Method for calculating adsorption force of electrostatic chuck of comb-tooth electrode structure

Abstract

The invention discloses a method for calculating the adsorption force of an electrostatic chuck of a comb-tooth electrode structure. The method comprises the following steps: establishing a surface charge density function on the surface of an electrostatic chuck electrode; calculating the potential coefficient of a certain point P in a three-dimensional space of the electrostatic chuck electrode based on a surface charge density function and according to the Coulomb's law, and further obtaining the potential of the point P; according to the potential gradient of the P point in the three-dimensional space, the electric field intensity of the P point in the x, y and z directions in the single-medium three-dimensional space is solved; according to the obtained electric field intensities of the P point in the x, y and z directions, the electric field intensity of the P point under the multi-medium condition is calculated through a charge mirror image method; according to the obtained electric field intensity of the P point under the multi-medium condition, the unit area electrostatic adsorption force borne by the adsorbed medium is calculated through a Maxwell stress tensor method. According to the method for calculating the adsorption force of the electrostatic chuck of the comb-tooth electrode structure, the problem that an existing theoretical model is relatively large in limitation when the electrostatic adsorption force is relatively large in difference is solved.

Description

Technical Field

[0001] The invention belongs to the technical field of electrostatic chucks, and in particular relates to a method for calculating the adsorption force of an electrostatic chuck with a comb-tooth electrode structure. Background Art

[0002] In recent years, with the rapid development of science and technology, industrial fields such as manufacturing and logistics have put forward higher requirements on the positioning and placement accuracy of objects. In this context, adsorption technology, as a key means to achieve precise manipulation of objects, has received widespread attention.

[0003] Electrostatic adsorption technology, with its low energy consumption, fast response and lightweight characteristics, has shown certain advantages in adsorbing porous, fragile, soft and other materials. It has been widely used in many fields such as robotics, variable stiffness technology, and tactile technology. However, despite its wide range of practical applications, theoretical research on electrostatic adsorption technology, that is, in-depth analysis of electrostatic chucks with different electrode structures, is relatively scarce. By establishing an algorithm for the electrostatic adsorption force of an electrostatic chuck, the adsorption performance of the electrostatic chuck can be effectively controlled.

[0004] In 2016, Guo proposed a simplified theoretical analytical model of comb-tooth electrostatic adsorption. The model assumes that the dielectric in the system is a uniform, linear and isotropic material. Based on these assumptions, Guo calculated the equivalent dielectric constants of various media and converted the three-dimensional problem into a more tractable two-dimensional problem to solve the distribution characteristics of the electric field. In 2021, researchers from Nanjing Agricultural University used Guo's method to calculate the magnitude of the electrostatic adsorption force. They analyzed the influence of parameters such as electrode width, electrode spacing, and dielectric constant on the adsorption performance of the comb-tooth electrostatic chuck, and verified the validity of the theoretical model through simulation. The theoretical models and simulation results proposed by the two can coincide with each other under certain specific parameter conditions, but under other parameter conditions, there are large differences in the electrostatic adsorption force, which shows that the theoretical model has limitations to a certain extent. Summary of the invention

[0005] The purpose of the present invention is to provide a method for calculating the adsorption force of an electrostatic chuck of a comb-tooth electrode structure, so as to solve the problem that the existing theoretical model has great limitations when there are large differences in the electrostatic adsorption forces.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is: a method for calculating the adsorption force of an electrostatic chuck of a comb-tooth electrode structure, which is specifically implemented according to the following steps:

[0007] Step 1, establishing a surface charge density function on the surface of the electrostatic chuck electrode, and obtaining the value of the unknown coefficient of the surface charge density function based on the potential of a certain point on the known electrode according to the charging voltage;

[0008] Step 2, based on the surface charge density function and according to Coulomb's law, the potential coefficient of a point P in the three-dimensional space of the electrostatic chuck electrode is calculated, and then the potential of the point P is obtained;

[0009] Step 3, according to the potential gradient of point P in the three-dimensional space, solve the electric field intensity in the x, y, and z directions of point P in the three-dimensional space of the single medium;

[0010] Step 4, according to the obtained electric field strength in the x, y, and z directions of point P, the electric field strength at point P under multi-medium conditions is calculated by the charge image method;

[0011] Step 5, according to the obtained electric field strength at point P under the multi-medium condition, the electrostatic adsorption force per unit area on the adsorbed medium is calculated using the Maxwell stress tensor method.

[0012] As a preferred technical solution of the present invention, the step 1 is specifically implemented as follows:

[0013] The surface charge density distribution function of the electrostatic chuck electrode surface is established as:

[0014]

[0015] Where: A, B and C are assumed unknown coefficients; c is the x coordinate of the center position of the electrode;

[0016] The potentials of three points on the known electrode are known according to the charging voltage. The values ​​of unknown coefficients A, B and C are obtained according to the total potential formula at point P, and the potential of point P is calculated as:

[0017]

[0018] As a preferred technical solution of the present invention, step 2 is specifically implemented in the following manner:

[0019] According to Coulomb's law, the potential coefficient of point P is:

[0020]

[0021]

[0022] Where: P + is the potential coefficient of point P relative to a point charge on the positive electrode; P - is the potential coefficient of point P relative to a point charge on the negative electrode; (x, y, z) is the coordinate of point P; (x0, y0, z0) is the coordinate of a point charge on the electrode; ε is the dielectric constant of the medium; q + is the total charge on the positive electrode surface; q - is the total charge on the surface of the negative electrode; is the total potential at point P; σ+ is the charge density distribution function on the positive electrode surface; σ - is the charge density distribution function on the negative electrode surface; s0 is the area of ​​the electrode surface; l is the length of the electrode; w is the width of the electrode; is the potential generated by the positive electrode at point P; is the potential generated by the negative potential at point P; is the total potential at point P.

[0023] As a preferred technical solution of the present invention, step 3 is specifically implemented in the following manner:

[0024] According to the gradient of the electric potential, the electric field strength in the x, y, and z directions at point P in space is solved as follows:

[0025]

[0026] Where: E Pz , E Pz , E Pz is the electric field strength in the x, y, and z directions at point P.

[0027] As a preferred technical solution of the present invention, step 4 is specifically implemented in the following manner:

[0028] The electric field strength at point P under multi-medium conditions is calculated by the charge image method:

[0029]

[0030] Where: ε1 is the dielectric constant of region 1 under multi-medium conditions; ε2 is the dielectric constant of region 2 under multi-medium conditions; h is the distance from the charge to the region boundary; k1, k2 are proportional coefficients; is the electric potential in region 1 and region 2.

[0031] As a preferred technical solution of the present invention, step 5 is specifically implemented in the following manner:

[0032] Maxwell stress tensor method is used to calculate the electrostatic adsorption force per unit area on the adsorbed medium:

[0033]

[0034] Where: f x f is the tangential electrostatic adsorption force per unit area of ​​the adsorbed medium; z is the normal electrostatic adsorption force per unit area of ​​the adsorbed medium; w+s is the width of the adsorbed medium; ε0 is the dielectric constant of vacuum; ε3 is the relative dielectric constant of the adsorbed medium;

[0035] When calculating the tangential electrostatic adsorption force per unit area, the electric field intensity Ex and E z The function property of is an even function and an odd function. The product of the two must be 0 under the constraint of integration, that is, the tangential electrostatic adsorption force is zero; however, in actual situations, the electrostatic adsorption forces between adjacent electrodes and the polarized charges generated by themselves are equal in magnitude and opposite in direction. They are all electrostatic adsorption forces that attract opposite charges to the critical surface, so it can be changed to:

[0036]

[0037] Where: f x1 f is the tangential electrostatic adsorption force per unit area of ​​the adsorbed medium; z2 is the normal electrostatic adsorption force per unit area of ​​the adsorbed medium; w+s is the width of the adsorbed medium; ε0 is the dielectric constant of vacuum; ε3 is the relative dielectric constant of the adsorbed medium.

[0038] The beneficial effects of the present invention are as follows: a method for calculating the adsorption force of an electrostatic chuck of a comb-tooth electrode structure of the present invention, by assuming the surface charge density function of the electrode surface, uses Coulomb's law and the mirror method to derive a theoretical analytical model of the electric field strength of the electrostatic chuck at any point in space under multi-medium conditions; further uses the Maxwell stress tensor method to establish a theoretical analytical model of the electrostatic adsorption force exerted on the adsorbed medium, and by establishing a method for calculating the electrostatic adsorption force of the electrostatic chuck, the adsorption performance of the electrostatic chuck can be effectively controlled. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0040] Figure 1 This is a structural diagram of a planar double-comb electrode involved in the present invention;

[0041] Figure 2 It is a two-dimensional cross-sectional view of an electrostatic chuck with a planar double-comb-teeth electrode structure involved in the present invention;

[0042] Figure 3 It is a curve comparison diagram of the tangential and normal electrostatic adsorption force algorithm theoretical values ​​and software simulation values ​​per unit area of ​​an electrostatic chuck adsorption force algorithm of a comb-tooth electrode structure involved in the present invention. DETAILED DESCRIPTION

[0043] The technical solution of the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0044] Example 1

[0045] The method for calculating the adsorption force of the electrostatic chuck of the comb-tooth electrode structure of the present invention is specifically implemented according to the following steps:

[0046] Step 1, establishing a surface charge density function on the surface of the electrostatic chuck electrode, and obtaining the value of the unknown coefficient of the surface charge density function based on the potential of a certain point on the known electrode according to the charging voltage;

[0047] The surface charge density distribution function of the electrostatic chuck electrode surface is established as:

[0048]

[0049] Where: A, B and C are assumed unknown coefficients; c is the x coordinate of the center position of the electrode;

[0050] The potentials of three points on the known electrode are known according to the charging voltage. The values ​​of unknown coefficients A, B and C are obtained according to the total potential formula at point P, and the potential of point P is calculated as:

[0051]

[0052] Step 2, based on the surface charge density function and according to Coulomb's law, the potential coefficient of a point P in the three-dimensional space of the electrostatic chuck electrode is calculated, and then the potential of the point P is obtained;

[0053] According to Coulomb's law, the potential coefficient of point P is:

[0054]

[0055]

[0056] Where: P + is the potential coefficient of point P relative to a point charge on the positive electrode; P - is the potential coefficient of point P relative to a point charge on the negative electrode; (x, y, z) is the coordinate of point P; (x0, y0, z0) is the coordinate of a point charge on the electrode; ε is the dielectric constant of the medium; q + is the total charge on the positive electrode surface; q - is the total charge on the surface of the negative electrode; is the total potential at point P; σ + is the charge density distribution function on the positive electrode surface; σ - is the charge density distribution function on the negative electrode surface; s0 is the area of ​​the electrode surface; l is the length of the electrode; w is the width of the electrode; is the potential generated by the positive electrode at point P; is the potential generated by the negative potential at point P; is the total potential at point P.

[0057] Step 3, according to the potential gradient of point P in the three-dimensional space, solve the electric field intensity in the x, y, and z directions of point P in the three-dimensional space of the single medium;

[0058] Step 4, according to the obtained electric field strength in the x, y, and z directions of point P, the electric field strength at point P under multi-medium conditions is calculated by the charge image method;

[0059] Step 5, according to the obtained electric field strength at point P under the multi-medium condition, the electrostatic adsorption force per unit area on the adsorbed medium is calculated using the Maxwell stress tensor method.

[0060] Example 2

[0061] Combination Figure 1 and Figure 2 , which is different from Example 1, in the method for calculating the adsorption force of an electrostatic chuck of a comb-tooth electrode structure of the present invention, the step 3 is specifically implemented in the following manner:

[0062] According to the gradient of the electric potential, the electric field strength in the x, y, and z directions at point P in space is solved as follows:

[0063]

[0064] Where: E Pz , E Pz , E Pz is the electric field strength in the x, y, and z directions at point P.

[0065] Step 4 is specifically implemented as follows:

[0066] The electric field strength at point P under multi-medium conditions is calculated by the charge image method:

[0067]

[0068] Where: ε1 is the dielectric constant in region 1 (air layer region) under multi-medium conditions; ε2 is the dielectric constant in region 2 (covering layer region) under multi-medium conditions; h is the distance from the charge to the region boundary; k1, k2 are proportional coefficients; is the electric potential in region 1 and region 2.

[0069] Example 3

[0070] Different from Example 1, in a method for calculating the adsorption force of an electrostatic chuck with a comb-tooth electrode structure of the present invention, the Maxwell stress tensor method is used to calculate the electrostatic adsorption force per unit area on the adsorbed medium:

[0071]

[0072] Where: f xf is the tangential electrostatic adsorption force per unit area of ​​the adsorbed medium; z is the normal electrostatic adsorption force per unit area of ​​the adsorbed medium; w+s is the width of the adsorbed medium; ε0 is the dielectric constant of vacuum; ε3 is the relative dielectric constant of the adsorbed medium;

[0073] When calculating the tangential electrostatic adsorption force per unit area, the electric field intensity E x and E z The function property of is an even function and an odd function. The product of the two must be 0 under the constraint of integration, that is, the tangential electrostatic adsorption force is zero; however, in actual situations, the electrostatic adsorption forces between adjacent electrodes and the polarized charges generated by themselves are equal in magnitude and opposite in direction. They are all electrostatic adsorption forces that attract opposite charges to the critical surface, so it can be changed to:

[0074]

[0075] Where: f x1 f is the tangential electrostatic adsorption force per unit area of ​​the adsorbed medium; z2 is the normal electrostatic adsorption force per unit area of ​​the adsorbed medium; w+s is the width of the adsorbed medium; ε0 is the dielectric constant of vacuum; ε3 is the relative dielectric constant of the adsorbed medium.

[0076] Combination Figure 3 , through this calculation method, the above parameters are substituted into the model of the comb-tooth electrostatic chuck. The theoretical value is the value calculated by the above calculation method, and the simulation value is the value output by the software simulation. The tangential and normal electrostatic adsorption forces per unit area of ​​the adsorbed medium can be calculated. The theoretical parameters of the comb-tooth electrostatic chuck are shown in Table 1.

[0077] Table 1 Theoretical parameters of comb-tooth electrostatic chuck

[0078] Charging voltage U(kV) 3 <![CDATA[Relative permittivity ε of the insulating dielectric r > 7.8 <![CDATA[Base layer thickness d1 (mm)]]> 0.2 <![CDATA[Thickness d2 of the intermediate layer (mm)]]> 0.15 <![CDATA[Thickness d3 of the covering layer (mm)]]> 0.2 <![CDATA[Width w1 (mm) of the first-layer electrode]]> 0.6 <![CDATA[Width w2 (mm) of the second layer electrode]]> 0.6 Tooth pitch(mm) 0.4 <![CDATA[Relative permittivity ε1 of the adsorption medium]]> 2.5

[0079] The present invention designs a theoretical analytical model for calculating the electric field strength and adsorption force of an electrostatic chuck with a comb-tooth electrode structure. By comparing the simulation results with the theoretical analytical values, the effectiveness of the theoretical analytical model of the electric field strength is verified, and the comb-tooth electrode structure electrostatic chuck is deeply analyzed. By establishing a calculation method for the electrostatic adsorption force of the electrostatic chuck, the adsorption performance of the electrostatic chuck can be effectively controlled.

[0080] The above description shows and describes several preferred embodiments of the invention, but as mentioned above, it should be understood that the invention is not limited to the form disclosed herein, and should not be regarded as excluding other embodiments, but can be used in various other combinations, modifications and environments, and can be modified within the scope of the invention concept described herein through the above teachings or the technology or knowledge of the relevant field. Changes and modifications made by those skilled in the art should be within the scope of protection of the claims attached to the invention without departing from the spirit and scope of the invention.

Claims

1. A method for calculating the adsorption force of an electrostatic chuck of a comb-tooth electrode structure, characterized in that: Follow the steps below to implement it: Step 1, establishing a surface charge density function on the surface of the electrostatic chuck electrode, and obtaining the value of the unknown coefficient of the surface charge density function based on the potential of a certain point on the known electrode according to the charging voltage; Step 2, based on the surface charge density function and according to Coulomb's law, the potential coefficient of a point P in the three-dimensional space of the electrostatic chuck electrode is calculated, and then the potential of the point P is obtained; Step 3, according to the potential gradient of point P in the three-dimensional space, solve the electric field intensity in the x, y, and z directions of point P in the three-dimensional space of the single medium; Step 4, according to the obtained electric field strength in the x, y, and z directions of point P, the electric field strength at point P under multi-medium conditions is calculated by the charge image method; Step 5, according to the obtained electric field strength at point P under the multi-medium condition, the electrostatic adsorption force per unit area on the adsorbed medium is calculated using the Maxwell stress tensor method.

2. The method for calculating the adsorption force of an electrostatic chuck of a comb-tooth electrode structure according to claim 1, characterized in that: The step 1 is specifically implemented in the following manner: The surface charge density distribution function of the electrostatic chuck electrode surface is established as: Where: A, B and C are assumed unknown coefficients; c is the x coordinate of the center position of the electrode; The potentials of three points on the known electrode are known according to the charging voltage. The values ​​of unknown coefficients A, B and C are obtained according to the total potential formula at point P, and the potential of point P is calculated as:

3. The method for calculating the adsorption force of an electrostatic chuck of a comb-tooth electrode structure according to claim 1, characterized in that: The step 2 is specifically implemented in the following manner: According to Coulomb's law, the potential coefficient of point P is: Where: P + is the potential coefficient of point P relative to a point charge on the positive electrode; P - is the potential coefficient of point P relative to a point charge on the negative electrode; (x, y, z) is the coordinate of point P; (x0, y0, z0) is the coordinate of a point charge on the electrode; ε is the dielectric constant of the medium; q + is the total charge on the positive electrode surface; q - is the total charge on the surface of the negative electrode; is the total potential at point P; σ + is the charge density distribution function on the positive electrode surface; σ - is the charge density distribution function on the negative electrode surface; s0 is the area of ​​the electrode surface; l is the length of the electrode; w is the width of the electrode; is the potential generated by the positive electrode at point P; is the potential generated by the negative potential at point P; is the total potential at point P.

4. The method for calculating the adsorption force of an electrostatic chuck of a comb-tooth electrode structure according to claim 1, characterized in that: The step 3 is specifically implemented in the following manner: According to the gradient of the electric potential, the electric field strength in the x, y, and z directions at point P in space is solved as follows: Where: E Pz , E Pz , E Pz is the electric field strength in the x, y, and z directions at point P.

5. The method for calculating the adsorption force of an electrostatic chuck of a comb-tooth electrode structure according to claim 1, characterized in that: The step 4 is specifically implemented in the following manner: The electric field strength at point P under multi-medium conditions is calculated by the charge image method: Where: ε1 is the dielectric constant of region 1 under multi-medium conditions; ε2 is the dielectric constant of region 2 under multi-medium conditions; h is the distance from the charge to the region boundary; k1, k2 are proportional coefficients; is the electric potential in region 1 and region 2.

6. The method for calculating the adsorption force of an electrostatic chuck of a comb-tooth electrode structure according to claim 1, characterized in that: The step 5 is specifically implemented in the following manner: Maxwell stress tensor method is used to calculate the electrostatic adsorption force per unit area on the adsorbed medium: Where: f x f is the tangential electrostatic adsorption force per unit area of ​​the adsorbed medium; z is the normal electrostatic adsorption force per unit area of ​​the adsorbed medium; w+s is the width of the adsorbed medium; ε0 is the dielectric constant of vacuum; ε3 is the relative dielectric constant of the adsorbed medium; When calculating the tangential electrostatic adsorption force per unit area, the electric field intensity E x and E z The function property of is an even function and an odd function. The product of the two must be 0 under the constraint of integration, that is, the tangential electrostatic adsorption force is zero; however, in actual situations, the electrostatic adsorption forces between adjacent electrodes and the polarized charges generated by themselves are equal in magnitude and opposite in direction. They are all electrostatic adsorption forces that attract opposite charges to the critical surface, so it can be changed to: Where: f x1 f is the tangential electrostatic adsorption force per unit area of ​​the adsorbed medium; z2 is the normal electrostatic adsorption force per unit area of ​​the adsorbed medium; w+s is the width of the adsorbed medium; ε0 is the dielectric constant of vacuum; ε3 is the relative dielectric constant of the adsorbed medium.

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