A method for printing a two-dimensional structure of thermosetting material with controllable surface roughness

By establishing a stable flow extrusion model and a fiber spreading and fusion dynamics model, the problems of flow extrusion instability and difficulty in adjusting surface roughness in the printing process of thermosetting materials are solved, realizing high-precision printing of different materials and substrates, which is suitable for the manufacture of flexible sensors and biomedical devices.

CN121316255BActive Publication Date: 2026-04-17ZHEJIANG UNIV
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Patent Information

Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
ZHEJIANG UNIV
Filing Date
2025-12-16
Publication Date
2026-04-17

AI Technical Summary

Technical Problem

Existing thermosetting material printing processes suffer from problems such as unstable flow extrusion, uncontrollable spreading and fusion behavior, and difficulty in adjusting surface roughness, resulting in deviations between the printed morphology and the design path, making it difficult to achieve high-precision deposition molding. Furthermore, there is a lack of programmable control methods applicable to various material systems and substrate types.

Method used

By establishing a stable flow extrusion model for thermosetting materials, a fiber spreading and fusion dynamics model, and an interface curing and locking mechanism, continuous programmable adjustment of surface roughness is achieved. This includes a stable flow extrusion stage, a printing parameter determination stage, a spreading and fusion control stage, and an interface curing and locking stage, which is applicable to any thermosetting material system and substrate.

Benefits of technology

It achieves unified process description and scalable printing control for different thermosetting material systems, enabling continuous adjustment of surface roughness on different substrates, improving the accuracy and adaptability of the printing process, and is suitable for flexible sensors, flexible actuators and biomedical functional interfaces.

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Abstract

This invention discloses a method for printing two-dimensional structures of thermosetting materials with controllable surface roughness, comprising: controlling the rheological properties of the extrusion behavior of a selected thermosetting material; calculating and determining the relationship between the deposition morphology of the thermosetting material and printing parameters, and plotting an H-V phase diagram to determine the critical printing range for achieving uniform deposition; establishing a single-fiber deposition morphology evolution model and a multi-fiber fusion geometric evolution model to provide a correlation between the degree of fusion of adjacent deposited fibers and the surface micro-roughness, thereby obtaining a preset surface microstructure morphology; and, according to the required surface roughness, solidifying and locking the interface through thermal curing at different fusion stages to obtain a continuously adjustable and structurally stable two-dimensional film of thermosetting material. This invention can realize the printing of two-dimensional structures of any thermosetting material system on any spread substrate, and can continuously and programmably adjust the surface roughness.
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Description

Technical Field

[0001] This invention relates to the field of additive manufacturing technology for thermosetting materials, and in particular to a method for printing two-dimensional structures of thermosetting materials with controllable surface roughness. Background Technology

[0002] Thermosetting silicone has wide applications in flexible electronics, soft robotics, and biomedical devices. Its excellent deformability, low modulus, and good interfacial compatibility make it an important basic material for realizing flexible functional structures. With the development of additive manufacturing technology, the processing method of thermosetting materials based on ink-to-ink extrusion printing has become the main means of constructing two-dimensional thin films and three-dimensional structures.

[0003] For example, Chinese patent document CN118181741A discloses a silicone layer-by-layer curing 3D printer and printing method; Chinese patent document CN107053662A discloses a photocurable silicone 3D printing device and printing method.

[0004] However, the following technical problems still exist in the existing thermosetting material printing process:

[0005] First, the rheological properties of silicone systems vary significantly. Different materials are prone to flow instability, filament breakage, or flow stagnation during the printing process, resulting in deviations between the printed morphology and the design path, making it difficult to achieve high-precision deposition molding.

[0006] Secondly, the dynamic behavior of deposited fibers during the spreading and fusing stages is influenced by multiple factors, including material viscosity, surface tension, substrate wettability, and ambient temperature. Consequently, the surface morphology and degree of fusing lack predictable and controllable mechanisms. Existing methods largely rely on empirical parameter adjustments, making it difficult to establish universally applicable parameter mapping models.

[0007] Furthermore, the surface roughness of two-dimensional silicone structures has a significant impact on device performance, affecting sensitivity in flexible sensors and cell adhesion and tissue compatibility in biomedical interfaces. However, in traditional printing processes, the fusion state of deposited fibers is usually fixed, lacking precise control over surface micro-roughness.

[0008] In summary, current technologies lack a printing method applicable to various thermosetting material systems, compatible with multiple substrate types, and capable of continuously adjustable surface roughness. Therefore, there is an urgent need to propose a universal and programmable thermosetting material printing strategy to achieve extrusion stability, spread controllability, and precise locking of the interface curing stage during the printing process, thereby obtaining a two-dimensional thermosetting material structure with adjustable roughness. Summary of the Invention

[0009] To overcome the problems of unstable flow extrusion, uncontrollable spreading and fusion behavior, and difficulty in adjusting surface roughness in the printing process of existing thermosetting materials, this invention provides a two-dimensional structure printing method for thermosetting materials with controllable surface roughness. It is mainly applicable to ink direct writing printing. By establishing a stable flow extrusion model of thermosetting materials, a fiber spreading and fusion dynamics model, and an interface curing and locking mechanism, it is possible to print two-dimensional structures of any thermosetting material system on any spreading substrate, and the surface roughness can be continuously and programmably adjusted.

[0010] A method for printing two-dimensional structures of thermosetting materials with controllable surface roughness includes the following steps:

[0011] (1) Stable flow extrusion stage: Select thermosetting materials and substrates, and regulate the extrusion rheological behavior of thermosetting materials to keep the thermosetting materials in a continuous and stable flow extrusion state during the printing process;

[0012] (2) Printing parameter determination stage: Determine the relationship between the deposition morphology of thermosetting materials and printing parameters, based on the dimensionless height parameter in the printing parameters. and dimensionless velocity parameters ,draw Phase diagrams are used to determine the critical printing range for achieving uniform deposition;

[0013] (3) Spreading and fusion control stage: Under the set printing path, printing parameters and substrate environment conditions, establish a single fiber deposition morphology evolution model and a multi-fiber fusion geometric evolution model to provide a correlation between the degree of fusion of adjacent deposited fibers and the surface micro-roughness, thereby obtaining the preset surface microstructure morphology.

[0014] (4) Interface curing and locking stage: According to the required surface roughness, the interface is cured and locked by thermal curing at different times in the fusion stage to obtain a thermosetting material two-dimensional film with continuously adjustable roughness and stable structure.

[0015] In step (1), the selected thermosetting material is a temperature-dependent material, such as silicone rubber, liquid metal, or hydrogel.

[0016] In step (1), the selected substrate is metal, glass, polymer film, superspreadable surface or biological tissue surface, etc.

[0017] In step (1), the extrusion rheological properties of the thermosetting material are controlled, specifically including adjusting at least one of the following: nozzle diameter, air pressure, temperature, shear rate, material ratio and solvent content.

[0018] In step (2), the dimensionless height parameter and dimensionless velocity parameters They are defined as follows:

[0019] ;

[0020] ;

[0021] In the formula, The height from the nozzle to the substrate. Nozzle diameter, For nozzle printing speed, This refers to the material extrusion speed.

[0022] In step (3), when establishing the single fiber deposition morphology evolution model, it is assumed that the fiber deposition on the substrate is in the form of an arc. The contact angle and baseline length of a single fiber are recorded from the moment it falls onto the spreading substrate, as well as the relationship between time and substrate environment, by means of contact angle measurement or CCD camera monitoring. The spreading coefficient is defined as the ratio of the baseline length at any time to the initial baseline length to characterize the spreading dynamic characteristics. The substrate environment includes temperature, humidity, substrate surface energy, substrate surface roughness, or light intensity, etc.

[0023] When selecting a temperature for the substrate environment, the evolution model of the depositional morphology of a single fiber is based on a three-dimensional function of viscosity-time-temperature, and the specific establishment process is as follows:

[0024] Viscosity evolution was fitted using the double Arrhenius equations, and the rheological solidification trajectory of the material under different conditions was obtained through numerical integration:

[0025] ;

[0026] in, It is temperature Under the condition of time Changing viscosity, yes The calculated viscosity, It is the activation energy of viscosity. for The calculated apparent dynamic factor, It is the ideal gas constant. It is the gelation activation energy;

[0027] Next, viscosity and dynamic contact angle were compared. Establish coupling relationship,

[0028] ;

[0029] in, Steady-state contact angle , It is the initial contact angle. , It is a time constant. The viscosity constant is used for dimensionless determination;

[0030] Further, the spreading coefficient over time was obtained. With temperature Evolutionary laws :

[0031] ;

[0032] Among them, the spreading coefficient Defined as the ratio of the baseline length at any given time to the baseline length at the initial time, it is used to characterize the dynamic properties of fiber spreading.

[0033] In step (3), a geometric evolution model for the fusion of multiple fibers is established, specifically including:

[0034] By characterizing the geometric morphology and descriptive parameters of multiple deposited fibers before and after fusion under different substrate environments and curing conditions, a model of the degree of fiber fusion and geometric evolution process is constructed.

[0035] By adjusting the printing path, substrate environment, and printing parameters such as nozzle movement speed, material extrusion speed, and / or external field control conditions, the fiber spreading coefficient and fusion degree can be controlled, thereby achieving controllable adjustment of surface roughness. The adjustment range of surface roughness includes various surface structures from continuous dense and smooth surfaces to those with periodic structures, ordered microstructures, or random micro-protrusion morphologies.

[0036] In step (4), the different fusion stages include: initial contact, equal arc fusion, equal height trapezoid and final flattening.

[0037] In this invention, the fiber fusion and interface curing process can be achieved through multi-physics field-assisted regulation, including but not limited to temperature, light, electric, or flow fields. Different physical fields acting on the deposited fibers during the printing process can alter their spreading dynamics and interface curing rate, thereby achieving spatiotemporal controllability of the fusion rate and surface roughness. The multi-physics field regulation methods can work independently or collaboratively to achieve adaptive printing control for different thermosetting material systems and different substrate environments.

[0038] Compared with the prior art, the present invention has the following beneficial effects:

[0039] 1. High versatility: Through dimensionless parameterization and spreading dynamics modeling, it achieves unified process description and scalable printing control for different thermosetting material systems.

[0040] 2. Programmable roughness control: Through dynamic control and interface curing locking strategies during the fiber fusion stage, continuous adjustment from smooth surfaces to microstructured rough surfaces can be achieved.

[0041] 3. High interface adaptability: The method does not depend on a specific substrate and can print stably under different wetting and temperature conditions.

[0042] 4. Predictable and Designable Process: By establishing the H-V phase diagram and spreading model, quantitative prediction of printing morphology and roughness is achieved.

[0043] 5. Wide range of applications: Applicable to the fabrication of flexible sensors, flexible actuators and biomedical functional interfaces, providing a new technical route for the manufacturing of multifunctional thermosetting material devices. Attached Figure Description

[0044] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0045] Figure 1 This is a model describing the extrusion behavior of ultra-soft silicone in an embodiment of the present invention.

[0046] Figure 2 This is a schematic diagram of curing degree fitting in an embodiment of the present invention.

[0047] Figure 3 This is a schematic diagram of dynamic contact angle fitting on a metal substrate in an embodiment of the present invention.

[0048] Figure 4 This is a schematic diagram of dynamic contact angle fitting on a silicone substrate in an embodiment of the present invention.

[0049] Figure 5 This is a schematic diagram of the fitting of the spreading coefficient on the metal substrate in an embodiment of the present invention.

[0050] Figure 6 This is a schematic diagram of the fitting of the spreading coefficient on the silicone substrate in an embodiment of the present invention.

[0051] Figure 7 This is a geometric evolution model for the fusion of parallel fibers into a thin film in an embodiment of the present invention.

[0052] Figure 8 This is a schematic diagram of the printing path and the film formation result in an embodiment of the present invention.

[0053] Figure 9 The roughness results are shown in the embodiments of the present invention. Detailed Implementation

[0054] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0055] It should be noted that, unless otherwise specified, the features in the following embodiments and implementation methods can be combined with each other.

[0056] A method for printing two-dimensional structures of thermosetting materials with controllable surface roughness includes the following steps:

[0057] (1) Steady flow extrusion stage

[0058] Representative low-modulus soft elastomers (such as Ecoflex, PDMS, hydrogels, or liquid metallic elastomers) were selected as printing materials (Ecoflex00-20 was selected in this embodiment). First, the viscosity-shear rate relationship of the material was determined through rheological testing, and its yield stress and thixotropic recovery characteristics were examined. Based on the test results, the temperature, shear rate, component ratio, and solvent content were adjusted to achieve a continuous and stable extrusion state during the printing process.

[0059] In practical operation, an extrusion nozzle system equipped with air pressure control and a rapid shut-off valve was used. By adjusting the nozzle diameter and air pressure, a phase diagram relating material extrusion behavior to process parameters was established. The results show that ( Figure 1 As shown, the nozzle diameter and extrusion pressure together determine the continuity of the fiber: too fine a nozzle or too low a pressure can easily lead to fiber breakage, while too high a pressure can cause material accumulation. By selecting a suitable nozzle size (such as 22G) and extrusion pressure (200–300 kPa), a stable and continuous cylindrical extruded fluid can be obtained, achieving stable flow extrusion.

[0060] (2) Printing parameter determination stage

[0061] After determining stable extrusion conditions, adjust the nozzle movement speed. Nozzle-base height Extrusion speed With nozzle diameter A model was constructed to establish the relationship between the deposition morphology of thermosetting materials and printing parameters.

[0062] A dimensionless height parameter was defined, and the printing states for different parameter ranges were classified using the H-V phase diagram. The results show that:

[0063] When V < 0.35, the material is prone to accumulation;

[0064] When H is small and V is too low, the fiber is dragged by the nozzle to produce a coating state;

[0065] When H and V increase simultaneously, a uniform and continuous depositional morphology can be obtained.

[0066] This determines the critical printing range for achieving continuous spreading and stable forming.

[0067] (3) Spreading and Integration Control Stage

[0068] When establishing a model for the evolution of depositional morphology of a single fiber, the relationship between the contact angle and baseline length of a single fiber and the substrate environment is recorded from the moment it falls onto the spreading substrate, using contact angle measurement or CCD camera monitoring. The spreading coefficient is defined as the ratio of the baseline length at any time to the initial baseline length to characterize the spreading dynamics. The substrate environment can be selected as temperature, humidity or light intensity.

[0069] In this embodiment, the substrate environment is selected based on temperature, and the evolution model of single fiber deposition morphology is based on a three-dimensional function of viscosity-time-temperature. The specific establishment process is as follows:

[0070] Viscosity evolution was fitted using the double Arrhenius equations, and the rheological solidification trajectory of the material under different conditions was obtained through numerical integration. Experimental data were obtained using temperature-scanning rheological tests.

[0071] ;

[0072] in, It is temperature Under the condition of time Changing viscosity, yes The calculated viscosity, It is the activation energy of viscosity. for The calculated apparent dynamic factor, It is the ideal gas constant. This is the gelation activation energy. A schematic diagram of the curing degree fitting is shown below. Figure 2 As shown.

[0073] Next, viscosity and dynamic contact angle were compared. Establish coupling relationship:

[0074] ;

[0075] in, It is the steady-state contact angle. It is the initial contact angle. It is a time constant. The viscosity constant is used for dimensionless conversion. In this embodiment, a schematic diagram of the dynamic contact angle fitting on the metal substrate is shown below. Figure 3 As shown in the figure, a schematic diagram of dynamic contact angle fitting on the silicone substrate is as follows. Figure 4 As shown.

[0076] Further, the spreading coefficient over time was obtained. With temperature Evolutionary laws :

[0077] ;

[0078] Among them, the spreading coefficient Defined as the ratio of the baseline length at any given time to the baseline length at the initial time, it is used to characterize the dynamic properties of fiber spreading. In this embodiment, a schematic diagram of the spreading coefficient fitting on the metal substrate is shown below. Figure 5 As shown in the figure, the schematic diagram of the spreading coefficient fitting on the silicone substrate is as follows. Figure 6 As shown.

[0079] When establishing a geometric evolution model for the fusion of multiple fibers, the fiber spreading coefficient and fusion degree are controlled by adjusting the nozzle moving speed, material extrusion speed, printing path and substrate environment, so as to achieve controllable adjustment of surface roughness. The roughness control range includes adjustable surface morphology from continuous dense and smooth surface to periodic or random micro-protrusion structure.

[0080] (4) Interface solidification and locking stage

[0081] Through dual-fiber fusion experiments, the morphological evolution of adjacent deposited fibers under different spacing and temperature conditions was observed, and a multi-stage fiber fusion geometric model was proposed, such as... Figure 7 As shown, the initial state is one of parallel fiber contact. Even if the fibers are not in contact initially, the shape at the time of fiber contact can be deduced based on the fiber morphology evolution model, and they will eventually become a state of fiber contact. The fusion process can be divided into four stages: initial contact, equal-arc fusion, equal-height trapezoid, and final flattening. For ease of calculation, this invention introduces two intermediate states ( Figure 7 In section ii and iii), the fibers begin to level and fuse upon contact, thus resulting in... Figure 7 In section ii, the sizes of S1 and S2 determine the height change after merging. After merging, it will become a circular arc of equal width, at which point the contact angle is defined as... Since there are currently two fiber assumptions, as the number of fibers increases, the upper surface of the arc can be approximated as an infinitely long straight line, thus approximating a trapezoid. Therefore, a trapezoid is used to describe the fused shape. After becoming a trapezoid, it will continue to flow, eventually reaching the final flattened trapezoidal state. Figure 7(vi) The model introduces a dimensionless fiber spacing S. When S > 1.8, the fibers only slightly contact to form a smooth surface; when S < 1.6, the fibers over-fusion occurs, and the surface tends to be flat; in the interval between the two, the surface roughness can be continuously adjusted by controlling S, H, V, and the curing rate. Taking the thin film structure as an example, the thickness of the third-stage film can be derived as follows:

[0082] ;

[0083] in, It is the third stage film thickness. It is the angle between the base and the hypotenuse of the trapezoid at this stage. The cross-sectional area of ​​the fiber is... This is the length of the base of the trapezoid at this stage.

[0084] The thickness of the film in the fourth stage is:

[0085] ;

[0086] in, It is the fourth stage of film thickness. It is the angle between the base and the hypotenuse of the trapezoid at this stage. This is the length of the base of the trapezoid at this stage. This represents the initial fiber length.

[0087] By employing thermal, optical, or chemical crosslinking techniques to achieve "interface locking" at different fusion stages, it is possible to obtain macroscopically different geometric profiles and continuously tunable surface microstructures ranging from smooth to rough. Experimental results show that smooth films with Ra of approximately 5 nm can be obtained under non-heating conditions, while controllable rough structures with Ra of approximately 1–10 μm can be achieved by heating to 60–80 °C or by delayed curing. Figure 8 This invention illustrates a schematic diagram of the printing path and the film formation results in an embodiment of the invention. Figure 9 The film roughness results in the embodiments of the present invention are shown.

[0088] The method of this invention is not dependent on a specific material system and is applicable to a variety of thermosetting materials such as silicone rubber, hydrogels, and liquid metals. The type of printing substrate is also unrestricted, and stable deposition and film formation can be achieved on metals, glass, polymer films, superhydrophobic / superhydrophilic surfaces, or biological tissue models.

[0089] For highly wettable substrates, temperature gradients or photocuring are used to control and suppress overspreading; for low-wetting substrates, spreading ability is enhanced by increasing temperature or introducing surface energy modifiers or superwetting surfaces. The contact angle evolution trend of different substrates can be uniformly regularized by controlling the external field, thereby ensuring predictable roughness output.

[0090] This invention can be applied to fields such as flexible sensors, flexible actuators, and functional modulation of biomedical implantable devices. For example, in the manufacture of flexible capacitive sensors, the dielectric layer thickness constructed using this printing method can be continuously adjusted between 80–1000 μm, with a surface roughness Ra of approximately 5 nm, effectively improving the sensitivity and stability of the device.

[0091] The embodiments described above provide a detailed explanation of the technical solutions and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, additions, and equivalent substitutions made within the scope of the principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for printing two-dimensional structures of thermosetting materials with controllable surface roughness, characterized in that, Includes the following steps: (1) Stable flow extrusion stage: Select thermosetting materials and substrates, and regulate the extrusion rheological behavior of thermosetting materials to keep the thermosetting materials in a continuous and stable flow extrusion state during the printing process; (2) Printing parameter determination stage: Determine the relationship between the deposition morphology of thermosetting materials and printing parameters, based on the dimensionless height parameter in the printing parameters. and dimensionless velocity parameters ,draw Phase diagrams were used to determine the critical printing range for achieving uniform deposition; dimensionless height parameters were also used. and dimensionless velocity parameters They are defined as follows: ; ; wherein is the nozzle to substrate height, is the nozzle diameter, is the nozzle print movement speed, is the material extrusion speed; (3) Spreading and fusion control stage: Under the set printing path, printing parameters and substrate environment conditions, establish a single fiber deposition morphology evolution model and a multi-fiber fusion geometric evolution model to provide a correlation between the degree of fusion of adjacent deposited fibers and the surface micro-roughness, thereby obtaining the preset surface microstructure morphology. (4) Interface curing and locking stage: According to the required surface roughness, the interface is cured and locked by thermal curing at different times in the fusion stage to obtain a thermosetting material two-dimensional film with continuously adjustable roughness and stable structure.

2. The method according to claim 1, wherein In step (1), the selected thermosetting material is silicone rubber, liquid metal or hydrogel.

3. The method according to claim 1, wherein In step (1), the selected substrate is metal, glass, polymer film, superspreadable surface or biological tissue surface.

4. The method according to claim 1, wherein In step (1), the extrusion rheological properties of the thermosetting material are controlled, specifically including adjusting at least one of the following: nozzle diameter, air pressure, temperature, shear rate, material ratio and solvent content.

5. The method according to claim 1, wherein In step (3), when establishing the single fiber deposition morphology evolution model, it is assumed that the fiber deposition on the substrate is in the shape of an arc. The contact angle and baseline length of a single fiber are recorded with time and substrate environment from the moment it falls onto the spreading substrate. The spreading coefficient is defined as the ratio of the baseline length at any time to the initial baseline length to characterize the spreading dynamic characteristics. The substrate environment includes temperature, humidity, substrate surface energy, substrate surface roughness or light intensity.

6. The method according to claim 5, wherein The substrate environment is selected based on temperature. The evolution model of single-fiber deposition morphology is based on a three-dimensional function of viscosity-time-temperature. The specific establishment process is as follows: Viscosity evolution was fitted using the double Arrhenius equations, and the rheological solidification trajectory of the material under different conditions was obtained through numerical integration: ; in, It is temperature Under the condition of time Changing viscosity, yes Calculated viscosity, It is the activation energy of viscosity. for The calculated apparent dynamic factor, It is the ideal gas constant. It is the gelation activation energy; The viscosity is then correlated with the dynamic contact angle Coupling relationships are established, ; in, Steady-state contact angle , It is the initial contact angle. , It is a time constant. The viscosity constant is used for dimensionless determination; Further, the spreading coefficient over time was obtained. With temperature Evolutionary laws : ; where the spreading coefficient defined as the ratio of the baseline length at any time to the baseline length at the initial time, is used to characterize the dynamic behavior of fiber spreading.

7. The method according to claim 5, wherein In step (3), a geometric evolution model for the fusion of multiple fibers is established, specifically including: By characterizing the geometric morphology and descriptive parameters of multiple deposited fibers before and after fusion under different substrate environments and curing conditions, a model of the degree of fiber fusion and geometric evolution process is constructed. By adjusting the printing path, substrate environment, and printing parameters such as nozzle movement speed, material extrusion speed, and / or external field control conditions, the fiber spreading coefficient and fusion degree can be controlled, thereby achieving controllable adjustment of surface roughness. The adjustment range of surface roughness includes various surface structures from continuous dense and smooth surfaces to those with periodic structures, ordered microstructures, or random micro-protrusion morphologies.

8. The method according to claim 1, wherein In step (4), the different fusion stages include: initial contact, equal arc fusion, equal height trapezoid and final flattening.

Citation Information

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