sub- and superdiffusive displacement laws in disordered media probed by nmr techniques
DESCRIPTION
Sub- and superdiffusive displacement laws in disordered media probed by NMR techniques Rainer Kimmich, Yujie Li, German Farrher, Nail Fatkullin, Markus Kehr Sektion Kernresonanzspektroskopie, Universität Ulm, Germany. R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000). Outline. - PowerPoint PPT PresentationTRANSCRIPT
Sub- and superdiffusive displacement lawsSub- and superdiffusive displacement lawsin disordered mediain disordered media
probed by NMR techniquesprobed by NMR techniques
Rainer Kimmich,Rainer Kimmich,Yujie Li, German Farrher, Nail Fatkullin, Markus Kehr Yujie Li, German Farrher, Nail Fatkullin, Markus Kehr
Sektion Kernresonanzspektroskopie,Sektion Kernresonanzspektroskopie,Universität Ulm, GermanyUniversität Ulm, Germany
2 2 ( )r r t
R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)
2r t
1: subdiff usive displacements
1: superdiff usive displacemen
0 : localized position
1: normal diff usion
2 : ballistic displacements
3 : turbulent displac
ts
ements
OutlineOutline
• perspectives of NMR techniques to measure <r2(t)> over many orders of magnitude of time
• Systems showing anomalous transport properties(fluids in confining geometries, porous media, polymer melts in bulk)
• Examples: polymer dynamics, hydrodynamic dispersion in porous media
G
(/2)x
time
Stim. Echo
1 1
(/2)x (/2)x
2
21 2 1 2
2
Attenuation of the stimulated echo
1exp ( ) exp 2
6S G Tr
B0maximumgradient
z
200 MHzGo = 60 T/m
NMR diffusometry in the fringe field NMR diffusometry in the fringe field of a superconducting magnetof a superconducting magnet
fringe field
Magnet
9.4 T magnet89 mm bore
Rapid MAGROFI DiffusometryRapid MAGROFI Diffusometry(magnetization grid rotating frame imaging)(magnetization grid rotating frame imaging)p
tprep. diffusion comp. imaging
op
B. Simon, R. Ki., H. Köstler, J. Magn. Reson. A 118 (1996) 78
Mz mapsafter
FT
AQ
p
B1 gradients (radiofrequency field) instead of B0 gradients:
x y z( )
4.5
mm
5 m
m
12
mm
6 mm
8 mm
sample
10-1 100 101 102 103
1,0x10-9
1,5x10-9
2,0x10-9
2,5x10-9
3,0x10-9
Bulk Water VitraPor #5 Bulk Water VitraPor #5
<r2 >
/6t
(m2 /s
)
t (ms)
MAGROFI
FFStE
Combination of fringe-field with rotating frame NMR diffusometryCombination of fringe-field with rotating frame NMR diffusometry(or likewise with the pulsed gradient spin echo (PGSE) variant)(or likewise with the pulsed gradient spin echo (PGSE) variant)
water in VitraPor (10-6 m pore size)
NMR relaxometrydue to intermoleculardipolar interactions
four decades of time
NMR imagingof interdiffusion of
isotopically labeled molecules
A. Klemm, R. Metzler, R. Ki.,Phys. Rev. E 65 (2002)
021112-1
InterIntermolecular interactions molecular interactions and relative displacementsand relative displacements
x
y
z, B0
molecule
molecularmotion
homonuclear dipole-dipole couplinghomonuclear dipole-dipole couplingdominates for dominates for I = 1/2 I = 1/2 (e.g. protons)(e.g. protons)
1
2
r
pair of nuclear dipoles
Spin-lattice relaxation by molecular motionsSpin-lattice relaxation by molecular motions
1
2
““inter”inter”
““intra”
intra”
““intra”: reorientationsintra”: reorientations““inter”: relative inter”: relative translationstranslations
z‘
y‘
x‘
r‘(0)
r‘(t) r‘(t)
k
l
*2, 2,( )
3 3
I ntermolecular
dipolar interactions:
correlation f unction
of the dipole pair ,
( ) (0) ( )
( ) (0)m mm
kl
k l
Y t YG t
r t r
3
probability that dipole
is still in ' (0)
around its initial position
l
V r2 2
mean squared displacement
relative to dipole
1
2)( (' )
k
r r tt
Evaluation of the relative intermolecular mean square displacement Evaluation of the relative intermolecular mean square displacement from field-cycling NMR relaxometry datafrom field-cycling NMR relaxometry data
32 4 2
spin0inter total intra
1 1 1
2 / 32 4 2 inter
/ 22
s n 1
0
2 pi
1 1 1 2 3 1 2 2
4 5
2 3 1 2
'
'24
5
1
r
r
T T T
T
t
• spin-lattice relaxation by dipolar coupling of protons • distinction of intra- and inter-molecular contributions • separable by mixtures of deuterated and undeuterated molecules
dilute solution of undeuterated molecules in deuterated matrix
undeuterated species
variation of the angular frequency 0B
F
0
1
1
f requency:
rel ( ) 4 (2 )
ax. rate:
spectr. dens
(
) ( ).:
t
B
C I IT
I G t
2 23 3
2 2
(0) ( )( )
(0)
dipolar correlation f unction:
quadrupolar corr. f unc
( )
( ) (
tio
( )
n
0)
:
m m
m m
Y Y tG t
r r t
G t Y Y t
Field-cycling NMR relaxometryField-cycling NMR relaxometry
3 8
2 7
ω10 Hz < <4×10 Hz2π
ω10 Hz < <6×10 Hz2π
1
2
H
H
t
relaxation
detection
polarization
t
B0/T0.5 ... 1.5
0~ms ~ms
RF
~s
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100
10-18
10-17
10-16
10-15
polyethylene oxideM
w= 460 000, T = 355 K
~ t 0.4
(renormalized Rouse model)
<r2 >
[m
²]
t [s]
fringe fieldNMR
diffusometry
field-cyclingNMR
relaxometry
IntraIntramolecular spin-lattice relaxationmolecular spin-lattice relaxationby chain modesby chain modes
also reflects the mean squared displacement behavioralso reflects the mean squared displacement behavior
three different model theories for polymer chain modespolymer chain modes three different experimental scenariosthree different experimental scenarios:
• Rouse modelRouse model (chain in a viscous medium; no hydrodynamic backflow; no “entanglements”, i.e. M < Mc)
• Renormalized Rouse formalismRenormalized Rouse formalism (“entanglements”, i.e. M > Mc, t << terminal)
• Tube/reptation conceptTube/reptation concept (chains confined in nanoscopic “tubes”)
Rouse modelRouse model: : Bead-and-spring chain in a viscous medium without Bead-and-spring chain in a viscous medium without backflowbackflow
2
21 1
equation of motion f or the -th bead:
02 nn n n
nn
nn
r r
n
rr r r
n tF FK K
t
2(entropic spring const. ; f rict 6ion coeff . ; random f orce3
) h nB
ba
kK F
T
Solution: Superposition of discrete Rouse relaxation modes with time constants
P. E. Rouse, J. Chem. Phys. 21 (1953) 1272
2 2
2 2 , where 13p
B
b Np N
kTp
nr
0x1nr
1nr
2 ha
2Kuhn length b x(M<Mc)
I. Laws for NMR measurands predicted by the Rouse model:
(polymer melts; M < Mc ; no “entanglements”)
T. N. Khazanovich, Polymer Sci. USSR 4 (1963) 727N. Fatkullin, R. Ki., H. W. Weber, Phys. Rev. E 47 (1993) 4600
( local segment fl uctuation time;
longest Rouse relaxation time)s
R
R
21 2
1 2
1 21
R
1 1 f or
1 1 1ln f or
1
1ln
ss s
s
C C
C C MT T
Trelaxation:
2 0 1/ 2
2 1
f or
f or
R
R
sR M t t
R M t t
diffusion:
II. Laws for NMR measurands predicted by the renormalized Rouse formalism:
(polymer melts; M > Mc ; “entanglements”; t << tterminal)
relaxation:
2 0 1/ 4
2 0 1/ 3 2/ 5
f or / 6 "high-mode number limit"
f or / 6 "low-mode number limit"
R M t p N
R M t p N
diffusion:
0 1/ 21 f or / 6 "high-mode number limit"( )s p NT M
0 1/ 5...1/ 31 f or / 6 "low-mode number limit"( )s p NT M
d
s
R
ed
III. Tube/reptation concept by Doi and Edwards
(definition of 4 characteristic time constants)
(I)DE
(II)DE
(III)DE
(IV)DE
limitsmean squared
segment displacementspin-lattice
relaxation time
,1/ es t
,1/e Rt
,1/R dt
,1/d
t
01 / ln( )sT M
0 3/ 41T M
1/ 2 1/ 21T M
01T M
2 0 1/ 2R M t
2 0 1/ 4R M t
2 1/ 2 1/ 2R M t
2 2 1R M t
special evaluation formalism needed!(N. Fatkullin, R. Ki., Phys. Rev. E 52 (1995) 3273)
Laws for NMR measurands predicted by the tube/reptation concept:
(polymer melts confined in mesoscopic pores)
Rouse
(I)DE
(II)DE
(III)DE
1/ 2( )t
1/ 4( )t
1/ 2( )t
crossover from ““Rouse”Rouse”
totoreptationreptation
chain dynamicswith decreasing tube diameter
A.Denissov, M.Kroutieva, N.Fatkullin, R. Ki.,J. Chem. Phys. 116 (2002) 5217
a) harmonic radial a) harmonic radial potential theorypotential theory
b) and Monte Carlo b) and Monte Carlo simulations of a simulations of a modified Stockmayermodified Stockmayer chain model in a chain model in a tube with hard walls )tube with hard walls )
1/T c
Rouse
reptation 3/ 4 1 1( )R e
s
experimental juxtapositionexperimental juxtaposition
of the three model scenariosof the three model scenarios
• M < MM < Mcc , bulk: , bulk: scen. Iscen. I• M > MM > Mcc , bulk: , bulk: scen. IIscen. II• M M arb., arb., confinedconfined:: scen. IIIscen. III
10-3 10-2 10-1 100 101 102 10310-3
10-2
10-1
100
T1 (s)
(MHz)
Rouse
bulk PEO 2000, Mw< Mc
Field-cycling NMR relaxometry at Field-cycling NMR relaxometry at 85°C85°C
10-3 10-2 10-1 100 101 102 10310-3
10-2
10-1
100
T1 (s)
(MHz)
Rouse
bulk PEO 2000, Mw<Mc
Ren. Rouse
bulk PEO 10 000Mw>Mc
Field-cycling NMR relaxometry at Field-cycling NMR relaxometry at 85°C85°C
1/ 5...1/ 31T
100 nm
100 nm
1 m
Linear polyethyleneoxide (PEO; MLinear polyethyleneoxide (PEO; Mww=6000) =6000) in solid cross-linked polyhydroxyethylmethacrylate (PHEMA)in solid cross-linked polyhydroxyethylmethacrylate (PHEMA)
TEM, replicaTEM, replica
pore width 10 nmpore width 10 nm
E. Fischer et al., Macromolecules 37 (2004) 3277
polymer melts confined in porespolymer melts confined in pores
10-3 10-2 10-1 100 101 102 10310-3
10-2
10-1
100
T1 (s)
(MHz)
Rouse
bulk PEO 2000, Mw<Mc
Ren. Rouse
bulk PEO 10 000Mw>Mc
Field-cycling NMR relaxometry at Field-cycling NMR relaxometry at 85°C85°C
PEO 2,000 to 10,000confined in nanopores from 8 to 60 nm
2 0 1/
1
4
0 3 / 4
corresponding to
reptation limit (II)
w
DE
e R
M t
t
T M
r1/ 5...1/ 3
1T
Evaluation of “tube” diameter effective on time scale 10Evaluation of “tube” diameter effective on time scale 10-9 -9 ... 10... 10-5-5 s: s: 0.6 nm0.6 nm
bulk
confi ned
103 104 105 106 107 108 109
10-2
10-1
PFPE in Vycor (4 nm)T = 313 K
T1~ 0.5
T1(s)
(Hz)
Mw= 1850
Mw= 2450
Mw= 3000
Mw=11 000
103 104
10-2
10-1
PFPE in Vycor (4 nm)T = 313 K
= 9.4 MHz = 4.3 MHz = 2.3 MHz = 0.46 MHz = 0.094 MHz
T1~ M
W
-0.5
T1(s)
Mw
DE
reptation
limit (I I I )
R dt
1/ 2 1/ 21
2 1/ 2 1/ 2
corresponding to
w
w
T M
r M t
Hydrodynamic dispersionHydrodynamic dispersion
Péclet number
1010 m/sinletv 910 m/sinletv 810 m/sinletv
simulation of hydrodynamic dispersionsimulation of hydrodynamic dispersion
magnetpolarizationbuffer comp.
poroussample
water supply
waterreservoir
2.4 l
HPLCpump
experimental set-upexperimental set-up
hydrodynamic dispersion:hydrodynamic dispersion:measurement of incoherent displacements measurement of incoherent displacements while coherent flow velocity is compensatedwhile coherent flow velocity is compensated
total displacement time: 2t
Δ Δ
90o 90o90o90o
τm
echo
t
90o
field gradient pulses
10-2 10-1 100
10-10
10-9
10-8
flow rate in ml/min 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
<Z
2 > /m
2
t / s
2 1.95z t
2 0.84
2.38
1.6w
f
z t
d
d
water flowing through VitraPor (10-4 m pore size)
SummarySummary
• NMR variants promise access to mean squared displacements in the time range 10-10 s … 100 s (... and beyond: magnetic resonance imaging of interdiffusionof isotopically labeled molecules)
• hydrodynamic dispersion shows cross-over from sub-to superdiffusive behavior with increasing Pe
• chain dynamics under mesoscopic confinement
reveals characteristic laws of the tube/reptation model
the group in summer …the group in summer …
… … and in winterand in winter
recent collaborators:recent collaborators:
Esteban AnoardoEsteban AnoardoIoan ArdeleanIoan ArdeleanBogdan BuhaiBogdan BuhaiGerman FarrherGerman FarrherNail FatkullinNail FatkullinElmar FischerElmar FischerRavinath KausikRavinath KausikMarkus KehrMarkus KehrElke KosselElke KosselRavinath KausikRavinath KausikYujie LiYujie LiCarlos MatteaCarlos Mattea……
Funding:Funding:
Deutsche ForschungsgemeinschaftDeutsche ForschungsgemeinschaftAlexander-von-Humboldt FoundationAlexander-von-Humboldt FoundationVolkswagen FoundationVolkswagen Foundation
Low-frequency surface relaxation: Low-frequency surface relaxation: Reorientation mediated by translational displacements (RMTD)Reorientation mediated by translational displacements (RMTD)
B0
initial final
reorientation determined bya) translational diffusionb) surface topology
02 0
0 02
2
4
22( )
123 18t
b D tD ts t
D t D tNN b
d
Nb
dt tube diameter(b , N, D0 known)
• mean square curvilinear segment displacements mean square curvilinear segment displacements
rs
s
2,segm
exp -center-of-mant ss se
, s c
s
s
c
c
iik r t
r
E k
k r t
r
k
ik
t
r
Dt
r tE k ee et
(wave vector )k G
NMR diffusometry and the tube/reptation conceptNMR diffusometry and the tube/reptation concept
2
ave
average over all
rage over all for a
243 / 2
giv
2
3
2
en
2 (4, exp erf
)(c
3 2 6 2
)
7s s
s
t tik tr rs
d s
s
ts
r
s
t s tsdd r
kk t dE dsk t e e
• anomalous segment diffusionanomalous segment diffusion
1s 1s
119.8 10 ss
1sT
/Hz
91.4 10 ss
1sT
/Hz
PIB, 4700,
357 KwM
T
PDMS, 5200, 293 KwM T
meltsM<Mc
““Rouse”Rouse”1
s
T
310 K
T
PIB, = 90 MHZ
/Hz
1sT
415% PDMS + 85% CCl ,
423 000, 293 KwM T
129.7 10 ss
1s
conc. solution
0.0 0.5 0.0 0.251 1
"high mode numbers" "low mode numbers"
region I : region I I : w wT M T M
1,
Hz
1 1,
s
T T
spin-latticerelaxation dispersionof polyisobutylenemelts Mw>Mc
(H.W. Weber, R. K., Macromolecules (26 (1993) 2597)
““RenormalizedRenormalized Rouse”Rouse”
1
s
T
Hz
spin-latticerelaxation dispersion
of polyethylene oxide melts
(R. K., N. Fatkullin, R.-O. Seitter, K. Gille, J. Chem. Phys. 98 (1998)
2173)
0 0.25 0 0.451 1
high mode numbers intersegment dipolar couplings
region I I : region I I I : w wT M T M
protons
deuterons
III
II1
s
Tpolyethyleneoxide
protons
deuterons
II
III
1
s
Tpolybutadiene
protons: intra- and intersegment interactionsdeuterons: only intrasegment interactions
R. Ki., N. Fatkullin, R.-O. Seitter, K. Gille, J. Chem. Phys. 98 (1998) 2173
102 103 104 105 106 107 10810-3
10-2
10-1
100
PEO-d4 (entangled)deuteron relaxation at 80oC
Mw = 7400
Mw = 17300
Mw = 43200
Mw = 460000
T1
/ s
/ Hz
102 103 104 105 106 107 10810-4
10-3
10-2
10-1
T2
PEO-d4 in porous PHEMA,deuteron relaxation at 80oC
Mw = 7400
Mw = 17300
Mw = 43200
T1
/ s
/ Hz
polymers confinedin pores
melts in bulk(“entangled“ polymers)
R. Ki., R. O. Seitter, U. Beginn, M. Möller, N. Fatkullin, Chem. Phys. Letters 307 (1999) 147
diffusometry, transverse relaxation,residual spin couplings
field cycling relaxometry
10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9
s
10-1
102 103 104 105 106 107 108 109 rad Hz
101
conv. relaxometryrot. frame relax.
NMRNMR
mobile linear polyethylene oxide:mobile linear polyethylene oxide:
PEO 2,000: PEO 2,000: RRF F = 4 nm= 4 nm
PEO 10,000: PEO 10,000: RRF F = 9 nm= 9 nm
nearest neighbor distancenearest neighbor distance0.5 nm0.5 nm
The corset effectThe corset effect
rigid crosslinked HEMA+DMA methacrylate matrix:
pore diameters from 8 to 60 nm
that is:… up to 122 PEO diameters… up to 15 PEO Flory radii
“tube”effective
for relaxation
random coilfor Mw=1665
(RF=N1/2b)
60 nm
“tube” effectivefor diffusion
FC-relaxometry and length scales:FC-relaxometry and length scales:
• polymer dynamics polymer dynamics the corset effect the corset effect
• surface relaxation mechanismssurface relaxation mechanisms the flow-relaxation effect the flow-relaxation effect
Crossoverto “bulk” ?
flow
Does flowinfluence T1 ?
0 1/ 21
high-mode number limit ( / 6 )
("regi
( )
on I ")sT M
p N
0 1/ 5 1/ 3
1
low-mode number limit ( / 6 )
("regio
(
n I I ")
)sT M
p N
spin-lattice relaxationspin-lattice relaxation
2
2 2
memory
f uncti
m
on
( ) random f orce3 ( ) entropic spring f orce
( )
( )
n
B n
n
nm
F tkT r tb n
r tt
t
0
atrix/ entanglement eff ects
( )f riction
tm
m
r t dt
Renormalized Rouse formalismRenormalized Rouse formalism
Rouse GeneralizedGeneralized
LangevinLangevinequationequation
diffusiondiffusion
2 0 1/ 4
high-mode number limit( / 6 )
R M t
p N
2 0 1/ 3 2/ 5
low-mode number limit( / 6 )
R M t
p N
103 104
10-2
10-1
PFPE in VycorT = 313 K
9.4MHz 4.3 MHz 2.34 MHz 0.46 MHz 0.094 MHz
T1~ M
W
-1/21/2
T1(s)
Mw
subdiffusive anomalous diffusion: subdiffusive anomalous diffusion: 2 ( 1)r t
R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)
Lévy
Brown
a) “(mutual) obstruction effect”;Gaussian propagator, D=D(t) (e.g. single-file diffusion in zeolites,Rouse mode based diffusion)
a) “trapping effect”;non-Gaussian propagator; waiting time distribution due to “traps”(e.g. random walk on fractals, reptation)
reptation:
“trapping effect” non-Gaussian propagators special evaluation theory for spin echo attenuation required! Elmar Fischer
Low-frequency surface relaxation: Low-frequency surface relaxation: Reorientation mediated by translational displacements (RMTD)Reorientation mediated by translational displacements (RMTD)
B0
initial final
reorientation determined bya) translational diffusionb) surface topology
2
2
f avorable if
a) short 100 s
b) small 100 nm
T
R
su
pe
rco
nd
uc
ting
co
il
su
pe
rco
nc
utin
gc
oil9.4 T,
400 MHz,10-5 T/m
damping buffers
4.7 T,
200 MHz,
60 T/m
sampleand RF coil
NMR diffusometry in the fringe field of a superconducting magnetNMR diffusometry in the fringe field of a superconducting magnet
0 1x1015 2x1015 3x1015 4x1015 5x1015 6x1015 7x1015
0.1
1PEO 11,200
in PHEMA at 353 K
Ediff
(k,t)
k2 [ 1/m2 ]
t / ms 10 15 30 60
echo attenuation formalism:(N. Fatkullin, R. Ki., Phys. Rev. E 52 (1995) 3273)
typical echo attenuation curves measured in typical echo attenuation curves measured in linear PEO (linear PEO (MMww=11,200) confined in PHEMA pores at 80°C=11,200) confined in PHEMA pores at 80°C(fringe field technique; 60 T/m; 200 MHz)(fringe field technique; 60 T/m; 200 MHz)
k G
1 1 fitting parameter:fitting parameter: pore diameter pore diameter ddporepore = (8+/-1) nm= (8+/-1) nm
( ) ( )sin( )x xS k x k x dx
“k-space signal” recorded as the FID amplitude immediately
after the B1(x) pulse “nutation frequency encoding”
p=kxx
B1
G1
B1
G1
B1
G1
B1
G1p
Rotating frame imaging:Rotating frame imaging:
( ) ( ) ( )x xkS x S k x F
FT
one-dimensional
imagex
pseudo-FID
p
1wavenumber x p
Bk
x
Rapid Rotating-Frame ImagingRapid Rotating-Frame Imaging
t
t
P. Maffei et al., J. Magn. Reson. A 107 (1994) 40K.R. Metz et al., J. Magn. Reson. B 103 (1994) 152
“stroboscopic acquisition”
Spin Echo
Stimulated Echo
2
2 22 1
6
SE
Tr G
S e e
1 2
2 1
2 2 21
6
STE
T Tr G
S e e e
2 1T T
G G
(/2)x
time
()x
Spin Echo
G G
(/2)x
time
Stimulated Echo
1 1
(/2)x (/2)x
2
Pulsed-gradient spin echo techniques Echo attenuation factors
Conventional field-gradient NMR diffusometryConventional field-gradient NMR diffusometry
2 21e.g.2 2
a) Gaussian propagator
with time dependent di
ff usion coeffi ci t
en :
Field- gradient NMR diff usometry
for anomalous diff usion:
e
ki tedif
Z
fA t e f Z e
("wavenumber" )
... special f ormalism (e.g. f or reptation)
... l
b) non-Gaussian propagato
ow wavenumber limit ( )
r
1
:
k G
kZ G Z
2
12 2 3 32
1 1 1
1! 2! 3! 2
(initi al decay slo pe)
kZ
ikZ ikZ k Z ik Ze k Z
x
y
z, B0
molecule
molecularmotion
nuclear dipole-dipole couplingnuclear dipole-dipole couplingdominates for dominates for I=1/2 I=1/2 (e.g. protons)(e.g. protons)
1
2
r
pair of nuclear dipoles
x
y
z, B0
FGT(molecule) I
QT(nucleus)
molecularmotion
nuclear quadrupole couplingnuclear quadrupole couplingto electric field gradientsto electric field gradientsfor for I>1/2 I>1/2 (e.g. deuterons)(e.g. deuterons)
a) Time dependent diffusion coefficientD=D(t) in a homogeneous medium (mutual obstructionobstruction)GaussianGaussian propagatorexample: single-file diffusion in straight cylindrical pores,
Rouse mode based diffusion
b) Diffusion under geometrical restrictions (waiting time distribution due to “trapstraps“)non-Gaussiannon-Gaussian propagatorexamples: reptation, random walk on fractals
2 classes of anomalous diffusion: 2 1r t
dtube ~ 0.6 nm(NMR relaxometry, 10-9 … 10-4 s)
dpore~ 8 … 60 nm(NMR diffusometry, 10 … 300 ms)
experimental findingsexperimental findingsfor different time scalesfor different time scales
the “corset effect”:
pore walls are sensed over more than 60 chain diameters or more than 7 Flory radii !
C. Mattea et al., Appl. Magn. Reson. 27 (2004) 371N. Fatkullin et al., ChemPhysChem 5 (2004) 884 and NJP 6 (2004) 46
polymer meltspolymer melts
many-chain problem
“tagged chain“ in a “matrix“
tagged chain1 2 3
m
n
N-1
N0
nr
mr
0
1
flow velocity map of a random-site percolation model objectrecorded with a NMR velocity mapping technique
6 cm
6 cm
polymer dynamics in general: local segment (and sidegroup) fluctuations
+ chain modeschain modes
+ global chain displacements
techniques to probe chain modes:
field-gradient NMR diffusometrydiffusometry + field-cycling NMR relaxometryrelaxometry
102 109NMR relaxometry /Hz
300
chain modes
(comp. B)
400center-of-massmotions(comp. C)
T/ K Mw104105
segmentfluctuations
(comp. A)
the corset effect - a finite size phenomenonthe corset effect - a finite size phenomenon
conformational changes require fluctuations of the free volume ~ fluctuations of the number of segments in the available volume
2 2 TmBn n k T n
compressibility
<n> small segments can only be displaced along the contour line of the chain
2tube m B Td b k T 10
1/33
pore F FB T
bd R Rk T
effective tube diameter bulk dynamics for
N. Fatkullin, R. Ki., E. Fischer, C. Mattea, U. Beginn, M. Kroutieva, New J. Phys. 6, 46 (2004)
102 103 104 105 106 10710-3
10-2
10-1
T1 ~ 0.5
con-fined
bulk
Time spent dipped in the polymer
12 hours 2 days 10 days
Rouse
PFPE in VycorM
w = 11000
T = 313 K
T1 (s)
(Hz)
solenoid
B1
< 1 mm
BB11 gradients gradients
strong gradients:… thin coils… high-power transmitters
coniccoil
B1
6 mm
G1 = 0.3 T/mo = 400 MHz
x y z( )
4.5
mm
5 m
m
12
mm
6 mm
8 mm
Larger confinementsLarger confinements
(i.e. (i.e. ddconfconf ~ 10 R ~ 10 RF F ~ ~ m)m)
crossover to bulk dynamics ?crossover to bulk dynamics ?
M. H. Sherwood, B. Schwickert, Polymer Preprints 2003
polyimide tape 7.5 µm (KAPTON)
motor driven axis
tensioned drum
dilute solution of perfluoropolyether in 2,3-dihydrodecafluoropentane
Final Sample ConfigurationRoll Coating Technique
3 2 n 2 3F[-CF(CF )-CF -O-] CF -CF
time
RFRF
GG
echo
echo
Hahn echo
stimulated echo
pulsed gradients
steady gradient(fringe field)
+
Field-gradient NMR diffusometryField-gradient NMR diffusometry
102 103 104 105 106 107 108 109
10-3
10-2
10-1
bulk in 1.6 m layers in 1.3 m layers in 0.8 m layers
fluorine (PFPE)
protons (Kapton)
PFPE in Kapton rollsM
w = 11000
T = 313 K
T1 (s)
(Hz)
Rouse
bulk 1s
distinction limits (II)DE < - - > (III)DE of the tube reptation model?
t < Rouse < - - > t > Rouse
diffusion:
2 0 1/ 4R M t2 1/ 2 1/ 2R M t
relaxation:
0 3/ 41T M 1/ 2 1/ 2
1T M 0 3/ 41T M
01 / ln( )sT M 0 3/ 4
1T M
crossover Rouse dynamics < - - > reptation dynamics (limit II)under mesoscopic confinements:
fluid
silica glass (Vycor)
4 nm
B)
PEO melt
solid methacrylate matrix
A)
10 … 60 nm
C) PFPE melt layer
Kapton foil ( 7.5 m )
~ 1 m