1 Introduction

One of the most important developments in concrete technology is to control the amount of water absorbed and maintained in the concrete mix Paiva et al. (2009). Superabsorbent polymers (SAP) are new, very promising multipurpose chemical admixtures for applications in concrete with a wide window of potential for innovation. SAPs can trap water within up to \(\sim 100\) times of their own weight Esteves (2011), change the rheological properties of fresh concrete mixture Paiva et al. (2009) and tune it’s autogenous and plastic shrinkage behavior during both fresh and hardened states through internal curing Shen et al. (2015); Pang et al. (2011); Al-Nasra and Daoud (2013). Such utilization as a self-curing agent saves water as the concrete dries to a large extent Lee et al. (2016); Justs et al. (2015); Mechtcherine and Reinhardt (2012). Typical chemical composition is sodium salt of poly-acrylate acid \([-CH_{2}-CH(CO_{2}Na)-]_{n}\), a crystal-like structure classified hydrogel when cross-linked Mejlhede Jensen (2013). SAPs are commonly used in diapers, sanitary napkins, biomedical purposes, agriculture etc.

The recent introduction of SAPs into concrete has lead to promising results in terms of physical and mechanical properties which directly affect pumping, placement and compressibility of the mortar as well as the corresponding reliability in both fresh and hardened states. During absorption, water is entrained and cured into the concrete leading to hydration. Subsequently, SAP particles can slowly release their inner water as humidity supply when decreased in cement paste. This process fills the pores and reduces the micro-cracks, relieving the autogenous and drying shrinkages Wang et al. (2009). Therefore, the release process of water as well as its magnitude is a critical parameter for investigation of internal curing and it can control pumping, placement and compressibility Han et al. (2014). Additionally, predicting and controlling the workability of SAP modified concretes during the mixing process is a significant factor for the practical use of SAP in concrete industry, which dramatically changes the workability of the cement mortar samples Mechtcherine et al. (2015); Toledo Filho et al. (2012); AzariJafari et al. (2016).

Studies have shown that the addition of \(0.4\%\) of a certain type of SAP relative to cement mass will lead to a decreasing of the w/c of 0.06, causing the increase in the yield stress and the plastic viscosity Jensen and Hansen (2001). Consequently, adding a certain amount of SAP additive per cement mass is equivalent to removing water from the concrete mix due to its high water absorbency. If extra water is not added to concrete mix to compensate for absorbed water, it is inevitable to observe an increase in the yield stress and plastic viscosity of cement mortar mixtures Jensen and Hansen (2002). As the result, a decrease in slump flow spread and an increase in flow time is observed.

According to the findings of previous studies, the utilization of SAP could increase or a decrease the mechanical performance of concrete based on the type of SAP and the amount of added water, which control the workability and the mixing procedure. The flexural and compressive strength typically reduce Mechtcherine et al. (2006); Craeye and De Schutter (2008); Igarashi and Watanabe (2006). The water-entrained concrete mixes with SAP inclusion has macro-pores added, which are expected to reduce the compressive strength, albeit the self-curing SAP particles, improve continuous hydration, leading to later increase in the ultimate compressive strength. It is known that a \(1\%\) increase in voids in the concrete, reduce the compressive strength up to \(5\%\) Popovics (1998). Such marginal reduction due to macro-pores suggest the efficacy of SAP when utilized in their maximum capacity. In particular, at a high water-to-binder ratio (\({\displaystyle \frac{w}{b}>0.45}\)), SAP addition affect the hydration negligibly and, therefore, reduces compressive strength. Vice versa at lower ratios (\({\displaystyle \frac{w}{b}>0.45}\)), SAP addition may increase the compressive strength Powers and Brownyard (1946).

In addition to water-binding effect, it is believed that a further increase in yield stress and plastic viscosity is achievable by the inclusion of the swollen SAP particles. The absorption of water and moisture from the fresh concrete mix reduces the slump of the concrete decreasing the workability and the water needed for curing Dudziak and Mechtcherine (2010). Therefore, internal curing water is needed to compensate the moisture absorbed via SAP, where the reduction of workability can be compensated up to an extent Schröfl et al. (2012). The SAP particles create voids in the hardened cement paste (HCP) when water is released for internal curing. The formed porous concrete reduces the resulting concrete strength KONG and ZHANG (2013, 2014). Nevertheless, very few results are available regarding the influence of SAP on the workability of concrete.

The main objective of this study is to characterize the fresh state performance of SAP-modified cementitious composites and particularly to propose an percolation-based image analysis for predicting the workability of the mortar mixtures before the casting process. For characterizing the transient behavior, the radial propagation of slump samples of mortar mixtures is measured and is correlated to the instantaneous viscosity \(\mu (t)\). Hence this method introduces a cost-effective method in the absence of the advanced and costly measurement devices such as the Rheometer. Additionally this method explains rather a time-dependent viscosity, which addresses both transient and steady-state behavior of the cementitious composite during the casting process. Such depiction allows the prediction and control the workability and plastic behavior during concrete casting.

2 Methodology

2.1 Materials and Design

A simple hydration test on mixing the water with the SAP shows that it can absorb up to 100g of water per gram of SAP. The SAP particles grow in volume, similar to the air entrainment technique utilized for improving the resistance of concrete to the freeze–thaw. The entrainment is homogenous and enhances hydration and hardening of concrete Jensen and Hansen (2002).

The control mixture with the water-to-cement ratio of \(w/c=0.4\) was prepared as reference. The additional SAP content is equivalent to removing water from the concrete mixture Paiva et al. (2006), which causes the thickening of the concrete paste, decreasing the flowability and increasing the rheological results. For SAP contents varying between \(0.3\%\) and \(0.4\%\) , related to the mass of cement, additional water of \(0.04\%\) to \(0.07\%\) needs to be added, relative to the mass of cement Dudziak and Mechtcherine (2009). Subsequently, the mixtures were papered by varying amounts of SAP-to-cement mass containments. Additional \(0.01\%\) water for each \(0.1\%\) of SAP (both by mass of cement) was added to compensate for the absorption of mixing water by the polymers and to create mixtures showing a similar workability to the reference with a water-to-cement ratio of 0.40. Such water allowance has previously been utilized to compensate the loss of the workability Mechtcherine et al. (2006).

A maximum aggregate size of 2mm for the fine aggregates of ordinary Portland cement with the grade CEM|42.5. The mortar samples were prepared in the room with temperature of \(20\pm 2^{oC}\), by mixing the solid powder with liquid for 2min at low speed of \(140\pm 5rev/min\) and a further 2 min at high speed of \(285\pm 10rev/min\). The SAPs was treated by sieves, and the SAP with particle size of \(150--200\mu m\) was selected. The detailed properties of SAP used in this study is given in Table 1. The tea-bag test method, as suggested by RILEM, was carried out for the experimental measurements of SAP absorption–desorption. The test was applied in the cement slurry solution using an ordinary tea-bag which is filled with SAP particles and immersed in the pure water.

Table 1 Properties of SAP-contained mortar mixtures.
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The mini slump flow was carried out by placing the cone in the middle of a flat piece of square glass and filled with mortar. The cone is then quickly lifted, allowing the mortar to flow. After certain enlargement the sample turns to steady shape. High resolution digital images (15frames/second) were also taken at regular time intervals during the mini slump flow test to evaluate the flow behavior of materials in the subsequent computations.

2.2 Image Processing

Subsequently, an original image processing percolation-based numerical method was developed and carried out based on the flowchart presented in Fig. 1. The detailed steps are described as below.

  1. 1.

    The bare images \(Im_{k}\) from the concrete, sample of which is given in Fig. 2a contains information from the red, green and blue values (i.e., \(\{R,G,B\}\in [0,255]\)) which can be transferred to a normalized gray-scale intensity image by averaging as

    $$\begin{aligned} I_{i,j}={\displaystyle \frac{R_{i,j}+G_{i,j}+B_{i,j}}{3\times 255}} \end{aligned}$$
    (1)

    where the \(0\le I_{i,j}\le 1\) is the intensity value of the obtained grayscale image. The concrete regions can be distinguished by establishing a grayness threshold, \(I_{c}\in [0,1]\), which classifies the elements into black and white classes \(\{B,W\}\in (0,1)\). This value is determined iteratively from Otsu’s method by minimizing the intra-class variance \(\sigma ^{2}\) as follows Otsu (1975); Aryanfar et al. (2019): minimize \(\sigma ^{2}\), such that

    $$\begin{aligned} \text {}{\left\{ \begin{array}{ll} \sigma ^{2}=\omega _{0}\sigma _{0}^{2}+\omega _{1}\sigma _{1}^{2}\\ \omega _{0}+\omega _{1}=1 \end{array}\right. } \end{aligned}$$
    (2)

    where \(\omega _{0}\) and \(\omega _{1}\) are the fractions of black/white portions and \(\sigma _{0}^{2}\) and \(\sigma _{1}^{2}\) are the corresponding variances for each classified zone. The obtained binarized image from the minimization of intra-class variance \(\sigma ^{2}\) ensures the closest proximity in the values of each chosen group (\( B \& W\)), and therefore, it ensures the best approximation of the original gray-scale image.

  2. 2.

    Starting from the center of each image in the concrete, start to percolate and propagate through the first-order neighbors (i.e., left, right, top, bottom) until no further progress is possible Aryanfar et al. (2019). Moving forward each new pixel is indexed in descending order. Continue the propagation until no additional progress is made and the boundary of the slump sample is reached. The computed circular region \(A_{k}\) is the effective slump area.

  3. 3.

    The difference between each successive images in the time span between the experimental images \(\left( t_{k},t_{k+1}\right) \) represents the incremental area \(\Delta A_{k}\) , which can be translated from the geometry to the incremental radius \(\delta r_{k}\). This has been visualized in Fig. 2b.

2.3 Modeling

Fig. 1
figure 1

Pseudo-chart of the computational modeling.

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The obtained radius \(r_{k}\) from Fig. 1 is normalized to it’s original value (i.e., \({\displaystyle \frac{r(t)}{r_{0}}}\)) and it’s transient behavior in time is given in Fig. 3a based on the SAP containment. In general the radius is time-dependent r(t) and it can be could be segmented in time to obtain the values of the the radial velocity (\(V_{k+1}={\displaystyle \frac{r_{k+1}-r_{k}}{\delta t}}\)). This gravity-driven flow generates a shear stress \(\tau (t)\) proportional to the shear rate (\({\displaystyle {\displaystyle {\dot{\gamma }}={\displaystyle {\displaystyle \frac{dv}{dr}}}}}\)) as

$$\begin{aligned} \tau (t)=\mu (t)\frac{dv}{dr} \end{aligned}$$
(3)

where \(\mu (t)\) is the instantaneous viscosity of the sample. In the absence of the external pressure gradient (\({\displaystyle \frac{dp}{dr}\approx 0}\)) and the variation of the velocity in the azimuthal direction (\({\displaystyle \frac{du}{d\theta }\approx 0}\)), the general Navier–Stokes relationship can be simplified as Munson (2013)

$$\begin{aligned} \rho _{mix}\left( \frac{dv}{dt}+u\frac{dv}{dr}\right) =\mu (t)\left( \frac{d^{2}v}{dr^{2}}+\frac{1}{r}\frac{\partial v}{\partial r}\right) \end{aligned}$$
(4)
Fig. 2
figure 2

Analysis.

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where the LHS represents the applied convective momentum applied to the sample from gravity and the RHS illustrates the resulted viscous momentum. Equation 4 can be solved using the finite-difference method and for simplicity the boundary of the concrete is focused-on. The boundary velocity V is defined at the given time \(t_{k}\) from the incremental variation in the outer radius r, such that

$$\begin{aligned} V_{k+1}=\frac{R_{k+1}-R_{k}}{\delta t} \end{aligned}$$
(5)

From Fig. 2a, the velocity value v(r, t) depends on the radial distance from the center, with the boundary condition, as below:

$$\begin{aligned} {\left\{ \begin{array}{ll} v(0,t)=0\\ v(R,t)=V \end{array}\right. } \end{aligned}$$
(6)

where u is the velocity in the boundary of the sample. Therefore, the velocity is obtained via linear approximation as

$$\begin{aligned} v(r,t)=\frac{r}{R}V(t) \end{aligned}$$
(7)

Therefore, the terms in Eq. 4 are obtained as

$$\begin{aligned} \frac{dv}{dr}&=\frac{1}{R}V(t) \end{aligned}$$

and

$$\begin{aligned} \frac{d^{2}v}{dr^{2}}=0 \end{aligned}$$

Therefore, the initial decomposition, and assuming \(v_{i}^{j}\) as the velocity in the time \(t^{j}\) and the radial distance \(r_{i}\) , adopting the scheme of forward move in time and space (FTFS) for segmentation of \(\delta t\) in time and \(\delta r\) in the space, Eq. 4 is discretized as

$$\begin{aligned} \rho _{mix}\left( \frac{v_{i}^{j+1}-v_{i}^{j}}{\delta t}+v_{i}^{j}\frac{v_{i+1}^{j}-v_{i}^{j}}{\delta r}\right) =\mu ^{j}\left( {\displaystyle \frac{1}{r_{i}}\frac{v_{i+1}^{j}-v_{i}^{j}}{\delta r}}\right) \end{aligned}$$
(8)

Thus, for the experimental samples in Fig. 2, the value of the viscosity \(\mu ^{j}\) is obtained as a function of SAP, in the given time \(t^{j}\) by re-arrangement as

$$\begin{aligned} \mu ^{j}=\rho _{mix}\frac{{\displaystyle \frac{v_{i}^{j+1}-v_{i}^{j}}{\delta t}+v_{i}^{j}\frac{v_{i+1}^{j}-v_{i}^{j}}{\delta r}}}{{\displaystyle \frac{1}{r_{i}}\frac{v_{i+1}^{j}-v_{i}^{j}}{\delta r}}} \end{aligned}$$
(9)

where \(\rho _{mix}\) depends on the concentration of the SAP utilized in the mortar mixture. If the weight concentration is \(\alpha \) (\(m_{w}=\alpha m_{c}\)) the total density \(\rho _{mix}\) will be obtained as

$$\begin{aligned} \rho _{mix}&=\frac{m_{tot}}{V_{tot}}=\frac{(1+\alpha )m_{c}}{{\displaystyle \frac{m_{c}}{\rho _{c}}}+\alpha {\displaystyle \frac{m_{c}}{\rho _{w}}}}\\&=(1+\alpha )\frac{\rho _{w}\rho _{c}}{\rho _{w}+\alpha \rho _{c}}\nonumber \end{aligned}$$
(10)

where the indices w and c represent the water and the cement, respectively. Equations 9 and 5 can be merged to address the viscosity \(\mu _{k}\) in terms of the variation in the radius \(r_{k}\) as below:

$$\begin{aligned} {\left\{ \begin{array}{ll} \mu ^{j}=\rho _{mix}\left( {\displaystyle \frac{dv}{dt}+v\frac{dv}{dr}}\right) \left( {\displaystyle \frac{1}{r}\frac{dv}{dr}}\right) ^{-1}\\ {\displaystyle \frac{dv}{dt}=\frac{v_{i}^{j+1}-v_{i}^{j}}{\delta t}}\\ {\displaystyle v\frac{dv}{dr}=v_{i}^{j}\frac{v_{i+1}^{j}-v_{i}^{j}}{\delta r}}\\ {\displaystyle {\displaystyle \frac{1}{r}\frac{dv}{dr}}=\frac{1}{r_{i}}\frac{v_{i+1}^{j}-v_{i}^{j}}{\delta r}} \end{array}\right. } \end{aligned}$$

The results of computations from the above equation is shown in Fig. 3b as the difference from the original viscosity \(\mu -\mu _{0}\), where the value of mixed density \(\rho _{mix}\) is calculated from Eq. 10. As well the normalized time is shown by \({\displaystyle \left( \frac{mg}{A_{0}\mu _{0}}\right) t}\), where m is the mass of mortar mixture, g is gravity and \(A_{0}\) and \(\mu _{0}\) are the initial area and viscosity of the mixture.

Fig. 3
figure 3

Evolution of the normalized radius \({\displaystyle \frac{r(t)}{r_{0}}}\) and the computed incremental viscosity \(\mu -\mu _{0}\) throughout the normalized time \({\displaystyle \frac{mg}{A_{0}\mu _{0}}}t\).

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3 Results and Discussion

The inclusion of the higher amount SAP in fact means that the remaining mortar mixture is getting less workable. Despite the additional amount of water in the mixtures due to initial water absorption by the SAP, the slump flow decreased and the viscosity correlates directly with the SAP content. Meanwhile, the viscosity gain indicates that the amount of extra added water was completely absorbed by the SAP particles and yet still insufficient to compensate for the workability loss.

The developed image processing method in the Flowchart 1 tends to quantitatively capture such behavior via computing the incremental area \(A_{k}\) - and, therefore, the radius \(r_{k}\) - between the time slots \(t_{k}\) and \(t_{k-1}\). Due to hardening behavior, such value is reduced in time as obtained in Fig. 2b, which is also obvious from the negative curvature of the normalized radius \({\displaystyle \frac{r(t)}{r_{0}}}\) versus time as shown in Fig. 3a. In fact the variation in the radius decreases with the amount of SAP containment, comprising the inverse effect of water absorption on the flowability Al-Nasra and Daoud (2013); Mechtcherine and Reinhardt (2012).

The finite difference computations from Eq. 3b reveals that the viscosity \(\mu (t)\) is higher in the presence of augmented SAP mixture, therefore

$$\begin{aligned} SAP\uparrow \sim { \mu \uparrow } \end{aligned}$$

which is related to the water entrainment into the SAP particles, leaving the mixtures more dry. Consequently, effect of time t is the most pronounced among all the parameters for the hardening behavior, where the viscosity is increasing exponentially. Therefore

$$\begin{aligned} t\uparrow \sim { \mu \uparrow \uparrow } \end{aligned}$$
Fig. 4
figure 4

Further analysis of the mortar samples.

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Having a closer look to the dynamics of flow in the slump test, one realizes that the radial velocity within the mortar mixture is variant. Based on Eq. 7, one gets

$$\begin{aligned} \frac{v}{V}\approx \frac{r}{R}<1 \end{aligned}$$
(11)

which means that the velocity increases linearly along the radial direction. Such velocity variation from Eq. 11 has been visualized in Fig. 4a. Since the outer-most region of the sample, with highest flowability is considered for viscosity calculation, therefore, the results show the most conservative case scenario and the viscosity in the inner regions is higher.

Viscosity results were experimentally measured for each mix by Brookfield Viscometer DV2T, as given in Fig. 4b (respective data table is available in the supplemental materials). As well the obtained rheology results for the mortar mixtures in the figure shows the direct correlation of the viscosity \(\mu \) with the SAP concentration, the order of which is consistent with the results in Fig. 3b. While the mortar mixtures subjected to the centrifugal force, they are initially get stretched during large deformations with high rates and reach to the stretching limit, where the obtained viscosity converges to a constant value. The transition to steady state viscosity re-emphasizes on role of the curing time during the transitory stretching of the mortar mixtures.

The comparison of the numerical result in this study with the literature in Fig. 5 reveals the increasing trend. The relative lower value of the viscosity is related to the lower containment of SAP particles (\(0-0.8\%\) versus \(10\%\) in Hu et al. (2016)) which absorbs less amount of water, leading to more more flowable (i.e., less viscous) medium.

Ultimately, adding a certain amount of SAP additive per cement mass is equivalent to removing water from the concrete mix due to its water absorbency. Therefore, an decrease in slump flow spread and flow time is observed (Fig. 3a). The later-on release process of the water from SAP particles, which can occur during the long run, leads to a formation of moisturized concrete with a fraction of water containment W. In such case, the viscosity of the mixture \(\mu _{mix}\) is described from classic theory as LaAHN (1949)

$$\begin{aligned} \log \mu _{mix}=N_{c}\log \mu _{c}+N_{w}\log \mu _{w} \end{aligned}$$
(12)

where \(N_{c}\) and \(N_{w}\) and \(\mu _{ct}\) and \(\mu _{w}\) represent the mole fraction and the viscosity for cement and water, respectively. The mole fraction N is defined as

$$\begin{aligned} N_{c}&=\frac{n_{c}}{n_{c}+n_{w}}\\&=\frac{{\displaystyle \frac{m_{c}}{M_{c}}}}{{\displaystyle \frac{m_{c}}{M_{c}}+\frac{m_{w}}{M_{w}}}} \end{aligned}$$
Fig. 5
figure 5

Obtained viscosities compared with reported data with \(W\approx 10\%\).

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where \(n_{c}\) and \(n_{w}\) and \(m_{c}\) and \(m_{w}\) and \(M_{c}\) and \(M_{w}\) are the number of moles, the respective mass and the molar mass of cement and water, respectively. Based on the release process, one has

$$\begin{aligned} W=\frac{m_{w}}{m_{c}+m_{w}} \end{aligned}$$

solving for the mass of cement \(m_{c}\) one has

$$\begin{aligned} m_{c}=m_{w}\left( \frac{1-W}{W}\right) \end{aligned}$$

the mole fraction of the cement \(N_{c}\) is obtained consequently as

$$\begin{aligned} N_{c}{\displaystyle =\frac{\left( 1-W\right) M_{W}}{WM_{c}+(1-W)M_{w}}} \end{aligned}$$

and, respectively, the mole fraction of water \(N_{w}\) is obtained as

$$\begin{aligned} N_{w}=\frac{WM_{c}}{WM_{c}+(1-W)M_{w}} \end{aligned}$$

Hence, the mixed viscosity \(\mu _{mix}\) is obtained subsequently as

$$\begin{aligned} \mu _{mix}=\exp \left( \frac{WM_{W}}{WM_{w}+(1-W)M_{c}}\log \mu _{c}+\frac{(1-W)M_{c}}{WM_{w}+(1-W)M_{c}}\log \mu _{w}\right) \end{aligned}$$
(13)

which is in fact an Arrhenius-type relationship Arrhenius (1887). In fact such exponential trend resonates very well with the obtained viscosity results in Fig. 3b and the linear trend obtained in the log scale in Fig. 5.

4 Conclusions

In this paper, an image-processing method has been developed as a powerful technique for computing the radial propagation, the velocity and ultimately the instantaneous viscosity of the mortar mixtures with the given concentration of the super absorbent polymer. While the method, which is supported by the supplemental rheological measurement and analytical elaboration, can be used for predicting the concrete’s workability before casting process, the quantified utilization of SAP contents can be used as determining parameters to tune the other flow properties of the concrete effectively. Further studies are on the way for characterizing the combination of SAP with superplasticizer (SP) on the fresh state rheological properties of cementitious mortars.