Modelling the effect of mortar constituents on rheological properties

Concrete is a widely used construction material whose potential has not yet been fully exploited due to the lack of knowledge about its rheological behaviour. The mix composition is assumed to play a key role in controlling the rheological properties of concrete. Due to the extremely high proportion of finer particles in concrete, the investigation of the rheological properties of the mortar and paste phases is of major importance in understanding the rheological behaviour of concrete. Therefore, this paper systematically investigates the impact of water‐to‐cement ratio, sand content, and maximum sand size on the rheology of flowable mortars. It also explores the applicability of existing models for modeling these mortars.


Introduction
Concrete is the most used building material in construction worldwide and is generally considered a multi-component suspension composed of cement, water, aggregates, mineral and chemical admixtures [1,2].For most processes, the fresh concrete must fulfil the following two requirements: a) segregation stability and b) pumpability [3,4].Thus, the mix composition of the concrete is a decisive factor, as it controls the rheological properties of the concrete.Due to the extremely high proportion of fine particles involving significant colloidal interactions, the rheological properties of the concrete are highly dependent on the mortar composition and paste phase [3,5].In recent decades, the effect of the individual parameters on the rheology of cement-based suspensions has generally been summarised into rheographs.A rheograph characterises and generalises the effect of the various components on the rheological properties viscosity and yield stress of a suspension [1,6].
When dealing with concrete or mortar as a polydisperse suspension, it is first essential to fully understand the mechanisms and parameters affecting the rheology.In general, the following parameters impact the rheology of suspensions and have to be investigated for modelling [1,[6][7][8]: Our research focused on the study of three parameters affecting the rheological behaviour of cement-based mortars.The first parameter is the w/c ratio, expressed as the ratio between the amount of water to the amount of cement used in cement paste (both by weight).The second parameter is the change in the amount of sand existing in the mortar, termed the sand solid content Φsand.This allows us to obtain key input data for the modelling.The third variable parameter is the maximum sand size dmax.The maximum sand size used in the mortar affects the PSD and the maximum solid content Φmax.

I.
Effect of the w/c ratio on the rheology of cementbased suspension In the modelling the rheology of cement-based suspensions, cement paste can be assumed as a fluid phase.When increasing the cement paste content in concrete, cement paste fills the gaps between the aggregates and the concrete becomes much more workable.More significantly, an increase in the w/c ratio results in a decrease in the yield stress and viscosity of the cement paste, resulting in a decrease of yield stress and viscosity in the concrete, see Figure 1 [1,6,9].Increasing the w/c ratio lowers the viscosity, as the suspension becomes more fluid and easier to flow.This is due to the fact that water molecules are able to fill the spaces between the cement particles and thus reduce their interactions with each other.Therefore, the viscosity of cement paste declines as the water content (the w/c ratio) increases.Furthermore, cement particles tend to aggregate and form clusters due to their surface chemistry.These clusters possess a high flow resistance and contribute to the yield stress of the material.The addition of water (increasing the w/c ratio) reduces the attractive forces between these particles and permits them to move more freely, thereby reducing the friction between the particles and the yield stress.In general, the addition of water amount to cement (increasing the w/c ratio) in cement paste turns it into more fluid and is much easier to cast or pump, however, it is possibly also more sensitive to segregation and bleeding [4].As the solid content of particles in the fluid phase (e.g.cement paste) increases, the interactions between particles may move from hydrodynamic forces (dominant at a low solid content) to frictional forces between particles (at a high solid content) [8,10].Such a transition happens because, at a low solid content, the particles are separated and interact mainly through the fluid phase, whereas at a higher solid content, the particles are coming into contact and direct particle-particle interactions start to play a dominant role.Furthermore, this transition may create a "particle network" in the fluid phase, in which the particles are in contact with each other and are carried by the fluid.The existence of these particle-particle interactions can strongly affect the rheological properties of the suspension, in particular its viscosity [10][11][12].Figure 2 presents the change in viscosity of monodisperse suspensions at a constant shear rate as the number of spheres in a constant volume, and thus the solid content of the particles, increases.

Figure 2
Change in viscosity of monodisperse, i.e. constant sphere size, sphere-based suspensions as a function of solid content at a constant shear rate; the dashed line indicates Newtonian behaviour; redrawn from [13,14] At a low solid content of 0.1 (-), the viscosity of the suspension starts to rise, however, the particle-particle interactions play no role, meaning that the flow behaviour is still controlled by the Newtonian behaviour of a fluid phase.
As the solid content reaches 0.3 (-), particle-particle interactions begin to accumulate and act as obstacles to flow.As a result, the flow behaviour turns into a non-Newtonian behaviour.Once the solid content of particles gets closer to maximum solid content Φmax (0.64 (-) in the case of monodisperse suspension), the particle-particle interactions become much more pronounced, resulting in a higher shear stress required to shear the suspension at a constant shear rate.

III. Maximum particle size dmax and its effect on maximum solid content Φmax
The Φmax is commonly used to assess the effect of particle concentration on the viscosity of concentrated suspensions.This parameter is tightly linked to the arrangement of the particles in the suspension and is affected by several factors, including, maximum particle size dmax, PSD and particle shape [15,16].For many real suspensions, the particle size is not uniform and is expressed most appropriately by a PSD.Moreover, the Φmax of a suspension is not only affected by the method of particle packing, rather it is also strongly dependent on the maximum particle size dmax and the PSD [17][18][19].To estimate this relationship, Hu and de Larrad [20] developed a semi-empirical model, Using percentile values (d10 and d90) obtained by a PSD, the Φmax can be predicted.In general, mixing various particle sizes in a suspension causes a broader PSD and hence a higher Φmax [8]. Figure 3 illustrates the effect of a monodisperse (fine particles) and a polydisperse (variation from fine to coarse particles) suspension on the maximum solid content Φmax.The monodisperse suspension has just finer particles and therefore a narrow PSD, leading to a low maximum solid content.As the polydispersity of the suspension rises (mixing finer and coarser particles), the PSD becomes broader and thus the Φmax increases.The reason for this is that a wider variety of particle sizes causes the creation of gaps or voids between particles during the processing of the filling in a suspension.These voids could result in a larger packing density and thus a higher Φmax, since the particles are not able to pack together as closely as in a suspension having a lower polydispersity.
Figure 3 Effect of monodispersity and polydispersity on maximum solid content Φmax.Polydisperse suspensions exhibit a higher maximum solid content Φmax than monodisperse suspensions, revised from [14] Not only the monodispersity or polydispersity of the suspension (expressed by PSD) is relevant, but also the maximum particle size present in the suspension matters for the characterisation of the rheology [8,21].Figure 4 reveals the effect of maximum particle size (e.g., percentage of finer particles contained in the suspension composition) on viscosity as the suspension turns from a monodisperse to a polydisperse suspension.
Figure 4 Effect of the maximum sand size (e.g., percentage of finer particles) on the viscosity of a suspension sheared at a constant shear rate and at a constant solid content; redrawn from [13] It is known that finer particles give a higher viscosity for a given shear rate and solid content [22].Thus, comparing the viscosity of a monodisperse suspension (consisting of finer particles) and a polydisperse suspension (raising the maximum sand size, incl.variable size from finer to coarser particles), the viscosity of the polydisperse suspension is lower compared to the monodisperse suspension due to the two main sources: (a) higher polydispersity (resulting on higher Φmax) and (b) less proportion of finer particles [8,23].

Aim and concept of investigations
The main objective of this work is, on the one hand, the fundamental systematic investigation of the effect of constituents on the rheological properties of flowable fresh mortars and, on the other hand, the verifying the applicability of existing models for the modelling of polydisperse cement-based mortars.To meet this goal, different cement-based mortars were prepared by varying three various parameters, namely (a) the w/c ratio, (b) the sand solid content Φsand (volume of sand/total volume) in mortar, and (c) the maximum sand size dmax (leading to variation in maximum solid content Φmax), and their rheological properties were measured using a wide gap vane-in-cup rheometer.The effect of each parameter mentioned above on the rheological properties of cement-based mortars was compared and discussed.In conclusion, a rheograph was presented enabling engineers to optimise the mix composition of cement-based suspensions according to the goals of various rheology-based applications.The next step involved modelling the relative viscosity of cement-based mortars using the existing methods developed so far.To further improve the accuracy of the model, the PSD (linked with the maximum solid content Φmax) needs to be included as a key parameter in the modelling.So, the models and their capabilities in modelling polydisperse cement-based mortars were discussed.

Materials and sample preparation
In this work, Portland cement CEM I 42.5 R (HeidelbergCement) was used with a density of 3.14 g/cm³ and a specific surface area of 3499.4 cm²/g.The median particle size of this cement d50 was 14.1 μm.The clinker phase was composed of 54.45% C3S, 18.07%C2S, 10.83% C3A and 5.17% C4AF.For a full characterisation of the cement, the reader is referred to [24].Deionised water without superplasticiser was used for the production of cement paste (CP) and cement-based mortar (M).Besides, dry quartz sands with variable sizes, namely (a) Fine: 0.2-1 mm; (b) Medium: 1-2 mm and (c) Coarse: 2-4 mm with a density of 2.65 g/cm³ were also used.Figure 5 represents the PSD of the different sand sizes used in this work, measured using dynamic image analysis.In the production of cement-based mortars, the w/c ratio changes from 0.44 (-) to 0.48 (-).In addition, the sand's solid content Φsand varies between 25 and 40 Vol-%.Furthermore, three different sand mixes varying maximum sand size from 1 to 4 mm were used in this work, see Table 1.They are termed Mdmm, where M specifies the mortar and d refers to the diameter of maximum sand size present in the mortar.In M1mm, only the f fraction was used in mortar.For M2mm, 40 Vol-% of the total sand solid content of M1mm was substituted by the m fraction.The composition of the sand in the case of M4mm is 60 Vol-% f, 20 Vol-% m and 20 Vol-% c in the total sand solid content in the mortar, see Table 1.For each experiment, 1.3 dm³ of cement paste or cementbased mortar was prepared.Therefore, cement and sand were stored at 20 °C.Deionised water was cooled at 10 °C.Water and cement were mixed in a mortar mixer according to DIN EN 196-1 [25].Table 2 specifies the mixing process with a total mixing time of 300 s, including the individual steps, as well as the mixing intensity and duration of each step.The temperature of the raw materials was controlled to ensure that the temperature of fresh cement paste or cement-based mortar measured immediately after mixing was kept within a range of 20 ± 1 °C for all samples.The cement-based mortars are labelled like Mdmm_w/c_Φsand, in which d shows the maximum sand size, w/c shows the water-to-cement ratio and Φsand indicates the sand's solid content.To minimise the influence of thixotropy on the determination of the rheological properties, the suspension was presheared in two separate steps before conducting the experiments.The first step was done using the mortar mixer, mixing intensity 285 min -1 for 60 s (starts 11 min.after the water and cement contact) and after pouring the material into the rheometer cup, then using a drill at 1700 min -1 for 60 s (begins 13 min.after the water and cement contact).
It is remarkable to note that the time between the end of the pre-shearing with the drill (1700 min -1 for 60 s) inside the rheometer cup and the beginning of the experiment (15 min.after the water and cement contact) was 60 s.

Rheological experiments
In order to verify the sedimentation stability of the cement pastes and cement-based mortars prepared, the test according to DIN EN ISO 10426-2 [26] was carried out once for each suspension.For rheological characterisation, the prepared suspension was pre-sheared with the mortar mixer 285 min -1 for 30 s (starting at 13:45 min.after water and cement contact) and the flow test was prepared according to DIN EN 1015-3 [27].The Haegermann cone (d1 = 70 mm, d2 = 100 mm, h = 60 mm according to DIN EN 1015-3) was lifted simultaneously when the rheometer experiment was started.
The rheological properties of the samples were determined using a wide gap vane-in-cup rheometer (Anton Paar MCR 502) with an effective gap size of 15 mm.This is a Searletype rheometer with a stationary outer cylinder and a rotating vane probe recording the torque data.24 vertical lamellae were fixed to the outer wall to eliminate the effect of wall slippage, see [28,29] for more details on the dimensions of the rheometer used in this work.
To identify the most appropriate velocity profile to evaluate the rheological properties, several velocity profiles were applied and tested.The key factors involved in choosing the most appropriate velocity profile are a) reaching a steady state flow at each velocity step and b) shortening the duration of the velocity profile to minimise the impact of sedimentation and shear-induced particle migration during experiments.Towards this end, the velocity profile shown in Figure 7 was used.First, the velocity was rapidly increased from 0 to 100 rpm within 15 s.The velocity was then kept at 100 rpm for 30 s to allow the measured torque value to reach a steady state.Finally, a stepwise velocity profile was continued, where the velocity was reduced from 80 rpm to 0.1 rpm, lasting each step 15 s, see Figure 7.Moreover, Figure 7 contains an example of the torque values recorded and reported by the rheometer.A steady state is obtained at each velocity step.Thus, these values are precise enough to determine the rheological properties of the suspensions in this study.To determine the rheological properties using the Bingham model, the velocities 100 -10 rpm and the corresponding torque values (average value for the last 2 s in each step; shown by circles) were used.

2.4
Analytical methods

Determination of rheological properties
Usually, the rheological properties of cement-based materials are expressed in terms of two different physical parameters, namely yield stress and plastic viscosity.The Bingham model was used in this work to calculate the flow curve of cement paste and cement-based mortars.As a first step, the raw data (velocity, torque) needs to be converted into a flow curve (shear rate, shear stress).Due to the complex material flow behaviour present in wide gap rheometers, such as non-linear shear rate distribution across the rheometer gap and the existence of plug flow region, the conversion of raw data to a flow curve is not as straightforward in wide gap rheometers as it is in small gap rheometers [28,29,30].However, the Reiner-Riwlin equation developed for the Bingham model was used for this conversion.For the conversion, the rotational velocities (100, 80, 60, 40, 20 and 10 rpm) from the velocity profile and the average for the last 2 s of the torque values in each step were used.First, the torque values were converted into shear stresses using equation 2. After assuming the initial values for yield stress and plastic viscosity, the analytical rotational speed (calculated according to equation 3) was fitted to the actual rotational speed of the rheometer at each data point.The non-linear iterative procedure continued until the calculated "mean error (mse)" is minimum.Then the final data obtained from the iterative process, i.e., plastic viscosity and yield stress, were expressed.Then the rotational speed was converted into the shear rate using known parameters, see equation 4.
where  is the shear stress (Pa) and T is the torque (Nm), h is the vane height (m), Ri is the vane radius (m), n (rps) is the rotational speed, μ p is the plastic viscosity (Pa•s), τ 0 is the yield stress (Pa), Rplug is the plug radius (m) and γ̇ is the shear rate (s -1 ).For additional information about the Reiner-Riwlin equation, the authors refer to [9,28,29,31].

Modelling the viscosity of suspensions
In the last decades, many theoretical and semi-empirical expressions have been proposed to link the solid content with the relative viscosity, which is defined in our case as: the division of the mortar viscosity μmortar (Pa•s) by the cement paste viscosity μcement paste (Pa•s) [32,33].Based on these relationships, the relative viscosity commonly depends only on the solid content (in our work Φsand) and the Φmax in suspensions.Hence, the general form of this correlation equals μrel = f (Φsand; Φmax).
Among these models, Krieger and Dougherty [16] proposed a viscosity model correlating the relative viscosity μrel of the low to highly-concentrated suspensions with the sand solid content Φsand considering K as a dimensionless fitting parameter.
where μrel (-) is the relative viscosity and Φsand/Φmax (-) represents a normalised solid content and K is a dimensionless fitting parameter.This K parameter can be expressed in different forms.Maron and Pierce [34], for example, found that K=2 is sufficient to model the relative viscosity of low to highly-concentrated suspension.As well, Krieger-Dougherty [16] defined K=q•Φmax, making it possible for the researchers to include the non-circularity of the particles in the calculations using q.For monodisperse hard spheres, q=2.5 is usually considered.In addition, Φmax is another key parameter correlating with the maximum particle size, i.e., variation in PSD of the particles used in suspensions.For monodisperse hard spheres, it is commonly considered as 0.64 (-).In general, for random packing of uniform spheres, K = q•Φmax = 2.5•0.64 = 1.6.In literature also varying values for K between 1.4 to 3 for monodisperse suspensions are found [7,21].
This model is a known equation for modelling the relative viscosity of cement-based suspensions.This is straightforward to use and allows an almost accurate estimation of the viscosity of suspensions.To use this model, only basic information about the Φmax (depending on the PSD) and the solid content of the suspension is required [11,16].This expression is considering the following assumptions: A Newtonian fluid with a constant viscosity is involved, independent of the shear rate.In addition, the particles are non-interacting, spherical and uniformly distributed in the fluid phase, with no agglomeration.Also, the particles do not deform during shear.It only applies to suspensions in which the solid content of the particles is less than or equal to the Φmax.Besides, this model is developed for monodisperse particle suspensions [11,16].However, it can be used for polydisperse suspensions whereby the PSD is not broad as long as the particles are all spherical and do not interact with each other.Polydisperse spherical particles suspended in a fluid phase differ from monodisperse suspensions in two aspects [7,11,15]: The Φmax may be higher since finer particles can fill the space between the coarser particles.

−
During flow conditions, the small particles behave as a lubricant helping the flow of the coarser particles, resulting in a reduction of the overall viscosity.
As stated in (a) and (b), the relative viscosity is mainly linked to the Φmax, indicating that Φmax is a key focus of interest for the modelling of the polydisperse suspension.Also, it highlights the role of the finer particles in a suspension on the overall viscosity of the To guarantee the flexibility of the Krieger-Dougherty model in modelling polydisperse suspensions having a broader PSD, Liu proposed a fitting parameter to modify this model as follows [11,15,35]: where b is the adjustable parameter referring to the PSD.Based on his experimental results and a link that exists between the PSD and the Φmax, he concluded that b shows a significant correlation with Φmax.Therefore, the viscosity model improves as follows [15,35]: This model considers the effect of PSD on the Φmax, thereby advancing the flexibility of the Krieger and Dougherty model in modelling the polydispersity of the suspension.In this study, the two models Krieger and Dougherty and Liu were fitted to the experimental data and their accuracy in modelling a cement-based polydisperse suspension was discussed.Since the PSD of the different sands was defined and the d10 and d90 are both known, the Φmax was calculated in this work using equation 1.The values of Φmax and normalised Φsand/Φmax are reported in Table 3.As shown, the shear rates converted from the velocity profile (10 to 100 rpm) lie in the range of 15 to 60 s -1 for all samples investigated.In general, both rheological properties, plastic viscosity μ p (slope of the curve) and yield stress τ 0 (intersection point with the y-axis, shown with full symbols at a shear rate γ̇ = 0), rise as the sand solid content Φsand in cement-based mortars increases.As the sand's solid content Φsand becomes higher, it is more likely that particles may collide.From a rheological point of view, a higher sand solid content Φsand in the mortar results in more physical particle-particle interactions.This phenomenon restricts the flowability of the mortar, whereby both the yield stress and the plastic viscosity of the mortar containing a higher sand content Φsand are greater than those of the mortar containing a lower sand content Φsand and of the cement paste.

Rheological characterisation of cement-based mortars
Figure 9 demonstrates the variation in plastic viscosity μ p of various cement-based mortars (M1mm to M4mm) at two different w/c ratios for different normalised sand solid contents Φsand/Φmax; the lines are drawn for better observation only.Φsand/Φmax represents the normalised solid contents, i.e., Φsand (in our case 25-40 Vol-%) divided by Φmax.At first glance, it is observed that an increase of the w/c ratio from 0.44 (-) to 0.48 (-) induces a decrease in the measured plastic viscosity of the mortar (by a factor that varies between 0.5 to 0.65), regardless of the sand composition used in the mortar M1mm to M4mm.In addition, raising normalised sand solid content Φsand/Φmax results in an increase in plastic viscosity in all cases, beginning with a linear increase followed by an exponential rise at higher normalised solid content Φsand/Φmax. Figure 10 provides the plastic viscosity derived using the Reiner-Riwlin equation for M1mm to M4mm for various sand solid contents Φsand 25-40 Vol-% at two different w/c ratios; (a) 0.44 (-) and (b) 0.48 (-).It is shown in Figure 10 (a) that the plastic viscosity for M1mm with a finer maximum sand size is higher than for M2mm and M4mm at all sand solid contents.This difference is more pronounced as the sand's solid content rises, e.g., the plastic viscosity of M1mm decreases by around 10% for Φsand 25 Vol-% and 66% for Φsand 40 Vol-% when the sand composition of M1mm changed to M2mm and M4mm.This reduction has two main sources: First, increasing maximum sand size (increasing polydispersity) exist in mortar yields a higher maximum solid content and consequently a decline in plastic viscosity.Secondly, the proportion of finer particles in the mortar mix.However, no significant difference was recorded in the plastic viscosities of M2mm and M4mm. Figure 10 (b) reveals that this reduction in plastic viscosity for M1mm is seen at higher sand solid contents of 35 and 40 Vol-%.At sand solid contents below 30 Vol-%, no effect on the plastic viscosity is observable.Again, as seen in case (a), no differences are found between the plastic viscosities of M2mm and M4mm.
The variation of the yield stress of several cement-based mortars composed of two different w/c ratios, 3 different sand compositions M1mm to M4mm, and different normalised sand solid contents Φsand/Φmax is plotted in Figure 11; the lines are drawn just to make observation easier.First, it is concluded that an increase in the w/c ratio from 0.44 (-) to 0.48 (-) yields a decline in the measured yield stress τ 0 of the mortar (by a factor between 0.6 and 0.72), regardless of the sand used in mortar M1mm to M4mm.Moreover, the yield stress is highly dependent on normalised sand solid content Φsand/Φmax and exhibits, based on our results, a nearly linear correlation between yield stress and Φsand/Φmax for both cases of w/c ratios.Curiously, an increase in maximum sand size M1mm to M4mm (rising polydispersity) revealed a very small effect on the yield stress, and almost the same quantity of yield stress was measured for all three cement-based mortars M1mm to M4mm.

Application of a rheograph
The results are summarised in a rheograph, a tool allowing engineers to predict the effect of constituent variations on the rheological properties of cement-based mortars.This rheograph, Figure 12  As presented in Figure 12, both the plastic viscosity and the yield stress tend to increase as more sand is added to the mortar mix (increase in Φsand/Φmax).This implies that both rheological properties, the yield stress τ 0 and the plastic viscosity μ p , must be included whenever a decision needs to be taken on the optimisation of the mix design in terms of the flowability and stability of mortar or concrete.Another conclusion of this rheograph is that the plastic viscosity declines when the maximum sand size dmax in the cement-based mortar composition becomes coarser (e.g., if a certain number of finer particles are substituted by coarser sands), whereas the yield stress remains almost constant.This underlines the effect of the maximum sand size dmax on the maximum solid content and, in addition, the effect of the finer sands and their contribution to the mix formulation as they allow us to control the plastic viscosity of our mortar.The third key insight concerns the impact of the cement paste recipe on the mortar or concrete suspension.The results reveal that an increase in the w/c ratio (accompanied by a decrease in the cement content) reduces the plastic viscosity and yield stress.Although the reduction of cement may be beneficial from an ecological point of view, it may enhance the risk of instability (segregation) of the suspension during the process if no other additives such as stabilisers are used.Compared to Krieger and Dougherty, the Liu model, equation 8, includes an additional Φmax in its formulation, considering the PSD in the modelling.Looking at Figure 14, the Liu model agrees quite well with the experimental data (R²≥0.99),even at higher normalised solid contents.This means that the consideration of an adjustable parameter specifying the effect of the PSD improves the flexibility and accuracy of the model while modelling polydisperse suspensions.Compared with the Krieger and Dougherty model, the K value increases to 1.5 -1.7 for mortar suspensions.In general, both of these models are accurate enough to predict the relative viscosity of polydisperse suspensions.Between these two models, Liu's model yields a more precise prediction as it includes Φmax in its formulation, which correlates with PSD.

Conclusion
The article explores the effect of w/c ratio, sand content and maximum grains size on the rheological properties of flowable fresh mortars and also verifies the applicability of two existing models for modelling polydisperse cementbased mortars.The results were merged into a rheograph  An increase in the w/c ratio in cement paste reduces the plastic viscosity and yield stress.Nevertheless, this parameter has to be varied very cautiously in a suspension, as it might enhance the risk of material instability.

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The design of the sand composition in cement-based mortars affects the plastic viscosity considerably.Varying maximum sand size dmax (mixing of a wider range of different sand sizes) raises the Φmax and thus reduces the plastic viscosity.

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The effect of the finer particle fraction matters in the modelling of polydisperse suspensions.The higher proportion of finer particles in a suspension produces a larger plastic viscosity of the mortar.
The available viscosity models correlating the relative viscosity with the solid content in suspensions derived relying on many assumptions as commonly used for monodisperse, non-interacting, spherical and uniformly distributed particles.According to the results, both models, the Krieger-Dougherty model and the Liu model, are capable of modelling polydisperse suspensions.Since Liu includes an additional Φmax in its formulation, incorporating the PSD of the sand used in the mortar, a closer comparability between the model and the experimental data points is observed.
of particles in suspension, − Particle size distribution (PSD), − Maximum solid content, − Shape and density of the particles, − Surface roughness, − Amount of water absorbed by the particle surface, − Forces acting between the particles, − Agglomeration or clustering, − Viscosity of fluid phase.

Figure 1
Figure 1 Effect of the w/c ratio in cement paste on rheological properties; viscosity and yield stress

Figure 5
Figure 5 PSD of the different sand sizes; Fine (f), Medium (m) and Coarse (c); measured by QICPIC

Figure 6
Figure 6  gives an overview of the variable parameters included in the cement-based mortar compositions.2 different w/c ratios (from 0.44 (-) to 0.48 (-)), 4 different sand solid contents Φsand (between 25 and 40 Vol-%), and 3 variations in sand composition and maximum sand size, labelled M1mm to M4mm, comprise 3 axes of the cement-based mortar matrix.

Figure 6
Figure 6 Summary of the experimental matrix of cement-based mortars

Figure 7
Figure 7 Velocity profile used to determine the rheological properties of cement paste and cement-based mortars

Figure 8
Figure 8 presents the flow curve of cement paste (CP) with a w/c ratio of 0.44 (-) and corresponding cement-based mortars having variable sand solid content (Φsand 25-40 Vol-%) for M1mm.

Figure 8
Figure 8 Flow curve of cement paste with a w/c ratio 0.44 (-) and respective cement-based mortars containing variable sand solid content Φsand 25-40 Vol-% for M1mm

̇Figure 9
Figure 9 Plastic viscosity of different cement-based mortars (M1mm to M4mm) at two different w/c ratios and various normalised sand solid contents Φsand/Φmax; the lines are drawn only for easier observation.

Figure 11
Figure 11 Yield stress of different cement-based mortars (M1mm to M4mm) designed at two different w/c ratios for various normalised sand solid contents Φsand/Φmax; lines are drawn for a better observation only.
, illustrates the change in plastic viscosity and yield stress obtained from the flow curve as (a) Φsand/Φmax increases, (b) the w/c ratio increases, and (c) the maximum sand size (substituting finer sands by coarser ones) rises.It is noted that the range of shear rate pastes and cement-based mortars remains constant in all tests.

Figure 12
Figure 12Rheograph; principal representations indicating the effect of adding or changing various constituents in a reference material (in our case: cement paste).

Figure 13
Figure 13 Comparison of the experimental relative viscosity data for polydisperse suspensions with the predictions of the Krieger and Dougherty (equation 6) for three different cement-based mortars M1mm-M4mm

Figure 14
Figure 14 Comparison of the experimental rel.viscosity data for polydisperse suspensions with the predictions of the Liu (equation 8) for three different cement-based mortars M1mm-M4mm

Table 1
Sand composition in various cement-based mortars

Table 2
Mixing procedure involving the individual steps, mixing intensity and duration of each step

Table 3
Maximum solid content Φmax calculated using equation 1 and normalised solid content Φsand/Φmax for three different sand mixes in cement-based mortars
Krieger-Dougherty model, equation 6, was also fitted to these experimental data points.The fitted model agrees quite closely with the experimental data (R²≥0.96).However, the Krieger and Dougherty model fails in modelling the experimental data in some Φsand/Φmax data points like 0.66 (-) for M1mm or 0.61 (-) for M2mm.The K values, representing the exponent of the model, vary between 0.95 and 1.2.
Liu modelproviding the potential to pinpoint your current location in modelling cement-based suspensions, where you intend to go, and what is the best path to achieve your goal.The highlights for the investigated materials are:−Rising normalised sand solid content (Φsand/Φmax) in the mortar causes an increase in both, plastic viscosity and yield stress.−