Inversion of electrical mobility measurements using bipolar or unipolar chargers for the arbitrary distribution of channels

Please ci e his a icle in p ess as: Doma , M., e al. In e sion o elec ical mobili y measu emen s using bipola o unipola cha ge s o he a bi a y dis ibu ion o channels. Pa icuology (2014), h p://dx.doi.o g/10.1016/j.pa ic.2014.08.007 ARTICLE IN PRESS G Model PARTIC 738 1–10 Pa icuology xxx (2014) xxx–xxx Con en s lis s a ailable a ScienceDi ec Pa icuology jou nal homepage: www.else ie .com/loca e/pa ic In e sion o elec ical mobili y measu emen s using bipola o unipola cha ge s o he a bi a y dis ibu ion o channels 1 2 M. Doma a,∗,F.E. K uisb,N.L. Azong-Wa ab,J.M. Fe nandez-Diaza Q1 3 aDepa men o Physics, Uni e si y o O iedo, C/ Cal o So elo, s/n, E-33007 O iedo, Spain4 bIns i u e o Technology o Nanos uc u es (NST) and Cen e o Nanoin eg a ion Duisbu g-Essen (CENIDE), Uni e si y o Duisbu g-Essen, Bisma cks . 81, D-47057 Duisbu g, Ge many 5 6 7 a icle in o 8 9 A icle his o y: 10 Recei ed 16 Ma ch 201411 Recei ed in e ised o m 1 Augus 201412 Accep ed 15 Augus 201413 14 Keywo ds:15 Tandem di e en ial mobili y analysis16 In e se p oblem17 Regula iza ion me hods18 Size dis ibu ion19 T ans e unc ion20 Unipola cha ge 21 abs ac The in e sion o he pa icle size dis ibu ion om elec ical mobili y measu emen s is analyzed. Th ee di e en me hods a e adap ed o a do -ma ix app oach o he p oblem, especially o non-squa e o singula ma ices, and applied o elec ical mobili y measu emen s om fixed o scanning ol ages. Mul- iply cha ged pa icles, di usion losses, a bi a y ol age s eps, and noise we e conside ed, which esul s in non-adjoining and o e lapping ans e unc ions. The indi idual con ibu ion o he ans e unc- ions in each size in e al was geome ically es ima ed, which equi es only i s cha ac e is ic mobili ies. The me hodology is applied o mobili y measu emen s om pa icles cha ged wi h unipola and bipo- la cha ge s. Howe e , he me hod can be ex apola ed o any cha ging me hod wi h a defined cha ge dis ibu ion, and e ie al o he singly cha ged pa icle dis ibu ion and mean cha ge om a andem di e en ial mobili y analysis configu a ion was success ully demons a ed. © 2014 Published by Else ie B.V. on behal o Chinese Socie y o Pa icuology and Ins i u e o P ocess Enginee ing, Chinese Academy o Sciences. 22 In oduc ion23 Q3 Knowledge o he pa icle size dis ibu ion (PSD) is a unda-24 men al equisi e o cha ac e ize an ae osol popula ion. Di e en ial25 mobili y analysis (DMA) is he mos common p inciple o measu e26 submic on PSDs, because i is able o cha ac e ize a wide ange o 27 sizes wi h high esolu ion in almos eal ime independen ly o he28 composi ion o he pa icles.29 Howe e , he PSD canno be di ec ly measu ed and mus be30 in e ed om ela ed mobili y measu emen s using in e sion ech- 31 niques. The fi s a emp a in e sion was by Knu son and Whi by 32 (1975a, 1975b) based on in eg a ion o he pa icle ajec o y equa-33 ion inside he DMA. The pa icle size was ep esen ed by he size34 Abb e ia ions: CN, condi ion numbe ; CPC, condensa ion pa icle coun e ; DMA, di e en ial mobili y analyze ; DMPS, di e en ial mobili y pa icle size ; FWHM, ull wid h a hal maximum; GSD, geome ic s anda d de ia ion; GCV, gene alized c oss- alida ion; LC, L-Cu e; LS, leas squa es; NNLS, non-nega i e leas squa es; NRMSE, no malized oo mean squa e e o ; PSD, pa icle size dis ibu ion; SMPS, scanning mobili y pa icle size ; SVD, singula alue decomposi ion; TDMA, andem di e en ial mobili y analysis; TF, ans e unc ion. ∗Co esponding au ho . Tel.: +34 659390180. Q2 E-mail add ess: [email p o ec ed] (M. Doma ). o a sphe e wi h he same mobili y Zpas he pa icle, which is called 35 he mobili y diame e (DZp): 36 Zp=qeCc(DZp) 3DZp ,(1) 37 whe e qis he numbe o elemen a y cha ges eon he pa icle, is 38 he dynamic iscosi y o he gas flow, and Cc(DZp) is he Cunning- 39 ham slip co ec ion ac o . 40 The pa icle size dis ibu ion N(Dp) and pa icle mobili y dis- 41 ibu ion M(DZp) a e ela ed h ough he non-nega i e Ke nel 42 unc ion Kq(Dp,D Zp) by a F edholm- ype in eg al equa ion ha also 43 conside s he unce ain y in he measu emen s ε(DZp): 44 M(DZp)=Dp max Dp min Kq(Dp,D Zp)N(Dp)dDp+ε(DZp).(2) 45 When he Ke nel is linea wi h espec o he size dis ibu ion, 46 Eq. (2) can be exp essed in ma ix o m as 47 Mm≈Kq mnNn+εm.(3) 48 The equa ion is app oxima e because o he e o εm∈Rmno 49 only in he da a, bu also in he app oach o he in eg al (Eq. (2)) oa 50 se o disc e e linea equa ions. The indices mand na e he numbe 51 o disc e e measu emen s and ou pu channels, espec i ely. 52 h p://dx.doi.o g/10.1016/j.pa ic.2014.08.007 1674-2001/© 2014 Published by Else ie B.V. on behal o Chinese Socie y o Pa icuology and Ins i u e o P ocess Enginee ing, Chinese Academy o Sciences. Please ci e his a icle in p ess as: Doma , M., e al. In e sion o elec ical mobili y measu emen s using bipola o unipola cha ge s o he a bi a y dis ibu ion o channels. Pa icuology (2014), h p://dx.doi.o g/10.1016/j.pa ic.2014.08.007 ARTICLE IN PRESS G Model PARTIC 738 1–10 2M. Doma e al. / Pa icuology xxx (2014) xxx–xxx Nomencla u e Aa ea o he ans e unc ion CcCunningham slip co ec ion ac o d a io be ween diame e s Dgap e ical dis ance om needle ip o ou pu o ioniz- ing egion in co ona Dppa icle diame e Dpg geome ic mean diame e DZpmobili y diame e eelemen a y uni o cha ge (1.602 ×10−19)C c ac ion o cha ged pa icles Hheigh o he fi ing TF iangle kBBol zmann’s cons an (1.380 ×10−23) J/K Kke nel ma ix Lleng h o DMA mnumbe o measu emen in e als Mpa icle mobili y dis ibu ion nnumbe o ou pu in e als Npa icle size dis ibu ion Pp essu e Pnpene a ion e ficiency Pe Pecle numbe qnumbe o elemen a y cha ges on a pa icle Qiion dilu ion flow Qaae osol flow Qex excess flow Qmae osol monodispe se flow Qsh shea h flow Ri,R0inne and ou e adius o he DMA elec odes esidence ime V0 ol age applied o he DMA cen al od Zccen oid mobili y o ans e unc ion Zl,Zulowe and uppe bounda ies om he mobili y TF in e als Z∗ l,Z∗ ulowe and uppe bounda ies om he size in e als Zpelec ical pa icle mobili y G eek le e s ˇ0 ela i e wid h o he TF εe o  egula iza ion pa ame e dynamic iscosi y o gases ggeome ic s anda d de ia ion scan ime  ans e unc ion (ma h.) One o he main d awbacks o his echnique is ha la ge pa -53 icles wi h mul iple cha ges can ha e he same mobili y as small54 singly cha ged pa icles. I is an ill-posed p oblem in he sense ha 55 a s able solu ion may no exis o may no be unique.56 In gene al, he ank o he con ol ma ix Kq mn de e mines he57 eliabili y o finding a solu ion. Fo de e mined sys ems (m=n), 58 he e may be a unique solu ion. I he sys em is unde -de e mined59 (m<n), i has many possible solu ions, whe eas i m>ni is an60 o e de e mined sys em and he e is no exac solu ion (Vou ilainen,61 2001; Talukda & Swiha , 2003).62 Da a in e sion echnique63 The ill-posedness o he p oblem can be ec ified by eplac-64 ing i wi h an app oxima e well-posed p oblem whose solu ion is65 assumed o be close o he ac ual PSD. The main me hodology o66 sol e Eq. (3) is based on me hods de i ed om he leas squa es 67 (LS): 68 NLS =a g min KNin −M .(4) 69 In his wo k, only de e minis ic echniques we e applied, such 70 as i e a i e (Alo s & Balakuma , 1982; Bazan & F ancisco, 2009; 71 Collins, Flagan, & Sein eld, 2002; C ump & Sein eld, 1981; Fiebig, 72 S ein, Sch öde , Feldpausch, & Pe zold, 2005; Hagen & Alo s, 1983; 73 P ei e e al., 2013; Rojas & S eihaug, 2002; Talukda & Swiha , 74 2003; Twomey, 1975), egula iza ion (Hansen & O’Lea y, 1993; 75 Talukda & Swiha , 2003), o linea in e sion me hods (Hagen & 76 Alo s, 1983), al hough ecen ly he s a is ical echniques ha e been 77 imp o ed (Dubey & Dhaniyala, 2013; Vou ilainen, 2001). 78 When he con ol ma ix is s ongly ill-condi ioned, small e o s 79 in he da a can be g ea ly magnified in he solu ion, e en allowing 80 nega i e alues o he PSD. One way o a oid nega i e alues is 81 o use he non-nega i e leas squa es (NNLS) algo i hm (Lawson & 82 Hanson, 1974), which is a e o mula ion o he LS solu ion in which 83 a dual ec o w=KT(M−KN) o ces he solu ion o be posi i e: 84 NN=a g min KNin −M ;NN≥0.(5) 85 The bigges d awback o his me hod is he con e gence: he e 86 is no single ac o iza ion and he esul s can con e ge o a di e en 87 local minimum. Mo eo e , i is only alid o sys ems whe e m≥n,88 excluding unde -de e mined p oblems. 89 The Tikhono egula iza ion (Willoughby, 1979) is widely used 90 o sol e p oblems o signal p ocessing such as noise educ ion, 91 image es o a ion, and da a in e sion. The me hod eplaces Eq. (4) 92 by a minimiza ion p oblem: 93 N␭=N∈Rna g min KN−M  2+ N  2,(6) 94 whe e is he egula iza ion pa ame e , which is a posi i e con- 95 s an ha balances he fi and smoo hness o he dis ibu ion and 96 is chosen such ha Nbecomes as close as possible o he noise- ee 97 solu ion. 98 The main ad an age o he Tikhono egula iza ion is ha diag- 99 nosis o he p oblem can be made h ough he singula alue 100 decomposi ion (SVD). I achie es a pseudo-in e se o he ke nel 101 ma ix, K+, by disagg ega ing i in o wo o hogonal ma ices, 102 U∈Rm×mand W∈Rn×n, and a diagonal ma ix o med by non- 103 nega i e elemen s, he singula alues,˙∈Rm×n:104 K+=W˙+UT= k wkuT k k .(7) 105 Exp essing Eq. (6) in ec o ized o m, he in e ed PSD is 106 N␭=(K+K+II)−1K+M=  k=1 k uT kM k wk,(8) 107 whe e ka e he fil e ac o s o Wiene weigh s: 108 k=2 k 2 k+.(9) 109 The L-Cu e (Hagen & Alo s, 1983; Hansen & O’Lea y, 1993; 110 Lloyd, Taylo , Lawson, & Shields, 1997) is a powe ul ool o es i- 111 ma ing he op imal egula iza ion pa ame e . In b ie , i consis s o 112 minimizing Eq. (8) o each alue o un il he poin o maximum 113 cu a u e is ound (Johns on & Gul ajani, 2000). Ano he egula - 114 iza ion me hod is based on gene alized c oss- alida ion (GCV). I 115 Please ci e his a icle in p ess as: Doma , M., e al. In e sion o elec ical mobili y measu emen s using bipola o unipola cha ge s o he a bi a y dis ibu ion o channels. Pa icuology (2014), h p://dx.doi.o g/10.1016/j.pa ic.2014.08.007 ARTICLE IN PRESS G Model PARTIC 738 1–10 M. Doma e al. / Pa icuology xxx (2014) xxx–xxx 3 de e mines a educed solu ion omi ing one da a poin and ecal-116 cula ing. The op imal alue is he minimize o he unc ion:117 NG=||KN␭−M||2 [ ace(II −K(K+K+II)−1K+)]2,(10)118 whe e ace()is he sum o he diagonal elemen s (C ump & 119 Sein eld, 1981).120 The sensi i i y o he solu ions o inaccu acies in he da a is121 iden ified by he condi ion numbe (CN), which is he a io o he122 la ges o he smalles singula alue o he han ze o. The la ge 123 he condi ion numbe , he g ea e he ill-condi ioning, which is124 infini e o singula ma ices. I s loga i hm is an es ima e o how125 many digi s a e los in he p ocess.126 Geome ical app oach o he DMA ans e unc ion127 The p obabili y o pa icles lea ing he DMA wi h he classified128 flow is de e mined by he DMA ans e unc ion (TF), ˝q mn. Fo an129 ideal case, he TF can be fi ed by a iangula shape cen e ed a 130 he mean mobili y Zc.Zcdepends on he ol age Vapplied o he131 DMA and on i s geome y, which is assumed o be cylind ical wi h 132 Riand R0as he inne and ou e adii o he DMA elec odes and L133 he leng h be ween he en ance sli o he ae osol flow Qaand he134 exi sli o he classified flow Qm:135 Zc=ln (R0Ri) 4L (Qsh +Qex) V.(11) 136 The ela i e wid h ˇ0o he TF only depends on he flow a es, 137 conside ing also he shea h Qsh and excess Qex flow a es, hen138 ˇ0=Zc Zc=Qa+Qm Qsh +Qex .(12)139 Fo he pa icle de ec o (i.e., CPC Model 3775) used in he140 p esen expe imen s (TSI Inc., 2007), he ae osol- o-shea h ai flow141 a io was kep cons an a ˇ0= 0.1, ensu ing su ficien esolu ion142 ᏾=ˇ−1 0.143 The DMA ol age can be a ied in s epping o scanning mode:144 DMPS o SMPS (di e en ial o scanning mobili y pa icle sizing).145 SMPS is exponen ially amped a fixed alues wi h a cons an a io146 V( )=V0e± /, whe e is he scan ime, is he esidence ime, and147 V0 he ol age applied o he DMA cen al od. A slow scan a es148 (ou case, wi h = 120 s and = 0.1567 s), he esul ing TF o he 149 SMPS sys em can be app oxima ed o he fixed TF (He & Dhaniyala,150 2013), al hough i will be shown ha he p oposed p ocedu e is151 equally alid o bo h me hods.152 The di usion o pa icles caused by B ownian mo ion enhances 153 pa icle losses, and he eby influences he shape o he TF, because154 i is b oadened and esolu ion is educed. An analy ical exp ession155 o he ela i e ull wid h a hal maximum (FWHM) o a iangula 156 TF assuming di usional b oadening, , was ob ained by Ka lsson157 and Ma insson (2003) o any DMA ype:158 =1+0.15 ˇ021 2 0−1−12 ,(13)159 whe e 0is he expe imen al b oadening exp ession o a Vienna160 ype DMA wi h ˇ0= 0.15 and Pe is he Pecle numbe :161 0=1.69 1−exp −Pe 105 0.33−0.69.(14)162 S olzenbu g (1988) p oposed a ealis ic bell-shaped TF model, 163 while S a mann, Kau eld , Hummes, and Fissan (1997) showed 164 ha he iangula shape is a good app oxima ion because he e o 165 in oduced is below he expe imen al unce ain y. A geome ical TF 166 Fig. 1. T ans e unc ion o singly cha ged pa icles calcula ed in his wo k. The mobili y in e als ma e ep esen ed by he cen al mobili y o each iangula TF, and he size in e als nby he e ical lines. Bo h in e als can be independen ly chosen, gi ing a di e en co e age o he TF a ea and hus di e en p ecision in he in e sion p ocess. can be calcula ed simply knowing he FWHM (namely, Zc) and he 167 heigh o he fi ing iangle H:168 Zc=ˇ0 Zc,(15) 169 H=Pn,(16) 170 whe e he pene a ion e ficiency Pn(Dp) was desc ibed in de ail by 171 Reis (1993). The maximum heigh o he ans e unc ion is ide- 172 ally 1 when pa icle losses a e igno ed. Howe e , as pa icle size is 173 educed, di usion enhances pa icle losses and he wid h o he TF 174 b oadens as i s heigh dec eases, as shown in Fig. 1.175 Es ima ion o he TF a ea wi hin e e y size egion (numbe ed 176 om I o VII in Fig. 2) is calcula ed compa ing he lowe and uppe 177 bounda ies om he TF, Zl(q,m) and Zu(q,m), and om he size 178 in e als, Z∗ l(q, n) and Z∗ u(q, n). Table 1 shows all o he final alues, 179 whe e calcula ion o sec ion “Resul s and discussion” is shown o 180 q= 1 as an example. In his pa icula case, he mobili ies ollow 181 Zc(m)≥Z∗ u(n)>Z ∗ l(n)>Z l(m), and he co esponding TF a ea is cal- 182 cula ed om he polygonal shape among he o al a ea: 183 A˝(m, n)=1 2a(b+c),(17) 184 Fig. 2. Example o he geome ical a ea es ima ion o he TF wi hin he di e en size in e als. Please ci e his a icle in p ess as: Doma , M., e al. In e sion o elec ical mobili y measu emen s using bipola o unipola cha ge s o he a bi a y dis ibu ion o channels. Pa icuology (2014), h p://dx.doi.o g/10.1016/j.pa ic.2014.08.007 ARTICLE IN PRESS G Model PARTIC 738 1–10 4M. Doma e al. / Pa icuology xxx (2014) xxx–xxx Table 1 Geome ical calcula ion o he TF a ea wi hin each size in e al. The dependence (m) o (n) wi h he in e al is omi ed and he a iable J=H/2Zcis defined o simpli y he equa ions. Sec ions A–C o igina ed when wide size in e als co e wide a eas o he TF, desc ibed he e as a sum o se e al sec ions o Fig.2. Sec ion Bound TF a ea IZ∗ u≤Zl0 II Zc≥Z∗ u>Z l≥Z∗ lJ(Z∗ u−Zl)2 III Zc≥Z∗ u>Z ∗ l≥ZlZ∗ pJ(Z∗ u+Z∗ l−2Zl) IV Zu>Z ∗ u>Z c>Z ∗ l>Z lJ[(Zu−Z∗ u)(Z∗ u−Zc)+ (Z∗ l−Zl)(Zc−Z∗ l)+ Z∗ pZc] VZu>Z ∗ u>Z ∗ l≥ZcZ∗ pJ(2Zu−Z∗ u−Z∗ l) VI Z∗ u≥Zu>Z ∗ l≥ZcJ(Zu−Z∗ l)2 VII Z∗ l≥Zu0 A: (II + III + IV) Zu>Z ∗ u>Z c>Z l≥Z∗ lJ(2Z2 c−(Zu−Z∗ u)2) B:(IV+V+VI) Z∗ u≥Zu>Z c>Z ∗ l>Z lJ(2Z2 c−(Z∗ l−Zl)2) C:(II+III+IV+V+VI) Z∗ u≥Zu>Z c>Z l≥Z∗ lHZc whe e a=Z∗ p(n)=Z∗ u(n)−Z∗ l(n) and he unknown heigh s band185 ca e 186 b=H(m) Zc(m)[Z∗ u(n)−Zl(m)] and c=H(m) Zc(m)[Z∗ l(n)−Zl(m)].(18)187 Subs i u ing Eq. (18) in o Eq. (17), he TF a ea is ob ained188 depending only on known mobili ies:189 A˝(m, n)=H(m)Z∗ p(n) 2Zc(m)(Z∗ u(n)+Z∗ l(n)−2Zl).(19)190 In many ins umen s, such as in TSI SMPS-3080 se ies de ices,191 he numbe o channels pe decade o bo h inpu and ou pu 192 da a a e fixed o a mul iple o 4, indica ing ha m=n.In he193 p esen wo k, he numbe o mobili y measu emen s and ou pu 194 size channels a e independen and can be chosen a bi a ily, gi ing 195 o e lapping TF in e als ha can co e mo e mobili ies when he196 TF is b oadened by di usion.197 The numbe o in e als is es ima ed h ough he numbe o 198 channels pe decade measu ed by he ins umen ; hey a e ela ed199 by he exp ession o he a io be ween diame e s o he PSD log-200 dis ibu ion, d= ln(Dp(i)/Dp(i−1)):201 d=ln (DpUp/DpLow) num Decades =ln(DpMax/DpMin) num In e als ,(20) 202 whe e DpUp and DpLow a e usually in a a io o DpUp/DpLow = 10 o 203 he decade, and DpMax and DpMin a e he bounda y diame e s o he204 PSD ange. Modi ying he numbe o in e als (mo n), he co e ed205 a ea o he TF will change acco dingly. 206 The TF is c ucial o con ol ma ix es ima ion, which also 207 depends on he ac ion o cha ge c(q,Dp) and he inpu o ou pu 208 flow a io. The a ia ion o hese alues, along wi h N(Dp), is small209 ac oss he wid h o he in eg a ion in e als. F om he geome ical 210 TF es ima ion, ˝(q,Zp,Zc)=q˝(Zp,Zc), Eq. (2) becomes211 dM(DZp) dln DZp=Qa Qm q q c(q, Dp)Pn(Dp)dN(Dp) dln Dp 212 ×Dp max Dp min ˝(Zp,Z c)dDp+ε(DZp).(21) 213 214 Changing he in eg a ion a iable simplifies he calcula ion o 215 he TF a ea: 216 Zp max Zp min ˝(Zp,Z c) dln Dp dZp dZp=A˝(m, n).(22)217 F omFig. 3, he p oposed geome ic me hod shows simila 218 esul s o he bell-shaped TF o he o al a ea, wi h a ela i e e o 219 Fig. 3. Compa ison o he TF a eas calcula ed wi h he bell-shaped model om S olzenbu g (1988) and he iangula shape o he ideal case (uppe se o cu es) and conside ing losses (lowe se o cu es), including he SMPS analysis es ima ed wi h he geome ic TF me hod. Bo h me hods co e p ac ically he same a ea wi h a di e ence o abou 1%. below 1% o pa icles >2 nm o bo h ideal (no losses) and eal 220 app oxima ions. 221 Fo SMPS measu emen s, he mobili y shi depends on he a io 222 be ween esidence and scanning imes, which was expe imen ally 223 de e mined by Dubey and Dhaniyala (2008) o a nano-DMA, whe e 224 he a ea o he iangula scanning TF is ime-dependen : 225 A˝( )=1 2 H =ˇ01+k1 +k2 2.(23) 226 The cons an s k1and k2depend on he flow a io, and he heigh 227 H=2A˝( )/ is he same o he empo al and mobili y based TFs. 228 Combined wi h he exp ession o he ol age scan a io, he em- 229 po al and mobili y-based a eas o he TF a e ela ed by 230 A˝(Zp)=A˝( )Zp ln(Zu/Zl).(24) 231 The esul s o he ideal and eal cases a e shown in Fig. 3. The 232 di e ences be ween he esul s om scanned and fixed ol age 233 a eas, which a e less han 2% o pa icle sizes abo e 15 nm when 234 losses a e conside ed, a e p obably because o he dependence o 235 he empo al a ea A˝( ) on he scan ime and he mobili y shi in 236 he con e sion o A˝(Zp). 237 Pa icle size and cha ge dis ibu ion e ie al om any 238 cha ging me hod 239 Calcula ion o he PSD equi es accu a e in o ma ion abou 240 he cha ge dis ibu ion o he incoming ae osol. Al hough bipo- 241 la adioac i e cha ge s ha e been adi ionally used because o 242 hei well-defined cha ge dis ibu ions, unipola di usion cha ge s 243 a ain highe cha ging e ficiencies while a oiding he se e e sa e y 244 es ic ions o bipola adioac i e cha ge s. Some ecen unipo- 245 la de ices a e based on X- ays (Kim, Kim, Kwon, & Pa k, 2011), 246 UV pho oioniza ion (Hon a˜ non & K uis, 2008), co ona discha ge 247 (Chien, Tsai, Chen, Lin, & Wu, 2011; Doma , K uis, & Fe nandez- 248 Diaz, 2014b; In a, 2012; Li & Chen, 2011; Qi, Chen, & Pui, 2007; 249 Tsai, Lin, Chen, Huang, & Alonso, 2010; Vi as, Hon a˜ non, & Schmid - 250 O , 2008)and o he s (Kwon, Fujimo o, Kuga, Saku ai, & Se o, 2005; 251 Kwon, Saku ai, & Se o, 2007). 252 Howe e , in e sion me hods a e seldom applied o unipo- 253 la cha ge s, possibly because hese de ices a e o en in he 254 expe imen al phase and he e is no homogeneous exp ession o 255 hei cha ge dis ibu ion, such as he e is o bipola adioac i e 256 Please ci e his a icle in p ess as: Doma , M., e al. In e sion o elec ical mobili y measu emen s using bipola o unipola cha ge s o he a bi a y dis ibu ion o channels. Pa icuology (2014), h p://dx.doi.o g/10.1016/j.pa ic.2014.08.007 ARTICLE IN PRESS G Model PARTIC 738 1–10 M. Doma e al. / Pa icuology xxx (2014) xxx–xxx 5 Table 2 Coe ficien s o he log-no mal fi ing o he cha ge ac ions om he co ona cha ge epo ed by Doma e al. (2014b). Dgap (mm) q 0(Dp)¯ Dpg (nm) gCR 2 3 1−0.02065 20.4725 1.6373 13.44060 0.98987 2 0.00192 33.6301 1.3967 12.86995 0.96662 3 0.00131 47.0735 1.2406 10.62972 0.99802 4 0.00125 69.9434 1.2823 16.75291 0.99868 5 6.69 ×10−5102.5040 1.3247 25.18727 0.99867 10 1−0.02375 66.9368 2.0618 48.98775 0.96978 2−2.44 ×10−484.9966 1.5394 26.87955 0.99990 3−7.78 ×10−5134.0929 1.5506 23.01109 0.99934 cha ge s (Wiedensohle e al., 2012). Fu he mo e, he a io o mul-257 iply cha ged pa icles in unipola cha ge s is highe han in bipola 258 cha ge s, complica ing he mul iple cha ge co ec ion in he decon- 259 olu ion.260 Once he cha ge dis ibu ion is known, i can be fi ed by a261 ep oducible exp ession (Kaminski e al., 2012). The e a e analy ic262 models o es ima e he cha ge ac ion, such as he bi h-and-263 dea h model (Adachi, Kousaka, & Okuyama, 1985; Biskos, Rea ell, & 264 Collings, 2005; Boisd on & B ock, 1970; Fuchs, 1964). Howe e , we 265 chose simple empi ical fi ing ( he log-no mal dis ibu ion shown266 in Eq. (25)) because analy ic models do no accoun o he loss o 267 pa icles and ions in a fi s app oxima ion and hese assump ions268 in oduce mo e unce ain y in he in e sion p ocess when fi ing269 o he expe imen al da a. 270 c(Dp)= 0(Dp)+C √2ln gDp exp −ln (Dp/¯ Dpg)2 2ln2 g,(25)271 whe e a possible o se is deno ed by 0(Dp), Cis he p obabili y o 272 acqui ing cha ges, ¯ Dpg is he geome ic mean diame e , and gis273 he geome ic s anda d de ia ion.274 Cha ge ac ions om a p e iously de eloped unipola co ona275 cha ge (Doma e al., 2014b) we e fi ed. The cha ge had sep-276 a a e ioniza ion and pa icle cha ging zones, and i s cha ging 277 pe o mance was ob ained expe imen ally wi h concen a ions o 278 monodispe se pa icles be ween 6 and 60 nm below 106cm−3. The279 discha ge elec ode was a needle moun ed on a mic ome e sc ew, 280 allowing he dis ance o he walls (namely, Dgap) o change, he eby281 a ying he Ni -p oduc in he cha ging zone be ween 2.4 ×1012 282 and 1.8 ×1013 s/m3(Doma , K uis, & Fe nandez-Diaz, 2014a). Thus,283 di e en cha ging le els can be a ained, wi h he aim o imp o ing 284 he concen a ion o singly cha ged pa icles. The fi ing pa ame-285 e s a e lis ed in Table 2 o 3 and 10 mm gap dis ances, along wi h286 he eg ession coe ficien R2.287 Mul iple cha ge co ec ion288 One o he di ficul ies o accu a e e ie al o he PSD is he289 mul iple cha ging o pa icles, which causes a p opo ion o he290 pa icles measu ed unde he same applied ol age o co espond291 o pa icles wi h di e en diame e s. Hoppel (1978) de eloped a 292 echnique in which Eq. (3) can be disagg ega ed depending on he293 cha ge con ibu ion:294 Mq m≈ n o  n=1(K1 mnN1 n+ε1 m)+ q o  q>1 (Kq mnNq n+εq m)≈M1 m+Mq>1 m.295 (26)296 297 The e o e, e ie al o he singly cha ged PSD would consis o 298 in e ing M1 m=Mq m−Mq>1 m ollowing some ini ial s eps. Fi s , es i-299 ma ion o M1 massuming only singly cha ged pa icles, leading o a300 fi s app oxima ion o N1 n ha will be used o calcula e he con i- 301 bu ion o he penul ima e channel o he mobili y measu emen s, 302 Mq>1 m−1.. The singly cha ged dis ibu ion is e-calcula ed conside ing 303 N1 n≈[K1 mn]+(Mq m−Mq>1 m−1), and his p ocess is epea ed o e e y 304 mobili y channel m o successi ely co ec he p esence o di e en 305 pa icle sizes. 306 The algo i hm o Hoppel supposes ha an impac o is placed o 307 emo e he la ge pa icles ha can con ibu e mul iple cha ges and 308 dis o he measu emen s. This algo i hm was ecen ly upda ed by 309 He and Dhaniyala (2013), who added an in e media e s ep a e he 310 fi s app oxima ion ha inc eases he accu acy when no impac o 311 is applied. They fi ed he la ge diame e s o he ze o h-o de singly 312 cha ged PSD by a Gumbel dis ibu ion unc ion, and he esul an 313 fi was used o co ec he mul iply cha ged con ibu ion. Howe e , 314 his app oach is only sui able o smoo h, unimodal dis ibu ions, 315 o he wise he Hoppel algo i hm is applied. 316 Re ie al o he cha ge dis ibu ion in he TDMA se up 317 Tandem di e en ial mobili y analysis (TDMA) (Liu e al., 1978; 318 Rade & McMu y, 1986) is a popula echnique o cha ac e ize 319 changes in he shape, size, concen a ion, and cha ge o pa icles 320 o o calib a e de ices. The a angemen consis s o wo iden i- 321 cal DMAs in se ies wi h a pa icle condi ione in be ween. In ou 322 me hod, we used he a o emen ioned co ona cha ge , whe e a fixed 323 ol age applied on he fi s DMA classifies pa icles o a ce ain size, 324 and he esul ing dis ibu ion is neu alized and hen echa ged 325 wi h he new cha ge o be measu ed wi h he second DMA. 326 The esponse a e he second DMA is analogous o he esponse 327 a e he fi s DMA (Eq. (21)). Howe e , his has o be exp essed 328 aking in o accoun he TF and o he cha ac e is ics o he p e ious 329 DMA sys em: 330 dM2(DZp) dln DZp=Qa1 Qm1 Qa2 Qm2  q q c2(q, Dp)Pn2(Dp)dN2(Dp) dln Dp 331 ×Dp max Dp min ˝1(Zp,Z c1)˝2(Zp,Z c2)dDp+ε(DZp).(27) 332 333 Usually, he cha ging p ope ies o he es cha ge a e 334 de e mined h ough a TDMA a angemen by compa ing he con- 335 cen a ions a di e en poin s o he se up (see Chien e al., 2011 336 o Doma e al., 2014b o de ails). Howe e , hese p ope ies a e 337 subjec ed o he unce ain ies o he concen a ion measu emen s. 338 Conside ing ha he mean cha ge can be es ima ed as 339 ¯ q2(Dp)= q q c2(q, Dp).(28) 340 A new me hod o calcula e he mean cha ge is p oposed, ha is, 341 h ough in e sion o he esponse om he second DMA (Eq. (27))342 once he PSD inpu in he fi s DMA is known. 343 Fo p ope in e p e a ion o he in e ed esul s, he known sys- 344 ema ic e o (up o 2%) be ween he mean mobili y diame e s 345 measu ed by DMA-1 and DMA-2 has o be conside ed (Rade & 346 McMu y, 1986). This e o can be pa ially co ec ed by adjus ing 347 he ol ages o each DMA (Alonso & Kousaka, 1996), al hough abou 348 1% o he shi is una oidable (Doma e al., 2014b). In addi ion, a i- 349 a ions o he concen a ion be ween DMAs because o di usion and 350 elec os a ic e ec s a e p oduced by he cha ging-neu aliza ion 351 p ocess: 352 dN2(Dp) dln Dp=Pn0(Dp)dN1(Dp) dln Dp .(29) 353 Please ci e his a icle in p ess as: Doma , M., e al. In e sion o elec ical mobili y measu emen s using bipola o unipola cha ge s o he a bi a y dis ibu ion o channels. Pa icuology (2014), h p://dx.doi.o g/10.1016/j.pa ic.2014.08.007 ARTICLE IN PRESS G Model PARTIC 738 1–10 6M. Doma e al. / Pa icuology xxx (2014) xxx–xxx Fig. 4. Simula ed and in e ed PSD o anges measu able wi h nano-DMA (a) and long-DMA (b), and ac ion o singly cha ged pa icles among he o al numbe o cha ged pa icles (c) cha ged o he bipola adioac i e dis ibu ion and eco e ed om he p e ious dis ibu ions wi h he mul iple cha ge co ec ion algo i hm. The losses acco ding o fi ings o he expe imen al da a o 3354 and 10 mm gap dis ances a e ep esen ed by Pn0, espec i ely, as355 Pn0 =0.83 −0.89e−0.09Dp(R2=0.984),(30)356 Pn0=0.93 −0.85e−0.13Dp(R2=0.962).(31) 357 Because he ubing o he pa icle coun e is he same o DMA-1358 and DMA-2, Pn1 =Pn2 (namely Pn), and Eq. (27) becomes 359 dM2(DZp) dln DZp=Qa1 Qm1 Qa2 Qm2 ¯ q2(Dp)Pn(Dp)Pn0(Dp)dN1(Dp) dln Dp 360 ×Dp max Dp min ˝1(Zp,Z c1)˝2(Zp,Z c2)dDp+ε(DZp).(32)361 362 Then, analogous o PSD in e sion, he mean cha ge can be 363 decon olu ed:364 Mm2≈˜ Kmn ¯ qn+εm,(33) 365 whe e ˜ Kis he new defini ion o he ke nel ma ix acco ding o Eq.366 (32).367 A simila p ocedu e has been used by Li, Li, and Chen (2006) and368 S olzenbu g and McMu y (2008) o decon olu e he TF o pa icle369 sizing de ices, and could also be applied he e. Howe e , because 370 no epo s o cha ge dis ibu ion decon olu ion we e ound, and 371 i is an impo an issue o PSD cha ac e iza ion, his me hod was372 chosen among he o he s.373 Resul s and discussion374 The p oposed in e sion heo y was es ed wi h wo ypes o 375 da a: simula ed and expe imen al. Simula ed da a we e gene a ed376 o es he alidi y o he echniques by compa ing he esul s wi h 377 e e ence dis ibu ions. Bo h he adioac i e and co ona cha ging 378 echniques we e es ed and in e ed by he NNLS and SVD me hods 379 compa ing wi h he egula iza ion pa ame e s om Sec ion “Da a 380 in e sion echnique”. 381 The accu acy o he simula ed in e sion was es ima ed by he 382 no malized oo -mean-squa ed e o (NRMSE), which gi es a single 383 alue o he mean o he s anda d de ia ion o he esiduals om 384 he in e sion be ween he o iginal (simula ed, N) and calcula ed 385 da a (N*). Tha is, 386 NRMSE =1 Nmax −Nmin     1 n n  i=l (Ni−N∗ i)2.(34) 387 On he o he hand, in expe imen al measu emen s, bo h he 388 unce ain y and he o iginal da a a e unknown. In hese cases, he 389 p ecision o he in e ed esul s was calcula ed by a e aging he 390 esul s o se e al es s om he same se o da a o gi e an es ima- 391 ion o he de ia ion om he o iginal da a. 392 Resul s om he simula ed da a 393 Two ep esen a i e se s o log-no mal PSD (Eq. (25) wi hou 394 o se ) cha ged wi h a bipola 85K cha ge we e es ed. One o a 395 geome ic mean diame e o 50 nm wi h 1.3 s anda d de ia ion and 396 a o al concen a ion o C= 3.5 ×1011 m−3, and he o he o a la ge 397 diame e ange ( ¯ Dpg =300 nm, g=1.5,C=4.5×1015 m−3),398 each wi hin he measu able anges o he nano- and long-DMA, 399 espec i ely. Some unce ain y was added o imi a e he mea- 400 su emen p ocess: a ound 5% o ambien ae osol and up o 10% 401 o gene a ed ae osol (Gysel, McFiggans, & Coe, 2009). Howe e , 402 Please ci e his a icle in p ess as: Doma , M., e al. In e sion o elec ical mobili y measu emen s using bipola o unipola cha ge s o he a bi a y dis ibu ion o channels. Pa icuology (2014), h p://dx.doi.o g/10.1016/j.pa ic.2014.08.007 ARTICLE IN PRESS G Model PARTIC 738 1–10 M. Doma e al. / Pa icuology xxx (2014) xxx–xxx 7 Fig. 5. (a) In e ed PSD o he unipola co ona cha ge a di e en dis ances be ween elec odes and (b) in e ed mean cha ge o he unipola and bipola es cha ge s in a TDMA configu a ion. phenomena such as coagula ion and agglome a ion we e no con-403 side ed.404 In Fig. 4(a) and (b), he in e sion o he de e mined p oblem405 wi h 64 channels/decade and 5% noise is shown. In he smalle 406 diame e ange (50 nm, Fig. 4(a)), bo h SVD and NNLS me hods407 lead o he same indis inguishable esul wi h iden ical de ia ion408 and an NRMSE below 0.5%. Fo he la ge diame e ange (300 nm,409 Fig. 4(b)), he in e sion pe o med by SVD was sligh ly be e han 410 ha by NNLS (NRMSESVD = 0.8%, NRMSENNLS = 1.2%). The L-Cu e411 (LC) and GCV egula iza ion me hods also ga e e y simila esul s,412 al hough he LC had a lowe NRMSE han GCV e en o he nois- 413 ies da a. Because he pe o mance wi h he di e en me hods is 414 easonable, only he SVD-LC will be shown in he ollowing.415 The esul s o he mul iple cha ge co ec ion algo i hm om416 Sec ion “Mul iple cha ge co ec ion” a e shown in Fig. 4(c). The 417 singly and mul iply cha ged dis ibu ions a e e y simila in he418 small pa icle ange, bu become di e en as he pa icle size419 inc eases and acqui es mo e cha ge. The heo e ical ac ion o 420 singly cha ged pa icles among he o al numbe o cha ged pa i- 421 cles (Wiedensohle , 1988; Wiedensohle e al., 2012) is compa ed422 wi h he esul s om p e ious plo s in Fig. 4(c). The e is good ag ee-423 men be ween he p edic ed and ob ained esul s, al hough o he 424 la ges diame e s ou p edic ion is clea ly unde es ima ed, p ob-425 ably because o he dec ease in he concen a ion in he ails o 426 he PSD and he possible con ibu ion o la ge pa icles wi h mo e 427 cha ges. 428 The mul imodal dis ibu ions a e shown in Fig. 5(a) ( ¯ Dpg =429 6,10,and 23 nm; g= 1.18, 1.3, and 1.05; C=(3.0, 5.0, and430 1.2) ×1010 m−3, espec i ely). The da a we e simula ed wi h431 unipola cha ging a wo di e en Ni alues gi en by he432 Fig. 6. P ecision o he esul s o (a) an inc ease in a ificial unce ain y du ing he measu emen s and (b) a a iable a io o mobili y measu emen (inpu , m) in e als o size (ou pu , n) in e als. The NRMSE and he loga i hm o he condi ion numbe CN gi e an indica ion o he accu acy o he in e sion p ocess. gap dis ance. The in e sion pe o med be e wi h D=10mm 433 (NRMSE = 1.6%) han wi h D= 3 mm gap (NRMSE = 3%), because a 434 highe Ni -p oduc , and hus a highe le el o pa icle cha ging, 435 leads o mo e singula i y in he es ima ion o he ke nel ma ix. 436 The mean cha ge o a es cha ge in a TDMA a angemen was 437 es ima ed om he eco e ed PSD cha ged in a fi s DMA sys- 438 em wi h a known bipola 85K cha ge dis ibu ion. In he second 439 DMA sys em, h ee ypes o cha ge dis ibu ion we e es ed: bipola 440 85K and unipola co ona cha ging wi h gaps a 3 and 10 mm. The 441 eco e ed esul s a e shown in Fig. 5(b). The bipola mean cha ge 442 decon olu ed wi h a NRMSE o 2%, while he unipola cha ging was 443 sligh ly unde es ima ed wi h a NRMSE o 8% o Dgap =10mmand 444 up o 20% NRMSE o Dgap = 3 mm. I is impo an o no ice om Eq. 445 (32) ha he e is a eco e ed unc ion only o he in e ed N1(Dp)446 alues ha a e no equal o ze o. Mo eo e , he unce ain ies in he 447 in e sion o he PSD will p opaga e o he in e sion o he mean 448 cha ge. 449 In Fig. 6(a), he a ia ion o he e o in he in e sion p ocess 450 wi h he a ificial noise is shown. I is expec ed ha he g ea e 451 he added noise, he g ea e he e o in he in e sion, as occu s in 452 PSD in e sion. Howe e , al hough he in e ed mean cha ge gene - 453 ally has highe e o , i is almos insensi i e o added noise, which 454 indica es ha he main con ibu o s o he esiduals in he second 455 in e sion a e he e o s om he fi s in e sion. 456 While he CN seems o be gene ally insensi i e o e o s and 457 noise du ing measu emen o he da a, as shown in he lowe 458 g aph o Fig. 6(a), he numbe o in e als di ec ly a ec s he size 459 Please ci e his a icle in p ess as: Doma , M., e al. In e sion o elec ical mobili y measu emen s using bipola o unipola cha ge s o he a bi a y dis ibu ion o channels. Pa icuology (2014), h p://dx.doi.o g/10.1016/j.pa ic.2014.08.007 ARTICLE IN PRESS G Model PARTIC 738 1–10 8M. Doma e al. / Pa icuology xxx (2014) xxx–xxx Fig. 7. Decon olu ion in le axis o he mobili y dis ibu ed da a om igh axis o he o e -de e mined, unde -de e mined, and de e mined p oblems. The da a we e acqui ed in a DMPS sys em o symme ic flows o QQ= 1.5 lpm and Qsh =15lpm wi h 64 channels pe decade. o he ma ix and hus he accu acy o he esul s, ollowing he460 expec ed end o being mo e p ecise o o e -de e mined han 461 unde -de e mined sys ems (Fig. 6(b)). The e o in he in e sion462 o he PSD dec eases o <2% when m/n≥1. The same occu s o 463 he in e ed mean cha ge, bu , as in p e ious cases, he e is an464 accumula ed unce ain y om he p e ious in e sion ha he sec-465 ond calcula ion canno o e come, eaching a lowe “limi ” o he 466 NRMSE.467 I mus be men ioned ha , al hough ex a equa ions in he468 o e -de e mined p oblem can be used as addi ional in o ma ion 469 o educe he high le els o unce ain y in he da a, hey can o e - 470 smoo h he e ie ed da a, losing in o ma ion abou na ow peaks.471 Mo eo e , i is common in eal measu emen s o ha e a limi ed472 numbe o channels o measu e, and i is he e o e impo an o be473 able o sol e unde -de e mined p oblems. Howe e , in he lowe 474 plo o Fig. 6(b), he loga i hm o he CN shows lowe p ecision loss475 o he unde -de e mined p oblems han o he de e mined p ob-476 lems, and fi s inc eases and hen dec eases wi h inc easing m/n.477 This can lead o con usion, bu i is caused because he mos unde -478 de e mined case has such a lack o in o ma ion ha he di e ence479 be ween he smalles and la ges singula alues is minimal.480 Resul s om expe imen al measu emen s481 The esul s om simula ed measu emen s a e use ul o de e -482 mine he applicabili y o he p oposed me hod o expe imen al da a 483 and o analyze he ac o s ha a ec he in e sion p ocess. The nex 484 s ep is o es expe imen al da a om bipola o unipola cha ging485 measu ed by s epping and slow-scanning ol ages in he DMA.486 The in e sion o a se o da a measu ed in a DMPS sys em wo k- 487 ing wi h a bipola adioac i e 85K cha ge is shown in Fig. 7. Th ee488 a ios o in e als (m/n) we e es ed. The unde -de e mined and489 de e mined p oblems ga e esul s wi h simila shapes and a eas o 490 he measu ed mobili y dis ibu ion. Howe e , he o e -de e mined 491 p oblem showed a di e en shaped cu e nea he ails, wi h wo492 small peaks p obably because o mul iply cha ged pa icles ha 493 en e ed wi h he selec ed mobili y in he classifie . This indica es 494 he impo ance o being able o modi y he a io o in e als.495 Measu emen s in a TDMA configu a ion a e always pe o med496 o monodispe se dis ibu ions, some imes wi h e y na ow peaks497 and low concen a ion o pa icles, which complica es he in e -498 sion. In Fig. 8, a se o monodispe se measu emen s used o499 calib a e he co ona cha ge (wi h Dgap =10 mm) is shown in a500 Fig. 8. Measu ed esponse and in e ed PSD o he se o monodispe se measu e- men s used o calib a e he co ona cha ge in he TDMA a angemen . TDMA configu a ion a slow-scan a es o a oid smea ing e ec s. 501 The monodispe se dis ibu ions we e ea ed oge he o o m a 502 wide dis ibu ion and o use he in o ma ion o each o he single 503 peaks in he in e sion p ocess o he esponse. The eco e ed esul 504 om he PSD in e sion is also shown in Fig. 8, and, al hough he 505 Fig. 9. (a) F ac ion o singly cha ged pa icles among he o al cha ged pa icles eco e ed wi h he mul iple cha ge co ec ion algo i hm and (b) in e ed mean cha ge o he unipola co ona cha ge wi h Dgap = 10 mm in a TDMA configu a ion. Please ci e his a icle in p ess as: Doma , M., e al. In e sion o elec ical mobili y measu emen s using bipola o unipola cha ge s o he a bi a y dis ibu ion o channels. Pa icuology (2014), h p://dx.doi.o g/10.1016/j.pa ic.2014.08.007 ARTICLE IN PRESS G Model PARTIC 738 1–10 M. Doma e al. / Pa icuology xxx (2014) xxx–xxx 9 singly cha ged pa icle dis ibu ion was calcula ed, i s ep esen a-506 ion esul s a e almos indis inguishable om he o al.507 Sepa a ely ep esen ing he ac ion o pa icles wi h cha ge508 uni y (Fig. 9(a)) indica es ha his ac ion is o e es ima ed509 compa ed wi h he expec ed alue o a simila ex en o he unde -510 es ima ion o he doubly cha ged pa icles. A eason o his could511 be ha i was only possible o measu e up o h ee cha ges in he512 cha ge dis ibu ion cha ac e iza ion, and he e o e highe pa icle513 cha ges a e p esen in he measu emen s ha a e no being aken514 in o accoun .515 The e ie al o he mean cha ge om he PSD eco e ed in516 Fig. 8 h ough he me hod p esen ed in Sec ion “Re ie al o he517 cha ge dis ibu ion in he TDMA se up” is shown in Fig. 9(b). Oscil-518 la ions due o e o s and unce ain ies in he da a appea below 519 20 nm, which o e es ima es he esul s. Howe e , o la ge pa -520 icle sizes, he di e ences wi h he expec ed alues a e educed,521 p obably because o a highe concen a ion and le el o cha ge o 522 he pa icles. 523 The diame e ange below 20 nm is always he mos sensi i e524 ange because o he o ces o which pa icles a e subjec ed, such525 as di usion and elec os a ic e ec s. These phenomena a e compa-526 able o he fluc ua ions o he dis ibu ion, and he algo i hm has527 o be able o dis inguish be ween in o ma ion om he eal dis-528 ibu ion and noise and unce ain ies. We ha e o also aken in o529 accoun ha hese pa icle anges usually ha e he lowes cha ging530 e ficiencies, and hus ha e low concen a ions. The e o e, in e sion531 o size anges whe e di usion plays an impo an ole always ha e 532 la ge unce ain y, which is e en la ge i he e o is able o p op- 533 aga e om he fi s o he second in e sion p ocedu es, and hus534 p ecise es ima ion equi es mo e e o .535 Conclusions536 A comp ehensi e s udy o he equi emen s o ob aining accu-537 a e in e sion o da a om mobili y measu emen s was conduc ed 538 based mainly on NNLS and SVD algo i hms, wi h he SVD algo i hm539 compa ing wo me hods o egula iza ion: L-Cu e and GCV. The540 me hodology was de eloped o be used wi h any ype o cha ge ,541 once i s cha ge dis ibu ion is defined, and an app oxima ion is 542 pe o med o use da a om scanning o s epping ol age config-543 u a ions in he DMA. G ea flexibili y is allowed in choosing he544 numbe o in e als and he numbe o da a channels, allowing545 a bi a y ol age s eps and o e lapping DMA ans e unc ions. 546 The ans e unc ion is es ima ed by a simple geome ical cal-547 cula ion om he a eas wi hin each ou pu in e al, equi ing only548 he mobili ies ha define he TF, which is an accu a e app oxima-549 ion compa ed wi h he bell-shaped app oxima ion. The analysis550 o he ke nel ma ix is a key o unde s anding he o igin o he551 ins abili ies, because i s ank condi ions he numbe o po en ial552 ma hema ical solu ions o he p oblem. 553 F om di ec compa ison o he SVD and NNLS me hods, i was554 ound ha he e o s we e simila and usually depended on he555 cha ac e is ics o he dis ibu ion. Howe e , he SVD me hod com-556 bined wi h he L-Cu e egula iza ion has he ad an age o g ea e 557 flexibili y in selec ing he numbe o channels, esul ing in be e 558 pe o mance because i is capable o accu a ely sol ing e en unde -559 de e mined sys ems. In e sion pe o med in his way ga e a good 560 fi wi h less han 1% e o o noise le els o 5%, which could be561 e en less when he in e als a e p ope ly chosen.562 A me hod o calcula e he con ibu ion o singly cha ged563 pa icles o he o al dis ibu ion was success ully applied o dis-564 ibu ions om unipola and bipola cha ging. Howe e , in he 565 ails o he dis ibu ions, whe e he concen a ion is low, he ac-566 ion o singly cha ged pa icles was mises ima ed. The echnique567 was also applied o he TDMA a angemen wi h he possibili y 568 o e ie ing he mean cha ge pe pa icle o he dis ibu ion. The 569 esul s ga e a good app oxima ion when he o iginal PSD was accu- 570 a ely es ima ed, al hough e o s can p opaga e and dis o he 571 in e ed mean cha ge and u he me hod de elopmen is equi ed 572 o educe he e o p opaga ion. 573 Acknowledgmen s 574 M. Doma and J.M. Fe nandez-Diaz hank he Minis e io de Edu- Q4575 cacion y Ciencia (MEC) o Spain o suppo unde he p ojec 576 MEC05CGL2005-05244/CLI and g an BES-2006-12469. 577 F.E. K uis is financially suppo ed by he Deu sche Fo schungs- 578 gemeinscha (DFG) in he amewo k o he join esea ch 579 p og am “Mul ipa ame e Cha ac e iza ion o Pa icle-based Func- 580 ional Ma e ials by Inno a i e Online Measu emen Technology” 581 (PAK688). 582 Re e ences 583 Adachi, M., Kousaka, Y., & Okuyama, K. (1985). Unipola and bipola di usion cha g- 584 ing o ul afine ae osol pa icles. Jou nal o Ae osol Science,16(2), 109–123. 585 Alo s, D. J., & Balakuma , P. (1982). In e sion o ob ain ae osol size dis ibu ions om 586 measu emen s wi h a di e en ial mobili y analyze . Jou nal o Ae osol Science,587 13(6), 513–527. 588 Alonso, M., & Kousaka, Y. (1996). Mobili y shi in he di e en ial mobili y analyze 589 due o B ownian di usion and space-cha ge e ec s. Jou nal o Ae osol Science,590 27(8), 1201–1225. 591 Bazan, F. S. V., & F ancisco, J. B. (2009). An imp o ed fixed-poin algo i hm o de e - 592 mining a Tikhono egula iza ion pa ame e . In e se P oblems,25(4), 045007. 593 Biskos, G., Rea ell, K., & Collings, N. (2005). Unipola di usion cha ging o ae osol 594 pa icles in he ansi ion egime. Jou nal o Ae osol Science,36(2), 247–265. 595 Boisd on, K., & B ock, J. R. (1970). On he s ochas ic na u e o he acquisi ion o elec- 596 ical cha ge and adioac i i y by ae osol pa icles. A mosphe ic En i onmen ,597 4(1), 35–50. 598 Chien, C. L., Tsai, C. J., Chen, H. L., Lin, G. Y., & Wu, J. S. (2011). Modeling and alida ion 599 o nanopa icle cha ging e ficiency o a single-wi e co ona unipola cha ge . 600 Ae osol Science and Technology,45, 1468–1479. 601 Collins, D. R., Flagan, R. C., & Sein eld, J. H. (2002). Imp o ed in e sion o scanning 602 DMA da a. Ae osol Science and Technology,36(1), 1–9. 603 C ump, J., & Sein eld, J. (1981). A new algo i hm o in e sion o ae osol size dis i- 604 bu ion da a. Ae osol Science and Technology,1(1), 15–34. 605 Doma , M., K uis, F., & Fe nandez-Diaz, J. (2014a). De e mina ion o he ele an 606 cha ging pa ame e s o he modelling o unipola cha ge s. Jou nal o Ae osol 607 Science,71, 16–28. 608 Doma , M., K uis, F., & Fe nandez-Diaz, J. (2014b). In es iga ions o he e ec o 609 elec ode gap on he pe o mance o a co ona cha ge ha ing sepa a ed co ona 610 and cha ging zones. Jou nal o Ae osol Science,68, 1–13. 611 Dubey, P., & Dhaniyala, S. (2008). Analysis o scanning DMA ans e unc ions. 612 Ae osol Science and Technology,42(7), 544–555. 613 Dubey, P., & Dhaniyala, S. (2013). Imp o ed in e sion o scanning elec ical mobili y 614 spec ome e da a using a new mul iscale expec a ion maximiza ion algo i hm. 615 Ae osol Science and Technology,47(1), 69–80. 616 Fiebig, M., S ein, C., Sch öde , F., Feldpausch, P., & Pe zold, A. (2005). In e sion o da a 617 con aining in o ma ion on he ae osol pa icle size dis ibu ion using mul iple 618 ins umen s. Jou nal o Ae osol Science,36(11), 1353–1372. 619 Fuchs, N. (1964). The mechanics o ae osols. New Yo k: Pe gamon P ess. 620 Gysel, M., McFiggans, G. B., & Coe, H. (2009). In e sion o andem di e en ial mobili y 621 analyse (TDMA) measu emen s. Jou nal o Ae osol Science,40(2), 134–151. 622 Hagen, D. E., & Alo s, D. J. (1983). Linea in e sion me hod o ob ain ae osol size 623 dis ibu ions om measu emen s wi h a di e en ial mobili y analyze . Ae osol 624 Science and Technology,2(4), 465–475. 625 Hansen, P. C., & O’Lea y, D. P. (1993). The use o he L-cu e in he egula iza ion o 626 disc e e ill-posed p oblems. SIAM Jou nal on Scien ific Compu ing,14, 1487–1503. 627 He, M., & Dhaniyala, S. (2013). A mul iple cha ging co ec ion algo i hm o scanning 628 elec ical mobili y spec ome e da a. Jou nal o Ae osol Science,61, 13–26. 629 Hon a˜ non, E., & K uis, F. E. (2008). Single cha ging o nanopa icles by UV pho oion- 630 iza ion a high flow a es. Ae osol Science and Technology,42, 310–323. 631 Hoppel, W. A. (1978). De e mina ion o he ae osol size dis ibu ion om he mobil- 632 i y dis ibu ion o he cha ged ac ion o ae osols. Jou nal o Ae osol Science,9(1), 633 41–54. 634 In a, P. (2012). Co ona discha ge in a cylind ical iode cha ge o unipola di usion 635 ae osol cha ging. Jou nal o Elec os a ics,70(1), 136–143. 636 Johns on, P. R., & Gul ajani, R. M. (2000). Selec ing he co ne in he L-cu e app oach 637 o Tikhono egula iza ion. IEEE T ansac ions on Biomedical Enginee ing,47,638 1293–1296. 639 Kaminski, H., Kuhlbusch, T. A. J., Fissan, H., Ra i, L., Ho n, H.-G., Han, H.-S., 640 e al. (2012). Ma hema ical desc ip ion o expe imen ally de e mined cha ge 641