Diffusion measurement via the Gradient EchoDiffusion measurement via the Gradient Echo Knowledge of the diffusion coefficient, D, can tell you about the viscosity of a solution, or

©A.‐F.Miller2010 page 1

DiffusionmeasurementviatheGradientEcho

Knowledge of the diffusion coefficient, D, can tell you about the viscosity of a solution, or if the viscosity is known it can reveal the molecular weight or hydrodynamic radius of molecules in solution. If the solvent is water, it provides a built-in diffusion standard. Other solvents whose diffusion coefficient is known can equally well be used, although this example uses water. This determination is broken into three portions: collecting the NMR data, -2- analyzing the results, -3- calculating the diffusion coefficient. The theory underlying equations used can be found in Stejskal and Tanner's original papers or in Pregosin, Martinez-Viviente and Kumar (2003) Dalton Trans. 2003, 4007-4014.

Collectingthedata

Inoneworkspace,collectabeautiful1H1Dandcalibrateapw90foryoursample.Also.ithelpstoknowtheT1approximately.

Inasecondworkspace,loadtheparameterfilewetPGEinvAFM.

AsshownonFigure2,thisissetuptorunthepulsesequencewetPGEinv_afm,whichisaPulsedGradientEchosequenceincorporatingthepossibilityofwetsolventsuppression(seethehandoutonthattopic).Asshown,theexperimentisnotimplementingthewetsolventsuppressioncapability[1](InAcquire>Wet,WETisturnedoff).

Thefirstpulse[2]isa90°excitationpulsethatbringsallmagnetizationintotheXYplane.Afieldgradient[3]isappliedforashorttime(2mshere)causingspinsatdifferentheightsintheNMRtubetogetoutofphasefromoneanother(dephasing).Next,thereareapairof90°pulsesappliedalongthesameaxis[4].Thesecanbethoughtofastwohalvesofa180°pulse,forthisexperiment.Thus,theexperimentcanberegardedasaspinechowithgradientsinthetwoflankingdelays[3]and[5].Howeverinbetweenthetwocentralpulses,adelayisallowedduringwhichmoleculeswilldiffusewithinthesampletube[6].Smallmoleculesthatdiffusefastmaymovesubstantiallyduringthistimeandthereforeendupatadifferentheightinthesample.Thesecondhalf‐180°returnsmagnetizationtotheXYplanewherethesecondgradient[5]canrefocusthedephasingproducedbythefirstgradient.Howeverrefocussingwillonlyperfectformoleculesthatareatthesamepositioninthetubeforbothgradients,becauseonlytheywillexperiencethesamefieldstrength.Thefurtheramoleculediffuses(onaverage)thelessmagnetizationisrefocussed.Thus,smallmoleculeswithhighdiffusioncoefficientswillloosemoreintensitythanlargermoleculeswithlowdiffusioncoefficients.Intensitycanalsobelostduetorelaxation.Inordertodistinguishbetweenlossestorelaxationvs.lossesduetodiffusion,wewillrepeattheexperimentwithaseriesofdifferentgradientpulsestrengths.Lossesduetorelaxationwillbeconstantwhereaslossesduetodiffusionwillincreaseinproportiontothesquareofthegradientstrength.


InAcquire>Acquisition(Figure3),enteryourvaluesforsw[1],at[2],d1[3]pw90[4]andtpwr[5]basedonyourbeautifulcalibrated1H1D.(Toenter,tofyouwillhavetogotoAcquire>Channels.)Notethatthispulsesequenceresemblesthesimple1Dpulsesequenceinthattheexcitationpulseisnotnecessarilysetto90°.Checkthevalueofthe'firstpulse'[6]andchangeittoyourpw90value.(Youcanalsodothisbytypingpw=pw90tomakeallthepulses90°pulses.Checkthe'Sequence'[7]displayandconfirmthatallthepulselengths[8]havethevalueofyourpw90.Alsorecallthattherelaxationdelayd1shouldbeappropriateforyoursample,(eg.d1≈1.3*T1).

Thediffusiondelayd5[9]shouldbeshorterthanT1,butwilldependprimarilyonthediffusioncoefficient.Wewillchooseitbasedonatestrun(morebelow).

Firstsetgzlvl=0bytypinginthecommandline[1](Figure4).WedonothaveaGUIsetupforthePGE‐specificparametersyet.Toconfirmthatyourvaluehasbeenaccepted,clickon'Sequence'againandseethattheheightofthegradientpulsesisnowzero[2].Also,viewthePGEparametersetbygoingtotheAcquire>Overviewpanel[3],[4].(Ihadd5=.5andgradientpulsedurationsof0.1secondswhenthisscreenshotwastaken.)

Collectaspectrumandcorrectthephase.Set'FutureActions'forwhentheexperimentfinishesto'wftdcds'toweightandFouriertransform(=wft),levelthespectrum(driftcorrect=dc)anddisplay(=ds).

Nowactivatethegradientpulsesagainbytypinginanarrayofvaluesforgzlvl(Figure5).Becausewewillwantthevaluesofgzlvl2tobeevenlyspaced,wewillenterourlistofgzlvlvaluesbyhandinsteadofusingthe'Arrays'button.Type(forexample)gzlvl=100,600,1100,1500,1800,2000[1].Youwillnotbeallowedtouseavaluelargerthan2047,inordersafeguardtheinstrument(2047isthemaximumonGORT,largevaluesapplyforT2JandT1J,whichhaveolderhardware.)

Click'ShowTime'[2],chooseanumberofscansthatwillgiveyousignal‐to‐noiseofatleast20:1(eg.nt=8).Acquire[3].

Fouriertransformallyourspectraanddisplaythemside‐by‐sideeitherusingthebuttonsinthe'ArrayedSpectra'panelontheleft‐hand‐side[4]andarrayinghorizontally[5],orbytypingdssh(displayspectrastackedhorizontally).Youseethatthestrongergradientsresultingreatersignallossforthesameamountofdiffusioninallcases(allcasesusethesamevalueofd5.)

Analyzingtheresults

Theintensityofthesignal,I,dependsonthegradientstrengthGanddiffusionconstantDasfollows,

€

ln IIo

= − γδ( )2G2 Δ −

δ3

D


whereIoisthesignalintensityobtainedfromthesameexperimentwhenthegradientstrengthiszero.γisthegyromagneticratioofthenucleusbeingobserved(for1Hitis4258Hz/G)andδandΔaretimeintervalsinthepulsesequence(Page6).

δ=gt

Δ=gt+d2+2pw+d5+d0

Rearrangingtheequation,weget:

ln(I)‐ln(Io)=‐(γδ)2G2(Δ‐δ/3)D=‐(γδ)2(Δ‐δ/3)DG2

ln(I)=ln(Io)‐(γδ)2(Δ‐δ/3)G2D

Predictsthatln[I]vs.G2shouldbeastraightlinewithaslopeof‐(γδ)2(Δ‐δ/3)D.

Wedon'tyetknowwhatourGvaluesare(seebelow),butweknowtheyareproportionaltogzlvl,sowepropose:G=grad_calxgzlvl,wheregrad_calistheconversionfactorfromVarianunitstoG/cmgradientstrengths.Takingthatintoaccountwepredictthataplotofthenaturallogofsignalintensity,ln[I]vs.gzlvl2shouldbeastraightlinewithaslopeof‐(γδ)2(Δ‐δ/3)grad_cal2D.

Fortheexampleofwater,anactualplotofln(amplitude)vs.gzlvl2isshowninFigure7.Thisshowsthatwedoindeedgetastraightlinewithanegativeslope.

Toexecutethisanalysisonyourowndatayouwillneedtoextractthepeakintensitiesfoeeachpeakineachspectrum.Oncethedatacollectioniscompletedisplaythefirstspectrumbytypingtheds(1)command.(Figure8).GototheProcess>Integrationpanel[1],activateintegraldisplay[2].Select'clearintegrals'[3]andthen'interactiveResets'[4].Nowplaceresetsoneithersideofeachpeakbyclickingwiththeleft‐mousebeginningontheleftsideofeachsignal.

Nowtypebctousetheregionsnotselectedasabasisforbaselinecorrection.Toapplythesamebaselinecorrectiontoeachoftheotherspectrainyourarray,typeds(2)bcthends(3)bc,thends(4)bc,etcuntilallthespectrahavecorrectedbaselines[5].

Toseeallyourspectraoneaboveanother,typedssa(Figure9)

Nowwewanttoextractpeakamplitudesforeachpeakineachspectrum.Displaythefirstspectrumbytypingtheds(1).commandresults.InProcess>Display(Figure10),selectthethresholdicon[1].Thenclickon'findpeaks'[2]toshowthemwiththeirppms.Figure11showstheresult.Makeanoteofwhichpeaksareofinterest.


Togetamplitudesforthese,gotoProcess>Cursors/LineLists(Figure12).ClickonDisplayLineList[1],checkthatyouseethepeaksyousawinfindpeaks.ThenclickonFindPeaksinArray[2].

Scrolldownwiththeright‐handscrollbar[3]torevealahorizontalbottom‐of‐windowscrollbar[4].Scrollthatallthewaytotherighttorevealtwoverticalscrollbars[5],theinneroneofwhichallowsyoutoseethepeakamplitudesforeachofyourlinesineachspectrum.

Usethemousetoselectallthedata,clickCtrl'c'tocopyit.(Figure13)UnderApplications>Office[1]gotoOpenOffice.org>Writer

Inthewindowthatopens,clickCtrl'v'topasteinallyourdata.Savethetextfilewithausefulnameinausefullocation,andcloseit.Transferthisfiletoyourowncomputerandreopenitinaspreadsheetpackagesuchasexcel.Keepthedataforagivenpeaktogether[2],andcopyittoacolumninyourspreadsheet.

Dothisforeachpeakthatinterestsyousothatyouhaveonecolumnforeachpeak[2]nexttocolumnsforgzlvlangzlvl2valuesusedineachspectrum[1](Figure14).

Youwillthenneedcorrespondingcolumnsforthenaturallogsofthepeakamplitudes[3].Plotthesevs.thesquareofgzlvl(Iusedgzlvl/100allsquaredinordertonothavehugenumbers)[4].

Ifyoursampleemployswaterasthesolvent,youcaneasilyuseinternalwaterasyourcalibrationbecauseitsdiffusioncoefficientisknown(below).Ifnot,weassumethatyouareusingapreviously‐determinedvalueforthecalibrationcoefficientgrad_cal.Alternately,youcanlookupthediffusioncoefficientofyoursolventatyourtemperature,anduseitinthesamemanneraswaterisusedbelow.

NowgenerateanXYchartorplotoftheln(amplitudes)vs.gzlvl2values(Figure15),andobtaintheslopesofallthelines.Ifthelinesarenotgenuinestraightlines,thenyourdatadonotfitthemodel.Re‐examineyouranalysis,yoursample,yourpulsesequenceetc,butdonotproceedunlessthelinesreallyarestraightlines.

Usingyourvaluesof‐(γδ)2(Δ‐δ/3)grad_cal2(beloworpriorcalibrationofgrad_cal),usetheslopestocalculateDforeachofthemoleculesinthesample,basedontheslopesobtainedfromsignalsfromeachmolecule(Figure16).(Recallthatslopedividedby‐(γδ)2(Δ‐δ/3)grad_cal2=D.Intheexamplesamplethereweretwospeciesdissolvedinwater.Onewasfoundtohaveadiffusionconstantaround8x10‐10M2s‐1,andalargeronewithaslightlyslowerdiffusionratewithD≈5x10‐10M2s‐1.

Shortcutifyoursolventiswater


Forsamplesdissolvedinwaterthereisnoneedtoactuallydeterminetheindividualconstants,sincewatercanbeaninternalstandardandtheentire‐(γδ)2(Δ‐δ/3)termisconstant.Recallthatslope=‐(γδ)2(Δ‐δ/3)grad_cal2xDandforwaterIhadaslope=.0114x10‐4=‐(γδ)2(Δ‐δ/3)grad_cal2x1.8x10‐9M2s‐1.Therefore,(γδ)2(Δ‐δ/3)grad_cal2=633inthiscase,forthedelaysIused.

However,forsamplesdissolvedinwater,thechallengeistocombinethediffusionmeasurementwithwater‐supression.

Gobacktoyourbeautiful1D,whichwillbedominatedbythesignalofwater.Inthatworkspace,youcanmakeashapedpulseforwatersuppression(seethehandoutonwetsolventsuppressionformoredetail).

Makeashapedpuseusing'Edit>Newpulse'(Figure17).

IntheAcquire>Wetpanel,checkthe'WET'box(Figure2).ThepulsesequencenowdisplaystheWETpre‐sequence(Figure18).

OptimizetheWETpulsepowerifneeded,withgzlvl=0.Thensetupyourgzlvlarrayasabove.

Gradientcalibration

Inordertoobtainnumericalvaluesforthediffusionrateofyourmolecules,youneedtoknowhowstrongthegradientsare.Inotherwords,weneedtoknowwhatgradientstrengthscorrespondtothevaluesofgzlvl.Wedeterminetheconversionfactorbyperformingourexperimentonamoleculewhosediffusioncoefficientisknown.Inthiscase,waterwasused,becauseat25°C,weknowthatthediffusioncoefficientofwaterisD=1.8x10‐9M2/s.

Wewillusetheslopeobtainedforwaterat25°CtocalculateaconversionfactorthatmultipliesourgzlvlvaluestogiveG.

Theslope=‐.0114x10‐4(G/cm)‐2=‐(γδ)2(Δ‐δ/3)Dgrad_cal2andln(Io)=4.65

D=1.8x10‐9M2/s

Inthiscased0=50us,gt=2ms,d2=50us,pw=14.75usandd5=200ms.

Thusδ=.002sandΔ=(.002+.00005+.0000295+.2+.00005)s=.2021295s

Δ‐δ/3=.201463s

slope=(4258Hz/G*.002s)2(.201463s)1.8x10‐9M2s‐1grad_cal2=0.0114x10‐4(takingintoaccountmyhavingdividedgzlvlby100beforesquaring.


72.5G‐2x.201463sx1.8x10‐9M2s‐1grad_cal2=0.0114x10‐4

2.63G‐2M2grad_cal2=114

grad_cal2=43.3G2/M2grad_cal=6.6G/M=.066G/cm

Thus,ourtopvalueforgzlvlof2000correspondsto132G/cmor13,200G/mgradient**soundslow**

PGE Diffusion Measurement Figure

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Calculation of diffusion constants via calibration on solvent water.

resonance slope x 10,000 D

3.97 ppm -0.0053 8.4 x 10-10 M2s-1

3.54 ppm -0.0050 7.9 x 10-10 M2s-1

0.72 ppm -0.0037 5.8 x 10-10 M2s-1

0.26 ppm -0.0030 4.7 x 10-10 M2s-1

vs. water, slope x 10,000 =-.0114 and D = 1.8 x 10-9 M2s-1


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Documents

Diffusion measurement via the Gradient EchoDiffusion measurement via the Gradient Echo Knowledge of the diffusion coefficient, D, can tell you about the viscosity of a solution, or