# Recent Progress in the Viscosity Modeling of Concentrated Suspensions of Unimodal Hard Spheres

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Dilute Suspensions of Hard Spheres

#### 2.2. Concentrated Suspensions of Hard Spheres

## 3. Recent Progress in the Viscosity Modeling of Concentrated Suspensions

#### 3.1. Cheng et al. Model (2002)

#### 3.2. Mendoza and Santamaria-Holek Model (2009)

#### 3.3. Brouwers Model (2010)

#### 3.4. Faroughi–Huber Model (2015)

#### 3.5. Pal Model (2015)

#### 3.6. Pal Model (2017)

#### 3.7. Pal Model (2020)

## 4. Comparisons of Model Predictions

- The Mooney model (M) generally predicts the highest values of relative viscosities.
- Pal model P2 generally predicts the lowest values of relative viscosities.
- The Krieger–Dougherty model (KD) and Pal model P2 predict similar values of relative viscosities.
- The Mendoza and Santamaria-Holek model (MS) generally predicts relative viscosities lower than that of Pal model P3.
- The Cheng et al. (C) model predicts unrealistically high values of relative viscosities when $\phi <0.25$. Also, the relative viscosity is greater than unity at $\phi =0.$
- For large values of particle volume fractions $(\phi >0.50)$, the predictions of the models are generally in the following order: $M>C>B>P1>P3>MS>KD>P2$.

## 5. Comparisons of Model Predictions with Experimental Data

#### 5.1. Experimental Data

^{−1}. While coarse suspensions were subject to mainly hydrodynamic interactions, nanosuspensions were subject to both hydrodynamic and Brownian interactions.

#### 5.2. Estimation of Maximum Packing Volume Fraction of Suspensions

#### 5.3. Model Predictions Versus Experimental Data

- The Mooney model (M) overpredicts the relative viscosity of suspensions.
- The Krieger–Dougherty model (KD) underpredicts the relative viscosity of suspensions.
- The Cheng et al. model (C) overpredicts the relative viscosity at low and high particle concentrations.
- The Mendoza and Santamaria-Holek model (MS) underpredicts the relative viscosity of suspensions.
- The Brouwers model (B) overpredicts the relative viscosity of suspensions.
- Pal model P1 overpredicts the relative viscosity of suspensions.
- Pal model P2 underpredicts the relative viscosity of suspensions.
- Pal model P3 predictions are close to experimental values.

## 6. Simulation of the Viscous Behavior of Concentrated Multimodal Suspensions

#### 6.1. Relative Viscosity of Bimodal Suspensions

#### 6.2. Minimum Relative Viscosity of Multimodal Suspensions

#### 6.3. Composition of Multimodal Suspensions at Minimum Relative Viscosity

## 7. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Greek Symbols | |

$\stackrel{\u0304}{\stackrel{\u0304}{\delta}}$ | Unit tensor |

$\eta $ | Viscosity |

${\eta}_{m}$ | Viscosity of matrix phase (suspending medium) |

${\eta}_{r}$ | Relative viscosity |

${\eta}_{r\infty}$ | High-frequency relative viscosity |

${\eta}_{r,exp}$ | Experimental value of relative viscosity |

${\eta}_{r,mod}$ | Relative viscosity predicted by the model |

$\u2206{\eta}_{o}$ | Non-hydrodynamic contribution to relative viscosity (see Equation (16)) |

$\stackrel{\u0304}{\stackrel{\u0304}{\sigma}}$ | Bulk stress tensor |

$\phi $ | Volume fraction of particles |

${\phi}_{eff}$ | Effective volume fraction of particles |

${\phi}_{i}$ | Volume fraction of ith set of particles in a multimodal suspension, defined in Equations (48) and (49) |

${\phi}_{m}$ | Maximum packing volume fraction of particles where the viscosity of suspension diverges |

${\phi}_{n}$ | Volume fraction of nth set of particles in a multimodal suspension, defined in Equations (48) and (49) |

${\phi}_{o}$ | Volume fraction of different-size particles in a multimodal suspension corresponding to minimum relative viscosity of suspension (see Equation (61)) |

${\phi}_{T}$ | Total volume fraction of particles in a multimodal suspension, defined in Equation (51) |

$\Omega $ | Self-crowding parameter, defined in Equation (28) |

Latin Symbols | |

a | Constant in Equation (42) |

APRE | Average percentage relative error |

b | Constant in Equation (42) |

c | Self-crowding parameter (see Equation (22)) |

$\stackrel{\u0304}{\stackrel{\u0304}{E}}$ | Bulk rate of strain tensor |

${\stackrel{\u0304}{\stackrel{\u0304}{E}}}_{\infty}$ | Rate of strain tensor far away from the particle |

${f}_{c}$ | Fraction (volume basis) of coarse particles in a bimodal or trimodal mixture of particles |

${f}_{f}$ | Fraction (volume basis) of fine particles in a bimodal or trimodal mixture of particles |

${f}_{m}$ | Fraction (volume basis) of medium-sized particles in a trimodal mixture of particles |

GT | Glass transition point |

H | Relative viscosity function |

k | Aggregation coefficient (see Equation (42)) |

n | Number of data points or nth set of particles in a multimodal suspension |

N | Number of different-size particle fractions in a multimodal suspension, same as modality |

P | Pressure |

R | Radius of particle |

RCP | Random close packing |

${\stackrel{\u0304}{\stackrel{\u0304}{S}}}^{0}$ | Dipole strength of a single particle in an infinite matrix |

${V}_{1}$ | Volume of smallest-size particles in a multimodal suspension |

${V}_{i}$ | Volume of ith set of particles in a multimodal suspension |

${V}_{L}$ | Volume of suspending medium of a multimodal suspension |

${V}_{n}$ | Volume of nth set of particles in a multimodal suspension |

${V}_{N}$ | Volume of largest-size particles in a multimodal suspension |

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**Figure 2.**Increase in ${\phi}_{eff}/\phi $ with an increase in $\phi $ for two different types of particle packing clusters.

**Figure 3.**Comparisons of model predictions for two different values of maximum packing concentrations: ${\phi}_{m}^{RCP}=0.637$ and ${\phi}_{m}^{GT}=0.58$. M refers to Mooney model (Equation (13)), C refers to Cheng et al. model (Equation (16)), B refers to Brouwers model (Equation (26)), P1 refers to Pal model P1 (Equation (35)), P2 refers to Pal model P2 (Equation (41)), P3 refers to Pal model P3 (Equation (46)), MS refers to Mendoza and Santamaria-Holek model (Equation (23)), and KD refers to Krieger–Dougherty model (Equation (15)).

**Figure 4.**Experimental relative viscosity versus particle volume fraction data for twenty-two sets of coarse suspensions.

**Figure 5.**Experimental relative viscosity versus particle volume fraction data for sixteen sets of nanosuspensions.

**Figure 6.**Experimental relative viscosity versus particle volume fraction data for six sets of SNP-thickened coarse suspensions.

**Figure 7.**Experimental relative viscosity versus particle volume fraction data for all the suspensions (44 sets), including coarse suspensions, nanosuspensions, and SNP-thickened coarse suspensions.

**Figure 19.**Master curve generated by Pal model P3 (Equation (46)). The experimental data for all the suspensions (coarse suspensions, nanosuspensions, and SNP-thickened coarse suspensions) overlap with each other, which is in agreement with Pal model P3.

**Figure 20.**Relative viscosity of bimodal suspensions as a function of coarse particle fraction of the particle mixture.

**Figure 21.**Minimum relative viscosity of multimodal suspensions as a function of total particle concentration, ${\phi}_{T}$.

Set No | $\mathbf{Range}\mathbf{of}\mathit{\phi}$ | Description | Source |
---|---|---|---|

1 | 0–0.50 | Spherical particles of glass 100–160 µm in diameter. | Vand [21] |

2–3 | 0–0.30 | Spherical non-colloidal particles made from methyl methacrylate. The ratio of large to small diameters in set 2 was 1.6:1. For set 3, the ratio was 3:1. | Ward and Whitmore [22] |

4 | 0–0.512 | Spherical particles of glass of average diameter 230 µm. | Ting and Luebbers [33] |

5 | 0–0.50 | The data were extracted from the plot representing the average of the experimental viscosity data of many non-colloidal suspensions of spherical particles. | Rutgers [35] |

6 | 0–0.57 | The data were extracted from the plot representing the average of the experimental viscosity data of many non-colloidal suspensions of spherical particles. | Thomas [36] |

7 | 0.50–0.576 | Spherical monomodal particles of glass with diameters in the range of 53.8 to 236 µm. | Chong et al. [38] |

8 | 0–0.40 | Monomodal spherical particles ranging in diameter from 51.8 to 240.3 µm. | |

9 | 0–0.397 | Spherical particles of glass 5–10 µm in diameter. | Lewis and Nielsen [39] |

10 | 0–0.410 | Spherical particles of glass 30–40 µm in diameter. | |

11 | 0–0.50 | Spherical particles of glass 45–60 µm in diameter. | |

12 | 0–0.45 | Spherical particles of glass 90–105 µm in diameter. | |

13–14 | 0–0.50 | Spherical monodisperse particles of glass. For set 13, average diameter was 26 µm. For set 14, average diameter was 61 µm. | Smith [42] |

15 | 0–0.5236 | Spherical monodisperse particles of glass with mean diameter of 125 µm. | Smith [42] |

16 | 0–0.55 | Spherical monodisperse particles of glass with mean diameter of 183 µm. | |

17 | 0–0.50 | Spherical monodisperse particles of glass with mean diameter of 221 µm. | |

18 | 0–0.398 | Spherical particles of polystyrene with mean diameter of 700 µm. | Ilic and Phan-Thien [54] |

19 | 0–0.50 | Spherical particles of glass with mean diameter 43 $\pm $ 5.7 µm. | Zarraga et al. [58] |

20 | 0.41–0.58 | Poly (methyl methacrylate) particles, diameter of 1100 µm; polystyrene particles, diameter of 580 µm. | Boyer et al. [74] |

21 | 0–0.45 | Spherical particles of polystyrene with mean diameter of 40 µm. | Tanner et al. [66] |

22 | 0–0.60 | Particles of limestone with average diameter 4.91 µm. | Wilms et al. [75] |

Set No | Type and Diameter of Nanoparticles | Temp (°C) | Reference |
---|---|---|---|

1–4 | Oil nanodroplets: set 1 (27.5 nm), set 2 (58.5 nm), set 3 (102 nm), set 4 (205 nm) | 20 | Van der Waarden [30] |

5 | Silica: 156 nm | 20 | de Kruif et al. [76] |

6–8 | Silica: set 6 (56 nm), set 7 (96 nm), set 8 (245 nm) | 20 | Van der Werff and De Kruif [77] |

9 | Silica: 50 nm | 20 | Jones et al. [52] |

10 | Polymer: 56 nm | 20 | Jones et al. [53] |

11–12 | Polystyrene latex: set 11 (282 nm), set 12 (168 nm) | 20 | Rodriguez et al. [78] |

13 | Silica of three different sizes: 113, 280, and 427 nm | −10 | Shikata and Pearson [79] |

14 | Polystyrene latex: 146 nm | 20 | Weiss et al. [56] |

15 | CuO: 29 nm | 22–25 | Nguyen et al. [80] |

16 | Al_{2}O_{3}: 36 nm | 22–25 | Nguyen et al. [81] |

Set No | SNP Concentration of Matrix Phase (wt%) | Type and Size of Solid Particles | Concentration Range of Solid Particles (vol%) | Reference |
---|---|---|---|---|

1–6 | Set 1 (9.89%), set 2 (14.83%), set 3 (19.75%), set 4 (24.71%), set 5 (29.67%), set 6 (34.60%) | Ceramic hollow spheres, 10 to 340 µm; Sauter mean diameter of 138 µm | Set 1 (0–55.08%), set 2 (0–54.54%), set 3 (0–56%), set 4 (0–53.44%), set 5 (0–51.93%), set 6 (0–50.36%) | Ghanaatpishehsanaei and Pal [82] |

Model Type | Coarse Suspensions | Nanosuspensions | SNP-Thickened Coarse Suspensions |
---|---|---|---|

Mooney model (M) | −8.98 × 10^{8}% (Overpredicts extremely) | −1.64 × 10^{18}% (Overpredicts extremely) | −3.88 × 10^{31}%(Overpredicts extremely) |

Krieger–Dougherty model (KD) | 27.55% (Underpredicts substantially) | 23.05% (Underpredicts substantially) | 47.73% (Underpredicts severely) |

Cheng et al. model (C) | −3531% (Overpredicts extremely) | −1.14 × 10^{7}% (Overpredicts extremely) | −2.16 × 10^{22}%(Overpredicts extremely) |

Mendoza and Santamaria-Holek model (MS) | 12.56% (Underpredicts substantially) | 10.43% (Underpredicts substantially) | 23.96% (Underpredicts substantially) |

Brouwers model (B) | −21.6% (Overpredicts substantially) | −32.43% (Overpredicts severely) | −43.4% (Overpredicts severely) |

Pal model one (P1) | −34.81% (Overpredicts severely) | −26.01% (Overpredicts severely) | −56.42% (Overpredicts severely) |

Pal model two (P2) | 24.85% (Underpredicts substantially) | 22.96% (Underpredicts substantially) | 42.64% (Underpredicts severely) |

Pal model three (P3) | 3.22% (Underpredicts slightly) | 3.51% (Underpredicts slightly) | 6.37% (Underpredicts moderately) |

${\mathit{\phi}}_{\mathit{T}}$ | ${\mathit{\phi}}_{\mathit{O}}$ | ${\mathit{\eta}}_{\mathit{r}}$ | ${\mathit{f}}_{\mathit{c}}$ | ${\mathit{f}}_{\mathit{m}}$ | ${\mathit{f}}_{\mathit{f}}$ |

0.50 | 0.21 | 8.18 | 0.41 | 0.33 | 0.26 |

0.55 | 0.23 | 12.07 | 0.42 | 0.33 | 0.25 |

0.60 | 0.26 | 19.10 | 0.44 | 0.32 | 0.24 |

0.65 | 0.30 | 33.26 | 0.45 | 0.32 | 0.23 |

0.70 | 0.33 | 66.28 | 0.47 | 0.32 | 0.21 |

0.75 | 0.37 | 161.98 | 0.49 | 0.31 | 0.20 |

0.80 | 0.42 | 555.34 | 0.52 | 0.30 | 0.18 |

0.85 | 0.47 | 3662.97 | 0.55 | 0.29 | 0.16 |

0.89 | 0.52 | 49,779.29 | 0.59 | 0.28 | 0.13 |

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**MDPI and ACS Style**

Pal, R.
Recent Progress in the Viscosity Modeling of Concentrated Suspensions of Unimodal Hard Spheres. *ChemEngineering* **2023**, *7*, 70.
https://doi.org/10.3390/chemengineering7040070

**AMA Style**

Pal R.
Recent Progress in the Viscosity Modeling of Concentrated Suspensions of Unimodal Hard Spheres. *ChemEngineering*. 2023; 7(4):70.
https://doi.org/10.3390/chemengineering7040070

**Chicago/Turabian Style**

Pal, Rajinder.
2023. "Recent Progress in the Viscosity Modeling of Concentrated Suspensions of Unimodal Hard Spheres" *ChemEngineering* 7, no. 4: 70.
https://doi.org/10.3390/chemengineering7040070