(12)

where is mass ratio of Na_{2}O per mass of binder, is molar mass of Na_{2}O (61.98 g・mol^{−1}), factor 2 comes from the fact that there are 2 moles of Na in 1 mol of Na_{2}O, and m_{binder} is amount of binder, 1409 × 10^{3} g per m^{3} of geopolymer material which was recalculated from binder density (for fly ash assumed here to be ρ_{binder} = 2600 kg m^{−3}) and water to powder ratio used for the mix design (w/b = 0.325 [2] ):

(13)

From the experimental results of the pore extraction analysis the concentration of Na^{+} is c_{0} = 612 mol/m^{3} and in this paper, it is proposed to obtain the corresponding equilibrated bound concentration from Equation (11), which yields C_{b}_{,0} to be 1.6580 mol/kg of geopolymer solids.

Introducing the initial binding isotherm experimental point for geopolymer material at leaching time t = 0 we can express K_{eq} as a function of C_{max}, according to Equation (10). Inserting the known c value (c = 612 mol/m^{3} of pore solution) measured by pore fluid extraction and the bound Na content (C_{b} = 1.6580 mol/kg of solids), the functional dependency between Langmuir parameters K_{eq} and C_{max} is then defined as:

(14)

Levenberg-Marquardt inverse analysis was performed to estimate D_{ef} and C_{max} parameters by minimising the sum of squared residuals between modelled and experimental leaching results. Within the model the K_{eq} is obtained from Equation (14).

5. Results

Comparison of modelled and experimental results is given in Figure 1. For a diffusion model (dotted curve) the initial amount of Na measured in the expressed pore solution was not enough to account for the amount of Na leached from the paste. This results in significant underestimation of the leached Na amount for times after around t^{0.5} = 400 s^{0.5}, because no release of bound alkali is made possible within the plain diffusion model. After that time, the model asymptotically approaches to a constant steady state value of the system. However, introducing the reaction (equilibrium) term for release of bound alkali enables to describe the experimental observations. Levenberg-Marquardt inverse analysis (fitted model) obtained by minimising the sum of squared residuals yielded following estimates: D_{ef} = 3.95 × 10^{−11} m^{2}/s; C_{max} = 1.663 mol/kg; K_{eq} = 0.5764 m^{3}・mol^{−}^{1}. The value obtained by fitting oversimplified diffusion model gave 22% lower estimate (D_{ef} = 4.8 × 10^{−11} m^{2}/s [2] ). More importantly the capacity to release bound alkali is discussed next. Figure 2 depicts modelled Na profile concentration (in same units of

Figure 1. Comparison of modelled and experimental results [2] . Levenberg-Mar- quardt inverse analysis (fitted model) obtained by minimising the sum of squared residuals resulted in estimates for D_{ef} = 3.95 × 10^{−11} m^{2}/s; C_{max} = 1.663 mol/kg; K_{eq} = 0.5764 m^{3}・mol^{−1}.

Figure 2. Modelled Na profile concentration across sample thickness for leaching time t^{0.5} = 655 s^{0.5}: dashed curve for total Na showing huge values on left ordinate compared to solid curve for relatively low amount of free Na on right ordinate (same units of mol/m^{3} of geopolymer paste).

mol/m^{3} of geopolymer paste) after a leaching time t^{0.5} = 655 s^{0.5}. The dashed curve represents total Na values on left ordinate and solid curve represents free Na amount on right ordinate. The huge difference between the two curves emphasises the importance for introducing the ion-exchange equilibrium reactions in modelling of alkali leaching process from geopolymer materials.

There are two extreme external (boundary) condition scenarios on which geopolymer material can be exposed to: 1) rapid leaching of the pore solution from the material, e.g. due to external flow of water or 2) acid attack that neutralises the pore solution. In both cases the drop in alkali concentration of the pore solution (and thus its pH) is very sudden when there is no source of the dissolved alkalies from the solids. Dissolution of alkalies by ion exchange mechanism with protons allows for a more gradual drop in pH of the pore solution, as compared to a sudden drop when considering only diffusion transport mechanism with no release of bound alkalis.

The rate of drop in alkali concentration of the pore solution (and thus its pH) is then governed by the rate of the dissolution (Equations (7) and (8)), which is obtained by differentiating the Langmuir binding isotherm Equation (6) with respect to c. Initially, the drop in pH is high due to low ion exchange capacity (Figure 3): represented by a right end point in Langmuir curve where a large drop in free alkalies gives negligible amount (rate) of dissolved alkalies. The dissolution rate starts to be significant only when concentrations of the free Na^{+} are below around 25 mol/m^{3} of pore solution (Figure 3). Below this threshold point, the binding sensitivity (i.e. its rate) is now reversed: a small drop in free Na^{+} concentration is accompanied by a high amount of dissolved alkalies. This threshold corresponds to a pH of 12.4 which is very close to

Figure 3. Langmuir isotherm (Equation (6); C_{max} = 1.663 mol/kg; K_{eq} = 0.5764 m^{3}・mol^{−1}) obtained by the inverse analysis. For better visibility logarithmic scale was used across abscissa.

the value of saturated Ca(OH)_{2} solution of pH = 12.5.

The degree of Na^{+} binding selectivity among various kinds of zeolite structures, and thus also geopolymers, varies significantly depending on the prevailing factors of their structural frameworks [2] [7] . Various geopolymers have different semi-crystal structures arising from variations in composition, distribution and ordering of -SiO_{4} tetrahedral units in linkages of their global structural (Al-O-Si-) framework [1] [7] . Primarily, variability in Si/Al ratio results in differences in the location, amount and distribution of negative charge density in the structural frameworks, cages or pores of different diameters, nature or absence of hydration water or other ligands and presence and position of charge compensating cations.

6. Conclusions

Plain diffusion model is inappropriate to describe the leaching of alkalies from geopolymer materials. For longer leaching times the model falsely starts to asymptotically approach a steady state condition with a significant underestimation of both the remaining alkali amount in pore solution as well as the amount leached into the environment. The newly proposed transport model is coupled to the reaction (equilibrium) term describing a release of bound alkalies and enables accurate predictions of the experimental observations. Bound alkalies provide the geopolymer paste with a large reservoir of exchangeable (soluble) alkali. The calculated huge difference between the simulated total and free alkali concentrations emphasises the importance for introducing the ion-exchange equilibrium reactions in modelling of alkali leaching process from geopolymer materials. Such a reservoir of potentially dissolvable alkalies allows for a more gradual drop in pH of the pore solution, as compared to a sudden drop when considering only diffusion transport mechanism with no release of bound alkalies. The bounded alkali concentration in the solid phase is calculated from Langmuir equilibrium relationship with the alkali concentration in the solution.

Levenberg-Marquardt inverse analysis was performed to estimate D_{ef} and C_{max} (one of the Langmuir) parameters by minimising the sum of squared residuals between modelled and experimental leaching results. To reduce the number of parameters to be determined simultaneously with the inverse analysis, it is proposed to use experimental data on the binding isotherm for the case of material’s initial condition. From the pore extraction experimental analysis one can obtain the free concentration of alkali initially available in pore solution, while the corresponding equilibrated bound concentration is calculated from the total concentration of alkali in the system, which is known from mix design.

The proposed model for alkali leaching that accounts for ion-exchange solid-liquid equilibrium presents a more reliable way to obtain long term durability predictions of geopolymer materials.

Cite this paper

Ukrainczyk, N. , Vogt, O. and Koenders, E. (2016) Reactive Transport Numerical Model for Durability of Geopolymer Materials.*Advances in Chemical Engineering and Science*, **6**, 355-363. doi: 10.4236/aces.2016.64036.

Ukrainczyk, N. , Vogt, O. and Koenders, E. (2016) Reactive Transport Numerical Model for Durability of Geopolymer Materials.

References

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http://dx.doi.org/10.1039/c0jm01254h

[2] Lloyd, R.R., Provis, J.L. and van Deventer, J.S.J. (2010) Pore Solution Composition and Alkali Diffusion in Inorganic Polymer Cement. Cement and Concrete Research, 40, 1386-1392.

http://dx.doi.org/10.1016/j.cemconres.2010.04.008

[3] Skvara, F., Smilauer, V., Hlavacek, P., Kopecky, L. and Cilova, Z. (2012) A Weak Alkali Bond in (N, K)-A-S-H Gels: Evidence from Leaching and Modeling. Ceramics-Silikaty, 56, 374-382.

[4] Schiesser, W.E. and Griffiths, G.W. (2009) A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511576270

[5] Skeel, R.D. and Berzins, M. (1990) A Method for the Spatial Discretization of Parabolic Equations. SIAM Journal on Scientific and Statistical Computing, 11, 1-32.

http://dx.doi.org/10.1137/0911001

[6] Shampine, L.F. and Reichelt, M.W. (1997) The MATLAB ODE Suite. SIAM Journal on Scientific and Statistical Computing, 18, 1-22.

http://dx.doi.org/10.1137/S1064827594276424

[7] Munthali, M.W., Elsheikh, M.A., Johan, E. and Matsue, N. (2014) Proton Adsorption Selectivity of Zeolites in Aqueous Media: Effect of Si/Al Ratio of Zeolites. Molecules, 19, 20468-20481.

http://dx.doi.org/10.3390/molecules191220468

[1] O’Connor, S.J., MacKenzie, K.J.D., Smith, M. and Hanna, J. (2010) Ion Exchange in the Charge Balancing Sites of Aluminosilicate Inorganic Polymers. Journal of Materials Chemistry, 20, 10234-10240.

http://dx.doi.org/10.1039/c0jm01254h

[2] Lloyd, R.R., Provis, J.L. and van Deventer, J.S.J. (2010) Pore Solution Composition and Alkali Diffusion in Inorganic Polymer Cement. Cement and Concrete Research, 40, 1386-1392.

http://dx.doi.org/10.1016/j.cemconres.2010.04.008

[3] Skvara, F., Smilauer, V., Hlavacek, P., Kopecky, L. and Cilova, Z. (2012) A Weak Alkali Bond in (N, K)-A-S-H Gels: Evidence from Leaching and Modeling. Ceramics-Silikaty, 56, 374-382.

[4] Schiesser, W.E. and Griffiths, G.W. (2009) A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511576270

[5] Skeel, R.D. and Berzins, M. (1990) A Method for the Spatial Discretization of Parabolic Equations. SIAM Journal on Scientific and Statistical Computing, 11, 1-32.

http://dx.doi.org/10.1137/0911001

[6] Shampine, L.F. and Reichelt, M.W. (1997) The MATLAB ODE Suite. SIAM Journal on Scientific and Statistical Computing, 18, 1-22.

http://dx.doi.org/10.1137/S1064827594276424

[7] Munthali, M.W., Elsheikh, M.A., Johan, E. and Matsue, N. (2014) Proton Adsorption Selectivity of Zeolites in Aqueous Media: Effect of Si/Al Ratio of Zeolites. Molecules, 19, 20468-20481.

http://dx.doi.org/10.3390/molecules191220468