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