Parametric model of capillary action under microscale probe sphere diameter and its design method

Abstract

The present application relates to the field of micro-nano measurement, and in particular to a parameterized model of capillary action under the diameter of a micro-scale probe ball and a design method thereof. The parameterized model of capillary action under the diameter of a micro-scale probe ball provided by the present application includes two types, which are a relative humidity sensitive parameterized model suitable for application in a measurement scene in which the relative humidity of a measurement environment fluctuates, and a surface roughness sensitive parameterized model suitable for application in a strategy scene in which the surface roughness of a measured interface in a measurement environment is not uniform. The former is obtained by first constructing a mapping relationship between a Kelvin radius and relative humidity, and then introducing a simplified model for secondary simplification, and the latter is obtained by first reconstructing a mathematical equation of a rough interface topography, and then introducing a simplified model for targeted simplification. The present application solves the problem that the precision of various capillary force measurement schemes in the prior art is too low at the micron scale, and the measurement schemes cannot adapt to measurement scenes in which the humidity changes or the surface roughness is not uniform.

Description

Technical Field

[0001] This invention relates to the field of micro-nano measurement, specifically to a parameterized model and design method of capillary action under the microscale probe sphere diameter. Background Technology

[0002] With the development of science and technology, people have focused their attention on the micro-nano field, such as microelectromechanical systems, femtosecond laser micro-nano fabrication technology, and micro-nano fiber optic technology. All of these applications rely heavily on micro-nano measurement technology. As the size of micro-nano devices continues to shrink, the microscopic forces between interfaces become increasingly significant. Based on their mechanisms, these microscopic forces can be categorized into van der Waals forces, electrostatic forces, and capillary forces. Capillary forces, as the dominant force component, significantly affect the detection accuracy and reliability of micro-nano measurement systems.

[0003] Existing research indicates that even in environments with low relative humidity, it is impossible to completely eliminate capillary action in contact areas. On the one hand, capillary forces can be harmful; for example, the adhesive forces generated by capillary condensation in MEMS devices can cause friction between micro-device interfaces, and friction has become a major cause of MEMS device failures. On the other hand, capillary forces can be applied to micro-grasping, enabling precise operations such as nano-assembly and nano-fabrication at the micro- and nanoscale. Therefore, in-depth research into the mechanism of capillary forces and the construction of accurate numerical models have significant scientific importance and application potential.

[0004] Currently, there are two theoretical modeling methods for capillary forces: one is based on a combination of the meniscus bridge profile and the Laplace-Young surface tension equation; the other is based on the total energy of the meniscus bridge. However, the mathematical model obtained by the former method is only applicable to interfacial contacts at the nanoscale and cannot be applied at the micrometer scale. The latter method approximates the contact angle as zero and assumes a very small and constant volume of the meniscus bridge, which also becomes inapplicable when the probe sphere diameter is on the order of micrometers. The laws governing capillary forces at the micrometer and nanoscales are different; therefore, both of the above schemes have significant errors when used to measure capillary forces at the micrometer scale.

[0005] In research reports on the influence of relative humidity on capillary forces, the behavior of capillary forces varies. This is due to significant differences in parameters such as sample surface characteristics, probe spherical diameter, and surface roughness used in the experiments. It is evident that the capillary force behavior differs under different material properties, probe spherical diameters, and surface roughness conditions. In studies on the influence of surface roughness on capillary forces, researchers typically use nanoscale probe spherical diameters for capillary force research, neglecting microscale spherical diameters. Furthermore, existing mechanical modeling methods often assume surface morphologies of cubes, hemispheres, or cones, which differ significantly from actual surface morphologies.

[0006] Therefore, incorporating the relative humidity of the measurement environment or the surface roughness of the measured interface into the measurement of capillary action is becoming a key to solving the inapplicability of existing measurement schemes at the micrometer scale. Summary of the Invention

[0007] To address the problem that existing capillary action measurement schemes have low accuracy at the micrometer scale and cannot adapt to measurement scenarios with varying humidity or uneven surface roughness, this invention provides a parameterized model and design method for capillary action at the microscale probe sphere diameter.

[0008] This invention is achieved using the following technical solution:

[0009] A parameterized model of capillary action under micrometer-scale probe spherical diameter is proposed, which is used as a data processing module in micro / nano measurement systems to measure the capillary force F between interfaces under micrometer-scale probe detection conditions. Cap The parametric model uses the relative humidity of the measurement environment as a key parameter, and its expression is as follows:

[0010]

[0011] In the above formula, γ L R represents the surface tension coefficient of the liquid between the probe and the measured interface; C represents the diameter of the sphere at the probe tip; α represents the mean cosine of the surface contact angle between the measured interface and the meniscus bridge; and α represents the angle between the central axis of the sphere at the probe tip and the boundary of the meniscus bridge between the measured interface and the measured interface. g The value represents the gas constant; T represents the ambient temperature on the Kelvin scale; R represents the relative humidity of the measurement environment; and D represents the distance between the probe and the interface being measured.

[0012] In another parameterized model of capillary action at a micrometer-scale probe sphere diameter provided by this invention, the surface roughness of the measured interface is used as a key parameter, and the corresponding expression of the parameterized model is as follows:

[0013]

[0014] In the above formula, γ L θ represents the surface tension coefficient of the liquid between the probe and the measured interface; R represents the diameter of the sphere at the probe tip; θ1 represents the surface contact angle formed by the sphere at the probe tip and the meniscus bridge; ω represents the wetting coefficient of the wetting region between the interfaces; S W S represents the wetted area between interfaces; D θ represents the projected wetted area between interfaces. e The equilibrium contact angle θ of a droplet on an ideal smooth surface. e V m The value represents the molar volume of the liquid in the meniscus bridge between the interfaces; D represents the distance between the probe and the interface being measured.

[0015] This invention also includes a method for designing a parameterized model of capillary action under a micrometer-scale probe spherical diameter. This method uses the relative humidity of the measurement environment as a variable key parameter to design the aforementioned parameterized model of capillary action under a micrometer-scale probe spherical diameter. The design method includes the following steps:

[0016] S1: Calculate the Laplace pressure difference ΔP at the meniscus cross-section formed by the continuous liquid medium using the Young-Laplace equation. L .

[0017] S2: The change in vapor pressure at the detection interface caused by the tortuous liquid-gas interface within the capillary is characterized by relative humidity (RH).

[0018] S3: Under thermodynamic equilibrium conditions, the mapping relationship between the Kelvin equilibrium radius r and the relative humidity RH is constructed, as shown in the following expression:

[0019]

[0020] In the above formula, γ L V represents the surface tension coefficient of the liquid between the probe and the measured interface; m R represents the molar volume of the liquid at the interface; g denoted by ; T represents the ambient temperature on the Kelvin scale; RH represents the relative humidity of the measurement environment; D represents the distance between the probe and the interface being measured; c represents the average cosine of the surface contact angle between the interface being measured and the meniscus bridge; r represents the Kelvin equilibrium radius.

[0021] S4: Capillary force F under the condition of establishing the ball-plate model Cap The general model is derived by ignoring some force components based on microscale scenes, resulting in the following simplified model:

[0022]

[0023] In the above formula, α represents the angle between the central axis of the sphere at the probe tip and the boundary of the meniscus bridge between the interface being measured.

[0024] S5: Introduce the mapping relationship between the Kelvin equilibrium radius r and the relative humidity RH established in step S3 into the simplified model in step S4 to obtain the required parameterized model that takes relative humidity into account.

[0025] As a further improvement of the present invention, in step S1, the Laplace pressure difference ΔP of the meniscus cross section formed by the continuous liquid medium is... L The calculation formula is as follows:

[0026]

[0027] In the above formula, r1 and r2 are any two orthogonal radii of curvature at a point on the surface where pressure is applied, where r1 is perpendicular to the direction of the plate and r2 is parallel to the direction of the plate; X is the azimuth radius, and satisfies:

[0028] X=R sin(α)-r[1-sin(θ1+α)];

[0029] In the above formula, θ1 represents the surface contact angle formed by the sphere at the tip of the probe and the meniscus bridge.

[0030] As a further improvement of the present invention, in step S2, the vapor pressure change caused by the tortuous liquid-gas interface inside the capillary can be calculated and measured using the Kelvin equation based on thermodynamic principles, as follows:

[0031]

[0032] In the above formula, P is the actual external vapor pressure, and P0 is the saturated vapor pressure inside the meniscus bridge. The ratio of the two can be expressed by the relative humidity RH, so:

[0033]

[0034] As a further improvement to the present invention, the process of establishing the simplified model in step S4 is as follows:

[0035] S41: Capillary force F under the condition of constructing a spherical plate model Cap The general model is:

[0036] F Cap =2πXγ L -πX 2 ΔP L

[0037] S42: Substituting the capillary force parameters into the upward equation yields:

[0038]

[0039] S43: Considering that R >> X >> r, D, and that angles α, θ1, and θ2 are extremely small, the 2sin(θ1+α) term in the expression of step S42 can be directly ignored, resulting in:

[0040]

[0041] S44: Since X >> r, then (1 / r - 1 / X) ≈ 1 / r. Item simplified to The simplified model is obtained as follows:

[0042]

[0043] The present invention also includes a method for designing a parameterized model of capillary action under a micrometer-scale probe sphere diameter, which uses the surface roughness of the interface under test as a variable key parameter to design a parameterized model of capillary action under a micrometer-scale probe sphere diameter as described above; the design method includes the following steps:

[0044] S01: Defines the equilibrium contact angle θ of an ideal smooth surface. e Apparent contact angle θ under rough conditions a A unified formula for constructing a roughness surface wetting model:

[0045] cosθ a =ωr RMS cosθ e +ω-1,

[0046] In the above formula, ω and (1-ω) represent the wetting coefficients of the wetted and dry regions within the projected area when the droplet partially contacts the rough surface, respectively; r RMS Roughness ratio, representing the ratio of actual wetted area to projected wetted area.

[0047] S02:

[0048] Define the cross-sectional type of the rough measured interface as a sinusoidal curve surface type, and obtain the equation of the cross-sectional height vector z that characterizes the surface roughness.

[0049] S03: Integrate the morphology curve of the rough surface to obtain the actual wetting area S of the interface. W and projected wetting area S D And calculate the roughness ratio r RMS .

[0050] S04: Capillary force F under the condition of establishing the ball-plate model Cap The general model is derived by ignoring some force components based on microscale scenes, resulting in the following simplified model:

[0051]

[0052] In the above formula, γ L R represents the surface tension coefficient of the liquid between the probe and the interface being measured; R represents the diameter of the sphere at the tip of the probe; α represents the angle between the central axis of the sphere at the tip of the probe and the meniscus bridge boundary between the interface being measured; and r represents the Kelvin equilibrium radius.

[0053] S05: Considering sin(α) << 1, for sin 2 (α) The following approximate calculation is used:

[0054]

[0055] Where c represents the average cosine value of the surface contact angle between the measured interface and the meniscus bridge.

[0056] A parameterized model of the capillary force function related to the apparent contact angle of the rough interface is obtained:

[0057]

[0058] In the above formula, θ1 represents the surface contact angle formed by the sphere at the probe tip and the meniscus bridge; θ2 represents the surface contact angle formed by the measured interface and the meniscus bridge; V m This represents the molar volume of the liquid at the interface.

[0059] S06: Finally, the roughness surface wetting model from step S1 is introduced into the parameterized model from step S05 to obtain the required parameterized model that takes into account the surface roughness of the measured interface.

[0060] As a further improvement of the present invention, in step S01, the contact angle θ is balanced. e The expression satisfies the following:

[0061]

[0062] In the above formula, γ sg γ represents the tension at the solid-gas interface. sl γ represents the tension at the solid-liquid interface. lg The apparent contact angle θ represents the surface tension at the liquid-gas interface. a The surface tension balance equation can be expressed as:

[0063] ωr RMS γ sg =γ lg cosθ a +ωr RMS γ sl +(1-ω)γ lg .

[0064] As a further improvement of the present invention, in step S02, in the rough surface with a cross-section type of sine curve, the equation for the cross-section height parameter is as follows:

[0065]

[0066] In the above formula, x and y are the horizontal and vertical coordinates of any position on the horizontal cross section, respectively; λ is the wavelength of the sine curve; and RMS is the surface roughness, used to characterize the amplitude parameter of the sine curve.

[0067] As a further improvement of the present invention, in step S03, the actual wetting area S W The calculation formula is as follows:

[0068]

[0069] In the above formula, a is the height coordinate of the liquid wetting trough, and b is the height coordinate of the liquid wetting peak.

[0070] Projected wetting area S D The calculation formula is as follows:

[0071]

[0072] In the above formula, c is the height coordinate of the trough of the liquid infiltration wave in the vertical projection area of ​​the meniscus bridge, and d is the height coordinate of the peak of the liquid infiltration wave in the vertical projection area of ​​the meniscus bridge.

[0073] The formula for calculating the roughness ratio is:

[0074]

[0075] The technical solution provided by this invention has the following beneficial effects:

[0076] 1. This invention constructs a parameterized model that can accurately characterize the mapping relationship between capillary force and relative humidity. In the improved parameterized model of this invention, the Kelvin radius changes accordingly with parameters such as the distance between the sphere and the plate and the relative humidity, thus better meeting the application requirements for accurate calculation of capillary force under different measurement environment conditions.

[0077] The optimized parameterized model proposed in this invention avoids the bias introduced by the assumption of a constant Kelvin radius and focuses on considering two variables: relative humidity and the distance between the probe and the probe plate. Based on the relative humidity parameterized model, the capillary force between the micrometer-scale probe head and the measured surface at different relative humidities can be accurately and effectively calculated, which is of great significance for the study of the relationship between capillary force and relative humidity parameters.

[0078] 2. This invention also constructs a parametric model that can accurately characterize the mapping relationship between capillary force and the surface roughness of the measured interface. This invention uses a sinusoidal surface roughness model to characterize the morphological changes of the actual processed surface, making the surface features more consistent with the peaks and valleys on a microscopic surface. Simultaneously, it uses the partial wetting state of surface grooves under rough surface conditions to characterize the change in contact angle. Using the contact angle parameter as a medium, an accurate functional model of surface roughness parameters and capillary force applicable to micrometer-scale probes can be constructed. This can more accurately characterize the surface morphology of the measured interface in actual applications and improve the measurement accuracy of capillary force under different surface roughness conditions, which is of great significance for the study of the relationship between capillary force and surface roughness parameters. Attached Figure Description

[0079] Figure 1 This diagram illustrates the principle of capillary forces between a ball and a plate (ball: refers to the spherical probe, plate: refers to the interface being measured) at a micrometer scale under different humidity conditions.

[0080] Figure 2 This is a schematic diagram of the cross-section of the meniscus bridge between the spheres with a micron-scale diameter in Embodiment 2 of the present invention.

[0081] Figure 3 This is a schematic diagram showing the variation of the Kelvin equilibrium radius under different ball-plate spacing conditions in Embodiment 2 of the present invention.

[0082] Figure 4 This is a schematic diagram of droplet wetting on a rough surface with a sinusoidal shape, used to describe the interface morphology in Scheme 2 of Embodiment 2 of the present invention.

[0083] Figure 5 This is a three-dimensional schematic diagram of the conical rough surface used in the traditional scheme.

[0084] Figure 6 This is a three-dimensional schematic diagram of the square wave rough surface used in the traditional scheme.

[0085] Figure 7 In the test experiment of Scheme 1, the parameterized models of the present invention and the control group showed the changes in capillary force and relative humidity under the conditions of constant contact angle, probe ball diameter and ball-plate spacing.

[0086] Figure 8 In the test experiment of Scheme 1, when the probe ball diameter is 1nm and the distance between the ball and the plate is 0.5nm, the measured curves of capillary force and relative humidity are shown.

[0087] Figure 9 The curves showing the change of capillary force on the micrometer-scale probe sphere diameter with the distance between the spheres and plates under different relative humidity conditions in the test experiment of Scheme 1 are shown.

[0088] Figure 10 The curves showing the change of capillary force on the diameter of the nanoscale probe ball with the distance between the balls and plates under different relative humidity conditions in the test experiment of Scheme 1.

[0089] Figure 11 In the test experiment of Scheme 1, under the same ball-plate spacing conditions, the actual measurement data and its fitted curve and theoretical curve are compared.

[0090] Figure 12 In the test experiment of Scheme 2, under the conditions of fixed probe ball diameter and the same test surface, the capillary force variation curve with the ball-plate spacing was plotted based on the surface roughness of different test interfaces.

[0091] Figure 13 The figure shows a comparison of the capillary force and surface roughness curves of three different parameterized models under a ball-plate spacing of 10 nm in the test experiment of Scheme 2.

[0092] Figure 14 The curve showing the relationship between capillary force and surface roughness under the condition of a probe ball diameter of 5 nm in the test experiment of Scheme 2. Detailed Implementation

[0093] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0094] Example 1

[0095] This embodiment provides a parameterized model of capillary action under different micrometer-scale probe spherical diameters. This parameterized model is mainly used as a data processing module in micro / nano measurement systems to measure the capillary force F between interfaces under micrometer-scale probe detection conditions. Cap Specifically, based on existing technologies, this embodiment provides a relative humidity-sensitive parameterized model and a surface roughness-sensitive parameterized model, respectively.

[0096] Option 1:

[0097] In Scheme 1 of this embodiment, based on existing parametric models, the actual characteristic that the Kelvin equilibrium radius changes accordingly with parameters such as the distance between the spheres and relative humidity is considered. The influence of relative humidity on capillary force under the micrometer-scale probe sphere diameter is studied, and relative humidity is introduced as a variable parameter into the parametric model, thus obtaining a parametric model of capillary force considering relative humidity. The parametric model provided by this scheme offers an important basis for in-depth research and application of microscopic forces, and is of great significance to modern precision measurement, processing, and assembly. Especially for scenarios where relative humidity varies at the micrometer scale, the parametric model of this embodiment has higher accuracy and practical value.

[0098] Specifically, the expression for the parameterized model that uses the relative humidity of the measurement environment as a key parameter is as follows:

[0099]

[0100] In the above formula, γ L R represents the surface tension coefficient of the liquid between the probe and the measured interface; C represents the diameter of the sphere at the probe tip; α represents the mean cosine of the surface contact angle between the measured interface and the meniscus bridge; and α represents the angle between the central axis of the sphere at the probe tip and the boundary of the meniscus bridge between the measured interface and the measured interface. g RH represents the gas constant; T represents the ambient temperature on the Kelvin scale; RH represents the relative humidity of the measurement environment; D represents the distance between the probe and the interface being measured.

[0101] Option 1:

[0102] Considering that the surface morphology of a rough interface has a significant impact on wetting state at the micrometer scale, the parameterized model for measuring capillary forces should incorporate the analysis of morphological characteristics. Existing conventional methods use square wave and triangular wavefront structures to construct rough surface morphologies, which differs considerably from actual surface morphologies. This is one of the reasons for the large discrepancy between theoretical models and actual measurement results. Therefore, Scheme 2 in this embodiment defines the surface roughness model cross-section of the interface under test as a sinusoidal curve. Using a sinusoidal curve can characterize the periodic changes in the morphology of the actual processed surface, and the fluctuations of the curve more closely resemble the peaks and valleys on a microscopic surface. Then, based on the roughness model cross-section, the wetting characteristics of different regions are analyzed, thereby establishing a parameterized model that can more accurately measure capillary action.

[0103] In the second embodiment of this invention, another parameterized model for capillary action under a micrometer-scale probe sphere diameter is provided, in which the surface roughness of the measured interface is used as a key parameter, and the corresponding expression of the parameterized model is as follows:

[0104]

[0105] In the above formula, γ L θ represents the surface tension coefficient of the liquid between the probe and the measured interface; R represents the diameter of the sphere at the probe tip; θ1 represents the surface contact angle formed by the sphere at the probe tip and the meniscus bridge; ω represents the wetting coefficient of the wetting region between the interfaces; S W S represents the wetted area between interfaces; D θ represents the projected wetted area between interfaces. e The equilibrium contact angle θ of a droplet on an ideal smooth surface. e V m The value represents the molar volume of the liquid in the meniscus bridge between the interfaces; D represents the distance between the probe and the interface being measured.

[0106] Example 2

[0107] This embodiment provides two methods for designing parameterized models of capillary action under micron-scale probe sphere diameters. One method uses the relative humidity of the measurement environment as a variable key parameter to design a parameterized model of capillary action under micron-scale probe sphere diameters, as shown in Scheme 1 of Embodiment 1. The other method uses the surface roughness of the measured interface as a variable key parameter to design a parameterized model of capillary action under micron-scale probe sphere diameters, as shown in Scheme 2 of Embodiment 1.

[0108] In this embodiment, the design method of the parameterized model of Scheme 1 includes the following steps:

[0109] S1: Figure 1 This diagram illustrates the principle of capillary forces between a spherical probe and a plate (the probe itself being measured) at a micrometer-scale diameter under different humidity conditions. In this state, the capillary force is caused by both surface tension and Laplace pressure. Surface tension is determined by the properties of the liquid itself, while the Laplace pressure is the macroscopic pressure difference at the curved gas-liquid interface when the meniscus bridge is in equilibrium. In this embodiment, the Laplace pressure difference ΔP of the meniscus cross-section formed by the continuous liquid medium is calculated using the Young-Laplace equation. L Its expression is as follows:

[0110]

[0111] In the above formula, γ L Let γ be the surface tension coefficient of the liquid. Under different ambient humidity conditions, assuming the liquid at the interface is water, then the surface tension coefficient γ of water is... L =72.8mJ / m 2 r1 and r2 are any two orthogonal radii of curvature at a point on the surface where pressure is applied, where r1 is perpendicular to the direction of the plate and r2 is parallel to the direction of the plate; r is the Kelvin equilibrium radius; X is the azimuth radius.

[0112] The azimuth radius X can be calculated using geometric relationships:

[0113] X=R sin(α)-r[1-sin(θ1+α)];

[0114] In the above formula, θ1 represents the surface contact angle formed by the sphere at the probe tip and the meniscus bridge; R represents the diameter of the sphere at the probe tip; and α represents the angle between the central axis of the sphere at the probe tip and the boundary of the meniscus bridge between the measured interface.

[0115] S2: Based on thermodynamic principles and using the Kelvin equation, the vapor pressure change caused by the tortuous liquid-gas interface inside the capillary can be calculated as follows:

[0116]

[0117] In the above formula, V m R represents the molar volume of the liquid at the interface; g denoted by ; T represents the ambient temperature on the Kelvin scale; P is the actual external vapor pressure, and P0 is the saturated vapor pressure inside the meniscus bridge. The ratio of the two (P / P0) can also be expressed by relative humidity RH.

[0118] S3: The parameterized model proposed in this embodiment reflects the relationship between the relative humidity parameter and capillary force under the micrometer-scale probe spherical diameter. Therefore, the relative humidity parameter RH is used to describe its influence on the Kelvin radius r. Under thermodynamic equilibrium conditions, the relationship between the Kelvin radius r and the relative humidity RH can be described as follows:

[0119]

[0120] Figure 2 This is a cross-sectional view of the meniscus bridge between the spherical plates at the micrometer scale in this embodiment; in this embodiment, the side curvature of the meniscus bridge is approximated as a circle, so the volume V of the meniscus bridge is... m The following can be determined based on the geometric relationships shown in the diagram:

[0121] V m =πR(4r 2 c 2 -D 2 )

[0122] In the above formula, D represents the distance between the probe and the interface being measured; c represents the average cosine value of the surface contact angle between the interface being measured and the meniscus bridge, and satisfies:

[0123] c = 1 / 2(cos(θ1+α)+cosθ2)

[0124] In the above formula, θ2 represents the surface contact angle formed by the measured interface and the meniscus bridge.

[0125] S4: Capillary force F under the condition of establishing the ball-plate model Cap A general model was derived, and a simplified model was obtained by ignoring some force components based on micro-scale scenes. Specifically, the process of establishing the simplified model is as follows:

[0126] S41: Capillary force F under the condition of constructing a spherical plate model Cap The general model is:

[0127] F Cap =2πXγ L -πX 2 ΔP L ;

[0128] In the above formula, the first term (2πXγ) L The first term represents the capillary force component caused by surface tension; the second term represents the area of ​​the cross-section, πX. 2 The capillary force component caused by the Laplace pressure of the meniscus bridge.

[0129] Since X > r, (1 / X - 1 / r) is negative, meaning the Laplace pressure difference ΔP L The value is negative. Therefore, the capillary force in the ball-plate model constructed in this embodiment exhibits attractive characteristics, and an adhesion effect occurs between the probe ball head and the plate.

[0130] S42: Considering that in the scenario of this embodiment, the surface tension component does not act perpendicularly to the measured surface, but contributes to the capillary force at the angle (θ1+α), the general model can be rewritten as follows by substituting the specific capillary force parameters into the upward equation:

[0131]

[0132] S43: The solution provided in this embodiment is mainly used to realize capillary force at the micrometer scale. Therefore, the distance D between the probe and the interface being measured, the spherical diameter R of the probe tip, the azimuth radius X, and the Kelvin equilibrium radius r satisfy the following relationship: R >> X >> r, D. Furthermore, the three angle values ​​α, θ1, and θ2 are also extremely small. Therefore, the value of the 2sin(θ1+α) term in the expression of step S42 is also extremely small. At this time, the capillary force component caused by surface tension can be directly ignored, thus simplifying the expression of step S42 to:

[0133]

[0134] S44: Since X >> r in this embodiment, then (1 / r - 1 / X) ≈ 1 / r; therefore, in the expression of step S43... The item can be further simplified to The simplified model is obtained as follows:

[0135]

[0136] S5: In step S3 of this embodiment, the mapping relationship between the Kelvin equilibrium radius r and the relative humidity RH has been established, that is:

[0137]

[0138] In the simplified model of step S4, the capillary force F Cap It is another function that is clearly negatively correlated with the Kelvin equilibrium radius r.

[0139] Therefore, substituting the mapping relationship from step S3 into the simplified model of step S4 yields the required parameterized model that considers relative humidity. The expression for this parameterized model is:

[0140]

[0141] In traditional approaches, technicians use a simplified model, similar to the one in this case, as a parameterized model of capillary force. In such models, the Kelvin radius is set as a constant or calculated based on geometric relationships. However, in real-world situations, such as... Figure 3 As shown, the Kelvin radius changes accordingly with parameters such as the distance between the probe and the measured interface in this spherical plate model and the relative humidity. Therefore, existing simplified models cannot effectively describe this law of capillary force, thus causing traditional parametric models to fail at the micrometer scale.

[0142] In the parameterized model provided in this example, a variable relative humidity parameter is introduced. The optimized parameterized model of capillary force includes two variables: relative humidity (RH) and the distance between the ball and plate (D). Other parameters can be considered constants, which effectively avoids the bias introduced by the assumption of a constant Kelvin radius. Furthermore, the optimized parameterized model in this embodiment can more directly characterize the changes in capillary force caused by the relative humidity parameter. Therefore, it has better adaptability and higher accuracy for measuring dynamic changes in relative humidity in the environment.

[0143] In this embodiment, the design method of the parameterized model of Scheme 2 includes the following steps:

[0144] S01: Defines the equilibrium contact angle θ of an ideal smooth surface. e Apparent contact angle θ under rough conditions a When a droplet reaches a stable state on an ideal smooth surface, the surface tensions at the solid-liquid-gas interface are in equilibrium, and the equilibrium contact angle θ eThe cosine value can be expressed as:

[0145]

[0146] In the above formula, γ sg γ represents the tension at the solid-gas interface. sl γ represents the tension at the solid-liquid interface. lg It represents the tension at the liquid-gas interface.

[0147] However, in reality, due to material properties, surface tension, and gravity, the liquid film will penetrate into the surface grooves of the interface being measured. Simultaneously, due to the presence of air in the grooves, they will not be completely filled by the liquid. Therefore, this embodiment employs... Figure 4 The transition state shown characterizes the wetting of a rough surface under normal conditions. Due to the roughness, the bottom of the droplet is not in complete contact with the solid, and the voids are filled with air. That is, the protrusions in the rough interface are completely wetted, while the depressions remain dry and do not wet. At this time, there are wetted and dry regions within the projected area of ​​the droplet, and the wetting coefficients of the wetted and dry regions are represented by ω and (1-ω), respectively. Assuming that the droplet radius is much larger than the scale of the rough surface structure, then the apparent contact angle θ... a The surface tension balance equation can be expressed as:

[0148] ωr RMS γ sg =γ lg cosθ a +ωr RMS γ sl +(1-ω)γ lg

[0149] In the above formula, r RMS Roughness ratio, representing the ratio of actual wetted area to projected wetted area.

[0150] Combining the above two equations, a unified formula for the surface roughness wetting model can be further constructed:

[0151] cosθ a =ωr RMS cosθ e +ω-1.

[0152] S02: To accurately describe the effect of surface roughness on the contact angle, a realistic surface morphology model must be constructed. Traditional methods typically use square wave and triangular wavefront structures to construct rough surface morphologies, resulting in rough surfaces with the following morphologies respectively: Figure 5 and Figure 6As shown. However, in practical applications, taking MEMS devices as an example, the roughness of the measured interface is usually caused by the non-uniformity of the deposition or etching process, and the morphology of this roughness is not... Figure 5 or Figure 6 It is not a state in the middle, but a state similar to water ripples containing regional peaks and regional valleys.

[0153] Therefore, in this embodiment, the function describing the morphological characteristics of the measured interface is rewritten. Specifically, in this embodiment, the cross-sectional type of the rough measured interface is defined as a sinusoidal curve surface. Figure 4 This corresponds to the wetting state diagram. Therefore, the equation for the cross-sectional height parameter z, which characterizes the surface roughness, can be obtained as follows:

[0154]

[0155] In the above formula, x and y are the horizontal and vertical coordinates of any position on the horizontal cross section, respectively; λ is the wavelength of the sine curve; and RMS is the surface roughness, used to characterize the amplitude parameter of the sine curve.

[0156] S03: In Figure 4 Under these conditions, the actual wetting area within a single cycle can be obtained by integrating the surface morphology curve model. Integrating the morphology curve of a rough surface yields the actual wetting area S at the interface. W and projected wetting area S D Among them, the actual wetting area S W The calculation formula is as follows:

[0157]

[0158] In the above formula, a is the height coordinate of the liquid wetting trough, and b is the height coordinate of the liquid wetting peak.

[0159]

[0160] In the above formula, c is the height coordinate of the liquid trough in the vertical projection area of ​​the meniscus bridge, and d is the height coordinate of the peak of the liquid trough in the vertical projection area of ​​the meniscus bridge.

[0161] Next, the roughness ratio r is calculated. RMS :

[0162]

[0163] S04: Using the same design approach as in Scheme 1, establish the capillary force F under the condition of the spherical plate model. Cap The general model is derived by ignoring some force components based on microscale scenes, resulting in the following simplified model:

[0164]

[0165] In the above formula, γ L R represents the surface tension coefficient of the liquid between the probe and the interface being measured; R represents the diameter of the sphere at the tip of the probe; α represents the angle between the central axis of the sphere at the tip of the probe and the meniscus bridge boundary between the interface being measured; and r represents the Kelvin equilibrium radius.

[0166] S05: This embodiment further simplifies the mathematical model of the morphological features of the created rough interface. Specifically, sin(α) in the simplified model can be calculated from the geometric relationship of the meniscus bridge; considering that sin(α) << 1, for sin 2 (α) The following approximate calculation is used:

[0167]

[0168] Where c represents the average cosine value of the surface contact angle between the measured interface and the meniscus bridge.

[0169] By using approximations to replace the relevant parameters in the simplified model, a parameterized model of the capillary force function related to the apparent contact angle of the rough interface can be obtained.

[0170]

[0171] In the above formula, θ1 represents the surface contact angle formed by the sphere at the probe tip and the meniscus bridge; θ2 represents the surface contact angle formed by the measured interface and the meniscus bridge; V m This represents the molar volume of the liquid at the interface.

[0172] In this parameterized model, capillary forces at the micrometer scale are directly related to the surface roughness of different measured interfaces.

[0173] S06: Finally, let the roughness surface wetting model cosθ from step S1 be... a =ωr RMS cosθ e The apparent contact angle θ in +ω-1 a =θ2, and substitute it into the parameterized model in step S05 to obtain the required parameterized model that considers the surface roughness of the measured interface, that is:

[0174]

[0175] The second scheme designed in this embodiment uses a sinusoidal curve to describe the surface morphology of the interface under test at a micron-scale probe sphere diameter, thereby obtaining a characterizing capillary force F. CapA parameterized model of the mapping relationship between surface roughness (RMS) and the measured surface roughness (RSM) is used. Simultaneously, this parameterized model retains the influence weight of the distance between the probe and the measured interface on the capillary force, as in the spherical plate model. Therefore, the solution in this embodiment is a method biased towards application in measurement scenarios where the surface roughness of the measured interface changes dynamically. In this specific scenario, the parameterized model in the designed solution has higher accuracy and better adaptability.

[0176] Example 3

[0177] This embodiment provides a micro / nano measurement system for measuring the capillary force between a probe ball and the measured interface at the micrometer scale. In different scenarios, the Wiener measurement system in this embodiment selects the parameterized model of capillary action at different micrometer-scale probe ball diameters from Scheme 1 or Scheme 2 in Embodiment 1.

[0178] Specifically, Option 1 is a relative humidity-sensitive parameterized model, suitable for measurement scenarios where the relative humidity of the measurement environment fluctuates. Option 2 is a surface roughness-sensitive parameterized model, suitable for measurement scenarios where the surface roughness of the measured interface is uneven.

[0179] Performance testing

[0180] To verify the accuracy of the parameterized models designed in Examples 1 and 2, the present inventors also used a simplified model as a control group to simulate and test the parameterized models of capillary action under the two micron-scale probe sphere diameters in Example 1 of this invention.

[0181] 1. Verification Experiment of Scheme 1

[0182] 1.1 Under the condition that the contact angle, probe ball diameter, and ball-plate spacing are constant, this embodiment first draws the following based on their respective parametric models: Figure 7 The theoretical curves showing the changes in capillary force and relative humidity at the micrometer scale are shown. Figure 7 In the diagram, the dashed lines represent the simulation curves obtained from the control group scheme, while the solid lines represent the simulation curves drawn from the parametric model of Scheme 1 in this embodiment.

[0183] Analysis of the two curves reveals that while both schemes exhibit a trend of increasing capillary force with increasing relative humidity, the optimized scheme in this embodiment shows a significant change in the capillary force value under medium-to-high humidity conditions. This indicates that the capillary force is more sensitive to changes in relative humidity under these conditions, and its slope continuously increases. In reality, capillary force is more sensitive to changes in relative humidity at higher levels, which aligns with the curve in this case. The slope change of this curve reflects its closer fit to the actual micrometer-scale results. In contrast, the theoretical model curve of the traditional scheme shows the opposite trend, highlighting the limitations of the traditional scheme, which is only applicable to the measurement of capillary force at the nanometer scale.

[0184] 1.2. In this experiment, the relationship between capillary force and relative humidity was measured and plotted when the probe sphere diameter was 1 nm and the distance between the sphere and the plate was 0.5 nm. The results are as follows: Figure 8 As shown. Comparison Figure 8 and Figure 7 It can be seen that the original model is suitable for calculating capillary forces at the nanoscale, while the interaction between capillary forces and relative humidity at the micrometer scale differs from that at the nanoscale. This further confirms the necessity of the parameterized model created in Example 1 of this invention for describing the relationship between capillary forces and relative humidity at the micrometer-scale probe sphere diameter.

[0185] 1.3 Under different relative humidity conditions, the force-distance curve of the capillary force on the probe sphere diameter at the micrometer scale in this embodiment as a function of the sphere-plate spacing is approximately as follows: Figure 9 As shown. The theoretical curves for traditional methods at the nanoscale are as follows. Figure 9 As shown. Analysis Figure 9 and Figure 10 It can be observed that the capillary force in both decreases as the distance between the ball and the plate decreases, showing the same trend. However, further comparison of the model curves in the two scenarios reveals that... Figure 9 The capillary force model curve at the micrometer scale shows a rapid decrease in capillary force in the initial section, when the distance between the spheres and plates is small, followed by a gradual decrease in the latter part. Figure 10 At the nanoscale, the value of capillary force tends to decrease linearly, indicating that the capillary force behavior is not the same at the two scales.

[0186] based on Figure 9 and Figure 10The curves lead to the conclusion that, in the optimized parameterized model provided in this embodiment, when other conditions remain constant, the capillary force increases with increasing relative humidity within the range of 20-60% RH. This is consistent with the findings of most researchers. It is particularly important to emphasize that the slope of the capillary force change curve in the optimized model of this embodiment increases simultaneously with increasing relative humidity, which better reflects actual phenomena. Therefore, the optimized parameterized model of this embodiment can more accurately describe the action law of capillary force at the micrometer scale.

[0187] 1.4 To verify the accuracy of the parameterized model proposed in this embodiment, this experiment also measured the capillary force between the micron-scale probe head and the measured surface using a micro-nano measurement system. The probe head material was 304 stainless steel, the substrate material was SiO2, and the relative humidity was set to 20%RH, 40%RH, and 60%RH, respectively.

[0188] Figure 11 The data presented are experimental measurements of capillary force under 20%RH, 40%RH, and 60%RH conditions when the distance between the spheres and plates is 30nm. The triangles represent the experimental measurements obtained under three different relative humidity conditions, the solid lines are the fitting curves obtained by fitting the experimental measurements, and the dashed lines are the theoretical curves of the model proposed in this invention.

[0189] analyze Figure 11 The data shows that the simulation curve of the parameterized model proposed in this embodiment has a high degree of overlap with the fitting curve of the measured data, which proves the accuracy of the theoretical optimization model, with only a very small deviation between the two.

[0190] 1. Verification Experiment of Scheme 2

[0191] 2.1 Under the conditions of fixed probe diameter and identical measured surfaces, this experiment adjusted the surface roughness of different measured interfaces and used the parametric model of Scheme 2 to draw the following... Figure 12 The curves showing the capillary force versus the ball-plate spacing are presented. Analysis of the data in the graphs reveals that, in the parameterized model of the scheme, observing each curve individually shows that the capillary force decreases as the ball-plate spacing increases. Comparing different curves shows that the rate of decrease in capillary force changes with increasing surface roughness. Because the ball-plate spacing and surface roughness have different effects, curves with different slopes intersect when the ball-plate spacing is around 210 nm.

[0192] 2.2 This experiment also plotted a comparison of the capillary force versus different surface roughness curves for different parameterized models under a ball-plate spacing of 10 nm. The long dashed line represents the Wenzel model curve, the short dashed line represents the Cassie-Boxter model curve, and the solid line represents the optimized model curve proposed in this invention; the results are as follows... Figure 13 As shown. Observation Figure 13 The data shows that, since the Cassie-Boxter model assumes a fully wetted state, surface roughness has no effect on capillary force, hence the Cassie-Boxter model curve is horizontal. The Wenzel model, however, considers a completely non-wetted state, so under this assumption, capillary force decreases linearly with relative humidity. These two models represent two extreme cases of wetting, which are not applicable in real-world scenarios. The overall curves of the Wenzel parameterized model and the improved model show the same trend, indicating that the capillary force decreases with increasing surface roughness. However, only the force curve of the optimized model in this invention does not have a fixed slope. This is more consistent with actual application scenarios.

[0193] 2.3. This experiment also measured and plotted the following: Figure 14 The curves showing the relationship between capillary force and surface roughness under the condition of a probe sphere diameter of 5 nm (nanoscale) are presented. Analysis Figure 14 and Figure 13 It can be seen that at the nanoscale, capillary forces decrease significantly with increasing roughness, which is consistent with... Figure 13 The interaction patterns at the micrometer scale differ significantly, thus further verifying the correctness and accuracy of the parameterized model of capillary force and surface roughness at the micrometer scale proposed in this invention.

[0194] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for designing a parameterized model of capillary action under a probe sphere diameter at the micrometer scale, characterized in that, It is used to design a parameterized model of capillary action under the micron-scale probe sphere diameter by taking the relative humidity of the measurement environment as a variable key parameter. A parameterized model of capillary action under micrometer-scale probe spherical diameter is used as a data processing module in micro / nano measurement systems to measure interfacial capillary forces under micrometer-scale probe detection conditions. F cap The parameterized model uses the relative humidity of the measurement environment as a key parameter, and the expression of the parameterized model is as follows: ； In the above formula, R represents the surface tension coefficient of the liquid between the probe and the interface being measured; C represents the ball diameter at the tip of the probe; and C represents the average cosine value of the surface contact angle between the interface being measured and the meniscus bridge. This indicates the angle between the central axis of the sphere at the probe tip and the boundary of the meniscus bridge between the interface being measured; R g The value represents the gas constant; T represents the ambient temperature on the Kelvin scale; RH represents the relative humidity of the measurement environment; and D represents the distance between the probe and the interface being measured. The design method includes the following steps: S1: Calculate the Laplace pressure difference across the meniscus section formed by the continuous liquid medium using the Young-Laplace equation. ; S2: The change in vapor pressure at the detection interface caused by the tortuous liquid-gas interface within the capillary is characterized by relative humidity (RH). S3: Under thermodynamic equilibrium conditions, the mapping relationship between the Kelvin equilibrium radius r and the relative humidity RH is constructed, as shown in the following expression: ； In the above formula, This represents the surface tension coefficient of the liquid between the probe and the interface being measured. V m This represents the molar volume of the liquid at the interface; R g denoted by ; T represents the ambient temperature on the Kelvin scale; RH represents the relative humidity of the measurement environment; D represents the distance between the probe and the interface being measured; c represents the average cosine value of the surface contact angle between the interface being measured and the meniscus bridge. r Indicates the Kelvin equilibrium radius; S4: Capillary force under the condition of establishing a ball-plate model F cap The general model is derived by ignoring some force components based on microscale scenes, resulting in the following simplified model: ； In the above formula, This indicates the angle between the central axis of the sphere at the probe tip and the boundary of the meniscus bridge between the interface being measured; S5: Calculate the Kelvin equilibrium radius established in step S3. r The mapping relationship between relative humidity (RH) is introduced into the simplified model in step S4 to obtain the required parameterized model that takes relative humidity into account.

2. The method for designing a parameterized model of capillary action under a micrometer-scale probe sphere diameter as described in claim 1, characterized in that: In step S1, the Laplace pressure difference of the meniscus cross section formed by the continuous liquid medium The calculation formula is as follows: ； In the above formula, r1 and r2 are any two orthogonal radii of curvature at a point on the surface where the pressure is applied, where r1 is perpendicular to the direction of the plate and r2 is parallel to the direction of the plate. X Let be the azimuth radius, and satisfy: ； In the above formula, This indicates the surface contact angle formed by the sphere at the probe tip and the meniscus bridge.

3. The method for designing a parameterized model of capillary action under a micrometer-scale probe spherical diameter as described in claim 2, characterized in that: In step S2, the vapor pressure change caused by the tortuous liquid-gas interface inside the capillary can be calculated based on thermodynamic principles using the Kelvin equation, as follows: ； In the above formula, P This is the actual external vapor pressure. P Let 0 be the saturated vapor pressure inside the meniscus bridge, and the ratio of the two be expressed in terms of relative humidity (RH). Then: 。 4. The method for designing a parameterized model of capillary action under a micrometer-scale probe spherical diameter as described in claim 3, characterized in that, In step S4, the process of establishing the simplified model is as follows: S41: Capillary Force under the Condition of Constructing a Spherical Plate Model F cap The general model is: ； S42: Substituting the capillary force parameters into the upward equation yields: ； S43: Considering ,and The angle is extremely small, and the expression in step S42 is... Ignoring the term, we get: ； S44: Due to ,but Then Item simplified to The simplified model is obtained as follows: 。 5. A method for designing a parameterized model of capillary action under a probe sphere diameter at the micrometer scale, characterized in that, It is used to design a parameterized model of capillary action under the micrometer-scale probe sphere diameter, with the surface roughness of the interface to be measured as a variable key parameter. A parameterized model of capillary action under micrometer-scale probe spherical diameter is used as a data processing module in micro / nano measurement systems to measure interfacial capillary forces under micrometer-scale probe detection conditions. F cap The parameterized model uses the surface roughness of the measured interface as a key parameter, and the expression of the parameterized model is as follows: ； In the above formula, This represents the surface tension coefficient of the liquid between the probe and the interface being measured. R represents the diameter of the ball at the probe tip; This indicates the surface contact angle formed by the sphere at the probe tip and the meniscus bridge; This represents the wetting coefficient of the wetting area between interfaces; S W Indicates the wetted area between interfaces; S D Indicates the projected wetted area between interfaces; This represents the equilibrium contact angle of a droplet on an ideal smooth surface; V m The value represents the molar volume of the liquid at the interface; D represents the distance between the probe and the interface being measured. The design method includes the following steps: S01: Defines the equilibrium contact angle of an ideal smooth surface Apparent contact angle under rough conditions A unified formula for constructing a roughness surface wetting model: ； In the above formula, and These represent the wetting coefficients of the wetted and dry regions within the projected area when a droplet comes into contact with a rough surface, respectively. Roughness ratio, representing the ratio of actual wetted area to projected wetted area; S02: Define the cross-sectional type of the rough measured interface as a sinusoidal curve surface to obtain the cross-sectional height vector characterizing the surface roughness. z The equation; S03: Perform integral calculations on the morphology curves of the rough surface to obtain the actual wetting area of ​​the interface. S W and projected wetting area S D And calculate the roughness ratio ; S04: Capillary Force under the Condition of Establishing a Ball-Plate Model F cap The general model is derived by ignoring some force components based on microscale scenes, resulting in the following simplified model: ； In the above formula, This represents the surface tension coefficient of the liquid between the probe and the interface being measured. R represents the diameter of the ball at the probe tip; This indicates the angle between the central axis of the sphere at the probe tip and the boundary of the meniscus bridge between the interface being measured; r Indicates the Kelvin equilibrium radius; S05: Considering ,right The following approximate calculation is used: ； Where c represents the average cosine value of the surface contact angle between the measured interface and the meniscus bridge; A parameterized model of the capillary force function related to the apparent contact angle of the rough interface is obtained: ； In the above formula, This indicates the surface contact angle formed by the sphere at the probe tip and the meniscus bridge; Indicates the surface contact angle formed between the measured interface and the meniscus bridge; V m This represents the molar volume of the liquid at the interface; S06: Introduce the roughness surface wetting model from step S1 into the parameterized model from step S05 to obtain the required parameterized model that considers the surface roughness of the measured interface.

6. The method for designing a parameterized model of capillary action under a micrometer-scale probe spherical diameter as described in claim 5, characterized in that, In step S01, the balanced contact angle The expression satisfies the following: ； In the above formula, This represents the tension at the solid-gas interface. It represents the tension at the solid-liquid interface. It represents the tension at the liquid-gas interface; The apparent contact angle The surface tension balance equation can be expressed as: 。 7. The method for designing a parameterized model of capillary action under a micrometer-scale probe spherical diameter as described in claim 5, characterized in that, In step S02, in a rough surface with a cross-section type of sine curve, the equation for the cross-section height parameter is as follows: ； In the above formula, x, y These are the x and y coordinates of any position on the horizontal cross section, respectively. The wavelength of the sine curve is . RMS Surface roughness is a parameter used to characterize the amplitude of a sine curve.

8. The method for designing a parameterized model of capillary action under a micrometer-scale probe spherical diameter as described in claim 5, characterized in that, In step S03, the actual wetting area S W The calculation formula is as follows: ； In the above formula, The height coordinates of the liquid wetting trough. The height coordinates of the crest of the liquid wetting wave; The projected wetting area S D The calculation formula is as follows: ； In the above formula, c is the height coordinate of the trough of the liquid infiltration wave in the vertical projection area of ​​the meniscus bridge, and d is the height coordinate of the peak of the liquid infiltration wave in the vertical projection area of ​​the meniscus bridge. The formula for calculating the roughness ratio is: 。

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