CALCULATION OF DIFFUSION COEFFICIENT OF ION IN MULTICOMPONENT SOLUTION FOR ION MOVEMENT IN CONCRETE

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1 CALCULATION OF DIFFUSION COEFFICIENT OF ION IN MULTICOMPONENT SOLUTION FOR ION MOVEMENT IN CONCRETE Worapatt RITTHICHAUY *1, Takafum SUGIYAMA * and Yukkazu TSUJI *3 ABSTRACT: A method for calculatng the mutual dffuson coeffcents of an on nfluenced by all other coexstng o n the multcomponent soluton for on movement n concrete s presented n ths paper. The method of calculaton s based on the generalzed form of Fck s frst law suggested by Oager, whch s composed of the Oager phenomenologcal coeffcent and the thermodynamc force between o that exerted by the gradent of electrochemcal potental n the soluton. As a result of verfcaton by comparson wth the expermental results by Sato H. et al., t s shown that by usng the presented method of calculaton, the calculated leachng amount of some o exhbted close to the expermental results. KEYWORDS: on mgraton, multcomponent soluton, mutual dffuson coeffcent, generalzed Fck s frst law, Oager phenomenologcal coeffcent, thermodynamc force, electrochemcal potental 1. INTRODUCTION Dffuson s the most mportant process of the on traport n a porous meda, for example, n the fractured rocks or structural concrete. It can be codered to be the most mportant factor causng the deteroraton of the concrete structure. In some envronmental condto for concrete structure such as n a salne envronment or nuclear waste-dsposal contanment barrers, the durablty of concrete structure must be sgnfcantly concerned. The ngress of aggressve o such as Cl - and SO 4 -, the leachng of Ca + or some metallc o from portlandte or calcum slcate hydrated gel n concrete pore structure are the man factors that affect the durablty of concrete structures. Therefore, t s necessary to develop a calculaton model for the dffuson traport, whch can be used for accurate predcton of such phenomenon n the global state of concrete structure. For the dffuson of o n concrete, the multcomponent system of the o n concrete pore soluton has to be manly concerned. The concentrato of the co-exstng o such as Ca +, Na +, K +, SO 4 -, etc. have the sgnfcant nfluence on the moblty of each on [1]. The researches of on traport n concrete concernng to the co-exstng o n the multcomponent soluton have been carred out by Otsuk N. et al. [], n whch, the flux of an on s calculated by the Nert-Plank Equaton, electro-neutralty cotrant and Debye-Hückel theory. The purpose of ths research s to apply the theores of the on traport n a multcomponent soluton for calculatng the matrx of dffuson coeffcents of every on exstng n the pore soluton n concrete. Ths dffuson coeffcent s desgnated as D or the mutual dffuson coeffcent of the th speces nfluenced by the concentraton of the th speces. The calculaton of ths model s based on a generalzed form of Fck s frst law that was suggested by Oager [3]. Ths generalzed form composed of the Oager phenomenologcal coeffcent [4] and the thermodynamc force between o exerted by the gradent of electrochemcal potental [1]. Coequently, from ths D, the flux and/or the concentraton profle of each on can be calculated from the generalzed Fck s law. The correctness of ths calculaton model for D was verfed by the comparson wth the expermental results of the leachng of some o (.e. Na + and K + ) from the mortar exposed to water performed by Sato H. et al. [5].. ELECTROLYTE DIFFUSION PROCESS It s well known that n a electrolyte dffuson process, the movement of an aqueous speces wll occur by the drvng forces that created from the concentraton gradent of that speces tself and by those *1 Department of Cvl Engneerng, Gunma Unversty, Graduate student, Member of JCI * Department of Cvl Engneerng, Gunma Unversty, Assocate professor, Ph.D., Member of JCI *3 Department of Cvl Engneerng, Gunma Unversty, Professor, Dr.E., Member of JCI

2 of the other speces [1,3]. Moreover, another Faster Cl - drvng force s the gradent of electrcal potental Cl - created by the dfference between moblty of caton and anon. The way n whch the electrcal potental or the electrostatc force has an nfluence on the electrolyte dffuson s llustrated Mutual schematcally n Fg.1. Assumng that n a bnary traport electrolyte soluton composed of NaCl, the faster Cl - and the slower Na + are cotraned by the electrostatc force, to move at the same rate. In Na + addton to ths, by the electro-neutralty cotrant Slower Na + these two o must mantan the same dffusve Fg.1 Electrolyte dffuson for system of Na + and Cl - flux through out the traport n the soluton. Because of these electrostatc requrements, the flux of NaCl s characterzed by a sngle dffuson coeffcent,.e. mutual dffuson coeffcent, whch s an average of the dffuson coeffcent of Na + and Cl -. Ths mutual dffuson coeffcent can be magnatvely codered as a leash tyng the Na + and Cl - together. The faster Cl - wll be gong to accelerate the slower Na +, whle at the same tme, the slower Na + wll decelerate the Cl -, keepng t from runnng away. All of ths acceleratng and deceleratng effect produces a mutual dffuson coeffcent. In the smlar acton for a multcomponent soluton system, tead of the sngle electrostatc leash tyng one caton and one anon, magne that there are a lot of leashes connected together by all charged speces exstng n the soluton system. If one on, e.g. OH -, whch s much more moble than the others, t can accelerate the movement of o wth opposte charge, or decelerate the dffuson of o wth smlar charge. In some cases, t can even cause the other o to move agat ther concentraton gradent. Imagnary electrostatc leash 3. DEVELOPMENT OF THE DIFFUSION COEFFICIENT CALCULATION MODEL The dffuson coeffcent of a speces n a multcomponent soluton can be calculated by coderng the drvng dffusve flux n the soluton. The forces drvng the dffuson of an aqueous speces are bascally orgnated from the thermodynamc force that can be codered as the gradent of electrochemcal potental (µ) [1]. The electrochemcal potental, whch composed of the chemcal part (µ chem ) and the electrcal potental (φ) part, can be shown n the followng expresson. µ = µ (chem.) + z Fφ (1) As mentoned before, accordng to Mller [3] the thermodynamc force on on (X ) s the gradent of electrochemcal potental gven by Eq.. X (chem) φ = = + z F () At ths pont a lnear relaton, whch relates the flux of the on to force exerted on the on, s assumed and expressed n Eq. 3. J n s (chem) φ = l X = l ( z F ) (3) =1 =1 In Eq. 3, l stands for the Oager phenomenologcal traport coeffcent [1,3,4] (hereafter Oager coeffcent) the sgnfcance of ths coeffcent on the traport n multcomponent soluton wll be dscussed later. The electrcal potental gradent ( φ ), whch s not readly measurable, can be elmnated from Eq. 3 by the electro-neutralty cotrant n the soluton ( n s z J = ). = 1 From ths step, the µ wll be used for representng the µ (chem) as for abbrevaton. By substtuton of

3 the flux equaton n Eq. 3 nto the electro-neutralty cotrant, one can derve to Eq. 4. l zklkl φ k = 1 l= 1 = F z z l k= 1 l= 1 By substtuton of Eq. 4 nto the flux equaton of Eq. 3, the flux equaton of th speces wll become as followng equaton. l n z = = = s kl kl k 1 l 1 x J l + z (5) n s = 1 zkzllkl k= 1 l= 1 At ths step of calculaton, determnng the Oager coeffcent (l ), whch s the functon of on concentrato n the soluton especally n relatvely dlute soluton [4], s necessary. However, n the system of porous meda, the off-dagonal terms of ths Oager coeffcent (l, ) can be approxmated to [3]. In the other words, t can be neglgble compared to ts on-dagonal term of Oager coeffcent (l, = ) wthout sgnfcant error [3, 4]. In addton, for a relatvely dlute soluton, the on-dagonal Oager coeffcent can be computed from the followng equaton. D C l = (6) RT where D stands for the tracer dffuson coeffcent n dlute soluton of th speces, R refers to the gas cotant, equal to JK -1 mol -1, T s the absolute temperature (K). Coequently, the gradent of the electrochemcal potental s converted to the gradent of concentraton by the followng equaton. = k The defnton of chemcal potental (µ ) can be extended to the followng expresson [3]. l kl (4) (7) µ = µ + RT ln C + RT ln γ (8) where γ refers to the mean actvty coeffcent of the th speces and µ refers to the chemcal potental n the standard state of the soluton, whch s a cotant number. Therefore, the followng equaton can be derved. 1 ln γ = RT + (9) C C ln C Coequently, the generalzed form of the Fck s frst law suggested by Oager [3,4] s shown n the followng expresson. J = D (1) = 1 where D s meant for the mutual dffuson coeffcent of the th speces n free lqud phase nfluenced by the concentraton gradent of the th speces ( C ). By substtuton all of expresso from Eq. 6 to Eq. 9 nto Eq. 5, and comparng wth the smlar flux term n Eq. 1, the expresson of mutual dffuson coeffcent can be derved as shown n Eq. 11 [3]. γ ln γ ln zd C D = δ D (1 + ) z D (1 + ) (11) n s lnc lnc zkdkck k = 1 where δ s desgnated for the Kronecker delta functon whch s equal to 1 f =, but equal to f, γ s the mean actvty coeffcent of the th speces and ts dervatve respected to the concentraton

4 can be calculated by the Debye-Hückel theory [6] as shown n followng equaton, ln γ.59 Z =.3 C ln C I( α I) where α s the on sze parameter (nm) of the th speces and I s the onc strength (mol/l) of the soluton. As mentoned before, the defnton of D s based on the assumpton that the mutual movement of separated o s caused by not only the electro-neutralty cotrant, but also the electrochemcal potental gradent occurred by the dfference between each on moblty. Therefore, the movement of o can be codered smlar to that of uncharged speces. In general, the matrx of mutual dffuson coeffcents s not symmetrc,.e. D D. The dagonal terms (D ) are generally large compared to the off-dagonal terms (D, ). By usng the ntal value quoted n the next secton as the nput parameters, the matrx of D n the unt m /s at partcular concentrato n a multcomponent soluton composed of 7 o s shown n Table 1. From the value shown n Table 1, the postve value represents the pullng effect between mutual movements of o, whle the negatve value represents the pushng effect of that. In addton, we can see that the [D], = Table 1. Example of calculated mutual dffuson coeffcents at partcular concentrato (D, 1-9 m /s) dagonal term s hgher than the off-dagonal terms for every on. Therefore, at each partcular concentrato and tme the matrx of dffuson coeffcents (D ) can be calculated. In the smulaton of on traport n concrete materals by applyng these dffuson coeffcents and the equaton of contnuty, the general form of Fck s second law can be wrtten as followng equaton. = Ca + Mg + Na + K + SO 4 - D,I = 1 t x where D,I desgnates the ntrc mutual dffuson coeffcent n the pore soluton of concrete. For a general porous meda, t s easer to determne the term of dffuson coeffcent n the terms of the average flux per unt area of the medum (rather than the lqud) leadng the defnton of ntrc dffuson coeffcent. The ntrc mutual dffuson coeffcent of concrete porosty ε and tortuosty τ can be formulated by the followng expresson [7]. C (1) Cl - OH - Ca Mg +.76 Na K SO Cl -.3. OH (13) ε D,I = D (14) τ 4. VERIFICATION OF THE CALCULATION MODEL In the present research, the correctness of the calculaton model for the D of the o n multcomponent soluton for predcton the traport of o n concrete s verfed by comparng wth the expermental results performed by Sato H. et al. [5]. In ths experment, the mortar specmen of 65% W/C wth 5 cm dameter and 3 cm thckness dsk Table. The parameters for calculaton of D matrx C, Concentraton n α, Ion sze D o, Tracer dffuson Ion pore soluton (mol/l) parameter (nm) coeffcent (m /s) Ca x 1-1 Mg x 1-1 Na +.3 (55 mg) x 1-9 K (86 mg) x SO x 1-9 Cl x 1-9 OH x 1-9 The number s referred from C. L. Page et al. [8]. The value calculated by leachng amount whch shown n parenthess. [5]

5 was prepared for predctng the deteroraton due to dssoluton of extremely amount of cement hydrate. In some parts of ths expermental work, the amount of Na + and K + that leached from the pore soluton of mortar exposed to water was nvestgated wth the exposure tme. As shown n Table, the ntal concentrato of Na + and K + n the pore soluton are determned from the total accumulatve amount of leachng o from the experment [5], though those of Ca + and SO - 4 are quoted from C.L.Page et al. [8], and that of OH - s. determned by the electro-neutralty cotrant. In addton, the other necessary parameters such as on sze parameter (α ) and the tracer dffuson coeffcent. (D ) for the calculaton of D are also shown n ths Start of exposure table. Therefore, the matrx of dffuson coeffcent (D ) of every on can be establshed accordng to the Dstance from exposure 3 mm concentraton at each tme step and poston. In the Accumulatve leachng amount (mg) Porosty of mortar End of exposure (7 days) Fg. Porosty change of mortar specmen due to dssolved Ca(OH) K + (Exp.) Na + (Exp.) K + (Cal.) Na + (Cal.) Exposure tme (days) calculaton of the concentraton profle or the leachng amount of o, the dssoluton of Ca + from pore structure of mortar dd not taken nto account for present study. However, from the expermental result, the dssoluton of Ca(OH) can approxmately account for the ncreased porosty by 1% n perod of years. The change of porosty for ths calculaton, whch based on the expermental result, s shown n Fg.. The porosty of exposure surface ncreased lnearly from. to. after 7 days of exposure, wth a cotant at 3 cm depth. The tortuosty (τ) of mortar for the calculaton of present research was assumed to be cotant and equal to 3. [9]. In ths way t s assumed that the effect of dssolvng Ca(OH) s of great sgnfcance wth respect to the change of the physcal characterstcs but lttle to the electrochemcal characterstcs for the partcular test. The comparso of the calculated results wth the expermental ones are shown n Fg.3. The expermental results n ths fgure are the accumulatve leachng amount of Na + and K + collected from the exposure soluton (water). At the 1 days of exposure, the leached on could be collected to the amount of 39 mg and 6 mg for Na + and K +, respectvely. Ths number would gradually ncrease wth the exposure tme and could be estmated to the possble maxmum value of 55 mg and 86 mg, respectvely [5]. From the same fgure, the calculated results exhbted preferable conformty wth the expermental results for Na + and wth slghtly hgher estmaton for K + at 1 days of exposure. Nevertheless, t 86 mg 55 mg Fg.3 Calculated and expermental results for leachng amount of o from mortar exposed to water Accumulatve leachng amount (x1-3 mol) Na + K Exposure tme (days) Fg.4 Calculated leachng amount n mol unt

6 can be concluded that the model for calculaton of D can provde the approprate calculaton of the on traport n cement-based materal. The rate of dffuson can be seen from Fg.4, whch shows the calculated accumulatve leachng amount of Na + and K + n mol unt. From ths fgure, we can see that n the begnnng, the dffuson rate of K + s hgher than that of Na + because of the hgher tracer dffuson coeffcent of K + that contrbutes to hgher mutual dffuson coeffcent of t. However, after the exposure tme beng passed, as a result of rapd decreasng n the concentraton of K +, ts mutual dffuson coeffcents also decrease. Therefore, n the later part of exposure tme, the rate of dffuson of Na + s slghtly hgher than that of K +. In other words, even though D of Na + s lower than that of K +, Na + wth hgher concentraton than that of K + shows hgher rate of dffuson. Coequently, we can state that not only the tracer dffuson coeffcent, but also the concentraton of the on tself has an nfluence on the rate of dffuson of these o. 5. CONCLUSIONS From the model for calculatng the mutual dffuson coeffcents of o n the multcomponent soluton, the on movement n concrete can be smulated. By comparson the calculated results wth some expermental results we can conclude that; 1. A new calculaton model s proposed for smulatng the on traport n cement-based materals. The model s establshed by the generalzed form of Fck s frst law, whch composed of the Oager phenomenologcal coeffcent and the thermodynamc force between o exerted by the gradent of electrochemcal potental.. By verfcaton of the model wth the expermental results, t can predct well the accumulatve leachng amount of Na + and K + from the mortar specmen exposed to water. However, the model of chemcal equlbrum such as the dssoluton of Ca + or the other metallc o that occurs de the pore soluton of a cement-based materal has also to be ncluded to the model for further study. REFERENCES 1. Andrew, R. Felmy and John, H. Weare, Calculaton of Multcomponent Ionc Dffuson from Zero to Hgh Concentraton: I. The System Na-K-Ca-Mg-Cl-SO 4 -H O at 5, Geochmca et Cosmochmca Acta, Vol.55, 1991, pp N. OTSUKI, S. MIYAZATO, H. MINAGAWA and S. HIRAYAMA, Theoretcal Smulaton of Ion Mgraton n Concrete, Concrete Research and Technology, Vol.1, No., May 1999, pp P. C. Lchtner, C. I. Steefel and E. H. Oelkers, Reactve Traport n Porous Meda: Revew n Mneralogy, Vol.34, The Mneralogcal Socety of Amerca, 1996, pp E. L. Cussler, Multcomponent dffuson, Elsever Scentfc Publshng Company, 1976, pp H. SAITO, S. NAKANE and A. FUJIWARA, Effects of Electrcal Potental Gradents on Dssoluton and Deteroraton of Cement Hydrate by Electrcal Acceleraton Test Method, Concrete Research and Technology, Vol.4, No., July 1993, pp L. Tang, Concentraton Dependence of Dffuson and Mgraton of Chlorde Io Part. Expermental Evaluato, Cement and Concrete Research, Vol.9, 1999, pp A. Atkon and A. K. Nckerson, The Dffuson of Io Through Water-Saturated Cement, Journal of Materals Scence, 1984, pp C. L. Page and Ø. Vennesland, Pore Soluton Composton and Chlorde Bndng Capacty of Slca-fume Cement Pastes, Materals and Structures Research and Testng, RILEM, Pars, Vol.16, No.91, 1983, pp R. T. Tenchev, L. Y. L and J. A. Purkss, Fnte Element Analyss of Coupled Heat and Mosture Trafer n Concrete Subected to Fre, Report Paper, School of Engneerng and Appled Scence, Aston Unversty, Brmngham, UK,.