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Reply to the discussion by Q. Zeng and S.L. Xu of the paper “Numerical simulation of moisture transport in concrete based on a pore size distribution model”
Cement and Concrete Research 73 (2015) 67–69
Contents lists available at ScienceDirect
Cement and Concrete Research journal homepage: http://ees.elsevier.com/CEMCON/default.asp
Discussion
Reply to the discussion by Q. Zeng and S.L. Xu of the paper “Numerical simulation of moisture transport in concrete based on a pore size distribution model” Zhilu Jiang 1, Xianglin Gu ⁎, Qinghua Huang 1, Weiping Zhang 1 Department of Structural Engineering, Tongji University, 1239 Siping Road, Shanghai, PR China
a r t i c l e
i n f o
Article history: Received 30 January 2015 Accepted 5 March 2015 Available online xxxx
a b s t r a c t This paper presents a reply to the discussion of the paper “Numerical simulation of moisture transport in concrete based on a pore size distribution model” by Q. Zeng and S.L. Xu. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Pore size distribution (B) Concrete (E)
1. Introduction We sincerely thank Zeng and Xu [1] for their comments on the paper by Huang et al. [2]. In the discussion the authors pointed out that an adsorbed liquid-like layer (ALLL) had an influence on the calculated results of the pore size distribution (PSD) of concrete based on the water vapour sorption isotherm (WVSI). Zeng and Xu developed an improved method and compared their results with ours [2]. The comparison indicated that the consideration of ALLL had a minor influence on the PSD at the macroscale while it became important at the micro- and meso-scales. However, some factors were still not considered in their proposed method. This reply will, therefore, focus on further improving the method for obtaining the micro- and mesopores based on the WVSI and clarifying the influence of ALLL mentioned in the discussion. 2. Excess surface work and disjoining pressure isotherm method (macro- and meso-pores) Condensation and the multilayer sorption of water molecules can occur at the same time in capillary pores. However, the amount of the adsorbed water was not considered for simplification in our model [2]. Zeng and Xu discussed the effect of neglecting the adsorbed water [1].
⁎ Corresponding author. Tel./fax: +86 21 65982928. E-mail addresses: [email protected] (Z. Jiang), [email protected] (X. Gu), [email protected] (Q. Huang), [email protected] (W. Zhang). 1 Tel./fax: +86 21 65982928.
http://dx.doi.org/10.1016/j.cemconres.2015.03.005 0008-8846/© 2015 Elsevier Ltd. All rights reserved.
In Section 3 we will further investigate the impact of different determination methods of the thickness of the adsorbed layer. In mesopores, the interaction due to van der Waals forces during adsorption process is much stronger than that in the macropores. Consequently, using Kelvin's equation without considering the interaction between the pore walls and the adsorbate may lead to an inaccurate result. However, Zeng and Xu only considered the thickness of the adsorbed layer but did not consider the interaction in calculating the capillary radius. More accurate results can possibly be obtained by using excess surface work and the disjoining pressure isotherm (ESW–DPI) method [3]. “Excess surface work” (ESW) can be used in modelling adsorption in porous materials and its advantage is that the relevant parameters have physical meanings. ESW is defined as the product of the adsorbate amount na and the change of chemical potential Δμ. ESW Φ increases when the adsorbate gradually covers the pore wall and it decreases after the pore wall is covered with one layer of the adsorbate due to the decreased interaction between the surface and the adsorbate, so the following relation is obtained [3]: ∂Φ ¼0 ∂na
at na ¼ nm
ð1Þ
where nm is the amount for monolayer of the adsorbate. Based on the relation between the chemical potential and the adsorbate amount proposed by Adolphs and Setzer [4], the following relation was obtained: Δμ t a ¼ −t m ln Δμ 0
ð2Þ
68
Z. Jiang et al. / Cement and Concrete Research 73 (2015) 67–69
where ta is the thickness of the adsorbed layer, tm is the thickness of the monolayer and Δμ0 is the chemical potential to start adsorption. According to thermodynamic equilibrium between vapour molecules and adsorbate, Δμ can be calculated by Δμ ¼ RT ln h
ð3Þ
3. Influence of different determination methods of thickness Different determination methods used in the calculation of PSD may have an influence on the results. Many methods have been proposed to determine the thickness of ALLL. The following equations to obtain the thickness ta (nm) were used in order to investigate the influence: t a ¼ 0:395−0:189 ln ð− ln hÞ [3]
where R is the gas constant, T is the temperature and h is the relative humidity. Due to the interaction between the adsorbed layer and the pore surface, the absorbed layer is subject to the pressure to disjoin the layer and the pore wall. Introducing the disjoining pressure Π, Δμ0 is the product of the disjoining pressure between an even film and the solid when adsorption starts Π0 and the adsorbate mole volume vm. The thermodynamic equilibrium between the adsorbate and water vapour leads to the following equation in cylindrical pores: −t a σ RT þ ¼− lnh Π 0 exp tm r−t a vm
ð4Þ
where σ is the surface tension of water, and r is the pore radius. If the left side of the equation reaches the minimum, the adsorbate becomes unstable and capillary condensation occurs. Consequently, the following relation is obtained at capillary condensation: −t a;c 1 σ ¼ Π 0 exp 2 tm tm r c −t a;c
ð5Þ
where ta,c and rc are the adsorbed layer thickness and radius of pores at condensation, respectively. Replacing the pore radius r and the thickness ta in Eq. (4) with rc and ta,c, respectively, rc could be eliminated by substituting Eq. (5) to Eq. (4). Assuming that Π0 = 474.6 MPa, tm = 2.8 × 10− 10 m, vm = 18 × 10− 6 mol/m3, σ = 0.0723 N/m, R = 8.314 J·mol− 1·K−1, and T = 293 K, the radius of pores at capillary condensation was then calculated under different relative humidity (RH) levels by solving the Eq. (4) using the Newton–Raphson method. The percentages of the disjoining and capillary pressures acting on the adsorbed layer in pores at condensation are shown in Fig. 1. The results indicated that the percentage of the disjoining pressure decreased significantly with increasing pore radius. For large capillary pores the disjoining pressure can be neglected during adsorption while Kelvin equation may not hold for small pores due to the significant influence of the disjoining pressure.
Fig. 1. Percentage of disjoining pressure and capillary pressure on the adsorbed water in pores with different radii at condensation.
ta ¼
0:3Ch ð1−hÞ½1 þ ðC−1Þh
[5]
t a ¼ 0:834h þ 0:0626 þ 0:02309=ð1:105−hÞ [6]
ð6Þ ð7Þ ð8Þ
where C is the BET constant given by C = exp(855/T). The PSD of the specimen A in [2] was calculated. The comparison in Fig. 2 shows that the influence of different methods of determining the ALLL thickness mainly lies in pores with a radius of less than about 6 nm. The results were similar on pore radii larger than 1 nm by using Eq. (8) and the ESW–DPI method, respectively. The PSD curves obtained by using the other equations moved leftward and had larger peak values, indicating that the sizes of the calcium–silicate–hydrate (C–S–H) and small capillary pores were smaller. In [2], we investigated the influence PSD on moisture transport in concrete by numerical simulations for specimens with different PSDs. The results indicated that large capillaries and microcracks had a greater influence on the moisture transport than the other pore components. The influence of C–S–H and small capillary pores may have less significant influence because liquid water permeability decreases linearly with the decreased pore size and their sizes are relatively small. Additionally, moisture capacity increased more significantly with the proportions of the C–S–H and the small capillary pores at a normal relative humidity (RH) range according to the analysis of the PSD and the critical pore radius at different RH levels [2]. 4. Micropore filling (micropores) In Zeng and Xu's results [1], the intensity of the porosity distribution density was zero when the pore radius reached about 0.7 nm because of the limitation of the application scope of Kelvin's equation. In [2], we obtained the parameters of a multi-Rayleigh–Ritz model for the PSD of the concrete based on the water vapour sorption isotherm (WVSI). The PSD of pores with radii less than 0.7 nm was actually the result of extrapolation. The result may be unauthentic, so we presented another method here to evaluate the PSD of the micropores in the concrete. At low RH, water molecules adsorbed on the pore walls in thin pores are subject to high forces due to the overlapped potential fields of two
Fig. 2. Influence of different methods of determining the adsorbed layer thickness on the pore size distribution of specimen A in [2] (ϕ is the ratio of porosity of pores with radius less than r to the total porosity).
Z. Jiang et al. / Cement and Concrete Research 73 (2015) 67–69
69
5. Conclusions The ESW–DPI method was used for calculating PSD based on WVSI to consider the interaction between the adsorbate and the pore walls. Different methods of determining the ALLL thickness had a great influence on the PSD at radii less than about 6 nm. However, these small pores may have a less significant influence on the moisture transport. Due to a different mechanism of water adsorption at low RH, the PSD of pores with radii of less than 0.5 nm was calculated by using the ta–Sr relation.
Acknowledgements
Fig. 3. The relationship between the thickness of the adsorbed water and the water saturation for specimen A in [2].
This research project was financially supported by the National Natural Science Foundation of China (Grant Nos. 51320105013 and 51109163).
References pore walls. In these pores, water molecules are not adsorbed layer by layer but may fill the pores by volume filling. The PSD of the thin pore can be obtained by using the relation between the adsorbed layer thickness ta (m) and the water saturation Sr. The ta–Sr relation for specimen A in [2] as shown in Fig. 3 indicates that at first the water content almost linearly increases with the thickness. However, the slope of the curve decreases gradually after the thickness reaches around 0.23 nm. The slope is equal to the surface area (m2) divided by the total pore volume (m3). The slope decreases because some pores are filled by the water and the area of the surface available to adsorb water molecules also decreases. In Fig. 3, the slopes of lines 1 and 2 are k1 and k2, respectively. Assuming the walls of the thin pores are two parallel plates, the ratio of the porosity of pores with radius r (r is approximately equal to (r1 + r2)/2) to the total porosity is given by ϕ12 ¼ ðk1 −k2 Þ r:
ð9Þ
We used the above method to evaluate the PSDs of micro-C–S–H gel pores in specimens A to D in [2]. The results in Fig. 4 indicate that the proportion of the C–S–H pores decreases with increasing water–cement ratio. Compared with the results in [2], the evaluated size of the gel pores was generally smaller due to different assumptions used in the methods. However, the difference may have a minor influence on the moisture transport of the concrete at normal environmental conditions because the size of the pores in C–S–H is too small to have an adverse effect on the permeability of the hydrated cement paste [7]. On the other hand, the water removal in these pores at strong drying conditions (i.e., below 11% RH) may contribute to drying shrinkage and creep [7].
Fig. 4. Pore size distribution of C–S–H pores for specimens A–D in [2].
[1] Q. Zeng, S.L. Xu, Discussion of “Numerical simulation of moisture transport in concrete based on a pore size distribution model”, Cem. Concr. Res. (2015) (in this issue). [2] Q.H. Huang, Z.L. Jiang, X.L. Gu, W.P. Zhang, B.H. Guo, Numerical simulation of moisture transport in concrete based on a pore size distribution model, Cem. Concr. Res. 67 (2015) 31–43. [3] R.M. Espinosa, L. Franke, Inkbottle pore-method: prediction of hygroscopic water content in hardened cement paste at variable climatic conditions, Cem. Concr. Res. 36 (2006) 1954–1968. [4] J. Adolphs, M.J. Setzer, A model to describe sorption isotherms, J. Colloid Interface Sci. 180 (1996) 70–76. [5] H. Ranaivomanana, J. Verdier, A. Sellier, X. Bourbon, Toward a better comprehension and modeling of hysteresis cycles in the water sorption–desorption process for cement based materials, Cem. Concr. Res. 41 (2011) 817–827. [6] J.J.R. Hagymassy, S. Brunauer, R.S.H. Mikhail, Pore structure analysis by water vapor adsorption: I. t-Curves for water vapor, J. Colloid Interface Sci. 29 (1969) 485–491. [7] P.K. Mehta, P.J.M. Monteiro, Concrete: microstructure, properties, and materials, Third ed. McGraw-Hill, New York, 2006.
Contents lists available at ScienceDirect
Cement and Concrete Research journal homepage: http://ees.elsevier.com/CEMCON/default.asp
Discussion
Reply to the discussion by Q. Zeng and S.L. Xu of the paper “Numerical simulation of moisture transport in concrete based on a pore size distribution model” Zhilu Jiang 1, Xianglin Gu ⁎, Qinghua Huang 1, Weiping Zhang 1 Department of Structural Engineering, Tongji University, 1239 Siping Road, Shanghai, PR China
a r t i c l e
i n f o
Article history: Received 30 January 2015 Accepted 5 March 2015 Available online xxxx
a b s t r a c t This paper presents a reply to the discussion of the paper “Numerical simulation of moisture transport in concrete based on a pore size distribution model” by Q. Zeng and S.L. Xu. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Pore size distribution (B) Concrete (E)
1. Introduction We sincerely thank Zeng and Xu [1] for their comments on the paper by Huang et al. [2]. In the discussion the authors pointed out that an adsorbed liquid-like layer (ALLL) had an influence on the calculated results of the pore size distribution (PSD) of concrete based on the water vapour sorption isotherm (WVSI). Zeng and Xu developed an improved method and compared their results with ours [2]. The comparison indicated that the consideration of ALLL had a minor influence on the PSD at the macroscale while it became important at the micro- and meso-scales. However, some factors were still not considered in their proposed method. This reply will, therefore, focus on further improving the method for obtaining the micro- and mesopores based on the WVSI and clarifying the influence of ALLL mentioned in the discussion. 2. Excess surface work and disjoining pressure isotherm method (macro- and meso-pores) Condensation and the multilayer sorption of water molecules can occur at the same time in capillary pores. However, the amount of the adsorbed water was not considered for simplification in our model [2]. Zeng and Xu discussed the effect of neglecting the adsorbed water [1].
⁎ Corresponding author. Tel./fax: +86 21 65982928. E-mail addresses: [email protected] (Z. Jiang), [email protected] (X. Gu), [email protected] (Q. Huang), [email protected] (W. Zhang). 1 Tel./fax: +86 21 65982928.
http://dx.doi.org/10.1016/j.cemconres.2015.03.005 0008-8846/© 2015 Elsevier Ltd. All rights reserved.
In Section 3 we will further investigate the impact of different determination methods of the thickness of the adsorbed layer. In mesopores, the interaction due to van der Waals forces during adsorption process is much stronger than that in the macropores. Consequently, using Kelvin's equation without considering the interaction between the pore walls and the adsorbate may lead to an inaccurate result. However, Zeng and Xu only considered the thickness of the adsorbed layer but did not consider the interaction in calculating the capillary radius. More accurate results can possibly be obtained by using excess surface work and the disjoining pressure isotherm (ESW–DPI) method [3]. “Excess surface work” (ESW) can be used in modelling adsorption in porous materials and its advantage is that the relevant parameters have physical meanings. ESW is defined as the product of the adsorbate amount na and the change of chemical potential Δμ. ESW Φ increases when the adsorbate gradually covers the pore wall and it decreases after the pore wall is covered with one layer of the adsorbate due to the decreased interaction between the surface and the adsorbate, so the following relation is obtained [3]: ∂Φ ¼0 ∂na
at na ¼ nm
ð1Þ
where nm is the amount for monolayer of the adsorbate. Based on the relation between the chemical potential and the adsorbate amount proposed by Adolphs and Setzer [4], the following relation was obtained: Δμ t a ¼ −t m ln Δμ 0
ð2Þ
68
Z. Jiang et al. / Cement and Concrete Research 73 (2015) 67–69
where ta is the thickness of the adsorbed layer, tm is the thickness of the monolayer and Δμ0 is the chemical potential to start adsorption. According to thermodynamic equilibrium between vapour molecules and adsorbate, Δμ can be calculated by Δμ ¼ RT ln h
ð3Þ
3. Influence of different determination methods of thickness Different determination methods used in the calculation of PSD may have an influence on the results. Many methods have been proposed to determine the thickness of ALLL. The following equations to obtain the thickness ta (nm) were used in order to investigate the influence: t a ¼ 0:395−0:189 ln ð− ln hÞ [3]
where R is the gas constant, T is the temperature and h is the relative humidity. Due to the interaction between the adsorbed layer and the pore surface, the absorbed layer is subject to the pressure to disjoin the layer and the pore wall. Introducing the disjoining pressure Π, Δμ0 is the product of the disjoining pressure between an even film and the solid when adsorption starts Π0 and the adsorbate mole volume vm. The thermodynamic equilibrium between the adsorbate and water vapour leads to the following equation in cylindrical pores: −t a σ RT þ ¼− lnh Π 0 exp tm r−t a vm
ð4Þ
where σ is the surface tension of water, and r is the pore radius. If the left side of the equation reaches the minimum, the adsorbate becomes unstable and capillary condensation occurs. Consequently, the following relation is obtained at capillary condensation: −t a;c 1 σ ¼ Π 0 exp 2 tm tm r c −t a;c
ð5Þ
where ta,c and rc are the adsorbed layer thickness and radius of pores at condensation, respectively. Replacing the pore radius r and the thickness ta in Eq. (4) with rc and ta,c, respectively, rc could be eliminated by substituting Eq. (5) to Eq. (4). Assuming that Π0 = 474.6 MPa, tm = 2.8 × 10− 10 m, vm = 18 × 10− 6 mol/m3, σ = 0.0723 N/m, R = 8.314 J·mol− 1·K−1, and T = 293 K, the radius of pores at capillary condensation was then calculated under different relative humidity (RH) levels by solving the Eq. (4) using the Newton–Raphson method. The percentages of the disjoining and capillary pressures acting on the adsorbed layer in pores at condensation are shown in Fig. 1. The results indicated that the percentage of the disjoining pressure decreased significantly with increasing pore radius. For large capillary pores the disjoining pressure can be neglected during adsorption while Kelvin equation may not hold for small pores due to the significant influence of the disjoining pressure.
Fig. 1. Percentage of disjoining pressure and capillary pressure on the adsorbed water in pores with different radii at condensation.
ta ¼
0:3Ch ð1−hÞ½1 þ ðC−1Þh
[5]
t a ¼ 0:834h þ 0:0626 þ 0:02309=ð1:105−hÞ [6]
ð6Þ ð7Þ ð8Þ
where C is the BET constant given by C = exp(855/T). The PSD of the specimen A in [2] was calculated. The comparison in Fig. 2 shows that the influence of different methods of determining the ALLL thickness mainly lies in pores with a radius of less than about 6 nm. The results were similar on pore radii larger than 1 nm by using Eq. (8) and the ESW–DPI method, respectively. The PSD curves obtained by using the other equations moved leftward and had larger peak values, indicating that the sizes of the calcium–silicate–hydrate (C–S–H) and small capillary pores were smaller. In [2], we investigated the influence PSD on moisture transport in concrete by numerical simulations for specimens with different PSDs. The results indicated that large capillaries and microcracks had a greater influence on the moisture transport than the other pore components. The influence of C–S–H and small capillary pores may have less significant influence because liquid water permeability decreases linearly with the decreased pore size and their sizes are relatively small. Additionally, moisture capacity increased more significantly with the proportions of the C–S–H and the small capillary pores at a normal relative humidity (RH) range according to the analysis of the PSD and the critical pore radius at different RH levels [2]. 4. Micropore filling (micropores) In Zeng and Xu's results [1], the intensity of the porosity distribution density was zero when the pore radius reached about 0.7 nm because of the limitation of the application scope of Kelvin's equation. In [2], we obtained the parameters of a multi-Rayleigh–Ritz model for the PSD of the concrete based on the water vapour sorption isotherm (WVSI). The PSD of pores with radii less than 0.7 nm was actually the result of extrapolation. The result may be unauthentic, so we presented another method here to evaluate the PSD of the micropores in the concrete. At low RH, water molecules adsorbed on the pore walls in thin pores are subject to high forces due to the overlapped potential fields of two
Fig. 2. Influence of different methods of determining the adsorbed layer thickness on the pore size distribution of specimen A in [2] (ϕ is the ratio of porosity of pores with radius less than r to the total porosity).
Z. Jiang et al. / Cement and Concrete Research 73 (2015) 67–69
69
5. Conclusions The ESW–DPI method was used for calculating PSD based on WVSI to consider the interaction between the adsorbate and the pore walls. Different methods of determining the ALLL thickness had a great influence on the PSD at radii less than about 6 nm. However, these small pores may have a less significant influence on the moisture transport. Due to a different mechanism of water adsorption at low RH, the PSD of pores with radii of less than 0.5 nm was calculated by using the ta–Sr relation.
Acknowledgements
Fig. 3. The relationship between the thickness of the adsorbed water and the water saturation for specimen A in [2].
This research project was financially supported by the National Natural Science Foundation of China (Grant Nos. 51320105013 and 51109163).
References pore walls. In these pores, water molecules are not adsorbed layer by layer but may fill the pores by volume filling. The PSD of the thin pore can be obtained by using the relation between the adsorbed layer thickness ta (m) and the water saturation Sr. The ta–Sr relation for specimen A in [2] as shown in Fig. 3 indicates that at first the water content almost linearly increases with the thickness. However, the slope of the curve decreases gradually after the thickness reaches around 0.23 nm. The slope is equal to the surface area (m2) divided by the total pore volume (m3). The slope decreases because some pores are filled by the water and the area of the surface available to adsorb water molecules also decreases. In Fig. 3, the slopes of lines 1 and 2 are k1 and k2, respectively. Assuming the walls of the thin pores are two parallel plates, the ratio of the porosity of pores with radius r (r is approximately equal to (r1 + r2)/2) to the total porosity is given by ϕ12 ¼ ðk1 −k2 Þ r:
ð9Þ
We used the above method to evaluate the PSDs of micro-C–S–H gel pores in specimens A to D in [2]. The results in Fig. 4 indicate that the proportion of the C–S–H pores decreases with increasing water–cement ratio. Compared with the results in [2], the evaluated size of the gel pores was generally smaller due to different assumptions used in the methods. However, the difference may have a minor influence on the moisture transport of the concrete at normal environmental conditions because the size of the pores in C–S–H is too small to have an adverse effect on the permeability of the hydrated cement paste [7]. On the other hand, the water removal in these pores at strong drying conditions (i.e., below 11% RH) may contribute to drying shrinkage and creep [7].
Fig. 4. Pore size distribution of C–S–H pores for specimens A–D in [2].
[1] Q. Zeng, S.L. Xu, Discussion of “Numerical simulation of moisture transport in concrete based on a pore size distribution model”, Cem. Concr. Res. (2015) (in this issue). [2] Q.H. Huang, Z.L. Jiang, X.L. Gu, W.P. Zhang, B.H. Guo, Numerical simulation of moisture transport in concrete based on a pore size distribution model, Cem. Concr. Res. 67 (2015) 31–43. [3] R.M. Espinosa, L. Franke, Inkbottle pore-method: prediction of hygroscopic water content in hardened cement paste at variable climatic conditions, Cem. Concr. Res. 36 (2006) 1954–1968. [4] J. Adolphs, M.J. Setzer, A model to describe sorption isotherms, J. Colloid Interface Sci. 180 (1996) 70–76. [5] H. Ranaivomanana, J. Verdier, A. Sellier, X. Bourbon, Toward a better comprehension and modeling of hysteresis cycles in the water sorption–desorption process for cement based materials, Cem. Concr. Res. 41 (2011) 817–827. [6] J.J.R. Hagymassy, S. Brunauer, R.S.H. Mikhail, Pore structure analysis by water vapor adsorption: I. t-Curves for water vapor, J. Colloid Interface Sci. 29 (1969) 485–491. [7] P.K. Mehta, P.J.M. Monteiro, Concrete: microstructure, properties, and materials, Third ed. McGraw-Hill, New York, 2006.