Analytical Method for Prediction of Water Permeability of Cement Paste

(ProQuest: ... denotes formulae omitted.)

INTRODUCTION

Permeability properties of concrete are a key factor for predicting its durability because it has been shown that corrosion, leaching, and carbonation are all related to the way a fluid can pour through the microstructure (Breysse and Gérard 1997). In view of their importance to the durability assessment and design of cement-based materials, considerable research has been undertaken, both experimentally and theoretically, on water transport in cement paste.

The conventional method for measuring water permeability is to establish a pressure gradient over a slab of material and to measure the flow through it. This can take days or even weeks for good quality concrete, however, and the sample can be changed during that time owing to hydration and/or leaching (El-Dieb and Hooton 1995). Moreover, there is a high risk of leaks under pressure that can be hard to detect (Scherer 2000). To circumvent these difficulties, a novel method was proposed recently by subjecting the sample to a fixed deflection in three-point bending and measuring the force as a function of time (Scherer 2000; Vichit-Vadakan and Scherer 2002). The novel method provides two primary advantages over conventional techniques. First, it can provide permeability results within a few minutes to a few hours. Second, there is no need to sustain a pressurized vessel during the measurement period so leaks are not a problem (Vichit-Vadakan and Scherer 2002). With these testing methods, the linking of structural to permeability properties of cement paste can be established. It was found that, at a given water-cement ratio (w/c), the water permeability of cement paste is well correlated with the time of hydration and the surface area. It was also found that there exists a broad-banded relationship between the water permeability and the hydraulic radius (Nyame and Illston 1981). The effects of various factors on water transport in cement paste have also been investigated. Hughes (1985) conducted tests on ordinary portland cement and ordinary portland cement/ pulverized fuel ash pastes cured for 1, 4, and 12 weeks in Ca(OH)2 solution. It was shown that the water permeability of both types is similar for those pastes cured for 4 and 12 weeks at 35 °C (96.8 °F), whereas after 1 week's curing, the ordinary portland cement/pulverised fuel ash paste is significantly more permeable. Based on experimental results, Banthia and Mindess (1989) concluded that the use of silica fume reduces the water permeability of cement paste, but it does not particularly depend on the amount of silica fume added. The effect of curing temperature on the porosity and water permeability of cement paste was also systematically examined (Goto and Roy 1981). Compared with samples cured at 27 °C (80.6 °F), the pore volume larger than 750 Å in those cured at 60 °C (140 °F) is larger and, as expected, higher permeabilities have been observed.

The theoretical prediction of the water permeability of cement paste, subject to varying environmental conditions, is a great challenge because of the difficulty in modeling fluid flow in random pore geometries (Martys and Hagedorn 2002). Early work expressed the water permeability as a function of hydraulic radius and such macroscopic properties as the total porosity (Scheidegger 1974). Later, work attempted to justify the Carmen-Kozeny equation on the basis of network or effective-medium models (Wong et al. 1984). These attempts fail to give satisfactory results because they do not recognize the fundamental physical significance and sensitivity of length scale in the water permeability prediction (Katz and Thompson 1986). Katz and Thompson (1986) derived a concise formula for the water permeability of rocks saturated with a single liquid phase. Whether this formula can be directly used to determine the water permeability of cement paste, however, is still an open issue (Christensen et al. 1996; Tumidajski and Lin 1998). Empirical formulas can directly be formulated by mathematical regression based on the measured experimental results, but no consensus of opinion has been achieved so far as to which pore structural parameter is most suitable for describing the water permeability (Nyame and Illston 1981; Bágel and ?ivica 1997). Moreover, the practical application of the existing empirical models for predicting the water permeability of cement paste remains very insufficient (Bágel and ?ivica 1997). Computer simulations are also regarded as a type of numerical experiment. With reconstructed cement paste, the water permeability can be determined by incorporating the finite element technique. However, simulated permeabilities are usually two or three orders of magnitude higher than the measured permeabilities (Pignat et al. 2005). The main reason is that the cement particles used in computer simulation are coarser than real cement particles owing to the limitations on computer memory and speed. Other reasons are that neither shrinkage, which could lead to closure of small pores, nor surface roughness of pores is considered (Pignat et al. 2005). The concept of fractality, introduced by Mandelbrot (1982) as a mathematical framework for studying irregular, complex shapes, is useful for providing a different way of understanding the underlying organization of porous cement paste. When the pore space is characterized approximately by a uniform fractal scaling and the internal conductances are assumed to be dependent only on the characteristic pore sizes, the water permeability of cement paste can be computed using mean field theory (Hansen and Muller 1992). Xu et al. (1997) proposed a renormalization method for computation of the transport properties of a porous medium modeled as a multiscale random network fully or partially saturated by one or two immiscible fluids. This method has two restrictive conditions: 1) no spatial organization must exist in the pore structure of the material, to allow for a modeling by random network; and 2) the pores must have no particular shape, to allow for a modeling by well-proportioned cylinders. The general effective media theory is a promising method for predicting the water permeability of cement paste (Cui and Cahyadi 2001). In this theory, the contributions of capillary and gel pores are both considered. When the cement paste is very porous, capillary pores form a continuous network to cover the whole and water mainly flows inside this capillary pore network. When the cement paste is less porous, part of the capillary pores are blocked by the hydration product and therefore water flows inside both capillary and gel pores to penetrate the material. The connectivity behavior of each phase constituent and the Hashin-Shtrikman bounds, however, are not considered. All of the aforementioned research clearly show that it is still highly desirable that an analytical method can be available, such as a formula for predicting the water permeability of cement paste more accurately and conveniently.

The intention of this paper is to present an analytical method for prediction of the water permeability of cement paste. In this method, cement paste is modeled as an isotropic two-phase composite material, with capillary and gel pores included in two separate phases. By introducing a hypothetical homogeneous medium of nonzero water permeability and applying the general effective medium theory, an analytical solution is derived for the water permeability of cement paste. The key parameter involved in the solution is then determined by experimental calibration. Finally, the derived analytical solution is verified with experimental results obtained from the literature.

RESEARCH SIGNIFICANCE

In predicting its water permeability, gel and capillary pores in cement paste should be treated as two separate phases due to their different roles in transporting water and importantly the volume fraction, water permeability, and connectivity behavior of each phase constituent; the interaction of the two phases; and the Hashin-Shtrikman bounds should all be taken into account. Unfortunately, existing analytical methods hardly consider all these factors in a proper way. This will inevitably affect the accuracy of the existing methods in predicting the water permeability of cement paste. It is also acknowledged that, although numerical methods exist that can be used to predict the water permeability of cement paste, the simulated results are usually two or three orders of magnitude higher than the measured ones due to the limitations on computer memory and speed. Moreover, numerical methods are often computationally difficult for practical use by engineers. It is therefore highly desirable that an analytical method can be available, with which the permeability of cement paste can be predicted more accurately and conveniently. In the present paper, an analytical solution is derived for the water permeability of cement paste as a two-phase composite material. The derived solution takes into account the morphological characteristics and physical properties of each phase constituent and the interaction of the two phases. In addition, because the derived solution satisfies the Hashin-Shtrikman bounds, better predictions can be expected.

TWO-PHASE COMPOSITE MATERIAL MODEL

In the hardened, mature state, cement paste is composed of cement gel and spaces that are penetrable by water (Hansen 1986). The cement gel is a dense substance and consists primarily of C-S-H and CH from a chemical point of view. Gel pores reside in C-S-H and have a minor effect on water transport. The space not filled with the cement gel is the called capillary pores. Unlike gel pores, capillary pores are assumed to have a major effect on water transport. Because the diameter of gel or capillary pores is much larger than that of water molecules, a continuum description of water transport through the two types of pores is appropriate and allowable (Pivonka et al. 2004). By taking the specific gravity of the portland cement as 3.15, Powers derived the respective volume fractions of capillary pores, unhydrated cement particles, and hydrated products (cement gel) as follows (Hansen 1986)

... (1)

... (2)

... (3)

where w/c is the water-cement ratio and α is the degree of hydration. It should be pointed out that, in deriving the volume fraction of capillary pores, only the space around the cement particle generated by the mixing water is considered. The entrapped air is neglected and therefore its effect on the water permeability is not taken into account in this paper as a preliminary study. In addition, because the addition of mineral admixtures will lead to different laws (Feldman and Huang 1985), Eq. (1) through (3) are only valid for cement pastes without mineral admixtures. Because gel and capillary pores play different roles in transporting water, they should be treated as two separate phases to predict the water permeability of cement paste in a proper way. For this purpose, cement paste is modeled as a two-phase composite material in this paper. The first phase is capillary pores, whereas the second consists of hydrated products and, if any, unhydrated cement particles. Thus, the volume fractions of Phases 1 and 2 can be obtained from Eq. (1) through (3) that

... (4)

... (5)

According to Cui and Cahyadi (2001), the water permeability of Phase 1, k^sub 1^, is given by

... (6)

where d^sub c^ is the critical pore diameter and V^sub 1,c^ is the critical volume fraction of Phase 1 or capillary pores. The critical pore diameter can be defined as follows. If the pores are added to the network one by one in order from largest to smallest, the critical pore diameter d^sub c^ is the diameter of the pore that just completes the first connected pore pathway (Katz and Thompson 1986; Cui and Cahyadi 2004). Experimentally, d^sub c^ can be determined by mercury intrusion porosimetry (MIP). Mercury, which is assumed to be an ideal nonwetting fluid, is forced into the evacuated pore space under quasistatic conditions. For each externally applied pressure, the diameter of the mercury-pore space interface is determined by the Washburn equation (Van Brakel et al. 1981). The pore dimension corresponding to the inflection point in the cumulative intruded-versus-pressure plot is taken as the critical pore diameter (Katz and Thompson 1986). Based on simulation results, Bentz and Garboczi (1991) found that the critical volume fraction of Phase 1 is almost independent of the w/c and is approximately equal to

V^sub 1,c^ = 0.18 (7)

Substitution of Eq. (7) into Eq. (6) yields

k^sub 1^ = 5.355 × 10^sup -3^d^sup 2^^sub c^ (8)

Because Phase 2 consists of permeable C-S-H, impermeable CH, and unhydrated cement particles, it is necessary to determine the volume fraction of C-S-H to compute the water permeability of Phase 2. It is well known that, for an isolated system, the net reaction for the cement hydration process can be expressed as

C^sub 3^S + 4H^sub 2^O [arrow right] C^sub 1.5^SH^sub 2.5^ + 1.5CH (9)

Although one cannot treat the volume elements as chemically isolated, simply because of ion fluxes going through the elements (Tzschichholz et al. 1996), the volume ratio of C-S-H to CH can be assumed to be equal to 1.66:0.63 as a preliminary study (Bentz and Garboczi 1991). Thus, the volume fraction of C-S-H can be obtained that

... (10)

According to Mclachlan (1988), the water permeability of Phase 2, k^sub 2^, is related to that of C-S-H, k^sub C-S-H^, by

... (11)

where V^sub C-S-H,c^ is the critical volume fraction of C-S-H. It has been shown by Bentz and Garboczi (1991) that the critical volume fraction of C-S-H is approximately equal to 0.17, almost independent of the w/c. In addition, the water permeability of C-S-H is approximately 7 × 10-23 m2 (7.5347 × 10-22 ft^sup 2^) (Powers 1958). Thus, kC-S-H can be expressed as

k^sub C-S-H^ = 7 × 10^sup -23^(12)

By substituting Eq. (12) into Eq. (11), k2 is given by

... (13)

WATER PERMEABILITY OF CEMENT PASTE

In deriving its water permeability, it is assumed in this paper that cement paste is an isotropic material. Because water transport in the porous material is mathematically similar to electron transport in the conductor, the general effective medium theory can be applied to the prediction of water permeability of cement paste. Introducing a hypothetical homogeneous medium of nonzero water permeability k0 and applying the general effective medium theory, one has (Koelman and de Kuijper 1997)

... (14)

By solving Eq. (14), kcp can be expressed in terms of k^sub 1^, k^sub 2^, k^sub 0^, V^sub 1^, and V^sub 2^ as

... (15)

Thus, once k^sub 0^ is known, the water permeability of cement paste can be determined from Eq. (15).

According to Koelman and de Kuijper (1997), the water permeability of the hypothetical homogeneous medium, k0, is of the form

k^sub 0^ = h^sub 1^k^sub 1^ + h^sub 2^k^sub 2^ (16)

where

h^sub 1^ = 0, h^sub 2^ = 0 (17)

h^sub 1^ + h^sub 2^ = 1 (18)

The conditions of Eq. (17) and (18) ensure that the water permeability of the cement paste satisfies the Hashin- Shtrikman bounds (Hashin and Shtrikman 1962). It is easily proven that when hi is equal to zero, Phase i is isolated; when hi is larger than zero, Phase i is percolating; when hi increases from zero to a positive number, Phase i is changed from an isolated state to a percolating one. Therefore, in loose terms, Eq. (16) amounts to parametrizing the connectivity behavior of the two phases. A particular choice for h1 and h2 that can take care of the conditions of Eq. (17) and (18) is given by (Koelman and de Kuijper 1997)

... (19)

where t is a parameter larger than or equal to zero, m is a percolation exponent, V^sub 2,c^ is the critical volume fraction of Phase 2, and the Heaviside step function H(V^sub i^ - V^sub i,c^) (i = 1, 2) is defined as

... (20)

Obviously, for hardened cement paste

V^sub 2,c^ = 0 (21)

It has been shown that, for a composite material consisting of an impermeable solid phase and a porous phase of a zero percolation threshold, the percolation exponent m varies from 3.0 to 4.0 (Wong et al. 1984). Thus in this paper, m takes the average, that is

m = 3.5 (22)

By substituting Eq. (7), (21), and (22) into Eq. (19), h1 and h^sub 2^ are given by

... (23)

Thus, once the parameter t involved in Eq. (23) is obtained by experimental calibration, the water permeability of the cement paste can be determined from Eq. (15).

DETERMINATION OF PARAMETER t

Before Eq. (15) can be used to predict the water permeability of cement paste, the parameter t involved in Eq. (23) needs to be determined beforehand. The determination of t is generally difficult, and experimental calibration seems to be the only feasible, practical method. This can be achieved as follows. According to the method of least squares, the bestfit curve of a given type is the curve that has the minimal sum of the deviations squared from a given set of data, that is

... (24)

where k^sub cp,i^ is the i-th (i = 1, 2, ... n) measured water permeability of cement paste at volume fractions of V^sub 1,i^ and V^sub 2,i^, k^sub 1,i^, k^sub 2,i^, and k0 are the water permeabilities of Phase 1, Phase 2, and the hypothetical homogeneous medium, respectively, and n is the total number of experimental data pairs. Thus, t can be determined from Eq. (24) for a given set of experimental data.

To investigate the effects of pore structure parameters on water transport in cement paste, Nyame and Illston (1981) conducted a cement paste experiment aimed at measuring the water permeability of cement pastes of different w/c and degrees of hydration. In their experiment, Type I portland cement and deionized water were used to prepare cement pastes at w/c of 0.47, 0.71, and 1.0, respectively. Pastes were mixed in a planetary mixer for 5 minutes before casting into 50 x 50 mm (1.97 x 1.97 in.) cylinders. Pastes with w/c = 0.71 and 1.0 were rotated for 4 hours after mixing to prevent sedimentation before casting. Samples were cured continuously at 100% relative humidity, followed by curing in saturated limewater for the duration of the test. A steady-state method of measuring the water permeability of cement pastes was used (Christensen et al. 1996; Nyame and Illston 1981). Samples for MIP were obtained by fracturing portions of the bulk paste specimens with a hammer and collecting the shards that were nearly spherical and had diameters close to 10 mm (0.39 in.). Five to 10 of these fragments were immersed in containers of isopropyl alcohol (IPA) for 48 hours, replaced with fresh IPA after 24 hours. Subsequently, fragments were removed from the IPA and dried for 24 hours. This was followed by 2 hours of oven-drying at 105 °C (221 °F). Samples were moved directly from the oven to a desiccator containing calcium sulfate desiccant and remained there until testing. One to two fragments were used for the intrusion experiment. The pressure in the porosimeter was increased and decreased at a constant rate of approximately 900 kPa/min (130 psi/min). The pore dimension corresponding to the inflection point in the cumulative intruded-versuspressure plot was taken as the critical pore diameter (Katz and Thompson 1986). The measured values of α and d^sub c^ of samples at different w/c are shown in Table 1. If Vt is used to denote the total porosity of cement paste, which can be expressed as (Hansen 1986)

... (25)

the measured water permeability is plotted against total porosity as shown in Fig. 1 through 3 for w/c = 0.47, 0.71, and 1.0, respectively.

With the three sets of experimental data, three optimized values of t corresponding to different w/c can be calculated from Eq. (24). The calculation results show that t increases with the increase of w/c. Specifically, when w/c = 0.47, 0.71, and 1.0, t is equal to 1.83, 9.15, and 100, respectively. For practical applications, it is desirable to obtain a quantitative relationship between t and w/c based on the three calculated values of t. This can be achieved by mathematical regression that

t = 94.39(w/c)^sup 5.885[/sup (26)

A comparison between the calculated and regressive values of t is made as shown in Fig. 4. As can be seen from Fig. 4, the regressive results are in good agreement with the calculated results with a correlation coefficient of 0.9989. It should be noted that, because the expression of t of Eq. (26) is derived from the experimental results of Nyame and Illston (1981), it is valid for cement pastes made with ordinary portland cement and cured for at least 1 day in saturated water at normal temperature. For other cases, Eq. (26) can provide an approximate estimate of t if the experimental results are not available. In addition, it can be seen from Eq. (4), (23), and (26) that, when t = 1.795, the capillary pore space always percolates for any given value of α. When t < 1.795, the capillary pore space percolates or not depending on whether α is smaller or larger than 1.052t0.1699 - 0.16. Therefore, t can be used to describe the percolation characteristics of capillary pores to a certain extent. With the calculated values of t, the analytical solution for the water permeability of cement paste can be obtained from Eq. (15). The results are shown in Fig. 1 through 3 at w/c = 0.47, 0.71, and 1.0, respectively. From Fig. 1 through 3, it can be seen that, at w/c = 0.47 and 1.0, the analytical results are in good agreement with the experimental results and that, at w/c = 0.71, the analytical results are slightly larger than the experimental results and the deviation increases with the decrease of the total porosity. This shows that t determined previously seems to be appropriate.

EXPERIMENTAL VERIFICATION

From the previous discussion, it should be noted that, although the derived analytical solution for the water permeability of cement paste is in good agreement with the experimental results of Nyame and Illston (1981), the value of t used in Eq. (23) is obtained by calibration based on the same experimental results. Therefore, it is necessary to further verify the derived analytical solution with other experimental results collected from the literature.

For the purpose of verification, the experimental results of Halamickova et al. (1995) are collected for comparison. In their experiment, portland cement meeting the requirements of ASTM Type I was used for all specimens. Two w/c of 0.40 and 0.50 by mass were chosen for all mixtures. The mixtures were cast into 100 x 200 mm (3.94 x 7.87 in.) cylinders and rotated for the first 20 to 24 hours to minimize segregation and bleeding. Following rotation, the samples were demolded and either stored in saturated limewater or prepared for testing, depending on the degree of hydration required. The pore structures of the cement paste at different degrees of hydration were measured by MIP. Specimens for MIP were cut from the cylinder cured in water at the time corresponding to the estimated degree of hydration, and then stored in IPA to remove the free water. The solvent was replaced after the first 3 days and storage in solvent was continued for several weeks. One month before the MIP measurements, the samples were removed from the IPA and dried to a constant mass in a vacuum oven at 55 °C (131 °F). The critical pore diameter was taken to be the inflection point on the volume intrusion versus diameter curve (Katz and Thompson 1986). The measured values of α and d^sub c^ of samples at different w/c are shown in Table 2. The water permeability was measured using the permeability cell designed and built by Hearn and Mills (1991). The measured water permeability is plotted against total porosity of cement paste as shown in Fig. 5 and 6 for w/c of 0.4 and 0.5, respectively. On the other hand, with the value of t computed from Eq. (26) for each w/c, the corresponding analytical solution for the water permeability of cement paste can be obtained from Eq. (15). The results are shown in Fig. 5 and 6. From Fig. 5 and 6, it can be seen that the analytical results are in good agreement with the experimental results. Therefore, the validity of the derived analytical solution for the water permeability of cement paste is verified.

To further verify the analytical solution, experimental results from Cui and Cahyadi (2001) are used. In their experiment, ordinary portland cement pastes were mixed with water at w/c = 0.4. Cement pastes were mixed in a pan mixer for 5 minutes before casting into a 150 x 150 x 150 mm (5.91 x 5.91 x 5.91 in.) mold for permeability testing. They were then cured in the moist condition for 7, 35, and 210 days. Thus, the degrees of hydration can be determined from the empirical formula proposed by Lam et al. (2000) as shown in Table 3. The input method was used to measure the permeability (Hedegaard and Hansen 1992). The testing surface was carefully polished with a steel brush to remove the surface skin. All samples were tested under a constant pressure, 700 kPa (102 psi), for 3 days. Some samples (210th day) with very low permeability were tested for 4 days. The pore size distribution was determined by 5 mm (0.2 in.) thick samples cutting from identically produced and cured cubes. The fragments were dipped into acetone for 5 minutes and then dried in a vacuum desiccator for 2 days. The pressure of MIP ranged from 0 to 414 MPa (60,000 psi). The measured critical pore diameters of samples at 7, 35, and 210 days are also shown in Table 3. With these data, the water permeability of cement paste can be evaluated by Eq. (15) and compared with the experimental results of Cui and Cahyadi (2001). The results are shown in Fig. 7. As can be seen, the analytical results are in good agreement with the experimental results. Therefore, the derived analytical solution is again corroborated.

CONCLUSIONS

An analytical method for predicting the water permeability of cement paste has been presented in this paper. In view of the different roles of gel and capillary pores in transporting water in cement paste, an isotropic two-phase composite material model has been constituted. Based on the general effective medium theory, an analytical solution has been derived for the water permeability of cement paste. The key parameter involved in the solution has then been determined by experimental calibration. The primary advantage of the proposed method is that the morphological characteristics and physical properties of each phase constituent, the interaction of the two phases, and the Hashin-Shtrikman bounds are all taken into account. Finally, the derived analytical solution has been verified with experimental results obtained from the literature. It can be concluded that, when the physical properties of the constituent materials, the w/c, and the degree of hydration are known, the solution derived in the paper can be used to predict the water permeability of cement paste.

ACKNOWLEDGMENTS

The financial support of the National Natural Science Foundation (No. 50578147) and the Natural Science Foundation of Zhejiang Province (No. Y107638), both of China, is greatly acknowledged. All reviewers' constructive comments and suggestions are gratefully appreciated.