Computers&Concrete, An Internatonal Journal, Vo. 4, No. 4, August 27 MECHANISMS OF SULFATE IONIC DIFFUSION IN POROUS CEMENT BASED COMPOSITES P. Gospodnov, M. Mronova 2, R. Kaandjev * Insttute of Mechancs, BAS png@mbm.bas.bg, robert@nfo.mbm.bas.bg 2 Central Laboratory of Physco-Chemcal Mechancs, mronova@clphchm.bas.bg Abstract The paper consders a theoretcal model for the study of the process of transfer of sulfate ons n saturated porous meda mneral compostes. In ts turn, the model treats dffuson of sulfate ons nto cement based compostes, accountng for smultaneous effects such as fllng of mcrocapllares wth ons and chemcal products and lqud push out of them. The proposed numercal algorthm enables one to account for those smultaneous effects, as well as to model the dffusve behavor of separate sectons of the consdered volume, such as nert fllers. The cases studed llustrate the capabltes of the proposed model and those of the algorthm developed to study dffuson, consderng the specmen complex confguraton. Computatons show that the theoretcal assumptons enable one to qualtatvely estmate the expermental evdence and the capabltes of the studed composte. The results found can be used to both assess the sulfate corroson n saturated systems and predct and estmate damage of structures bult of cement-based mneral compostes. Keywords Dffuson, Sulfate Attack, Cement Paste, Corroson, Mathematcal Model, Numercal Analyss. Introducton Mneral cement-based compostes such as concrete are wdely appled n structure buldng. They are often exposed to the nfluence of aggressve lqud meda. The sulfate corroson of real concrete structures s a popular phenomenon. To analye ts destructve effects, on needs to study the behavor of the most mportant concrete component the cement paste (cement stone), mmersed n a lqud aggressve sulfate medum. The mpact of the aggressve medum s realed by a penetraton of sulfate ons nto the lqud that flls the pores of the cement paste. Snce ons are charged partcles, an electrc potental occurs when they enter the pore-fllng lqud. Possbltes of modelng the transfer of ons n unsaturated systems are dscussed n Marchand et al.22, and n saturated systems n Marchand et al. 2. The on transfer, accountng for the acton of electrostatc forces, s followed by employng the Nernst-Plank model, (Stansh et al. 24, Samson et al. 999c), whle n cases of strong onc soluton and especally under the strong nteracton of several on speces, an extended model that regards chemcal actvty s used n Samson and Marchand 999. Consderng saturated system, the mathematcal model needs to be completed wth Posson s equaton n order to fnd the electrc potental, whle relatons for the calculaton of the coeffcents of ons chemcal actvty n electrolytc solutons are part of the extended model (Samson et al. 999a). Samson et al. 999b proposes a homogenaton technque whch allows for averagng the results for the concentraton of the porefllng soluton over the entre materal volume. Our studes consder the acton of an aggressve sulfate medum n a saturated system, assumng that the effect of other on speces can be dsregarded. The pore space of the nvestgated cement paste prsmatc specmen has been prevously flled wth water, and the specmen has been mmersed and kept n water for 28 days after castng. It s also assumed that on moton can only occur n the lqud phase. * Correspondng author
2. Mathematcal model Ion transfer n the lqud phase that flls the pores of a saturated system s generally due to two mechansms dffuson transfer and mgraton owng to electrostatc forces. On the other hand, lqud moton wthn the pores s due to fllng them wth products of the developed heterogeneous chemcal reacton. That reacton takes place on the pore wall as a result of the nteracton of ons wth the cement stone. The pores are formally treated as capllares (mcro-capllares), shaped as straght crcular cylnders and wth symmetry axes parallel to the coordnate axes. The present study accounts for the effect of the sulfate ons, only, whle ther nteracton wth other on speces s dsregarded. Consder cement stone capllares n partcular and the materal balance of an elementary volume of the area under control. Then, the followng equaton of on transport can be wrtten, accountng for the heterogeneous chemcal reactons that develop on the capllary walls: c = dv ( J T ) kc () t or n the one-dmensonal case c = ( J T ) kc (2) t x where J J T = J dff + J U (3) As stated, snce ons are electrcally charged partcles, an electrc potental occurs when they enter the soluton that flls the capllares. Thus, the flux of on transfer J T has two components the frst one J dff s due to the dffuson force whle the second one J U to the drvng force of the electrc potental U (Stansh et al. 24 ).. Quantty c ( x, t) n Eqs. ()-(2) s the current concentraton value, q ( x, t) s the quantty of chemcally reacted ons at a pont wth coordnates x and at a moment t, and k s the coeffcent of chemcal reacton rate. The frst term n the rght-hand sde (RHS) of Eq. (3) can be expressed by Fck s frst law: c J dff = k dff (4) x whle the second one has the followng form accordng to the smplfed model of Nernst-Plank, (Samson et al. 999c): VF kdff U J U = c (5) RT x Multpler k dff n Eq. (4) s the coeffcent of dffuson of sulfate ons wthn the whole volume, consstng of a cement matrx and capllares flled wth lqud. It reflects the materal porosty, structure and capllary shape. V s the valence of sulfate on, F s Faraday s constant, U s the dffuson potental, R s the unversal gas constant, T s the temperature. The mathematcal model should be completed wth an equaton for determnng the dffuson potental U. It has the form (Samson et al. 999c):. 2 ρ U + = (6) ε where ρ s densty of the electrc charge and ε s the medum delectrc constant. The consderaton of the on flux due to the drvng force of the dffuson potental s sgnfcant n two cases, manly when an external electrc potental s appled or a smultaneous transfer of two and more on speces s n operaton. When consderng the transfer of sulfate ons, only, under sothermal condtons, the account for the effect of Nernst-Plank yelds model complcatons. On the other hand, the model proposed by Eqs. ()-(6) does not account for some characterstc features of the process,.e. change of the capllary volume as a result of the precptaton of chemcal reacton products on the capllary walls, as well as the resultng push out of the capllary-fllng lqud. Note also that lqud moton exports part of the ons out of the capllares and n a drecton opposte to that of flux J.
In prevous papers (Gospodnov et al. 999), (Mronova et al.22) we propose a model of dffuson transfer consderng heterogeneous chemcal reacton and lqud push out of the capllares resultng from the capllary fllng. The model can be presented n the followng form:: c 2 = dv( Deff gradc) dv( V c) k( k q) c (7) t where the spatal operators dv ( ) and grad ( ) are of the type K K dv() () grad() j () (8) = x = x For one dmensonal case - K=. The last term n the RHS of Eq. (7) s a source term whch models the heterogeneous chemcal reacton between the sulfate ons n the soluton and the capllary walls. The term also reflects the change of the capllary reactng surface. Projectons V, =, K, of velocty V n Eq. (7) denote the lqud mean velocty wthn the cross secton of the capllary along axes x, =, K. The effectve dffuson coeffcent wthn the whole volume has the form (Gospodnov et al 999) : ( k ) 2 Deff = D q, (9) where D = k exp β c x,..., xr, t. c. () [ ( ( ) )] dff 5 Constant β s a parameter and c s a characterstc value of the concentraton the concentraton of the soluton whch the specmen s mmersed n, k denotes the coeffcent of capllary fllng and the quantty of chemcally reacted ons q s found from q t ( x x, t) = k c( x,..., x, τ) d,,..., R R τ () o For fxed values of the rest of the coordnates x m, m, the velocty component V s calculated by means of the followng ntegral along the capllary length, wthn boundares [, x ], (Mronova et al. 22). V η G = = η= U 2kk ( k q) c dη, U = L / 2 x, G = L / 2, =, R Snce the orgn of the coordnate system s the center of the specmen symmetry, the velocty s ero for x = and t s maxmal for x = / 2. The velocty drecton concdes wth the postve drecton L of axs Ox. V s found calculatng the ntegral Eq. (6). The lower ntegraton lmt s L / 2 x, whle the upper one - L / 2. Thus, velocty V ( x,..., xr, t) can be found at each pont of the area consdered. Velocty components Vm ( x,..., xr, t), m, m [,..., R] are smlarly found. The model s completed by specfyng a ero ntal dstrbuton of the concentraton, condtons at the specmen-soluton nterface and symmetry condtons, Mronova et al. 22. Note however that the model outlned above descrbes the dstrbuton of sulfate ons wthn the lqud phase that flls the capllary space of the cement paste. 3. Numercal soluton The equatons of transfer Eqs.(7) - (8), together wth the ntal and boundary condtons pose a nonsteady boundary value problem. It s completed by the ntegral relatons Eq. (2), gvng the quantty of chemcally reacted ons, as well as by the ntegral relatons Eq. (3), necessary for the calculaton of the velocty feld of the lqud pushed out of the cement stone capllares. (2)
(a) L 2 /2 y (b) +N*M +N L 3 /2 O y - + x L /2 x -N -N*M Fg. Coordnate system and dscretaton scheme The problem s solved usng an mplct dfference scheme and t s reduced to solvng a sparse system of lnear algebrac equatons wth a non-symmetrc band matrx. The algorthm used s descrbed n detal n Gospodnov 25. The dscretaton scheme s shown n Fg..a. Numbers of N, M and L cross sectons wth planes perpendcular to the coordnate axes x, y and, respectvely, are ntroduced along each axs, startng from the coordnate system orgn. Let x, y and are numbers of the dscretaton sectons (Fg. ) perpendcular to axes х, y and. Obvously, N, M, L. The number of a grd knot for fxed values of x, y and s determned by the followng relaton: ( ) * N + ( ) * N M x = + (3). x y * Thus, the unknown values of the quanttes sought at each knot of the dskreted area can be numbered usng one ndex, only. Besdes, the soluton of the ntal boundary value problem outlned above s reduced to the soluton of a dfference value problem for a lneared system of (N*M*L) algebrac equatons wth a dagonal and weakly flled matrx. The algorthm thus developed s completed by a numercal procedure for calculatng the ntegrals of type Eq. (3) along the three coordnate axes. The specfc features of cases of sub-areas wth conductvty, dstnctly dfferent from that of the matrx (e.g. nert fller), are dscussed n Gospodnov 25. The model thus desgned can be effectvely appled to dsclose the character of a number of complex phenomena observed durng the sulfate aggresson of cement paste. 4 Results and dscusson Havng set forth the mathematcal model of dffuson transfer of sulfate ons n cement stone, we proceed wth the dscusson of some expermental evdence and calculaton results. We have consdered n our prevous study (Mronova et al. 22), sulfate ons dffuson n prsmatc specmens, thus modelng the development of the sulfate attack on real structural elements, whch s a problem of essental practcal mportance. The specmens tested are long prsms wth cross secton.8 x -2 x.5 x -2 m. As found, they are dvded nto two parts a thn external corrupted layer and an ntact cement core, Fg. 2. y Fg.2. Prsmatc cement paste specmen, layered after a sulfate attack from 5 % sulfate soluton; tme of mmerson 9 months. The specmens tested are long prsms wth cross secton.8 x -2 x.5 x -2 m.
However, the study of the mechansm of formaton of the external corrupted layer as a result of the sulfate attack s a dffcult task. To model that process, we have studed the development of sulfate ons dffuson n thn cement paste specmens, mmersed n aqueous solutons of Na 2 SO 4. The specmens have been fabrcated form SRP cement, type 35, n complance wth the Bulgaran standard. The materal chemcal composton s specfed n Mronova et al. 22. Specmens have been soldfyng for 28 days n pure water after preparaton and pror to mmerson. Then, they have been mmersed and kept n water solutons of Na 2 SO 4 of dfferent concentraton and for dfferent perods of tme. Fg. 3 shows the sequence of cement stone degradaton n tme, dependng on the soluton sulfate concentraton. It summares the expermental fnds of our prevous studes (Mronova et al. 22, Kaandjev et al. 25). Immerson duraton 864 hours (2 months); lateral specmen vew a) concentraton of Na 2 SO 4 n the soluton % Immerson duraton 864 hours (2 months); lateral specmen vew b) concentraton of Na 2 SO 4 n the soluton 5% Immerson duraton 864 hours (2 months); lateral specmen vew c) concentraton of Na 2 SO 4 n the soluton % Fg. 3. Cement paste degradaton n tme, regardng dfferent soluton sulfate concentratons As seen n Fg. 3 and specfcally for the hgher concentratons of the sulfate soluton, partal destructon of the specmen takes place after a perod of 2 months of mmerson, and dffuson and ts accompanyng effects (fllng of the cement stone pores wth ons and chemcal products, mcrocrack formaton etc.) change the materal structure. Hence, a perod shorter than 2 months of mmerson can be taken as a reasonable tme lmt for the cement stone saturaton wth sulfate ons from the hgher sulfate solutons, where the materal dffusve behavor can be taken as smlar to that of the external corrupted layer of the bulk specmen shown n Fg. 2. Thus, the analyss of ons dffuson n thn specmens, based on the model proposed proves useful for analyng the on transfer n bulk specmens and n real structural elements. In what follows, we gve some calculaton results found usng the dffuson model desgned.
We consder sulfate ons dffuson nto thn cement paste specmens wth dmensons.5x -2.5x -2.2 =2 m, beng mmersed n 5% aqueous soluton of Na 2 SO 4. The calculaton are performed usng the followng values of the parameters, partcpatng n the model ()-() and found analyng the expermental evdence of Gospodnov et al. 999.: - coeffcent of on dffuson n the water soluton k dff =.34 x -3 m 2 /s - fttng parameter partcpatng n Eq. (): β =.297 m 3 /kg; - constant of the chemcal reacton rate: k =.35 x -7 s - - coeffcent of pore fllng: k =.5 m 3 /kg - content of the sulfate ons n the 5% sulfate soluton - 33.83 kg/m 3 Fgures 4, 5 and 6 present the numercal results of the study of D dffuson of sulfate ons wthn the specmen. Due to symmetry, ons penetraton from x= to the symmetry lne x=. x -2 m s followed. Concentraton profle, c Concentraton profle, c 3 3 6 4 5 c [kg/m 3 ] 2 2 3 c [kg/m 3 ] 2.25.75. x* -2.25.75. [m] x* -2 [m] Fg.4 Dstrbuton of the ons concentraton D case q [kg/m 3 ] 2.5.5 Chemcally reacted quantty, q 3 2 q [kg/m 3 ] 2 8 6 4 2 Chemcally reacted quantty, q 6 5 4.25 x* -2 [m].75..25.75. x* -2 [m] Fg.5 Dstrbuton of the chemcally reacted quantty of ons- D case D eff [m 2 /s] x -2 Effectve dffuson coeffcent.2.8.6.4 2 3 D eff [m 2 /s] 7 x -3 6 5 4 3 2 Effectve dffuson coeffcent 4 5.2 6.25.75. x* -2 [m].25.75. x* -2 [m]
Fg.6. Dstrbuton of the effectve dffuson coeffcent D case Notatons of the curves n Fg.4, 5 and 6 correspond to the followng perods of mmerson: - 2 hours, 2 36 hours, 3 6 hours, 4 26 hours (3 month), 5 432 hours (6 months), 6 648 hours (9 months). The calculaton results show that ons penetraton s comparatvely ntensve at the start of process, whle the current concentraton attans a maxmal value on the symmetry lne for a 3-months of mmerson and then t starts decreasng but wth weaker ntensty Fg. 4. Ths s due to the proceedng chemcal reacton, where part of the penetratng ons precptate on the capllary walls as partcpatng n chemcal compounds, and to the nverse convectve transfer owng to the occurrng lqud push as a result of the capllary partal fllng. Besdes, the process of on connecton due to the chemcal reactons s more ntensve than the transfer of ons ncomng from the surroundng soluton. Ths s confrmed by the ncrease n tme of the quantty of chemcally reacted ons Fg. 5. The character of the process s addtonally elucdated by the results found for the change of the effectve dffuson coeffcent Fg. 6. At the process start and near the nterface, the coeffcent values are the hghest ones, and the coeffcent vares wth the largest gradent. Then, due to the pore fllng and decrease of the concentraton mpetus, the dstrbuton curve tends to a straght lne and the mean value of the effectve coeffcent of dffuson tends to decrease along the coordnate axs x. The model proposed, together wth the numercal algorthm desgned, allows consderng more complex 3D cases of ons dffuson, as s that shown n Fg. 7. The specmen under consderaton s a prsm wth dmensons L =.5 m, L2 =.8 m, L3 =.9 m. It s assumed that two cylndrcal bodes of nert fller are located n /8 th of the specmen volume. Snce the specmen s symmetrc wth respect to the coordnate axes x= y= и =, the spatal dstrbuton of the ons concentraton, the quantty of chemcally reacted ons and the dstrbuton of the effectve dffuson coeffcent are plotted for /8-th of the specmen volume n the fgures below. Ths confguraton of the specmen s chosen to llustrate the capabltes of the numercal soluton. Fg.7. Schematc representaton of the nert fller sub-area n /8 th volume:: R =. m; R2 =.85 m; xc =.25 m; =.25 m; c of the specmen y =.5 m ; c The features of the concentraton front dstrbuton, outlned n the analyss of the case of D dffuson, are also characterstc for the 3D case. Besdes, for values of the concentraton closer to the concentraton of the surroundng lqud ths specfcty s more dstnct as compared to the case of hgher concentratons. Fg. 8 a, b, c shows comparson of the change n tme of two sosurfaces c const = [kg/m 3 ] and c const = 25 [kg/m 3 ]
Fg.8 Isosurfaces of the ons concentraton. Tme of mmerson a) month b) 4 months c) months The sosurface of the lower concentraton value ( kg/m 3 ) penetrates wthn the specmen, and the change of ts locaton between the 4-th and the -th month of mmerson s relatvely weak. The sosurface of the hgher concentraton value (25 kg/m 3 ) penetrates wthn the specmen for about 4 months of mmerson, and then draws back. Besdes, t s observed that the sosurface locaton s closer to the specmen external surface for months of mmerson as compared to ts locaton for month of mmerson. These results confrm the concluson drawn for the D case and further clarfy the role of the effects, accompanyng the sulfate ons dffuson,.e. capllary fllng and lqud push out of them. As a further llustraton, Fg. 9 shows the subsequent change n tme of the quantty of chemcally reacted ons.
Fg.9 Isosurfaces of the chemcally reacted quantty of ons Tme of mmerson a) month, b) 2 months, c) 4 months, d) months Fg. llustrate the change of the dffuson coeffcent. The shft n tme of the sosurface of D eff to the external nterface cеment stone/soluton s due partlly to the capllary fllng wth products of the chemcal reacton and partally to the decrease of the sulfate ons concentraton wthn the volume. The results shown n Fg. 8 Fg. also llustrate the effect of nert fllers of the ons transfer, and the capabltes of the numercal algorthm when treatng a specmen wth a complex shape. Fg.. Isosurfaces of the effectve dffuson coeffcent Tme of mmerson a) month b) 2 months c) 4 months, d) months
Conclusons The relaton for the effectve dffuson coeffcent used allows also to follow how the current sulfate concentraton wthn the volume affects the rate of ons transfer. The comparson of the theoretcal results wth the expermental data shows qualtatve agreement. It may be concluded that modelng the ons transfer as a dffuson process gves plausble quanttatve and qualtatve descrpton of the phenomena. The results found can be used to both assess the sulfate corroson n saturated systems and predct and estmate damage of structures bult of cement-based mneral compostes. References Gospodnov P. (25), Numercal smulaton of 3D sulfate on dffuson and lqud push out of the materal capllares n cement compostes, Cement and Concrete Research, 35 (3), 52-526 Gospodnov P., Kaandjev R., Partaln T., Mronova M., (999), Dffuson of sulfate ons nto cement stone regardng smultaneous chemcal reactons and resultng effects, Cement and Concrete Research. 29 (), 59-596 Kaandjev R., Gospodnov P., Mronova M., (25), Assessment of the mechancal characterstcs of cement stone subjected to sulfate attack of varous ntensty, Proceedngs of the -th Nat. Congress on Theor. and Appl. Mechancs, Varna 25,, 236-24 Marchand J., Samson E. and Maltas Y., (2), Modelng onc dffuson mechansms n saturated cement-based materals an overvew, Materal Scence of Concrete Ion and Mass Transport n Cement-Based Materals, Amercan Ceram. Socety (USA), D.Hooton et al. eds., 97- Marchand J., Samson E., Beaudon J.J., (22), Modelng on transport mechansms n unsaturated porous meda, Encyclopeda of Surface and Colod Scence, New York, USA, M. Dekker Mronova M., Gospodnov P., Kaandjev R., (22), The effect of lqud push out of the materal capllares under sulfate on dffuson n cement compostes, Cement and Concrete Research, 32 (), 9-5 Samson E., Lemare G., Marchand J. and Beaudon J.J, (999a), Modelng chemcal actvty effects n strong onc solutons, Computatonal Materal Scence 5, 285-294 Samson E. and Marchand J., (999), Numercal soluton of the extended Nernst Planck model, Journal of Collod amd Interface Scence, 25, -8 Samson E., Marchand J., Beaudon J.J., (999b), Descrbng on dffuson mechansms n cementbased materals usng the homogenaton technque, Cement and Concrete Research, 29 (), 34-345 Samson E., Marchand J., Robert J.-L., Bournael J.-P., (999c), Modellng on dffuson mechansms n porous meda, Int. J. Numer.Mech. Engng. 46, 243-26 Stansh K., Hooton R.D., Thomas M.D.A., (24), A novel method for descrbng chlorde on transport due to an electrcal gradent n concrete: Part. Theoretcal descrpton, Cement and Concrete Research 34 (), 43 49