Experimental Measurement of CO2 Diffusion Coefficient in ...
Diffusion measurement via the Gradient EchoDiffusion measurement via the Gradient Echo Knowledge of...
Transcript of Diffusion measurement via the Gradient EchoDiffusion measurement via the Gradient Echo Knowledge of...
©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.
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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
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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.
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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
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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.
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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**
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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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