CN116705198B - Liquid bridge full-range calculation method for water-soluble KDP crystal element surface microdefect DPN repair process - Google Patents
Liquid bridge full-range calculation method for water-soluble KDP crystal element surface microdefect DPN repair process Download PDFInfo
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Abstract
The invention provides a liquid bridge full-range calculation method for a water-soluble KDP crystal element surface microdefect DPN repair process, and belongs to the technical field of micro-nano manufacturing. In order to solve the extreme cases that the existing method is not suitable for the situation that a liquid bridge can cover and extend to a conical main body area under high environmental humidity, and the slope at the contact point of a liquid bridge appearance curve and the surface of an element is close to infinity, the phenomenon of double solution exists in an error curve for calculating the nanometer-scale liquid bridge appearance, and an error result is easy to obtain. According to the invention, an AFM needle point model is constructed as a needle point ball head and a conical body, and a parameterized ordinary differential equation of a liquid bridge morphology curve and a geometric equation of a probe needle point compound contour are constructed according to the geometric relationship of the probe needle point, a KDP crystal element and the liquid bridge morphology curve; the method comprises the steps of rough root searching and fine root searching, and a dichotomy method is combined to solve the appearance curve of the liquid bridge. The method is more suitable for calculating the liquid bridge morphology of the DPN repair process of the water-soluble KDP crystal element under the condition of high ambient humidity.
Description
Technical Field
The invention relates to the technical field of micro-nano manufacturing, in particular to a liquid bridge full-range calculation method for a water-soluble KDP crystal element surface micro-defect DPN repair process.
Background
Laser fusion devices are key to solving the energy problem in the future. Among them, KDP (potassium dihydrogen phosphate) crystal is used to make photoelectric switch and frequency doubling device in laser fusion device. The KDP crystal is soft and crisp and is deliquescent, and is one of the most difficult materials accepted internationally. Currently, SPDT (single point diamond fly cutter milling) is the mainstream ultra-precision machining technology for KDP crystals. After SPDT processing, micro cracks, micro pits, etc. remain on the crystal surface, these defects interact with the incident laser to cause laser damage, and then the damage point increases rapidly to cause component rejection. The problem of laser damage to the KDP crystal element and its growth has become a bottleneck problem that limits the performance improvement of laser fusion devices. For example, the united states "national ignition" proposes an increase in output energy from 1.8MJ to 3.0MJ, which requires a higher resistance to laser damage for the KDP crystal element. At this time, the nm-order defect on the surface of the KDP crystal element may cause serious laser damage. Research shows that advanced manufacturing technology is adopted to repair micro defects on the surface of the element in advance, so that the laser damage problem can be greatly relieved. Therefore, the local repair technology of the micro defects on the surface of the KDP crystal element has important engineering application value.
At present, local repair of microdefects on the surface of a KDP crystal element is mainly carried out through micro milling. The method removes the defects by processing the controllable repair contour on the surface of the element, and has good stability. But the method is mainly aimed at surface defects of tens to hundreds of micrometers, and can leave mm-level repair pits, resulting in large laser flux loss and optical field enhancement. With the introduction of the ignition energy of 3.0MJ, the repair requirement of the nm-level defect on the surface of the element is urgent. DPN (dip pen nanolithography) is the only surface treatment technology capable of realizing the local repair of nm-level defects on the surface of a KDP crystal element at present based on the water-soluble property of the material. The technology is matched with a 3.0MJ ignition target, and has strong engineering application prospect. However, the research on the technology on the repair of micro defects on the surface of a KDP crystal element is basically in a blank state. DPN utilizes a nano-scale liquid bridge formed between an AFM probe and the KDP crystal surface to dissolve and molecularly reconstruct materials in a defect area, and the core comprises: AFM probe, substrate and liquid bridge. The liquid bridge not only can dissolve and transfer KDP crystal molecules, but also affects acting force between the probe and the crystal, probe movement, imaging precision and the like. Thus, the liquid bridge plays a decisive role in the repair process.
However, the current method for calculating the morphology of the liquid bridge has the following disadvantages:
(1) The study objects are mainly focused on liquid bridges formed among large-size spherical contours, spherical contours and planes, conical contours and planes. The method is mainly concentrated in the research fields of flow of the granular material, friction between surfaces and the like, the calculation scale of the method is at the level of mu m/mm, and the method is not suitable for a nano-scale AFM needle point used in the DPN repair process of a water-soluble KDP crystal element.
(2) When researching a liquid bridge formed between an AFM probe and a plane, the needle point is generally approximate to a sphere or a cone, and the needle point sphere is approximate to the condition that the environment humidity is low and the dimensions of the liquid bridge and the needle point ball head are equivalent. It is not applicable to cases where the coverage of the liquid bridge on the probe surface extends to the cone-shaped body area at high ambient humidity. However, the liquid bridge topography error tends to be larger when the tip taper approximates the case where the diameter of the tip arc is larger. In addition, the conical needle tip can fail due to small coverage area of the liquid bridge on the surface of the probe under low environmental humidity, which is unfavorable for researching the optimization process of the DPN water-soluble repair process parameters.
(3) The analytical method, the iterative method, the approximation method and the numerical method are commonly used methods for solving the appearance of the liquid bridge at present. The resolution method can obtain the accurate solution of the liquid bridge morphology, but the solution process involves elliptic integral, which is unfavorable for the rapid analysis of DPN water-soluble repair process parameters. The iterative method is used for solving the meridian radius and the azimuth radius alternately through the geometric relationship, and the algorithm is simple. However, for the DPN water-soluble repair process, the matrix is a plane, the contact angle between the liquid bridge curve and the surface of the KDP crystal element is close to 0 degrees, and the iterative method cannot normally solve the meridian radius. The approximation method approximates the liquid bridge morphology to an arc. The method can obtain more accurate results only when the radius of the arc of the needle point is larger than 1 mu m, and the size of the needle point of the AFM is far larger than that of the needle point of the AFM. And the higher the relative humidity, the greater the approximation error. Therefore, the approximation method is not applicable to DPN water-soluble repair processes. The main stream numerical method is mainly used for solving the appearance of a liquid bridge with a fixed volume between large-scale spherical contours, ignoring the Kelvin effect and is not suitable for solving a capillary condensation liquid bridge in the DPN water-soluble restoration process. In addition, the control equation of the method is established under a rectangular coordinate system, and the method can not be normally solved for a high humidity environment in the DPN water-soluble repair process similar to an iterative method.
(4) Because the AFM probe tip scale used in the DPN repair process of the water-soluble KDP crystal is in the nanometer scale, the formed liquid bridge scale is also in the nanometer scale basically. The error Qu Xianji is easy to generate double solution phenomenon when solving the appearance of the liquid bridge under the scale. However, the existing calculation method does not avoid the problem, so that an error solution is very easy to obtain when calculating the liquid bridge in the DPN water-soluble repair process, and an error liquid bridge morphology is obtained.
Research shows that the liquid bridge is closely related to repair process parameters such as relative humidity, temperature, surface tension, surface wetting characteristics, probe structure, probe lift-off height and the like. Therefore, establishing the mapping relation between the liquid bridge morphology features and the repairing process parameters can provide important support for researching the DPN water-soluble repairing technology, optimizing the process parameters and realizing the effective repairing of the surface micro defects of the water-soluble KDP crystal element.
Disclosure of Invention
The invention aims to solve the technical problems that:
the existing method is not suitable for the extreme cases that the liquid bridge can cover and extend to the conical main body area under high environmental humidity, the slope at the contact point of the liquid bridge appearance curve and the element surface is close to infinity, and meanwhile, the double solution phenomenon exists in the error curve of the calculation of the nano-scale liquid bridge appearance, so that the error result is very easy to obtain.
The invention adopts the technical scheme for solving the technical problems:
the invention provides a liquid bridge full-range calculation method for a water-soluble KDP crystal element surface microdefect DPN repair process, which comprises the following steps:
s1, constructing a DPN water-soluble repair KDP crystal element model, wherein the DPN water-soluble repair KDP crystal element model comprises an AFM probe tip, a KDP crystal element and a geometric relationship of a liquid bridge morphology curve under a two-dimensional rectangular coordinate system, and the AFM probe tip comprises a tip ball head and a conical body;
s2, constructing a parameterized ordinary differential equation of the liquid bridge morphology curve based on an included angle between a tangent of the liquid bridge morphology curve in the model and the surface of the KDP crystal element, and determining a value range of the included angle between the tangent of the liquid bridge morphology curve and the KDP crystal element;
constructing a geometric equation of the probe tip composite contour based on the model;
s3, determining a rough root-finding interval and dispersing, solving the coordinates of discrete points by adopting a geometric equation of a probe tip composite contour, solving the liquid bridge topography curve by assuming each point as an initial value of a parameterized ordinary differential equation of the liquid bridge topography curve, calculating errors of each curve, drawing an error curve, and analyzing the error curve to determine an interval of accurate solution;
s4, targeting the liquid bridge morphology curve by adopting a dichotomy in an accurate solution interval until a desired result is reached, and obtaining a final liquid bridge morphology curve;
s5, calculating the geometric characteristic parameters of the liquid bridge according to the final appearance curve of the liquid bridge.
Further, in S2, assuming that the central axis of the AFM probe tip is perpendicular to the surface of the KDP crystal element, a rectangular coordinate system is established by taking the central axis of the AFM probe tip as the Z axis and any straight line passing through the Z axis of the surface of the KDP crystal element as the X axis, and a parameterized ordinary differential equation for constructing a liquid bridge morphology curve is:
wherein delta represents an included angle between a tangent line of the liquid bridge morphology curve and the surface of the KDP crystal element; the value range is as follows:
wherein r represents Kelvin radius, unit is nm, rm represents universal gas constant, vm represents molecular molar volume of liquid water, T represents temperature, gamma represents surface tension of interface between water and air, θ 1 Represents the contact angle theta between the liquid bridge curve and the KDP crystal element surface 2 Representing the contact angle between the liquid bridge curve and the surface of the probe, RH representing the relative humidity of the environment, R representing the radius of the ball head of the probe tip, xc representing the abscissa of a critical transition point, wherein the critical transition point is the tangential point of the ball head of the probe tip and the conical body.
Further, the geometric equation of the probe tip profile in S2 is:
wherein,represents the half angle of the cone angle of the conical body of the AFM probe, d represents the distance alpha between the tip of the AFM probe and the surface of the KDP crystal element max Is the angle of coverage of the liquid bridge on the probe tip sphere when the liquid bridge topography curve intersects the probe surface at the critical transition point.
Further, S3 includes the following steps:
s31, selecting a rough root finding solving interval and discretizing the rough root finding solving interval;
s32, solving the coordinates of a series of discrete points by adopting a geometric equation of the contour of the probe tip;
s33, calculating a solving interval of a liquid bridge morphology curve equation according to the relation between the coordinates solved in the S32 and the critical conversion point coordinates, and solving the liquid bridge morphology curve by taking each coordinate solved in the S32 as an initial value of a parameterized ordinary differential equation of the liquid bridge morphology;
s34, calculating the position deviation between each curve end point and the KDP crystal element surface as a curve error, drawing an error curve, and if the error curve does not meet the conditions, adjusting a rough root finding solving interval; if the error curve meets the conditions, determining the interval of the accurate solution and the error corresponding to the interval boundary according to the error curve.
Further, the error curve is obtained by calculating the difference between the end point Z value of each curve and the surface Z value of the KDP crystal element, and the basis for judging whether the error curve meets the condition is as follows: when the error curve has no minimum value, has a minimum value smaller than zero but has no solution, has a single solution but has a negative slope of the position error curve, the range of the solution interval of the rough root finding should be enlarged, the existence of the minimum value larger than zero indicates that a liquid bridge cannot be formed under the condition of the DPN water-soluble repair process parameter, the existence of the minimum value equal to zero indicates that the DPN water-soluble repair process parameter is a critical condition for forming the liquid bridge, the interval of the accurate solution should be determined according to the solution with a larger value when the double solution exists, the interval of the accurate solution should be determined according to the solution when the single solution exists and the slope of the position error curve is positive.
Further, S4 includes the following steps:
s41, taking the midpoint of the interval of the accurate solution, and solving the coordinates of the midpoint through a geometrical equation of the needle point contour;
s42, calculating a solving interval of a liquid bridge morphology curve equation according to the position relation between the solved midpoint coordinates and the critical conversion point coordinates, and solving the liquid bridge morphology curve by taking the midpoint coordinates as an initial value of a parameterized ordinary differential equation parameterized by the liquid bridge morphology curve;
s43, calculating the position deviation between the curve end point and the KDP crystal element surface as a curve error, and dividing the interval of the accurate solution into two parts by utilizing the curve midpoint according to the dichotomy iteration principle until the obtained interval is shortened to be below a set tolerance or the iteration times reach a preset value, so as to obtain a final liquid bridge curve.
Further, S5, the final liquid bridge morphology curve is rotated for 360 degrees along the Z axis to obtain the liquid bridge morphology of the water-soluble KDP crystal element DPN repairing process, and the geometric characteristic parameters of the liquid bridge under specific repairing process parameters are further calculated through the liquid bridge morphology.
Compared with the prior art, the invention has the beneficial effects that:
the invention relates to a liquid bridge full-range calculation method for a water-soluble KDP crystal element surface micro defect DPN repair process, which comprises (1) approximating an AFM probe tip structure to a composite contour of a tip ball head and a conical body, defining a critical transition point to control a solving process. (2) The method can solve the limit condition that the contact angle between the liquid bridge curve and the surface of the KDP crystal element is 0 degree when the matrix contour is a plane by adopting the liquid bridge morphology curve equation established under the arc length coordinate system. The method is more suitable for calculating the liquid bridge morphology of the DPN repair process of the water-soluble KDP crystal element under the condition of high ambient humidity. (3) The method comprises two steps of rough root searching and fine root searching, and can lock the interval where the accurate solution is located through rough root searching, reduce the iteration times of the fine root searching process and improve the calculation efficiency. Meanwhile, the precise root searching process is always carried out in the locking interval, so that the problem that an error curve has double solutions to obtain an error liquid bridge shape can be avoided. (4) The method does not relate to dimensionless equation, does not need to introduce variable conversion boundaries, does not need to rewrite control equations and boundary conditions, does not relate to Newton method iteration, does not need to solve the derivative of the equations, and has simple calculation process.
Drawings
FIG. 1 is a flow chart of a liquid bridge full-range calculation method for a water-soluble KDP crystal element surface microdefect DPN repair process in an embodiment of the invention;
FIG. 2 is a schematic diagram of a DPN water-soluble repair KDP crystal unit in accordance with an embodiment of the invention;
FIG. 3 is a graph of error and anomaly error for a method of the present invention in an embodiment of the present invention;
FIG. 4 is a graph showing the comparison of the results of the liquid bridge morphology obtained by the method of the present invention and the prior probe approximation method in the embodiment of the present invention;
FIG. 5 is a graph of the verification result of the method of the present invention in an embodiment of the present invention; wherein, (a) is a liquid bridge test result diagram, and (b) is a comparison diagram of the liquid bridge calculation result and the test result.
Detailed Description
In the description of the present invention, it should be noted that the terms "first," "second," and "third" mentioned in the embodiments of the present invention are used for descriptive purposes only and are not to be construed as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defining "a first", "a second", or a third "may explicitly or implicitly include one or more such feature.
In order that the above objects, features and advantages of the invention will be readily understood, a more particular description of the invention will be rendered by reference to specific embodiments thereof which are illustrated in the appended drawings.
Referring to fig. 1 to 2, the invention provides a liquid bridge full-range calculation method for a water-soluble KDP crystal element surface microdefect DPN repair process, which comprises the following steps:
s1, as shown in FIG. 2, constructing a DPN water-soluble repair KDP crystal element model, wherein the DPN water-soluble repair KDP crystal element model comprises a geometric relationship of an AFM probe tip, a KDP crystal element and a liquid bridge morphology curve under a two-dimensional rectangular coordinate system, and the AFM probe tip is of a composite structure comprising a tip ball head and a conical body.
S2, constructing a parameterized ordinary differential equation of the liquid bridge morphology curve based on an included angle between a tangent of the liquid bridge morphology curve in the model and the surface of the KDP crystal element, and determining a value range of the included angle between the tangent of the liquid bridge morphology curve and the KDP crystal element.
Assuming that the central axis of the AFM probe tip is perpendicular to the surface of the KDP crystal element, establishing a rectangular coordinate system by taking the central axis of the AFM probe tip as a Z axis and taking any straight line passing through the Z axis of the surface of the KDP crystal element as an X axis, and constructing a parameterized ordinary differential equation of a liquid bridge morphology curve, wherein the parameterized ordinary differential equation is as follows:
wherein delta represents an included angle between a tangent line of the liquid bridge morphology curve and the surface of the KDP crystal element; the value range is as follows:
wherein r represents Kelvin radius, unit is nm, rm represents universal gas constant, vm represents molecular molar volume of liquid water, T represents temperature, gamma represents surface tension of interface between water and air, θ 1 Represents the contact angle theta between water and KDP crystal element surface 2 Representing the contact angle between water and the surface of the probe, RH representing the relative humidity of the environment, R representing the radius of the ball head of the probe tip, xc representing the abscissa of a critical transition point, wherein the critical transition point is the tangential point of the ball head of the tip and the conical body.
S2, based on the model, according to the diameter of the probe tip ball head and the tip coneCorner angleConstructing a geometric equation Z (X) of the composite profile of the probe tip:
wherein,represents the half angle of the taper angle of the conical body of the probe, d represents the distance alpha between the tip of the AFM probe and the surface of the KDP crystal element max Is the angle of coverage of the liquid bridge on the probe tip sphere when the liquid bridge topography curve intersects the probe surface at the critical transition point.
Determining a rough root-finding interval and dispersing, solving discrete point coordinates by adopting a geometric equation of a probe tip composite contour, solving a liquid bridge topography curve by assuming each point as an initial value of a parameterized ordinary differential equation of the liquid bridge topography curve, calculating errors of each curve, drawing an error curve, and analyzing the error curve to determine an accurate solution interval;
s3, determining a rough root-finding interval (0, xr) and dispersing, solving the coordinates of discrete points by adopting a geometric equation of a probe tip composite contour, and assuming each point as an initial value of a parameterized ordinary differential equation of a liquid bridge morphology curve to solve the liquid bridge morphology curve:
s31, selecting a coarse root finding solving interval (0, xr) and discretizing the interval;
s32, solving the coordinates of a series of discrete points by adopting a probe tip contour geometrical equation to obtain discrete points (X1, Z1), (X2, Z2), …, (Xi, zi), …, (Xr, zr) on the surface of the probe;
s33, calculating a solving interval of a liquid bridge morphology curve equation according to the relation between the coordinates solved in the S32 and the coordinates of the critical conversion points, and solving the liquid bridge morphology curve by taking each coordinate solved in the S32 as an initial value of a parameterized ordinary differential equation of the liquid bridge morphology in the interval;
s34, calculating the position deviation between each curve end point and the KDP crystal element surface as a curve error, namely calculating the targeting heuristic solution Z (theta) 1 ) Taking the difference value with Z=0 as a curve error, drawing an error curve, and adjusting a rough root-finding solving interval (0, xr if the error curve does not meet the condition]The method comprises the steps of carrying out a first treatment on the surface of the If the error curve meets the condition, determining the interval [ Xa, xb ] of the accurate solution according to the error curve]And errors Erra and Errb corresponding to interval boundaries.
The error curves are obtained by calculating the difference value between the end point Z value of each curve and the surface Z value of the KDP crystal element, and the basis for judging whether the error curves meet the conditions is as follows: when the error curve has no minimum value, has a minimum value smaller than zero and has no solution, and has a single solution but the slope of the position error curve is negative, the range of the solution interval of the rough root finding should be enlarged, the existence of the minimum value larger than zero indicates that a liquid bridge cannot be formed under the condition of the DPN water-soluble repair process parameter, the existence of the minimum value equal to zero indicates that the DPN water-soluble repair process parameter is a critical condition for forming the liquid bridge, the existence of the double solution should determine the interval [ Xa, xb ] of the precise solution according to the solution with larger value, the existence of the single solution and the slope of the position error curve should be positive, and the interval [ Xa, xb ] of the precise solution should be determined according to the solution.
And S4, targeting the liquid bridge morphology curve by adopting a dichotomy in the accurate solution interval until a desired result is reached, and obtaining the liquid bridge morphology curve. The method comprises the following steps:
s41, taking the midpoint Xm of the interval of the accurate solution, and solving the coordinates (Xm, zm) of the midpoint Xm by a geometrical equation of the needle tip contour;
s42, calculating a solving interval of a liquid bridge morphology curve equation according to the position relation between the solved midpoint coordinates and the critical conversion point coordinates, and solving the liquid bridge morphology curve by taking the midpoint coordinates as an initial value of a parameterized ordinary differential equation of the liquid bridge morphology curve;
s43, calculating the position deviation between the curve end point and the KDP crystal element surface as a curve error Errm, and dividing the interval [ Xa, xb ] of the accurate solution into two parts by utilizing the curve midpoint Xm according to the dichotomy iteration principle until the obtained interval is shortened to be below a set tolerance Tol or the iteration times reach a preset value, and simultaneously enabling the error of the targeting heuristic solution and the target value to be converged rapidly to obtain the final liquid bridge curve.
S5, calculating the geometric characteristic parameters of the liquid bridge according to the final appearance curve of the liquid bridge.
And rotating the final liquid bridge morphology curve by 360 degrees along the Z axis to obtain the liquid bridge morphology of the water-soluble KDP crystal element DPN repairing process, and further calculating geometric characteristic parameters such as the volume, the height, the diameter and the like of the liquid bridge under specific repairing process parameters through the liquid bridge morphology.
Example 1
TABLE 1
DPN repair parameters shown in Table 1 are input into the formulas (1) - (6), and a parameterized ordinary differential equation of a liquid bridge morphology curve and a geometric equation of a probe tip composite contour are constructed.
The method is used for calculating the error of the curve and drawing an error curve, 2 solutions exist for the error curve, and according to the basis of whether the error meets the condition or not, namely, the accurate solution interval is determined according to the solution with larger values when double solutions exist, the accurate solution interval is determined to be [20,23] when the error curve exists as shown in fig. 3 (b), but the slope of the position error curve is negative, the solution interval (0.2, Z (0.2)), (0.4, Z (0.4)), …, (Xi, zi), …, (50, Z (50)) is required to be increased, as shown in fig. 3 (a), the error curve is calculated by adopting the method, the error curve is drawn, 2 solutions exist according to the basis of whether the error meets the condition, namely, the accurate solution interval is determined according to the solution with larger values when double solutions exist, the accurate solution interval is determined to be [20,23] when the error curve exists as shown in fig. 3 (b), the error curve exists as the error curve exists in the single solution, the situation that the slope of the position error curve is negative, the error curve needs to be enlarged, and the error curve (0, xr) is larger than the error curve exists under the condition that the surface of the DPP is unable to be repaired, and the error curve exists under the condition of the liquid bridge is larger than the liquid, and the error curve exists and the error curve is not larger than the DPP is formed.
Setting the tolerance Tol=1E-6, updating the interval according to the principle of dichotomy iteration in the accurate solution interval [20,23], and finally iterating 22 times to converge the interval length below the set tolerance 1E-6, so as to obtain the error of 2.196E-7. The principle of the dichotomy iteration is as follows:
the precise root searching process is always carried out in the interval locked by the rough root searching process, and the phenomenon that an error liquid bridge shape is obtained when double solutions exist can be avoided.
And (3) obtaining points (Xm, zm) after the binary iteration cycle is finished as initial points of a liquid bridge morphology curve, wherein the points (Xm, zm) are on a conical body of the AFM probe tip, a series of discrete points obtained by taking (Xm, zm) as initial values are used for obtaining a final liquid bridge morphology curve, and a three-dimensional liquid bridge morphology of a final water-soluble KDP crystal element DPN repairing process obtained after the curve is rotated 360 degrees around a Z axis is shown in fig. 4 (a).
Fig. 4 (b) and (c) are liquid bridge morphologies under the approximation of other AFM probe tip profiles under the same DPN water-soluble repair process parameters. The spherical approximation of the tip of FIG. 4 (b) is that the liquid bridge is oversized in the coverage area of the AFM probe tip ball head due to the high relative humidity of the environment beyond the critical transition point defined by the present invention, and the contact point of the liquid bridge topography curve with the AFM probe tip surface extends into the interior of the actual profile of the AFM probe because the spherical approximation does not take this into account. Meanwhile, the liquid bridge morphology error can be increased along with the reduction of the diameter of the AFM probe tip ball head. The conical approximation of the tip shown in fig. 4 (c), although the liquid bridge topography error is relatively small compared to the spherical approximation, because it does not consider the tip ball head diameter, results in a higher conical approximation profile of the AFM probe, and the ball head diameter will increase with wear during use of the AFM probe, and the deviation of the conical approximation profile will increase. Meanwhile, the cone approximation method can fail under low environmental humidity due to the fact that the liquid bridge is smaller in the coverage area of the AFM probe tip, and therefore research on the optimal process of DPN water-soluble repair process parameters is not facilitated.
To further verify the effectiveness of the present invention, the results of the calculations of the present invention were compared to the test values. The Lawrence Lifromo national laboratory in the United states adhered a glass substrate supporting the AFM probe cantilever to a silicon substrate via an epoxy resin, and brought the AFM probe tip into direct contact with the silicon substrate. After such special treatment, the conditions are similar to those in the case where the process parameter d=0 nm for repairing the water-soluble KDP crystal element DPN. The liquid bridge was then imaged by a field emission environmental scanning electron microscope at an ambient temperature T of 278K (corresponding surface tension gamma of the water-gas interface of approximately 74.97mJ/m 2 ) The relative humidity was near saturation, i.e., RH was near 100%, and the result was shown in fig. 5 (a). The AFM probe tip taper angle of about 70℃can be obtained by measuring FIG. 5 (a))，θ 1 About 20 DEG, theta 2 About 12. The radius R of the bulb of the AFM probe tip is 10nm (typical value). The model parameters (high humidity environment) are substituted into a parameterized ordinary differential equation of a liquid bridge morphology curve in the DPN repair process of the water-soluble KDP crystal element and a geometric equation of a probe tip compound profile, so that the liquid bridge curve shown in the figure 5 (b) is calculated, and the result and a test value have good consistency. The accuracy and the effectiveness of the liquid bridge morphology of the invention under the condition of high environmental humidity are fully proved.
The method realizes the accurate calculation of the liquid bridge morphology between the AFM probe tip and the surface of the KDP crystal element in the DPN repair process of the water-soluble KDP crystal element, provides an important basis for researching the DPN repair process of the microdefect on the surface of the water-soluble crystal element, and guides the optimization of the DPN water-soluble repair process parameters.
Although the present disclosure is disclosed above, the scope of the present disclosure is not limited thereto. Various changes and modifications may be made by one skilled in the art without departing from the spirit and scope of the disclosure, and such changes and modifications would be within the scope of the disclosure.
Claims (6)
1. A liquid bridge full-range calculation method for a water-soluble KDP crystal element surface microdefect DPN repair process is characterized by comprising the following steps:
s1, constructing a DPN water-soluble repair KDP crystal element model, wherein the DPN water-soluble repair KDP crystal element model comprises an AFM probe tip, a KDP crystal element and a geometric relationship of a liquid bridge morphology curve under a two-dimensional rectangular coordinate system, and the AFM probe tip comprises a tip ball head and a conical body;
s2, constructing a parameterized ordinary differential equation of the liquid bridge morphology curve based on an included angle between a tangent of the liquid bridge morphology curve in the model and the surface of the KDP crystal element, and determining a value range of the included angle between the tangent of the liquid bridge morphology curve and the KDP crystal element;
constructing a geometric equation of the probe tip composite contour based on the model;
s3, determining a rough root-finding interval and dispersing, solving the coordinates of discrete points by adopting a geometric equation of a probe tip composite contour, solving the liquid bridge topography curve by assuming each point as an initial value of a parameterized ordinary differential equation of the liquid bridge topography curve, calculating errors of each curve, drawing an error curve, and analyzing the error curve to determine an interval of accurate solution;
s4, targeting the liquid bridge morphology curve by adopting a dichotomy in an accurate solution interval until a desired result is reached, and obtaining a final liquid bridge morphology curve;
s5, calculating geometric characteristic parameters of the liquid bridge according to the final appearance curve of the liquid bridge;
in the S2, assuming that the central axis of the AFM probe tip is vertical to the surface of the KDP crystal element, establishing a rectangular coordinate system by taking the central axis of the AFM probe tip as a Z axis and any straight line passing through the Z axis of the surface of the KDP crystal element as an X axis, and constructing a parameterized ordinary differential equation of a liquid bridge morphology curve, wherein the parameterized ordinary differential equation is as follows:
wherein delta represents an included angle between a tangent line of the liquid bridge morphology curve and the surface of the KDP crystal element; the value range is as follows:
wherein r represents Kelvin radius, unit is nm, rm represents universal gas constant, vm represents molecular molar volume of liquid water, T represents temperature, gamma represents surface tension of interface between water and air, θ 1 Represents the contact angle theta between the liquid bridge curve and the KDP crystal element surface 2 Representing the contact angle between the liquid bridge curve and the surface of the probe, RH representing the relative humidity of the environment, R representing the radius of the ball head of the probe tip, xc representing the abscissa of a critical transition point, wherein the critical transition point is the tangential point of the ball head of the probe tip and the conical body.
2. The method for calculating the full range of the liquid bridge in the process of repairing the surface microdefect DPN of the water-soluble KDP crystal element according to claim 1, wherein the geometric equation of the probe tip compound profile in S2 is as follows:
wherein,represents the half angle of the cone angle of the conical body of the AFM probe, d represents the distance alpha between the tip of the AFM probe and the surface of the KDP crystal element max Is the angle of coverage of the liquid bridge on the probe tip sphere when the liquid bridge topography curve intersects the probe surface at the critical transition point.
3. The liquid bridge full-range calculation method for the surface micro-defect DPN repair process of the water-soluble KDP crystal element according to claim 2, wherein S3 comprises the steps of:
s31, selecting a rough root finding solving interval and discretizing the rough root finding solving interval;
s32, solving the coordinates of a series of discrete points by adopting a geometric equation of the probe tip composite contour;
s33, calculating a solving interval of a liquid bridge morphology curve equation according to the relation between the coordinates solved in the S32 and the critical conversion point coordinates, and solving the liquid bridge morphology curve by taking each coordinate solved in the S32 as an initial value of a parameterized ordinary differential equation of the liquid bridge morphology;
s34, calculating the position deviation between each curve end point and the KDP crystal element surface as a curve error, drawing an error curve, and if the error curve does not meet the conditions, adjusting a rough root finding solving interval; if the error curve meets the conditions, determining the interval of the accurate solution and the error corresponding to the interval boundary according to the error curve.
4. The method for calculating the full range of a liquid bridge in a process of repairing a microdefect DPN on a surface of a water-soluble KDP crystal element according to claim 3, wherein the error curve is obtained by calculating a difference between an end point Z value of each curve and a surface Z value of the KDP crystal element, and the basis for determining whether the error curve satisfies the condition is as follows: when the error curve has no minimum value, has a minimum value smaller than zero but has no solution, has a single solution but has a negative slope, the range of the solution interval of the rough root finding should be enlarged, the existence of the minimum value larger than zero indicates that a liquid bridge cannot be formed under the condition of the DPN water-soluble repair process parameter, the existence of the minimum value equal to zero indicates that the DPN water-soluble repair process parameter is a critical condition of the liquid bridge formation, the interval of the precise solution should be determined according to the solution with a larger value when the double solution exists, the interval of the precise solution should be determined according to the solution when the single solution exists and the slope of the error curve is positive.
5. The liquid bridge full-range calculation method for the surface micro-defect DPN repair process of the water-soluble KDP crystal element according to claim 4, wherein S4 comprises the following steps:
s41, taking the midpoint of an accurately solved interval, and solving the coordinates of the interval by using a geometric equation of the composite contour of the probe tip;
s42, calculating a solving interval of a liquid bridge morphology curve equation according to the position relation between the solved midpoint coordinates and the critical conversion point coordinates, and solving the liquid bridge morphology curve by taking the midpoint coordinates as an initial value of a parameterized ordinary differential equation parameterized by the liquid bridge morphology curve;
s43, calculating the position deviation between the curve end point and the KDP crystal element surface as a curve error, and dividing the interval of the accurate solution into two parts by utilizing the curve midpoint according to the dichotomy iteration principle until the obtained interval is shortened to be below a set tolerance or the iteration times reach a preset value, so as to obtain a final liquid bridge curve.
6. The method for calculating the full range of the liquid bridge in the process of repairing the surface microdefect DPN of the water-soluble KDP crystal element according to claim 5, wherein S5 is characterized in that the final liquid bridge morphology curve is rotated 360 degrees along the Z axis to obtain the liquid bridge morphology in the process of repairing the water-soluble KDP crystal element, and the geometric characteristic parameters of the liquid bridge under the repairing process parameters are further calculated through the liquid bridge morphology.
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