# On the Calculation of Van der Waals Force between Clay Particles

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}TSW and O

_{2}XYZ moving with plate1 and plate2, respectively, are defined, was simplified into that between a finite plate and an infinite plate which is referenced to as a wall as shown in Figure 2. According to the ideal plate-wall model shown in Figure 2 where Z-axes and W-axes are assumed to be parallel to each other, a close-form formula for van der Waals force was obtained [17], which has been used widely in the later DEM simulations of clays [13,14,15,18,19] due to the high efficiency of close form solution.

^{−20}J, c is a constant dependent on the characteristic length of interaction (49.363 nm is taken throughout the present study following [17]), w

_{1}, d

_{1}and L

_{1}are the width, thickness, and length of plate1, respectively, d

_{2}is the thickness of the wall (i.e., plate2 in Figure 2), and d and $\alpha $ are the minimum distance and angle between plate1 and the wall, respectively. It was also shown that the force F

_{pw}in Equation (1) is in the direction of Y-axes, i.e., perpendicular to the selected wall shown in Figure 2 [17]. Anandarajah and Chen further gave the similar formulas of the coordinates of action of F

_{pw}on plate1, which are not presented here for space reasons.

_{1}, V

_{2}denote the volume of plate1 and plate2, r is the distance between an arbitrary molecule A(x,y,z) in plate2 and B(t,s,w) in plate1, ${\beta}_{i}$ denotes the direction of the vector AB with respect to O

_{2}XYZ coordinate system, D is the minimum distance between the two plates as shown in Figure 3, d

_{2}is the thickness of plate2, and l

_{x}and l

_{z}are the coordinates of O

_{1}in the O

_{2}XYZ system. The expressions of the locations of total force on the two plates C

_{1}and C

_{2}as shown in Figure 3 were also obtained [16], which are not present here for space reasons.

## 2. New Rigorous Plate-Plate Model and Its Numerical Solution

#### 2.1. Van der Waals Force for the New Rigorous Plate-Plate Model

_{B}, y

_{B}, z

_{B}) in plate1 with respect to the O

_{2}XYZ system are related to the counterpart B(t,s,w) in the O

_{1}TSW system by a rotational matrix instead of only one angle α as described in Chen’s plate-plate model.

_{1}with respect to O

_{2}XYZ system,

**R**

_{12}is the rotational matrix, and the element of R

_{12}, r

_{ij}denotes the directional cosine between i axes and j axes. Accordingly, the distance between an arbitrary point A(x,y,z) in plate2 and B(t,s,w) in plate1 is

_{2}XYZ coordinate system is

#### 2.2. Numerical Solution of van der Waals Force of the New Model

_{i},b

_{i}] is divided into 4m

_{i}equal subintervals, then each subinterval can be divided into five equal parts repeatedly until the desired precision has been reached. The corresponding numerical integral formula is

#### 2.3. Verification of the Obtained Numerical Solution

_{1}= d

_{2}= 0.01, D = 0.001, l

_{1}= l

_{2}= w

_{1}= w

_{2}= 0.1, l

_{x}= l

_{z}= 0, α = 0 (the unit μm is omitted for brevity).

_{ppy}and F

_{pwy}to denote the calculated force based on the plate-plate and plate-wall model, respectively. All the forces are along the Y-axes since the two plates are parallel to each other.

_{ppy}to F

_{pwy}is about 0.3, which means that the ideal plate-wall model overestimates the total van der Waals force between neighboring clay particles.

_{ppy}to F

_{pwy}is about 0.93 when the area of plate2 is 10

^{6}times that of plate1.

## 3. Effect of the Choice of Ideal Wall

#### 3.1. Face-Face Type

_{1}= w

_{1}= 0.1, l

_{2}= w

_{2}= 0.2, d

_{1}= d

_{2}= 0.01. l

_{x}and l

_{z}are shown in Figure 6. The distance between the two plates, D, increases from 1 nm to 10 nm.

_{pwy}to F

_{ppy}of the two specific cases shown in Figure 6. For each case, two different choices of the ideal wall were made. It can be seen that the calculated F

_{pwy}corresponding to the choice of treating plate1 as the ideal wall is larger than that of plate2 as the ideal wall, and the ratio of F

_{pwy}to F

_{ppy}corresponding to case (a) is larger than that to case (b). The reason why the calculated F

_{pwy}corresponding to plate1 being the wall is much larger is that plate2 has a larger area. Therefore, it seems like the choice of the ideal wall has a remarkable effect on the calculated force depending on the geometric configuration of the two plates.

#### 3.2. Edge-Edge Type

_{1}= w

_{1}= 0.1, l

_{2}= w

_{2}= 0.2, d

_{1}= d

_{2}= 0.01. Two specific cases of relative positions of the two plates are shown in Figure 11 where various distance between plates is along Y direction in case (a) and X direction in case (b). According to the previous observation, the calculated force for these cases will be small since the corresponding projection area is small.

_{ee}, to that of case (a) of face-face type shown in Figure 6, F

_{ff}, while keeping the same distance between two plates. It can be seen that the van der Waals forces in the case of the edge-edge type are much smaller than those of face-face type under the same conditions of plate size and distance between plates. So, it seems that van der Waals force for the case of edge-edge type can be ignored in DEM simulations. Therefore, the effect of the choice of the ideal wall will not be analyzed further.

#### 3.3. Edge-Face Type

#### 3.3.1. Magnitude of the Calculated Forces

_{1}= w

_{1}= 0.1, l

_{2}= w

_{2}= 0.2, d

_{1}= d

_{2}= 0.01 in case (a); l

_{1}= 0.2, w

_{1}= 0.1, l

_{2}= w

_{2}= 0.2, d

_{1}= d

_{2}= 0.01 in case (b).

#### 3.3.2. Direction and Location of the Calculated Forces

_{pp}and X

_{pw}to denote the X-coordinate of the acting point based on the plate-plate and plate-wall model, respectively. The other variables in Table 3 denote the relevant quantity in a similar way. It can be seen that the difference in the X-coordinate between the plate-plate and plate-wall model is most significant, the Z-coordinate is smallest, and the Y-coordinate is in between. This difference in the X-coordinate increases nonlinearly with the increasing angle between two particles. However, the projection length and the calculated van der Waals force becomes small with the increasing angle between two particles. Therefore, the influence of the above difference in the location of calculated force on the calculated moment will not be as large as the difference itself shown in Table 3.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Comparison of numerical solutions based on Chen’s plate-plate model and the present model.

**Figure 16.**Comparison of the calculated forces of different choices for the cases of edge-face type.

**Table 1.**Comparison of van der Waals force calculated based on plate-wall model and the present model.

α = 0, d_{1} = d_{2} = 0.01, D = 0.001, l_{1} = w_{1} = 0.1, l_{2} = w_{2} | |||
---|---|---|---|

l_{2}/l_{1} | F_{pwy}/10^{−9}N | F_{ppy}/10^{−9}N | F_{ppy}/F_{pwy} |

1(l_{x} = l_{z} = 0, m = 27.5) | 8.10 | 2.84 | 0.35 |

2(l_{x} = l_{z} = 0.05, m = 27.5) | 3.49 | 0.43 | |

3(l_{x} = l_{z} = 0.1, m = 27.5) | 3.87 | 0.48 | |

10(l_{x} = l_{z} = 0.45, m = 27.5) | 4.58 | 0.57 | |

100(l_{x} = l_{z} = 4.95, m = 27.5) | 5.91 | 0.73 | |

1000(l_{x} = l_{z}= 0, m = 49.95, m = 37.5) | 7.56 | 0.93 |

**Table 2.**Comparison of the directions of Van der Waals forces based on plate-plate and plate-wall model.

α | F_{ppx}/10^{−9}N | F_{ppy}/10^{−9}N | F_{ppz}/10^{−9}N | F_{ppx}/F_{ppy} | F_{pwx}/F_{pwy} |
---|---|---|---|---|---|

15° | −4.02 × 10^{−3} | 2.86 × 10^{−2} | 5.85 × 10^{−5} | −0.141 | −0.268 |

30° | −2.78 × 10^{−3} | 1.37 × 10^{−2} | 2.58 × 10^{−5} | −0.203 | −0.577 |

45° | −2.32 × 10^{−3} | 9.60 × 10^{−3} | 1.75 × 10^{−5} | −0.241 | −1.0 |

60° | −2.33 × 10^{−3} | 8.15 × 10^{−3} | 1.54 × 10^{−5} | −0.286 | −1.732 |

75° | −2.80 × 10^{−3} | 7.86 × 10^{−3} | 1.62 × 10^{−5} | −0.357 | −3.732 |

**Table 3.**Comparison of the locations of Van der Waals forces based on plate-plate and plate-wall model.

α | X_{pw}/X_{pp} | Y_{pw}/Y_{pp} | Z_{pw}/Z_{pp} |
---|---|---|---|

15° | 0.845424352 | 0.987905290 | 1.000105178 |

30° | 1.067378334 | 0.989210020 | 1.000041638 |

45° | 1.715227995 | 0.966569502 | 1.000008018 |

60° | 3.244969855 | 0.879831892 | 1.00000768 |

75° | 7.722029849 | 0.696792073 | 1.000036639 |

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**MDPI and ACS Style**

Shang, X.-Y.; Zhao, K.; Qian, W.-X.; Zhu, Q.-Y.; Zhou, G.-Q.
On the Calculation of Van der Waals Force between Clay Particles. *Minerals* **2020**, *10*, 993.
https://doi.org/10.3390/min10110993

**AMA Style**

Shang X-Y, Zhao K, Qian W-X, Zhu Q-Y, Zhou G-Q.
On the Calculation of Van der Waals Force between Clay Particles. *Minerals*. 2020; 10(11):993.
https://doi.org/10.3390/min10110993

**Chicago/Turabian Style**

Shang, Xiang-Yu, Kang Zhao, Wan-Xiong Qian, Qi-Yin Zhu, and Guo-Qing Zhou.
2020. "On the Calculation of Van der Waals Force between Clay Particles" *Minerals* 10, no. 11: 993.
https://doi.org/10.3390/min10110993