A simulation method for morphology evolution of DPN aqueous solution repair of micro-nano defects on surface of KDP crystal

Abstract

The application provides a KDP crystal surface micro-nano defect DPN water-soluble repair morphology evolution simulation method, and relates to the technical field of micro-nano manufacturing, and aims to solve the problem that there is no quantitative method to simulate the evolution process of the KDP crystal surface micro-nano defect DPN water-soluble repair morphology in the prior art. The method comprises the following steps: step one, constructing a defect local growth mathematical model; step two, performing dimension conversion on the defect depth information, and performing dimension reduction processing on the model; step three, converting the defect local growth mathematical model into a standard partial differential equation form; step four, obtaining the initial value of the defect local growth mathematical model, and setting the model boundary condition; step five, performing parameterization scanning on the undetermined coefficient, and determining the undetermined coefficient value; and step six, simulating the KDP crystal surface micro-nano defect DPN water-soluble repair morphology evolution process. Through dimension conversion, the dimension reduction processing of the model is realized, and finally the simulation of the KDP crystal surface micro-nano defect DPN water-soluble repair morphology evolution process is realized.

Description

Technical Field

[0001] This invention relates to the field of micro-nano manufacturing technology, and more specifically, to a method for simulating the morphological evolution of DPN water-soluble repair of micro-nano defects on the surface of KDP crystals. Background Technology

[0002] Inertial confinement fusion energy is an ideal clean energy source, offering an effective solution to the future energy crisis. Giant laser devices are currently the mainstream method for driving inertial confinement fusion, such as the National Ignition Facility (NIF) in the United States, the megajoule laser facility in France, and the giant laser device in China. Potassium dihydrogen phosphate (KDP) crystals, due to their unique photoelectric properties and high nonlinear coefficients, are used to fabricate large-aperture core optical components such as the Phelps cell and frequency multiplier. Giant laser fusion devices place extremely high demands on the processing quality of KDP crystal components (nm-level full-frequency domain error control). However, KDP crystals are soft and brittle, sensitive to temperature changes, prone to deliquescence, and highly anisotropic, making them one of the internationally recognized difficult-to-machine materials. Currently, single-point diamond fly-cut milling is the mainstream ultra-precision machining technology for KDP crystals, but the surface of the machined components will retain defects such as microcracks, micro-pits, and scratches. These defects interact with the incident laser, causing laser damage, and the size of these damage points grows rapidly, leading to component failure. Currently, laser damage and its growth in KDP crystal elements have become a bottleneck limiting the increase in output energy of laser fusion devices. To address this challenge, the Lawrence Livermore National Laboratory in the United States proposed a "circular strategy" for optical elements. This strategy significantly improves their resistance to laser damage by detecting surface damage and promptly performing micro-repair treatments on the surface of optical elements. According to recent reports, the National Institute for Space Technology (NIF) has been able to output 2.05 MJ of ultraviolet laser energy to drive 3.15 MJ of fusion energy, achieving laboratory fusion ignition. However, to ensure a continuous and stable fusion energy output, NIF has increased its output energy requirement to 3.0 MJ, which poses a greater challenge to the laser damage resistance of KDP crystal elements. At this point, even nanometer-level defects on the surface of KDP crystal elements can cause serious laser damage. Research shows that if advanced surface treatment technologies can be used to repair micro-defects on the surface of KDP crystal elements in advance, the problem of laser-induced damage can be greatly alleviated. Therefore, local repair technology for micro-defects on the surface of KDP crystal elements has significant engineering application value.

[0003] Currently, precision micromilling is the mainstream international method for repairing micro-defects on the surface of KDP crystal devices. This method uses a micromilling cutter to machine specific contours on the surface of the KDP crystal device to replace micro-defects and achieve defect repair. Micromilling can completely repair micro-defects on the surface of KDP crystal devices, with controllable repair contours, high surface quality, and good stability. However, this method is mainly used for machining micro-defects or laser-induced damage points on surfaces with sizes ranging from tens to hundreds of μm. After repair, millimeter-scale repair pits remain on the surface of the device, causing significant laser flux loss (limited repair cycles) and enhanced optical field modulation. With the proposed "3.0 MJ" fusion ignition goal, repairing nm-scale defects on the surface of KDP crystal devices has become urgent. Based on the water-soluble properties of KDP crystals, DPN (dipping pen nanolithography) technology is currently the only surface treatment technology that can achieve local repair of nm-scale defects on the surface of KDP crystal devices. This technology aligns with the "3.0 MJ" fusion ignition goal and has strong engineering application prospects. DPN (Dihydroparticle Neural Network) technology utilizes spontaneously formed nanometer-scale liquid bridges between an AFM (Anti-Fluorescent Micrometer) probe and the surface of a KDP (Kinsey & Company) crystal element to locally dissolve and molecularly reconstruct materials in defect regions, thereby achieving the repair of micro- and nano-scale defects. Theoretically, this is not limited by the number of repair attempts. During the DPN water-soluble repair process, the transport of material on the KDP crystal surface is mainly controlled by the Gibbs-Thomson law, and the surface curvature at the defect gradually decreases, achieving surface smoothing and defect repair. However, there is currently a lack of quantitative methods to simulate the evolution of the morphology of micro- and nano-defects on the KDP crystal surface during DPN water-soluble repair, posing a significant challenge to the research on DPN water-soluble repair technology and its engineering applications for KDP crystal elements. Summary of the Invention

[0004] The technical problem to be solved by this invention is:

[0005] The existing technology lacks a quantitative method to simulate the evolution process of DPN water-soluble repair morphology of micro-nano defects on the KDP crystal surface.

[0006] The technical solution adopted by the present invention to solve the above-mentioned technical problems is as follows:

[0007] This invention provides a method for simulating the morphological evolution of DPN water-soluble repair of micro / nano defects on the surface of KDP crystals, comprising the following steps:

[0008] Step 1: Establish a coordinate system and construct a mathematical model for the local growth of defects;

[0009] Step 2: Transform the defect depth information to reduce the dimensionality of the model;

[0010] Step 3: Convert the mathematical model of local defect growth into standard partial differential equation form;

[0011] Step 4: Obtain the initial values ​​of the mathematical model for local defect growth and set the boundary conditions of the model;

[0012] Step 5: Perform parametric scanning on the undetermined coefficients, fit the simulation results and experimental data, and determine the undetermined coefficients of the growth mathematical model based on the error between the simulation and experimental data;

[0013] Step 6: Substitute the determined equation coefficients into the mathematical model of local defect growth to simulate the morphological evolution process of DPN water-soluble repair of micro-nano defects on the KDP crystal surface.

[0014] Furthermore, the mathematical model for local growth of defects described in step one is based on the morphology of the defects. It assumes that the curvature of the defect surface will change the local solution equilibrium concentration, that the AFM probe will increase the local solution concentration, and that the solution concentration in the water film layer adsorbed on the surface of the KDP crystal element is always in an equilibrium saturation state during the DPN water-soluble repair process.

[0015] Furthermore, the mathematical model for local growth of defects described in step one is based on the three-dimensional morphology of the defects. The X and Y axes of the model coordinate system are parallel to the crystal surface, and the Z axis is the direction of the defect depth.

[0016] Furthermore, the mathematical model for local growth of defects described in step one is based on the two-dimensional morphology of the defects. The X-axis of the model coordinate system is parallel to the crystal surface, and the Z-axis is the direction of the defect depth.

[0017] Furthermore, the mathematical model for local defect growth, based on the Gibbs-Thomson law, assumes that the surface curvature of the KDP crystal defect will change the local equilibrium concentration of the solution, and the supersaturation degree corrected for the local curvature is:

[0018]

[0019] Where ω represents the molecular volume of the KDP crystal, in nm. 3 α represents surface free energy, with units of J / nm. 2 k represents Boltzmann's constant, in J / K, and T represents temperature, in K.

[0020] R c The radius of curvature of the local micro / nano defects on the KDP crystal surface, in nm, is as follows:

[0021]

[0022] If the model is based on the two-dimensional morphology of the defect, then If the model is based on the three-dimensional morphology of the defect, then

[0023] The equilibrium concentration C of KDP crystal solution corrected by the local curvature of micro / nano defect surfaces ec for:

[0024]

[0025] Where C e0 This represents the equilibrium saturation concentration of the KDP crystal solution.

[0026] The concentration of the solution in the water film layer adsorbed on the surface of the KDP crystal element is always in a state of equilibrium saturation.

[0027] Assuming the AFM probe causes a local increase in solution concentration, resulting in supersaturation λ, the final supersaturation of the AFM probe's action region is constructed as follows:

[0028]

[0029] According to crystal growth theory, the growth and repair rate at any point on the micro / nano defect profile of the KDP crystal surface along a direction parallel to the crystal surface is:

[0030] v=Aσ final (5)

[0031] Where A is the step dynamics coefficient of the growth mound on the surface of the KDP crystal, in nm / s;

[0032] The growth and repair rate at this point along the Z-axis is:

[0033]

[0034] Where p is the slope of the KDP crystal growth mound.

[0035] Furthermore, the defect depth information mentioned in step two undergoes dimensional transformation. Specifically, the defect depth information, which is measured in terms of length, is converted into other physical quantities. At the same time, the units of other coordinate values ​​are scaled to bring the physical quantities to the same order of magnitude, so as to maintain the consistency of the equation's dimensions.

[0036] Furthermore, the defect depth information mentioned in step two undergoes a dimensional transformation. Specifically, the defect depth information, which is in the dimension of length, is converted to temperature U, the unit of the depth information is converted to K, and the unit of other coordinate values ​​X or (X,Y) is expanded to m.

[0037] Furthermore, in step three, COMSOL Multiphysics is used as the solver to transform the dimensionally transformed formula (6) into the standard partial differential equation form shown in formula (7):

[0038]

[0039] Where C is the diffusion coefficient and F is the source term. The dimensions of C and F are adjusted synchronously according to the dimension conversion of the defect contour depth information Z. The unit of C is nm. 2 / s is converted to W / (m·K), and the unit of F is converted from nm / s to W / m. 3 .

[0040] Furthermore, in step four, the model boundary condition is set to zero flux, that is:

[0041]

[0042] Where n is the normal vector.

[0043] Furthermore, in step five, the least squares method is used to interpolate and fit the simulation results and experimental data, and the values ​​of the undetermined coefficients of the growth mathematical model are determined based on the minimum error between the simulation and experimental data.

[0044] Compared with the prior art, the beneficial effects of the present invention are:

[0045] This invention provides a simulation method for the morphological evolution of DPN water-soluble repair of micro-nano defects on the KDP crystal surface. Based on the Gibbs-Thomson law and crystal growth theory, a mathematical model of the local growth of micro-nano defects on the KDP crystal surface is constructed. The model is reduced in dimension by dimensional transformation and converted into a standard partial differential equation form. Furthermore, the morphological evolution process of DPN water-soluble repair of micro-nano defects on the KDP crystal surface is simulated through numerical solution and parameter fitting.

[0046] The method of this invention is applicable to the simulation of the evolution of micro-nano defect repair morphology on the surface of KDP crystal elements with various complex morphologies, and provides important theoretical support for the optimization of DPN water-soluble repair process and the study of repair mechanism for micro-nano defects on the surface of KDP crystals. Attached Figure Description

[0047] Figure 1 This is a flowchart illustrating the morphology evolution simulation method for DPN water-soluble repair of micro-nano defects on the KDP crystal surface in this embodiment of the invention.

[0048] Figure 2 This is a schematic diagram of the coordinate system on which the mathematical model for the local growth of micro-nano defects in this embodiment of the invention is based;

[0049] Figure 3 This is a schematic diagram of the cross-sectional profile of the KDP crystal surface before DPN water-soluble repair of micro-nano scratch defects in an embodiment of the present invention.

[0050] Figure 4 This is a fitting error graph corresponding to different undetermined coefficients C in the embodiments of the present invention;

[0051] Figure 5This is a fitting effect diagram corresponding to the undetermined coefficient C = 42 in the embodiment of the present invention;

[0052] Figure 6 This is a comparison chart of simulation and experimental results of the evolution of KDP crystal surface defects at different repair times in the embodiments of the present invention. Detailed Implementation

[0053] In the description of this invention, it should be noted that the terms "first," "second," and "third" mentioned in the embodiments of this invention are for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Therefore, a feature defined with "first," "second," and "third" may explicitly or implicitly include one or more of that feature.

[0054] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0055] Specific Implementation Plan 1: Combining Figures 1 to 6 As shown, this invention provides a method for simulating the morphological evolution of DPN water-soluble repair of micro / nano defects on the surface of KDP crystals, comprising the following steps:

[0056] Step 1: Establish a coordinate system and construct a mathematical model for the local growth of defects;

[0057] Step 2: Transform the defect depth information to reduce the dimensionality of the model;

[0058] Step 3: Convert the mathematical model of local defect growth into standard partial differential equation form;

[0059] Step 4: Obtain the initial values ​​of the mathematical model for local defect growth and set the boundary conditions of the model;

[0060] Step 5: Perform parametric scanning on the undetermined coefficients, fit the simulation results and experimental data, and determine the undetermined coefficients of the growth mathematical model based on the error between the simulation and experimental data;

[0061] Step 6: Substitute the determined equation coefficients into the mathematical model of local defect growth to simulate the morphological evolution process of DPN water-soluble repair of micro-nano defects on the KDP crystal surface.

[0062] Specific Implementation Scheme Two: The mathematical model for local defect growth described in Step One is based on the defect morphology. It assumes that the surface curvature of the defect changes the local solution equilibrium concentration, that the AFM probe increases the local solution concentration, and that the solution concentration within the water film layer adsorbed on the KDP crystal element surface remains in equilibrium saturation during the DPN water-soluble repair process. Through these assumptions, a quantitative mathematical description of the complex local chemical growth process of micro / nano defects on the KDP crystal surface can be achieved. All other aspects of this implementation scheme are the same as Specific Implementation Scheme One.

[0063] Specific Implementation Plan Three: (e.g.) Figure 2 As shown, the mathematical model for local defect growth described in step one is based on the three-dimensional morphology of the defect. The X and Y axes of the model coordinate system are parallel to the crystal surface, and the Z axis represents the defect depth direction. Through the constructed three-dimensional coordinate system, the three-dimensional morphology of micro / nano defects on the KDP crystal surface can be mapped to spatial geometric coordinates (X, Y, Z) in a three-dimensional Cartesian coordinate system. Other aspects of this implementation scheme are the same as in specific implementation scheme two.

[0064] Specific Implementation Plan Four: (e.g.) Figure 2 As shown, the mathematical model for local defect growth described in step one is based on the two-dimensional morphology of the defect. The model coordinate system has its X-axis parallel to the crystal surface and its Z-axis representing the defect depth direction. Through this constructed two-dimensional coordinate system, the two-dimensional cross-sectional morphology of micro / nano defects on the KDP crystal surface can be mapped to spatial geometric coordinates (X, Z) in a two-dimensional Cartesian coordinate system. Other aspects of this implementation scheme are the same as in specific implementation scheme two.

[0065] Specific Implementation Scheme Five: The mathematical model for local defect growth, based on the Gibbs-Thomson law, assumes that the surface curvature of the KDP crystal defect will change the local equilibrium concentration of the solution, and the supersaturation degree of the local curvature correction is:

[0066]

[0067] Where ω represents the molecular volume of the KDP crystal, in nm. 3 α represents surface free energy, with units of J / nm. 2 k represents Boltzmann's constant, in J / K, and T represents temperature, in K.

[0068] R c The radius of curvature of the local micro / nano defects on the KDP crystal surface, in nm, is as follows:

[0069]

[0070] If the model is based on the two-dimensional morphology of the defect, then If the model is based on the three-dimensional morphology of the defect, then

[0071] The equilibrium concentration C of KDP crystal solution corrected by the local curvature of micro / nano defect surfaces ec for:

[0072]

[0073] Where C e0 The equilibrium saturation concentration of the KDP crystal solution is represented by the equation (3). Through the equation (3), a quantitative mathematical description of the equilibrium saturation concentration of the solution in the water film on the surface of the KDP crystal can be realized.

[0074] Because KDP crystals have high solubility in water, it is assumed that the solution concentration within the water film layer adsorbed on the surface of the KDP crystal element remains in equilibrium saturation during the DPN water-soluble remediation process. Simultaneously, considering the adsorption of free dihydrogen phosphate ions and potassium ions (generated by microscopic dissolution of the crystal) on the AFM probe surface, it is assumed that the AFM probe will cause a local increase in solution concentration, forming a supersaturation λ. The final supersaturation of the AFM probe's action area is constructed as follows:

[0075]

[0076] Formula (4) provides a quantitative mathematical description of the supersaturation within the local action region of the AFM probe. According to crystal growth theory, the growth and repair rate at any point (X,Z) / (X,Y,Z) on the micro / nano defect profile of the KDP crystal surface, along the direction parallel to the crystal surface, is:

[0077] v=Aσ final (5)

[0078] Where A is the step dynamics coefficient of the growth mound on the surface of the KDP crystal, in nm / s;

[0079] The growth and repair rate at this point along the Z-axis is:

[0080]

[0081] Where p is the slope of the KDP crystal growth mound. Formula (6) is the first analytical expression for the evolution of local micro-nano defect morphology on the KDP crystal surface during DPN water-soluble repair, realizing a quantitative description of the evolution process of micro-nano defect morphology on the KDP crystal surface during DPN water-soluble repair. Other aspects of this implementation scheme are the same as those in specific implementation schemes three or four.

[0082] Specific Implementation Scheme Six: The defect depth information mentioned in Step Two undergoes dimensional transformation. Specifically, the defect depth information, which is measured in length, is converted into other physical quantities. At the same time, the units of other coordinate values ​​are scaled to ensure that the physical quantities are on the same order of magnitude, thus maintaining the consistency of the equation's dimensions. In Equation (6), the (X,Y,Z) coordinates are coupled and cannot be solved directly using existing methods. Through dimensional transformation, the micro-nano defect depth information (Z coordinate) on the KDP crystal surface can be decoupled from the (X,Y) coordinates, providing a possibility for solving Equation (6). This implementation scheme is otherwise the same as Specific Implementation Scheme Five.

[0083] Specific Implementation Scheme Seven: The defect depth information mentioned in Step Two undergoes dimensional transformation. Specifically, the defect depth information, which is based on length, is converted into temperature U. At this point, the change in defect contour depth with the KDP crystal surface coordinates is transformed into the change in temperature with the KDP crystal surface coordinates. The geometric dimension of formula (6) is reduced by one dimension, and the unit of depth information is converted to K. At the same time, the unit of other coordinate values ​​X or (X,Y) is expanded to m. After performing the temperature dimensional transformation, the two-dimensional distribution of the micro-nano defect depth information on the KDP crystal surface is transformed into a two-dimensional distribution of temperature U, thereby achieving the decoupling of the micro-nano defect depth information (Z coordinate) and (X,Y) coordinates on the KDP crystal surface. The rest of this implementation scheme is the same as that of Specific Implementation Scheme Six.

[0084] Specific implementation plan eight: In step three, COMSOL Multiphysics is used as the solver to transform the dimensionally converted formula (6) into the standard partial differential equation form shown in formula (7). Formula (7) can be regarded as a transformation of the standard heat equation:

[0085]

[0086]

[0087] Where C is the diffusion coefficient and F is the source term. The dimensions of C and F are adjusted synchronously according to the dimension conversion of the defect contour depth information Z. The unit of C is nm. 2 / s is converted to W / (m·K), and the unit of F is converted from nm / s to W / m. 3 The F term in formula (7) can be considered as the source term, which will cause an overall shift in the defect depth but will not affect the change in the relative depth of the defect. Therefore, the effect of the F term can be ignored in the simulation process of the morphological evolution of DPN water-soluble repair of micro-nano defects on the KDP crystal surface. After formula (6) is converted into the standard thermal equation form shown in formula (7), formula (6) can be solved inversely by solving formula (7), thereby solving the problem that formula (6) cannot be solved directly. The rest of this implementation scheme is the same as that of the specific implementation scheme seven.

[0088] Specific implementation plan nine: In step four, the model boundary condition is set to zero flux, that is:

[0089]

[0090] Where n is the normal vector. The zero-flux boundary condition ensures that the temperature gradient at the boundary of the local growth mathematical model constructed in this invention is zero, avoiding distortion of the temperature value at the model boundary. The initial depth information of the defect is converted to the appropriate dimension and used as the initial value of the local growth mathematical model of the defect. Other aspects of this implementation scheme are the same as those in specific implementation scheme eight.

[0091] Specific Implementation Scheme Ten: In step five, the least squares method is used to interpolate and fit the simulation results and experimental data. The undetermined coefficients of the growth mathematical model are determined based on the minimum error between the simulation and experimental data. During fitting, it is assumed that the entire nanoscale scratch defect region on the KDP crystal surface simultaneously experiences the action of the AFM probe. The rest of this implementation scheme is the same as Specific Implementation Scheme One.

[0092] Example 1

[0093] To further illustrate the effectiveness of the method of the present invention, the following embodiments are used to explain the effects of the present invention.

[0094] A nanoscale scratch defect was selected on the surface of the KDP crystal, and then AFM scanning was used to obtain the cross-sectional contour information of the scratch, such as... Figure 3 As shown, the dimensions of the vertical and horizontal axes are transformed: the depth information is converted to temperature U, and the unit is adjusted from nm to K; the unit of the horizontal axis is adjusted from nm to m. The defect contour depth information after dimension transformation is used as the initial value of the aforementioned growth mathematical model.

[0095] Assuming the undetermined coefficient C ranges from [1, 80] W / (m·K), a parametric scanning solution was performed on the growth mathematical model, with a scanning step size of 1 W / (m·K). Defect depth information was then obtained through a DPN water-soluble repair experiment. The relative humidity was set to 75%, the scanning range to 2.5 μm × 2.5 μm, the AFM probe and the KDP crystal surface were in contact mode with a contact force of 1.5 μN, and the scanning frequency to 1.5 Hz. Repair data was collected approximately every 3 minutes, and the minimum depth of the repaired morphology cross-section was recorded. The parametric scanning results obtained from the numerical simulation were then interpolated and fitted using the experimental data. During fitting, it was assumed that the entire nanoscale scratch defect area on the KDP crystal surface simultaneously experienced the action of the AFM probe. The error between the interpolated theoretical data points and the experimental data points was evaluated using the least squares method. The fitting errors obtained with different undetermined coefficients C are shown below. Figure 4 As shown, the fitting error reached its minimum value of 0.0125534 when C=42.

[0096] Substituting C = 42W / (m·K) into the mathematical model for local defect growth, and comparing the obtained numerical simulation results with experimental results, such as... Figure 5 As shown, the results demonstrate that the defect local growth model proposed in this invention can simulate the evolution process of the DPN water-soluble repair morphology of micro- and nano-defects on the KDP crystal surface. Figure 6 As shown, the simulation results and experimental results of the cross-sectional profile of the scratch defect on the KDP crystal surface at different times during the DPN water-soluble repair process are compared. The two results show good consistency, which further proves the effectiveness of the present invention in simulating the morphological evolution of micro-nano defects on the KDP crystal surface during DPN water-soluble repair.

[0097] While the present invention has been disclosed above, its scope of protection is not limited thereto. Those skilled in the art can make various changes and modifications without departing from the spirit and scope of the present invention, and all such changes and modifications will fall within the scope of protection of the present invention.

Claims

1. A method for simulating the morphological evolution of DPN water-soluble repair of micro / nano defects on the surface of KDP crystals, characterized in that, Includes the following steps: Step 1: Establish a coordinate system and construct a mathematical model for the local growth of defects; Step 2: Transform the defect depth information to reduce the dimensionality of the model; Step 3: Convert the mathematical model of local defect growth into standard partial differential equation form; Step 4: Obtain the initial values ​​of the mathematical model for local defect growth and set the boundary conditions of the model; Step 5: Perform parametric scanning on the undetermined coefficients, fit the simulation results and experimental data, and determine the undetermined coefficients of the growth mathematical model based on the error between the simulation and experimental data; Step 6: Substitute the determined equation coefficients into the mathematical model of local defect growth to simulate the morphological evolution process of DPN water-soluble repair of micro-nano defects on the KDP crystal surface. The mathematical model for local defect growth, based on the Gibbs-Thomson law, assumes that the surface curvature of the KDP crystal defect will change the local equilibrium concentration of the solution, and that the supersaturation degree corrected for the local curvature is: （1） Where ω represents the molecular volume of the KDP crystal, in nm. 3 α represents surface free energy, with units of J / nm. 2 k represents Boltzmann's constant, in J / K, and T represents temperature, in K. R c The radius of curvature of the local micro / nano defects on the KDP crystal surface, in nm, is as follows: （2） If the model is based on the two-dimensional morphology of the defect, then If the model is based on the three-dimensional morphology of the defect, then ; The equilibrium concentration C of KDP crystal solution corrected by the local curvature of micro / nano defect surfaces ec for: （3） Where C e0 This represents the equilibrium saturation concentration of the KDP crystal solution. The concentration of the solution in the water film layer adsorbed on the surface of the KDP crystal element is always in a state of equilibrium saturation. Assuming the AFM probe causes a local increase in solution concentration, resulting in supersaturation λ, the final supersaturation of the AFM probe's action region is constructed as follows: （4） According to crystal growth theory, the growth and repair rate at any point on the micro / nano defect profile of the KDP crystal surface along a direction parallel to the crystal surface is: （5） Where A is the step dynamics coefficient of the growth mound on the surface of the KDP crystal, in nm / s; The growth and repair rate at this point along the Z-axis is: （6） Where p is the slope of the KDP crystal growth mound.

2. The method for simulating the morphological evolution of DPN water-soluble repair of micro / nano defects on the KDP crystal surface according to claim 1, characterized in that, The mathematical model for local growth of defects described in step one is based on the morphology of the defects. It assumes that the curvature of the defect surface will change the local solution equilibrium concentration, that the AFM probe will increase the local solution concentration, and that the solution concentration in the water film layer adsorbed on the surface of the KDP crystal element is always in equilibrium saturation during the DPN water-soluble repair process.

3. The method for simulating the morphological evolution of DPN water-soluble repair of micro / nano defects on the KDP crystal surface according to claim 2, characterized in that, The mathematical model for local growth of defects described in step one is based on the three-dimensional morphology of the defects. The X and Y axes of the model coordinate system are parallel to the crystal surface, and the Z axis is the direction of the defect depth.

4. The method for simulating the morphological evolution of DPN water-soluble repair of micro / nano defects on the KDP crystal surface according to claim 2, characterized in that, The mathematical model for local growth of defects described in step one is based on the two-dimensional morphology of the defects. The X-axis of the model coordinate system is parallel to the crystal surface, and the Z-axis is the direction of the defect depth.

5. The method for simulating the morphological evolution of DPN water-soluble repair of micro / nano defects on the KDP crystal surface according to claim 4, characterized in that, The defect depth information mentioned in step two is subjected to dimensional transformation. Specifically, the defect depth information with length as the dimension is converted into other physical quantities, and the units of other coordinate values ​​are scaled to make the physical quantities of the same order of magnitude so as to maintain the consistency of the equation dimensions.

6. The method for simulating the morphological evolution of DPN water-soluble repair of micro / nano defects on the KDP crystal surface according to claim 5, characterized in that, In step two, the defect depth information is transformed in terms of dimensions. Specifically, the defect depth information, which is in the dimension of length, is converted to temperature U, the unit of depth information is converted to K, and the unit of other coordinate values ​​X or (X, Y) is expanded to m.

7. The method for simulating the morphological evolution of DPN water-soluble repair of micro / nano defects on the KDP crystal surface according to claim 6, characterized in that, In step three, COMSOL Multiphysics is used as the solver to transform the dimensionally transformed formula (6) into the standard partial differential equation form shown in formula (7): （7） （8） Where C is the diffusion coefficient and F is the source term. The dimensions of C and F are adjusted synchronously according to the dimension conversion of the defect contour depth information Z. The unit of C is nm. 2 / s is converted to W / (m·K), and the unit of F is converted from nm / s to W / m. 3 .

8. The method for simulating the morphological evolution of DPN water-soluble repair of micro / nano defects on the KDP crystal surface according to claim 7, characterized in that, In step four, the model boundary condition is set to zero flux, that is: （9） Where n is the normal vector.

9. The method for simulating the morphological evolution of DPN water-soluble repair of micro / nano defects on the KDP crystal surface according to claim 1, characterized in that, In step five, the least squares method is used to interpolate and fit the simulation results and experimental data, and the values ​​of the undetermined coefficients of the growth mathematical model are determined based on the minimum error between the simulation and experimental data.

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