polymer dynamics & field cycling NMR
- glass transition
- dynamics in polymer melts
- analyzing T1 relaxation
E.A. Rössler
Experimentalphysik II, Universität Bayreuth, Germany
a rheological crossover
in the viscosity
of polymer melts
M < Mc:   M1
M > Mc:   M 3.4-3.7
(
Mc: „entanglement M“
Berry and Fox 1968
two microscopic theories
for explaining two rheological regimes
11
10
D~M
-1

D
D~M
Mc
-2
1
10
~M
~M
(3.4-3.7)
1
Mc
M
-6
10
M
rheological behavior changes at
entanglement molecular weight Mc
M < Mc: Rouse theory (1953)
M > Mc: tube reptation theory
(deGennes 1979, Doi/Edwards 1987)
Rouse model (M < Mc):
bead & spring model in a viscous medium
entropic force constant:
Rouse model II: calculating correlation
function of normal mode coordinates
neglecting inertia effects,
(overdamped oscillations
due to viscous medium)
and continuous chain

X p  k p [ X p  f pL (t )]
t
decoupled relaxation
(„breathing“) modes
Rouse model III – results
a.) mean square
segment displacement
b.) correlation function of
tangent vector b
 r (t )
b n
n
Rouse
1
-1/2
<R2>
R
 
 b (0)b (t )  
1/2
N 1
2
p
 C p (t )
p 1
lgt
lgt
Rouse modell IV
scaling arguments yielding  r   t
2
1/ 2
 r 2   D(t ) t
D  1 / N (t )
N (t )  t
1/ 2
r  t
2
more and more segments
are dragged
1/ 2
cf. T. Springer, IFF-Ferienkurs 1994
glassy & Rouse dynamics
mean square segmental displacement <r2(t)> = g1(t)
MD simulations (Binder et al. 2003)
- ballistic regime:  t2
Rouse
- cage regime  to
- sub-diffusive Rouse
regime:  t1/2
- hydrodynamic regime:
(t  ): <r2> = 6 D t
accounting for entanglement effects:
deGennes´ idea: tupe-reptation model
forecast:
M
3
D  M -2
(exp.:   M
3.60.2
(exp.: D  M -2.2)
)
from Rouse to entanglement dynamics
hierachy of power-laws is expected
mean square
displacement
rank-two
orientational correlation function
(courtesy K. Saalwächter)
regime II and III:
C(t )  r 2 (t )1
shear modulus
lg G(t)
What mechanical relaxations do we expect
in polymers?
Rouse
simple liquid =
„glassy dynamics“
(-process)
entanglement
lg t
M <MR M >MR
M >Me
bead-and-spring model
JCP
0
I
II
III
IV
1
2
log <r(t) >
t
t
cage
t
0.25
0.5
t 0.5
2
s
e
log t
R
d
evidence of tube
coming from short times: neutron scattering
0
I
II
III
IV
1
2
log <r(t) >
t
t
cage
t
0.25
0.5
t 0.5
2
s
e
log t
R
d
coming from long times: field gradient NMR
0
I
II
III
IV
1
2
log <r(t) >
t
t
t
cage
t
t
0.5
0.25
0.5
2
s
e
log t
R
d
fast or electronic field cycling NMR:
powerful tool for investigating polymer dynamics
since 2005 in Bayreuth (EPII)
Doi, M.; Edwards, S.F. The Theory of Polymer Dynamics; Oxford 1986.
Binder, K., Baschnagel, J., Paul, W. Progr. Polym. Sci. 28, 115 (2003)
Kimmich, R., Fatkullin, N. Adv. Polym. Sci. 170, 1 (2004)
Greassley, W.W. Polymer Liquids & Ntworks: Dynamics and Rheology, Garland 2008
Kruk D.; Herrmann, A.; Rössler, E.A.Progr..NMR Spectroscopy 63,33 (2012)
Bayreuth (EP II; FFC NMR and DS):
Kariyo et al. Phys. Rev. Lett. 97, 207803 (2006)
Kariyo et al. Macromolecules 41, 5313 dito 41, 5322 (2008)
Herrmann et al. Macromolecules 42, 2063 dito 42, 5236 (2009)
Abou Elfadl et al. Macromolecules 42, 6816 (2009) dito 43, 3340 (2010)
Herrman et al. Macromolecules 45, 1408 dito 45, 6516 (2012)
Hofmann et al. Macromoleces 45, 2390 (2012)
Kruk, Meier, Rössler, J. Phys. Chem. B115, 951 (2011)
Meier, Kruk, Rössler, J. Chem. Phys.136, 034508 (2012)
Kruk, Meier, Rössler, PRE 85, 020201 (2012)
Meier et al. ACS MacroLetters 2, 96 (2013)
short introduction to field cycling NMR
nuclear spin in external magnetic field B0
Hˆ z  μB0  B0 Iˆz
Em   hB0mI
Zeeman operator
energyeigen values
2I  1 energylevels
I  1 / 2 (e.g., 1H ) : two level system
N-0
E  E  E  h L  hL
with L  B0 (Larmorfreq.)
N+0
nuclear magnetization (in therm. eq.)
N 0
E / kT

e
N 0

1
2
m
c

1
0
0
nucl B0
M 0  M z   hm Nm 
V m
3kT
M x0  M y0  0
Curie law
 
M0 || B0
impact of pulses
in resonanceand rotatingframe:

 
M
  M x B1
t
RF probe pulse (B1 field) turns Mz into x,y-plane
after pulse: magnetization precesses with L and decays
freely: free induction decay (FID)
FID amplitude  M0
signal detection
precessing magnetization induces NMR signal S(t)
in RF coil perpendicular to Bz=B0
M (t )
S (t ) 
t
S (0)  L B0  L2 (amplitude)
spin-lattice relaxation
o
after90 pulse :
M z  0 (saturation)
What re-establishes Mz and Boltzmann distribution of Ni?
no spontaneous emission
( As  )
3
relevance of internal relaxation processes
due to coupling of spins with “lattice”
M z 1
 ( M z0  M z )  M z  M z0 (1  e
t
T1
T1 : spin  latticerelaxationtime

t
T1
) Mz
M0
M0 (1- e-1)
T1
t
source of spin-lattice interaction:
dipolarly coupled spins
simplest case: pair of spins fixed by distance r
Eklass  1z 2 z (1  3 cos2  ) / r 3
fluctuation of orientation () produces Bloc(t) driving
transitions in 4-level-system of spin pairs
T1 relevant: intensity of fluctuations at L and 2L
Bloembergen, Purcell, Pound theory (BPP)
1
 CDD[ J1 (L )  4 J 2 (2L )]
T1
(BPP)

J m ( )  1 / 2  Cm (t )eimt dt spectraldensity

Cm (t ) : time correlation fct.
isotropic liquid: C 2(t )  C1(t )  C0(t )  Y20 (0) Y20 (t )  / | Y20 (0) |2 
1/T1 probes spectral density Ji() at L=B0 and 2L
a noise probing experiment
correlation function CAA(t)
describes equilibrium
fluctuations of the quantity A
stationary process:
T
CAA( )  A(0) A( )   limT  1 / T  A(t ) A(t   )dt
0
lim   A(0) A( )  A(0)  A( )  A 2
 A(0) A(0)  A2 
FAA ( )  A(0)A( ) /(A2 ) with A  A(t )  A 
normalizedcorrelation fct.(  t)
correlation function, spectral density,
susceptibility

J AA ( )  1 / 2  FAA (t )eit dt spectraldensity

a measure of the fluctuation of A with frequency 
0
 AA"()   AA
 J AA () / kT susceptibility
FAA (t ), J AA ( ) and  AA" ( )
equivalentformsof describing fluctuations
What is a susceptibilty?
relaxation experiment:
equilibrium disturbed by applying AC field and response
measured
field :
ˆF(t )  F e it
0
response:
Bˆ (t )  ˆ () Fˆ (t )
measured:
B  Re[ˆ ( ) F (t )]
 F0 [  ( ) cost   ( ) sin t ]
ˆ ()   ()  i ()
fluctuation-dissipation theorem
linear response theory
0


 AA()   AA  J AA() / kT
dissipation
(with exciting field)
fluctuations in equilibrium
(without field)
dW
  ( )
dt
 :  static susceptibility
0
AA
some important properties
FAA(t  0)  1
„normalized“


0
0
JAA(  0)  lim0  FAA (t ) cos t dt   FAA (t ) dt   AA

J
AA
( )d 
0

 [  () / 
AA

0
AA
„time constant“

2
] d ln  

2
simplest case: „Debye relaxation“
solution of rotational diffusion equation
FAA (t )  e
t / 
with (T )

JAA( ) 
1  ( ) 2
AA" ( ) / 
0
AA

 JAA( ) 
1  ( ) 2
0
1
10
J()
 = 10
=1
-1
 = 0.1
10
 = 1  = 0.1
 = 0.01
 = 10
0
10
10
-1
10
"()
-2
10
-2
10
 = 0.01
-3
10
-3
10
-3
10
-2
10
-1
10
0
10

1
10
2
10
3
10
-3
10
-2
10
-1
10
0
10
1

10
2
10
3
10
4
10
dielectric spectroscopy probing susceptibility
 ()   1" ()
related to reorientational correlation function C1(t)
3
10
glycerol
0
 / s
10
DS
NMR Bayreuth
NMR Darmstadt
LS
-3
10
-6
10
-9
10
-12
10
200
250
300
T/K
350
400
frequency-temperature superposition (FTS)
spectral shape is independent of  or T
0



F (t )  F (t /  ) or  () /  A  f ( )
0
10
glycerol, Tg = 186 K
-1
'' scaled
10
-2
10
-3
10
DS: 290 K - 184 K
-4
10
-5
10 -2
10
0
10
2
10
4

10
6
10
8
10
feature commonly observed for glassy and polymer dynamics
1H
NMR in polymers: probing fluctuation of
bond vector
segmental correlation function C2(t)

 
1
   b (0)b (t ) 2 eiwt dt
T1 
 
 b (0)b (t )  
N 1
2
p
 C p (t )
p 1
polybutadiene ( PB)
principle of field cycling NMR
Larmor frequency becomes
variable  =  Bo
Mz
polarization
detection
relaxation
Mz
T1
t
Mz
T1
t
T1
relaxation T1
Mz
T1
frequency
M0(Bp)
magnetization
Mz()
Bp
Baq
B-field
Br

p

90
o
FID
RF
Mz() = M0(Br) + [M0(Bp) – M0(Br)] exp {- 
/T1(Br)}
Bp
Bd
Br
90o
Tx
Acq
switching time of Bo ~ 1.5 ms
Bo/2 =  = 10 kHz – 20 MHz
FFC 2000 relaxometer (STELAR)
operating in Bayreuth since 2005
basic FFC sequence
transforming to susceptibility representation
"
 / T1   NMR
()   ()  2 (2) with  ()  J ()
T[K]
393
373
363
353
343
333
323
313
303
296
283
273
263
OTP
8
10
0
10
10 T1/ s
7
-1
10

-2
"NMR / T1 [s ]
T [K]
6
10
-2
10

5
10

-3
4
10
10
5
4
10
10
4
10
7
6
NMR
10
10 susceptibility
 / Hz
- a simple6 glassformer
o-terphenyl (OTP)
5
10
 [Hz]
10
7
10
263
273
283
296
303
313
323
333
343
353
363
373
393
similar relaxation behavior as revealed by
dielectric spectroscopy
7
10
6
10

5
10
4

NMR susceptibility
10
4
10
6
5
10
 [Hz]
10
7
10
263
273
283
296
303
313
323
333
343
353
363
373
393
0
10
144
145
147
149
151
153 155 157
158
-1
10
146
148
-2
10
150
152
''
OTP
8
10

-2
"NMR / T1 [s ]
T [K]
154
156
-3
10
-4
10
DS
PDMS21600
-5
10
-4
10
-2
10
0
10
2
10
 / Hz
behavior typical of glass formers
4
10
6
10
applying FTS yields
master curves
T[K]
263
273
283
296
303
313
323
333
343
353
363
373
393
"NMR( in Hz
2
8
10
7
10
6
10
5
10
OTP
4
10
4
10
5
10
6
 in Hz
10
10
''
0
10
-1
10
-2
10
-3
10
-4
10
-5


OTP
(263 - 393 K)
Kohlrausch Fit, K = 0.61)
-6
7
-5
-4
-3
-2
-1
0
1
2
3
4
10 10 10 10 10 10 10 10 10 10 10
10

5 - 6 decades in amplitude & frequency
accessible at  << 1
  1:
1


 ()   „simple liquid limit“
master curve yields time constant (T)
2
10
OTP (NMR)
OTP (other techniques)
0
10
-2
 [s]
10
tristyrene (NMR)
tristyrene (DS)
-4
10
-6
10
FFC NMR:
10-11 s – 10-6 s
NMR
-8
10
-10
10
2.5
3.0
3.5
4.0
-1
1000/T [K ]
agreement with results from other techniques
non-Arrhenius behavior typical of glass transition
crossover from simple liquid to polymer melt
10
''
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
PB355
-6
10
-5
10
-4
10
-3
10
-2

10
-1
10
0
10
1
master curves of a series of polybutadienes
crossover from simple liquid to polymer melt
10
''
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
PB355
PB466
-6
10
-5
10
-4
10
-3
10
-2

10
-1
10
0
10
1
crossover from simple liquid to polymer melt
10
''
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
OTP
GG fit
PB355
PB466
-6
10
-5
10
-4
10
-3
10
-2

10
-1
10
0
10
1
PB355 and PB466 show no polymer effect (like OTP)
crossover from simple liquid to polymer melt
10
''
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
OTP
GG fit
PB355
PB466
PB777
-6
10
-5
10
-4
10
-3
10
-2

10
-1
10
0
10
first polymer effect for PB777
1
crossover from simple liquid to polymer melt
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
''
10
OTP
GG fit
PB355
PB466
PB777
PB1450
-6
10
-5
10
-4
10
-3
10
-2

10
-1
10
0
10
1
crossover from simple liquid to polymer melt
10
''
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
OTP
GG fit
PB355
PB466
PB777
PB1450
PB2020
-6
10
-5
10
-4
10
-3
10
-2

10
-1
10
0
10
1
crossover from simple liquid to polymer melt
10
''
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
OTP
GG fit
PB355
PB466
PB777
PB1450
PB2020
PB2760
-6
10
-5
10
-4
10
-3
10
-2

10
-1
10
0
10
1
note:   s (segmental time) assumed
crossover from simple liquid to polymer melt
10
''
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
OTP
GG fit
PB355
PB466
PB777
PB1450
PB2020
PB2760
PB4600
-6
10
-5
10
-4
10
-3
10
-2

10
-1
10
0
10
1
crossover from simple liquid to polymer melt
10
''
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
OTP
GG fit
PB355
PB466
PB777
PB1450
PB2020
PB2760
PB4600
PB11400
-6
10
-5
10
-4
10
-3
10
-2

10
-1
10
0
10
1
increasing contribution at low frequencies (<<1)
crossover from simple liquid to polymer melt
10
''
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
OTP
GG fit
PB355
PB466
PB777
PB1450
PB2020
PB2760
PB4600
PB11400
PB56500
PB87000
PB314000
PB817000
-6
10
-5
10
-4
10
-3
10
-2

10
-1
10
0
10
1
saturation due to entanglement (M >> Me)
crossover from simple liquid to polymer melt
10
10
''
0
OTP
GG fit
PB355
PB466
PB777
PB1450
PB2020
PB2760
PB4600
PB11400
PB56500
PB87000
PB314000
PB817000
-1
10
-2
10
-3
10
-4
10
-5
high M
limit
M
10
1
simple
liquid
limit

-6
10
-5
10
-4
10
-3
10
-2

10
-1
10
0
10
1
dipolar correlation function obtained from
Fourier transforming master curves  ( )
0
0
10
I
t
log g (t)
glassy
-1
(2)
10
PB466
PB777
PB2760
PB4600
PB9470
PB11400
PB18000
CDD(t/s)
-2
10
-3
10
-4
10
-2
10
-1
10
-0.25
t
s
-0.5
e
R
d
log t
bimodal decay due to
entanglement and
terminal relaxation
(Kohlrausch)
-3
t
Rouse
terminal
relaxation
low-M
10
IV
high-M
-6
10
III
constraint
Rouse
-5
10
II
-1
0
10
1
10
2
10
3
10
4
10
5
10
t / s
6 vs 8 decades!
Rouse & entanglement
retarded relaxation
Fourier transforming NMR data to get
segmental correlation function
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
1x10
0
PB355
PB1450
PB355
PB1450PB11400
Rouse
PB11400
F2(t/)
simple
liquid
F2(t/)
PB 18800
PB 18800
bimodal shape
onset of
entanglement
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
t/
10
-2
10
-1
10
0
10
1
10
t/
2
10
3
10
4
10
5
glassy (local) and Rouse regime identified
but no crossover to reptation (regime II)
not observed at Z = M/Mc ~ 5
0
10
Rouse (I)
 t -1
glassy
dynamics (0)
-1
10
regime (II)
?
CDD(t/s)
-2
10
-3
PB466
PB777
PB2760
PB4600
PB9470
PB11400
PB18000
10
-4
10
-5
10
 t -0.25
 t -0.5
-6
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
t / s
crossover protracted, appearing only at M >> Mc?!
longer times/lower frequencies needed!
need for lower fields/frequencies
compensating earth and stray fields!
cooperation with Fujara, Privalov, Kresse (Darmstadt)
3
10
-1
1 / T1 [s ]
2
10
Darmstadt
1
10
compensation
BT
PB 87500, T = 223 - 393 K
0
10
2
10
3
10
4
10
5
10
6
10
7
10
 [Hz]
2 decades gained in frequency by homebuilt spectrometer
latest results from measuring at lower frequencies
0
10
T = 223 K - 408 K
PB355
PB466
PB777
PB816
PB1450
PB2020
PB2760
PB4600
PB9470
PB11400
PB18000
PB35300
PB56500
PB87000
PB314000
PB817000
-1
~''
[a. u.]
10
-2
10
new
measurements
-3
10
-4
10
-5
10
-6
10
-9
10
-7
10
-5
10
-3
10

-1
10
1
10
NMR (microscopic) vs.
rheology (macroscopic)
PB
T = 223 K - 408 K
-1
10
-2
DD'' [a. u.]
10
with
compensation
-3
M [g/mol]
355
466
777
816
1450
2020
2760
4600
9470
11400
18000
18200
10
0.32

-4
10
-5
10
1

-6
10
23600
24300
35300
47000
56500
87000
87500
143000
196000
314000
817000
982000
-7
10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
10 10 10 10 10 10 10 10 10 10 10 10
s
similar evolution with M
similar frequency range covered
extension to extremely low fields/long times
0
10
-1
10
-2
10
C
DD
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
-0.85
-1
10
0
10
-0.25
t
PB
466
2020
9470
11400
18000
24300
35300
47000
56500
87500
143000
196000
441000
t
-(M)
t
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
t/s
8 decades in amplitude, 10 decades in time
exponent (M) of regime II decreases further at high M
crossover to reptation highly protracted
exponent (M) of regime II
0.8
polybutadiene
this work
1
from DQ H NMR
0.6
agreement

with DQ NMR
0.4
(K. Saalwächter, Halle)
DE II limit
0.2
4
10
5
10
6
10
M [g/mol]
reaching (almost) Doi-Edwards (DE) limit only at M/Mc > 50
yet, there is a problem . . .
1H
NMR (dipolar coupling):
1, intra
1 / T11/T 1R~1, R
intra
 R1, inter
R1,intra: reorientation: C2(t)
R1,inter: translation
separation by isotope dilution:
PB-h6 in PB-d6
plus FC 2H NMR C2(t) ~ Cintra(t)
isotope dilution experiment
(a)
(b)
PB 24300
1.0
PDMS 21600
0.5
0.5
lg s = -7, -6, ..., -1
0
0
0.5
xH
1.0 0
lg s = -6, -5, ..., -2
0.5
xH
1.0
0
''(xH) / ''(1)
''(xH) / ''(1)
1.0
C(
-4
10
10dipolar-,
-5
-6
10
-7
10
2
H-NMR
22800-d6
(a)
intra- and inter-correlation function
from isotope dilution experiment
PB 24300, T = 228 - 393 K
100%  = 0.32
intra  = 0.49
inter  = 0.28
-1
10
C(t/s)
-2
10

-3
10
-4
10
-5
10
C2(t) ~ Cintra (t)
2
H-NMR
191000-d6
PB 196000, T = 223 - 393 K
0
1
2
3
4
5
6
7
8
10 10 10 10 10 10 10 10 10
t / s
exponent  increased in
intra correlation function
C2(t), no agreement
with Doi-Edwards theory
excursus: returning to simple liquids: glycerol
263 270
248
237
9
10
8
7
R1 = 1/T1 [s ]
10
-1
-2
·R1 [s ]
10
6
10
278K
283K
288K
296K
301K
306K
263K
268K
270K
273K
1000
360
100
glycerol
5
10
4
5
10
10
glycerol
10
10
6
7
 [Hz]
4
10
4
10
5
10
6
10
7
10
 [Hz]
low-frequency shoulder?
T [K]
360
350
340
330
320
310
301
288
278
270
263
248
237
"
NMR
() allows for comparing
NMR and dielectric spectroscopy(DS)
'' / 
208
203
213
246
223
262
273
-1
10
-2
10
1
FC H NMR
CD
-3
10
glycerol
1
10
3
10
DS
5
10
7
10
9
10
 / Hz
similar time constant
but NMR indicates a low-frequency shoulder
master curves
applying frequency-temperature superposition
9
10
shifting data along  axis
NMR
8
-2
/T1 [s ]
10
9
10
8
-1
T1 [s-2]
10
DS
7
10
6
10
7
10
glycerol-h8
6
5
10
10
200 - 400 K
CD fit
5
10
4
10
5
10
6
 [Hz]
10
7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
rot
0
10
1
10
indeed: low-frequency shoulder seen by NMR but not by DS
2
10
low-frequency shoulder: other examples
0
10
0
-1
master curves
10
-1
10
NMR()
-2
10
threitol
sorbitol
xylitol
3-fluoroaniline
propylene glycol
2,3-butandiol
m-toluidine
Debye
-2
10
''
"NMR · f
10
-3
-3
10
10
threitol
-4
10
-4
10 -4
10
-5
10
-4
10
-3
10
-3
10
-2
10
-2
10
-1
10
0
10
 10-1
rot
1
10
2
10
0
10
3
10
1
10
strength of low-frequency shoulder varying
hypothesis: low-frequency shoulder
originates from intermolecular 1H relaxation
9
10
intra
rotation
8
-2
/T1 [s ]
10
inter
translation
7
10
6
10
glycerol-h8
5
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
rot
0
10
1
10
2
10
translational and rotational contribution are spectrally separated
intra- and intermolecular 1H relaxation
R1  R1,intra  R1,inter
inter
intra
rotation
Cintra (t )  Cole  Davidson
"rot "
translation (+ rotation?)
D02,m  0 0D02,  m  t  t 
~
C inter t  
r 3 0r 3 t 
"trans "
ultimate proof: isotope dilution experiment
R1  R1,intra  x  R1,inter
9
10
glycerol-h5 in glycerol-d8
8
-2
/T1 [s ]
10
7
10
x=1
x=0.87
x=0.56
x=0.30
x=0.22
x 0
Cole-Davidson fit
6
10
5
10
4
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10

low-frequency shoulder disappears - as expected
intra- and intermolecular relaxation separated
9
10
8
-2
/T1 [s ]
10
rotation
translation
7
10
6
10
glycerol-h5
5
10
/T1
intra
intra
/T1
inter
inter
Cole-Davidson fit
4
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
rot
ρ
- inter /T1: contains translation and rotation
„eccentricity effect“
I
d  2r
I
- intra /T1: low-frequency contribution disappeared (CD fits)
low-frequency part of T1,inter modeled by
“force-free-hard-sphere model”
Fick diffusion with appropriate boundary conditions
spins centered in molecule
I
I
trans
 trans
d2

2D
d  2r

u 2 trans
3
u2
J inter    72 
du
2
4
6
2
4
trans
4 0 81 9u  2u  u u   trans 
J trans    J (0)  b trans
Hwang, Freed J. Chem. Phys. 1975
Ayant, Belorizky, Alizon, Gallice J. Phys. (Paris) 1975
low-frequency limit
(no extreme narrowing)
rotation and translation in a single experiment
glycerol
-9
-6
10
10
11
Ref.NMR
FG
-10
10
DS vs NMR
-8
glycerol-h5
-11
10
glycerol-h3
2
[s]
D [m /s]
10
glycerol-h8

rot
glycerol
-10
10
-12
10
-13
10
rotation
translation
-14
10
-12
10
2.5
3.0
3.5
1000/T [1/K]
4.0
4.5
2.5
3.0
3.5
4.0
4.5
1000/T [1/K]
agreement with DS and field-gradient (FG) NMR, respectively
facit: low-frequency dispersion allows
discriminating between translation and rotation
J trans    J (0)  b trans

J rot,Debye     rot 1  rot 
2
m
R1    R1 0   3 / 2 
D

with m  N
1.0
rotation (Debye)
J()/J(0)
0.8
0.6
0.4
translation
D model independently
accessible
rotation (CD)
0.2
0.0
0.0
0.5
1.0
()
1.5
1/2
2.0
experiment
1H NMR (total) vs. dielectric spectroscopy
R1(rot) / R1(0)
1.0
glycerol
0.8
dielectric loss
(pure rotation)
0.6
0.4
0.2
1
H NMR (total)
(translation & rotation)
0.0
0.4
0.2
0.0
(rot)
0.6
0.8
1.0
1/2
 1H NMR (total) dominated by translation
experiment
1H NMR (total & intra) vs. dielectric spectroscopy
R1(rot) / R1(0)
1.0
glycerol
0.8
R1(intra)
DS
0.6
0.4
0.2
1
H NMR
(translation & rotation)
0.0
0.0
0.2
0.4
(rot)
0.6
0.8
1.0
1/2
 1H NMR (intra) dominated by rotation
analyzing low-frequency dispersion yields D(T)
1000
100
glycerol-h5
80
800
600
40
20
0
400
0
1000
2000

1/2
3000
4000
1/2
[Hz ]
200
R1 (1/ 2 )
-10
0
1000
2000

1/2
[s
3000
-1/2
]
4000
glycerol-h5
-11
10
2
0
fitting linear part of
10
D [m /s]
-1
R1 [s ]
-1
R1 [s ]
60
273K
275K
278K
283K
288K
298K
308K
328K
348K
-12
10
-13
10
from low  behavior
FG NMR
3.0
3.5
-1
1000/T [K ]
FC 1H NMR compared to field-gradient NMR
diffusion coefficient D(T)
glycerol h8,
-9
10
xylitol
DL-threitol
o-terphenyl,
2-ethyl-1-hexanol
propylene glycol
2,3-butanol
sorbitol
small symbols: FG NMR
-10
10
-11
2
D [m /s]
10
-12
10
-13
10
-14
10
h5,
2.5
3.0
3.5
4.0
-1
1000/T [K ]
4.5
h3
universal low-frequency limit of T1 dispersion
allows for new master curve (1H NMR)
1/ 2

R1 / R1(0)  1  a ( )
1.0
1.0
xylitol
sorbitol
DL-threitol
glycerol-h8
0.8
R1() /R1()
glycerol-h5
glycerol-h3
0.6
propylene glycol
2,3 butanol
o-terphenyl
tristyrene
0.4
H NMR
polybutadiene
0.6
M [g/mol] =
4800
2760
2020
1400
0.4
0.2
0.2
0.0
0.0
R1() /R1()
0.8
1
0.5
1.0
a · (')
1.5
1/2
2.0
0.0
0.0
0.5
1.0
a (')
1.5
1/2
higher frequencies: interplay of rotation and translation
2.0
square root dispersion law
reflects long-time power-law t-3/2 in time domain
characteristic of free diffusion
0
10
M=
PDMS
-1
860
5940
41400
10
-2
CDD(t)
10
propylene
glycol
-3
10
-4
10
-5
10
-6
10
-7
10
-1
10
0
1
2
3
4
5
6
7
8
10 10 10 10 10 10 10 10 10
t/s
diffusion coefficient in polymers
-9
10
Mc = 2300
-10
10
-1.2
T = 100°C
-11
10
2
D [m /s]
~M
-12
10
polybutadiene
-2.3
~M
-13
10
Fleischer & Appelt 1995
FG NMR
-14
10
100
1000
10000
M [g/mol]
crossover Rouse to entanglement dynamics identified
agreement with field gradient NMR
conclusions
• CDD(t), Cintra(t) & Cinter(t) covered over ten decades in time,
eight in amplitude, not achieved by any other technique
• C2(t) ~ Cintra(t) at variance with tube-reptation model
• intermolecular relaxation must not be ignored
- dominates at low-frequency
- universal dispersion law
- simple determination of D (and rot)
•
alternative to FG NMR
“molecular rheology” by FC NMR