Time-Resolved Mass Sensing of a Molecular Adsorbate Nonuniformly Distributed Along a Nanomechnical String

0.97 MB·English

by  T. S. Biswas

Time-ResolvedMassSensingofaMolecularAdsorbateNonuniformlyDistributed
AlongaNanomechnicalString
T.S.Biswas,1 JinXu,1 N.Miriyala,2 C.Doolin,1 T.Thundat,2 J.P.Davis,1,∗ andK.S.D.Beach3,†
1Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2E1
2DepartmentofChemicalEngineering, UniversityofAlberta, Edmonton, Alberta, CanadaT6G2E1
3DepartmentofPhysicsandAstronomy,TheUniversityofMississippi,University,Mississippi38677,USA
(Dated:June3,2015)
Weshowthattheparticulardistributionofmassdepositedonthesurfaceofananomechanicalresonatorcan
beestimatedbytrackingtheevolutionofthedevice’sresonancefrequenciesduringtheprocessofdesorption.
5 Thetechnique,whichreliesonanalyticalmodelswehavedevelopedforthemultimodalresponseofthesystem,
1
enables mass sensing at much higher levels of accuracy than is typically achieved with a single frequency-
0
shift measurement and no rigorous knowledge of the mass profile. We report on a series of demonstration
2
experiments, in which the explosive molecule 1,3,5-trinitroperhydro-1,3,5-triazine (RDX) is vapor deposited

n alongthelengthofasiliconnitridenanostringtocreateadense,randomcoveringofRDXcrystallitesonthe
u surface. Insomecases,thedepositionisbiasedtoproducedistributionswithaslightexcessordeficitofmass
J at the string midpoint. The added mass is then allowed to sublimate away under vacuum conditions, with

3 the device returning to its original state over about 4h (and the resonance frequencies, measured via optical
2 interferometry,relaxingbacktotheirpre-mass-depositionvalues). Ourclaimisthatthedetailedtimetraceof
observedfrequencyshiftsisrichininformation—notonlyaboutthequantityofRDXinitiallydepositedbutalso

] aboutitsspatialarrangementalongthenanostring. Thedataalsorevealthatsublimationinthiscasefollowsa
l
l nontrivialratelaw,consistentwithmasslossoccurringattheexposedsurfaceareaoftheRDXcrystallites.
a
h
-
s I. INTRODUCTION complex fabrication [13]. An alternative approach has been
e toemploymultimodemeasurements[14–20],whichcanpro-
m
Nanoscale mechanical resonators have proven to be use- videsomedegreeofspatialresolution(limitedbythepropaga-
t. ful tools for chemical detection [1–3] thanks to their incred- tionofexperimentaluncertaintiesthroughthe“inversionker-
a ibly high sensitivity to added mass [4–7] and the ease with nel”[21]). Thesimplestexampleisthatboththesizeandlo-
m which their vibrational frequencies can be measured to great cationofapointmass—situatedalongaresonatorwithextent
- accuracy. For a device of tens or hundreds of picograms, primarilyinonedimension—canbedeterminedfromasimul-
d
molecules adsorbed onto the surface at the scale of fem- taneousmeasurementoftworesonancefrequencyshifts[22–
n
o tograms or smaller are detectable as shifts in the resonance 25]. Inaslightlydifferentcontext,multimodemeasurements
c frequencies[8–10]. Aseriouslimitation,however,isthatthe have been shown to provide single-resonator discrimination
[ linear relationship between the amount of mass added and for the mass added along an array of strongly coupled de-

the change in the observed resonances involves a sometimes vices[26].
2
v difficult-to-determine constant of proportionality. This fact, The work reported in this paper also takes advantage of a
7 whichgoesunacknowledgedinmuchoftheliterature,isama- multimodeframework(describedinSec.III),andwefindthat
4 jorimpedimenttohigh-accuracynanomechanicalmasssens- silicon nitride nanostrings [27, 28] are an excellent platform
6 ing. for our technique. Indeed, although the original experiment
5
Foraperfectlyuniformdistributionofaddedmass,thecon- by Dohn and collaborators to determine the mass and loca-
0
stant of proportionality is straightforward to compute: it is a tionofapointlikeobjectwasperformedoncantilevers[22],it
.
1 simplegeometricfactor, givenbytheratiooftheresonator’s wassoonrecognizedthatthesinusoidalmodeshapescharac-
0 mode-specific “effective mass” to its true inertial mass [11]. teristicofnanomechanicalstringsgreatlysimplifytheanaly-
5
Even if the deposition is nonuniform, the situation is still sis [23]. As an added benefit, for materials such as stoichio-
1
manageable so long as the mass distribution is well charac- metric silicon nitride, the high internal tension that draws a
:
v terized(andnottooconcentratednearthenodesofthedetec- doublyclampedbeamtautenoughtobecomestringlikeleads
Xi tionmode). Moretypicalofasensingapplication,though,is to correspondingly high mechanical quality factors. These
thatthedistributionisofarbitraryformandmoreorlessun- highvaluesaidindetection[29–31]andaremaintainedeven
r
a known. In that case, there is no reliable way to extract the inthepresenceofametallicoverlayeraddedtothedevicefor
totaladsorbedmassfromfrequency-shiftmeasurements[12], thepurposeoffunctionalization[32,33].
exceptatthelevelofanorder-of-magnitudeestimate. In a previous study on nanostrings, real-time mass sens-
To avoid this problem, efforts have been made to concen- ingwasmimickedbycarryingoutsequentialfrequency-shift
trate mass adsorption to specific sites on a device through measurementsofthefirsttwoharmonicsinconjunctionwith
pick-and-place deposition of a single microparticle [23]. In
ourwork,genuinereal-timeobservationsaremade: wemea-
sure the first, third, and fifth harmonics simultaneously as a
∗[email protected] function of time while mass sublimates from the device sur-
†[email protected] face. Thisispossiblebecauseweareabletoresolveallthree
2
of these modes using optical interferometry at the midpoint Weexpectittohaveparticularimportanceforsingle-molecule
ofthestringwiththedeviceentirelyunderthermomechanical massspectroscopy[12].
actuation.
We show that measurements of any pair of modes reveal
not only the instantaneous total mass of molecules adsorbed II. EXPERIMENTALMETHOD
but also their distribution—at least in the regime where the
distribution is smooth, slowly varying, and roughly symmet-
The nanomechanical devices under study are of a simple,
ricabouttheresonatormidpoint.Weprovideexplicitformulas
doublyclampedbeamarchitecture: 250-nm-thickribbons,2–
(derivedinSec.IIIandpresentedinTableII)forthemassco-
3µm wide and 100–300µm long. Their out-of-plane vibra-
efficients of the uniform and nonuniform components of the
tional modes oscillate in the MHz range and exhibit quality
adsorbatedistributionintermsofallcombinationsofthefre- factorsexceeding105.Theresonatorsarefabricatedfromsto-
quency shifts measured in modes 1, 3, and 5—including the
ichiometricsiliconnitridegrownonasacrificialsilicondiox-
pseudoinverseresultinwhichthreemeasurementsdetermine
idelayerontopofasiliconhandle. Theyarepatternedusing
thetwocoefficientsintheleastsquaressense—andweshow
standardopticallithographyandreactiveionetchingthrough
thatthevarianceamongtheseestimatescanbeusedasanef-
the nitrideand thenreleased usinga buffered oxideetch. (A
fectivetooltojudgethecombineduncertaintyduetomeasure-
detailed description of the fabrication process is provided in
menterrorandanalyticalassumptions.
Refs.26,31,and32.)Theresultinggeometryestablisheshigh
Themultimodemeasurementtechniquealsotellsusabout tensilestrainalongthelengthofthebeam;asaconsequence,
thedesorptioncharacteristicsoftheaddedmolecule,herethe the device acts as a classical string, with its sinusoidal mode
explosive1,3,5-trinitroperhydro-1,3,5-triazine(RDX).Inpar- shapes [31] enabling the analytical treatments that appear in
ticular, our analytical model is able to discriminate between Sec. III. Molecules are adsorbed onto the nanostrings by va-
twosituations:(i)whereRDXmoleculesareboundtotheres- pordepositionofRDXthathasbeenheatedandcarriedtothe
onator surface and thus display first-order or Langmuir-like sample chip via a flow of nitrogen gas. It is observed that
rate kinetics; and (ii) where the RDX molecules are bound the RDX preferentially adsorbs onto the silicon nitride and
to each other in crystalline formations (born from randomly aggregates in clusters, visible in Fig. 1. We employ various
distributed nucleation sites), and sublimation occurs primar- masks to gently bias adsorption around the midpoint of the
ily from the crystal interface. As we discuss in Sec. IV, our string. Theweakthermalactuationofthestringhasnoeffect
frequency-shiftdatasupportthelatterinterpretation.Thisisin onwheretheRDXsettles.
keepingwithourexpectations,sinceatomicforcemicroscopy
Measurements are performed using optical interferometry.
studies have already established that RDX forms nanoscale
Light from a HeNe laser (632.8-nm wavelength) is focused
crystallitesonthesiliconnitridesurface(seethelowerpanel
onto the nanostrings, with part of the light reflecting from
ofFig.1inRef.34).
the surface of the nanostring and part from the silicon sub-
Our combined experimental and theoretical study of mass strate underneath the string. The interference of light there-
distributionandsublimationmaybeusefulnotonlyinthede- fore encodes information on the relative separation between
sign of future explosive sensors [35, 36] but also in a wide thenanostringandthesubstrateasamodulationoftheoptical
rangeofnanomechanicalmass-sensingapplicationswhereac- intensity. The laser power incident on the string is no more
curatedeterminationofadsorbedmolecularmassisrelevant. than120µW,andwehaveshownelsewhere[31]thatthereis
noheatingeffect.
Wefocusthelaserlighthalfwayalongthenanostring. The
even-numbered, out-of-plane vibrational modes have a node
there,andsoourmeasurementsarenotsensitivetothem[11].
But for the odd-numbered harmonics that we can detect, the
string center is always the point of maximum amplitude;
henceweareabletopositionthelaserspottohighaccuracy,
andthesignal-to-noiseratioofthemeasurementisatitslocal
maximum.
Thenanostringchipisplacedintoanoptical-accessvacuum
chamber,reducingtheviscousdampingontheresonators[28]
and enabling thermomechanical measurements of the nanos-
trings [11]. The resulting vacuum (∼10−4torr) causes sub-
limation of the RDX from the nanostrings, which is not ob-
served under ambient conditions [35, 36]. The HeNe detec-
tion laser is not resonant with any of the internal levels of
RDXandthereforeitslightisnotadsorbedbythemolecule.
A high-frequency Zurich lock-in amplifier (model HF2LI,
FIG. 1. Optical microscope image showing a 101.7µm nanostring capableofdemodulatingasmanyassixindependentfrequen-
withalargenumberofRDXcrystallitesphysisorbedontothedevice cies) is used to isolate up to three harmonics that have large
surface. thermomechanical displacement at the device midpoint. We
3
tracktheseharmonicsasafunctionoftimeinordertoextract coveringcaninducesurfacestressesthatsignificantlyalterthe
usefulinformationaboutthedistributionofmassonthedevice bending stiffness [38] (which, in the case of cantilevers, can
surfaceandaboutthecharacteristicsoftheRDXsublimation itselfbethebasisofadetectionscheme[39,40]). Additional
process. In practice, the continuous time trace of the optical masscanalsoprovideanewpathwayforenergydissipation,
interferometrymeasurement,z(t),isusedtogenerateapower sothatshiftsintheresonantfrequenciescomeaboutindirectly
spectraldensity(PSD), through changes in the mechanical quality factor. Neither of
these effects is relevant to the devices we are studying. Our
S(ω;t)= 1 (cid:90) Tdτeiωτ(cid:90) t−τdτ(cid:48)z(τ(cid:48))z(τ(cid:48)+τ), (1) experimentexploitstwopropertiesofnanostrings:(i)theylive
T 0 t−T inahigh-tensionlimitwherethebendingtermsarenegligible
andthevibrationalmodesarealmostperfectlyharmonic,and
computedoveraslidingtimewindowofdurationT ≈0.85s.
(ii)theyexhibithighmechanicalQthatisquiterobusttothe
The “instantaneous” resonance frequency ω (t) is obtained
n presenceofanymolecularoverlayer.
at each time t by fitting the PSD in the vicinity of its
For a high-tension string, the mode shape is a sinusoid
nth peak to the usual (nearly-Lorentzian) damped harmonic u (x)=(2/L)1/2sin(nπx/L);and,hence
oscillator lineshape, S(ω;t) = A ω (t)/(cid:8)(cid:2)ω2(t)−ω2(cid:3)2+ n
n n n
(cid:2)ωωn(t)/Qn(cid:3)2(cid:9), with the quality factor Qn and the overall −2δωn = (cid:82)0Ldxδµ(x)sin2(nπx/L). (5)
amplitude An also optimized as part of the fit [11, 31]. This ωn (cid:82)0Ldxµ(x)sin2(nπx/L)
analysisissimpleenoughthatitcanbedoneconcurrentlywith
thedataacquisition. [Thereflectionsymmetryofu2 imposesthefundamentallim-
n
itation that δµ(x) cannot be distinguished from δµ(L−x)
using frequency-shift measurements alone.] If the unper-
III. FRAMEWORKFORANALYSIS turbed string has a uniform mass distribution µ(x)=M/L,
thenEq.(5)specializesto
A. Frequencyshiftinresponsetoaddedmass
δω 1 (cid:90) L (cid:18)nπx(cid:19)
− n = dxδµ(x)sin2 . (6)
We begin by considering the total mechanical energy (ki- ωn M 0 L
neticpluspotential)ofastringoflengthLvibratinginoneof
Therearetwolimitsworthemphasizing. Inthecaseofauni-
itsnormalmodes:
formmassdepositionprofile,withδµ(x)=m/Lleadingto
(cid:90) L
En= dxµ(x)ωn2u2n(x). (2) −δωn = m , (7)
0 ω 2M
n
Here, x measuresthedistancealongthestring; ω andu (x)
n n themassaddedmcanbedeterminedfromasinglefrequency-
are the angular frequency and displacement profile of mode
shift measurement in any mode. A strongly peaked profile
n;andµ(x)isthemassperunitlengthatpositionx. Asmall
represents the extreme opposite case. A point mass m de-
mass perturbation, arising from deposition of a distribution
positedatpositionx leadsto
m
ofmoleculesonthesurfaceofthestring,leadstoamodified
massdistribution µ(x)→µ(x)+δµ(x)andacorresponding δω m (cid:18)nπx (cid:19)
frequencyshiftω →ω +δω . Toleadingorder, thevaria- − n = sin2 m , (8)
n n n ω M L
n
tions δµ(x) and δω leave the energy stationary (dE =0);
n n
hence and hence the unknown values of m and x must be deter-
m
minedfromtwofrequency-shiftmeasurementsinanypairof
−2δωn = (cid:82)0Ldxδµ(x)u2n(x). (3) modes: specifically,
ωn (cid:82)0Ldxµ(x)u2n(x) m δω (cid:18) 1δω /ω (cid:19)−1
− = 1 1− 2 2 , (9)
Equation (3) can be understood as a linear relationship M ω 4δω /ω
1 1 1
δω /ω ∝m/M betweentherelativefrequencyshiftandthe
ratinoofnthemassadded,m=(cid:82)0Ldxδµ(x),tothatoftheorigi- xm= Larcsin(cid:18)1−1δω2/ω2(cid:19)1/2 (10)
naldevice,M=(cid:82)Ldxµ(x). Theconstantofproportionality π 4δω1/ω1
0
(cid:32) (cid:33)(cid:32) (cid:33) formodes1and2(firstderivedinRef.23);and
1 (cid:82)Ldxδµ(x)u2(x) (cid:82)Ldxµ(x)
−2 0(cid:82)0Ldxδµ(xn) (cid:82)0Ld0xµ(x)u2n(x) (4) −m = δω1(cid:18)3±1δω3/ω3(cid:19)−1, (11)
M ω 4 4δω /ω
1 1 1
is unique to each mode and depends on how the resonator’s (cid:32) (cid:115) (cid:33)1/2
ownmassandthemassadsorbedonthesurfacearearranged. x = Larcsin 3±1 δω3/ω3 (12)
m
A variety of other effects can come into play when mass π 4 4 δω /ω
1 1
impinges on a resonator [37]. In devices whose vibrational
modesaredominatedbyYoung’smodulusterms,amolecular formodes1and3.
4
Thegeneralmass-sensingproblem,however,ismuchmore lessusefulfordistinguishingthecomponentsofthemassdis-
difficultthaneitherofthesetwolimitsandhasthecharacter- tribution, −δω /ω →(m +m )/2M, independent of n, as
n n 0 1
isticsofanill-posedinverseproblem. Theconstantofpropor- n→∞.
tionalityintherelationshipδω /ω ∝m/Misundetermined, Arevealing,preliminaryapplicationofthisstyleofanalysis
n n
andthereisnowaytorecoveranarbitrarymassprofileδµ(x), isprovidedinTableI,whereEq.(17)isenlistedtocharacter-
except from an infinite number of error-free frequency-shift izethemassdistributedonsevennanostringsoffourdifferent
measurements(or,inpractice,fromknowledgeofδω /ω to lengths following various levels of RDX exposure. In each
n n
sufficient accuracy for all n=1,2,3,... up to a cutoff corre- instance,theresonantfrequenciesofmodes1and3aremea-
spondingtothedesiredspatialresolution). sured before and after RDX vapor deposition. We find that
We propose to extract the information from the time evo- the estimates for m exhibit both positive and negative sign
1
lutionoftheresonancefrequenciesinahandfulofmodesby (meaning convex and concave mass profiles), and in some
makingsomereasonableassumptionsaboutδµ(x). cases show magnitude |m1| comparable to m0 itself. This is
evidenceforaquitesubstantialvariationinhowRDXsettles
alongthelengthofthestringfromoneexperimenttothenext,
B. Two-parametermassdistributionansatz and it confirms that a mass distribution close to uniform is
only rarely the outcome of our sample preparation. The re-
sultsinTableIarefullyconsistentwithourintentionalbiasing
We suggest a simple, two-parameter form to represent the
ofthevapor-depositionprocess.
distributionofmassdepositedonthestring:
How confident should we be in this analysis? One issue
(cid:18) (cid:19) thatarisesishowtoquantifytheuncertaintyarisingfromour
m πm πx
δµ(x)= 0 + 1sin . (13) choiceofansatz. ThecoefficientsappearinginEqs.(16)and
L 2L L
(17)arespecifictoourchoiceofthesinefunctioninEq.(13).
Asatestoftherobustnessofourassumption,itwillbehelpful
Equation(13)isconstructedsothatm andm haveunitsof
0 1
tochecktheresultsagainstanalternativefunctionalform. We
mass,andthetotalmasssittingonthestringis
thereforealsotryasymmetricpolynomial
(cid:90) Ldxδµ(x)= m0L+πm12L =m +m ≡m. (14) x(cid:18) x(cid:19)
0 L 2L π 0 1 δµ(x)=m0+6m1 1− . (18)
L L
m isnonnegative, butm maybeofeithersign. m >0de-
0 1 1
Itisalsoimportantistolookforconsistencybetweenthere-
scribesaconvexmassdistribution,whereas−2m /π <m <
0 1
sultsachievedwithmeasurementsondifferentsetsofmodes.
0describesaconcaveone.Thelowerboundonm ,whichfol-
1
AsdiscussedinSec.II,thelaserinouropticaldetectionsys-
lowsfromδµ(L/2)∼m +πm /2>0,ensuresthatthemass
0 1
temisparkedatthestringmidpoint,soonlytheodd-numbered
distributionremainseverywherenonnegative.
modesaremeasured. TableIIlistsallthepossiblemassdeter-
TheassumptionsbehindEq.(13)arethat(i)thedistribution
minations using frequency shifts in modes 1, 3, and 5 (exact
issymmetricunderreflectionaboutthemidpointofthestring
determinationswiththethreeavailablepairsandoneoverde-
and(ii)thedistributionissmoothandslowlyvaryingenough
terminationwiththefulltripletofmodes),undertheassump-
thatitcanbeapproximatedbyonecomponentthatisuniform
tionofamassprofilefollowingeitherEq.(13)orEq.(18).
across the string and another that places mass preferentially
Areasonablefinalvalueofthefractionalmassaddedcomes
toward(orawayfrom,whenm <0)itscenter.
1 fromaveragingthoseeightestimates:
Putting Eq. (13) into Eq. (5) gives an expression for the
frequencyshiftsinthevariousmodes: m m +m
= 0 1
M M
δω 1 (cid:18)m 2n2m (cid:19)
− ω n = M 20 +4n2−11 . (15) =0.108031δω1 −0.534596δω3 −1.573434δω5.
n ω ω ω
1 3 5
(19)
Invertingtherelationship,wefindthatthevaluesofm andm
0 1
can be estimated from frequency-shift measurements on any (cid:0) (cid:2) (cid:3)2 (cid:2) (cid:3)2(cid:1)1/2
The total uncertainty, ∆ m/M +∆ m/M , is taken
pairofmodes. Forinstance,inthecaseofmodes1and2and e a
to be a combination of the experimental errors arising from
modes1and3,wefindthat
imperfectknowledgeoftherelativefrequencyshifts,
(cid:18) (cid:19)
m δω δω m 15 δω δω
M0 =8 ω11 −10 ω22, M1 = 2 ω22 − ω11 ; (16) ∆ (cid:20)m(cid:21)2=0.0117∆ (cid:20)δω1(cid:21)2+0.2858∆ (cid:20)δω3(cid:21)2
m0 = 27δω1 −35δω3, m1 = 105(cid:18)δω3 −δω1(cid:19). e M e ω1 e ω3
M 4 ω 4 ω M 16 ω ω (cid:20)δω (cid:21)2
1 3 3 1 +2.4757∆ 5 , (20)
(17) e ω
5
According to Eq. (15), higher modes (larger n) exhibit de- andtheuncertaintyduetothechoiceofansatz,whichweap-
creasing differentiation in the m coefficient and so become proximate by the spread around the mean estimate given in
1
5
Length Width M Trial f1 f3 δf1 δf3 m0/M m1/M m=m0+m1 mu mu/m
(µm) (µm) (pg) (MHz) (MHz) (kHz) (kHz) (pg) (pg)
101.07† 2.83 218.10 1 2.5823 7.7928 −71.083 −210.511 0.0506 0.00340 11.8 12.0 1.02
172.60 2.05 269.80 1 1.4871 4.4745 −0.100 −0.237 9.56×10−6 93.7×10−6 0.0279 0.0363 1.30
2 1.4885 4.4785 −1.708 −11.103 0.0139 −0.00874 1.41 0.619 0.441
3 1.4898 4.4821 −13.045 −48.873 0.0363 −0.0141 5.99 4.73 0.788
4 1.4934 4.4931 −2.115 −7.752 0.00554 −0.00203 0.947 0.764 0.807
216.34 2.56 422.30 1 1.2104 3.6304 −9.891 −26.318 0.00827 0.00605 6.05 6.90 1.14
309.49 2.72 641.88 1 0.8345 2.5067 −2.163 −5.641 0.00219 0.00224 2.85 3.33 1.17
TABLEI. Results ofasequenceofRDXdepositionsontonanostrings spanningfourdifferentgeometries, organizedbystring lengthand
depositiontrial.Misthetotalpredepositionmassinferredfromthematerialdensityanddevicevolume.Experimentallydeterminedfrequency
values are reported in columns 5–8, viz., the initial mode frequencies of the first and third harmonics, f and f , and the shifts in those
1 3
frequenciesafterthemassdeposition,δf1 andδf3. Theratiosm0/M andm1/M areobtainedfromEq.(17). Traditionally,onewoulduse
theshiftofthefirstharmonicaloneinconjunctionwithEq.(7)todeterminetheadsorbedmass;welabelthisquantitymu,sinceitisbased
ontheassumptionofauniformmassdistribution. Comparisonofmuwiththecorrespondingvaluem=m0+m1,comingfromourimproved
two-componentanalysis,revealsthatthetwoestimatescandifferquitesubstantially.Fluctuationsintheratiomu/mindicatethatthetraditional
approachroutinelyover-orunderestimatestheadsorbedmassintherangeof10%to50%. Thisquantitativecomparisondemonstratesthe
absoluteimportanceofamultimodeanalysisforaccuratemasssensing.Thedagger(†)markstheoneexperimentinwhichtheRDXdeposition
processiscompletelyunbiased;theothersinvolvesomedegreeofmaskinginordertoengineeranonuniformmassdistribution.
Eq.(19): As a consequence, there are two rate constants, given by the
eigenvalues of the matrix. When γ =0, the eigenvalues are
∆a(cid:20)Mm(cid:21)2=0.00791(cid:18)δωω1(cid:19)2+1.7344(cid:18)δωω3(cid:19)2 λra1te=α0+aβndtoλ2a=steαad+yβst;atheewnciteh, mthve/smys=temα/eβq.uiOlinbrtahteesotahtear
1 3
hand, when γ is a fast rate (i.e., larger than both α and β,
(cid:18) (cid:19)2
+1.5138 δω5 −0.2285δω1δω3 whichistheexperimentallyrelevantsituation)wefindthat
ω ω ω
5 1 3
αβ αβ(α−β)
+0.2127δω1δω5 −3.2403δω3δω5. (21) λ1=−α+ γ + γ2 +O(γ−3),
ω1 ω5 ω3 ω5 (23)
αβ αβ(α−β)
λ =−γ−β− − +O(γ−3).
Thenotation∆e[δωn/ωn]inEq.(20)ismeanttoindicatethe 2 γ γ2
errorintherelativefrequencyshift,understoodastheratioof
twoinexactquantitiescomputedaccordingto∆e[δωn/ωn]2= Thefulltimedependenceofmandmvisthengivenby
(∆ [δω ]/ω )2+(∆ [ω ]δω /ω2)2.
e n n e n n n
(cid:18) (cid:19) (cid:18) (cid:19)
m(t) = γm(0)+βmv(0) γ e−(α−αβ/γ)t
mv(t) γ2+αβ α
C. Sublimationmodel (cid:18) (cid:19)
+γmv(0)−αm(0) −β e−(γ+β+αβ/γ)t. (24)
γ2+αβ γ
The preliminary results appearing in Table I are obtained
from two discrete measurements of the devices’ resonance
For timest (cid:29)(γ+β+αβ/γ)−1, the mass on the string de-
frequencies,beforeandafterRDXexposure. Wehavetheca-
caysaccordingto
pability,however,tomakeongoingmeasurementsoftheres-
onancefrequencies;thesechangecontinuouslyintimeasthe m(0)+(β/γ)m (0)
adsorbatemoleculesareremovedfromthenanostringbecause m(t)= v e−α(1−β/γ)t. (25)
1+αβ/γ2
ofthevacuumenvironment. Themainthrustofthispaperis
toshowhowwecanexploitthismuchricherdataset. Moreover, if the chamber is being very aggressively evacu-
Thereforeitisusefultosketchoutamodelofhowthemass ated,thenthebehaviorlookslike
onthestringevolveswithtimeinourexperimentalsetup.The
basic assumption is that the molecules are either residing on m(t)=m(0)e−αefft, (26)
thestring(m)orexistingasvaporinthechamber(m ). RDX
v
sublimates from the string surface at a uniform rate α, and with αeff =α−αβ/γ ≈α close to the intrinsic sublimation
thereisanincomingfluxofmoleculesreturningtothesurface rateoftheadsorbate,andwecansafelyproceedasifmv≈0
duetointermolecularcollisions,denotedbyβ. Finally,there atalltimesafterthepumpisactivated.
is γ, the rate at which the chamber is being evacuated. This Arefinementofthismodelistoconsiderthepossibilitythat
pictureleadstoacoupledpairofrateequations: therateofmasslossgoesasafractionalpowerofthecurrent
mass load. This may be appropriate here since the RDX on
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
d m −α β m the surface is known to aggregate, rather than arranging in a
= . (22)
dt mv α −(β+γ) mv smooth monolayer. Sublimation in that case is likely to be
6
Ansatz Modes m /M m /M
0 1
Eq.(13) 1,3 27δω1−35δω3 105(cid:0)−δω1+δω3(cid:1)
4 ω1 4 ω3 16 ω1 ω3
1,5 25δω1−33δω5 99(cid:0)−δω1+δω5(cid:1)
4 ω1 4 ω5 16 ω1 ω5
3,5 875δω3−891δω5 3465(cid:0)−δω3+δω5(cid:1)
8 ω3 8 ω5 32 ω3 ω5
1,3,5 15007δω1−36365δω3−128205δω5 −58905δω1+107415δω3+42207δω5
2318 ω1 37088 ω3 37088 ω5 9272 ω1 9272 ω3 9272 ω5
Eq.(18) 1,3 1+3π2δω1−9+3π2δω3 3π2(cid:0)−δω1+δω3(cid:1)
4 ω1 4 ω3 4 ω1 ω3
1,5 3+25π2δω1−25(3+π2)δω5 25π2(cid:0)−δω1+δω5(cid:1)
36 ω1 36 ω5 36 ω1 ω5
3,5 9+75π2δω3−25(1+3π2)δω5 −75π2(cid:0)−δω3+δω5(cid:1)
8 ω3 8 ω5 8 ω3 ω5
1,3,5 217+975π2δω1−5607+1725π2δω3−6125+2175π2δω5 75π2(cid:0)−52δω1+23δω3+29δω5(cid:1)
1358 ω1 5432 ω3 5432 ω5 5432 ω1 ω3 ω5
TABLEII.Estimatesoftheuniform(m )andnonuniform(m )masscomponentsasafunctionoftherelativefrequencyshifts. Asolutionto
0 1
thesystemofequationsisgivenforeachpairofmodes;alsoshownistheleast-squares,pseudoinversesolutioninvolvingallthreemodes.
at least partially controlled by loss from the surface area of time:
the RDX crystallites (A∼m2/3), which puts a lower bound (cid:20) (cid:21)
1 m(t) 1 t
of2/3ontheeffectivescalingexponent. Ifthedistributionof − ln =− ln 1−
t m(0) (1−φ)t t
crystallitesalongthedeviceisroughlyuniform, thenwecan r
accountforthissituationasfollows: 1 (cid:20) t t2 (cid:21)
= 1+ + +···
(1−φ)t 2t 3t2
r r r
dm =−αm(cid:63)(m/m(cid:63))φ. (27) =α(cid:18) m(cid:63) (cid:19)1−φ(cid:20)1+(1−φ)αt(cid:18) m(cid:63) (cid:19)1−φ
dt m(0) 2 m(0)
(cid:21)
This description requires that we introduce a new material- +··· .
specific mass scale m and a phenomenological exponent φ.
(cid:63) (30)
We anticipate a value 2/3≤φ ≤1, with conventional expo-
nentialdecayrecoveredattheupperendofthatrange. When Finally, a more complete description must account for a
φ (cid:54)=1,wefindthat mass distribution that varies along the length of the string.
We treat the deposited mass δµ(x,t) as a space- and time-
dependentfieldsubjecttothegoverningequation
dm 1
= d(m1−φ)=−αm1−φdt.
mφ 1−φ (cid:63) ∂ (cid:18)δµ(x,t)(cid:19)φ
δµ(x,t)=−αµ . (31)
(cid:63)
∂t µ
(cid:63)
Thisleadsto
Here, µ isastand-inform /L. Inthepreviouslyconsidered
(cid:63) (cid:63)
situations, whereeitherφ =1ortheinitialmassdistribution
(cid:20) t (cid:21)1/(1−φ) is uniform, the mass distribution simply shrinks away while
m(t)=m(0) 1− , (28) preservingitsoverallshape. Thatisnotgenerallytrue:
t
r
(cid:104) (cid:105)1/(1−φ)
δµ(x,t)= δµ(x,0)1−φ−(1−φ)αµ1−φt . (32)
(cid:63)
where,t =[m(0)/m ]1−φ/α(1−φ)isthefiniteremovaltime
r (cid:63)
afterwhichallRDXhasleftthenanostring. Equation (32) also makes clear that, unlike in Eq. (28), δµ
reaches zero at a removal time that is different at each point
Wenowemphasizeanimportantfeaturethatdistinguishes
alongthestring.
betweenthetwokindsofbehavior. Whenφ =1,thedecayis
With Eq. (32) in hand, it is straightforward to obtain the
exponentialataconstantrate
time-dependenttotalmass
(cid:90) L
1 m(t) m(t)= dxδµ(x,t) (33)
− ln =α. (29)
t m(0) 0
oranyoftherelativefrequencyshifts
Foranontrivialvalueoftheexponent,however,themassloss −δωn(t) = 1 (cid:90) Ldxδµ(x,t)sin2(cid:18)nπx(cid:19). (34)
ischaracterizedbyanapparentdecayratethatincreasesover ωn M 0 L
7
ThelatterissimplyEq.(6)withδµ(x)replacedbyδµ(x,t). integrals are carried out numerically with Eq. (32) serving
as the model for δµ(x,t). In the spirit of our original two-
componentansatz, wechoosetoexpresstheinitialmassdis-
IV. RESULTS tributioninthisslightlymoreexpressiveform:
m m (cid:20)x(cid:18) x(cid:19)(cid:21)p
We now apply the models developed in Sec. III to our ex- δµ(x,0)= 0 + 1N(p) 1− . (35)
L L L L
perimentally acquired frequency shifts. The results reported
herearetakenfrommeasurementsonasingledeviceoflength Theexponentpaddssomeadditionalflexibilitywithregardto
310µmandmass641.88pg. Thedevice’sfirst,third,andfifth theshapeofthenonuniformcontribution[andpermitsarough
modesaretrackedoverthecourseof4h. interpolationbetweenEqs.(13)and(18)]. Thenormalization
Our approach is to match Eq. (34), for the cases of n= factor
1,3,5, to our time traces of the resonance frequencies. The 21+2θ Γ(3/2+p)
N(p)= √
π Γ(1+p)
0.0010
(cid:82)
is chosen to preserve the property that m= dxδµ(x,0)=
+0.0005
0.0005 m +m .
hift 0 +0.000 25 0The 1set of independent variational parameters m0, m1, p,
cy s −0.0005 φ, and C =(1−φ)αµ(cid:63)1−φ is simultaneously optimized us-
n ing the nonlinear least-squares Levenberg-Marquardt algo-
e
qu −0.0010 rithm [41, 42]. The number of degrees of freedom—only
e
e fr −0.0015 Theory: five—is remarkably small relative to the large number (≈
elativ −0.0020 3str×in7g0i0s0m) 0of+dmat1a=po2i.n8t7s.1(W2)epfignadndthtahtatthtehetodtaelcamyasesxpoonnethnet
R
−0.0025 Experiment: has a value φ =0.826(6). The resulting high-quality fit is
displayed in the upper panel of Fig. 2. The lower panel of
−0.0030
0 2000 4000 6000 8000 10 000 12 000 14 000 thesamefigureshowsthemassextractedateachinstantfrom
Time (s) frequency-shiftmeasurementsaccordingtoEq.(19),wither-
3 rorbarsonthedatapointsobtainedfromthequadraticmean
ofEqs.(20)and(21). Thereareahandfulofpointsintheplot
2.5 wheretheuncertaintyestimateblowsup,butthisamountsto
g) Two-component ansatz justafewblipsinthemorethan7000datapoints. Thesolid
p 2
s ( Sublimation model linesuperimposedonthedatapointsisproducedbyintegrat-
s
a ing[viaEq.(33)]theevolvingmassprofilethatemergesfrom
m 1.5
d thefittingprocedure[Eq.(32)withgloballyoptimizedparam-
e
rb 1 eters]. The consistency between these two approaches gives
o
s us confidence that our estimate of the total adsorbed mass is
d
A 0.5 highlyreliable.
Figure 3 shows snapshots of the mass profile at various
0
times in the experiment. Such a determination of the real-
0 2000 4000 6000 8000 10 000 12 000 14 000 timemassdistributionmayhaveimportantapplications,such
Time (s) asthestudyofdiffusionofmoleculesalongthesurface[43],
oratomiclayerreconstruction[44],ofananomechanicalres-
onator.
FIG. 2. Multimode frequency shifts as a function of elapsed time Figure 4 indicates the effective instantaneous decay rate,
and the resulting RDX mass extraction. In the upper panel, each calculatedasperEqs.(29)and(30). Onecanseeclearlythat
experimental data point represents a relative shift in the resonance the rate is not constant in time—and hence it is incompati-
frequencyofoneofmodes1,3,or5,obtainedfromthefinite-time- blewithpureexponentialdecay. Rather,itseemstoincrease
windowPSD[asperEq.(1)]. (Thetime-windowresolutionisfine
steadily.Moreover,between8000sand11000s,itturnsupin
enoughthattheexperimentalpointsbleedtogetherintonearlycon-
awaythatisconsistentwiththefractionalpowerofφ found
tinuous curves.) For visual clarity, the mode 3 and mode 5 data
inourfittothefrequency-shiftmeasurements.Itisworthreit-
setsaretranslatedupwardby0.00025and0.0005,respectively.The
eratingthatthisimpliesthatsublimationoccursfromthesur-
threesolidlinesaretheresultofmatchingourtheoreticalmodel[viz.,
Eq.(34)incombinationwithEqs.(32)and(35)]totheexperimental face of crystallites—and not uniformly from the surface of
observations. Thelowerpanelshowsthecorrespondingmass. Two thedevice—inagreementwithoptical(seeFig.1)andatomic
estimates are presented: one is a discrete set of values calculated force(seeRef.34)microscopy.
ateachinstantaccordingtoEqs.(19–21); theotherisacontinuous Something further we have checked is whether the fitting
curve[producednumericallywithEq.(33)]basedontheglobalfit canbemeaningfullyimprovedbyallowingα →α(x)tovary
achieved in the upper panel. The two estimates are found to be in along the length of the string. One might imagine, for in-
excellentagreement.
stance, that the sublimation rate has a uniform contribution
8
V. CONCLUSIONS
) 0.012 t = 0, m = 2.87 pg
m
μ
/ 0.010 Taking advantage of the sinusoidal mode shape of silicon
g
(p 1000 s, 2.02 pg nitride nanostrings under high tension, we develop analyti-
sity 0.008 calmodelsforthemultimodefrequencyshiftsthatoccurasa
n
e 2000 s, 1.40pg result of mass deposited onto such devices. We apply those
d 0.006
s modelstorealresonatorsinthelabanddemonstrateourabil-
s
a
m 0.004 itytomakereliableestimatesofthetotalamountanddistribu-
ar 4000 s, 612 fg tionofadsorbedmolecules. Thisworkprovidesexperimental
e
n 0.002 determination of a nonuniform and nonpointlike mass distri-
Li
8000 s, 74.9 fg bution.
0 Ourmultimodetechniqueproducesreliable,time-evolving
0 50 100 150 200 250 300
estimates of the total mass adsorbed on the resonator with a
Position along the nanostring (μm)
sensitivityontheorderofafewfemtograms, evenintheab-
sence of detailed knowledge of the initial mass distribution.
FIG. 3. Profiles of the mass distributed along the nanostring of The time trace of frequency measurements in a handful of
length310µmandmass641.88pgatrepresentativemomentsintime. low-lyingnormalmodesissufficienttoreconstructthatmiss-
ShownhereareplotsofEq.(32)evaluatedwithoptimizedvariational inginformationandthusfixtheotherwiseunknownconstant
parametersattheappropriatetimet. Accompanyingeachdistribu- ofproportionality[appearingasEq.(4)].
tionisalabelgivingthetimeofthesnapshotandintegratedweight Notably,theestimateofthemassadded,asdeterminedby
underthecurve. ouranalysis,isoftenfoundtodisagreewiththevaluearrived
at by the more traditional approach, which relies on a fre-
0.0012 quencyshiftinasinglemodeandincorporatesnounderstand-
ing of how the added mass is arranged on the device. This
reveals the extreme importance of such an analysis to accu-
rate 0.0010 ratemasssensingwithnanomechanicalresonators.
y
a Inaddition,becausetheadsorbedmoleculessublimatefrom
c
de 0.0008 thesurfaceofthenanostringsinourexperimentalsystem,we
s
u have had an opportunity to analyze their desorption charac-
o
ne 0.0006 teristics. Real-timemeasurementsallowforsignificantlyim-
a
nt provedestimatesofthemassdistributionsandrevealtheevo-
a
st 0.0004 lutionofthosedistributionsovertime. Thismayproveause-
n
I fultoolinstudyingmoleculardiffusion[43]ortheformation
ofmonolayersontonanomechanicalresonators[44].
0.0002
0 2000 4000 6000 8000 10 000 12 000 14 000 Finally, multimode analysis also leads to insights into the
Time (s) form in which the molecules are removed from the nanos-
tring surface by the vacuum. Specifically, we are able to de-
velop sublimation models that account for uniform-rate loss
FIG. 4. The instantaneous rate at which RDX desorbs from fromthesurfaceandlocal-mass-dependentlossfromthesur-
the nanostring clearly increases over time. The data shown are a faceareaofacrystallite. Onlythelatterisconsistentwiththe
logarithmic rescaling of the data appearing in the lower panel of observedfrequencyshiftsandthemicroscopyofthedevices.
Fig. 2. A linear fit over the time interval [250s, 7750s] produces This may have important applications in real-world sensing
α(t)=α0+α1t =0.00032865(6)+1.445(1)×t. Substantially of explosive molecules such as RDX, for example in airport
better agreement is achieved with a function having the form of passengerandluggagescreening.
Eq.(30)withexponentφ =0.8278(fullyconsistentwiththevalue
φ =0.826(6) used in the upper panel of Fig. 2) and removal time
tr=17247s. ACKNOWLEDGMENTS
The authors acknowledge support from the Natural Sci-
controlled by the partial pressure of RDX in the evacuated ences and Engineering Research Council of Canada; the
chamber and a locally varying component that comes about Canada Foundation for Innovation; Alberta Innovates Tech-
insomeunspecifiedwayfromtheinfluenceofthemechanical nologyFutures;GrandChallengesCanada;AlbertaInnovates
motion. We have carefully considered various ad hoc mod- Health Solutions; and the Office of Research and Sponsored
els,butwefindconsistentlythatallowingspatialvariationin Programs of the University of Mississippi. One of the au-
α(x)doesnotimprovethefitenoughtojustifytheadditional thors (K.S.D.B.) benefitted from a stay at the Aspen Center
variationaldegreesoffreedom[45]. forPhysicsunderNSFGrantNo.1066293.
9
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