Which Is an Appropriate Quadratic Rheological Model of Fresh Paste, the Modified Bingham Model or the Parabolic Model?

Abstract

The physical meaning and calculating process of the rheological parameters of two nonlinear rheological models, the parabolic model and the modified Bingham model, were compared. The fluidity test and a rheological experiment on cementitious materials were performed. The Couette inverse problem is a key issue in measuring and solving the rheological parameters of fresh cementitious materials. The solution of the Couette inverse problem based on the modified Bingham model is discontinuous when the coefficient of the quadratic term is equal to zero, resulting in a large deviation between the fitting curve and the rheological experimental data. The credibility of the rheological parameters of the pastes calculated based on the modified Bingham model is low. The formulas for calculating yield stress, fiducial differential viscosity and the degree of shear thickening or shear thinning of the parabolic model have been developed. The credibility of the rheological parameters of the pastes calculated based on the parabolic model is high. The flow performance of the paste can be clearly characterized by the rheological parameters calculated with the parabolic model.

1. Introduction

Rheology, as a science to describe the flow and deformation of fluid, can be used not only in the study of the properties of hardened concrete such as creep [1], but also in the evaluation on the performance of fresh cementitious materials [2,3,4]. The rheological properties of fresh cementitious materials can be described by an appropriate rheological model. At present, the Bingham model is the most commonly utilized linear rheological model. The rheological expression of the Bingham model is simple. Two parameters in the Bingham model represent the yield stress and plastic viscosity of fresh cementitious materials [5,6]. With the comprehensive use of mineral admixtures and chemical admixtures, the rheological behavior of fresh cementitious materials deviates gradually from those described by the Bingham model [7,8,9,10,11,12]. In recent years, many studies [13,14,15,16] have shown that fresh cementitious materials exhibit shear thickening or shear thinning. Shear thickening or shear thinning describes the increase or decrease, respectively, in the apparent viscosity of a fluid with the increase in the shear stress [6,17]. The apparent viscosity of fresh cementitious materials with shear thickening or shear thinning is not constant, which corresponds to a nonlinear relationship between the shear stress and shear strain rate under the steady flow state. The expression of the Bingham model must be modified to accurately describe the nonlinear rheological behavior of fresh cementitious materials. The modified Bingham model introduces a quadratic term, and Herschel–Bulkley model introduces an exponential term, and these are the two most famous nonlinear rheological models [18,19,20,21,22].

How to calculate the rheological parameters based on these two nonlinear rheological models is the core problem of rheological research and the application of cementitious materials. The Couette inverse problem deals with how to transform the relationship between the shear strain rate and shear stress in the rheological models into a relationship between the measured rotational speed and torque of a fluid [23,24]. The solution of the Couette inverse problem connects the theoretical rheological model with the measurement and calculation of the rheological parameters of a fluid. The solution of the Couette inverse problem for the rheological models of various cementitious materials has been solved [23]. It has been proven that the solution of the Couette inverse problem based on Herschel–Bulkley model does not have an analytical solution in the real range. This means that the rheological parameters based on Herschel–Bulkley model cannot be calculated. In order to study the nonlinear rheological properties of cementitious materials, a new exponential rheological model was proposed to replace Herschel–Bulkley model [13].

The rheological expression of the modified Bingham model is shown in Equation (1) [21,22,25,26]. In order to be distinguished from the parameters of the parabolic model in this paper, the letter “s” is used as the coefficient of a second-order correction term in the shear strain rate in the modified Bingham model. Shear thickening occurs when the parameter s is higher than zero. Shear thinning occurs when the parameter s is less than zero. The Bingham model is obtained when the parameter s equals zero [27]. The solution of the Couette inverse problem based on the modified Bingham model exists. When the coefficient “s” of the quadratic term is not equal to zero, its analytical solution of the Couette inverse problem is shown in Equation (2) [23]. When the coefficient “s” of the quadratic term is equal to zero, the modified Bingham model is the Bingham model, and its analytical solution of the Couette inverse problem is the Reiner–Riwlin equation, as shown in Equation (3) [23,24,27,28,29].

$$\begin{array}{cc}\tau ={\tau }_0+\mu \cdot \frac{d\gamma }{dt}+s\cdot {\left(\frac{d\gamma }{dt}\right)}^2& \left(\tau \ge {\tau }_0\right)\\ \frac{d\gamma }{dt}=0& (0\le \tau <{\tau }_0)\end{array}$$

(1)

$$\begin{array}{l}\Omega =\frac{\mu }{2s}\mathit{ln}\frac{R_1}{R_2}+\frac{1}{2s}\cdot (\sqrt{{\mu }^2-4s{\tau }_0+2s\frac{T}{\pi h{R}_1^2}}\\ -\sqrt{{\mu }^2-4s{\tau }_0+2s\frac{T}{\pi h{R}_2^2}}\\ +\sqrt{4s{\tau }_0-{\mu }^2}\cdot \mathit{arctan}\sqrt{\frac{{\mu }^2-4s{\tau }_0+2s\frac{T}{\pi h{R}_1^2}}{4s{\tau }_0-{\mu }^2}}\\ -\sqrt{4s{\tau }_0-{\mu }^2}\cdot \mathit{arctan}\sqrt{\frac{{\mu }^2-4s{\tau }_0+2s\frac{T}{\pi h{R}_2^2}}{4s{\tau }_0-{\mu }^2}})\end{array}$$

(2)

$$\Omega =\frac{1}{4\pi h\mu }\left(\frac{1}{R_1^2}-\frac{1}{R_2^2}\right)\cdot T-\frac{{\tau }_0}{\mu }\mathit{ln}\frac{R_2}{R_1}$$

(3)

Since the analytical solution of the Couette inverse problem exists, can the modified Bingham model well characterize the rheological properties of cementitious materials with the performance of shear thickening or shear thinning? If so, how to calculate the rheological parameters? How accurate are the calculated rheological parameters? If not, are there other quadratic rheological models that can characterize the nonlinear rheological properties of cementitious materials? In order to answer these problems, the characteristics of two quadratic rheological models were analyzed and compared based on the rheological experiment of cement–fly ash–silica fume pastes.

2. Measurement and Calculation of Rheological Parameters

2.1. The Rheological Scheme for Measurement of Rheological Parameters

The rheological parameters represent the rheological behavior of a fluid under the steady flow state [30,31]. Therefore, it is necessary to create a steady flow state when establishing a rheological scheme. The rheological scheme suitable for measuring the rheological parameters of fresh cementitious materials is the one recommended by Wallevik [32], which is similar to that shown in Figure 1 [12,33,34,35]. The number of steps in Figure 1 can be appropriately adjusted as needed. A steady flow state is created through a fixed rotational speed on the steps. According to this rheological scheme, the rotational speed of the rheometer is stabilized immediately, but the stabilization of the measured torque on a step takes a little time [33,36,37]. Fresh cementitious materials in the rheological experiment are thought to be in a steady flow state only when both the rotational speed and torque are stabilized simultaneously. The stabilized rotational speed and torque are valid data to calculate the rheological parameters of fresh cementitious materials. At each step, an effective data pair of rotational speed and torque can be obtained under a steady flow state. As shown in Figure 1, the more steps that have been set in the rheological scheme, the more data pairs that will be obtained. In the “rotational speed-torque” diagram, these valid data pairs are discrete points.

2.2. Importance of the Solution of the Couette Inverse Problem

As the simplest rheometer, the coaxial cylinder rheometer is very representative in the calculation of the rheological parameters. There are three ideas which can be used to calculate the rheological parameters of fresh cementitious materials with the data measured by a coaxial cylinder rheometer.

Rheometer can only directly measure the rotational speed and torque of a fluid but not the shear stress and shear strain rate. However, most of the rheometers can offer the value of shear stress and shear strain rate corresponding to the measured rotational speed and torque at each measurement moment. The first idea is to use the shear stress and shear strain rate directly fed back by the rheometer to calculate the rheological parameters. Therefore, it is necessary to understand the internal program and calculation formula of a rheometer to calculate the shear stress and shear strain rate. For example, the shear stress and shear strain rate are calculated according to the formulas (as shown in Equations (4) and (5)) in Appendix A of the standard in [38] in some rheometers. This is a standard for determining the viscosity of liquid plastic, polymer, resin and similar fluid materials, which is not applicable to the calculation of the rheological parameters of fresh cementitious materials. Fresh cementitious materials usually display a yield stress, whereas the standard [38] is for fluids without a yield stress. In fact, there is no standard for measuring and calculating the rheological parameters of fresh cementitious materials at present.

$${\tau }_{\mathrm{rep}}=\frac{1+{\delta }^2}{2{\delta }^2}\times \frac{T}{2\pi h{R}_1^2{C}_h}$$

(4)

$$\frac{d{\gamma }_{\mathrm{rep}}}{dt}=\omega \times \frac{1+{\delta }^2}{{\delta }^2-1}$$

(5)

In general, when using a coaxial cylinder rheometer to measure the rheological parameters of fresh cementitious materials, both the shear stress and shear strain rate have clear calculation formulas, as shown in Equations (6) and (7), respectively [23,24]. It is easy to calculate the shear stress according to Equation (6), but it cannot directly calculate the shear strain rate according to Equation (7), and the reason for this is that the reason that the valid data pairs of rotational speed and torque measured by the rheometer are discontinuous from each other and there are no derivatives or differentials in the discontinuous data. Some literature studies [27,39] have shown that some approximate calculation formulas can be used to directly calculate the value of the shear strain rate through the measured rotational speed, which is the second idea of the rheological parameters calculation. For example, in a coaxial cylinder rheometer, if R₁/R₂ ≥ 0.99, the shear strain rate can be approximately calculated by Equation (8). If R₁/R₂ < 0.99, the shear stress rate can be approximately calculated by using Equations (9) and (10). However, Equation (8) can only be used when the ratio of the yield stress to the plastic viscosity is relatively small. Because of the restriction of R₁/R₂ ≥ 0.99, Equation (8) can only be used to calculate the rheological parameters of fresh paste, but it cannot be used to calculate the rheological parameters of fresh mortar and concrete. In addition, Equations (9) and (10) cannot be directly used to calculate the shear strain rate of a fluid with a yield stress since the parameter n in Equation (10) cannot be calculated because the valid data pairs of rotational speed and torque measured are discontinuous from each other [27].

$$\tau =\frac{T}{2\pi h{r}^2}$$

(6)

$$\frac{d\gamma }{dt}=r\frac{d\omega }{dr}$$

(7)

$$\frac{d{\gamma }_1}{dt}=\frac{2\Omega R_2^2}{R_2^2-R_1^2}$$

(8)

$$\frac{d{\gamma }_1}{dt}=\frac{2\Omega }{n\left[1-{\left(\frac{R_1}{R_2}\right)}^{2/n}\right]}$$

(9)

$$n=\frac{d\mathit{ln}T}{d\mathit{ln}\Omega }$$

(10)

In fact, the most suitable method to calculate the rheological parameters of fresh cementitious materials is to use the solution of the Couette inverse problem [23,24,27,28], which is the third idea of the rheological parameters calculation. For example, if the rheological properties of fresh cementitious materials can be described by the Bingham model, the rheological parameters can be solved by using Reiner–Riwlin equation (Equation (3)) [23,24,27,28,29], which is the solution of the Couette inverse problem based on the Bingham model. If the rheological properties of fresh cementitious materials need to be described by a nonlinear rheological model, the rheological parameters can be solved by using the solution of the Couette inverse problem based on the nonlinear rheological models. A general method for solving the Couette inverse problem of some common rheological models is given in [23]. It is helpful for solving the rheological parameters of fresh cementitious materials.

2.3. The Reasonable Calculation Process of Rheological Parameters

A reasonable process for the calculation of the rheological parameters is as follows [33,35]. First, the relationship between the rotational speed and the torque can be obtained according to existing rheological models describing the rheological behavior of a fluid, which is the process of solving the Couette inverse problem. Second, the calculated relationship between the rotational speed and the torque is compared with the value of the rotational speed and the torque measured using a rheometer to determine the appropriate rheological model for the fluid. Finally, the rheological parameters are calculated according to the appropriate rheological model. Therefore, the solution of the Couette inverse problem is the key to determine the method of measurement and calculation of the rheological parameters.

3. The Parabolic Model

3.1. The Parabolic Model and the Solution of the Couette Inverse Problem

The parabolic model is also one of the rheological models of fresh cementitious materials [23,40]. However, this rheological model is far less commonly utilized than the modified Bingham model is to characterize the nonlinear rheological behavior of fresh cementitious materials. The attention to the parabolic model is insufficient in the civil engineering field.

The expression of the parabolic model is shown in Equation (11). This rheological model is also a quadratic rheological model with three parameters [23,40]. The shear stress in the parabolic model is an independent variable, whereas the shear strain rate is the dependent variable.

$$\begin{array}{cc}\frac{d\gamma }{dt}=a+b\cdot \tau +c\cdot {\tau }^2& \left(\tau \ge {\tau }_0\right)\\ \frac{d\gamma }{dt}=0& (0\le \tau <{\tau }_0)\end{array}$$

(11)

The solution of the Couette inverse problem based on the parabolic model is shown in Equation (12) [23], which is much simpler than the one based on the modified Bingham model (Equation (2)). The parameters of the parabolic model can be easily calculated by combining the method of nonlinear fitting and the solution of the Couette inverse problem based on the parabolic model.

$$\Omega =a\mathit{ln}\frac{R_2}{R_1}+\frac{b}{4\pi h}\left(\frac{1}{R_1^2}-\frac{1}{R_2^2}\right)\cdot T+\frac{c}{16{\pi }^2{h}^2}\left(\frac{1}{R_1^4}-\frac{1}{R_2^4}\right)\cdot T^2$$

(12)

3.2. The Rheological Parameters in the Parabolic Model

To study the rheological behavior of fresh cementitious materials using the parabolic model, the significance of the rheological parameters in the parabolic model should be analyzed. The parabolic model has two rheological parameters to describe the yield stress and viscosity of a fluid, which is similar to the Bingham model [5,6], and the third parameter describes the shear thickening or shear thinning. Therefore, three rheological parameters in the parabolic model need to be analyzed.

3.2.1. Yield Stress

The yield stress in a rheological model refers to the minimum stress required to maintain fluid flow. Its expression can be shown in Equation (13) [41,42].

$$\begin{array}{cc}{\tau }_0=\tau & \left(\frac{d\gamma }{dt}=0\right)\end{array}$$

(13)

According to Equation (13), the yield stress is the shear stress when the shear strain rate is equal to zero. The following equation can be gained by combining Equations (11) and (13).

$$a+b\cdot {\tau }_0+c\cdot {\tau }_0^2=0$$

(14)

When parameter c is equal to zero, the parabolic model turns into the Bingham model. When parameter c is not zero, Equations (15) and (16) can be obtained according to Equation (14).

$${\tau }_0=\frac{-b+\sqrt{b^2-4ac}}{2c}$$

(15)

$${\tau }_0=\frac{-b-\sqrt{b^2-4ac}}{2c}$$

(16)

Because the root of Equation (14) exists, the following expression can be gained.

$$b^2-4ac\ge 0$$

(17)

Because the yield stress of a fluid is unique, it is necessary to select either Equation (15) or (16) as the unique value for the yield stress.

When the shear stress is not less than the yield stress is, the expression of the parabolic model can be as shown in Equation (18). Generally, when the rheological parameters of fresh cementitious materials are measured, the greater the shear stress is, the greater the shear strain rate is. In other words, the function shown in Equation (18) is a monotonically increasing function, which means that the derivative function of Equation (18) should not be less than zero. The derivative function of Equation (18) can be expressed in Equation (19). According to Equation (19), the derivative function based on the yield stress can be expressed as in Equation (20).

$$\begin{array}{cc}\frac{d\gamma }{dt}=f\left(\tau \right)=a+b\cdot \tau +c\cdot {\tau }^2& \left(\tau \ge {\tau }_0\right)\end{array}$$

(18)

$$\begin{array}{cc}f\prime \left(\tau \right)=b+2c\cdot \tau & \left(\tau \ge {\tau }_0\right)\end{array}$$

(19)

$$f\prime \left({\tau }_0\right)=b+2c\cdot {\tau }_0\ge 0$$

(20)

By replacing Equation (20) with Equation (15), the following relationships can be gained.

$$\begin{array}{l}f\prime \left({\tau }_0\right)=b+2c\cdot {\tau }_0\\ =b+2c\cdot \frac{-b+\sqrt{b^2-4ac}}{2c}\\ =\sqrt{b^2-4ac}\ge 0\end{array}$$

(21)

By replacing Equation (20) with Equation (16), the following relationships can be gained.

$$\begin{array}{l}f\prime \left({\tau }_0\right)=b+2c\cdot {\tau }_0\\ =b+2c\cdot \frac{-b-\sqrt{b^2-4ac}}{2c}\\ =-\sqrt{b^2-4ac}\le 0\end{array}$$

(22)

It can be seen from Equations (21) and (22) that Equation (15) can be used as the calculation formula for the yield stress of the parabolic model under the condition that parameter c is not zero.

3.2.2. Fiducial Differential Viscosity

The parabolic model can describe the properties of a fluid with shear thickening or shear thinning. The viscosity of the fluid with shear thickening or shear thinning is not constant. Therefore, the concept of plastic viscosity in the Bingham model cannot be used to describe viscosity in the parabolic model.

In the nonlinear rheological model, there are two concepts describing viscosity: apparent viscosity and differential viscosity [6,17]. Apparent viscosity indicates the slope of the line drawn from any point on the rheological curve to the origin of the coordinate. Differential viscosity indicates the slope of the tangent at each point on the rheological curve. The parabolic model can be used to describe the yield stress of the fluid, which means that the rheological curve does not pass through the coordinate origin. It is appropriate to use the concept of differential viscosity to study the viscosity of fresh cementitious materials in the parabolic model.

According to the definition for differential viscosity and the expression of the parabolic model, the differential viscosity under the condition that the shear stress is no less than the yield stress can be expressed as follows.

$${\eta }_d=\frac{d\tau }{\left(\frac{d\gamma }{dt}\right)}=\frac{1}{\frac{\left(d\gamma /dt\right)}{d\tau }}=\frac{1}{b+2c\tau }$$

(23)

The differential viscosity shown in Equation (23) is expressed as a function of the shear stress other than the shear strain rate. The shear stress can be easily calculated according to Equation (6) based on the rheological experimental data.

According to Equation (23), the differential viscosity is a variable related to the shear stress, which reflects the nonlinear characteristics of the parabolic model. Only when the rheological parameters based on the parabolic model are constants can they be easily measured by the rheological experiments and compared with other fluids. As shown in Equation (23), the differential viscosity shown in the parabolic model is independent of the shear stress when parameter c is equal to zero. Under this situation, the parabolic model turns into the Bingham model, and it is no longer capable of characterizing the properties of the shear thickening or shear thinning of a fluid. Therefore, the fiducial differential viscosity can be defined as the differential viscosity when parameter c equals zero. The fiducial differential viscosity can be regarded as an independent physical parameter of the parabolic model, and it can be calculated according to Equation (24).

$${\eta }_{d,0}=\frac{1}{b}$$

(24)

3.2.3. The Degree of Shear Thickening or Shear Thinning

It can be seen from Equation (23) that parameter c is closely related to the degree of shear thickening or shear thinning. When parameter c is positive, the differential viscosity of the parabolic model decreases with an increase in the shear stress. This reflects the characteristics of shear thinning. The higher the value of parameter c is, the higher the degree of shear thinning. When parameter c is negative, the differential viscosity of the parabolic model increases with an increase in shear stress, which reflects the characteristics of shear thickening. The smaller the value of parameter c is, the higher the degree of shear thickening is. When parameter c is equal to zero, the differential viscosity of the parabolic model is unaffected by shear stress. In this situation, the fluid does not exhibit shear thickening or shear thinning.

3.2.4. Dimension of Rheological Parameters

According to Equation (11) and the method of dimensional analysis, the dimension of parameter a is s⁻¹, the dimension of parameter b is Pa⁻¹ s⁻¹ and the dimension of parameter c is Pa⁻² s⁻¹.

According to Equation (15), the dimension of the yield stress in the parabolic model is Pa. According to Equation (24), the dimension of the fiducial differential viscosity in the parabolic model is Pa·s. The dimension of the degree of shear thickening or shear thinning in the parabolic model is Pa⁻² s⁻¹.

4. Experimental Study on Rheological Parameters of Pastes Based on Two Rheological Models

4.1. Raw Materials and Mix Proportion

PI 42.5 cement (Cement), Class II fly ash (FA) and silica fume (SF) were chosen to be used in this study. The specific surface areas of Cement and SF were 352 m²/kg (Blaine) and 23.40 m²/g (BET), respectively. The 45 µm square hole screen residual of FA was 30.8%. The oxide compositions of raw materials obtained from X-ray fluorescence (XRF) are listed in

Table 1. Chemical composition of raw materials/%.

Composition	SiO2	Al2O3	Fe2O3	CaO	MgO	SO3	Na2Oeq	Total
Cement	14.52	3.39	4.32	71.66	1.30	2.30	0.73	98.24
FA	46.52	36.52	6.37	4.58	0.46	1.02	1.04	96.51
SF	96.28	0.30	0.05	0.57	0.34	1.02	0.93	99.49

Note: Na₂O_eq includes Na₂O and 0.658K₂O.

. The morphologies of the powders determined by scanning electron microscopy (SEM) are shown in Figure 2.

A polycarboxylate superplasticizer (SP) was utilized in the study. The solid content of SP is 56.90%.

The mix proportions of the pastes are displayed in

Table 2. Mix proportions of pastes.

Series No.	Composition of Binders/%			w/b	SP/%
	Cement	FA	SF
P1-1	90.91	0.00	9.09	0.26	1.60
P1-2	83.01	8.69	8.30	0.26	1.60
P1-3	75.12	17.37	7.51	0.26	1.60
P1-4	67.22	26.06	6.72	0.26	1.60
P1-5	59.32	34.75	5.93	0.26	1.60
P1-6	51.42	43.43	5.14	0.26	1.60
P1-7	43.53	52.12	4.35	0.26	1.60
P2-1	63.06	36.94	0.00	0.26	1.60
P2-2	62.13	36.39	1.48	0.26	1.60
P2-3	61.19	35.84	2.97	0.26	1.60
P2-4	60.26	35.29	4.45	0.26	1.60
P2-5	59.32	34.75	5.93	0.26	1.60
P2-6	58.39	34.20	7.41	0.26	1.60
P2-7	57.45	33.65	8.90	0.26	1.60

Note: w/b is the water/binder weight ratios. SP/% is the proportion of polycarboxylate superplasticizer to the total amount of cement, FA and SF.

. Thirteen mix proportions of pastes were prepared since P1-5 and P2-5 have same composition. They were used as a reference. The water/binder ratio and the dosage of SP of each paste were 0.26 and 1.60%, respectively. According to the single factor control method, the mix proportions of pastes P1-1 to P1-7 were determined by changing the content of FA in the cementitious materials under the condition of taking cement and SF as a whole. The mix proportions of pastes P2-1 to P2-7 were determined by changing the content of SF in the cementitious materials under the condition of taking cement and FA as a whole. The descriptions in brackets in Table 2 indicate the proportional relationship between the FA or SF content in each group and that in the control group. The paste was mixed according to the Chinese national standard GB/T 1346-2011. The mixing time of each group remained the same.

4.2. Instruments and Test Procedure

A coaxial cylinder rheometer was utilized in this research. The diameter of the outer cylinder was 60 mm. The effective height and the diameter of the inner cylinder were 40 mm and 20 mm, respectively. During the experiment, the outer cylinder was stationary, and the inner cylinder rotated. The utilized rheological scheme proposed by Wallevik [12,32,33,34,35] is shown in Figure 1. As described in Section 2.1, the effective data pairs collected after the rheological experiment are plotted in the “rotational speed-torque” diagram, as shown in Figure 3.

The fluidity of the paste was tested according to the Chinese national standard GB/T 8077-2000.

4.3. Rheological Parameter Fitting Calculation Method

4.3.1. Modified Bingham Model

The solution of the Couette inverse problem based on the modified Bingham model under the condition that the coefficient “s” of the quadratic term is not equal to zero (Equation (2)) is very complex. In order to ensure the smooth progression of the fitting calculation process, some variables were replaced in Equation (2). The variable replacement equation is shown in Equation (25). After the variable replacement, Equation (2) can be expressed as Equation (26). By substituting the rotor size parameters of the rheometer described in Section 4.2 into Equation (26), Equation (27) can be obtained.

$$\begin{array}{l}\frac{2s}{\pi h{R}_2^2}=A\\ \frac{\mu }{2s}\mathit{ln}\frac{R_2}{R_1}=B\\ \begin{array}{cc}\sqrt{4s{\tau }_0-{\mu }^2}=C& \left(C\ge 0\right)\end{array}\end{array}$$

(25)

$$\begin{array}{l}\Omega =-B+\frac{1}{\pi h{R}_2^2}\cdot \frac{1}{A}\left(\sqrt{\frac{R_2^2}{R_1^2}AT-C^2}-\sqrt{AT-C^2}\right)\\ +\frac{1}{\pi h{R}_2^2}\cdot \frac{C}{A}\left(\mathit{arctan}\frac{\sqrt{\frac{R_2^2}{R_1^2}AT-C^2}}{C}-\mathit{arctan}\frac{\sqrt{AT-C^2}}{C}\right)\end{array}$$

(26)

$$\begin{array}{l}\Omega =-B+\frac{250000}{9\pi }\cdot \frac{1}{A}\left(\sqrt{9AT-C^2}-\sqrt{AT-C^2}\right)\\ +\frac{250000}{9\pi }\cdot \frac{C}{A}\left(\mathit{arctan}\frac{\sqrt{9AT-C^2}}{C}-\mathit{arctan}\frac{\sqrt{AT-C^2}}{C}\right)\end{array}$$

(27)

The initial values for the parameters of nonlinear fitting need to be given before the iterative calculation can be performed. In order to reduce the number of iterations and improve the accuracy of the calculation results, the initial values were determined according to the following analysis and the rheological test results in Figure 3.

Generally, pastes have a yield stress, so in Equation (1) is greater than zero. As the shear stress increases, the shear strain rate also increases, so μ in Equation (1) is greater than zero. The results shown in Figure 3 show that the pastes show the performance of shear thickening to a certain extent, so s in Equation (1) is greater than zero. Therefore, the initial values, μ and s in the fitting calculation can be set equal to 1, and their units are international units. The initial values of A, B and C can be calculated according to the rotor size parameters of the rheometer and Equation (25). The results are shown in Equation (28).

$$\begin{array}{l}{\tau }_0=1\\ \mu =1\\ s=1\\ A\approx 1.769\times {10}^4\\ B\approx 0.5493\\ C\approx 1.732\end{array}$$

(28)

4.3.2. The Parabolic Model

Similarly, variable substitution is also performed for rheological model expression (Equation (12)), as shown in Equation (29). After the variable replacement, Equation (12) can be expressed as Equation (30).

$$\begin{array}{l}a\mathit{ln}\frac{R_2}{R_1}=A\\ \frac{b}{4\pi h}\left(\frac{1}{R_1^2}-\frac{1}{R_2^2}\right)=B\\ \frac{c}{16{\pi }^2{h}^2}\left(\frac{1}{R_1^4}-\frac{1}{R_2^4}\right)=C\end{array}$$

(29)

$$\Omega =A+B\cdot T+C\cdot T^2$$

(30)

Similarly, pastes have a yield stress, so a in Equation (11) is less than zero. As the shear stress increases, the shear strain rate also increases, so b in Equation (1) is greater than zero. The results shown in Figure 3 show that the pastes show the performance of shear thickening to a certain extent, so c in Equation (1) is less than zero. Therefore, the initial values a, b and c in the fitting calculation can be set to −1, 1, −1, respectively, and their units are international units. The initial values of A, B and C can be calculated according to the rotor size parameters of the rheometer and Equation (29). The results are shown in Equation (31).

$$\begin{array}{l}a=-1\\ b=1\\ c=-1\\ A\approx -1.732\\ B\approx 1.768\times {10}^4\\ C\approx -3.909\times {10}^8\end{array}$$

(31)

4.4. Results and Discussion

4.4.1. Comparison of Fitting Calculation Results Based on Two Rheological Models

The modified Bingham model and the parabolic model are used to fit the rheological results (Figure 3) of the pastes. Figure 4 shows the comparison of the fitting results of P1 between modified Bingham model and parabolic model. In Figure 4, the scatter points are the measured values; the red curve is the fitting result based on the modified Bingham model, and the blue curve is the fitting result based on the parabolic model. It can be clearly seen that the fitting results based on the modified Bingham model of pastes P1-1, P1-2, P1-4 and P1-5 have a significant deviation from the rheological experimental data. The deviation between the fitting results based on modified Bingham model and the rheological experimental data of pastes P1-3, P1-6 and P1-7 is small. The deviations between the fitting results based on the parabolic model and the rheological test data are relatively small for all of the pastes.

Table 3. Fitting data of P1 based on modified Bingham model and parabolic model.

Series No.	Model	Modified Bingham Model		Parabolic Model
	Fitting Data	Value	Standard Error	Value	Standard Error
P1-1	Reduced Chi-Sqr	0.4178	/	0.0011	/
	Adj. R-Square	0.8615	/	0.9996	/
	Parameter A	1.802 × 104	3.155 × 104	−9.017 × 10−1	5.944 × 10−2
	Parameter B	1.196	8.526	3.515 × 103	8.003 × 101
	Parameter C	−7.096 × 10−6	2.909 × 106	−3.329 × 105	2.375 × 104
P1-2	Reduced Chi-Sqr	0.6408	/	0.0038	/
	Adj. R-Square	0.7876	/	0.9987	/
	Parameter A	1.708 × 104	3.082 × 104	−8.756 × 10−1	1.056 × 10−1
	Parameter B	8.566 × 10−1	8.661	3.931 × 103	1.536 × 102
	Parameter C	7.332 × 10−5	2.517 × 105	−4.620 × 105	4.873 × 104
P1-3	Reduced Chi-Sqr	0.0370	/	0.0069	/
	Adj. R-Square	0.9877	/	0.9977	/
	Parameter A	1.200 × 104	9.411 × 102	−5.822 × 10−1	1.166 × 10−1
	Parameter B	1.954	2.635 × 10−1	4.269 × 103	1.841 × 102
	Parameter C	8.365 × 10−7	2.636 × 105	−6.328 × 105	6.150 × 104
P1-4	Reduced Chi-Sqr	0.3277	/	0.0075	/
	Adj. R-Square	0.8914	/	0.9975	/
	Parameter A	1.560 × 104	1.554 × 104	−4.355 × 10−1	1.132 × 10−1
	Parameter B	5.542 × 10−1	4.246	5.064 × 103	2.178 × 102
	Parameter C	1.463 × 10−5	4.499 × 105	−9.190 × 105	8.780 × 104
P1-5	Reduced Chi-Sqr	0.3550	/	0.0070	/
	Adj. R-Square	0.8823	/	0.9977	/
	Parameter A	1.376 × 104	1.487 × 104	−3.184 × 10−1	1.071 × 10−1
	Parameter B	3.636 × 10−1	4.253	6.244 × 103	2.689 × 102
	Parameter C	−3.585 × 10−5	1.324 × 105	−1.377 × 106	1.422 × 105
P1-6	Reduced Chi-Sqr	0.0332	/	0.0132	/
	Adj. R-Square	0.9890	/	0.9956	/
	Parameter A	1.026 × 104	1.299 × 103	1.377 × 10−1	1.239 × 10−1
	Parameter B	1.282	5.083 × 10−1	4.980 × 103	2.845 × 102
	Parameter C	8.857 × 10−7	3.697 × 105	−9.491 × 105	1.333 × 105
P1-7	Reduced Chi-Sqr	0.0476	/	0.0047	/
	Adj. R-Square	0.9842	/	0.9984	/
	Parameter A	1.158 × 104	1.465 × 103	1.556 × 10−1	6.937 × 10−2
	Parameter B	1.067	4.073 × 10−1	4.797 × 103	1.465 × 102
	Parameter C	4.333 × 10−7	6.176 × 105	−9.072 × 105	6.327 × 104

exhibits the fitting data of P1 based on the modified Bingham model and the parabolic model. Taking paste P1-3 as an example, the adj. R-Square (an index to evaluate the degree of fitting) of the fitting curve in Figure 4c based on the modified Bingham model is 0.9877, which is smaller than the adj. R-Square value 0.9977 of the fitting curve in Figure 4c based on the parabolic model. In addition, the maximum standard error of the three parameters calculated based on the modified Bingham model is larger than four times than that calculated based on the parabolic model. These results indicate that the fitting effect of the parabolic model on the rheological results of paste P1-3 is better than the modified Bingham model is. Similarly, the fitting effect of the parabolic model on the rheological results of pastes P1-6 and P1-7 is better than that of the modified Bingham model based on the fitting results in Figure 4f,g.

As mentioned above, the deviations between the fitting results based on the modified Bingham model and the rheological experimental data of pastes P1-1, P1-2, P1-4 and P1-5 are more significant. Taking paste P1-1 as an example, the adj. R-Square of the fitting curve in Figure 4a based on the modified Bingham model is 0.8615 (see row 4 of Table 3), which is smaller than the adj. R-Square value 0.9996 (see row 4 of Table 3) of the fitting curve based on the parabolic model. In addition, the maximum standard error of the three parameters calculated with the modified Bingham model is larger than 100 times of that which was calculated with the parabolic model, as shown in Table 3. Similar conclusions are obtained for pastes P1-2, P1-4 and P1-5. Therefore, the modified Bingham model cannot well fit to characterize the rheological properties of the paste.

The adj. R-Square of the fitting curves based on the parabolic model of all of the pastes (P1) in Figure 4 are higher than 0.9950 (Table 3), which also indicates that the rheological properties of the pastes can be well fitted by the parabolic model.

Figure 5 shows the comparison of fitting results of P2 between the modified Bingham model and the parabolic model.

Table 4. Fitting data of P2 based on modified Bingham model and parabolic model.

Series No.	Model	Modified Bingham Model		Parabolic Model
	Fitting Data	Value	Standard Error	Value	Standard Error
P2-1	Reduced Chi-Sqr	0.0126	/	0.0016	/
	Adj. R-Square	0.9958	/	0.9995	/
	Parameter A	1.033 × 104	1.294 × 103	−7.614 × 10−1	6.093 × 10−2
	Parameter B	2.626	7.140 × 10−1	4.081 × 103	9.041 × 101
	Parameter C	2.565 × 10−6	2.649 × 105	−5.372 × 105	2.902 × 104
P2-2	Reduced Chi-Sqr	0.0715	/	0.0097	/
	Adj. R-Square	0.9763	/	0.9968	/
	Parameter A	1.904 × 104	4.027 × 103	−1.374 × 10−1	1.145 × 10−1
	Parameter B	1.176	1.727	5.383 × 103	2.532 × 102
	Parameter C	9.490 × 10−6	1.509 × 105	−1.089 × 106	1.156 × 105
P2-3	Reduced Chi-Sqr	0.3619	/	0.0091	/
	Adj. R-Square	0.8800	/	0.9970	/
	Parameter A	1.352 × 104	1.501 × 104	−2.495 × 10−1	1.183 × 10−1
	Parameter B	2.262 × 10−1	4.197	6.901 × 103	3.306 × 102
	Parameter C	2.192 × 10−5	1.963 × 105	−1.739 × 106	1.920 × 105
P2-4	Reduced Chi-Sqr	0.3184	/	0.0078	/
	Adj. R-Square	0.8944	/	0.9974	/
	Parameter A	1.370 × 104	1.357 × 104	−1.898 × 10−1	1.087 × 10−1
	Parameter B	3.230 × 10−1	3.791	6.367 × 103	2.896 × 102
	Parameter C	−2.797 × 10−5	1.450 × 105	−1.469 × 106	1.605 × 105
P2-5	Reduced Chi-Sqr	0.3550	/	0.0070	/
	Adj. R-Square	0.8823	/	0.9977	/
	Parameter A	1.376 × 104	1.487 × 104	−3.184 × 10−1	1.071 × 10−1
	Parameter B	3.636 × 10−1	4.253	6.244 × 103	2.689 × 102
	Parameter C	−3.585 × 10−5	1.324 × 105	−1.377 × 106	1.422 × 105
P2-6	Reduced Chi-Sqr	0.3464	/	0.0073	/
	Adj. R-Square	0.8851	/	0.9976	/
	Parameter A	1.500 × 104	1.585 × 104	−3.068 × 10−1	1.112 × 10−1
	Parameter B	4.870 × 10−1	4.351	5.137 × 103	2.358 × 102
	Parameter C	−2.116 × 10−5	2.853 × 105	−9.083 × 105	1.057 × 105
P2-7	Reduced Chi-Sqr	0.3595	/	0.0031	/
	Adj. R-Square	0.8808	/	0.9990	/
	Parameter A	1.520 × 104	1.765 × 104	−3.965 × 10−1	7.691 × 10−2
	Parameter B	5.853 × 10−1	4.827	4.978 × 103	1.580 × 102
	Parameter C	−1.223 × 10−5	5.719 × 105	−8.081 × 105	6.935 × 104

exhibits the fitting data of P2 based on modified the Bingham model and the parabolic model. The parabolic model can be better fitted to characterize the rheological properties of the pastes than the modified Bingham model, which is the same finding as in the fitting result of P1 in Figure 4 and Table 3.

4.4.2. Comparison of Yield Stress Calculated Based on Two Rheological Models

Except for the yield stress τ₀, the physical meaning of other rheological parameters μ and s in the modified Bingham model has not been unanimously accepted [15,16,27], and the calculation formulas of these parameters are also different from those of the parameters of the parabolic model. The comparison is not meaningful. However, the physical meaning of yield stress in two models is clear and consistent. It refer to the minimum shear stress required to maintain fluid flow [40,41]. Therefore, the yield stresses calculated based on two rheological models can be compared and analyzed. Figure 6 shows the comparison of the yield stresses calculated based on two models. It can be seen that there is a significant difference between the yield stresses calculated with the modified Bingham model and the parabolic model. Taking paste P1-1 as an example, the yield stress calculated with parabolic model is larger than three times of that which was calculated with the modified Bingham model.

Generally, fluidity is closely related to the yield stress of paste. The higher that the fluidity of the paste is, the lower the yield stress of the paste is. Figure 7 exhibits the experimental results of fluidity of pastes. With the increase in the fly ash content, the fluidity of paste increases. Compared with paste P1-5, the fluidity of pastes P1-6 and P1-7 increase significantly. With the increase in the silica fume content, the fluidity of the paste initially increases and subsequently decreases. From the pastes P2-1 to P2-7, the fluidity of paste P2-2 is the highest. According to Figure 7 and the relationship between fluidity and the yield stress of the paste, the yield stress of the paste decreases with the increase in the fly ash content. The yield stress of the paste initially decreases and subsequently increases with the increase in the silica fume content. The above conclusions are not consistent with the effect of the fly ash content and the silica fume content on the yield stress calculated based on the modified Bingham model (Figure 6). The effect of the fly ash content and the silica fume content on the yield stress calculated based on the parabolic model (Figure 6) is well correlated with the effect on the fluidity of the paste (Figure 7). It is proven that the credibility of the rheological parameters calculated with the modified Bingham model is low.

Why are the rheological parameters of the paste calculated with the modified Bingham model less reliable? Equations (2) and (3) represent the solutions of the Couette inverse problem when the coefficient s of the quadratic term of the shear strain rate of the modified Bingham model is not equal to zero and is equal to zero, respectively. It can be seen that when the parameter s in Equation (2) tends to zero, Equation (2) does not tend to Equation (3). On the contrary, Equation (2) has no practical meaning when the parameter s approaches zero indefinitely. In the modified Bingham model, the parameter s is used to characterize the shear thickening or shear thinning performance of the pastes. When the shear thickening degree of paste is not high, the value of the parameter s is close to zero. In this case, Equation (2) is difficult to accurately describe the rheological properties of the paste because the denominator in Equation (2) is close to zero. The solution of the Couette inverse problem based on the modified Bingham model is discontinuous whether the coefficient of the quadratic term is equal to zero. It results in a large deviation between the fitting curve and the experimental data.

There is no problem with the parabolic model in this respect. The solution of the Couette inverse problem based on the parabolic model is expressed as Equation (12). When c, the parameter representing the performance of shear thickening or shear thinning, is equal to zero, Equation (12) becomes Equation (3), which is the solution of the Couette inverse problem based on the Bingham model. This means that the solution of the Couette inverse problem based on the parabolic model is continuous when the quadratic term coefficient is equal to zero. Therefore, the rheological properties of the paste can be well described by the parabolic model when the shear thickening or shear thinning are significant or not.

It can be clearly seen in Figure 6 that the yield stress calculated based on the parabolic model of pastes P1-6 and P1-7 is negative. It is obviously incorrect. In fact, this problem is not unique to the parabolic model. The linear Bingham model also produces the situation in which the yield stress calculated is negative [40,41]. This phenomenon may be due to the following reasons. On the one hand, in the experimental process itself, errors exist. On the other hand, when the yield stress of the tested paste is close to zero, the small absolute value of yield stress calculated through the fitting method has a fitting error. If the yield stress with a large absolute value calculated based on a rheological model is negative, then either there is an error in the experimental process or the used rheological model is not suitable to describe the rheological behavior of the fluid. It is verified that there is no problem in the rheological experiment process, data collection and processing, and the absolute value of the yield stress calculated based on the parabolic model for pastes P1-6 and P1-7 is small. The fluidity of pastes P1-6 and P1-7 is much higher than the fluidity of pastes P1-1 to P1-5 in Figure 7, which also proves that yield stress of pastes P1-6 and P1-7 is low. The negative yield stress is the fitting result based on the parabolic model. Therefore, the parabolic model is deficient to describe the rheological properties of the paste under the condition of very low yield stress.

4.4.3. The Relationship of Flowing Performance of Paste and Rheological Parameters Based on Parabolic Model

The plowing performance of the paste is shown in Figure 3. It can be seen that the torque increases faster than the rotational speed does. All of the pastes show the performance of shear thickening. It also can be found in Figure 8 that the degree of shear thickening or shear thinning calculated based on the parabolic model of all of the pastes is negative, which means that all of the pastes show the performance of shear thickening. The phenomenon of shear thickening can be explained by the cluster formation theory [13,43,44,45]. When the shear stress exceeds the critical ones, the high hydrodynamic forces among the dispersed particles in the fresh paste will exceed the repulsive force including the Brownian force, the electrostatic force and the spatial force. At this point, the particles will gather and form clusters, thus, showing an increase in the viscosity of the paste. The whole process is reversible. When the shear stress decreases and is less than the critical value, the repulsive force will exceed the hydrodynamic forces, and the cluster will disappear.

Figure 3a shows the relationship between the rotational speed and the torque of pastes with different fly ash contents under the steady flow state. Under the condition of constant rotational speed, the torque initially decreases, and it subsequently increases with the increase in the fly ash content. That is, with the increase in the fly ash content, the flowing performance of paste is initially improved and subsequently deteriorated. As can be seen from Figure 2, the particle size of the fly ash is between the particle sizes of the cement and silica fume. The increase in the fly ash content is helpful to improve the particle size distribution. In addition, the spherical degree of the fly ash particles is higher than that of cement. Increasing the content of fly ash is helpful to increase the lubrication effect between the particles and reduce the internal friction. Therefore, the increase in the fly ash content is helpful to improve the flowing performance of the paste. At the same time, the particle size of fly ash particles is smaller than that of cement particles. When the increasing degree of fly ash content is much higher than the decreasing degree of silica fume content, the total specific surface area of powder is increased. It decreases the thickness of the water film on the surface of particles to increase the internal friction of the paste. In other words, the flowing performance of the paste decreases with the continuous increase in the content of fly ash. Therefore, with the increase in the fly ash content, the flowing performance of the paste is initially improved and subsequently deteriorated.

Figure 8a shows that the yield stress of the paste decreases continuously, and both the fiducial differential viscosity and the degree of shear thickening of paste initially decrease and subsequently increase with the increase in the fly ash content. Therefore, the flowing performance of the paste is firstly improved with the increase in the fly ash content, which is related to the decrease in the yield stress, fiducial differential viscosity and shear thickening degree of the paste. Then, the flowing performance of the paste deteriorates with the further increase in the fly ash content, which is closely related to the increase in the fiducial differential viscosity and the degree of shear thickening of the paste.

Figure 3b shows the relationship between the rotational speed and the torque of the pastes with different silica fume contents under the steady flow state. Under the condition of constant rotational speed, the torque of pastes initially decreases and subsequently increases with the increase in the silica fume content. The flowing performance of the paste is initially improved and subsequently deteriorated with the increase in the silica fume content. Figure 2 shows that the spherical degree of silica fume is higher than that of cement, and its particle size is smaller than that of cement and fly ash. Increasing the dosage of silica fume can improve the flowing performance of paste by improving the particle size distribution and the lubrication effect among the particles. However, the water demand of the paste increases significantly with the continuous increase in the silica fume content to keep a suitable thickness of water film on the surface of the particles. The flowing performance of the paste decreases correspondingly. Since the particle size of silica fume is much smaller than that of fly ash, the influence of increase in the silica fume content on the particle size distribution and water demand of pastes is more significant than that of fly ash.

Figure 8b shows all of the yield stresses; the fiducial differential viscosity and the degree of shear thickening of paste initially decrease and subsequently increase with the increase in the silica fume content. The flowing performance of paste is initially improved and subsequently deteriorated with the increase in the silica fume content, which is closely related to the similar changes of the yield stress, fiducial differential viscosity and shear thickening degree of the paste.

Therefore, the flowing performance of the paste can be clearly characterized by the rheological parameters calculated with the parabolic model.

5. Conclusions

(1)

The solution of the Couette inverse problem, which is the most suitable tool to calculate the rheological parameters of fresh cementitious materials, is the key to determine the method of the measurement and calculation of the rheological parameters.

(2)

The fiducial differential viscosity can be regarded as a rheological parameter of the parabolic model that is independent of the shear thickening or shear thinning of the fluid. The calculation formulas of the yield stress, the fiducial differential viscosity and the degree of shear thickening or shear thinning of fresh cementitious materials based on the parabolic model were developed.

(3)

The solution of the Couette inverse problem based on the modified Bingham model is discontinuous when the coefficient of the quadratic term is equal to zero, resulting in a large deviation between the fitting curve and the rheological experimental data. The credibility of the rheological parameters of the pastes calculated based on the modified Bingham model is low.

(4)

The rheological properties of the paste can be well described by the parabolic model when the shear thickening or shear thinning are significant or not. The credibility of the rheological parameters of the pastes calculated based on the parabolic model is high. At the same time, the yield stress calculated may be negative when it is describing the rheological behavior of the paste with a lower yield stress. The flowing performance of the paste can be clearly characterized by the rheological parameters calculated with the parabolic model.