Universal Handheld Micropipette Supplemental Information

Transcription

1 Submtted to Revew of Scentfc Instuments Unvesl Hndheld Mcoppette Supplementl Infomton Justn Beoz Mechnosynthess Goup Deptment of Mechncl Engneeng Msschusetts Insttute of Technology 77 Msschusetts Ave, 5-, Cmbdge, MA , jbeoz@mt.edu A. John Ht Mechnosynthess Goup Deptment of Mechncl Engneeng Msschusetts Insttute of Technology 77 Msschusetts Ave, 5-, Cmbdge, MA , jht@mt.edu Coespondng Autho

2 I. DERIVATION OF THE DIAPHRAGM S DEFLECTED SHAPE AND STIFFNESS Deflecton of the dphgm s teted n the followng wy: Dphgm z x ξ. We estmte tht the to of the dphgm s bendng enegy ove n-plne stetchng enegy s, especlly gven the pe-stetch. Theefoe we tet the dphgm s thn n the sense tht stess nd stn vy neglgbly coss ts thckness, so t s suffcent to smply dscuss the knemtcs of the mddle plne. Notce ths enfoces neglgble bendng stffness snce genetng esstng moment eques vton n stess coss the dphgm s thckness.. Stetch of mtel elements n the dphgm s eveywhee <., nd so the fee enegy densty s ppoxmtely neo-hooken.. The to of the elstome s she modulus ove bulk modulus s, nd so the dphgm s defomton s ppoxmtely ncompessble; ths s typcl of elstomes. Knemtcs The deflecton of the dphgm fom n ppled pessue dffeence P s zmuthlly symmetc (.e., ndependent of ), whch educes the poblem to D n the z, coodnte system (Fg. S). The sptl pofle of the mddle plne s of the fom, x, whee s functon only of the coodnte. A ptcle t x ognted t some pont x fo, P ; the dsplcement vecto connectng these two ponts s u x x u : u,, (S) whee u s lso functon only of the coodnte. We now fnd the stns t x n the medn nd pllel dectons. By zmuthl symmety s smply u. (S) b Fo, consde mtel lne element tht extends fom x dffeentl length dl long the medn, theeby temntng t x dl P dx dx dx, dx dx, dx dx, FIG. S. () Coss-secton pofle of the dphgm deflected fom n ppled pessue dffeence P coss t. The mgntude of deflecton s exggeted fo clty nd not to scle; (b) depcton of mtel element tht () begns undefomed, s then () pestetched dung dphgm ssembly, nd () defomed futhe wth dphgm deflecton., ; (S) s the unt vecto pontng long the medn t x nd the pme supescpt ndctes ptl devtve wth espect to. The ptcles t ponts x nd x dl ognted t the espectve ponts x () x nd x( x dl) x( x) x ( x) dl (S4) coespondng to P,.e., when the dphgm ws undeflected. The ppoxmton to st -ode n dl s suffcent snce dl s of nfntesml length. We my lso dentfy these ptcles n the x coodnte system, espectvely, s x nd dl x, (S5) whee s the unt vecto pontng long the mtel lne element s ognl oentton nd dl s ts ognl length. These e equvlent descptons of the sme ptcles. Defnng dl dl s the stetch long the medn, t follows fom Eq. s S4 nd S5 tht x dx = dx Justn Beoz

3 dl x( x) x( x). (S6) dl Clcultng explctly: u u u (S7) Theefoe the stn to ledng ode n u nd s u (S8) Those fml wth contnuum mechncs nd ts stndd lnguge wll ecognze tht x ( ) x s the nvese defomton gdent tenso nd x ( x) x ( x ) s the nvese ght Cuchy-Geen defomton tenso. Consde now the complete pocedue of pestetchng the ubbe sheet befoe clmpng t to fom the dphgm. The mtel elements compsng the dphgm expeence no she dung the homogeneous pe-stetch, s well s dung deflecton of the dphgm. Hence the stetchng of mtel lne element wth ntl length dx to pe-stetched length dx nd fnlly to length deflecton eltes to the totl pncpl stetch follows (Fg. Sb): : dx dx dx dx dung s dx dx dx (no sum). the dphgm t x pulled by lne tensons N nd N n the medn nd pllel dectons, espectvely. The foce blnce pojected n the decton s N d cos N sn( d ) d cos... N N d ddcos, (S) whee ctn s the ngle of the medn t x wth espect to hozontl. To st ode n d nd d we hve N N N. (S) Now consde the pt of the dphgm bove the pllel concdng wth x ; ts foce blnce pojected n the z decton s P sn. (S) N Neo-hooken elstome The consttutve elton between pncpl stesses nd pncpl stetches fo n ncompessble neo-hooken sold s p, (S) whee the only mtel pmete s the she modulus, nd p s scl feld ntoduced s Lgnge multple to enfoce ncompessblty. The knemtcs of the homogeneous n-plne pestetch e z, (S4) notng tht by ncompessblty. z dx :, dx dx : (S9) dx Algebc substtuton of Eq s S9 nd S4 nto the consttutve eltonshp Eq. S gves Foce blnce We wll now pefom foce blnces n the nd z dectons (the decton s tvl by symmety). By ou tetment of the dphgm s thn, the net lne tenson coss the coss the dphgm s thckness N s smply the stess multpled by the thckness. Fst consde the dffeentl segment of N t t t t N t t t t t t ; (S5) Justn Beoz

4 notng tht z, t t z nd, gn by ncompessblty,. z Summy of equtons. The equtons deved bove e summzed below. The knemtc nd foce blnce equtons e stndd. The pme supescpt denotes ptl devtve wth espect to coodnte (Fg. S). Knemtcs: u u (S6) Foce blnce: N N N N sn P (S7) Consttutve eltons: N t t t N t t t t t (S8). (S) The foce blnce n the vetcl decton (Eq. S7) to ledng nonlne ode n s theefoe A B A P. (S) O, by mtchng the coeffcents of the Tylo sees fo the functons P () nd (), my be wtten B AP P A 6A 4. (S) The hghe-ode tem compses contbuton fom geomety nd fom stetchng of the mddle plne, nd these coespond espectvely to A nd B n the dv numeto. Notng tht d d nd k : P, d dv t follows fom Eq. S tht to ledng nonlne ode: P k 4 ( ) P k 4 k d Soluton to ledng non-lne ode To st ode n, the consttutve eltons (Eq. S8) become V d k k (S) P P (S4) 8k A N A B C. (S9) N A C B 6 t 6, B t 6, Ct The boundy condtons on the dl dsplcement u of ptcles n the dphgm e u( ) u( ). Obvously to st ode n, u ( ). To nd ode n we hve, howeve t cn be u() shown tht only stsfes the foce blnce n the decton (Eq. S7). Bsed on ths we ssume u ( ), so the knemtcs become k 6 8t 6 4 k k k k, k, P P. (S5) k 6 6 t 6 II. ADAPTOR ASSEMBLY The pts of the dpto e shown n exploded vew n Fg. S. The djustble heght h of the cylndcl chmbe (Fg. ) s detemned by the dstnce pt 5 s nseted nto pt ; ths cn be mesued extenlly wth clpes. The tp ntefce s emovble nd ntechngeble; nd we desgned septe ones fo the ppettng nges. μl, μl Justn Beoz 4

5 nd μl s shown n Fg. Sb. Fo the nges μl nd μl, the coespondng commecl ppette tps ttch n the usul wy. Fo the nge. μl, we use only the poton of μl ppette tp cut dstnce cm fom the tp s ofce s shown; the dotted lne epesents the wte column heght fo μl volume. Ths ws done becuse usng the ente ppette tp seves only to dd mpecson n the dwn volumes dung the ppettng expements due to lge V c. Tp ntefce 4 5 b ml ppette Ou ssembly pocedue fo the dphgm follows the sequentl steps n Fg. Sc:. A poton of neopene ubbe sheet (McMste 9K45) s cut fom oll nd then stetched homogeneously n-plne usng custom fxtue (Fg Sd). c Rubbe sheet. μl μl μl. Pts nd mte though opposte sdes of the stetched ubbe sheet, whch foms the ccul dphgm wth dus nd homogeneous nplne stetch λ. We hve sevel vesons of pts nd tht dffe n dus. d 4. The emnde of the stetched ubbe sheet s cut wy nd pts nd of e fstened togethe by pts nd 4. Dphgm Thoughout ths pocedue, the mount of stetch s montoed usng dgtl mcoscope (Celeston 44) to ecod the chnge n dmete of chlk ng stmped onto the secton of neopene tht ultmtely becomes the dphgm. Bsed on nlyss of mges tken befoe nd fte ssembly, the pestns n ou ssembled dphgms,.e.,, e homogeneous wthn % nd t s the vege of mesuements tken n pependcul dectons tht e epoted n Tble nd used fo dwng the model cuves n ll elevnt fgues. Povdng the pe-stetch cme bout n ou expementton fst fom the pctcl necessty to ensue no slck n the dphgm whle n ts undeflected poston. Howeve, the model nd expements demonstte tht tunng s convenent nd necessy fo optmzng the de-mplfcton to of the ppette. Stetched ubbe sheet FIG. S. () exploded vew of ll pts compsng the dpto component; (b) vew of ech tp ntefce pt the. μl pt s shown wth the cut poton of. μl commecl ppette tp ttched s used n the ppettng expements; (c) the sequentl pocedue fo ssemblng the dphgm fom ubbe sheet wth (d) mges tken t steps nd. III. BENCHTOP MEASUREMENT SETUP DETAILS As shown n Fg., the stoke V p s povded by synge wth ts pston (Ø8.9mm) ttched to motozed mcomete stge (Physk Instumente M- 9.6S, ±µm poston pecson). The synge s connected n flud communcton wth pessue senso (Omeg PX9-5G5V, -/tm guge pessue nge wth ±.5% FSO ccucy) s well s the dpto component. As shown n the fgue, the tp ntefce pt s emoved fom the dpto to expose the dphgm. The dphgm s deflected pofle () s mesued fom ths extenl sde wth Justn Beoz 5

6 lse poflomete (Keyence LJ-V78, mesuement pecson s.5µm n nd µm n ) whch ecods the hozontl lne pofle, vsble s the lne of blue lght, coss the dphgm s dmete. The poflomete s mounted on vetcl mcomete stge fo postonng. Dect mesuements of the volume V nsde the pptus s mpctcl becuse of the complcted ntenl geometes of the senso nd fttngs. Howeve the combnton of motozed pston, lse poflomete nd pessue senso povde dect mesuement of the s chnge n volume nd pessue. Bsed on these mesuements we clculte V ccodng to the del gs lw (Eq. 4) nd fnd excellent geement nd consstency between these clculted vlues wthn fcton of pecent. It s ths vlue of V tht s epoted fo expements pefomed wth ths setup. IV. HEAT TRANSFER IN THE AIR VOLUME Usng the mesuement setup, we ecoded the pessue P of the dung nd fte compesson fom pston stoke V p. We confgued the dpto component wth gd plug nsted of dphgm so tht when the pston comes to est t the end of ts pogmmed stoke, the volume V s fxed nd ny subsequent chnge n P eflects chnge n the s tempetue T, snce by del gs lw PV Nk T nd so we hve P ~ T. Fg S shows B the esults fo n mpulse V p t t compessng the 7% wth the dpto s cylndcl chmbe dmensons set to hr 5.4mm ; tnsence lstng lmost seconds s ppent when the chmbe contns only, nd ths tnsence dsppes upon ddng n the ppopte mount of steel wool. The pessue spke t t s pessue shock fom the pd moton nd bupt stoppge of the pston; t s nconsequentl wth espect to the tmescle of het tnsfe tht follows nd s nonexstent fo the smooth contolled pston moton equed fo ccute ppettng. The pd pston mpulse n these ptcul expements llowed us to show the pessue befoe nd fte t on one plot P P [kp] 8 4 b P P [kp] - Tme [sec] 8 4 FIG. S. Usng the mesuement setup, pessue ws ecoded t Hz fo mpulse pston dsplcements () wthout steel wool nd (b) wth. volume fcton of steel wool. The gy dshed lnes ndcte the themlly equlbted pessue t long tmes. wth tmescle ppopte fo vewng the tnsence fom het tnsfe. As dscussed n the mn text, the tnsence fom het tnsfe consttutes n mpecson fo the ppette becuse t cuses the dphgm to deflect wth the pessue chnge. Sd dffeently, the het tnsfe chnges the vlue of n Eq. 4, whch ffects the s compessblty s:. Volume fcton steel wool. Volume fcton steel wool - Tme [sec] VP VP. VP (S6) Note tht fo :.4, chnges by 8% nd the coespondng deflecton of the dphgm, ccodng to Eq., cn be sml. Tmescle of het tnsfe Let the tme t concde wth the pston comng to est t the end of ts stoke. In the ntevl t thee Justn Beoz 6

7 s no longe ny sgnfcnt souce of het geneton n the, so the het equton becomes -μl ppette T t u T T ; (S7) Pessue equlzton tb 5 whee u s the s velocty nd.9 m /s ts theml dffusvty. The tmescle of het tnsfe n Fg. S cn be tonlzed the followng wy. The dpto s cylndcl chmbe hs the lgest theml dffuson length n the mesuement setup, whch sets the ovell tmescle of het tnsfe. Snce hr 5.4mm, the dffuson length s L.7mm n ll coodnte dectons. The pesence of theml convecton cts to homogenze the tempetue, whch nceses the te of het tnsfe out of the volume by effectvely shotenng L. Theefoe n uppe estmte of the het tnsfe tmescle coesponds to u,.e., the L dffuson-lmted tmescle ~.sec. Ths s n geement wth the expement; the obseved tnsent tme s, pesumbly due to u. Addng steel wool shotens L by ove n ode of mgntude, so the tnsence fom het tnsfe becomes neglgble. The steel wool lso ensues tht the tmescle of convecton of the s neglgble snce the momentum dffuses fom the on tmescle ~ L ; beng the s knemtc vscosty nd, oughly, s. Volume fcton of steel wool Povded the volume does not lek when compessed (.e., the numbe of molecules N compsng t s constnt), t s ntenl enegy E cn be shown to be functon of only ts tempetue T. Consevton of enegy fo dffeentl chnge de s de dq P dv ; (S8) whee dq nd PdV e the het nd wok eneges expended nto the. In the lmt of sotheml compesson de, theefoe dq P dv nd the totl het dsspted fom the ove the couse of compesson s V Q PdV. In the lmt of dbtc compesson dq, theefoe de PdV nd the totl chnge n the s ntenl enegy fom compesson s V E P dv. Howeve E s pecsely the V mount of het dsspted fom the n the poceedng theml equlbton, so gn V Q P dv. Substtutng n Eq. 4 nd V ntegtng gves q Adpto Tp 5 5 P P,.4 dbtc ln, sotheml fo the het densty Tp ntefce (emovble) FIG. S4. Ppettng expements wee pefomed wth n dpto ftted between commecl μl ppette nd tp; the dpto ntefces wth the commecl ppette nd mesuement setup n the sme wy. V (S9) Q q to be bsobed by the V V V steel wool fo compesson to. The V R V h Rubbe sheet Justn Beoz 7

8 ctul mount of het bsobed s bounded by these dbtc nd sotheml lmts. A volume fcton Vsteel V of steel wool dded to the wll expeence tempetue dffeence q T c ; (S) steel p, steel whee the mss densty nd specfc het cpcty of steel e, espectvely, ρ steel = 785 kg/m nd c p,steel =.49 kj/kg K. Fo.,. gves T K, whch s suffcent fo the to compess sothemlly s shown n Fg. Sb. In the ppettng expements, we desgned fo. to be the mxmum compesson of the, nd so the cylndcl chmbes n the dptos ll contn. of steel wool. We chose to use steel wool fo convenence nd cost, howeve othe fbous/poous mtels my wok just s well. V. ADAPTORS PREPARED FOR PIPETTING EXPERIMENTS The dpto ntefces wth commecl µl ppette (VWR VE) s shown n Fg. S4. Fo the dptos used n the ppettng expements (Fg. nd Tble ), the coespondng model cuves (Eq. ) e plotted n compson to the del demplfcton tos n Fg. S5. The del demplfctons (ed lnes) closely ovely the model cuves (sold blck lnes) n the left-hnd column; the dffeences e plotted n the ght-hnd column nd consttute n eo n the ppette s ccucy. Compng to Fg. c, the mgntude of ths nccucy fo ech dpto s wthn the mpecson of the ppetted volumes, so fom pctcl pont of vew the de-mplfcton of the µl ppette ppes lne. Also plotted n Fg. S5 e the model cuves fo whch the nonlne behvo of the nd dphgm e neglected,.e., k (dshed lnes, left-hnd column). These cuves e sublne becuse the volume chnges s de-mplfcton / / / [mm] t [mm] [MP] V [ml] P [kp].4..9 T [C]..9. % RH TABLE S. System nd envonment pmetes dung mesuement wth the benchtop setup. V V Vp fom the epostonng of the pston nsde the ppette. Fo ech dpto, the dffeence between these cuves nd the del de-mplfcton s % t mxmum V p, ndctng, s n Fg., the mpotnce of ccountng fo ledng-ode nonlne behvo. In ptcul, the stffenng of the plys fvoble ole n chevng ppoxmte lne de-mplfcton hee. Afte ssemblng the dptos wth the dphgms, we tested ech n the mesuement setup; the pmetes e lsted n Tble S nd the esults e shown column wse n Fg. S6. Note tht V n these mesuements e not the sme s the optmzed vlues lsted n Tble nd used n the ppettng expements. Ths s becuse the ntenl geometes of the mesuement setup nd the µl ppette e dffeent. We pogmmed the pston so tht the end poston of ll stokes ws the sme n ode to mmc the opeton of the ppette. The model cuves ft wthn mesuement eo fo ech dpto. Fo the / de-mplfcton dpto, the mgntude of the dphgm s deflecton () s compble to the sufce oughness of the ubbe sheet (Fg. S6) nd theefoe the mesuement mpecson n V s eltvely lge. The model stll ppes to be ppoxmte hee even though ts vldty s not gunteed. d Justn Beoz 8

9 .4 b c Dphgm dsplcement Vd [μl] Eo [μl] Dphgm dsplcement Vd [μl] Dphgm dsplcement Vd [μl] Pston dsplcement V p [μl] Eo [μl] Eo [μl] -.4 Pston dsplcement V p [μl].5 x- Pston dsplcement V p [μl] -.5 Pston dsplcement V p [μl] 6 x-4 Pston dsplcement V p [μl] -6 Pston dsplcement V p [μl] FIG. S5. Model cuves coespondng to the dptos nd pmetes lsted n Tble ; the de-mplfctons e () /, (b) /, nd (c) /. The del de-mplfcton (ed lnes) nd model cuve (sold blck lnes, Eq. ) closely ovely n the left-hnd column plots nd the dffeence s the eo shown n the ght-hnd column plots. The dotted lnes e model cuves fo whch k = β = β =. Justn Beoz 9

10 / de-mplfcton / de-mplfcton / de-mplfcton.6 - ω() [mm]. ω() [mm] ω() [mm].4 [mm] -5 [mm] - 5 [mm] b 9 Vd [μl] P P [kp] P P [kp] 4 P P [kp] 4 Vd [μl] 4 Vd [μl]. c 4 9 P [kp] P [kp] P [kp] 4 Compesson to. Compesson to. Compesson to.8 d Vp [μl] Vd [μl] Vd [μl] 4 Vd [μl] Vp [μl] Vp [μl] 5 FIG. S6. Mesuements tken wth the mesuement setup fo ech dpto used n the ppettng expements e dsplyed column wse; the system nd envonment pmetes e lsted n Tble S.The sold lnes e model cuves whch coespond to () Eq. b, (b) ntegted fom of Eq., (c) ntegted fom of Eq. 5, nd (d) Eq.. Repesenttve eo bs e gven n the coodnte dectons whee the mesuement pecson s lge thn the mke sze. Justn Beoz

11 VI. PIPETTING EXPERIMENTS: MATERIALS AND METHODS All ppettng expements epoted n Fg. wee pefomed n humdty-contolled oom set t 5%RH. We dew deonzed wte fom beke usng the stndd fowd ppettng technque depcted n Fg. b. Fo volume nges µl, µl nd µl, we used VWR dsposble ppette tps (ctlog numbes 87, 8979 nd 466, espectvely) nd dspensed the dwn volumes nto glss contne estng on mcoblnce (OHAUS DV5CD,.µ mesuement esoluton). The volume ws nfeed bsed on weght. Fo the µl nge we lso enclosed the weghng pltfom nd glss contne wth shm stock to mnmze mesuement mpecson fom cuents, wte evpoton nd sttc chge (Fg. S7). The wte volumes wee dspensed though sngle hole n the suspended pece of shm stock nd nto the glss contne below. Fo the. µl nge we used the µl VWR ppette tp (ctlog numbe 466), cut s shown n Fg. Sb. Dwn volumes wee mesued optclly wth CMOS cme (Edmund Optcs, EO-C) nd X telecentc objectve (Mtutoyo, ) s shown n Fg. S7b. The ppette ws fxed on spngloded vetcl slde so tht t could be loweed mnully to dw wte volume fom pet dsh, nd then sed bck to consstent poston wthn b Cme Ppette stnd Illumnto Ppettng hole Suspended shm stock.μl Glss contne FIG. S7. The dwn wte volumes n Fg. wee mesued () gvmetclly fo μl, μl nd μl nges; nd (b) optclly fo the.-μl nge. the mcoscope s vew. The ppette tp ws bcklt by collmted lght souce n ode to vew the heght of the wte column. Ceful mesuements wee tken of the ppette tp s extenl nd ntenl dmensons so tht the volume of the dwn wte column ws known bsed on ts heght. μl Justn Beoz

12 TABLE S. Detled lstng of the ccucy nd pecson fo ll ppetted volumes n Fg.. De-mp. ISO ± 8. (.) -.58 (.9) (.5).4 (.77).47 (.49) (.68) -.8 (.785) -.4 (.49) -.4 (.65).6 (.546).976 (.68) / ±.8 (.) -.7 (.4). (.9) -.58 (.).8 (.9).6 (.49).4 (.96). (.44).87 (.66).85 (.7).7 (.79) / ±. (.8) -. (.4).4 (.) -. (.4).8 (.6).5 (.).4 (.54).6 (.6).6 (.). (.5).6 (.7) / ±.5 (.5).4 (.) -.7 (.8). (.4).7 (.) -. (.6) -.7 (.4) -.4 (.4) -. (.8) -.5 (.). (.) The entes undeneth the ttles n the top ow e the followng: the De-mp column lsts the de-mplfcton to of the dpto fom Tble ; the ISO column lsts the llowed ± devton of the men nd (stndd devton) by ISO 8655 n mcoltes; the numbes to e the settngs fo V s n mcoltes, nd ech column lsts the coespondng devton of the men nd (stndd devton) fo 5 expementlly mesued lqud volumes n mcoltes. Justn Beoz