al PRL 1999Bouchaud et al 2006 Cipelletti et al 2006
![Page 40: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/40.jpg)
Variance of C(t)
2 (C)
![Page 41: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/41.jpg)
Local non-contact dielectric spectroscopy –
PVAc shows a small reduction in Tg and narrowing of the distribution of relaxation times in 20 nm free
surface layer. No suppression of glassy dielectric response
Spatio-temporal fluctuation images
Quantitative agreement with equilibrium thermal noise will allow study of local FDR violations.
Various x-t correlation functions can be studied
Summary
![Page 42: Experimental tests of the Fluctuation- Dissipation-Relation in aging glassy systems collaborators: Hassan Oukris Phil Crider Matt Majewski Northeastern](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d0c5503460f949dff77/html5/thumbnails/42.jpg)
Acknowledgements:
P. S. CriderH. Oukris M. E. MajewskiJ. ZhangT. S. GrigeraE. Vidal RussellNSF-DMR-ACS-PRF