Adhesion force during microcontact interaction between cylindrical-segment-on-flat and flat bodies

Abstract

Adhesion force including its both attractive and repulsive components during microcontact interaction between cylindrical-segment-on-flat and flat bodies is formulated using the volumetric integration of interactions among the atoms. Thenceforth the corresponding formulation for cylinder-on-flat and flat-on-flat is developed and compared with the available counterpart attractive force and they are in excellent agreement with each other. However, the present study provides also the repulsive component of adhesion force for these contact geometries. The formulation is also extended for a cylindrical segment with two radii to model the elastic–plastic deformation. Several phenomena related to adhesion during microcontact are then studied leading to the following salient observations. The adhesion force during microcontact interaction can be of the same order as the applied force, and thus causes the plastic deformation. The attractive force from the cylindrical segment is the major component of adhesion force for its radius/height ratios smaller than 1000 while that from the bulk body is the major for the larger ratios, i.e. when the cylindrical segment is similar to an asperity. There is considerable difference in the attractive force between a cylinder and a cylindrical-segment-on-flat for the smaller values of radius. The difference between attractive and total adhesion force is significant (∼60%) when adhesion force is the maximum, but it rapidly decreases with increasing separation distance and can be neglected at the separation distance larger than the twice of inter-atomic equilibrium distance. The equilibrium separation distance is about 60% of the inter-atomic equilibrium distance and adhesion force is maximum at about 80% of the inter-atomic equilibrium separation distance for both deformed and undeformed configurations. The adhesion force also increased with the deformation of cylindrical segment.

Appendix A: Expressions for attractive and repulsive terms of adhesion force

1.1 A.1 Undeformed bodies

$$ \begin{aligned} F_{attr}^{11} =\;& L\cdot \frac{H_{nr} }{\pi }\cdot \left( \frac{3D^{2}R_1 K_{31} }{2K_{21}^3 }+\frac{D^{2}R_1 K_{31} }{2K_{21}^3 }+\frac{2D^{2}R_1^2 K_{41} }{2K_{21}^3 }-\frac{R_1^2 K_{31}^3 }{2K_{21}^3 K_{11} }+\frac{DR_1^3 K_{41} }{2K_{21}^3 }\right. \\& +\frac{DK_{31}^3 }{2K_{21}^2 K_{11}^2 }+\frac{DR_1 K_{41} }{K_{21}^3 }+\frac{3DR_1^2 K_{31} }{2K_{21}^3 }+\frac{DR_1^2 K_{41} }{K_{21}^3 }+\frac{3DR_1^2 K_{31} }{2K_{21}^3 }+\frac{DR_1^2 K_{31} }{K_{21}^3 }+\frac{R_1 K_{31}^3 }{2K_{21}^2 K_{11}^2 }-\frac{2D^{2}R_1 K_{51} }{2K_{21}^{5/2} }\\ & +\frac{2D^{3}R_1 K_{41} }{K_{21}^3 }-\frac{DR_1 K_{31}^3 }{K_{21}^3 K_{11} }+\frac{D^{2}K_{41} }{2K_{21}^2 }+\frac{R_1 K_{31} }{2K_{21}^2 }+\frac{D^{4}K_{41} }{2K_{21}^3 }-\frac{3DR_1^2 K_{51} }{2K_{21}^{5/2} }+\frac{DK_{31} }{2K_{21}^3 }-\frac{R_1^3 K_{31} }{2K_{21}^3 }\\ & \left.-\frac{R_1^3 \cdot K_{51} }{2\cdot K_{21}^{5/2} }-\frac{K_{31}^3 }{3\cdot K_{21} \cdot K_{11}^3 }-\frac{R_1^3 \cdot K_{41} }{2\cdot K_{21}^3 }+\frac{D^{3}\cdot K_{31} }{2\cdot K_{21}^3 }-\frac{D^{3}\cdot K_{51} }{2\cdot K_{21}^{5/2} }-\frac{R_1 \cdot K_{51} }{2\cdot K_{21}^{3/2} }-\frac{DK_{51} }{2K_{21}^{5/2} }-\frac{D^{2}K_{31}^{3/2} }{2K_{21}^3 K_{11} }\right) \end{aligned} $$

(A1)

$$ \begin{aligned}F_{attr}^{12} =\;& L\frac{H_{nr} }{\pi }\left (\frac{3D^{2}R_1 K_{32} }{2K_{22}^3 }-\frac{D^{2}H_0 K_{32} }{2K_{22}^3 }+\frac{D^{2}R_1 K_{32} }{2K_{22}^3 }+\frac{2D^{2}R_1^2 K_{42}}{2K_{22}^3 }-\frac{R_1^2 K_{32}^3 }{2K_{22}^3 K_{12} }+\frac{DR_1^3 K_{42} }{2K_{22}^3 }-\frac{R_2^2 H_0 K_{32}}{2K_{22}^3}\right. \\ & +\frac{DK_{32}^3 }{2K_{22}^2 K_{12}^2 }+\frac{DR_1 K_{42}}{K_{22}^3 }+\frac{3DR_1^2 K_{32} }{2K_{22}^3 }+\frac{DR_1^2 K_{42}}{K_{22}^3 }+\frac{3DR_1^2 K_{32} }{2K_{22}^3 }+\frac{DR_1^2 K_{32}}{K_{22}^3 }+\frac{R_1 K_{32}^3 }{2K_{22}^2 K_{12}^2}-\frac{2D^{2}R_1 K_{52} }{2K_{22}^{5/2}}\\ & +\frac{2D^{3}R_1 K_{42} }{K_{22}^3 }-\frac{DR_1 K_{32}^3 }{K_{22}^3 K_{12} }-\frac{DR_1 H_0 K_{32} }{K_{22}^3 }+\frac{D^{2}K_{42} }{2K_{22}^2 }+\frac{R_1 K_{32} }{2K_{22}^2 }+\frac{D^{4}K_{42} }{2K_{22}^3 }-\frac{3DR_1^2 K_{52} }{2K_{22}^{5/2} }+\frac{DK_{32} }{2K_{22}^3 }-\frac{R_1^3 K_{32} }{2K_{22}^3 }\\ & \left.-\frac{R_1^3 K_{52} }{2K_{22}^{5/2} }-\frac{K_{32}^3 }{3K_{22} K_{12}^3 }-\frac{R_1^3 K_{42} }{2K_{22}^3 }+\frac{D^{3}K_{32} }{2K_{22}^3 }-\frac{D^{3}K_{52} }{2K_{22}^{5/2} }-\frac{R_1 K_{52} }{2K_{22}^{3/2} }-\frac{DK_{52} }{2K_{22}^{5/2} }-\frac{D^{2}K_{32}^{3/2} }{2K_{22}^3 K_{12}}\right) \end{aligned} $$

(A2)

$$ F_{attr}^{16} =\frac{2c_2 LH_{nr} \left( {-3D^{2}H_0 -3DH_0^2 -H_0^3 +3D^{2}H_4 +3DH_4^2 +H_4^3 } \right)}{6\pi (D+H_4 )^{3}(D+H_0 )^{3}} $$

(A3)

where

$$ K_{11}=D $$

(A4)

$$ K_{21} =-D(D+2R_1) $$

(A5)

$$ K_{31} =[-K_{11}^2 +(2R_1 +2D)\cdot K_{11} -D^{2}-2R_1 D]^{1/2} $$

(A6)

$$ K_{41} =\arctan \frac{-R_1 }{K_{31}} $$

(A7)

$$ K_{51}=\hbox{arctanh}\;\frac{-2D^{2}-4R_1 D+(2D+2R_1 )\cdot K_{11} }{2\cdot K_{21}^{1/2} \cdot K_{31}} $$

(A8)

$$ K_{12} =D+H_0 $$

(A9)

$$ K_{22} =-D(D+2R_1) $$

(A10)

$$ K_{32} =[-K_{12}^2 +(2R_1 +2D)\cdot K_{12} -D^{2}-2R_1 D]^{1/2} $$

(A11)

$$ K_{42} =\arctan \frac{H_0 -R_1 }{K_{32}} $$

(A12)

$$ K_{52}=\hbox{arctanh}\frac{-2D^{2}-4R_1 D+(2D+2R_1 )\cdot K_{12} }{2\cdot K_{22}^{1/2}\cdot K_{32}} $$

(A13)

$$ F_{rep}=(F_{rep}^{12} -F_{rep}^{11} )+F_{rep}^{16} $$

(A14)

where

$$ F_{rep}^{12} -F_{rep}^{11} =-L\frac{2H_{nr} \sigma ^{6}}{5\pi }\int\limits_0^{H_0 } {\frac{\sqrt{2R_1 h-h^{2}}}{(D+h)^{10}}}{\rm d}h $$

(A15)

$$ F_{rep}^{16} =-\frac{2c_2 LH_{nr} \sigma ^{6}}{5\pi }\int\limits_{H_0 }^{H_4 } {\frac{{\rm d}h}{(D+h)^{10}}} $$

(A16)

1.2 A.2 Deformed bodies

$$ \begin{aligned}F_{attr}^{17}=\;& L\cdot \frac{H_{nr} }{\pi }\cdot \left( \frac{3D^{2}R_2 K_{37} }{2K_{27}^3 }+\frac{D^{2}R_2 K_{37} }{2K_{27}^3 }+\frac{2D^{2}R_2^2 K_{47} }{2K_{27}^3 }-\frac{R_2^2 K_{37}^3 }{2K_{27}^3 K_{17} }+\frac{DR_2^3 K_{47} }{2K_{27}^3 }\right. \\ & +\frac{DK_{37}^3 }{2K_{27}^2 K_{17}^2 }+\frac{DR_2 K_{47} }{K_{27}^3 }+\frac{3DR_2^2 K_{37} }{2K_{27}^3 }+\frac{DR_2^2 K_{47} }{K_{27}^3 }+\frac{3DR_2^2 K_{37} }{2K_{27}^3 }+\frac{DR_2^2 K_{37} }{K_{27}^3 }+\frac{R_2 K_{37}^3 }{2K_{27}^2 K_{17}^2 }-\frac{2D^{2}R_2 K_{57} }{2K_{27}^{5/2} }\\ & +\frac{2D^{3}R_2 K_{47} }{K_{27}^3 }-\frac{DR_2 K_{37}^3 }{K_{27}^3 K_{17} }+\frac{D^{2}K_{47} }{2K_{27}^2 }+\frac{R_2 K_{37} }{2K_{27}^2 }+\frac{D^{4}K_{47} }{2K_{27}^3 }-\frac{3DR_2^2 K_{57} }{2K_{27}^{5/2} }+\frac{DK_{37} }{2K_{27}^3 }-\frac{R_2^3 K_{37} }{2K_{27}^3 }\\ & \left. -\frac{R_2^3 \cdot K_{57} }{2\cdot K_{27}^{5/2} }-\frac{K_{37}^3 }{3\cdot K_{27} \cdot K_{17}^3 }-\frac{R_2^3 \cdot K_{47} }{2\cdot K_{27}^3 }+\frac{D^{3}\cdot K_{37} }{2\cdot K_{27}^3 }-\frac{D^{3}\cdot K_{57} }{2\cdot K_{27}^{5/2} }-\frac{R_2 \cdot K_{57} }{2\cdot K_{27}^{3/2} }-\frac{DK_{57} }{2K_{27}^{5/2} }-\frac{D^{2}K_{37}^{3/2} }{2K_{27}^3 K_{17}}\right) \end{aligned} $$

(A17)

$$ \begin{aligned} F_{attr}^{18}=\;& L\frac{H_{nr} }{\pi }\left (\frac{3D^{2}R_2 K_{38} }{2K_{28}^3 }-\frac{D^{2}H_1 K_{38} }{2K_{28}^3 }+\frac{D^{2}R_2 K_{38} }{2K_{28}^3 }+\frac{2D^{2}R_2^2 K_{48} }{2K_{28}^3 }-\frac{R_2^2 K_{38}^3 }{2K_{28}^3 K_{18} }+\frac{DR_2^3 K_{48} }{2K_{28}^3 }-\frac{R_2^2 H_1 K_{38}}{2K_{28}^3}\right. \\ & +\frac{DK_{38}^3 }{2K_{28}^2 K_{18}^2 }+\frac{DR_2 K_{48} }{K_{28}^3 }+\frac{3DR_2^2 K_{38} }{2K_{28}^3 }+\frac{DR_2^2 K_{48} }{K_{28}^3 }+\frac{3DR_2^2 K_{38} }{2K_{28}^3 }+\frac{DR_2^2 K_{38} }{K_{28}^3 }+\frac{R_2 K_{38}^3 }{2K_{28}^2 K_{18}^2 }-\frac{2D^{2}R_2 K_{58} }{2K_{28}^{5/2} }\\ & +\frac{2D^{3}R_2 K_{48} }{K_{28}^3 }-\frac{DR_2 K_{38}^3 }{K_{28}^3 K_{18} }-\frac{DR_2 H_1 K_{38} }{K_{28}^3 }+\frac{D^{2}K_{48} }{2K_{28}^2 }+\frac{R_2 K_{38} }{2K_{28}^2 }+\frac{D^{4}K_{48} }{2K_{28}^3 }-\frac{3DR_2^2 K_{58} }{2K_{28}^{5/2} }+\frac{DK_{38} }{2K_{28}^3 }-\frac{R_2^3 K_{38} }{2K_{28}^3 }\\ & \left. -\frac{R_2^3 K_{58} }{2K_{28}^{5/2} }-\frac{K_{38}^3 }{3K_{28} K_{18}^3 }-\frac{R_2^3 K_{48} }{2K_{28}^3 }+\frac{D^{3}K_{38} }{2K_{28}^3 }-\frac{D^{3}K_{58} }{2K_{28}^{5/2} }-\frac{R_2 K_{58} }{2K_{28}^{3/2} }-\frac{DK_{58} }{2K_{28}^{5/2} }-\frac{D^{2}K_{38}^{3/2} }{2K_{28}^3 K_{18}}\right) \end{aligned} $$

(A18)

$$ \begin{aligned}F_{attr}^{13} =\;& L\frac{H_{nr} }{\pi } \left (\frac{3D^{2}R_1 K_{33} }{2K_{23}^3 }-\frac{D^{2}H_1 K_{33} }{2K_{23}^3 }+\frac{D^{2}R_1 K_{33} }{2K_{23}^3 }+\frac{2D^{2}R_1^2 K_{43} }{2K_{23}^3 }-\frac{R_1^2 K_{33}^3 }{2K_{23}^3 K_{13} }+\frac{DR_1^3 K_{43} }{2K_{23}^3 }-\frac{R_1^2 H_1 K_{33}}{2K_{23}^3}\right. \\ & +\frac{DK_{33}^3 }{2K_{23}^2 K_{13}^2 }+\frac{DR_1 K_{43} }{K_{23}^3 }+\frac{3DR_1^2 K_{33} }{2K_{23}^3 }+\frac{DR_1^2 K_{43} }{K_{23}^3 }+\frac{3DR_1^2 K_{33} }{2K_{23}^3 }+\frac{DR_1^2 K_{33} }{K_{23}^3 }+\frac{R_1 K_{33}^3 }{2K_{23}^2 K_{13}^2 }-\frac{2D^{2}R_1 K_{53} }{2K_{23}^{5/2} }\\ & +\frac{2D^{3}R_1 K_{43} }{K_{23}^3 }-\frac{DR_1 K_{33}^3 }{K_{23}^3 K_{13} }-\frac{DR_1 H_1 K_{33} }{K_{23}^3 }+\frac{D^{2}K_{43} }{2K_{23}^2 }+\frac{R_1 K_{33} }{2K_{23}^2 }+\frac{D^{4}K_{43} }{2K_{23}^3 }-\frac{3DR_1^2 K_{53} }{2K_{23}^{5/2} }+\frac{DK_{33} }{2K_{23}^3 }-\frac{R_1^3 K_{33} }{2K_{23}^3 }\\ & \left. -\frac{R_1^3 K_{53} }{2K_{23}^{5/2} }-\frac{K_{33}^3 }{3K_{23} K_{13}^3 }-\frac{R_1^3 K_{43} }{2K_{23}^3 }+\frac{D^{3}K_{33} }{2K_{23}^3 }-\frac{D^{3}K_{53} }{2K_{23}^{5/2} }-\frac{R_1 K_{53} }{2K_{23}^{3/2} }-\frac{DK_{53} }{2K_{23}^{5/2} }-\frac{D^{2}K_{33}^{3/2} }{2K_{23}^3 K_{13}}\right) \end{aligned} $$

(A19)

$$ \begin{aligned} F_{attr}^{14} =\;& L\frac{H_{nr} }{\pi }\left (\frac{3D^{2}R_1 K_{34} }{2K_{24}^3 }-\frac{D^{2}H_2 K_{34} }{2K_{24}^3 }+\frac{D^{2}R_1 K_{34} }{2K_{24}^3 }+\frac{2D^{2}R_1^2 K_{44} }{2K_{24}^3 }-\frac{R_1^2 K_{34}^3 }{2K_{24}^3 K_{14} }+\frac{DR_1^3 K_{44} }{2K_{24}^3 }-\frac{R_1^2 H_2 K_{34}}{2K_{24}^3 }\right. \\ & +\frac{DK_{34}^3 }{2K_{24}^2 K_{14}^2 }+\frac{DR_1 K_{44} }{K_{24}^3 }+\frac{3DR_1^2 K_{34} }{2K_{24}^3 }+\frac{DR_1^2 K_{44} }{K_{24}^3 }+\frac{3DR_1^2 K_{34} }{2K_{24}^3 }+\frac{DR_1^2 K_{34} }{K_{24}^3 }+\frac{R_1 K_{34}^3 }{2K_{24}^2 K_{14}^2 }-\frac{2D^{2}R_1 K_{54} }{2K_{24}^{5/2} }\\ & +\frac{2D^{3}R_1 K_{44} }{K_{24}^3 }-\frac{DR_1 K_{34}^3 }{K_{24}^3 K_{14} }-\frac{DR_1 H_2 K_{34} }{K_{24}^3 }+\frac{D^{2}K_{44} }{2K_{24}^2 }+\frac{R_1 K_{34} }{2K_{24}^2 }+\frac{D^{4}K_{44} }{2K_{24}^3 }-\frac{3DR_1^2 K_{54} }{2K_{24}^{5/2} }+\frac{DK_{34} }{2K_{24}^3 }-\frac{R_1^3 K_{34} }{2K_{24}^3 }\\ & \left. -\frac{R_1^3 K_{54} }{2K_{24}^{5/2} }-\frac{K_{34}^3 }{3K_{24} K_{14}^3 }-\frac{R_1^3 K_{44} }{2K_{24}^3 }+\frac{D^{3}K_{34} }{2K_{24}^3 }-\frac{D^{3}K_{54} }{2K_{24}^{5/2} }-\frac{R_1 K_{54} }{2K_{24}^{3/2} }-\frac{DK_{54} }{2K_{24}^{5/2} }-\frac{D^{2}K_{34}^{3/2} }{2K_{24}^3 K_{14}}\right) \end{aligned} $$

(A20)

$$ F_{attr}^{15} =\frac{LH_{nr} }{\pi }\int\limits_{H_2 }^{H_4 } \frac{c_2 {\rm d}h}{(D+h)^{3}}=\frac{2c_2 LH_{nr} \left( {-3D^{2}H_2 -3DH_2^2-H_2^3 +3D^{2}H_4 +3DH_4^2 +H_4^3 } \right)}{6\pi (D+H_4 )^{3}(D+H_2)^{3}} $$

(A21)

where

$$ K_{17}=D $$

(A22)

$$ K_{27} =-D(D+2R_2) $$

(A23)

$$ K_{37} =[-K_{17}^2 +(2R_2 +2D)\cdot K_{17} -D^{2}-2R_2 D]^{1/2} $$

(A24)

$$ K_{47} =\arctan \frac{-R_2 }{K_{37}} $$

(A25)

$$ K_{57}=\hbox{arctanh}\frac{-2D^{2}-4R_2 D+(2D+2R_2 )\cdot K_{17} }{2\cdot K_{27}^{1/2}\cdot K_{37}} $$

(A26)

$$ K_{18} =D+H_1 $$

(A27)

$$ K_{28} =-D(D+2R_2) $$

(A28)

$$ K_{38} =[-K_{18}^2 +(2R_2 +2D)\cdot K_{18} -D^{2}-2R_2 D]^{1/2} $$

(A29)

$$ K_{48} =\arctan \frac{H_1 -R_2 }{K_{38}} $$

(A30)

$$ K_{58}=\hbox{arctanh}\frac{-2D^{2}-4R_2 D+(2D+2R_2 )\cdot K_{18} }{2\cdot K_{28}^{1/2} \cdot K_{38}} $$

(A31)

$$ K_{13} =D+H_1 $$

(A32)

$$ K_{23} =-D(D+2R_1) $$

(A33)

$$ K_{33} =[-K_{13}^2 +(2R_1 +2D)\cdot K_{13} -D^{2}-2R_1 D]^{1/2} $$

(A34)

$$ K_{43} =\arctan \frac{H_1 -R_1 }{K_{33}} $$

(A35)

$$ K_{53}=\hbox{arctanh}\frac{-2D^{2}-4R_1 D+(2D+2R_1 )\cdot K_{13} }{2\cdot K_{23}^{1/2}\cdot K_{33}} $$

(A36)

$$ K_{14} =D+H_2 $$

(A37)

$$ K_{24} =-D(D+2R_1) $$

(A38)

$$ K_{34} =[-K_{14}^2 +(2R_1 +2D)\cdot K_{14} -D^{2}-2R_1 D]^{1/2} $$

(A39)

$$ K_{44} =\arctan \frac{H_2 -R_1 }{K_{34}} $$

(A40)

$$ K_{54}=\hbox{arctanh}\frac{-2D^{2}-4R_1 D+(2D+2R_1 )\cdot K_{14} }{2\cdot K_{24}^{1/2}\cdot K_{34}} $$

(A41)

The repulsive force for loaded, plastically deformed and unloaded bodies is expressed as

$$ F_{rep}=(F_{rep}^{18} -F_{rep}^{17} )+(F_{rep}^{14} -F_{rep}^{13} )+F_{rep}^{15} $$

(A42)

where

$$ F_{rep}^{18} -F_{rep}^{17} =-L\frac{2H_{nr} \sigma ^{6}}{5\pi }\int\limits_0^{H_1 } {\frac{\sqrt{2R_2 h-h^{2}}}{(D+h)^{10}}} {\rm d}h $$

(A43)

$$ F_{rep}^{14} -F_{rep}^{13} =-L\frac{2H_{nr} \sigma ^{6}}{5\pi }\int\limits_{H_1 }^{H_2 } {\frac{\sqrt{2R_1 h-h^{2}}}{(D+h)^{10}}} {\rm d}h $$

(A44)

$$ F_{rep}^{15} =-\frac{2c_2 LH_{nr} \sigma ^{6}}{5\pi }\int\limits_{H_2 }^{H_4 } {\frac{{\rm d}h}{(D+h)^{10}}} $$

(A45)

For the sake of brevity, the following repulsive force expressions can be written as:

$$ \begin{aligned} F_{rep}^{1i} =& -L\frac{2H_{nr} \sigma ^{6}}{5\pi }\cdot \left (-\frac{16K_{3i}^3 }{315K_{2i}^4 K_{1i}^3 }+\frac{693R_k^5 K_{3i} }{64K_{2i}^7 }-\frac{15015R_k^2 D^{4}K_{3i}^3 }{128K_{2i}^8 }+\frac{5005R_k^2 D^{3}K_{3i}^3 }{64K_{2i}^7 K_{1i}^2 }-\sigma \right. \\ & -\frac{15015R_k^4 D^{3}K_{3i}^3 }{128K_{2i}^8 K_{1i}}-\frac{143R_k^3 DK_{3i}^3 }{8K_{2i}^6 K_{1i}^3 }-\frac{5005R_k^3 D^{3}K_{3i} H_j }{32K_{2i}^8 }-\frac{5005R_k^3 D^{3}K_{3i}^3 }{32K_{2i}^8 K_{1i}}+ \frac{1155R_k^2 DK_{3i}^3 }{128K_{2i}^6 K_{1i}^2 }\\ & +\frac{3003R_k^5 DK_{3i}^3 }{64K_{2i}^8 K_{1i}}-\frac{105R_k^3 K_{5i} }{32K_{2i}^{11/2} }-\frac{1155R_k^2 D^{2}K_{3i}^3 }{64K_{2i}^7 K_{1i} }+\frac{35R_k^2 K_{4i}}{128K_{2i}^5 }+\frac{1001R_k^6 K_{4i}}{128K_{2i}^7 }\\ & +\frac{385R_k^4 K_{4i}}{128K_{2i}^6 }-\frac{429R_k^2 D^{2}K_{3i}^3 }{16K_{2i}^6 K_{1i}^3 }+\frac{5005R_k^4 DK_{3i}^3 }{128K_{2i}^7 K_{1i}^2 }+\frac{385R_k^3 DK_{3i}^3 }{32K_{2i}^7 K_{1i} }+\frac{715R_k^2 DK_{3i}^3 }{336K_{2i}^4 K_{1i}^6 }\\ & -\frac{5005R_k^2 D^{6}K_{3i}^3 }{32K_{2i}^9 K_{1i} }+\frac{15015R_k^2 D^{5}K_{3i}^3 }{128K_{2i}^8 K_{1i}^2 }+\frac{715R_k^8 K_{4i} }{128K_{2i}^8 K_{1i}^2 }-\frac{3003R_k^5 DH_j K_{3i} }{64K_{2i}^8 }+\frac{385R_k^5 K_{3i} }{128K_{2i}^7 }\\ & -\frac{15015R_k^4 D^{2}H_j K_{3i} }{128K_{2i}^8 }-\frac{5005R_k^6 D^{2}H_j K_{3i}^3 }{32K_{2i}^9 K_{1i} }+\frac{7293R_k^2 DK_{3i}^3 }{1120K_{2i}^5 K_{1i}^4 }+\frac{25025R_k^4 D^{3}K_{3i}^3 }{128K_{2i}^8 K_{1i}^2 }+\frac{715R_k^2 D^{3}K_{3i}^3 }{32K_{2i}^6 K_{1i}^4} \\ & -\frac{715R_k^9 K_{3i} }{128K_{2i}^9} +\frac{429R_k^7 K_{3i}}{32K_{2i}^8 }+\frac{1001R_k^7 K_{3i}}{128K_{2i}^8 }+\frac{25025R_k^3 D^{4}K_{3i}^3}{128K_{2i}^8 K_{1i}^2}-\frac{5005R_k^2 D^{6}H_j K_{3i}}{32K_{2i}^9 } \\ & -\frac{715R_k^7 DH_j K_{3i} }{16K_{2i}^9 }+\frac{5005R_k^6 DK_{3i}^3 }{128K_{2i}^8 K_{1i}^2 }-\frac{5005R_k^6 D^{2}H_j K_{3i}}{32K_{2i}^9 }+\frac{45045R_k^3 D^{5}K_{4i}}{64K_{2i}^8 }-\frac{715R_k^7 DK_{3i}^3 }{16K_{2i}^9 K_{1i} }\\ & +\frac{35R_k^3 K_{3i} }{128K_{2i}^6 }+\frac{5005R_k^5 D^{3}H_j K_{3i}}{16K_{2i}^9 }-\frac{715R_k^5 DK_{3i}^3 }{32K_{2i}^7 K_{1i}^3 }-\frac{25025R_k^4 D^{4}K_{3i}^3 }{64K_{2i}^9 K_{1i} }+\frac{35R_k K_{3i}^3 }{128K_{2i}^5 K_{1i}^2 }\\ & +\frac{35D^{2}K_{4i} }{128K_{2i}^5 }-\frac{2871D^{2}K_{3i} }{2240K_{2i}^5 }-\frac{715R_k^5 DK_{3i}^3 }{32K_{2i}^7 K_{1i}^3 }-\frac{3575R_k^4 D^{2}K_{3i}^3 }{64K_{2i}^7 K_{1i}^3 }-\frac{715R_k^9 K_{5i}}{128K_{2i}^{17/2} }-\frac{5005R_k^5 D^{3}K_{3i}^3 }{16K_{2i}^9 K_{1i}}\\ & +\frac{15015R_k^5 D^{2}K_{3i}^3 }{128K_{2i}^8 K_{1i}^2 }+\frac{1155R_k^4 D^{2}K_{4i}}{8K_{2i}^7 }+\frac{847R_k^5 DK_{4i}}{16K_{2i}^7 }-\frac{429D^{7}K_{5i}}{32K_{2i}^7 K_{1i}^3 }-\frac{715R_k^9 K_{5i}}{128K_{2i}^{15/2} }+\frac{385R_k^2 D^{3}K_{3i}}{32K_{2i}^7 }\\ & -\frac{15015R_k^3 D^{4}K_{5i}}{32K_{2i}^{15/2} }+\frac{715R_k^3 D^{2}K_{3i}^3 }{32K_{2i}^6 K_{1i}^4 }-\frac{385R_k^4 K_{3i}^3 }{128K_{2i}^7 K_{1i}}+\frac{385R_k^4 DK_{3i}}{32K_{2i}^7 }+\frac{1155R_k^3 D^{2}K_{3i}}{64K_{2i}^7 }+\frac{1001R_k^5 K_{3i}^3 }{128K_{2i}^7 K_{1i}^2 }\\ & +\frac{385R_k^3 K_{3i}^3 }{128K_{2i}^6 K_{1i}^2 }-\frac{143R_k^3 DK_{3i}^3 }{28K_{2i}^5 K_{1i}^5 }-\frac{385R_k^4 H_j K_{3i}}{128K_{2i}^7 }+\frac{5005R_k^7 D^{2}K_{3i}}{32K_{2i}^9 }-\frac{3575R_k^3 D^{3}K_{3i}^3 }{48K_{2i}^7 K_{1i}^3 }-\frac{9009R_k^2 D^{5}K_{5i}}{32K_{2i}^{15/2} K_{1i}^2}\\ & +\frac{715R_k^4 DK_{3i}^3 }{64K_{2i}^6 K_{1i}^4 }-\frac{315R_k^2 DK_{5i}}{32K_{2i}^{11/2} }-\frac{1001R_k^6 K_{3i}^3 }{128K_{2i}^8 K_{1i} }-\frac{6435R_k^2 D^{7}K_{5i}}{32K_{2i}^{17/2} }-\frac{1573R_k^2 K_{3i}^3 }{1680K_{2i}^4 K_{1i}^5 }+\frac{805R_k^3 DK_{4i}}{64K_{2i}^6 }\\ & -\frac{429R_k^2 D^{2}K_{3i}^3 }{56K_{2i}^5 K_{1i}^5 }-\frac{5005R_k^3 D^{5}H_j K_{3i}}{16K_{2i}^9}-\frac{5005R_k^3 D^{5}K_{3i}^3 }{16K_{2i}^9 K_{1i}}-\frac{25025R_k^4 D^{4}H_j K_{3i}}{64K_{2i}^9 }-\frac{6435R_k^7 D^{2}K_{5i}}{32K_{2i}^{17/2} }\\ & +\frac{15015R_k^3 D^{4}K_{3i}}{128K_{2i}^8 }-\frac{693D^{5}K_{5i} }{64K_{2i}^{13/2}}-\frac{143R_k^4 K_{3i}^3 }{32K_{2i}^6 K_{1i}^3 }+\frac{15015R_k^6 D^{3}K_{3i}}{32K_{2i}^9 }-\frac{6435R_k^8 DK_{5i} }{128K_{2i}^{17/2}}-\frac{3465R_k^2 D^{3}K_{5i} }{32K_{2i}^{13/2}}\\ & +\frac{3465R_k^2 D^{3}K_{3i}}{32K_{2i}^7 }+\frac{105D^{4}K_{4i}}{32K_{2i}^6 }-\frac{1001R_k^6 H_j K_{3i}}{128K_{2i}^8}-\frac{45045R_k^4 D^{5}K_{5i}}{64K_{2i}^{17/2} }-\frac{45045R_k^5 D^{4}K_{5i}}{64K_{2i}^{17/2} }\\ & -\frac{15015R_k^3 D^{6}K_{5i}}{32K_{2i}^{17/2} }-\frac{2871R_k^2 K_{3i}^3 }{2240K_{2i}^5 K_{1i}^3 }-\frac{35D^{2}K_{3i}^3 }{128K_{2i}^6 K_{1i} }+\frac{715R_k^9 K_{3i}}{128K_{2i}^9 }-\frac{15015R_k^4 D^{3}K_{3i}}{32K_{2i}^8 }+\frac{315R_k^2 DK_{3i} }{32K_{2i}^6 }\\ & -\frac{15015R_k^6 D^{3}K_{5i}}{32K_{2i}^{17/2} }+\frac{1515R_k^3 D^{6}K_{3i}}{32K_{2i}^9 }+\frac{105105R_k^4 D^{4}K_{4i}}{128K_{2i}^8 }+\frac{9009R_k^5 D^{2}K_{3i}}{32K_{2i}^8 }-\frac{715R_k^8 H_j K_{3i}}{128K_{2i}^9 }\\ & +\frac{2431R_k^3 K_{3i}^3 }{1120K_{2i}^5 K_{1i}^4 }+\frac{1515R_k^3 D^{4}K_{3i}}{32K_{2i}^8 }+\frac{3003R_k^6 DK_{3i}}{64K_{2i}^8 }+\frac{35DK_{3i}}{128K_{2i}^5 }-\frac{693R_k^5 K_{5i}}{64K_{2i}^{13/2} }+\frac{5005R_k^3 D^{7}K_{4i} }{8K_{2i}^9 }\\ & -\frac{35R_k^2 K_{3i}^3 }{128K_{2i}^6 K_{1i}}+\frac{9009R_k^2 D^{5}K_{3i}}{32K_{2i}^8 }-\frac{3465R_k D^{4}K_{5i}}{64K_{2i}^{13/2} }+\frac{693D^{5}K_{3i}}{64K_{2i}^7 }+\frac{20405R_k^2 D^{4}K_{4i}}{128K_{2i}^7 }\\ & +\frac{15015R_k^5 D^{2}K_{3i}}{128K_{2i}^8 }+\frac{429D^{8}K_{4i}}{32K_{2i}^8 }+\frac{6435R_k^7 D^{2}K_{3i}}{32K_{2i}^9 }+\frac{47047R_k^2 D^{6}K_{4i}}{128K_{2i}^8 }+\frac{35R_k^2 DK_{3i}}{64K_{2i}^6 }\\ & +\frac{3003R_k^2 D^{5}K_{3i}}{64K_{2i}^8 }+\frac{2485R_k^2 D^{2}K_{4i}}{128K_{2i}^6 }-\frac{3861R_k^7 DK_{4i}}{64K_{2i}^8 }-\frac{9009R_k^5 D^{2}K_{5i}}{32K_{2i}^{15/2} }-\frac{65D^{2}K_{3i}^3 }{168K_{2i}^3 K_{1i}^7 }-\frac{35R_k^2 H_j K_{3i} }{128K_{2i}^6 }\\ & -\frac{429R_k^7 K_{5i}}{32K_{2i}^{15/2} }+\frac{19019R_k^5 D^{3}K_{4i}}{32K_{2i}^8 }+\frac{5005R_k^4 D^{3}K_{3i}}{32K_{2i}^8 }-\frac{385D^{4}K_{3i}^3 }{128K_{2i}^7 K_{1i} }-\frac{385D^{4}H_j K_{3i}}{128K_{2i}^7 }-\frac{15015R_k^4 D^{3}K_{5i} }{32K_{2i}^{15/2} }\\ & +\frac{385D^{3}K_{3i}^3 }{128K_{2i}^6 K_{1i}^2 }-\frac{105D^{3}K_{5i}}{32K_{2i}^{11/2} }+\frac{35R_k D^{2}K_{3i}}{128K_{2i}^6 }+\frac{385R_k D^{4}K_{3i}}{128K_{2i}^7 }+\frac{6435R_k^2 D^{7}K_{3i}}{32K_{2i}^9 }+\frac{33033R_k^6 D^{2}K_{4i} }{128K_{2i}^8 }\\ & -\frac{143D^{4}K_{3i}^3 }{112K_{2i}^5 K_{1i}^5 }-\frac{3465R_k^4 DK_{5i}}{64K_{2i}^{13/2} }-\frac{715D^{6}K_{3i}^3 }{192K_{2i}^7 K_{1i}^3 }+\frac{715D^{7}K_{3i}^3 }{128K_{2i}^8 K_{1i}^2 }-\frac{715D^{8}H_j K_{3i}}{128K_{2i}^9 }-\frac{35D^{2}H_j K_{3i} }{128K_{2i}^6 }\\ & +\frac{143D^{5}K_{3i}^3 }{64K_{2i}^6 K_{1i}^4 }-\frac{3575R_k^2 D^{4}K_{3i}^3 }{64K_{2i}^7 K_{1i}^3 }-\frac{1573D^{2}K_{3i}^3 }{1680K_{2i}^4 K_{1i}^5 }+\frac{2651DK_{3i}^3 }{6720K_{2i}^4 K_{1i}^4 }+\frac{715R_k D^{8}K_{3i}}{128K_{2i}^9 }-\frac{715D^{8}K_{3i}^3 }{128K_{2i}^9 K_{1i} }\\ & +\frac{5005R_k D^{4}K_{3i}^3 }{128K_{2i}^7 K_{1i}^2 }+\frac{12155R_k^8 D^{2}K_{4i}}{128K_{2i}^9 }-\frac{3465R_k^3 D^{2}K_{5i}}{32K_{2i}^{13/2} }+\frac{5DK_{3i}^3 }{24K_{2i}^2 K_{1i}^8 }+\frac{39DK_{3i}}{112K_{2i}^3 K_{1i}^6 }+\frac{25025R_k^5 D^{4}K_{3i}}{64K_{2i}^9 }\\ & +\frac{3003R_k^6 DK_{3i}}{32K_{2i}^8 }+\frac{5005R_k^3 D^{6}K_{3i}}{32K_{2i}^9 }+\frac{35DK_{3i}^3 }{128K_{2i}^5 K_{1i}^2 }-\frac{65R_k^2 K_{3i}^3 }{168K_{2i}^3 K_{1i}^7 }-\frac{715D^{9}K_{5i}}{128K_{2i}^{17/2} }-\frac{3003R_k D^{5}H_j K_{3i}}{64K_{2i}^8 }\\ & +\frac{2079R_k D^{5}K_{4i}}{32K_{2i}^7 }-\frac{143R_k D^{3}K_{3i}^3 }{8K_{2i}^6 K_{1i}^3 }+\frac{105D^{3}K_{3i}}{32K_{2i}^6 }-\frac{2871R_k DK_{3i}^3 }{1120K_{2i}^5 K_{1i}^3 }-\frac{715R_k^8 K_{3i}^3 }{128K_{2i}^9 K_{1i}}+\frac{7293R_k D^{2}K_{3i}^3 }{1120K_{2i}^5 K_{1i}^4 }\\ & -\frac{1573R_k DK_{3i}^3 }{840K_{2i}^4 K_{1i}^5 }+\frac{25025R_k^6 D^{4}K_{4i}}{32K_{2i}^9 }+\frac{39R_k K_{3i}^3 }{112K_{2i}^3 K_{1i}^6 }+\frac{3575R_k D^{9}K_{4i}}{64K_{2i}^9 }+\frac{7865R_k^2 D^{8}K_{4i}}{32K_{2i}^9 }\\ & +\frac{6435R_k D^{8}K_{3i}}{128K_{2i}^9 }+\frac{715R_k^2 D^{7}K_{3i}}{16K_{2i}^9 }+\frac{5R_k K_{3i}^3 }{24K_{2i}^2 K_{1i}^8 }+\frac{715D^{3}K_{3i}^3 }{1008K_{2i}^4 K_{1i}^6 }-\frac{35R^k K_{5i}}{128K_{2i}^{9/2} }-\frac{715R_k D^{7}K_{3i}^3 }{16K_{2i}^9 K_{1i}}\\ & +\frac{65065R_k^4 D^{6}K_{4i}}{64K_{2i}^9 }-\frac{715R_k D^{5}K_{3i}^3 }{32K_{2i}^7 K_{1i}^3 }+\frac{5005R_k D^{6}K_{3i}^3 }{128K_{2i}^8 K_{1i}^2 }-\frac{715R_k^6 K_{3i}^3 }{192K_{2i}^7 K_{1i}^3 }-\frac{385R_k D^{3}K_{3i}^3 }{32K_{2i}^7 K_{1i} }+\frac{715D^{10}K_{4i}}{128K_{2i}^9 }\\ & -\frac{3003R_k D^{6}K_{5i}}{32K_{2i}^{15/2} }-\frac{3003R_k D^{5}K_{3i}^3 }{64K_{2i}^8 K_{1i}}+\frac{715D^{9}K_{3i}}{128K_{2i}^9 }-\frac{315R_k D^{2}K_{5i}}{32K_{2i}^{11/2} }-\frac{35DK_{5i}}{128K_{2i}^9 }+\frac{6435R_k^8 DK_{3i}}{128K_{2i}^9 }\\ & -\frac{65R_k DK_{3i}^3 }{84K_{2i}^3 K_{1i}^7 }+\frac{1155R_k D^{2}K_{3i}^3 }{128K_{2i}^6 K_{1i}^2 }+\frac{143R_k^5 K_{3i}^3 }{64K_{2i}^6 K_{1i}^4 }+\frac{3003R_k D^{6}K_{3i}}{32K_{2i}^8 }-\frac{1155R_k^2 D^{2}H_j K_{3i}}{64K_{2i}^7 }+\frac{429D^{7}K_{3i}}{32K_{2i}^8 }\\ & +\frac{2651R_k K_{3i}^3 }{6720K_{2i}^4 K_{1i}^4 }+\frac{429R_k D^{7}K_{4i}}{4K_{2i}^8 }+\frac{715R_k^8 DK_{3i}}{16K_{2i}^9 }-\frac{35R_k DK_{3i}^3 }{64K_{2i}^6 K_{1i} }-\frac{35R_k DH_j K_{3i}}{64K_{2i}^6 }+\frac{5005R_k^6 D^{3}K_{3i}}{16K_{2i}^9 }\\ & -\frac{385R_k D^{3}H_j K_{3i}}{32K_{2i}^7 }+\frac{105R_k D^{3}K_{4i}}{8K_{2i}^6 }+\frac{45045R_k^4 D^{5}K_{3i}}{64K_{2i}^9 }+\frac{1001D^{5}K_{3i}^3 }{128K_{2i}^7 K_{1i}^2 }-\frac{1001D^{6}H_j K_{3i}}{128K_{2i}^8 }-\frac{143D^{4}K_{3i}^3 }{32K_{2i}^6 K_{1i}^3 }\\ & +\frac{2431D^{3}K_{3i}^3 }{1120K_{2i}^5 K_{1i}^4 }-\frac{K_{3i}^3 }{9K_{2i} K_{1i}^9 }-\frac{143R_k D^{3}K_{3i}^3 }{28K_{2i}^5 K_{1i}^5 }-\frac{715R_k D^{7}H_j K_{3i}}{16K_{2i}^9 }-\frac{8K_{3i}^3 }{105K_{2i}^3 K_{1i}^5 }+\frac{693D^{6}K_{4i}}{64K_{2i}^7}\\ & +\frac{105R_k^3 K_{3i}}{32K_{2i}^6 }-\frac{6435R_k D^{8}K_{5i}}{128K_{2i}^{17/2} }-\frac{2K_{3i}^3 }{21K_{2i}^2 K_{1i}^7 }-\frac{385R_k^3 DH_j K_{3i}}{32K_{2i}^7 }+\frac{1001R_k D^{6}K_{3i}}{128K_{2i}^8 }-\frac{15015R_k^2 D^{4}H_j K_{3i}}{128K_{2i}^8 }\\ & +\frac{35R_k K_{3i}}{128K_{2i}^5 }+\frac{315R_k D^{2}K_{3i}}{32K_{2i}^6 }+\frac{6545R_k^3 D^{3}K_{4i}}{32K_{2i}^7 }+\frac{715R_k^7 D^{3}K_{4i}}{2K_{2i}^9 }-\frac{143R_k^4 K_{3i}^3 }{112K_{2i}^5 K_{1i}^5 }+\frac{45045R_k^5 D^{4}K_{3i}}{64K_{2i}^9 }\\ & +\frac{5005R_k^4 D^{5}K_{3i}}{16K_{2i}^9 }+\frac{3465R_k D^{4}K_{3i}}{64K_{2i}^7 }+\frac{715R_k^9 DK_{4i}}{64K_{2i}^9 }+\frac{3465R_k^4 DK_{3i}}{64K_{2i}^7 }+\frac{35035R_k^5 D^{5}K_{4i}}{32K_{2i}^9 }\\ & +\frac{3465R_k^3 D^{2}K_{3i}}{32K_{2i}^7 }+\frac{715R_k D^{4}K_{3i}^3 }{64K_{2i}^6 }+\frac{715R_k D^{2}K_{3i}^3 }{336K_{2i}^4 K_{1i}^6 }+\frac{35R_k DK_{4i}}{64K_{2i}^5 }-\frac{1001D^{6}K_{3i}^3 }{128K_{2i}^8 }+\frac{715R_k^7 K_{3i}^3 }{128K_{2i}^8 K_{1i}^2 }\\ & \left. -\frac{3003R_k^6 DK_{5i}}{32K_{2i}^{15/2} }+\frac{715R_k^3 K_{3i}^3 }{1008K_{2i}^4 K_{1i}^6}\right) \end{aligned} $$

(A46)

Here i = 1, 2, 3, 4, 7 or 8.

$$ \begin{aligned}F_{rep}^{16}= & \frac{-2c_2 LH_{nr} \sigma ^{6}}{45\pi (D+H_4)^{9}(D+H_0 )^{9}}\cdot \left (-9D^{8}H_0 -36D^{7}H_0^2 -84D^{6}H_0^3 -126D^{5}H_0^4 -126D^{4}H_0^5\right. \\ & -84D^{3}H_0^6 -36D^{2}H_0^7 -9DH_0^8 -H_0^9 +9D^{8}H_4 +36D^{7}H_4^2 +84D^{6}H_4^3 +\;126D^{5}H_4^4 \\ & \left. +126D^{4}H_4^5 +84D^{3}H_4^6 +36D^{2}H_4^7 +9DH_4^8 +H_4^9\right) \end{aligned} $$

(A47)

$$ \begin{aligned}F_{rep}^{15} = & \frac{-2c_2 LH_{nr} \sigma ^{6}}{45\pi (D+H_4 )^{9}(D+H_2 )^{9}}\cdot\left(-9D^{8}H_2 -36D^{7}H_2^2 -84D^{6}H_2^3 -126D^{5}H_2^4 -126D^{4}H_2^5\right. \\ & -84D^{3}H_2^6 -36D^{2}H_2^7 -9DH_2^8 -H_2^9 +9D^{8}H_4 +36D^{7}H_4^2 +84D^{6}H_4^3 +\;126D^{5}H_4^4 \\ & \left. +126D^{4}H_4^5 +84D^{3}H_4^6 +36D^{2}H_4^7 +9DH_4^8 +H_4^9\right) \end{aligned} $$

(A48)