Molecular and granular translational dynamics measured by a novel NMR technique

Transcription

1 Molecular and granular translatonal dynamcs measured by a novel NMR technque Janez Stepšnk Unversty of and J. Stefan Insttute, Slovena Unversty of Slovena 1

2 Measurement of the velocty autocorrelaton by CPMG sequence and constant magnetc feld gradent CPMG sequence as a modulated gradent spn echo to measure spectrum of molecular velocty autocorrelaton functon Unversty of Slovena RF π x / π y π y π y π y π y AQ Applcatons to obtan the spectrum of self-dffuson of flud n porous meda self-dffuson n gels and water and moton of fludzed granular systems π y Velocty autocorrelaton functon <v(t).v() >

3 Spn moton by two-pulse gradent spn echo G(t) q(t) π/ π Spn echo G E q Encodng of the spn moton ( ) e t () t = γ q G eff t. v t ( t' ) dt' dt G Phase gratng Unversty of Slovena 3

4 Molecular random moton and spn echo: Torrey s approaches H.C. Torrey. Bloch equatons wth dffuson terms, Phys. Rev., 14, , 1956 E Unversty of Slovena () γ q e Mesurement n the short tme-nterval dffuson can be tme dependent dfferent spn sub-ensembles can experence dfferent dffuson properte t. v t dt m m t () r e = γ B r E q( t). D ( ) (, t) m + D() r () r ( )m m=magnetzaton densty D=self-dffuson tensor. q( t) dt m j j dr () t q( t). D j. q( t) dt e Summaton over sub ensembles wth dfferent D j (t) 4

5 Molecular moton and spn echo: average propagator approach Short Pulsed gradent spn echo π/ π Spn echo q(t) tme E () e γ q t. v t dt e qr Probablty functon qr ( R, ) e dr r ( ) r ( ) = ( ) = P R J. Karger and W. Henk. The propagator representaton of molecular transport n mcroporous crystalltes, J. Magn. Reson. 51, 1-7 (1983) Unversty of Slovena R s stochastc varables ( R, ) qr = P e dr Average propagator n the lmt of long 5

6 Probablty dstrbuton of flow n porous meda E ( q, ) = P( r, r, ) q e r r dr q-space Fourer transform of spn echo can provde probablty dstrbuton of flow dsperson Short PGSE G(t) π/ π Spn echo q(t) tme Unversty of Slovena 6

7 Stochastc processes and spn echo as a characterstc functon N. G. van Kampen, Stochastc Processes n Physcs and Chemstry, North-Holland, Amsterdam, Random process of the stochastc varable r s characterzed by the probabty dstrbuton or by ts Fourer transform, whch s named the characterstc functon of random process Φ f r ( f ) = e P() r r = stochastc = varable f = parameter P() r = probablty functom e f r dr Sgnal of the short pulse gradent spn echo has the form of characterstc functonal wth the spn dsplacement R as a stochastc varable and the gradent dephasng q=γδg as the Unversty of Slovena parameter ( ) E, q e = Characterstc functon ( ) P( R, ) qr qr e dr Probablty dstrbuton 7

8 Stochastc processes and characterstc functonal Whenever the stochastc varable depends on another stochastc varable through relaton: the process s descrbed by the characterstc functonal Φ = e ( f, ) f ()() t' v t' dt' r () t = v( t' ) t where dt' f ( t) = arbtrary functon The characterstc functonal can be expanded nto the cumulant seres: e Unversty of Slovena f t. v t dt 1 f t. v t v t'.f c t' dt ' dt 6 f t.v t f t'.v t' f t'' v t'' dt. dt' dt' ' +... c 8

9 E f R ( ) = Spn echo as a characterstc functonal and the Gaussan phase approxmaton J. Stepšnk, Analyss of NMR selfdffuson measurements by densty matrx calculaton, Physca B, 14, 35-64, (1981) e () t q() t t q () t v ( t' ) () t () t dt'.v dt Spn dephasng = Velocty as the stochastc varable e G(t) q(t) π/ π Spn echo tme no lmts wth respect to the gradent pulse or gradent waveform 1 q() t. v () t dt (). () ('). () ' ' c q t v t v t q t dt dt Gaussan phase approxmaton (GPA) 6 e q t. v t q t'.v t' v t' '. q c t'' dt dt' dt'' +... Unversty of Slovena 9

10 Two-pulse gradent spn echo and GPA G(t) π/ π Spn echo Condton for the Gausan phase approxmaton (GPA): v λ < c 1 q < 1 q Characterstc lenght of moton s smaller than the phase gratng created by appled gradent. q(t) Phase gratng Unversty of Slovena 1

11 Gaussan phase approxmaton of spn echo E ( ) = () t. v () t dt q() t. v () t dt q. v () t v ( t' ) q e = e t v = v v ( t). q dt' dt.. Two gradent pulse spn echo Modulated gradent spn echo (MGSE) G(t) q(t) π/ π Spn echo tme rf G(t) q(t) π/ π π π π π π π π π π π π π π π T π tme Unversty of Slovena no lmts wth respect to the gradent pulse or gradent waveform 11

12 E( ) = β = Modulated Gradent Spn Echo (MGSE): tme-doman nto frequency doman X( ) β ( ) ϕ e 1 q () t. v () t v ( t' ). q() t' q ( ω, ) = q() t e dt D ( ω) = v () t v ( t' ) e dt dt' dt ωt 1 π Velocty autocorrelaton spectrum q ( ω, ) D( ω) dω ωt rf G(t) π/ π π π π π π π π π π π π π π π π q(t) T tme ω m =π/t frequency Long-tme measurement: Durng the applcaton of sequence each spn can roam over all characterstc vods of structure and D (ω) s dentcal for all spns. Unversty of Slovena E ( ) α D e ω m ω m = π T 1

13 Velocty autocorrelaton of restrcted moton box velocty autocorrelaton Free dffuson v( t ).v ( ) D( ω) t c ~ s autocorrelaton spectrum NMR range neutron scatterng range tme Restrcted dffuson- reflecton at boundares frequency Langevne equaton dv +α v = f dt () t velocty Autocorrel. tme autocorrelaton spectrum frequency Unversty of Slovena D rest ω = D + D k b k k 1+ ω k ω 13

14 Velocty correlaton spectrum of flows n a porous meda measured by MGSE Gradent mm rf G(t) q(t) (beads 1-3 µm) π/ π π π π π π π π π π π π π π π T π Flow tme Deff.1-9 -Log(E/E)/[ (1- /3 )] Flow of water through an on-exchange resn 5-1 mesh (1-3µm) MSSE-SS 5 ml/h MSSE-SS 5 ml/h MSSE-SS 1 ml/h MSSE-SS ml/h E Unversty of Slovena ( ) j α D e j ω m ω m = π T Modulaton frequency 1/T [Hz] Dsperson spectra for water flow through on-exchangeresn bead pack at dfferent veloctes as measured by MGSE sequence (δ=7µs). P.T. Callaghan, J. Stepšnk, J. of Mag. Res. A 117, 118-1(1995) 14

15 Dffuson spectra n porous medum Gradent mm autocorrelaton spectrum frequency Frequency range to 1.5 khz Pore sze ~1µm Unversty of Slovena 15

16 Restrcted dffuson n emulson droplets Frequency range to.5-1 khz Unversty of Slovena 16

17 CPMG sequence wth the constant gradent as MGSE sequence Carr-Purcell-Meboom-Gll sequence of the tran of π-rf pulses and the constant gradent RF π y π x / π y π y π y π y π y AQ T =ΝΤ β γ = π ( ) D ( ω) q( ω, ) dω G eff ω m =π/t = 8γ G π ω m NT D ( ω ) m Frequency range to above 1 khz J. G. Seland, G.H. Sørland, K. Zck and B.Hafskjold: Dffuson Measurements at Long Observaton Tmes n the Presence of Spatally Varable Internal Magnetc Feld Gradents, J of Mag. Res. 146, (), Unversty of Slovena 17

18 Resonance offsetî coherence pathways RF π y π x / π y π y π y π y π y AQ q ( ω) Ν=4 ω m =π/t frequency ω ω 1 γ G z = π δ << 1 Unversty of Slovena 18

19 CPMG measurement of dffuson G1 G G4 G8 G16 G3 G Unversty of Slovena 19

20 Velocty autocorrelaton spectrum of flud n porous structure [m s -1 ] RF π y π x / π y π y π y π y π y AQ D D(ω) 1.5x T =ΝΤ.5 D rest ω = D + D k b k k 1+ ω k ω ω [ π 1 3 s -1 ] [m s -1 ] D(ω).8 D α.4 ϕ tg () ϕ ( ) 4 pore sze ω [ π 1 3 s -1 ] Unversty of Slovena

21 Mean squared dsplacement from VAS [m s -1 ] 1.5x cos π ω [ () t z( ) ] = D( ω) z ( ωt) d ω 1 D(ω) ω [ π 1 3 s -1 ] Unversty of Slovena 1

22 Tme dependent dffuson constant D t π () = D( ω) sn ω ( ωt) d ω [m s -1 ] 1.5x D(ω) ω [ π 1 3 s -1 ] Unversty of Slovena

23 Surface-to-volume rato from VA spectrum D rest ω = D + D k b k k 1+ ω k ω Calculatons Sphercal Cylndrcal Planar Unversty of Slovena 3

24 MRI of polenta after mnutes of cookng T -weghted D-weghted PGSE a b c D-weghted by MGSE 15Hz.5 mm /s Igor Serša, Urša Mkac, J. Stefan Insttute Unversty of Slovena 4

25 Self-dffuson spectra of gels Dffuson spectra of gels T=3 O C 7,E-9 6,E-9 D( ν) m/s 5,E-9 4,E-9 3,E-9,E-9 gel N1% gel T4% Ntrobenzen Toluene 1,E-9,E Frequency [Hz] Unversty of Slovena 5

26 Self-dffuson of water Dffuson spectrum of H O T=4 O C, T=3 O C PGSE measurement 3,E-9,5E-9 D(ν) [m /s],e-9 1,5E-9 1,E Frequency [Hz] Unversty of Slovena 6

27 Anomalous self-dffuson of water Water n ntromethane Hydrogen bondng Unversty of Slovena 7

28 Models of water clusters Hypothess tll 195 from 198: On a very short tme scale (about a pcosecond), water s more lke a "gel" consstng of a sngle, huge hydrogen-bonded cluster. Unversty of Slovena 8

29 Restrcted self-dffuson of water? Dffuson spectrum of water T=4 O C.6E-9.5E-9.4E-9 D(ν) [m /s].3e-9.e-9.1e-9 E-9 1.9E-9 1.8E-9 1.7E Unversty of Slovena frequency -1/ [Hz -1/ ] 9

30 Ar-fludzed granular system Unversty of Slovena Sand dunes, gran slos, buldng materals, catalytc beds, fltraton towers, rverbeds snowfelds, many foods. Fludzed granular systems: catalyss of gas-phase reactons transport of powders combuston of ores The ar-fludzed granular system made up of mustard seeds wth the hgh speed camera 1frame/1 ms Measurng technques: hgh speed photography, CCD camera, dffusng-wave spectroscopy, PGSE (MIT, Harvard, Aachen..) 3

31 Velocty autocorrelaton spectrum of fludzed granular system by CPMG RF π y π x / π y π y π y π y π y AQ T =ΝΤ [ mm 8 / s] 6 D D(ω) ω [ π 1 3 ] ω [ π 1 3 ] / c Unversty of Slovena 31

32 Dependence of VA spectra on the gran densty and the ar-flow rate - 6 D( ν ) [1 m /s] 7 A ν [khz] - 6 D( ν ) [1 m /s] 7 B ν [khz] Unversty of Slovena Emprc formula D dffuson-lke constant c collson correlaton tme ξ free path 3

33 VAS to velocty autocorrelaton functon Unversty of Slovena 33

34 Mean squared dsplacement of gran Z 4 π () t = ( z() t z( ) ) = [ 1 cos( ω t) ] D ( ω ) d ω ω Unversty of Slovena 34

35 Conclusons RF π y π x / π y π y π y π y π y AQ T =ΝΤ Spectral analyss of moton to about 1-1 khz and, thus, the study of molecular dffuson close to nano lenght-scale. -Applcaton wth non-unform NMR magnets (one sded, NMR mouse etc.) -Use of ntrnsc susceptblty gradents to measure dffuson D(ω) NMR range neutron scatterng range Unversty of Slovena Frequency [Hz] 35

36 Aknowledgment Samo Lasč, Unversty of Ljubljana Aleš Mohorč, Unversty of Ljubljana Igor Serša, J. Stefan Insttute Antal Zupanc, Unversty of Ljubljana Ana Sepe, J. Stefan Insttute Gorazd Plannšč, Unversty o f Ljubljana E. Fukushma, Albequerque, USA H. Van As, Wagenngen, Holland O. Söderman, Lund, Sweden P.T. Callaghan, Wellngton, New Zealand Thank you for your attenton! Unversty of Slovena 36