Theoretical Background

A large part of concrete shrinkage during drying (external or internal) is controlled by capillary pressure developing in the partially water-filled pores, as illustrated schematically in Fig. 1. For a cylindrical pore of radius r that is partially filled with pore solution, the Kelvin equation ⁸ describes the relationship between the pore radius and the equilibrium vapor pressure (relative humidity) above the meniscus:

(1)

where RH is the relative humidity (with values between 0 and 1), γ is the surface tension of the pore solution (N/m), V_m is its molar volume (m³/mol), r is the pore radius (m), R is the universal gas constant [8.314 J/(mol·K)], and T is absolute temperature in degrees K. Equation 1 assumes a contact angle of 0º (complete wetting of the pore walls by the liquid). According to equation 1, as smaller and smaller pores are emptied in the concrete, whether due to self-desiccation or to external drying, the internal RH will decrease. There will also be a capillary tension created within the pore liquid, given by:

(2)

where σ_cap is the capillary tension (Pa). As an example using equations 1 and 2, when emptying pores with a radius of 50 nm, the internal RH will decrease to about 98 % (neglecting any further depression due to the alkalis in the pore solution) and a capillary stress on the order of 3 MPa will be generated. These capillary stresses will cause a deformation of the porous concrete or mortar that can be approximately described by an equation based on a slight modification ⁹ of the one originally provided by MacKenzie ¹⁰:

(3)

where ε is the linear strain or shrinkage, S is the saturation (fraction with values between 0 and 1) or fraction of water-filled porosity, K is the bulk modulus of the porous material (Pa) with empty pores (dry), and K_s is the bulk modulus of the solid framework within the porous material (Pa). This equation is strictly valid for a fully saturated linear elastic material and is only an approximation for partial saturation and for viscoelastic cement-based materials. It should be further noted that both S and σ_cap will be strong functions of the pore size distribution of the hardening cement-based material. A material with finer pores will either maintain a higher saturation at a given equilibrium RH or produce much larger capillary stresses at a given fixed saturation. In either case, for fixed values of the bulk and solid framework moduli (last terms in equation 3), the material with finer pores would exhibit higher shrinkage.

Looking carefully at equations 1 to 3 leads to an interesting hypothesis concerning drying shrinkage and SRAs. In a normal drying shrinkage exposure (in the laboratory), the external RH is typically fixed at some value below 100 % RH and the resulting shrinkage over time is monitored, as the specimens' internal RH will slowly approach the external value. Equation 2 indicates that for a given fixed relative humidity and temperature exposure, the equilibrium capillary stress in the pore solution will be the same, and would not be a function of the surface tension of the pore solution. However, for such a fixed RH (and fixed capillary stress level), a lower surface tension will influence the size of the pores that are emptied under drying once (and if) equilibrium is finally achieved (the r term in equation 2). According to equation 2, for a fixed equilibrium RH, a reduction in the surface tension of the pore solution γ will result in a reduction in the size of the pores that are empty when equilibrium is ultimately achieved. If smaller pores are emptied in the presence of an SRA, the bulk equilibrium mass loss of the specimens containing the SRA should be greater than specimens with no SRA, as has indeed been observed for cement paste specimens.³

As a consequence of this, in equation 3, the incorporation of an SRA would actually influence the saturation fraction S and not the capillary stress term σ_cap, which would itself simply be proportional to −ln(RH), regardless of the presence or absence of the SRA. Since mass loss is often not measured during a drying shrinkage experiment (and equilibrium mass loss may not even be achieved during the length of a typical experiment¹¹), there only exists a limited amount of data on which to evaluate this hypothesis. However, one of the earliest data sets, obtained by Sato et al.³ , does provide both mass loss and shrinkage measurements for the exposure of cement paste specimens (prepared with four different levels of SRA addition) to 12 different RH levels. In that study, eight weeks were allowed for the cement paste specimens to come to equilibrium with the various external RH conditions. ³ According to equation 3, shrinkage should be linearly related to the product of saturation and capillary stress. From equation 2, capillary stress will in turn be in direct proportion to the negative natural log of the internal RH. Analyzing the data of Sato et al.³ based on equation 3 produces the plot shown in Fig. 2, in which a basically linear relationship between their measured shrinkage values and a computed factor that should be directly proportional to the stress level is indeed observed. For this analysis, the equilibrium saturation, S, of each specimen at each RH was assumed to be proportional to the estimated water content remaining in the specimens after 8 weeks, based on their mass loss measurements, and using the maximum 0 % RH eight-week (evaporable water) mass loss to estimate a water content for the specimens at complete saturation (as 325 mg/mL paste). Thus, the quantity (325 − the measured equilibrium mass loss) will represent the saturation value for each individual specimen.

Fig. 2: Shrinkage of pastes with various SRA (CSR = 0, 2, 4, and 8) contents3 vs. proposed form of shrinkage stress (proportional to the first two terms on the right hand side of equation 3). Solid line indicates best-fit to experimental data that passes through the origin (R ² = 0.87).

In addition to considering the equilibrium drying shrinkage based on equations 1 to 3, the kinetics of water evaporation (drying) and resultant shrinkage must be considered.¹¹ Generally, mortars and cement pastes with SRA additions exhibit a slower drying and thus less mass loss at equal drying times than the corresponding material with no SRA addition. ^{6, 7} Thus, looking at equation 3, at equal times, the saturation S will often be greater in the specimens with the SRA. Still, their measured shrinkage could be less as the capillary stress σ _cap would be expected to be significantly reduced for two reasons, the reduction in surface tension γ due to the presence of the SRA and the increase in radius r of the largest partially water-filled pores due to the higher saturation. However, this implies that equilibrium has not yet been reached in the drying specimens and that their current internal RH is far from what would be the final equilibrium value that may actually take several years to achieve depending on the drying conditions, the specimen geometry, and the specimen mixture proportions.¹¹

While self-desiccation involves different moisture transport boundary conditions than drying, equations 1 to 3 can still be applied to examining the influence of surface tension reduction on resultant autogenous deformation.¹² In this case, it is not the equilibrium RH that is fixed, but rather most likely the volume of empty porosity created during self-desiccation. For materials with a similar pore size distribution, one would expect this to correspond to a fixed value of r in equations 1 and 2, and a fixed value of S in equation 3. Thus, in the presence of an SRA, the reduction in γ will result in a direct reduction in the capillary stress and an increase in the equilibrium internal RH. Direct measurements of the internal RH of both cement pastes and mortars cured under sealed conditions with and without the addition of an SRA have indeed indicated that less of an RH reduction occurs when an SRA is present in the mixture.⁷ Mixtures with an SRA have also exhibited substantially less autogenous shrinkage during sealed curing, ^{6, 7} as would be expected from consideration of equations 1 to 3.

One other influence of shrinkage-reducing admixtures on early age properties is that in addition to altering the drying kinetics, they also change the shape of the drying profile within fresh cement pastes and mortars.⁷ Normally, the top 10 mm to 20 mm of an exposed fresh paste or mortar is observed to dry out fairly uniformly throughout its thickness, as the largest pores everywhere are emptied first, regardless of their depth within the specimen.¹³ However, with the incorporation of an SRA into the pore solution, a fairly sharp drying front forms at the specimen's exposed surface.⁷ This can result in a dramatic difference in the water distribution in the specimens with and without an SRA, as illustrated in Fig. 3, which contrasts a microstructure with "uniform" drying with one where drying occurs from the top surface only. In the latter case, as the SRA is concentrated in the pore solution remaining at the top surface during the initial drying, the surface tension of that local solution is further decreased such that it can no longer easily imbibe pore solution upward from deeper within the specimen. The specimen will thus enter a so-called "capillary regime" of drying, with a concurrent decrease in its drying rate relative to that achieved in the "evaporative regime". ¹⁴ As mentioned previously, this results in the drying rates of a fresh cement paste or mortar being significantly reduced by the addition of an SRA, even though the drying kinetics of bulk solutions (where all "drying" would occur in the evaporative regime) are accelerated by the reduction in surface tension caused by SRA addition.⁷ The influences of a reduction in surface tension of the pore solution on drying shrinkage, autogenous shrinkage, and early age evaporative water loss are summarized in Table 1.

The observation that the shape of the drying profile changes when the surface tension of the pore solution is reduced suggests that it might be possible to formulate a curing protocol based on the application of a solution containing an SRA to the top (exposed) surface of mortar or concrete specimens. Here, as a first step in investigating this possibility, the evaporative water loss and achieved degree of hydration of mortar specimens treated with two different concentrations of SRA solution will be contrasted against those treated with only (distilled) water. Previously, Nmai et al.⁵ have presented the topical application of an SRA as an effective method for reducing 28 d drying shrinkage for concretes with w/c of 0.45 and 0.65 and thicknesses of 50 mm and 100 mm for application rates of 200 mL/m² and 300 mL/m². In the present study, reduced (diluted with water) dosages on the order of 50 mL/m² to 100 mL/m² will be employed, and the initial evaporation rates and achieved hydration will be evaluated, as opposed to the drying shrinkage. It should be noted that this approach of applying an SRA solution to the mortar or concrete surface is mechanistically different from that of applying a conventional evaporation reducer, such as a long chain aliphatic alcohol, that reduces evaporation by forming a (monolayer) film on the surface of the mortar or concrete.¹⁵

Fig 3: Two-dimensional microstructure images from three-dimensional computer simulations illustrating drying/evaporation in cement paste specimens with "uniform" drying (left) and with drying from the top surface only (right). ⁷ Red indicates the solid cement particles, white the water-filled porosity, and black the empty (dry) porosity.