Moisture Diffusion in Unsaturated Self-Desiccating Concrete with Humidity-Dependent Permeability and Nonlinear Sorption Isotherm

Home , Desiccation

Saeed Rahimi-Aghdam1; Mohammad Rasoolinejad2; and ZdeněkP.Bažant, Hon.M.ASCE3

Abstract: A nonlinear diffusion model for the drying of concrete, previously developed at Northwestern University and embedded in some design codes, was improved and calibrated on the basis of recent more extensive experimental data from the literature as well as theoretical considerations. The improvements include a new equation for the dependence of the self-desiccation rate on pore humidity and hydration degree; an updated equation for the decrease of moisture permeability at decreasing pore humidity; new equations to predict the permeability and diffusivity parameters from the water-cement and aggregate-cement ratios, hydration degree, the type of concrete; and new equations to capture the nonlinearity of the sorption isotherm as a function of pore humidity and water-cement ratio. Furthermore, the recent idea that the pore humidity drop is the driving force, rather than a side effect, of the autogenous shrinkage is verified. DOI: 10.1061/(ASCE)EM.1943- 7889.0001591. © 2019 American Society of Civil Engineers. Author keywords: Drying; Water transport; Self-desiccation; Aging; Cement hydration; Diffusivity; Pore pressure; Numerical analysis; Experimental verification.

Introduction The Bažant-Najjar (BN) nonlinear diffusion model worked quite well for concretes half a century ago, and has been embed- By now it is generally accepted that correct calculation of the spa- ded in various recommendations and design codes, e.g., in the tial distribution and time evolution of the pore humidity in concrete Model Code of fib (2012). Recently, though, it became clear that structures is necessary for realistic prediction of creep, shrinkage, a generalization and improvement of the BN model is necessary and thermal dilatation, along with their effects on deflections, crack to meet several broader objectives: formation, and various degradation processes (Bažant and Jirásek 1. For modern concretes, the foremost objective is to take into 2017; Bažant and Donmez 2016; Rahimi-Aghdam et al. 2019). account the self-desiccation, which drives the autogenous shrink- The relative humidity in the pores has a major effect on the mois- age. The self-desiccation was negligible in old concretes but be- ture permeability, the rate of hydration (or aging), and the rate of came significant in modern high-strength and high-performance strength-degrading chemical reactions such as the alkali-silica re- concretes. The causes are (1) a low water-cement ratio (w=c), action (ASR). It plays a large role in the assessments of durability, and (2) admixtures such as silica fume (SF) and blast furnace fire resistance, and radiation shielding. It matters for the age-old slag (Baroghel-Bouny et al. 2006; Jiang et al. 2006). Baroghel- question of uplift in dams, and it must be considered in formulating Bouny et al. (2006) showed that, in sealed specimens of the stress-strain law for concrete. high-strength concrete with a high content of silica fume, the Studies of nuclear reactor structures in the 1960s led to the dis- self-desiccation can decrease the relative humidity in the pores ž covery (Ba ant and Najjar 1971, 1972) that the diffusion equation to as low as 65%. Such a humidity decrease may be beneficial for concrete is highly nonlinear because the permeability and the because most of the deteriorative reactions in concrete stop or diffusivity decrease by more than an order of magnitude as the rel- drastically slow when the relative humidity in pores drops below ative humidity, h, in the pores decreases. This means that the linear 80%. For instance, the destructive alkali-silica reaction virtually diffusion theory is unable to fit the measurements of pore relative stops at h < 75% (Bažant and Rahimi-Aghdam 2016; Rahimi- humidity, which is, for example, documented by the poor optimum Aghdam et al. 2017a). During the last 2 decades, several self- fits of the experimental data in Fig. 1. desiccation models were proposed (Bentz 1997; van Breugel 1997; Hubler et al. 2015b), but they fit long-term data poorly, and were complicated by a number of empirical parameters re- 1Graduate Research Assistant, Dept. of Civil and Environmental Engi-

Downloaded from ascelibrary.org by Northwestern University Library on 03/06/19. Copyright ASCE. For personal use only; all rights reserved. quiring experimental calibration for different concretes, or were neering, Northwestern Univ., Evanston, IL 60208. 2Graduate Research Assistant, Dept. of Civil and Environmental Engi- computationally too expensive for use in finite-element analysis. neering, Northwestern Univ., Evanston, IL 60208. In addition, the existing models were not conceived as part of a 3McCormick Institute Professor and W.P. Murphy Professor of general model for water transport in concrete. Such a model must Civil and Mechanical Engineering and Materials Science, Northwestern combine self-desiccation with a simultaneous exposure to drying Univ., 2145 Sheridan Rd., CEE/A135, Evanston, IL 60208 (corresponding environment, and this is the main goal of this study. author). Email: [email protected] 2. The second objective is to use the existing database, significantly Note. This manuscript was submitted on March 11, 2018; approved on extended since 1972, to recalibrate the decrease of permeability October 3, 2018; published online on March 6, 2019. Discussion period open until August 6, 2019; separate discussions must be submitted with decreasing h. Better fits of test data can be obtained when for individual papers. This paper is part of the Journal of Engineering the permeability decrease with h is more gradual than in the Mechanics, © ASCE, ISSN 0733-9399. original BN model, and continues even for h < 50%.

J. Eng. Mech., 2019, 145(5): 04019032 (a) (b)

Fig. 1. Best possible fits of midcylinder evolutions and transverse distributions of relative humidity in pores, achievable with linear diffusion theory (characterized by constant permeability and diffusivity). y = distance from drying surface; and t 0 = drying exposure time. [Humidity data measured by (a) Kim and Lee 1999; (b) Abrams and Monfore 1965; (c) Abrams and Orals 1965; (d) Hanson 1968.]

3. The third objective is to improve the desorption isotherm. phenomena such as ettringite formation, portlandite formation, Its slope in the BN model was considered to be constant for h crystallization pressure, C-S-H shells pushing each other apart, greater than about 30%, but recent tests (Nilsson 2002; Nielsen disjoining pressure, and nanoscale creep. 1991; Xi et al. 1994) showed that this is not true for high w=c. 2. Like the drying shrinkage (caused by external drying), the auto- Models to capture it exist (Xi et al. 1994; Daian 1988) but ap- genous shrinkage also is caused by a decrease of pore humidity pear to be unnecessarily complicated, inaccurate, and computa- (or of the corresponding chemical potential of water) (Hua et al. tionally demanding. It is sufficiently accurate and simpler to 1995; Lura et al. 2003; Gawin et al. 2006; Grasley and Leung consider the isotherm to consist of two straight-line segments, 2011; Luan and Ishida 2013). Therefore, both must follow the the upper one being much steeper than the lower one and having same law. a length depending on the type of concrete, particularly on the When external drying occurs, the self-desiccation cannot be w=c. The steepness and length of the initial straight segment of modeled independently. Rather, it must be considered as coupled. the isotherm near h ¼ 1 are particularly important for calculat- For clarification, consider the right half of a concrete slab of width ing the self-desiccation. 2 L, exposed to drying at environmental humidity henv that is below 4. The fourth objective is to propose a simple approximate formula the maximum possible humidity decrease due to self-desiccation. for predicting the four parameters of the equation giving the All four sides of the slab are assumed to be sealed, so that the moisture permeability as a function of h. This is needed to en- drying would be unidirectional. able realistic calculation of the pore humidity field in a structure, Before exposure to drying, the relative humidity in the pores which must be known for predicting autogenous shrinkage, dry- decreases uniformly due to self-desiccation [Figs. 2(a and e)]. In ing shrinkage, creep, rate of aging, and various deteriorative re- this study, the initial humidity value minus the humidity decrease actions such as the ASR. Furthermore, this is needed to extract due to self-desiccation is called the self-desiccation humidity. After more useful information from the existing creep and shrinkage drying exposure, the humidity profile of drying propagates into data, for example, from the worldwide database of about 4,000 the slab, whereas the self-desiccation humidity decreases only in creep and shrinkage tests assembled at Northwestern University that portion of cross section where its relative humidity has not ž (Ba ant and Li 2008; Hubler et al. 2015a). yet reached the relative humidity at which hydration stops, hst In the literature, the autogenous shrinkage and the drying [Figs. 2(b, c, e, and f)]. Eventually, the drying front reaches the

Downloaded from ascelibrary.org by Northwestern University Library on 03/06/19. Copyright ASCE. For personal use only; all rights reserved. shrinkage due to environmental exposure have generally been re- slab center [Figs. 2(c, d, and h)]. After that, the humidity due to garded as separate additive phenomena, the former caused directly external drying separates from the self-desiccation humidity and by a volume change in the chemical reaction. But were this true, it decreases slower than the drying humidity profile, which causes would be impossible to explain, e.g., that (1) the lowest rather than a growing separation of the profiles of self-desiccation and of the highest w=c leads to the greatest autogenous shrinkage, or that external drying [Figs. 2(d and h)]. (2) under water immersion, concrete swells. From such arguments The area between the aforementioned two profiles, multiplied it was concluded (Bažant et al. 2015) that by the inverse isotherm slope (and weighted for axisymmetry), rep- 1. Contrary to prevailing opinion, the hydration reaction per se resents the amount of water lost to the environment, and the area is always expansive, beginning as soon as the solid nanoskele- above the upper profile, multiplied and weighted similarly, repre- ton of C-S-H is formed (i.e., after the first day of hydration). sents the amount of water consumed by the hydration reaction. – Swelling can be due to several hydration reaction related When henv is not smaller than the minimum possible humidity

J. Eng. Mech., 2019, 145(5): 04019032 (a) (b) (c) (d)

(e) (f) (g) (h)

Fig. 2. Schematic evolution of profiles of relative humidity (RH) caused by self-desiccation and by drying in the right half of a thin specimen: (a) uniform profile prior to drying exposure; (b) soon after that; (c) later; and (d) very long after drying exposure; and by drying in the right half of a thick specimen: (e) uniform profile prior to drying exposure; (f) soon after that; (g) later; and (h) very long after drying exposure.

reachable by self-desiccation, hst, the stages in Figs. 2(c, d, and h) 3. It can be verified and calibrated more directly, because of the are never reached. recent development of good pore humidity gauges (Nguyen et al. 2018). The Bažant-Najjar model, which has been embodied in the Moisture Transport Model Code 2010 (Fédération internationale du béton 2012), directly postulates that, under uniform temperature, the total mois- Thermodynamic equilibrium of moisture in concrete is achieved ture flux is when the chemical potential of water, μ (i.e., the Gibbs free energy ¼ − ð Þ∇ ð1Þ per unit mass), is the same in all phases of water and throughout jw cp h h the structure volume. Therefore, in principle, the mass flux of where c = moisture permeability (kg=m · s); and j = mass flux of water, j , should be proportional to grad μ. Such a diffusion p w w water (kg=m2 · s). model, however, seems inconvenient and apparently has never To ensure water mass conservation, the rate of change of mois- been developed. ture content, w˙ , must balance the influx of water mass, that is Until the 1960s, one approach was to consider the flux of mois- t ture to obey Fick’s law and to be proportional to grad w , where w ˙ ˙ e e wt ¼ −∇ · jw þ ws ð2Þ is the evaporable water mass per unit volume of concrete. Bažant

Downloaded from ascelibrary.org by Northwestern University Library on 03/06/19. Copyright ASCE. For personal use only; all rights reserved. ž ˙ and Najjar (1971) proposed and Ba ant and Najjar (1972) elabo- where ws = water mass that has been withdrawn from the pores ˙ rated a model in which the pore relative humidity, h, is considered and combined chemically by the hydration process; and ws = as the primary unknown, whose gradient drives the diffusion (and is distributed sink and is the cause of self-desiccation. proportional to both grad μ and grad pv times a factor dependent on Under local isothermal equilibrium conditions, the pore humid- h, where pv is the vapor pressure). ity rate is a function of the rate of specific evaporable water content, 3 The use of grad h has three advantages wt (kg=m concrete). This function differs between drying and 1. It is simpler, especially when the self-desiccation is mild, for wetting conditions, due to the capillary and hindered adsorption dealing with the effect of hydration, which acts as a sink with- hysteresis. In the literature, the functions describing the dependence drawing a great deal of evaporable water from the pores; of h on wt for drying and wetting are called the desorption and 2. It is more direct for formulating the environmental boundary resorption isotherms. Only the desorption isotherm is considered condition; and here, and is formulated as

J. Eng. Mech., 2019, 145(5): 04019032 (a) (b)

Fig. 3. Permeability dependence on pore relative humidity: (a) as used in Bažant-Najjar (1972) model; and (b) proposed model.

˙ ˙ = h ¼ kðα; hÞwt ð3Þ w c, for which the pores are larger, the permeability begins to decrease at a higher relative humidity. The hydration reaction causes where kðα; hÞ = reciprocal moisture capacity (m3=kg), i.e., the in- the pore sizes to decrease as a function of age or, more precisely, the verse slope of the isotherm, whose evolution depends on the pore hydration degree. The data fitting led to the following empirical relative humidity and hydration degree, α (characterizing the age of relation: concrete); and k varies with h, more so for concretes with high w=c α (see section “Sorption Isotherm”). 1=2 u hc ¼ 0.77 þ 0.22ðw=c − 0.17Þ þ 0.15 − 1 ; hc < 0.99 Incorporating Eqs. (1), (2), and (3) gives the governing moisture α diffusion equation for concrete ð7Þ ∂ ∂ h hs The curing condition can also affect the value of h , whereas ¼ kðα; hÞ∇ · ðc ∇hÞþ ð4Þ c ∂t p ∂t Eq. (7) is calibrated for the sealed curing condition. This value should be modified if the specimen is cured in a fog room or im- where the last term on the right-hand side is a distributed sink rep- mersed in water, because in the immersed condition, for the same resenting the self-desiccation. It can be calculated using Rahimi- hydration degree (and approximately the same pore structure), the ’ Aghdam et al. s(2017b) hydration model, and it can continue relative humidity is higher. In this study, for simplicity, for water for many years. As in the Bažant-Najjar (1972) model, the depend- immersion, the hc value is increased by the same amount as the h ence of moisture permeability cp on h may be expressed as follows increase at the start of drying, compared with the sealed condition. [Fig. 3(a)]: For instance, if, at a certain drying time, the relative humidity at 1 − β sealed condition is 0.96 and for water immersion is 1, the value ð ; αÞ¼ β þ ð5Þ 1 − 0 96 cp h c1 1− of hc is also increased 0.04 ( . ). This modification is more 1 þ h r = = 1−hc pronounced for lower w c and is almost negligible for high w c. Parameter β gives the maximum relative decrease of permeabil- β although the values of parameters c1, , hc, and r are somewhat ity as h decreases. Based on more limited data, Bažant and Najjar different, because they are obtained by the fitting of more-extensive (1972) assumed its value as 0.05, which was reached as h decreased experimental data. to approximately 0.7. Later experiments, however, indicated that The data needed to calibrate Eq. (5) should ideally include the permeability decreases further as h decreases below 0.7 (Fig. 4). time evolution of relative humidity in the pores and of the humidity Therefore, β is now changed to a variable parameter defined as profiles throughout the cross section at subsequent times, for vari- follows: ous specimen sizes, various environmental humidities, and various β ¼ = ð8Þ ages at exposure to drying atmosphere. Such comprehensive data, cf c1 unfortunately, do not usually exist for one concrete, and the param- 0 ¼ 60½1 þ 12ð = − 0 17Þ2α=α ð9Þ eters change significantly from one concrete to another. To be able cf w c . u to extract information from data for different concretes, the equa- ( 0 ð > Þ tions for the dependence of unknown material parameters in Eq. (5) cf h hs c ¼ ð10Þ had to be judiciously assumed and then empirically calibrated. f 0 1 0 þ 0 9 0 ð = Þ4 ð Þ . cf . cf h hs h < hs The basic parameter is c1, i.e., the permeability at saturation ¼ 1 (h ). It is mainly a function of the pore size distribution. Two where αu = ultimate hydration degree in sealed concrete, which is a Downloaded from ascelibrary.org by Northwestern University Library on 03/06/19. Copyright ASCE. For personal use only; all rights reserved. factors that affect the distribution most are the water-cement ratio, function of w=c; it is estimated as w=c, and the hydration degree, α. The fitting test data led to the 0.6 following empirical relation: αu ¼ 0.46 þ 0.95ðw=c − 0.17Þ ; αu < 1 ð11Þ

2 c1 ¼ 60½1 þ 12ðw=c − 0.17Þ α=αu ð6Þ The last parameter to define is r; it determines how steeply the permeability decreases with h. Analysis of the data available in The second important parameter is the transitional humidity, hc, 1972 indicated exponent r to be between 6 and 12, which gave at which the steep permeability decrease is centered. When w=c a rather steep permeability decrease (Fig. 3). However, based increases, the pore-size distribution shifts toward larger pores. on newer and more extensive data, r ¼ 2, i.e., the permeability As the drying starts, the larger pores are emptied first (because decrease is much smoother. Fig. 4 compares the old and new per- larger pores empty at a higher humidity). Therefore, for a higher meability equations.

J. Eng. Mech., 2019, 145(5): 04019032 (a) (b)

Fig. 4. Dependence of permeability on relative humidity. [Data from (a) Sørensen et al. 1979; (b) Nilsson 1980.]

(a) (b)

Fig. 5. Bažant-Najjar model for estimating humidity evolution and shrinkage using two different sets of calibration parameters: (a) predicted relative humidity; and (b) predicted shrinkage.

The predictions of pore humidity and drying shrinkage are quite in a dimensionless form by the saturation degree, S ¼ we=ws,asa sensitive to the shape of permeability law [Eq. (5)]. Figs. 5(a and b) function of h (where ws is the we at h ¼ 1). presents the humidity profiles and shrinkage strain evolutions, in The desorption isotherm is nonlinear (Nilsson 2002; Nielsen which two of the four parameters of permeability law are changed 1991; Xi et al. 1994; Bažant and Jirásek 2017), and highly so when in correspondence to the values commonly used in the literature. h is near 0 and near 1. Its shape depends on the concrete type. This makes a large difference. The most important parameters are the water-cement ratio and the age. Some tests showed that certain admixtures affect the desorption isotherm significantly. Adding silica fume fills the nanopores Sorption Isotherm and thus makes the matrix more uniform, which in turn makes the isotherm slope near h ¼ 1 less steep (Baroghel-Bouny et al. 2006; The sorption isotherm is defined as the dependence of specific Jiang et al. 2006). Various methods to measure the desorption iso- 3 evaporable water content, we (kg=m concrete at constant temper- therm have been proposed, but they either are too complicated to ature), on the relative pore humidity, h. Drying depends on the use or necessitate several calibration parameters that are hard to desorption isotherm. Because of pronounced capillary hysteresis, determine. the resorption (or adsorption) isotherm, which governs wetting, lies Based on numerous experimental studies, the desorption iso- significantly below the desorption isotherm. We focus on drying therm may be approximated by a linear asymptote dropping from and seek a simple equation for the desorption isotherm. To be appli- h ¼ 1 with a steep slope, m1, followed by a long linear segment cable to concretes of different porosities, the isotherm is described of milder slope m2 [Fig. 6(a)], extending down to about h ¼ 50%. Downloaded from ascelibrary.org by Northwestern University Library on 03/06/19. Copyright ASCE. For personal use only; all rights reserved.

(a) (b)

Fig. 6. Desorption isotherm: (a) considered simplified desorption isotherm; and (b) experimental versus simulated desorption isotherm (experimental data from Nilsson 2002). Lines = authors’ simulated results; and points = experimental results.

J. Eng. Mech., 2019, 145(5): 04019032 The proposed isotherm is a smooth transition between two linear Table 1. Molar volume and density of calcium-silicate phases present in asymptotes [Fig. 6(a)] cement paste 3 3 dh Phase Molar volume (cm =mol) Density (g=cm ) dS ¼ ð12Þ kðh; αÞ C3S 72.9 3.15 C2S 52.7 3.28 1 m1 − m2 H2O 18.0 1.0 m2 þ ð13Þ ð ; αÞ 1−h 2 C-S-H 110.1 2.05 k h 1 þ 1− h CH 33.1 2.24 where kðh; αÞ = inverse slope of the desorption isotherm; and h = intersection point of the two linear asymptotes. These parameters can be empirically estimated using experimental desorption iso- chemistry notation in which C, S, A, and F denote, respectively, therms. To do so, we use the tests of Nilsson (2002), who reported CaO, SiO2,Al2O3, and Fe2O3 (Taylor 1997)]. C3S and C2S are the desorption isotherm for various w=c ratios at different ages. two major components of OPC, and their ratio depends on the ce- Their fitting by the present model yields the following empirical ment type. To quantify the amount of water consumed by hydra- relations: tion, we consider, for simplicity, only C3S and C2S and we assume the cement to be composed of 80% C3S and 20% C2S. Chemically, fα 1.5 m1 ¼ ½2.4 þð5.26 w=c − 0.68Þ ð14Þ the hydration reaction of C2S and C3S is summarized as follows cSF (Qomi et al. 2014): ¼ 0.51 18ð = Þ0.4 ð15Þ m2 cSF . w c þ 2 1 → þ 0 3 C2S . H C1.7SH1.8 . CH

0.03 C3S þ 3.1H → C1 7SH1 8 þ 1.3CH ð19Þ h ¼ 1 − fc ð16Þ . . SF ðw=cÞ2 where C1.7SH1.8 and H constitute the typical C-S-H and water αu − α0 found in OPC pastes. Table 1 summarizes the molar volumes and fα ¼ ð17Þ α − α0 densities of different components in the hydration reactions. When analyzing the aforementioned hydration reactions, the C-S-H pores Parameter cSF accounts for the silica fume, in the absence of are assumed to be empty, and so it is necessary to include additional which cSF ¼ 1. More generally, for a finite specific content ξSF of water that is trapped in the nanopores of C-S-H. silica fume (mass of silica fume/mass of cement) Constantinides et al. (2003), Jennings (2000), and Tennis and Jennings (2000) established that the hydration reaction produces ¼ 1 þ 2ξ ð18Þ cSF SF two types of C-S-H: low-density C-S-H (LD), with porosity 36%; and high-density C-S-H (HD), with porosity 26%. Tennis and Fig. 6(b) compares the preceding equations (solid curves) with = ’ Jennings (2000) showed that, for different w c values, the ratio of Nilsson s(2002) test data. The agreement was satisfactory. The these two C-S-H types varies. For instance, their model predicted effect of temperature on the isotherm reported in some studies that, for w=c ¼ 0.45, the proportions are 50% HD and 50% LD (McLaughlin and Magee 1998; Poyet 2009) was minor, except at and, for w=c ¼ 0.25, the proportions are 80% HD and 20% LD. high temperatures, and is here neglected. Therefore, the average porosity of C-S-H, ϕgp, depends mainly on w=c. Here, for simplicity, we assume a linear relation limited by lower and upper bounds, as follows: Self-Desiccation 0.27 < ϕgp ¼ 0.28 þ 0.20ðw=c − 0.3Þ < 0.35 ð20Þ According to RILEM TC 196-ICC (Jensen 2007), the self- desiccation is defined as the reduction in the internal relative hu- The lower and upper bounds on ϕgp are introduced because it midity of sealed concrete. It is explained by removal of water for does not suffice to consider only the LD or HD phases. hydration and doubtless leads to enlargement of already existing Having quantified the hydration relation and calculated the vapor-filled pore space (as the capillary menisci recede into nar- porosity of C-S-H, we can calculate the total volume of water, ξwc, ower pores) rather than cavitation of new vapor bubbles within per unit volume of cement that is consumed by hydration to pro- liquid water. The volume and size of the empty pores depends duce a C-S-H gel with the empty pores, ξbwc, and the volume of chiefly on the water-cement ratio, the degree of hydration, the water that fills the gel pores, ξfwc particle-size distribution of cement, and the type of admixtures. ξ ¼ ξ þ ξ ð21Þ For lower w=c and finer cements, the microstructure is denser wc bwc fwc and the average pore size is smaller. A smaller pore size intensifies self-desiccation, i.e., causes a higher rate and a lower final value ξ ¼ ξC2S þ ξC3S ð22Þ bwc wC2S bwc wC3S bwc Downloaded from ascelibrary.org by Northwestern University Library on 03/06/19. Copyright ASCE. For personal use only; all rights reserved. of relative humidity, hst. The reason is that, in smaller pores, the meniscus curvature is higher and thus the pressure, given by the ξ ¼ ϕ ξ S ð23Þ Laplace equation, is higher, which makes the desorption isotherm fwc gp gc gp at high relative humidity values less steep. The decrease of pore where ξ = volume of C-S-H gel produced (per unit volume of size also explains why adding fine admixtures, such as silica fume, gc consumed cement); S = saturation degree of gel pores; w intensifies the self-desiccation. gp C2S To calculate the self-desiccation correctly, one should begin and wC3S = volume ratios of C2S and C3S, respectively; and ξC2S ξC3S with the amount of water consumed by hydration. Ordinary Port- bwc and bwc = volumes of water that is consumed to hydrate a unit land cement (OPC) consists of various phases such as alite (C3S), volume of C2S and C3S, respectively, and to produce the C-S-H gel belite (C2S), calcium aluminate (C3A), tetracalcium aluminoferrite with empty pores. For the typical case of cement consisting of 80% (C4AF), and gypsum, as well as minor other phases [we use cement C3S and 20% C2S

J. Eng. Mech., 2019, 145(5): 04019032 (a) (b)

Fig. 7. Experimental versus predicted hydration degree. Lines = present simulated results; and points = experimental results. [Experimental data from (a) Bentz 2006; (b) Danielson 1962.]

ξbwc ¼ 0.2ð37.8=52.7Þþ0.8ð55.8=72.9Þ¼0.755 account the temperature and cement fineness. The algorithm of this new model, with the aforementioned modification, is presented in ξgc ¼ 0.2ð110=52.7Þþ0.8ð110=72.9Þ¼0.487 the Appendix. Fig. 7 shows the predicted evolution of hydration degree using Because the gel pores are much smaller than the capillary pores, the presently modified version of Rahimi-Aghdam et al.’s(2017b) in the literature they have always been considered as saturated. hydration model. The model predicts the hydration degree well, However, Rahimi-Aghdam et al. (2017b) showed that this cannot both short-term and long-term. In addition, this model, unlike pre- be true and that at least some of the gel pores must be considered as vious models, predicts the hydration reaction to be a long-lived phe- unsaturated. Because gel pores are smaller than capillary pores, nomenon decaying roughly logarithmically (even for decades, if h their average saturation degree should be higher than that of normal does not decrease). This agrees with the long-term self-desiccation capillary pores. To calculate the average saturation degree of gel experiments. pores, we assume 33% of gel pores to have the same degree of Fig. 8 demonstrates the experimental versus simulated compar- saturation as the capillary pores, and 67% to have the same degree isons for the relative humidity change due to the hydration reaction of saturation as the nanopores (which, at high relative humidities, in sealed specimens. The results again agree with the experiments are always saturated). Accordingly, for a high relative humidity, we quite well. By virtue of the varying slope of the desorption isotherm, assume the saturation degree of gel pores to be they predict correctly the initial relative humidity, which is impor- S ¼ 0.67 þ 0.33S ð24Þ tant for autogenous shrinkage and swelling. The calculated relative gp cp humidity change due to hydration depends strongly on the w=c and is greater for concretes with a low water content. On the other hand, the self-desiccation of concretes with a high w=c at early ages is Hydration Model predicted to be mild, as expected. Although several experimental studies (Halamičková et al. 1995; The aforementioned self-desiccation curves are true predictions; Ji 2005) demonstrated a significant effect of aging on concrete per- no parameter had to be calibrated. In addition, considering various = meability, this effect has usually been omitted from the equations C-S-H gel porosities for concretes with different w c helped sig- for drying. This omission may be acceptable when the environmen- nificantly in matching the observed self-desiccation. tal exposure begins at a late age and high hydration degree, but it As the preceding figures show, the proposed hydration model can cause a considerable error for concretes that begin to dry at an predicts well both the hydration degree and the self-desiccation. early age. Because the present model is time consuming to code, and because Therefore, the effect of the age at drying exposure is here taken precise hydration predictions often are not necessary for obtaining into account. Although several hydration models took the age effect the correct humidity profiles, simplified empirical relations are here into account (Jennings and Johnson 1986; Bentz 1997; van Breugel proposed and calibrated using the experimental results of Bentz 1997; Lin and Meyer 2009), they were, unfortunately, too compli- (2006), as follows: cated and computationally demanding for use in finite-element 2 dα ¼ c ðα − αÞ ðh − h Þdt ð25Þ structural analysis. In addition, the previous hydration models ap- a u f plied only to concrete that is sealed or immersed in water. α − α nwc The only model that is computationally undemanding and usable ¼ f set ð26Þ hs hs α − α for all environmental conditions and ages appears to be the model of u set Rahimi-Aghdam et al. (2017b). This was achieved by considering Downloaded from ascelibrary.org by Northwestern University Library on 03/06/19. Copyright ASCE. For personal use only; all rights reserved. the relative pore humidity, h, or the corresponding chemical poten- where nwc ¼ 25ðwc − 0.1Þ; tial μ ¼ μ0 þðRT=MÞ ln h, as the driving force of the hydration f ¼ 0 5ð = Þ−0.26 − 0 53; process (where T is absolute temperature, M is molecular mass hs . w c . and μ 0.6 of water in moles, R is the universal gas constant, and 0 is reference αu ¼ 0.46 þ 0.95ðw=c − 0.17Þ < 1 ð27Þ chemical potential). In addition, this model is able to predict the self- desiccation without any extra calibration parameter. where hf = relative humidity at which the hydration reaction stops, ¼ 0 75 α Therefore, we adopt here the model by Rahimi-Aghdam et al. and for normal concretes it can be set to hf . ; and set = (2017b), albeit with a slight modification for multidecade hydra- hydration degree at the time of set, which can be calculated as α ¼ 0 05 þ 0 2ð = − 0 3Þ tion, which is needed to describe multidecade autogenous shrink- set . . w c . (according to the hydration model, age and swelling. The model is general, can be applied to different for normal temperature and normal cement). These empirical rela- cement types upon adjusting only a few parameters, and takes into tions are calibrated for concretes without admixtures, the presence

J. Eng. Mech., 2019, 145(5): 04019032 (a) (b)

Fig. 8. Experimental versus predicted self-desiccation. Lines = present simulated results; and points = experimental data. [Experimental data from (a) Jiang et al. 2006; (b) Kim and Lee 1999.]

(a) (b) (c)

Fig. 9. Predicted hydration degree and self-desiccation using simplified hydration model: (a) predicted versus experimental hydration degree for experiment by Bentz (2006); and predicted versus experimental self-desiccation for experiments by (b) Jiang et al. (2006); and (c) Kim and Lee (1999). Lines = present simulated results; and points = experimental results.

of which changes the rates of hydration degree and of self- (2005) used a boundary condition of slightly different form, ð Þ½∂ =∂ ¼ ð − Þ desiccation significantly. C h h r fs h hcavity , where the surface factor, fs, was The proposed simple equations can be integrated numerically by considered to be approximately 10−8 − 10−7 m=s. This is roughly any method and suffice for capturing the effect of relative humidity equivalent to an evaporation rate of 1.1 × 10−6 to 11× 10−6 kg=m2s. on the rate of hydration. Fig. 9 shows that these equations predict Parameter η has a rather uncertain value. Furthermore, it correctly the evolutions of hydration degree and self-desiccation. adds complexity and hardly leads to better predictions. Therefore, However, they do not suffice for predicting the autogenous shrink- Bažant and Najjar (1972) proposed replacing Eq. (28) for the boun- age. The reason is that the autogenous shrinkage is highly sensitive dary condition with a certain equivalent extra surface thickness, to an initial decrease of relative humidity, and a small error in pre- δheq, of an imagined added layer of concrete on whose surface dicting the self-desiccation and hydration can cause a large error in the pore humidity equals the environmental value. Comparing ana- predicting the autogenous shrinkage (Bažant and Rahimi-Aghdam lytical and experimental results, they adopted the value δheq≈ 2018; Rasoolinejad et al. 2018). To avoid major errors, the com- 0.75 mm. This method also was used to calculate the shape factors plete hydration model must be used for that purpose. for concrete shrinkage and drying creep (Dönmez and Bažant 2016). For simplicity, δheq is also used here, but with a different value of about 4 mm, as estimated from the present calculations. Boundary Conditions The error of this approximation was found to have little effect and decreases as drying continues. Unless the specimen or structure is very large, the surface moisture transmission between the concrete and environment cannot be considered as instantaneous. The relative humidities in the environment and in the concrete pores at the surface are different. In terms of Comparison of Numerical and Experimental Results Downloaded from ascelibrary.org by Northwestern University Library on 03/06/19. Copyright ASCE. For personal use only; all rights reserved. relative humidity, the boundary condition is (Bažant and Najjar The numerical results of the present model were first compared 1972) with the humidity profiles from the classical tests originally used ¼ ηð − Þð28Þ ž jw hc henv for calibration by Ba ant and Najjar (1972) (Figs. 10 and 11), and had a rather close agreement. The properties of the concrete used in η where jw = moisture flux across the surface; and = surface these tests are given in Table 2. In Fig. 11, the humidity profile 2 emissivity (kg=m s). Various values of η are found in the literature. indicated a decreasing humidity near the center. These experiments −6 2 For instance, Bažant and Najjar (1972) suggested 85× 10 kg=m s were performed under unidirectional drying (two parallel faces ex- −6 2 for indoor (or still) air and 350 × 10 kg=m s for outdoor (moving) posed and four faces sealed). The decrease of relative humidity near air. Ali and Urgessa (2014)used6 × 10−6 kg=m2s for indoors the center was probably caused by breakage of the seal near the and 9 × 10−6 kg=m2s for an isolated space. West and Holmes center.

J. Eng. Mech., 2019, 145(5): 04019032 (a) (b) (c)

Fig. 10. Relative humidity profiles in the cross-section for different drying times (t 0): (a) t 0 ¼ 3, 28, and 365 days; (b) t 0 ¼ 7, 90, and 730 days; and (c) t 0 ¼ 14, 180, and 1,100 days. Lines = present simulated results; and points = experimental results from Hanson (1968).

(a) (b)

Fig. 11. Relative humidity profiles in the cross-section for different drying times (t 0): (a) t 0 ¼ 40 and 130 days; and (b) t 0 ¼ 50 and 270 days. Lines = present simulated results; and points = experimental results from Abrams and Monfore (1965).

Table 2. Material parameters for analyzed experiments Thickness or Exposure Environmental = = 0 Figure Study w cac Shape diameter time, t humidity, henv Remarks Fig. 10 Hanson (1968) 0.66 6.95 Cylinder 6 in. 7 days 0.5 Elgin sand and gravel Fig. 11 Abrams and Monfore (1965) 0.45 4.36 Slab 12 in. 7 days 0.1 7 bags cement=yd3 Fig. 12 Kim and Lee (1999) 0.28, 0.4, 0.68 3.15, 4.14, 5.6 Slab 20 cm 3 and 28 days 0.6 Moist cured Fig. 13 Laurens et al. (2002) 0.66 6.95 Slab 7 cm 2 days 0.55 Small aggregates Fig. 14 Grasley and Lange (2004) 0.44 0.44 Slab 20 cm 1 day 0.5 Silica fume has added

Next, we consider the comprehensive experiments of Kim and different in some reported experiments, as seen from the time Lee (1999). Concrete specimens were exposed to drying at ages of evolution of pore humidity in, e.g., Laurens et al.’s(2002) experi- 3 and 28 days. Fig. 12 demonstrates again a satisfactory agreement ments. Fortunately, to fit other data, it is not necessary to change all with the pore humidity profiles for different depths below the dry- the coefficient values. Mere scaling of the permeability law is suf- ing surface, y, and for both exposure times. The specimens were ficient. The data of Laurens et al. (2002) can be closely matched if cured under water immersion before drying exposure, and thus the the permeability is reduced by 35% compared with the preceding relative humidity prior to drying exposure was higher than in the simulations (Fig. 13). sealed specimens. The equation proposed in this study for calcu- Furthermore, consider pore humidity predictions for concretes lating hc, Eq. (7), was calibrated for the sealed condition. However, with silica fume (SF) as an admixture, which has a major effect on

Downloaded from ascelibrary.org by Northwestern University Library on 03/06/19. Copyright ASCE. For personal use only; all rights reserved. in the case of curing under water immersion, it needs to be modi- permeability (Jiang et al. 2006; Baroghel-Bouny et al. 2006)by fied. Here, for simplicity, the value of hc in Eq. (7) was simply refining the pore structure due to filling the pores and pozzolanic increased by an amount equal to the humidity difference between effect (Bentz et al. 2000). Refining pores causes the pore humidity the sealed and immersed conditions. This change was negligible for to decrease faster and to lower values. In addition, SF reacts with drying that began 3 days after mixing, but for drying that began portlandite, producing more C-S-H and a denser structure, which after 28 days, the initial increase of hc at 28 days was about results in a faster decrease of pore relative humidity. Fig. 14(a) 2% for the finite elements near the surface. compared to the self-desiccation for concretes with and without the All the preceding simulations of experimental data used the per- SF. The SF effect on the self-desiccation was predicted correctly. meability law [Eq. (5)] with the same coefficient values. However, The SF effect is modified in specimens exposed to the environ- permeability varies widely, depending on the aggregate size and ment. This is evident from the plots of predicted versus measured type, cement type, casting technique, and so on. It was clearly quite relative humidity in Fig. 14. They indicate a good agreement.

J. Eng. Mech., 2019, 145(5): 04019032 (a) (b) (c)

(d) (e) (f)

Fig. 12. Evolution of relative humidity versus drying time (t 0) for different depths below the drying surface (y): (a–c) specimens immersed in water for 3 days and then exposed to drying; and (d–f) specimens immersed in water for 28 days and then exposed to drying. Lines = present simulated results; and points = experimental results from Kim and Lee (1999).

(a) (b)

Fig. 13. Relative humidity profiles in the cross section for different drying times (t 0): (a) t 0 ¼ 15, 48, and 139 days; and (b) t 0 ¼ 26, 100, and 187 days. Lines = simulated results; and points = experimental results from Laurens et al. (2002). Downloaded from ascelibrary.org by Northwestern University Library on 03/06/19. Copyright ASCE. For personal use only; all rights reserved.

(a) (b)

Fig. 14. Evolution of relative humidity versus drying time (t 0): (a) comparison of self-desiccation for specimens with and without silica fume (experiment by Jiang et al. 2006); and (b) evolution of relative humidity versus drying time for specimens at different depths from the drying surface (y), all with silica fume. Lines = simulated results; points = experimental results from Grasley and Lange (2004).

J. Eng. Mech., 2019, 145(5): 04019032 (a) (b)

Fig. 15. Predictions of humidity evolution in exposed specimens using: (a) complete hydration model; and (b) simplified hydration model. Lines = simulated results; and points = experimental results from Kim and Lee (1999).

Some experimenters (e.g., Jiang et al. 2006) studied admixtures 2. Calculate the average cement particle size (i.e., particle radius) other than the SF. However, their effect was found to be very small, a0, based on the cement type. In this study, the Blaine fineness 2 and is here neglected. of cement, fbl, 350 m =kg, was considered to correspond to a Finally, the validity of the simplified hydration model for particle radius of 6.5 μm. Then calculate the number of cement exposed specimens needs to be checked. This model was able to particles, ng, per unit volume of cement predict closely the self-desiccation and hydration degree in sealed 350 concrete (Fig. 9). Checking the model for exposed concrete, the a0 ¼ 6.5ðμmÞ ð31Þ simplified hydration model yielded almost the same predictions fbl as the full model, except for a small error during the initial days c (Fig. 15). The reason for this error was the slope change of the ¼ V0 ð32Þ n 4 3 desorption isotherm due to the drying process. This change cannot 3 πa0 be captured by the simplified linear isotherm. Overall, however, the predictions were quite good, and thus the present simplified hydra- 3. Choose a reasonable hydration degree for setting the time α α tion relation appears to be satisfactory. set, and the time c at which the C-S-H barrier will be com- pleted, i.e., the critical hydration degree at which a complete C-S-H barrier will form around the anhydrous cement grains n Summary and Conclusions (about 1 day). For a normal cement with a0 ¼ 6.5 μm, w=c ¼ 0 3 ¼ 20 α0 ¼ 0 05 α ¼ 0 36 . , and T °C, the values set . and c . are 1. An improved equation for the dependence of self-desiccation good approximations. For specimens with different T, w=c rate on pore humidity and hydration degree was here formulated and cement type, reasonable values may be calculated as and calibrated by experimental data; follows: 2. The 1972 Bažant-Najjar model for the strong decrease of moist- 0 ure permeability with decreasing pore humidity was improved αc ¼ αcf1ðaÞf2ðw=cÞf3ðTÞ < 0.65 ð33Þ and calibrated with recent experimental data; n a0 3. Equations to predict the four parameters for the permeability f1ðaÞ¼ ð34Þ variation from the material composition, water-cement ratio, a

and hydration degree were formulated and calibrated with f2ðw=cÞ¼1 þ 2.5ðw=c − 0.3Þð35Þ experimental data; and 4. The desorption isotherm as a function of pore relative humidity 1 1 ð Þ¼ Eα − ð36Þ was improved by a nonlinear isotherm with a steep top segment f3 T exp R 273 þ T0 273 þ T whose slope and length depend on the water-cement ratio, α α according to experimental calibration. set ¼ c ð37Þ α0 α0 set c

c Appendix. Algorithm for Evolution of Hydration, 4. Calculate the volume fraction of cement, Vset; portlandite, CH g Autogenous Shrinkage, and Swelling Vset ; and gel (C-S-H plus ettringite), Vset. Using these fractions, calculate the radius of the anhydrous cement remnants, 1. For cement paste or concrete with known water-cement and aset, and the outer radius of C-S-H barrier, zset. To describe the aggregate-cement ratios, calculate the initial volume fraction ζ ¼ Downloaded from ascelibrary.org by Northwestern University Library on 03/06/19. Copyright ASCE. For personal use only; all rights reserved. chemical reaction of hydration, use the volume ratios gc of cement Vc and water Vw 0 0 1.52 and ζCHc ¼ 0.59 in the following equations: ρ ρ c ¼ a w ð29Þ c ¼ð1 − α Þ c CH ¼ ζ α c g ¼ ζ α c V0 ρ ρ þ ρ ρ = þ ρ ρ = Vset set V0 Vset CHc setV0 Vset gc setV0 a w c wa c c aw c ð38Þ ρ ρ w=c w a c g V0 ¼ ð30Þ c 1=3 c þ 1=3 ρ ρ þ ρ ρ a=c þ ρ ρ w=c ¼ Vset ; ¼ Vset Vset ð39Þ a w c w c a aset 4 zset 4 3 πng 3 πng ρ ρ ρ 1,000 = 3 where w, c, and a = specific mass of water ( kg m ), cement (here considered as 3,150 kg=m3), and aggregates 5. In each time step, use the hydration degree α and humidity h 3 (1,600 kg=m for gravel and sand combined), respectively. from previous step to calculate the water diffusivity Beff

J. Eng. Mech., 2019, 145(5): 04019032 ¼ ð Þ ðαÞð40Þ Beff B0f0 hp f4 10. At each time step calculate the increment of gel barrier dzt g þ c 1 − dVt dVt cf z ¼ α > α ð48Þ ¼ þ ð41Þ d t 4πˆ2 for t c f0 cf 1− p n z 1 þ ht h 1−h 11. Finally, calculate the self-desiccation increment of relative s cap ðαÞ¼γ −γ α ≤ α ð42aÞ humidity, dht , of saturation degree, dSt , and of interparticle f4 e for cap porosity, dϕt ðβ=α Þm ðαÞ¼ðβ=α Þm s α > α ð42bÞ cap cap f4 s e for dVcðζ þ ϕ ζ Þ − dϕ S s ¼ t bw np gc t t ð49Þ dht Kh ϕcap where t m α where γ ¼ ; β ¼ α − α þ α αs=αmax ð43Þ αmax cap g CH c dϕt ¼ −ðdVt þ dVt þ dVt Þþðϕgp − ϕnpÞζgc ð50Þ Use αs ¼ 0.3, αmax ¼ αc=2, α ¼ 0.75αc, and m ¼ 1.8. where ϕnp = nanopore part of gel porosity, in which the pores Furthermore, c, h , nh, cf are empirical parameters. This study used n ¼ 8, h ¼ 0.88, and c ¼ 0. are too small to obey the Kelvin relation. These pores are h f 2=3 6. In each step, calculate the radius of the equivalent contact-free assumed to be always saturated. In this study, of gel pores C-S-H shells, zˆ, that give the same free surface area as would were assumed to be nanopores. the actual shell radius z if the shell surfaces were free, with no contacts Acknowledgments zt zˆt ¼ − ð44Þ 1 þ zt a0 5 u Partial financial support from the US Department of Transporta- tion, provided through Grant No. 20778 from the Infrastructure ¼ =6 4 where u a0 . . This relation was been modified from that Technology Institute of Northwestern University, and from the NSF in Rahimi-Aghdam et al. (2017b) in order to represent a multi- under Grant No. CMMI-1129449, are gratefully appreciated. decade continuation hydration better (this change has no effect on model performance up to 10 years). h 7. In each time step, using pore humidity , the unhydrous References cement remnant size a, and the C-S-H barrier outer radius z from the previous time step, as well as latest diffusivity value, Abrams, M., and D. Orals. 1965. “Concrete drying methods and their 1 calculate the water discharge Qt effect on fire resistance.” In Moisture in materials in relation to fire tests. Reston, VA: ASTM. h − h ˆ2 1 ¼ 4π ðα; Þ p c z ð45Þ Abrams, M. S., and G. Monfore. 1965. Application of a small probe- Qt atztBeff hp − 2 zt at zt type relative humidity gage to research on fire resistance of concrete. Report. Skokie, IL: Portland Cement Association. ˆ2= 2 The last factor on the right-hand side, z zt , serves to Ali, W., and G. Urgessa. 2014. “Computational model for internal relative consider the reduction of C-S-H shell surfaces due to mutual humidity distributions in concrete.” J. Comput. Eng. 2014: 1–7. https:// contacts. doi.org/10.1155/2014/539850. 1 8. In each step, using the calculated water discharge, Qt , calcu- Baroghel-Bouny, V., P. Mounanga, A. Khelidj, A. Loukili, and N. Rafa. c 2006. “Autogenous deformations of cement pastes. Part II: W/C effects, late the increment of cement volume, dVt , of portlandite, CH g micro–macro correlations, and threshold values.” Cem. Concr. Res. dVt , and cement gel dVt 36 (1): 123–136. https://doi.org/10.1016/j.cemconres.2004.10.020. c ¼ c þ c ¼ c − 1ζ ð46aÞ Bažant, Z., A. Donmez, E. Masoero, and S. R. Aghdam. 2015. “Interaction Vtþdt Vt dVt Vt ngQt cwdt of concrete creep, shrinkage and swelling with water, hydration, and ” – g ¼ g þ g ¼ g þ 1ζ ð46bÞ damage: Nano-macro-chemo. In Proc., CONCREEP 10,1 12. Reston, Vtþdt Vt dVt Vt ngQt gwdt VA: ASCE. Bažant, Z. P., and A. Donmez. 2016. “Extrapolation of short-time drying CH ¼ CH þ CH ¼ CH þ 1ζ ð46cÞ Vtþdt Vt dVt Vt ngQt CHwdt shrinkage tests based on measured diffusion size effect: Concept and reality.” Mater. 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