Modeling of diffusion and chemical reactions in ion-exchange resins during swelling and shrinking

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1 Modelng of dffuson and chemcal reactons n on-exchange resns durng swellng and shrnkng Tuomo ano * and Erkk Paatero, Laeenranta Unversty of Technology, Laboratory of Industral Chemstry, knnarlankatu 34, FIN Laeenranta, Fnland * Corresondng author. E-mal: tuomo.sano@lut.f, Tel: , fax: Abstract Ion-exchange resns used n chromatograhc searaton, on exchange, and heterogeneous catalyss are elastc olymerc materals. The extent of swellng of the olymer, and thus the dmensons of resn artcles, change n resonse to changes n, for examle, the solvent comoston and salt content of the surroundng lqud. Ths s reflected to the ntraartcle mass transfer rates through the mesh wdth of the aertures n the olymer network. Mass transfer and chemcal reactons n sulfonated P DVB on exchange resns durng shrnkng and swellng of the artcles were nvestgated exermentally and wth comuter smulatons. A secally constructed flow-through cell and an otcal mcroscoe were used to montor the dffuson nduced volume changes of resn artcles. The alcablty of artcle dffuson models based on the Fck's law and the Maxwell tefan aroach, as well as two-ont aroxmatons of these, was nvestgated. In the Maxwell tefan aroach, the chemcal otentals of the moble seces were calculated wth models derved from thermodynamcs of olymer solutons and gels by takng nto account the elastc nature of the olymer. The effect of swellng rato on the dffuson coeffcents was descrbed wth a geometrc obstructon model for the sake of smlcty. The governng equatons were wrtten and solved n olymer mass coordnates, whch enables faster and more accurate numercal soluton. The choce of the mass transfer model was found to have a large mact on the redcted swellng behavor of the resns. It was shown that, owng to the elastc nature of the resns, dffuson couled wth volume changes of artcles was best descrbed wth a model that exlctly takes the resence of the cross-lnks n the olymer nto account. Introducton everal aroaches to modelng dffuson of solvents n cross-lnked olymers have been resented n the lterature. The essence of the Tanaka Fllmore aroach [1, 2] s that lquds do not dffuse nto the olymer network but the olymer network dffuses nto a stagnant flud. The dffuson coeffcent of the olymer network s defned usng the bulk modulus of the network and a solvent olymer frcton coeffcent. The resent work focuses on dffusonnduced swellng of gels n lqud mxtures wth two or more comonents. To the best of our

2 knowledge, the Tanaka Fllmore aroach has not been extended to such systems, and only models descrbng enetraton of lquds nto the olymer network are consdered n ths work. In rgorous treatments of the roblem [3 5], the resence of the cross-lnks, and the resultng lmted exansblty of the olymer network, s accounted for by ntroducng stress tensors n the equatons of moton. Consequently, the volumetrc flux nto a volume element of the artcle s reduced f t results n an ncrease n the stress of the olymer network. An alternatve way s to embed the effect of the cross-lnks nto the drvng force va the solvent actvty [6], whch deends on the extent of swellng. Dffuson n gel-tye on-exchange resns s analyzed n ths study by usng two smlfed aroaches (.e., wthout solvng the equatons of moton): Fck s law and the generalzed Maxwell tefan formalsm. Mass balance equatons are gven for shercal artcles. The local swellng rato, and hence the volume of the artcle, s allowed to change wth tme. The mass transfer rates deend strongly on the olymer volume fracton and ths effect s ncluded by means of a smle obstructon model. In addton, smlfed methods for calculatng the average concentraton n a artcle as a functon of tme wthout solvng the artal dfferental equaton system are resented. The alcablty of these models n descrbng swellng and shrnkng rates of on-exchange resn artcles s dscussed. Materals and methods A sulfonated, gel-tye oly(styrene-co-dvnylbenze) resn C16G (Fnex Oy, Fnland) was used n the H + form. The resn has a cross-lnk densty of 8 wt-%, and an on-exchange caacty of 5.2 equv/g. Deonzed water and reagent grade acetc acd were used as the lqud comonents. The swellng and shrnkng knetcs of the resns n resonse to changes n the comoston of the surroundng lqud were montored n a secally constructed flow-through cell. The volume of the cell was aroxmately 0.1 cm 3, and the volumetrc flow rate of the solvent mxture was tycally 6.0 cm 3 mn -1. All exerments were erformed at room temerature. One or more resn beads was recondtoned n a solvent mxture and laced n the cell. A ste change was made n the solvent comoston, and the resultng swellng or shrnkng rocess was recorded wth a dgtal camera through an otcal mcroscoe. The samlng rate was between 1 and 30 seconds, deendng on the rate of the volume change of the artcle. The artcle szes were determned off-lne by usng mage analyss. The shrnkng knetcs measurements were started from water-swollen resn. After the new equlbrum state was reached, the swellng knetcs exerments were erformed by dslacng the aqueous acetc acd soluton n the cell wth water. Model develoment Coordnate system Polymer mass coordnates were chosen as the frame of reference n the calculatons [5, 7]. In ths coordnate system, the mass of the dry olymer ncreases lnearly along the

3 coordnate axs. If the locaton of a calculaton element s defned wth resect to the mass coordnate system, t remans fxed although the volume of the artcle, and that of the calculaton element, may change. Mathematcally, the transformaton between the two coordnate systems s defned through the dfferental oerators as shown n Eq. (1). W A( W ) ρm = (1) R θ W ( ) The mass coordnate system has also another roerty that s of ractcal mortance: when fnte-dfference methods wth equdstant grd sacng are used for satal dscretzaton of the resultng artal dfferental equaton system, the grd onts n the mass-coordnate system are automatcally concentrated n the vcnty of the artcle surface. It s ths regon where the steeest concentraton gradents are to be exected, and a denser grd mroves comutatonal accuracy and stablty. nce a smaller number of grd onts s requred, the numercal soluton s also faster. Fckan dffuson model In the mass coordnate system, the dfferental materal balance descrbng the Fckan dffuson becomes as shown n Eqs. (2) and (3), where θ s the local swellng rato and ρ m s the densty of the olymer. The numercal value of the dffuson coeffcent s not affected by the coordnate transformaton. It should be noted, however, that the assumton of neglgble molar average velocty of the mxture wth resect to the coordnate system has not been removed. When ractcal s referred to rgorous, Eqs. (2) and (3) can be used as an aroxmaton even for concentrated solutons and multcomonent systems. 2 2 C ρm D A C C = + νr t θ W θ W θ t (2) 2 2 D A C = Vm, ρm + νr θ t W θ W (3) Maxwell tefan aroach Use of a thermodynamc correcton factor n Fck s law of dffuson rovdes a better descrton of dffuson rates n non-deal systems, but does not remove the assumton of neglgble molar average velocty of the mxture. An alternatve and more general treatment of multcomonent mass transfer emloys the generalzed Maxwell tefan equatons. When the chemcal otental s the only drvng force, the force balance n the mass coordnate system may be wrtten as shown n Eq. (4). nce molar quanttes are not well defned for cross-lnked olymers, the frcton forces are here assumed to be roortonal to volume fractons. It should be noted that the seces veloctes are evaluated relatve to the olymer mass. The bootstra relaton s obtaned from the fact that the velocty of the olymer W wth resect to tself s zero,.e. u N+1 = 0. The ndex N+1 n the summatons refers to the olymer.

4 µ θ RT ( u u ) (4) 2 N + 1 g W W = φ 2 2 W A ρm = 1 D The materal balance for a volume element durng multcomonent dffuson controlled volume changes may be wrtten as n Eqs. (5) and (6). The molar fluxes are obtaned from a set of force balance equatons by usng the usual methods of lnear algebra. ( NA) C ρ m C = + νr t θ W θ t (5) N ( NA) = V m, ρm + νr t = 1 W (6) Estmates for the lqud lqud Maxwell tefan dffusvtes were calculated wth the nterolaton formula shown n Eq. (7) [8]. 1 φ + φ 1+ φ φ ( ) ( D ) D = D (7) Lnear drvng force aroxmaton In the sold hase lnear drvng force aroxmaton (LDF), the dffuson drvng force s aroxmated by the dearture of the average concentraton n the artcle from hase equlbrum, rather than by evaluatng the local concentraton gradent at the artcle surface. In the case of volume changes of the artcles, the temoral dervatve of the average sold hase concentraton becomes as shown n Eq. (8). An over-bar mles a quantty averaged over the artcle volume, and a hat denotes a quantty evaluated at the artcle surface n equlbrum wth the lqud hase. ( C C ) C 60 D ˆ C t d = 2 + νr θ t ( ˆ ) ρ V 10πd D C C θr νv (9) t W m = m, + m, (8) LDF aroxmaton n the Maxwell tefan formalsm The lnear drvng force model may also be nterreted such that mass transfer n the artcle occurs across a hyothetcal flm that comrses the relevant roertes of the sold hase. When the fnte dfference form of the Maxwell tefan model (.e., Eq. (4) wth µ W µ W ) s aled to dffuson across such a flm, one obtans Eq. (10). A tlde denotes a mean value n the flm. Eqs. (10) (12) together wll be here referred to as LDFM model. 1 ˆ d N + N µ µ φ Vm, d Vm, = N RT 10 φd φ 10 D φ g N ( ) ( ) = 1 = 1 (10)

5 C ρ t W ρ t W Na νr m = + θ C θ t N m = NaVm, + θr νvm, = 1 (11) (12) Effect of extent of swellng on the dffuson coeffcents The extent of swellng of an on-exchange resn affects the mesh wdth of the free sace as well as the tortuosty of the olymer network. Extensve exermental nvestgaton of sorton knetcs s beyond the scoe of the resent work, and a smle correlaton s desred. For ths urose the obstructon model shown n Eq. (13) was chosen. The model s based on that roosed by Macke and Meares [9] for descrbng dffuson of electrolytes n on-exchange membranes. ( 1 ) ( ) 2 1,ref D = D φ + φ (13) nce the orgnal model does not take account of sze or shae of the dffusng seces, t only ales to homogeneous materals wth a large mesh wdth relatve to the hydrodynamc dameter of the dffusng seces. The reference state n the orgnal model s free lqud. In ths work, however, the reference state was chosen as a hyothetcal, nfntely dlute soluton wth resect to the olymer, and the dffuson coeffcents n the reference state, D, are regarded as adustable arameters. The lattce model used by Macke and Meares n the dervaton of the model lends tself readly to the Maxwell tefan aroach as well. The correlaton n Eq. (13) s therefore used n both the Fckan dffuson model and the Maxwell tefan equatons. In the latter case, the Fckan dffuson coeffcents are relaced wth the Maxwell tefan dffusvtes D and D.,ref Boundary condtons for artcle dffuson models The lqud flm mass transfer resstance was neglected n the calculatons. The boundary condton at the artcle surface was therefore obtaned from sorton sotherms of the lqud comonents (see ref. [10] for sorton and swellng data and correlaton). A condton of zero flux through the center of the artcle yelds boundary condtons C W = 0 and W = 0,ref Results and dscusson The alcablty of the dffuson models resented above was tested wth data obtaned from non-reactve exerments wth water acetc acd mxtures. The shrnkng and swellng of C16G(H + ) as the external hase s changed from water to a 50 mol-% aqueous acetc acd mxture and vce versa are dslayed n Fgure 1. The swellng rato s lotted aganst a modfed tme varable n order to enable comarson of the results wth artcles of dfferent sze. When focusng frst on the shrnkng data, t s observed that the swellng rato decreases radly, asses through a mnmum, and ncreases slowly towards the new

6 equlbrum level. The exstence of the mnmum s due to the relatvely large dfference n the dffusvtes of water and acetc acd: the volumetrc flux of water out of the resn s larger than that of the acd nto the resn. At the mnmum, the volumetrc fluxes of the two seces have become equal. wellng dynamcs of the C16G when movng from 50 mol-% water HOAc mxture to ure water shows ooste behavor there exsts a maxmum n the extent of swellng. The data n Fgure 1 also shows that the fnal state s attaned more slowly durng shrnkng than swellng. Ths henomenon was already observed n the early works wth onexchange resns and other cross-lnked olymers [11,. 293]. nce shrnkng starts from the surface of the artcle, the solvents have to dffuse through a dense regon, the thckness of whch only ncreases wth tme. The calculated shrnkng and swellng knetc curves are lotted n Fgure 1 and the dffuson coeffcent values at the reference state used n the calculatons are gven n Table 1. As seen n Fgure 1, all models are caable of reroducng the basc shae of the curves but, as exected, the artcle dffuson models erform better than the lnear drvng force aroxmatons. The Maxwell tefan aroach yelded a hgher ndex of determnaton (R 2 = 98.9) than the Fckan model (R 2 = 94.0). As usual, the LDF and LDFM models underestmate the dffuson rates at the early stages of the rocess, but aroach the wellng rato t 1/2 d 1 x 10 3, s 1/2 m 1 Fgure 1. wellng and shrnkng knetcs of C16G(H + ) at room temerature. Intal and fnal states: 50 mol-% HOAc water mxture and ure water. Intal artcle sze: ( ) 984 m; ( ) 903 m. old lnes are calculated wth artcle dffuson models usng Fck s law ( ) and the generalzed Maxwell tefan equatons ( ). Dashed lnes are calculated wth the aroxmate LDF ( ) and LDFM ( ) models.

7 ,ref Table 1. Fckan ( D ) and Maxwell tefan ( D ) dffuson coeffcents at the reference state of,ref water and acetc acd n C16G(H + ). Estmated from swellng and shrnkng data n Fgure 1. Dffusant D, m 2 s 1 D,ref, m 2 s 1,ref, Water Acetc acd , equlbrum state sooner than the artcle dffuson models. The LDF model redcts larger changes n the extent of swellng than the Fckan model n both shrnkng and swellng exerments. In fact, a volume-average swellng rato of 3.5, redcted by the LDF model, s hardly hyscally realstc for the C16G resn under any condtons. The LDFM model (.e., Maxwell tefan formalsm aled to the LDF aroxmaton) redcts consderably lower swellng ratos than the LDF model n the swellng exerment, but only slghtly lower swellng ratos at the mnmum n the shrnkng exerment. Ths may aear surrsng at frst sght, esecally recallng that the mnma and maxma n the volume change data stem from dfferences n the dffuson rates of the lquds, and consderng that the rato of the D values of water and HOAc s more than three tmes larger than that of the,ref,,ref corresondng D values (see Table 1). The exlanaton s that the Maxwell tefan model accounts for the fnte exansblty of the olymer network through the chemcal otental of the dffusants. Conclusons Mathematcal models for couled dffuson and reacton n swellng and shrnkng olymer artcles were wrtten wthout emloyng the equatons of moton. Polymer mass coordnates were used as the frame of reference. Two artcle dffuson models based on the Fck s law of dffuson and the generalzed Maxwell tefan aroach, as well as two-ont aroxmatons of these, were comared. It was shown that dffuson nduced swellng and shrnkng of on-exchange resn artcles s best descrbed wth a model based on the Maxwell tefan aroach because t takes the fnte exansblty of the olymer network nto account. The LDF model redcted unrealstcally large swellng of the resn. Nomenclature A cross-sectonal area, m 2 a surface area to volume rato, m 1 C resn hase concentraton, mol m 3 D Fckan dffuson coeffcent, m 2 s 1 D Maxwell tefan dffuson coeffcent, m 2 s 1

8 d artcle dameter, m N flux, mol m 2 s 1 R radal coordnate, m R g gas constant, J mol 1 K 1 r reacton rate, mol m 3 s 1 T temerature, K t tme, s W u seces velocty, kg s 1 V m molar volume, m 3 mol 1 W mass coordnate, kg mass of a sngle resn artcle, kg W φ volume fracton, µ chemcal otental, J mol 1 ν stochometrc coeffcent, ρ m densty of the olymer, kg m 3 θ swellng rato, References 1. Tanaka, T., Hocker, L., Benedek, G. B., ectrum of Lght cattered from a Vscoelastc Gel, J. Chem. Phys., 59(1973), Tanaka, T., Fllmore, D. J., Knetcs of wellng of Gels, J. Chem. Phys., 70(1979), Carbonell, R. G., art, G. C., Couled Deformaton and Mass-Transort Processes n old Polymers, Ind. Eng. Chem. Res., 29(1990), El Aff, A., Grmela, M., Non-Fckan Mass Transort n Polymers, J. Rheol., 46(2002), Bllovts, G. F., Durnng, C. J., Polymer Materal Coordnates for Mutual Dffuson n Polymer Penetrant ystems, Chem. Eng. Commun. 82(1989), Bsschos, M. A. T., Luyben, K. C. A. M., van der Welen, L. A. M., Generalzed Maxwell-tefan Aroach for wellng Knetcs of Dextran Gels, Ind. Eng. Chem. Res., 37(1998), ano, T., Thonen, J., Paatero, E., Mass Transfer Couled Wth Volume Changes n Ion-Exchange Resn Beads, IEX 2004 conference, , Cambrdge, England 8. Xu, X., Cu, Z. F., Modelng of the Co-Transort of Cryorotectve Agents n Porous Medum as a Model Tssue, Botechnol. Progr., 19(2003), Macke, J.., Meares, P., The Dffuson of Electrolytes n a Caton-Exchange Resn Membrane. I. Theoretcal., Proc. Roy. oc. (London). A232(1955), ano, T., Ion-Exchange Resns as tatonary Phase n Reactve Chromatograhy, Dss. Laeenranta Unversty of Technology, Acta Unverstats Laeenrantaenss 218, Laeenranta, Helfferch, F., Ion-exchange, Dover Publcatons, New York, 1995