A chemical kinetic model for reactive transformations of aerosol particles

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1 00. Chpter 4 A chemicl kinetic model for rective trnsformtions of erosol prticles Previous models of heterogeneous interctions hve focused on rective trce gs depletion in cloud or erosol prticles nd droplets.,,3 However, recent lbortory studies hve demonstrted tht rective trce gs uptke cn significntly trnsform the chemicl composition of condensed phse species, s is evident from recent work on orgnic erosols. 4,5 This model focuses on trnsformtion rtes within the prticle itself. In limiting cses, the model leds to simple nlyticl expressions for the condensed phse species depletion s function of erosol/gs interction time. The model tkes into ccount gs phse diffusion, mss ccommodtion, bulk phse chemicl rections, surfce rections nd prticle phse rectnt diffusion from the erosol interior towrd the surfce. In this wy, five limiting regimes of erosol chemicl kinetic processing re identified, quntified, nd united to form single comprehensive mthemticl model of prticle phse species kinetics. The presenttion stresses the informtion to be gined from observing the size dependence of the prticle phse decy rte. Indeed, decy rtes exhibiting size dependence spnning prticle rdius, r 0 through r - re theoreticlly fesible, nd rtes going s r 0 nd r - hve been experimentlly observed,4. In ddition, n ppendix is provided in which existing chrcteristic times re summrized or new ones re derived for ech of the five relevnt regimes of chemicl processing. Chrcteristic times cn be quickly compred to both time-scles of observed chnges in lbortory or field studies nd pprent size dependence of rection rtes. Such time 54

2 00. scles offer relible indictor of the fundmentl processes most likely to govern the overll rection rte. They lso revel wht combintions of physico-chemicl prmeters will be obtined in studying the time dependence of tht process. In this wy they form useful compliment to the resistor model described below. The method of chrcteristic times pplied to cloud droplets ws first described by Schwrtz Modeling interctions between erosols nd rective trce gses An erosol of known size nd initil composition is exposed to rective trce gs for known period of time, t. The experimentlly observed quntity to be modeled here is the loss rte of prticle phse species, s opposed to the disppernce of gs phse rectnts, the focus of previous studies. We ssume the chemicl rection between the trce gs X nd prticle phse species Y is irreversible; tht is, X + Y Z, where Z represents one or more product species. The method presented here cn nonetheless be extended to more complicted chemicl kinetic scenrios. In generl, in the bsence of convective mixing, the overll kinetic processing will depend on the chemicl rection rtes s well s the mss trnsport of both prticle nd gs phse species. Mny prts of the model presented here re tken from our recent pper on the sme subject 7. The conceptul strtegy Our gol is to mthemticlly describe generl physicl system for which ll of the relevnt physics (in the form of prtil differentil equtions) is known, but tht cnnot be represented by ny known nlyticl solution. To our knowledge, even numericl solutions hve not been developed over the full rnge of conditions considered here. However, the system is well described t vrious limits. In such limits, the prtil differentil equtions reduce to forms tht 55

3 00. cn be solved nlyticlly by stndrd methods such s seprtion of vribles or Fourier trnsforms. Our strtegy is to construct mthemticl bsis set from collection of such physiclly limiting regimes, which we conceive s resistnces. To obtin resistnce, we solve the equtions describing system s if ll other processes were turned off. Once we hve cycled through ech of the physiclly limiting processes, with ll other processes turned off, we will hve formed the bsis set of resistnces required to model the overll process. The fundmentl ssumption is tht we my closely pproximte description of physicl relity (lmost lwys involving some degree of nonliner inter-couplings of processes) through simple decoupled superposition of the bsis resistnces. In the resistor nlogy, used here for mss trnsfer nd chemicl rection, processes cn dd either in series or in prllel. Series processes re those tht vnish if ny one element vnishes, wheres prllel processes occur long simultneous chnnels. For exmple, gs phse diffusion nd mss ccommodtion (see below) re in series with one nother since either one s bsence mkes mss trnsfer into the prticle impossible. This formultion hs been tested in previous studies, lthough not encompssing combintions of ll the vrious resistnces considered here. Solutions to limiting processes such s gs phse diffusion, mss ccommodtion, solubility, nd chemicl rections hve been presented in the literture 8,9. Using such vilble nlyticl solutions to form pproprite resistnces, pproximte expressions for overll processes hve been formulted. This decoupled formultion hs been shown to be in good greement (within bout 5%) with nlyticlly coupled solutions where these re vilble,0. 56

4 The model We envision five fundmentl processes tht form the bsis elements of the model. ) The gs phse molecule must first diffuse to the erosol surfce where it cn collide with nd thermlly ccommodte onto the surfce with probbility S clled the dsorption or therml ccommodtion coefficient. The dsorbed molecule my desorb t rte governed by the first order rte constnt k des or, in the cse of liquid, it my enter the bulk liquid, t rte governed by the rte constnt k solv. These two rtes together with therml ccommodtion coefficient, S determine the mss ccommodtion coefficient, α, which is the probbility tht n tom or molecule striking the surfce enters into the bulk liquid phse. ) Rections of the trce gs my lso occur specificlly on the surfce of the erosol with rtes tht my be different thn those found in bulk. While sturting the surfce, the trce species cn diffuse into the erosol bulk s it rects with it. As the gs enters the prticle, frction my evporte bck into gs phse due to the limited solubility of the species in the liquid (Henry's lw). It is this entrnce of the gs species into the prticle (dependent on gs solubility, diffusivity, nd rectivity) tht estblishes concentrtion profile within the prticle. 3) In the cse of slow rection or fst diffusion it will diffuse deep into the erosol bulk, filling it to sturtion. 4) In the cse of fst rection or slow diffusion it will tend to be confined to thin shell ner the surfce. 5) In this cse, the prticle phse species must diffuse to the surfce region in order to rect. In reltively lrge or highly viscous erosols, one my lso hve to tke into ccount diffusive trnsport of prticle-phse rectnt towrd the region ner the erosol surfce where rections re occurring. Ech of the ltter individul processes cn be described by independently solving the relevnt prtil differentil eqution with pproprite boundry nd initil conditions. 57

5 00. In experiments subject to ny of these processes nd/or limittions, the mesured flux, J of trce gs into the surfce is expressed in terms of mesured uptke coefficient, γ mes, s: J n x ( g) cγ mes = (4.) 4 Here nx(g) is the bulk concentrtion of the trce gs molecules (species X) nd c is their verge therml speed. Of course, γ mes is lwys smller thn or equl to since the flux cnnot exceed the collision rte per unit re, nx(g) c /4. The model ensures tht this is the cse, s will be seen below. In the cse where trce gs concentrtion profile hs been estblished within the prticle (shown below to be typiclly much fster thn other processes), ll the rective gs molecules in the flux, J (s in eq. 4.) through the erosol surfce rect with the prticle phse species Y. Therefore, mss blnce llows us to write the volume-verged prticle phse loss rte [Y & ] = d[y]/dt within n erosol of rdius, s in: n x ( g ) cγ 4 mes 4π = 4 π 3 3 [ Y& ] such tht, 3 n x ( g) cγ mes [ Y& ] = (4.) 4 In generl,,?γ mes, nd nx(g) will vry with time s result of gs-prticle interctions. The gol then is to formulte n expression for γ mes by dding the bsis resistnces (tken from the vrious limiting processes) ppropritely. In this tretment, the resistnce of the i th process is given by /Γ i, where Γ i represents the normlized rte of process i. More specificlly, Γ i is the rtio of the flux of trce gs to the kinetic collision frequency, ssuming 58

6 00. Mxwell-Boltzmnn distribution previls ner the erosol surfce. In the bsence of solubility limittions, the overll resistnce eqution from which one obtins γ mes is written s: γ mes = Γdiff + S + Γs + S α + Sα Γrxn + p Γ diff (4.3) nd shown in fig. 4-. Becuse the dsorption coefficient S, is the mximum vlue of γ mes is limited to unity. There is no limit to the vlue of the Γ terms. The /Γ terms in eq. 4.3 (nd fig. 4-) represent gs phse diffusion (Γ diff ), surfce rection (Γ s ), mss ccommodtion (α), chemicl rection within in the prticle (Γ rxn ) nd diffusion limited mixing within the prticle (Γ p diff). These terms ccount for process rtes (nd concentrtion grdients) in the gs phse, t the interfce nd within the prticle. Tken together, they form the bsis of the generl model. The term Γ p diff, which ccounts for the diffusion limited grdient of the rectnt Y within the prticle, is presented here for the first time. This formultion ssumes tht the erosol consists of single component Y. We note tht in modeling multi-component erosols, seprte Γ rxn, Γ s, nd Γ p diff need to be formulted for ech component. When one is concerned with the depletion of gs phse species or when solubility dynmics re expected to influence the overll rte, time dependent solubility term /Γ sol is dded in prllel with the /Γ rxn term. Alterntively (see below) this cn be included in more complicted form of /Γ rxn, for which n nlyticl solution hs been derived in the literture. We will show below why the term /Γ sol is not expected to ffect prticle species rective loss. In the limit of high Γ, the resistnce goes to zero. 59

7 00. Also, the trce gs (n x ) tht is simply solvted in the prticle phse does not result in prticle species loss. Therefore, since the purpose of this presenttion is to develop expressions for prticle species rection rte [Y & ], the term /Γ sol hs been omitted from eq Γ s Γ diff S S α Sα Γ rxn p Γ diff α Figure 4-: Electricl circuit nlogue for the gs uptke process governed by gs-phse diffusion, mss ccommodtion, surfce rections, nd prticle phse rection nd diffusion. 4.. Gs phse diffusion nd mss ccomodtion When gs phse diffusion or mss ccomodtion is limiting, the rte of processing within the erosol will be governed by how quickly the gs diffuses to nd psses through the surfce of the erosol. An empiricl formultion of isotherml diffusive trnsport tht is in good greement with experimentl dt cn be written s the resistnce /Γ 3 diff : Kn = Γdiff Kn( + Kn) (4.4) where Kn is the Knudsen number defined s λ/ nd λ is the gs phse men free pth. The men free pth is here expressed s λ=3dx(g)/ c, where Dx(g) is the gs phse diffusion coefficient of 60

8 00. the trce species. As shown schemticlly in fig. 4-, /Γ diff represents the resistnce of gs phse diffusion to the gs uptke, ccounting for the grdient in gs phse concentrtion due to the flux of gs into the prticle. As described in detil previously, interfcil resistnce to mss trnsport cn be formulted in terms of the mss ccommodtion coefficient, α, which cn be expressed s: α = kdes S + Sksol or kdes Sksol S α = (4.5) Sα The rtio k sol / k des is the true mesure of the interfcil kinetic resistnce, though α is often quoted s the collision probbility coefficient for trce gs uptke into the (liquid) prticle. 4.. Surfce rections Once the trce gs X hs diffused to the surfce, it cn rect with erosol species Y directly t the surfce vi mechnism tht is kineticlly seprble from rection within the prticle. As shown in fig., this corresponds to resistnce term between /S nd k des /Sk sol = (S-α)/Sα (see eq. 4.5), since such surfce rectivity competes with mss ccommodtion of X into the bulk prticle through the gs/surfce interfce. We use the sme pproch here to find Γ s s in the cse of gs phse diffusion. We clculte gs phse flux into the prticle, ssuming tht this specific process is solely the one governing the uptke. Here we suggest the following expression for the surfce rection rte between species X nd Y per unit re per second: n x( g) 4 cγ s = k s [ X ] s [ Y ] s = k s n x( g) H s RT[ Y ] s (4.6) In eq. 4.6 we hve ssumed tht the surfce concentrtion of trce species is in equilibrium with the gs phse, nlogous to Henry's lw equilibrium for bulk condensed phses. All surfce 6

9 00. concentrtions re in moles cm - nd H s nd k s re surfce nlogs to their bulk counterprts. Alternte formultions of eq. 4.6 my be more convenient in some cses (see Hu et l., [995] 4 ). Using the bove formultion with the further ssumption tht [Y] s = K s [Y], we my write the uptke coefficient Γ s from eq. 4.6: s 4k H s RTK s [ Y ] Γ s = (4.7) c Here, K s is n equilibrium constnt (with units of L cm - ) linking the surfce concentrtion [Y] s to the bulk concentrtion [Y]. The equilibrium prmeters H s nd K s used in eq. 4.7 cn t times be obtined experimentlly from surfce tension mesurements for exmple, or cn be estimted by extrpoltion from bulk phse equilibrium vlues Rections within the prticle interior For the simultneous rection nd diffusion of the trce gs into the erosol, we hold the trce species concentrtion constnt t the surfce, while llowing the trce gs to rect s it diffuses into the erosol. Becuse this process is expected to be importnt in lbortory nd field studies, we will derive it here in detil. We begin with the diffusion eqution including n dditionl term for rection within the prticle: D x ( p ) [ X ]( r, t) k[ X ]( r, t)[ Y ] = [ X& ]( r, t) (4.8) This expression tkes into ccount the full dynmics of trce gs species diffusion, solubility, nd rection within the prticle. It does not explicitly ccount for the loss of Y due to rection with X, which would require mirror eqution of eq. 4.8 for Y. For now we will mke simplifying ssumption tht we will justify below: The initil profile [X](r) is estblished so 6

10 00. quickly, tht we cn ssume the loss rte of Y to be governed by the pseudo-stedy stte profile of X within the prticle. In other words, fter some very brief moment, [X](r) develops, estblishing volume-verged loss rte of prticle species given by: 3 [ Y& ] = k[ X ]( r, t)[ Y]( t)4πr dr (4.9) 3 4π 0 In this expression we write [X](r,t), explicitly function of time, not due to the dynmics in eq 4.3, but becuse depletion of Y chnges the stedy stte profile of X. Mking the ssumption tht eq. 4.3 is t stedy stte llows us to write it s: D x ( p ) [ X ]( r) k[ X ]( r)[ Y ] = 0 (4.0) or in sphericl coordintes: d [ X ]( r) d[ X ]( r) k + [ X ]( r) = 0 dr r dr D x( p) (4.) Here, k = k [Y]. The nlyticl solution, describing the trce-gs concentrtion within the prticle s function of rdil position nd prticle species concentrtion, cn be shown to be 5 : sinh( r / l) Dx( p) [ X ]( r) = [ X ]( ) where l (4.) r sinh( / l) k [ Y] The prmeter l, is the recto-diffusive length representing the depth to which solvted trce species X diffuses while recting with Y. This depth reflects the distnce rectnt X will diffuse in one rective chrcteristic time (/k [Y]). We note tht the concentrtion profile of X depends on the concentrtion of Y within the prticle; X will diffuse more deeply on verge into the prticle the less Y there is with which it cn rect. It is for this reson tht we refer to [X](r) s pseudo-stedy stte profile. 63

11 00. At ny instnt in time, the net rte t which species X enters the prticle equls the rte t which species Y is depleted. Also, by definition this must be equl to the volume-verged rection rte within the prticle itself, given by eq Using Fick s lw to ccess the net flux of rectnt gs molecules t the surfce we hve: J x d[ X ]( r) n x( g ) cγrxn = Dx( g) ) r= = (4.3) dr 4 such tht we my write Γ rxn s: 4HRT Γ rxn = Dx( p) k[ Y ][ coth( / l) ( l / ) ] (4.4) c In this expression, we hve replced [X]() with the Henry s lw vlue, HP x =Hn x(g) /RT. Here D x(p) is the diffusion coefficient of the solvted trce species X in the prticle phse, H is its Henry's lw coefficient (H is in units M tm - L -, R is the gs constnt in mtching units), nd k is the second order rte coefficient for the rection of X with Y. In fig. 4-, we show how this function depends functionlly on l nd. This is the sme expression s presented in Hnson nd Lovejoy, where it ws first pplied to erosol kinetics. 64

12 00. G rxn l Figure 4-: Generl functionl dependence of rective uptke coefficient on prticle rdius, nd recto-diffusive depth, l Prticle phse diffusion As we hve seen in the cse of rpid rection, it is possible tht the trce gs diffuse to depth mny times smller tht the prticle rdius. Most of the erosol chemicl processing will occur within this recto-diffusive length, l. If diffusion of prticle species to this rection region is sufficiently slow, the overll rte of processing cn be expected to decrese due to depletion of prticle species concentrtion in this region. 65

13 00. Here we derive n exct expression for the resistnce when the diffusion of the prticle phse species is mny times slower thn rection ner the surfce. Thus the rte t which prticle phse molecules dispper from the prticle will be governed entirely by how quickly they cn diffuse to the surfce where they immeditely rect wy. In this cse the surfce concentrtion of the prticle phse species is zero by construction; the prticle phse molecule is destroyed by rection the very instnt it reches the surfce region. This process is tken into ccount by the coefficient Γ p diff, the lst term shown in eq. 4.3 nd fig. 4-. We derive n expression for Γ p diff here for the first time. We ssume tht the rective gs phse flux into the erosol is governed by the diffusion rte of Y from the prticle interior. Mtching the trce gs flux through the surfce to the flux of Y towrd the surfce we hve: p nx( g) cγ diff 4 = Dy [ Y ]( r, t) ) r= r (4.5) A solution to the diffusion eqution in sphericl coordintes cn be found through seprtion of vribles. Subject to the boundry condition tht [Y] = 0 t the prticle surfce nd [Y] = [Y] 0 initilly everywhere within the prticle, n exct expression for [Y](r,t) is found to be 6 (see fig. 4-3): [ Y ]( r, t) [ Y0 ] n+ ( ) nπr = sin exp( Dyn π t / ) πr n= n (4.6) Through eq. 4., (with some work) we my now write: 66

14 00. p Γ diff 8D y[ Y ] 0 idyπ t = Exp( ) n xc i= (4.7) A significntly simpler expression for Γ p diff cn be obtined by ssuming liner concentrtion profile for Y, s [Y](r) = [Y c ]( - r/), where [Y c ] = [Y](r=0). Therefore, d[y](r)/dr = -[Y c ]/. From the definition of the verge concentrtion, we hve: 3 [ Y Y Y r r dr = c ] [ ] [ ]( )4π 3 4π 4 0 (4.8) Combining expressions, the simplified expression for Γ p diff emerges s, p 6D y [ Y ] Γ diff = nx( g) c (4.9) This simple expression is in form consistent with the other uptke coefficients nd lends itself to fcile computtion. The ccurcy of this expression will be discussed lter. The uptke coefficient Γ p diff becomes importnt (i.e. /Γ p diff is lrge) in the limit of lrge, highly viscous prticles, where diffusion of Y from within the bulk of the prticle must compete with surfce or ner surfce rection, leding to grdient in [Y](r) within the prticle. 67

15 Prticle phse rectnt profiles for prticle processing in the prticle phse diffusion limited regime [Y](r/)/[Y] Chrcteristic time for prticle phse diffusion = 0.5 s Time (s) r/ Figure 4-3: Prticle phse concentrtion profiles s function of time in prticle recting in the prticle-phse diffusion limited regime. 68

16 00. Limiting process Gmm Tu Size dependence of rte d[y]/dt Criteri for importnce Gs phse diffusion Kn( + Kn) Kn π Dx( g) ~ 0-8 s to Very lrge prticles Mss ccomodtion α H πmrt 3α ~ 0-7 to Lrge prticles, soluble trce gs, low α Surfce rection s 4 k Hs RTKs[ Y] c 3k s HsPx Ks See resistor model Thin-shell rection 4HRT c Dx( p) k[ Y] [ Y] 3HP x kd x ( p) ~ 0 0 D p x, << k[ Y] Bulk rection 4HRT k[ Y] 3c k HP x ~ 0-0 D, p x >> k[ Y ] Prticle phse diffusion 6D y [ Y] nx( g) c π Dy ~ 0 - Fst rection, highly soluble trce gs nd/ or very lrge prticles Tble 4-: Summry of gmms (inverse resistnces) nd chrcteristic times (see ppendix) relevnt in erosol chemicl trnsformtions. 4.3 Prticle phse loss in limiting cses We will now pply eq. 4. to clculte the frction of prticle phse species Y remining ([Y](t)/[Y] 0 ) given the initil concentrtion [Y] 0, due to interction with gs phse rective species X in specific limiting cses. In these clcultions we will ssume tht the overll gs phse density n x(g) remins constnt throughout the rection, since tmospheric erosol mss lodings re smll, (typiclly on the order 0 µg m -3 ) nd therefore only smll frction of the trce gs is expected to rect. Also, while prticle size chnge due to gs-prticle interctions cn be ccounted for, here we will ssume tht the prticle size remins constnt. In ech cse shown below the functionl dependence on physico-chemicl prmeters, prticle size, gs phse 69

17 00. density n x(g), nd initil prticle phse concentrtion is clerly evident. Therefore, it is possible to estblish the rte limiting process through comprison with dt from controlled experiment or well-monitored field study Prticle phse loss limited by gs phse diffusion nd/or mss ccommodtion of X This is the cse for very rpid (non-surfce) X-Y rection, (i.e. Γ s << α nd Γ rxn >> α). In other words, the species X is consumed s rpidly s it enters the bulk nd therefore γ mes?= (/Γ diff + /α) -. Upon integrtion of eq. 4. we obtin: Y ]( t) nx( g) c 3 = (/ Γdiff + / α ) t (4.0) [ Y ] 4[ Y ] [ 0 0 The loss of Y is prticle size dependent in generl. In the cse of kinetic flux to the prticle surfce, the incoming mss flow rte goes s the prticle surfce re (flux independent of rdius), wheres the rection tkes plce throughout the bulk of the erosol. In the cse of continuum flux into the prticle, the mss flow rte goes s the rdius (flux going s the inverse rdius). Eq. 4.4 will hve vrying size dependence, depending on the regime pplicble, nd this will mnifest in the size dependence of the prticle phse loss rte (eq. 4.0) Prticle phse loss limited by surfce rection This my reflect very efficient liquid surfce rectivity or be effectively true when concentrtions ner the surfce re lrger thn their bulk vlues (i.e. Gibb s surfce excess). In this cse γ mes = Γ s s given by eq. 4. nd vi integrtion of eq. 4.: s [ Y ]( t) 3H s RTnx( g) K s k ln = t (4.) [ Y ] 0 70

18 Prticle phse loss limited by rection within short recto-diffusive length, l << In this cse, rection of X nd Y occurs in thin shell of depth l below the erosol surfce. When l<<, the coth term in eq. 4.4 pproches, nd we re left with: 4HRT Γ rxn = Dx( p) k[ Y] (4.) c Now with γ mes?= Γ rxn integrtion of eq. 4. yields: [ Y]( t) [ Y ] 3RTnx( g) H Dx( p) k = t (4.3) [ Y 0 ] 0 We note tht this expression breks down in the limit where [Y]->0, since here l is no longer much less thn. Relizing this physicl limittion obvites concerns of the long time solution (leding to incresing [Y](t)), clerly beyond the domin of pplicbility of the expression Prticle phse loss limited by rection within long recto-diffusive length, l >> In the limit where l >>, the coth term pproches l/ + /3l. This is consistent with the trce gs hving very quickly filled the prticle to its Henry s lw concentrtion (tht is, [X] = n x(g) HRT). In this cse eq. 4.4 reduces to: 4HRT Γ rxn = k [ Y] (4.4) 3c Now with γ mes?= Γ rxn, integrtion of eq. 4. yields: [ Y ]( t) ln = HRTnx( g) kt (4.5) [ Y ] 0 7

19 Prticle phse loss limited by diffusion of Y from the erosol interior towrd the surfce In this cse γ mes = Γ p diff. Using the simplified expression for Γ p diff given by eq. 4.9, upon integrting eq. 4. one obtins: [ Y ]( t) D y ln = t (4.6) [ Y ] 0 Dependence on prticle size is here second order. Further, this is the only cse where the frction remining is independent of n x(g). We nticipte this physiclly since dding more trce gs will not ffect the processing rte when mss trnsfer within the prticle is solely rte limiting. Eq. 4.6 reflects the mximum decy rte of Y, dependent only on the rte of diffusion of Y within the prticle. Thus, in the bsence of convective mixing within the prticle, eq. 4.6 sets n overll upper limit for the chemicl trnsformtion rte within erosols. This cse is rte controlling in the limit of lrge or highly viscous prticles nd in the limit of fst X +Y rection tht loclly depletes [Y] t or below the surfce of the prticle. A solution for prticle phse concentrtion using the more exct expression for Γ p diff (eq. 4.7) cn lso be derived 7 : [ Y ]( t) 6 = [ Y] 0 i= π i i π Dyt exp( ) (4.7) However, smooth function requires the summtion of bout 00 terms. Eq. 4.6 overestimtes [Y]/[Y] 0 t 0.50 nd 0.0 frction remining by 3% nd 0% respectively. At 0.05 frction remining eq. 4.6 is within % of the exct solution. This is shown in fig

20 00. Frction Remining from stright-line profile pproximtion from nlyticl solution with 00 terms Prticle phse diffusion limited kinetics in dimensionless form chrcteristic time,τ = rdius / π D y t/τ Figure 4-5: Comprison of kinetics bsed on stright-line concentrtion profile nd those bsed on nlyticl solution in the cse of the prticle phse diffusion limit. 4.4 Simultneous rection nd prticle phse diffusion The generl problem of combined rection nd diffusion of both the trce gs nd the prticle phse species within the prticle is mthemticlly described through the following system of prtil differentil equtions: D x ( p ) [ X ]( r, t) k[ X ]( r, t)[ Y ]( r, t) = [ X& ]( r, t) D y ( p ) [ Y ]( r, t) k[ X ]( r, t)[ Y ]( r, t) = [ Y& ]( r, t) (4.8) Tken with four boundry conditions (to stisfy the four sptil derivtives) nd two initil conditions (to stisfy the two time derivtives) eqs. 4.8 fully define the physicl process. However, no known nlyticl solution exists for this system. We will therefore pply the 73

21 00. method of resistnces to obtin n pproximte solution. We will couple prticle phse diffusion with rection chrcterized by short recto-diffusive length. Once in possession of solution we will compre it to n nlyticl solution tht is expected to pproximte the ctul solution closely. A more rigorous comprison cn be pursued using dvnced numericl techniques. Prticle phse diffusion nd thin shell rection re expected to dominte prticle phse loss simultneously when their chrcteristic times re within n order of mgnitude of one nother while ech is t lest two orders of mgnitude higher (slower) thn those for ll other processes. Mthemticlly we expect: τ rxn, l<< p τ diff = π Dy [ Y ] 3 HPx kdx( p) ~ (4.9) It is therefore expected to be importnt for lrge prticles whose self-diffusion coefficient is very low, in the presence of highly soluble nd highly rective trce gs. In cses where the bove rtio is ner unity, we will need to use the resistor model to pproximte the overll prticle phse loss rte. We develop n expression for γ mes by combining Γ rxn (with l<<, eq. 4.34) nd Γ p diff (eq. 4.30). This enbles us to use eq. 4. nd write: [ Y ] D y Px H Dx( p) k [ Y& ] = (4.30) (4D [ Y ] + P H D k ) y x x( p) Integrtion of this expression t the limits recovers eq. 4.6 nd eq. 4.3 s expected. The generl nlyticl solution of eq is quite cumbersome 3, but solving it through numericl integrtion using either Runge-Kutt or Euler schemes is strightforwrd. Comprison of resistor model result with n nlyticl solution In our resistor model formultion, we hve llowed for the fct tht s the rection ensues, the recto-diffusive length increses. Though generl nlyticl solution for eqs. 4.8 does not 74

22 00. exist, in the cse where we hold l constnt solution cn be found by ppropritely chnging the boundry conditions to previously solved problem. Since we confine the rection to thin shell ner the surfce, in the prticle interior we hve simple diffusion, or: [ Y ]( r, t) [ Y ]( r, t) [ Y ]( r, t) D y ( + ) = r r r t (4.3) At the boundry, which we for convenience define s the surfce lying constnt depth l 0 beneth the surfce, we require tht the flux of prticle phse species from the bulk mtch the rte of disppernce due to rection in the shell. Our first boundry condition is therefore: [ Y ] ) [ ]( ) D x, p Dy r~ = HPx k Y l0 l0 (4.3) r k[ Y ] 0 The second boundry condition requires the concentrtion grdient vnish t r = 0: [ Y] ) r r= 0 = 0 (4.33) The initil condition is: [ Y ]( r,0) = [ Y (4.34) ] 0 These conditions re nlogous to those in the surfce evportion cse for which solution is provided in the literture, nd solution cn be dpted to our cse of thin shell rection 8 : 6L Y ] n y HPx k l ( t) = L n n { n L L } 0 ] 0 = β β + ( ) Dy [ [ Y exp( β D t / ) (4.35) Here, the β n must be solved for numericlly (i.e. using bisection or the Newton-Rphson technique) from: β cot + L = 0 (4.36) n β n In fig. 4-6 we compre this nlyticl result to the resistor model pproximtion. 75

23 00. Frction remining resistor model (l constnt) nlyticl solution ( l constnt) resistor model (l chnging) Coupled rection nd prticle phse diffusion; resistor model vs nlyticl Tu (s) Thin shell rection 5.9 Prticle phse diffusion Time (s) Figure 4-6: Comprison of resistor model with nlyticl solution in the cse of constnt rectodiffusive length. The chnging recto-diffusive length result is lso shown for comprison. Clcultions were performed on 0 micron prticle with physico-chemicl prmeters yielding chrcteristic times shown. From fig. 4-6 it is evident tht the resistor model result very closely mtches the more complicted nlyticl solution for the cse where we hold the recto-diffusive length constnt. The resistor model result for l chnging mtches the nlyticl solution t erly times, but delivers smller frctions remining t longer times. This reflects the fct tht s the rectodiffusive depth increses, the rection tkes plce throughout lrger volume, nd thus results in higher overll loss rte. 76

24 Appliction of results The interest in erosol prticle-phse chemicl trnsformtion is reltively recent. To our knowledge there re only few studies illustrting the rective regimes discussed bove. In lbortory studies of sub-micron (0. to 0.6 µm) erosol ozonolysis, the rective depletion of oleic cid clerly mtched Cse (see ch. 5, below), exhibiting the predicted rectnt concentrtion nd erosol size dependence 5. In similr experiment with lrger oleic cid erosols (0.68 to.5 µm), the Miller reserch group t the University of North Crolin observed trnsition from Cse to Cse s the erosol size incresed (submitted for publiction). Both experiments yielded oleic cid/ozone trnsformtion rtes much fster thn deduced from tmospheric field studies 9, indicting tht tmospheric rectivity depends on the morphology of complex prticles. In n erlier study focusing on rective gs uptke, Hnson nd Lovejoy [995] observed uptke kinetics governed by Cses 4.3. nd With the dvent of single prticle erosol mss spectrometers, field dt quntifying the time nd/or distnce dependence of erosol chemicl composition will become much more common, nd models incorporting prticle phse rections using the formultion presented here will be required to interpret mny of these studies. 77

25 00. Chpter 4 Appendix A4. Chrcteristic times for limiting processes Obtining chrcteristic times for these processes is pursued in mnner similr to tht used in deriving individul resistnces. One solves the prtil differentil equtions relevnt to the process ssuming it lone is cting. The solution will typiclly contin n exponentil summtion of the form: j = exp ( jf(p i )t) (A4.) Here f(p i ) is function of the physico-chemicl prmeters, p i (i=,,3 ) chrcterizing the prticulr process. The chrcteristic time is then expressed s b τ = /f(p i ). It follows tht for t >> τ, the exponentil terms re vnishingly smll, nd the dynmics re expectedly very slow (very long times re required to see very smll chnges in the system). The system here is described s being relxed. On the other hnd, for t << τ, the system exhibits very lrge chnges in very smll increments of time. The chrcteristic time then gives us n pproximtion of the relxtion time of the system, dividing the regions of fst nd slow dynmics. Given estimtes of the chrcteristic times for vrious processes, the process with the slowest chrcteristic time is the one most likely to govern the overll rte. A4.. Gs phse diffusion nd mss ccomodtion The chrcteristic time for gs phse diffusion in the continuum regime (Kn <<) is tken from the time-dependent solution of the diffusion eqution nd pproximtes the time for the concentrtion profile round the erosol to chieves stedy stte vlue: b For systems of sufficient simplicity it my not be necessry to solve for eq. A4.. For exmple, first order ordinry differentil equtions describing mny kinetic phenomen hve chrcteristic times given by τ=[y]/[y & ]. 78

26 00. τ g, diff = (A4.) π Dx( g) In this expression D x(g) is the diffusion coefficient (cm s - ) of the trce gs round the prticle. The chrcteristic time for mss ccomodtion (sometimes clled the chrcteristic time for interfcil equilibrium) indictes the timescle over which the species tken up by the surfce obtins its equilibrium concentrtion vlue there. We re describing cse where n erosol initilly devoid of trce gs is suddenly plunged into known concentrtion of trce gs, held constnt during the uptke process. The chrcteristic time c emerging from the time-dependence of the nlyticl solution is 0 : H πm x RT τie = 3α (A4.3) In the specil cse of highly insoluble trce gs, the chrcteristic time becomes: τ ie, insol = (A4.4) π Dx( p) In these expressions, H is Henry s lw solubility coefficient (M tm - ), R is the gs constnt (tm M - K - ), M x is the moleculr weight of the trce species (g mole - ) nd D x(p) is the diffusivity of the trce gs in the prticles. 4.. Surfce rections In this cse we hve only to solve n ordinry differentil eqution, nd there is only one term in the form given in eq. A4.. From the solution (see eq. 4.), the chrcteristic time for surfce rection is: c The sme expression cn be derived much more simply by ssuming constnt flux to the surfce nd clculting the time required to fill the prticle to its Henry s lw concentrtion. 79

27 00. τ s = (A4.5) s 3k H s RTK snx( g) 4..3 Rections within the prticle interior Here we cn go bck to eq. 4.8 nd tke chrcteristic times from the more generl solution, describing the pproch to stedy stte. From the solution to eq. 4.8 in sphericl coordintes, it cn be shown tht the chrcteristic time s defined through eq. A4. is given by: τ rxn, diff = π D x( p) k[ Y] + (A4.6) This is the time required for X to pproch pseudo-stedy stte profile within the prticle. It cn therefore be thought of s recto-diffusive solubility time. An importnt question is whether the time it tkes to form the profile in X cn ever be expected to ffect the overll rte of mss trnsfer within tmospheric prticles. We will return to this question below. In the cse of very short recto-diffusive length, l<<, we cn derive chrcteristic time for rection of Y within the prticle using the expression in footnote () bove. Since rection is confined to within the shell of thickness l, the volume verged loss rte of Y cn be pproximted s: [ Y& ] = k[ X ][ Y ]4π l (A4.7) 4 π 3 3 Tking the trce species concentrtion in the shell to be ner its Henry s lw vlue, this leds to n estimted chrcteristic time for depletion of the liquid species within the liquid shell s: 80

28 00. [ Y ] [ Y ] τ rxn, l<< = = ~ (A4.8) [ Y& ] 3k HPx l 3HPx Dx( p) k Since the loss rte goes s the inverse of the chrcteristic time, this expression indictes tht prticle phse loss limited by rection within the recto-diffusive length will exhibit inverse rdius dependence. A similr pproch cn be used to derive the chrcteristic time in the cse where the trce gs fills the entire prticle (or by inspection of eq below). One finds: τ rxn, l>> = (A4.9) k HPx In this cse, the loss rte is independent of size. From considertion of the chrcteristic times in the limiting regimes of bulk nd thin shell rection, it is pprent tht when Γ rxn governs the overll rte, the size dependence spns 0 to - depending on the extent to which the recting trce gs penetrtes into the prticle interior. We my now return to the question of the importnce of rective solubility of the trce gs in determining the overll rte of chemicl trnsformtion within prticles. In the limit of very smll prticle rdii, the chrcteristic time for rective solubility, eq. A4.6, becomes: π Dx( p) τ rxn, diff << D k [ (A4.0) π Y ] x( p) Wheres in the limit of very lrge rdii, we obtin: π Dx( p) τ rxn, diff >> (A4.) k[ Y] k[ Y ] In the smll prticle limit, we my compre eq. A4.0 nd eq. A4. below. Since it is unlikely tht the trce gs diffusion coefficient within the prticle be less thn the self-diffusion 8

29 00. coefficient of the prticle species, it is unlikely tht this process limit the overll rte for smll sizes, since prticle phse diffusion will limit insted. For very lrge prticles, consider expressions A4. nd A4.9 bove. Since it is very unlikely tht the trce species concentrtion ever surpss tht of the prticle phse, it is unlikely tht this process ever limit the rte for lrge prticles. We my infer from these limits tht the sme will hold for intermediry sizes, such tht solubility of the rectnt trce gs is unlikely to limit the overll processing rte in erosols of tmospheric or lbortory interest Prticle phse diffusion Inspection of eq. 4.7 yields the chrcteristic time for prticle phse diffusion: p τ diff = (A4.) π Dy We note tht chrcteristic times llow one to quickly ssess the reltive importnce of the five rtes involved in erosol processing, while indicting wht physicl prmeters will likely be gined from dynmic informtion of the uptke process (Fig. A4-). If for exmple the time for rection of the liquid species in thin shell is the gretest of ll the chrcteristic times, one cn expect the overll rection rte to depend on the size (slower for lrger prticles) nd to extrct fundmentl informtion in the form of H(D x(p) k ) /. Tble 4- summrizes the results presented here. 8

30 00. Chrcteristic time (s) P x ~ 0 - torr rection of prticle phse within thin shell diffusion of prticle phse to surfce mss ccomodtion of trce gs (interfcil equilibrium) rective solubility of trce gs gs-phse diffusion of trce gs rection of prticle phse throughout bulk Prticle rdius ( µm) Physico-chemicl prmeters: Dx(p) ~ 0-6 cm /s H ~ M/tm Dx(g) ~ 0 - cm /s Dy ~ 0-7 cm /s k ~ 0 6 M - s - Figure A4-. Chrcteristic time plot to ssess competing rtes. Rection within thin shell chrcteristic of the trce gs recto-diffusive length governs the overll rte of rection from 0. to 5 micron rdius prticles with nticipted size dependence. 83

31 00. Chpter 4 References D. R. Hnson nd E. R. Lovejoy, Science, 67, 36 (995). Q. Shi, P. Dvidovits, J.T. Jyne, D. R. Worsnop, nd C. E. Kolb, J. Chem. Phys. A, 03, 88 (999). 3 C. E. Kolb, P. Dvidovits, J. T. Jyne, Q. Shi, nd D. R. Worsnop, Progress in Rection Kin. nd Mech., 7, -46 (00). 4 J. A. de Gouw nd E. R. Lovejoy, Geophys. Res. Lett., 5, 93 (998). 5 J. W. Morris, P. Dvidovits, J. T. Jyne, Q. Shi, C. E. Kolb, D. R. Worsnop, W. S. Brney, J. Jiminez, nd G. R. Css, Geophys. Res. Lett., 0 (00). 6 S. E. Schwrtz nd J. E. Freiberg, Mss-trnsport considertions pertinent to queous phse rections of gses in liquid-wter clouds, in Chemistry of Multiphse Atmospheric Systems, (Springer, Heidelberg, 984). 7 D. R. Worsnop, J. W. Morris, P. Dvidovits, Q. Shi, nd C. E. Kolb, Geophysicl Reserch Letters, October, P.V. Dnckwerts, Gs-Liquid Rections, (McGrw Hill, New York, 970). 9 J. Crnk, The Mthemtics of Diffusion, (Clrendon Press, 997). 0 D. R. Hnson, J. Phys. Chem. B, 0, 4998 (997). N. A. Fuchs nd A. G. Sutugin, Highly Dispersed Aerosols, (Ann Arbor Science Publishers, Ann Arbor, 970). J. F. Widmnn nd E. J. Dvis, J. Aerosol Sci., 8, 87 (997). 3 D. R. Hnson, A. R. Rvishnkr, nd E. R. Lovejoy, J. Geophys. Res., 0, 9063 (996). 84

32 00. 4 J. H. Hu, Q. Shi, P. Dvidovits, D. R. Worsnop, M. S. Zhniser, nd C. E. Kolb, J. Phys. Chem., 99, (995). 5 MATHEMATICA 4, Student Version. 6 Crnk, p Crnk, p Crnk, p W. F. Rogge, M. L. Hildemnn, M. A. Mzurek, G. R. Css, nd B. R. T. Simoneit, Environ. Sci. Technol., 5, (99). 0 S. Kumr, Atmos. Env., 3, 99 (989). S. E. Schwrtz nd J. E. Freiberg, Atmos. Environ., 5,9 (98). 85

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