A Unified and Versatile Model Study for Moisture Diffusion

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1 2017 IEEE 67th Electronc Comonents and Technology Conference A Unfed and Versatle Model Study for Mosture Dffuson Langbao Chen, Jenny Zhou, sng-e Chu, Xuejun Fan Deartment of Mechancal Engneerng Lamar Unversty eaumont, TX, USA xuejun.fan@lamar.edu Abstract All electronc ackages nvolve th a multmateral system, n hch many of the materals or flms are suscetble to mosture absorton. Deste dssmlar materal roertes, mosture transorts n a mult-materal system from a hgh ater actvty regon to a lo one, here ater actvty s a measure of ater energy n a secfc substance. Ths, hoever, has not been ell recognzed n electronc ackagng ndustry. Furthermore, mosture concentraton gradent s often deemed as the drvng force for mosture transort, hch nevtably oses a challengng dscontnuty ssue at nterface for mosture dffuson n mult-materal systems. Even though several normalzaton schemes have been develoed n the lterature, much confuson has exsted on the fundamental rncle of mosture dffuson. Ths aer derved an actvty-based dffuson model usng the concets of chemcal otental and ater actvty. We shoed that the contnuty of ater actvty at nterface n dssmlar materals s arranted, and furthermore, many nonlnear ater sorton sotherms can be aled n the actvty-based model by usng a ne roerty called generalzed solublty. The actvty-based model thus s caable to study comlex mosture dffuson n multmateral system. Moreover, n ths aer, the actvty-based model as used to unfy the dfferent normalzaton theores, such as solublty-based and the so-called etness normalzaton aroaches. We also dscussed ho ater sorton sotherm ould affect the conventonal mosture dffusvty, fndng that only for some lmtng cases (e.g., enry sorton sotherm), the effectve mosture dffusvty becomes ndeendent of mosture concentraton. We onted out that the generalzed solublty that are needed to solve the dffuson can be obtaned usng conventonal terms such as saturated mosture concentraton and solublty. As demonstraton, a numercal examle as erformed n commercal fnte element softare to study the mosture dffuson through a b-materal nterface under dynamc temerature and humdty condtons. The results from dfferent nonlnear sorton sotherms ere comared to demonstrate the caablty and versatlty of the model. We concluded that the actvty-based mosture dffuson model s a unfed and versatle aroach to study and understand the mosture dffuson mechansm n IC ackages. eyords-mosture dffusson model; mult-materal systems; mcrolectroncs; ater actvty;nonlnear sorton sotherms I. INTRODUCTION Polymerc materals, such as eoxy moldng comound and adhesve thn flms, lay an mortant role n manufacturng electronc ackages. One undesred characterstc of these olymerc materals s ther tendency to absorb mosture from the ambent, hch may cause many negatve effects ncludng hydroscoc sellng [1], electrochemcal mgraton [2], and even ocorn falures under solderng reflo [3-5]. Therefore, t s crucal to redct the mosture behavors n electronc ackages that nvolve th dssmlar materals. The dely-used theory for ater dffuson s based on Fck s la, hch assumes that ater flux s drven by the gradent of mosture concentraton C. The model s C DC t (1 ) here D s dffusvty coeffcent. oever, n multmateral systems lke electronc devces, the mosture concentraton s dscontnuous at the materal nterface due to the fact that dfferent materals have dstnct absorton caabltes. Ths ell-knon ssue has rohbted the alcaton of Eq. 1 n the commercal fnte element analyss (FEA) ackages for solvng mosture dffuson n mult-materal systems [6-11]. To remove the dscontnuty n the Fckan dffuson theory, several normalzaton schemes have been roosed, ncludng the etness aroach [12], the artal ressure technque [13, 14], and advanced normalzed concentraton scheme [15]. These methods dffer from each other by usng dfferent normalzed feld varables and dfferent assumtons, but they all based on the lnear Fck s la shon n Eq. 1 [16]. On the other hand, t has been shon that the lnear Fckan model s bascally a lmtng case of general mass dffuson that s based on the true drvng force the gradent of chemcal otental [17]. The chemcal otental of ater, n the meantme, s a functon of temerature and ater actvty, so ater actvty can be used as an mortant concet to understand ater dffuson. As a matter of fact, hen studyng ater dffuson n food and membrane systems, researchers have shon that ater transorts from the regons of hgh ater actvty to the regons of lo ater actvty [18-22]. Ths fundamental rncle of ater dffuson, hoever, s not ell recognzed n mcroelectroncs ndustry, and the /17 $ IEEE DOI /ECTC

2 concentraton-based dffuson theory s stll dely adoted. In ths aer, a ne ater dffuson model s roosed usng the concet of ater actvty and the fundamental dffuson rncle. The man objectve of the aer s to develo a unfed and versatle dffuson model to solve comlex ater transortaton n mult-materal systems thout normalzaton. To acheve ths, e adot ater actvty, hch s a contnuous state varable, as the feld varable, and derve the dffuson model based on the fundamental equatons of ater flux. In addton, a ne materal roerty called generalzed solublty s ntroduced to relate ater actvty th ater concentraton, so that the ater concentraton can be obtaned once ater actvty s solved. To make the ne model versatle, nonlnear sorton and dffuson behavors are also consdered durng the model develoment. The rest of the aer s organzed as follos. The model dervaton along th several key concets are resented n Secton II. A comarson beteen the ne model and dfferent normalzed schemes for lnear concentraton-based model s made n Secton III. Partcularly, the effectve dffusvty s dscussed n the context of nondeal ater sorton behavors. In Secton IV, the model s mlemented n commercal FEA ackage to model the mosture behavors n a b-materal system subjected to dynamc humdty and temerature condtons. oth lnear and nonlnear sorton behavors are consdered to demonstrate the versatlty of the roosed model. Fnally, the conclusons are dran n Secton V. II. MODEL DEVELOPMENT A. General Equatons Consder ater dffuson n a mult-materal system subjected to dynamc humdty and temerature condtons, as shon n Fg. 2. The system contans multle sold solvents th dfferent chemcal comostons and roertes. Each solvent s able to absorb ater from the ambent. For clarty and smlcty, e assume an sothermal condton throughout the system, so that thermal dffuson s not consdered and unform heatng can be aled. Fundamentally, the equlbrum of the system requres that the chemcal otental of ater must be the same everyhere. The chemcal otental of ater at an arbtrary tme and oston can be exressed n terms of to state varables, temerature T and ater actvty a. For solvent, e have[17, 23], 0, RT ln a, (2 ) th 0 s the standard chemcal otental and R s unversal gas constant. Eq. 2 ndcates that the establshment of system equlbrum s acheved by the movement of ater from the regons of hgh ater actvty Fgure 1. A mult-solvent system subected to dyamc temerature and humdty condtons. The chemcal otental of ater n the solvent s a functon of ater actvty and temerature. to the regons of lo ater actvty. At equlbrum, both the chemcal otental and ater actvty becomes unform n the system. The equlbrum ater actvty, a,eq, can be aroxmated by [19, 20, 24]. a, eq amb / g R (3) here amb s the artal ater vaor ressure of the system, g s saturated ater vaor ressure, and the relatve humdty R=/ g 100%. Eq. 3 states that the ater actvty of an equlbrum system can be accurately determned from R. Note that the chemcal otental, temerature, and ater actvty are alays contnuous, so the materal subscrt for these varables ll be droed n the dervaton belo. To develo a versatle model, the fundamental equaton for ater flux, Jm, s used, hch s [17] J m CM here C (kg/m 3 ) s ater concentraton and M (m 2 mol/j/s) s called moblty coeffcent. Substtutng Eq. 2 nto 4 yelds the ater flux based on ater actvty, as J m RT CM a a D a C a M RT a (4 ) (5 ) n hch D MRT and =C /a. ere, D s often referred as to tracer or ntrnsc dffusvty, and (kg/m 3 ) s ne arameter ntroduced as generalzed solublty n ths aer. From Eq. 5, the arameter s used to establsh the relatonsh beteen mosture concentraton and ater actvty, as C a. ( 6 ) Recall the contnuty equaton for the conservaton of ater mass, hch s C J m. ( 7 ) t 1661

3 Substtutng Eqs. 5 and 6 nto the contnuty equaton yelds a t D a (8 ) Eq. 8 s an actvty-based model th the gradent of ater actvty as the drvng force of net ater dffuson. To solve Eq. 8, must be determned for each solvent. Notce that at equlbrum ater actvty s equal to R accordng to Eq. 3, and the corresondng ater concentraton s tycally called saturated mosture concentraton, or C sat. Therefore, alyng Eq. 6 for an equlbrum state yelds the follong formula for : Csat, ( R, T) ( R, T). ( 9 ) R here both and C sat can be a functon of R and T. Eq. 9 gves a smle relatonsh beteen and C sat, and also rovdes the means of determnng by erformng ater sorton tests at varous relatve humdty and temeratures. Note that the relatonsh beteen C sat and R s n fact the ater sorton sotherm. One of the smle sorton sotherm s based on enry s la, hch s C sat S ( T) S ( T) ( T) R ( 10 ) amb here S s enry s solublty. Comarng Eq. 9 and 10 yelds the follong relatonsh: g S ( T) g ( T) ( 11 ) here denotes the generalzed solublty for enry s la. Therefore, can be obtaned once enry s solublty s knon. It can be seen that s only a functon of temerature. Generally, for a nonlnear sorton sotherm (e.g., Langmur tye), ll be a functon of both ater actvty and temerature. Eq. 9 ndcates that s generally a functon of temerature and ater actvty, so Eq. 8 can be further rtten as a T a Da a a ( 12 ) t T t hch s a general actvty-based dffuson model. ecause ater actvty s contnuous state varable lke temerature, Eq. 12 s ndeed analogous to heat transfer equaton that has the source term (.e., the second term on the rght-hand sde of Eq. 12). Usng the temerature-actvty analogy, the actvty-based model can be mlemented n commercal fnte-element-analyss (FEA) ackages to solvng ater dffuson n mult-materal systems thout dscontnuty ssues. Note that ater actvty s not derved by the normalzaton of concentraton, hch s fundamental dfference from other normalzed technques for mult-materal systems. Once the actvty-model s solved, the solutons of mosture concentraton can be obtaned usng Eq. 6.. Secal Cases One secal case of the actvty-based model can be derved by assumng all the materals have enry sorton behavors. In ths case, ll only deend on temerature accordng to Eq. 11, so Eq. 12 becomes a t T D a a T ; a ( 13 ) here s used as the ndeendence symbol. Note that there s stll a source term n Eq. 13 due to the rate of heatng or coolng. Ths s smlar to the Fckan model that as normalzed by enry s solublty n several orks (hch s also called the artal ressure technque) [7, 8, 15, 16]. Another secal case s assumng zero heatng rate ( T 0 ) or temerature-ndeendent ( / T 0 ), so that the source term ll vansh. The corresondng model becomes a a D a a ; t T or T =0. ( 14 ) At last, the smlest case can be obtaned hen s a constant for each materal and does not deend on ater actvty or temerature, leadng to a t D a, const ( 15 ) hch mathematcally resembles to the form of lnear Fckan model n Eq. 1. Therefore, the lnear Fckan dffuson model s n fact the lmtng case of the actvtybased model. C. Intal and oundary Condtons Wth Eq. 3, t s straghtforard to defne the ntal condtons for the actvty-based model, as a ( t 0) R ( 16 ) here R0 s ntal relatve humdty condton. Also, the ater actvty at the boundary can be nned th the ambent relatve humdty, so that a( t) bc R( t) ( 17 )

4 here R (t) changes th tme to reresent a dynamc humdty condton. In the case of zero ater flux at the boundary, e can have a 0 ( 18 ) bc III. COMPARISON WIT DIFFERENT NORMALIZED SCEMES AND NONLINEAR MODELS ased on the dervatons n Secton II, there are to man features of the actvty-based model. Frst, t s drectly derved from the fundamental ater dffuson and dscontnuty equatons. Second, ater actvty s a contnuous state varable that determnes the chemcal otental and s not the result of normalzaton. These to features make the actvty-based model fundamentally dfferent from the exstng normalzed models hch are smly based on the normalzaton of the classcal Fckan model (Eq. 1). Realzng that the classcal Fckan model may be the lmtng cases of the actvty-based model, e can examne the exstng models by comarng dfferent defntons of normalzed concentraton th ater actvty. We ll also dscuss ho the nondealty affects the effectve dffusvty n ths secton. A. Examaton of the Wetness Theory The concet of etness has been roosed to normalze Eq. 1 and to remove the concentraton dscontnuty at the materal nterface [12, 25]. The etness s defned as C / Csat, ( 19 ) Substtutng Eqs. 6 and 9 nto Eq. 19 yelds a ( 20 ) R R Eq. 20 ndcates that the etness s ndeed a functon of ater actvty. Wth Eq. 20, the contnuty of etness s met by satsfyng the follong condton: j, j ( 21 ) R R j If all the materals follos enry sorton behavors, Eq. 21 ll be satsfed, because ll not be a functon of ater actvty or R and the rato / (R) s equal to 1.0. On the other hand, f dfferent materals have dstnct and nonlnear sorton behavors, Eq. 21 may not be vald, resultng n dscontnuous etness at the nterface. In general, there ll be lmtaton n usng the etness aroach to deal th the mult-systems here the contnuty requrement n Eq. 21 s not met. Therefore, the etness aroach may not be a unversal method for studyng mosture dffuson n mult-materal systems.. Comarson to the Partal Pressure Technque The artal ressure technque s another often-used normalzed aroach for ater dffuson n mult-materal systems. It as frst roosed by Wadak [13], here the normalzed concentraton as defned as a ressure term C / S ( T, ) ( 22 ) here S s the solublty coeffcent, hch may be a functon of the ressure term and T. Smlar equaton to Eq. 22 as also derved by Wong et al. [16]. In many orks [8, 14], S as relaced by enry s solublty S, so that the equaton can be smlfed. Follong Eq. 11 for enry s la, e can also have a general relatonsh beteen and S, as S ( T, ) ( 23 ) Therefore, the ressure can be rtten as a / S ( T, ) a g g ( 24 ) Eq. 24 states that the ressure s ndeed the saturated ater vaor ressure multled by ater actvty. At equlbrum, becomes the ambent artal ressure of ater vaor accordng to Eq. 24. Snce a s alays contnuous and g does not deend on solvents, s also contnuous. In the lterature, the artal ressure technque as aled to the classcal lnear Fckan model, and thus the resulted models are the secal cases of the actvtybased model (e.g., Eq. 13 or 15). To obtan a general ressure-based model, one need to substtute Eq. 24 nto the general actvty-based model n Eq. 12 (the corresondng model s omtted here due to ts comlexty). C. Advanced Normalzed Concentraton Aroach Jang et al. [15] roosed an advanced normalzed concentraton (ANC) model based on enry s la, here a normalzed varable as defned as C / M ( 25 ) here M S, called the modfed solublty. The constants and S, are used n the Arrhenus equatons to calculate the saturated vaor ressure and enry solublty S, resectvely, as ( T) ex E RT g / ( 26 ) S S, ex Es, / RT ( 27 ) 1663

5 here E and E s, are the energy constants. y erformng least-square fttng of the steam table [26], e can obtan and E = J/mol. In the ANC model, t as assumed that E E s,, and thus for, g S S,. Substtutng Eq. 6 for C and Eq, 11 can be exressed as ( S ) a a g a th E E M S, s, ( 28 ) Eq. 29 mles that s actually a lmtng case of ater actvty based on to assumtons: 1) the sorton sotherm follos enry s la; and 2) the sorton sotherm s temerature-ndeendent th =M as constants hen E E s, ). When a constant, the corresondng dffuson model s the smlest case as gven n Eq. 15, hch agrees th the model used n Jang et al [15]. D. Effectve Dffusvty for Nondeal Dffuson In the lterature, nonlnear concentraton-based model has also been used for nondeal mass dffuson [17, 23, 27]. The nondeal concentraton-based model can also be derved from the actvty-based model usng the follong equaton [17] a ( C, T) C ( 29 ) here (m 3 /kg) s defned as actvty coeffcent. Comarng Eq. 6 and 29, one can see that s the recrocal of. Substtutng Eq. 29 nto Eq. 5 to relace a by C yelds 1 1 J m D C D C C C ln D C C D C C 1 C ln hch s the exact formulas used n the lterature [17, 23, 27]. Eq. 30 ndcates that the effectve dffusvty (D eff) may deend on concentraton f the term ln / ln C s not a constant. For the secal case of enry s la, both and are constants, and thus ln / ln C becomes zero. In that case, the effectve dffusvty ll not deend on concentraton. For other nonlnear sotherms, the effectve dffusvty should be evaluated so that the effect of nondealty can be better understood. Theoretcally, the term ln / ln C may be calculated by knong and ater actvty, hch ll be the future ork. D eff IV. NUMERICAL APPLICATIONS To demonstrate the caablty of the actvty-based model n descrbng nonlnear and non-deal dffuson, a tycal roblem nvolvng a b-materal system s consdered, as shon n Fg. 2. The b-materal system ncludes to materals th the same thckness of 1mm. The dth of each materal s far greater than the thckness so that the dffuson can be smlfed as a 1-D roblem. Fgure 2. A tycal mosture dffuson roblem n a b-materal sytem for valdatng and demostratng the actvty-based model. At the begnnng, the system s n equlbrum th the ambent at R 0=100% and T 0=25 o C. Then the envronmental loadng s aled by changng the temerature of the system at a rate of 1 o C/mn, but artal ressure of ater vaor s ket the same as the value of g at 25 o C. The corresondng equatons for R and temerature are exressed as o g (25 C) o R ( t) ; T 25 t /60 C ( 31 ) ( T ) g It s assumed that the dffusvty coeffcents for both materals follo the Arrhenus equaton, as D T) D ex E RT D, (, D, / ( 32 ) here and E D, are materal constants. To use the actvty-based model, the general solublty must be defned. For Mat-A, s defned by the enry solublty usng Eq. 11, as S S E RT A g A g, A ex s, / ( 33 ) For Mat-, to dfferent cases are consdered, hch are gs, b (1 ba ex Es, / RT E E m ex ) RT RT ref,case I ( 34 ),Case II here b, m, and E are model arameters, and T ref s the reference temerature (T ref=25 o C n ths examle). ascally, Case I s corresondng th enry sotherm, 1664

6 hle Case II s corresondng to Langmur sotherm th temerature terms. Note that n Case II, s a functon of ater actvty. Aarently, Case II s relatvely comlcated and s dffcult to be consdered by tradtonal models. The requred materal roertes for solvng the actvty-based model are gven Table 1. TALE I. MATERIAL PROPERTIES FOR I-MATERIAL SYSTEM [8] Materals Parameters A enry enry Langmur D (m2/s) E D (J/mol) S (kg/m/ Pa) / E s(j/mol) a / b / / 7.2 m (kg/m 3 ) / / 8.5 E (J/mol) / / a. Es s loer than the value used n the lterature to sho more thermal effects the solutons of ater actvty are obtaned. To valdate AAQUS mlementaton, fnte dfference method (FDM) s also used as reference. The calculated dstrbutons of ater actvty and mosture concentraton are lotted n Fg. 4 and 5, resectvely. It can be seen that the same results are obtaned from FEA and FDM, valdatng the mlementaton of the actvty-based model n AAQUS. The results also sho that the sorton sotherm has a great mact on the dffuson behavors. At t=1800 s, the Case- II, hch s based on temerature-deendent Langmur sotherm, yelds much loer ater actvty than the Case I for enry sotherm n the regon near Mat- (x=0~1 mm). Ths attern s more obvous at t=3600 s, as shon n Fg. 4. The rofles of ater concentraton are also lotted n Fg. 5. Generally, ater concentraton s dscontnuous at the materal nterface, th C A generally greater than C. At t=1800 s, the rofles of concentraton based on Case II s closed to that of Case I, hch agrees th the sotherms gven n Fg. 3 hen the ater actvty s stll hgh. oever, at t=3600 s, ater concentraton calculated from Case II s sgnfcantly dfferent from Case I. The numercal results sho that the sorton sotherm has a great mact on the dffuson behavors. Fgure 3. Sorton sotherm at dfferent temeratures based on Eq. 29. (a) For llustraton, Fg. 3 comares the to dfferent relatonshs beteen C sat and a, hch are resulted from the to cases of gven by Eq. 34. The values of C sat s calculated by Eq. 9. Generally, C sat s a functon of temerature for both cases. oever, for Case I, C sat has a lnear relatonsh th ater actvty or R for a gven temerature; for Case II, the relatonsh s nonlnear. Note that the to dfferent sotherms are urosely constructed to have a smlar C sat at a =1.0. The dffuson roblem s solved by mlementng the actvty-based model n commercal FEA ackage AAQUS usng the temerature-actvty analogy. The second term on the rght-hand sde of Eq. 12 s evaluated as a heat source, as the roblem nvolves th dynamc temerature condton and temerature-deendent. Water concentraton ll be calculated usng Eq. 6 once (b) Fgure 4. Calucated ater actvty at a) t=1800 s and b) 3600 s for to dfferent cases gven n Eq

7 (a) (b) Fgure 5. Calucated mosure concentraton at a) t=1800 s and b) 3600 s for to dfferent cases gven n Eq. 34. V. DISCUSSIONS AND CONCLUSIONS Mosture dffuson n electronc devces s crtcal due to mosture-nduced relablty ssues. Snce all electronc devces nvolve mult-materal systems and each materal has dfferent ater sorton caabltes, t has been a challenge to study mosture dffuson usng the classcal Fckan model. Even though many normalzaton schemes have been roosed to remove the dscontnuty, they are based on dfferent assumtons and a unfed model s stll lackng. Moreover, n the conventonal Fckan dffuson model as ell as the normalzed models, the drvng force s stll consdered to be the gradent of mosture concentraton, hch, nevertheless, s a lmtng case of the fundamental mass dffuson based on the chemcal otental. In ths aer, a unfed and versatle mosture dffuson model, hch s called the actvty-based model, as derved. Usng the fundamental equaton for mass dffuson rncle, e shoed that the drvng force of mass dffuson n chemcally nhomogeneous system s the gradent of ater actvty, hch agrees th the lterature regardng general mass dffuson. The actvty-based model ndeed s a theoretcal extenson of the alcaton of ater actvty concets n food and membrane scence. We found that the contnuty of ater actvty, hch s a state varable that determnes the chemcal otental of ater, s naturally arranted and thus fundamentally dfferent the tradtonal normalzaton varables based on ater concentraton. Also, snce ater actvty at equlbrum or at the boundary s equal to the ambent R, the ntal and boundary condtons of the actvty-based model can be easly defned. In the actvty-based model, e roosed a ne materal roerty, the generalzed solublty, to relate ater actvty th ater concentraton. We shoed that the generalzed solublty s a key arameter to defne ater sorton sotherms, so that many nonlnear sotherms can be consdered n the actvty-based model. We have establshed the relatonsh beteen the generalzed solublty and the saturated mosture content, hch allos the determnaton of the ne roerty by erformng ater sorton tests under varous temerature and humdty condtons. Furthermore, e shoed that the actvtybased model can be used to unfy the exstng normalzed models, ncludng the etness aroach, the artal ressure technque, as ell as the advanced normalzed method. It as found that the exstng models are the lmtng cases of the actvty-based model. Usng the fundamental dffuson rncle, e also onted out that the effectve dffusvty may be deendent on ater concentraton or actvty, unless the lmtng cases such as enry s la are assumed. Wth the above-mentoned consderatons ncororated nto the actvty-based model, e mlemented the model n commercal FEA ackage usng the temerature-actvty analogy. y solvng the dffuson n a tycal b-materal system, e demonstrated the caablty of the roosed model n modelng comlex dffuson behavors under dynamc temerature and humdty condtons. The results ndcated that ater sorton sotherms have sgnfcant effect on the dffuson behavors. ased on our study, e concluded that the actvty-based model s ndeed a unfed and versatle model to study and understand the mosture dffuson mechansm n IC ackages and other multmateral systems. REFERENCES [1] X. J. Fan, Lee SWR, an Q. Exermental nvestgatons and model study of mosture behavors n olymerc materals. Mcroelectroncs Relablty 49, [2] X. J. Fan and E. Suhr, Mosture Senstvty of Plastc Packages of IC Devces. Ne York: Srnger, [3] C. G. Shrley, "Pocorn Cavty Pressure," IEEE Transactons on Devce and Materals Relablty, vol. 14, , [4] X.J. Fan, Zhang GQ, van Drel WD, Ernst LJ. Interfacal delamnaton mechansms durng solderng reflo th mosture recondtonng, IEEE Transactons on Comonents and Packagng Technologes 31(2), , [5] A. A. Gallo and R. Munamarty, "Pocornng: a falure mechansm n lastc-encasulated mcrocrcuts," IEEE Transactons on Relablty, vol. 44, ,

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