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−	<span id='_GoBack'></span>	+	<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
		+	<big>'''Quantum dynamics of dissipative Chern insulator'''</big></div>
−	==Title Page==	+	<div id="OLE_LINK26" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
−		+	Jilian Zhonga<sup> *</sup>, Xiaoyue Li<sup>a</sup></div>
−	<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">	+
−	<big>'''Fragmentation to the Holistic: A Study of the Inner Logic of Smart Governance of Public Cultural Space'''</big></div>	+
		+	''<sup>a</sup> Department of Physics, Jiangsu University, Zhenjiang 212013, People’s Republic of China''
		+	<span id='OLE_LINK28'></span>''<sup>*</sup>Corresponding address: [mailto:zhongjilian0505@163.com zhongjilian0505@163.com]''
	-->==Abstract==		-->==Abstract==
−	For open quantum systems, a short-time evolution is usually well described by the effective non-Hermitian Hamiltonians, while long-time dynamics requires the Lindblad master equation, in which the Liouvillian superoperators characterize the time evolution. In this paper, we constructed an open system by adding suitable gain and loss operators to the Chern insulator to investigate the time evolution of quantum states at long times by numerical simulations. Finally, we also propose a topolectrical circuits to realize the dissipative system for experimental observation. It is found that the opening and closing of the Liouvillian gap leads to different damping behaviours of the system and that the presence of non-Hermitian skin effects leads to a phenomenon of chiral damping with sharp wavefronts. Our study deepens the understanding of quantum dynamics of dissipative system.	+	This research is aimed at Laser processing of micro holes there are recast layer and microcracks, analyzed the laser processing of recast layer formation mechanism and surface geometry features not, and for the laser processing of the formation of the recast layer in the process of analog simulation, for electrochemical finishing processing to remove the recast layer to provide a theoretical basis. The laser-electrolysis asynchronous composite processing experimental system was developed, and the electrolytic removal of recast layer on the wall of laser perforated holes and the process research of aperture trimming were carried out, which mainly analyzed the effects of electrolysis voltage, electrolyte and pulse electrolysis parameters on the rate of electrolysis and the effect of the removal of the recast layer. The cathode lifting method is also proposed to lift the tool cathode for orifice shaping after effective removal of the recast layer. The experimental results show that: when the current density is greater than 10.09 A/cm<sup>2</sup>, the electrolysis rate is proportional to the voltage; when the constant voltage electrolysis, the mixed solution of NaNO<sub>3</sub> and NaCl enhances the effect of the recast layer removal significantly, and relative to the single solution with the same electrical conductivity, the value of the change in the surface roughness of the workpiece before and after the finishing process is enhanced more significantly, and there is no significant difference in the processing efficiency; the low-frequency wide pulse and high-frequency narrow pulse in the pulse electrolysis processing is favorable for the removal of recast layer. Pulse, high-frequency narrow pulse is conducive to the improvement of electrolysis rate, and the frequency of 1 kHz, the duty cycle of 40% to 60% of the electrolysis rate is the largest; at the same time, the use of cathodic enhancement method of aperture trimming processing, the use of the truncation effect, through the appropriate electrolytic processing parameters, to achieve the complete removal of the laser hole hole wall recast layer at the same time to complete the aperture trimming processing, to meet the demand for the use of performance of the small holes to further enhance the processing efficiency and small holes, and to improve the processing efficiency of the laser hole. Enhance the processing efficiency and functionality of small holes.
−	'''Keywords''': Open quantum system, chiral damping, topolectrical circuits	+	'''Keywords''': Laser processing, recast layer simulation, electrochemical finishing, cathodic lift
−		+
−	Jilian Zhong  Department of Physics, Jiangsu University, Zhenjiang 212013, People’s Republic of China zhongjilian0532@163.com	+
−		+
−	Xiaoyue Li  Department of Physics, Jiangsu University, Zhenjiang 212013, People’s Republic of China	+
−		+
−	DOI: 10.23967/j.rimni.2024.05.008	+
	==1. Introduction==		==1. Introduction==
−	With the laboratory advances in modulating dissipation and quantum coherence,the theory of open and nonequilibrium systems has received renewed attention [1,2]. Non-Hermitian Hamiltonians have been used to describe a large number of non-conservative systems, such as classical waves with gain and loss [3-8], solids with finite quasiparticles lifetimes [9-11], and open quantum systems [12-14]. The unique features of non-Hermitian systems have been recognized in a variety of physical settings, in particular the non-Hermitian skin effect (NHSE) [15,16], where the eigenstates of the system are exponentially localized on the boundary. In recent years, the impact of NHSE has been extensively studied [17-29].	+	In the field of modern machinery and equipment, the application of microporous in enhancing the use of equipment performance and reliability and other aspects of the value is increasingly apparent. For example, in the key high-temperature components of aircraft engines, the application of microporous promotes the development of air film cooling technology, and effectively make up for the limitations of high-temperature-resistant materials to enhance the reliability and durability of the engine; in the field of internal combustion engines, the application of microporous design of injector nozzles not only significantly improves the effect of fuel atomization, but also plays an important role in improving combustion efficiency and effectively reducing harmful emissions. Therefore, the application of microporous not only promotes the development of greening technology, but also promotes the innovation of microporous machining technology and process, which makes the special machining technology such as electric discharge technology (EDM) [1], electrochemical machining (ECM) [2], and laser machining (LM) [3] become the mainstream microporous machining methods.
−	NHSE was also found in open quantum systems [30]. For open quantum systems, the non-Hermitian effective Hamiltonian describes the time evolution of the wavefunction under post-selection conditions, while the time evolution of the density matrix (without post-selection) is driven by the Liouvillian superoperator in the master equation [2,31-33]. It has been found that the Liouvillian superoperator can also exhibit non-Hermitian skin effects and that such effects can significantly affect the dynamical behaviour of the system at long times [30,34-43]. In a large class of open quantum systems, the quantum state in the long time limit converges to the steady state by algebraic damping under periodic boundary conditions and exponential damping under open boundary conditions [30].	+	However, micro-hole machining technology based on special machining methods such as EDM, laser machining and electron beam is also facing some technical challenges. Due to the existence of the recast layer, the changes in its composition, structure, hardness and strength affect the performance of micro-holes in use, so scholars at home and abroad have carried out a wide range of white research in the removal of the recast layer and its effects. On the one hand, some scholars have deeply studied the formation mechanism of the recast layer, and while revealing the physical and chemical processes of the formation of the recast layer, they have also proposed the introduction of new processing media (e.g., nano-fluid, carnauba oil, etc.) [4,5] to adjust the processing environment [6,7] (e.g., temperature, pressure, etc.) and the development of new machining tools [8,9] and other means, which have effectively reduced the thickness of the recast layer; on the other hand, some scholars have reduced the formation and thickness of the recast layer by optimizing the process parameters [10,11] (e.g., current, voltage, and processing speed, etc.); in addition, some scholars have utilized post-processing techniques (e.g., chemical etching [12], ion-beam polishing [13], and electron-beam evaporation [14], as well as femtosecond laser technology, etc.) to achieve efficient removal of the recast layer and improve the microporosity. efficient removal and improved processing quality of micro-holes.
−	In recent years, it has been discovered that topolectrical circuits can be used as platform to simulate the lattice systems, thus enabling the study of topological states in topolectrical circuits and gradually developing the field of topological circuitry [44-46]. Some of the early experiments and theories were extensively studied in Hermitian systems [45,47]. Since the phenomena of non-Hermitian systems are more rich than that of Hermitian systems, increasing attentions are contributed into the non-Hermitian physics, and some interesting phenomena have also been realized by topolectrical circuits [48-52].	+	Despite this, some major problems remain. The formation of recast layers involves complex physical and chemical processes, including melting, solidification, and phase transformation of the material. These processes are influenced by a variety of factors, such as processing temperature, pressure, material composition etc. Their complexity makes it difficult in controlling the formation of recast layers. During micro-hole machining, the optimization of machining parameters is crucial to reduce the recast layer. However, finding the best combination of parameters is a challenging task due to the diversity of machining parameters and the coupling of interactions. In addition, the optimal parameters may be different for different materials and machining conditions, which further increases the difficulty of parameter optimization. Limitations of post-processing technologies: chemical etching may introduce new surface defects or contaminants, while ion and electron beams may be limited by equipment cost and operability, resulting in restricted form geometry of the processed object. The problem of the balance between processing efficiency and quality. In the process of micro-hole machining, there is a certain contradiction between improving the processing efficiency and guaranteeing the processing quality, in order to reduce the formation of recast layer, it may be necessary to reduce the processing speed or processing steps, but this will sacrifice the processing efficiency. Therefore, based on the characteristics of electrochemical light finishing processing, this paper proposes the concept of using electrochemical anodic dissolution to remove the recast layer, and the process method of electrochemical removal of recast layer is studied. Finally, a two-step lifting process method is proposed to complete the micro-hole orifice trimming processing on the basis of satisfying the high quality removal of the recast layer, so that it can meet the specific use performance requirements.
−		+	==2. Experimental programming==
−	Previous studies on open quantum dynamics and topolectrical circuits have mainly focused on one-dimensional non-Hermitian models, and relatively few studies on higher-dimensional non-Hermitian models. In this paper, we consider a two-dimensional open quantum system based on Chern insulators. Following the method developed in Song et al. [30], we study the dynamics of this system in terms of the damping matrix derived from the Liouvillian superoperator, and give a model of topolectrical circuit realization of the damping matrix based on Kirchhoff’s theory. It is found that due to the NHSE of the damping matrix, the long-time dynamics of the system under open boundary conditions is significantly different from that under periodic boundary conditions.	+	===2.1 Formation mechanism and simulation of recast layer and its surface geometric characteristics===
−		+	====2.1.1 Formation mechanism and simulation of recast layer====
−	Our paper is organized as follows: in section 2, we briefly review the general framework on how to convert Liouvillian operators with linear jumps to non-Hermitian damping matrix. In sections 3 and 4, we compute and numerically simulate the long-time evolution of the model. In section 5, we give the circuit model of the non-Hermitian damping matrix . Finally, we conclude in section 6.	+	Laser processing is the result of the superposition of multiple single molten pools, so the recast layer formed by laser processing is inextricably linked to the formation of the single molten pool recast layer, the basic process of which is shown in [[Draft Seryazdan 514767809#img-1|Figure 1]](a). The material under a single pulsed laser irradiation is rapidly heated to the melting and boiling points, and melting and vaporization occur, forming a solid, liquid, and vapor three-phase coexistence. The melt flows along the walls of the processed area under a combination of vaporizing pressure and thermal influences, resulting in the formation of a concave contour of the melt pool that resembles a replica of the laser energy distribution curve. At the end of a single laser pulse, the vaporized metal in the center of the melt pool releases a large amount of thermal energy, which causes the melt attached to the walls of the melt pool and the curved melt at the bottom to be subjected to the recoil pressure of the metal vapor jet directly, which leads to the splashing of melt in the melt pool. Among them, part of the melt splashes out of the melt pool to form melt spatter, and accumulates at the edge of the melt pool to form a flying edge or burr, the simulation results are shown in [[Draft Seryazdan 514767809#img-1|Figure 1]](b); the other part of the melt will be deposited and attached to the surface of the inner wall of the melt pool,, and under the action of the combined coupling force, the melt will flow downward along the wall surface and reach a static equilibrium to form a smooth and excessive linear melt. After rapid cooling, the surface of the single molten pool forms a recast layer covered by a fused cladding after being subjected to a pressure gradient and a temperature gradient. The final laser process consists of the superposition of individual melt pools in the axial direction of the processing area and the formation of a recast layer with a periodic stream-like distribution on the surface of the hole wall.<div id="img-1"></div>
−	==2. General formalism of damping matrix==	+	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
−	An open quantum system undergoing Markovian damping satisfies the Lindblad master equation	+	|- style="background:white;"
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+	| style="text-align: center;padding:10px;" | [[Image:Draft_Seryazdan_636830651-image1.png|486px|link=https://www.scipedia.com/public/File:Draft_Seryazdan_636830651-image1.png]]
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
	|-		|-
−	|   <math>\frac{d\rho }{dt}=-i[H,\rho ]+\sum \left(2L_{\mu }\rho L_{\mu }^{\dagger }-\right. </math><math>\left. \lbrace L_{\mu }^{\dagger }L_{\mu },\rho \rbrace \right),</math>	+	| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;" | '''Figure 1'''. Schematic diagram of the simulation of recast layer formation
	|}		|}
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (1)	+	====2.1.2 Surface geometry of the recast layer====
−	|}where  <math>\rho </math> is the density matrix of the system,  <math>H</math> is the Hamiltonian that represents unitary evolution of the system, and  <math>L_{\mu }</math> are Lindblad dissipation operators describing the quantum jumps induced by the coupling to the environment. The above equation can be abbreviated as  <math>\frac{d\rho }{dt}=L\rho </math>, where  <math>L</math> is called the Liouvillian superoperator. By regarding the density matrix  <math>\rho </math> as a vector that consists of matrix elements  <math>{\rho }_{i,j}</math>,  <math>L</math> is represented as a matrix whose elements are given by [53]	+	As shown in [[Draft Seryazdan 514767809#img-2|Figure 2]](a), at the surface of the initial melt pool, due to the large temperature gradient, the overflowed melt is mainly subject to Marangoni force, surface tension and gravity. The Marangoni force affects the flow of the melt, thus forming the shape of a domed dome; the surface tension and gravity in turn cause the overflowed melt to shrink outside the hole to form a ring-shaped accumulation.
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>L_{ij,kl}=\sum_{\mu }2L_{\mu ;i,k}L_{\mu ;l,j}^{\dagger }-i{\left(H-i\sum_{\mu }L_{\mu }^{\dagger }L_{\mu }\right)}_{i,k}{\delta }_{l,j}+i{\left(H+i\sum_{\mu }L_{\mu }^{\dagger }L_{\mu }\right)}_{l,j}{\delta }_{ik.}</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (2)	+
−	|}These representations enable one to treat the Lindblad equation as a linear equation. In other words, the dynamics of the system can be understood in terms of the eigenvalue problem of the Liouvillian matrix: <math>L{\rho }^{\left(i\right)}={\lambda }_i{\rho }^{\left(i\right)}.</math> The Hamiltonian and dissipators can be expressed in terms of 2n Majorana fermions [54]	+
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>H=\sum_{i,j=1}^{2n}{\gamma }_iH_{{}_{ij}}^M{\gamma }_j, \quad L_{\mu }=\sum_{i=1}^{2n}l_{{}_{\mu ,i}}^M{\gamma }_i</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (3)	+
−	|}where  <math>{\gamma }_i</math> are Majorana fermions satisfying  <math>\lbrace {\gamma }_i,{\gamma }_j\rbrace =2{\delta }_{ij}</math>. The matrix  <math>H^M</math> is chosen to be an antisymmetric matrix, <math>{\left(H^M\right)}^T=-H^M</math>. Defining  <math>M_{ij}={\sum }_{\mu }l_{\mu ,i}^\ast l_{\mu ,j}^{}</math>,  <math>M_{{}_{ij}}^M={\sum }_{\mu }{\left(l_{\mu ,i}^M\right)}^\ast l_{\mu ,j}^M</math>, we have  <math>M_{{}_{}}^M=\frac{1}{4}M\otimes (1+{\sigma }_y)</math>. Under the third quantization [54,55], the Liouvillian superoperator is expressed as a quadratic form of the 2n complex fermions (4n Majorana fermions)	+
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>L\mbox{=}\frac{2}{i}\left(\begin{array}{cc}	+
−	c^{\dagger } & c	+
−	\end{array}\right)\left(\begin{array}{cc}	+
−	-Z^T & Y\\	+
−	0 & Z	+
−	\end{array}\right)\left(\begin{array}{c}	+
−	c\\	+
−	c^{\dagger }	+
−	\end{array}\right),</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (4)	+
−	|}where <math>Z=H^M+iRe{\left(M^M\right)}^T,Y=2Im{\left(M^M\right)}^T</math>, and  <math>c=(c_1,c_2,...,c_{2n})</math> are third quantized complex fermions. Through the above expression, we can obtain the Liouvillian eigenspectrum [54,55]	+
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>\lambda ={\sum }_iE_iv_i</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (5)	+
−	|}with  <math>v_i\in \lbrace 0,1\rbrace </math>, where  <math>\left\{E_i\right\}</math> is the eigenspectrum of  <math>4iZ</math>. Here  <math>\lambda </math> contains valuable information of the full density-matrix dynamics, and it can be easily obtained from the damping matrix <math>X</math> with  <math>X_{ij}=ih_{ji}-{\sum }_{\mu }l_{\mu ,j}^\ast l_{\mu ,i}^{}</math> [36]. Rewriting <math>M</math> as  <math>M=M_r+iM_i</math>, where  <math>M_r,M_i</math> are real matrices, we have  <math>M^M=\frac{1}{4}(M_r+iM_i)\otimes (1+{\sigma }_y)</math>.  <math>Z</math> can be further written as	+
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>\begin{align}	+
−	Z=&\frac{1}{4}(h_r\otimes {\sigma }_y+ih_i\otimes 1)+i\frac{1}{4}(M_r^T\otimes 1-iM_i^T\otimes {\sigma }_y)\\	+
−	=&\frac{1}{4}(h_r+M_i^T)\otimes {\sigma }_y+i\frac{1}{4}(h_i+M_r^T)\otimes 1	+
−	\end{align}</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (6)	+
−	|}Therefore,	+
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>\begin{array}{c}	+
−	\det(4iZ-\lambda E)=\det\left(\begin{array}{cc}	+
−	-(h_i+M_r^T)-\lambda  & h_r+M_i^T\\	+
−	-(h_r+M_i^T) & -(h_i+M_r^T)-\lambda	+
−	\end{array}\right)\\	+
−	=\det(X-\lambda E)\det(X^\ast -\lambda E)	+
−	\end{array}</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (7)	+
−	|}The eigenvalue of <math>4iZ</math> are the union of the eigenvalues of <math>X</math> and <math>X^\ast </math>, which gives the Liouvillian eigenspectrum.	+
−	Then we outline the general form of the Lindblad damping matrix in open quantum systems [30]. We consider tight-binding models whose Hamiltonian can generally be written as  <math>H=\sum {}_{ij}h_{ij}c_i^{\dagger }c_j</math>, where  <math>c_i^{\dagger },c_i</math> are the creation and annihilation operators on lattice site  <math>i</math>, and <math>h_{ij}=h_{ij}^\ast </math> is the hopping amplitude between the lattice points of the system (<math>i\not =j</math>) or onsite potential (<math>i=j</math>). It is convenient to define the single-particle correlation function  <math>{\Delta }_{ij}(t)=Tr[c_i^{\dagger }c_j\rho (t)]</math> to observe the time evolution of the density matrix. Each cell is coupled to the environment through the gain jump operator  <math>L_{\mu }^g=\sum {}_iD_{\mu i}^gc_i^{\dagger }</math> and loss jump operator <math>L_{\mu }^l=\sum {}_iD_{\mu j}^lc_i</math>. Substituting the Lindblad quantum master equation into the time evolution of the single-particle correlation function, we can obtained	+	As can be seen in [[Draft Seryazdan 514767809#img-2|Figure 2]](b) below, in the upper part of the inner wall of the micro-hole, the recast layer has a smooth surface and relatively low thickness but is unevenly distributed along the periphery of the hole. Because the steam action is the strongest here, the melt pool is shallow, the metal vapor expansion rate is unevenly distributed, and a large amount of molten metal is randomly discharged, but under the action of the Marangoni force, it shows a smooth and excessive gully shape. In the 1/3 position from the top of the orifice, the recast layer surface formed a turbulent ripple-like undulation, randomly distributed in the circumferential direction, the thickness of a relative increase in the molten metal was a localized layer accumulation. Because with the non-stop stacking of the molten pool, the pressure gradient and the molten liquid velocity increase, so that the molten liquid layer of the explosion, the small size of the fast particles and the large size of the liquid particles with small velocity successively excluded by spattering, the vapor flow still has a high recoil force but weakened in relation to the orifice. In the middle and lower part of the micro-hole, the flow on the surface of the recast layer tends to stabilize, and the flow direction is more consistent, but the thickness reaches the maximum; at the same time, the layer spacing decreases, and a large number of overlapping melts blown down by the gas stream appear in the cross-section. Because here the melt pool flow rate has been lower, the melt by gravity and surface tension to reach equilibrium, the formation of a large number of droplets of solute accumulation, and at the same time because of the convection of the melt pool so that the melt can not be quickly discharged and thus re-melting and condensation to form a thicker recast layer. At the bottom of the micro-hole, the surface of the recast layer shows an obvious insufficient flow, because the steam effect here is weakened, and the melt sputtering becomes the main mode of discharge to increase the viscosity of the melt, which leads to the formation of localized agglomerates. Overall, the surface geometry of the laser-processed recast layer shows a stable and continuous periodic stripe distribution, which is characterized by an orderly wave-like microstructure on the macroscopic level and a spherical dome distribution of "peaks and valleys" on the microscopic level.<div id="img-2"></div>
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
		+	|- style="background:white;"
		+	| style="text-align: center;padding:10px;" | [[Image:Draft_Seryazdan_636830651-image2.png|600px|link=https://www.scipedia.com/public/File:Draft_Seryazdan_636830651-image2.png]]
	|-		|-
−	|	+	| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;" | '''Figure 2'''. Simulation of surface geometric features of recast layer with SEM
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>\frac{d\Delta (\mbox{t})}{dt}=X\Delta (\mbox{t})+\Delta (\mbox{t})X^{\dagger }+2M_g,</math>	+
	|}		|}
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (8)	+	====2.2 Design of experimental platforms====
−	|}where  <math>X=ih^T-(M_l^T+M_g)</math> is the damping matrix with <math>{\left(M_g\right)}_i{}_j=\sum {}_{\mu }D_{\mu i}^{g\ast }D_{\mu j}^g</math> and <math>{\left(M_l\right)}_i{}_j=\sum {}_{\mu }D_{\mu i}^{l\ast }D_{\mu j}^l</math>. The steady state correlation  <math>{\Delta }_s=\Delta (\infty )</math>, to which the long-time evolution of any initial state converges, is determined by  <math>d{\Delta }_s/dt=0</math> or  <math>X{\Delta }_s+{\Delta }_sX^{\dagger }+2M_g=0</math>. Focusing on the deviation towards the steady state  <math>\tilde{\Delta }(t)=\Delta (t)-{\Delta }_s</math>, whose time evolution is  <math>d\tilde{\Delta }(t)/dt=X\tilde{\Delta }(t)+\tilde{\Delta }(t)X^{\dagger }</math>, we can integrate it with Eq. (1) to obtain	+	Experimentally used electrolytic combination processing device shown in [[Draft Seryazdan 514767809#img-3|Figure 3]], the device has electrochemical processing module and three-coordinate displacement platform, the platform is equipped with <math display="inline">  X, Y, Z</math> three axes, three-axis linkage can be realized. <math display="inline"> z </math>-axis is equipped with electrochemical machining device, <math display="inline"> XY </math> motion platform is equipped with a work box, electrochemical removal processing in the work box. The working box is made of plexiglass and resin materials, which not only realizes the electrical insulation from the machine tool, but also facilitates the observation of the experimental process.<div id="img-3"></div>
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
		+	|- style="background:white;"
		+	| style="text-align: center;padding:10px;" | [[Image:Draft_Seryazdan_636830651-picture- 1.svg|center|600px|link=https://www.scipedia.com/public/File:Draft_Seryazdan_636830651-picture-_1.svg]]
	|-		|-
−	|	+	| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;" | '''Figure 3'''. Electrochemical light finishing processing experimental platform
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>\tilde{\Delta }(\mbox{t})=e^{Xt}\tilde{\Delta }(\mbox{0})e^{X\dagger t}.</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (9)	+
−	|}Therefore, the dynamical behaviour of the system can be characterized by the damping matrix.	+
−	==3. Model==	+
−	In this paper, we consider the Chern insulator model with the Hamiltonian in momentum space as	+
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>h(k)=l_x\sin k_x{\sigma }_x+l_y\sin k_y{\sigma }_y+{\epsilon }_k{\sigma }_z,</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (10)	+
−	|}where  <math>{\epsilon }_k=m+t_x\cos k_x+t_y\cos k_y</math>. Let each unit cell contain a single loss and gain dissipator,	+
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>\begin{align}	+
−	L_x^l= & \frac{\sqrt{2\gamma }}{2}\left(e^{-\displaystyle\frac{\pi }{4}i}c_{xA}+e^{\displaystyle\frac{\pi }{4}i}c_{xB}\right)\\	+
−	L_x^g= & \frac{\sqrt{2\gamma }}{2}\left(e^{\displaystyle\frac{\pi }{4}i}c_{xA}^{\dagger }+e^{-\displaystyle\frac{\pi }{4}i}c_{xB}\right),	+
−	\end{align}</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (11)	+
−	|}where  <math>x</math> denotes the lattice site,  <math>A,B</math> refer to the sublattice. The Fourier transformation of  <math>X</math> is  <math>X(k)=ih^T(-k)-M_l^T(-k)-M_g(k)</math>. The gain and loss dissipators are intra-cell, so these  <math>M(k)</math> matrices are independent of  <math>k</math>,  <math>M_l(k)=\frac{\sqrt{2}}{2}\lambda +\frac{1}{2}{\sigma }_x-</math><math>\frac{1}{2}{\sigma }_y,M_g(k)=\frac{\sqrt{2}}{2}\lambda +</math><math>\frac{1}{2}{\sigma }_x+\frac{1}{2}{\sigma }_y</math>. Then, the damping matrix in momentum space is	+
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>X(k)=i[l_x\sin k_x{\sigma }_x+l_y\sin k_y{\sigma }_y+{\epsilon }_k{\sigma }_z+</math><math>i[\lambda {\sigma }_x+\lambda {\sigma }_y]-\sqrt{2}\lambda .</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (12)	+
−	|}It can be written in the form of left and right eigenvectors,	+
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>X=\sum_n{\lambda }_n\vert u_{Rn}\rangle \langle u_{Ln}\vert ,</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (13)	+
−	|}where  <math>X^{\dagger }\vert u_{Ln}\rangle ={\lambda }_n^\ast \vert u_{Ln}\rangle ,X\vert u_{Rn}\rangle =</math><math>{\lambda }_n\vert u_{Rn}\rangle</math>. It is worth noting that our  <math>M_l</math> and  <math>M_g</math> satisfy  <math>M_l^T+M_g=2M_g</math>, guaranteeing that  <math>{\Delta }_S=\frac{1}{2}I_{2L\times 2L}</math> is a steady state solution, where  <math>L=N_x\times N_y</math>, <math display="inline"> L </math> is the system size, and  <math>N_x,N_y</math> are the size in <math display="inline"> x,y </math> direction, respectively. We assume that the initial state of the system is the completely filled state, i.e., <math>\Delta (0)</math> is an identity matrix. Therefore, Eq.(9) can be re-expressed as	+
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>\begin{aligned}	+
−	\tilde{\Delta}(\mathrm{t})& =\frac{1}{2}\displaystyle\sum_{n,n^{'},l}\exp[(\lambda_{n}+\lambda_{n^{'}}^{*})t]\widetilde{u_{R}}(i,n)\widetilde{u_{L}}(l,n)\widetilde{u_{L}^{*}}(l,n^{'})\widetilde{u_{R}^{*}}(j,n^{'})  \\	+
−	&=\displaystyle\frac{1}{2}\displaystyle\sum_{n,n^{'}}\displaystyle\frac{\displaystyle\sum_{l}\exp[(\lambda_{n}+\lambda_{n^{'}}^{*})t]u_{R}(i,n)u_{L}(l,n)u_{L}^{*}(l,n^{'})u_{R}^{*}(j,n^{'})}{\displaystyle\sum_{k}u_{R}(k,n)u_{L}(k,n)\displaystyle\sum_{m}u_{L}^{*}(m,n^{'})u_{R}^{*}(m,n^{'})}	+
−	\end{aligned}</math>	+
	|}		|}
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (14)	+	===2.3 Design of experimental methodology===
−	|}According to the dissipative property,  <math>Re\lbrace {\lambda }_n\rbrace \leq 0</math> always holds. The Liouvillian gap  <math>\Lambda =\min[2Re(-{\lambda }_n)]</math> plays a decisive role in long-time dynamics. The opening gap (<math>\Lambda \not =0</math>) implies an exponential rate of convergence to the steady state, while the closing gap (<math>\Lambda =0</math>) implies algebraic convergence [34].	+	====2.3.1 Experimental flow design and preparation of samples====
−	==4. Chiral damping==	+	The basic idea of the experiment, the laser was first used to process the prefabricated holes in the 2 mm thick metal plate, and then electrochemical processing was used to remove the recast layer on the hole wall, the specific process is shown in [[Draft Seryazdan 514767809#img-4|Figure 4]]. Laser parameters for the optimized parameters: pulse width of 0.2ms, peak power of 16kW, repetition frequency of 70Hz, drilling process along the laser head sprayed 0.6MPa argon to accelerate the exclusion of molten material, to reduce the burr, melt spattering and residual in the hole wall of the recast layer. Electrochemical processing, the use of motion control systems to insert the electrode wire centered in the pre-fabricated holes, detect whether the short circuit and set the pulse power parameters. First turn on the electrolyte pump, wait for the electrolyte to flow evenly through the hole and see a clear and stable liquid column at the exit, turn on the pulse power supply to start electrolysis, observe the experimental phenomena and power supply parameter changes, and turn off the power supply according to the set electrolysis time. The electrolyte is a neutral solution configured with pure water, and the dynamic liquid flushing method is used. In the experiment, considering the experimental error, 3 holes were processed under each parameter, and after the completion of the processing, the samples were cleaned, polished and metallurgically corroded, and the micro-hole morphology was observed with a scanning electron microscope, and the pore diameter was observed and measured with an optical microscope, and the average value of the 3 holes' diameters was finally taken under the same parameter. The micro-hole morphology of the sample is shown in [[Draft Seryazdan 514767809#img-5|Figure 5]].<div id="img-4"></div>
−	For simplicity, the parameters of our model are taken as  <math>l_x=l_y=1</math>, <math>t_x=t_y=-1</math>. We first study the dynamical behaviour under the periodic boundary conditions. Diagonalizing  <math>X(k)</math>, we obtain the energy spectrum as shown in [[Zhong Li 2024a#img-1|Figure 1]]. It is found that the Liouvillian gap vanishes at  <math>m=1.5</math>, while the gap opens at  <math>m=2.5</math>. So we expect the damping rate to be algebraic and exponential in each case, respectively.<div id="img-1"></div>	+	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
−	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:75%;"	+
	|- style="background:white;"		|- style="background:white;"
−	| style="text-align: center;padding:10px;" | [[Image:Draft_Zhong_847600978-image84.png|600px|link=https://www.scipedia.com/public/File:Draft_Zhong_847600978-image84.png]]	+	| style="text-align: center;padding:10px;" | [[Image:Draft_Seryazdan_636830651-image5-c.png|504px|link=https://www.scipedia.com/public/File:Draft_Seryazdan_636830651-image5-c.png]]
	|-		|-
−	| style="background:#efefef;text-align:justify;padding:10px;font-size: 85%;" | '''Figure 1'''. Eigenvalues of the damping matrix X. Blue: periodic boundary; Red: open boundary . The Liouvillian gap under periodic boundary condition vanishes for (a) and (b), while it is nonzero for (c) and (d). Under open boundary condition, the Liouvillian gap is nonzero in all four cases. This significant difference between open and periodic boundary comes from the NHSE of <math display="inline">  X</math>. (a) <math>\lambda =0.1,m=1.5</math>. (b) <math>\lambda =0.5,m=1.5</math>. (c)  <math>\lambda =0.1,m=2.5</math>. (d) <math>\lambda =0.5,</math>  <math>m=2.5</math>	+	| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;" | '''Figure 4'''. Micro-hole machining process flowchart
−	|}To verify this, we define the site-averaged fermion number deviation from the steady state  <math>R(t)=\sqrt{\frac{1}{N_XN_Y}{\sum_xR_x(t)}^2}=\sqrt{\frac{1}{N_XN_Y}{\sum_x\left(\frac{n_x(t)-n_x(t-{\delta }_t)}{{\delta }_t}\right)}^2}</math>, where  <math>R_x(t)=n_x(t)-n_x(\infty )</math>, and <math>n_x(t)={\Delta }_{xA,xA}(t)+{\Delta }_{xB,xB}(t)</math>. The numerical results are shown in [[Zhong Li 2024a#img-2|Figure 2]]. As anticipated, it is observed that the damping of <math display="inline">  R(t)</math> is algebraic for cases black and red lines with <math display="inline">m=1.5</math>, while exponential for blue and green lines with <math display="inline">m=2.5</math> under the periodic boundary condition.<div id="img-2"></div>	+	|}<div id="img-5"></div>
−	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:55%;"	+	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
	|- style="background:white;"		|- style="background:white;"
−	| style="text-align: center;padding:10px;" | [[Image:Draft_Zhong_847600978-image93.png|348px|link=https://www.scipedia.com/public/File:Draft_Zhong_847600978-image93.png]]	+	| style="text-align: center;padding:10px;" | [[Image:Draft_Seryazdan_636830651-image6.png|600px|link=https://www.scipedia.com/public/File:Draft_Seryazdan_636830651-image6.png]]
	|-		|-
−	| style="background:#efefef;text-align:justify;padding:10px;font-size: 85%;" | '''Figure 2'''. Damping of site-averaged fermion number towards the steady state under periodic boundary condition with size  <math>L=30\times 30</math>. <math display="inline">m=1.5</math> (black and red) exhibits a slow algebraic damping, while <math display="inline">m=2.5</math> (blue and green) is an exponential damping. The  initial state is the completely filled state	+	| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;" | '''Figure 5'''. Laser processing sample drawing
−	|}Next we turn to the open boundary conditions. Since the damping matrix <math display="inline"> X </math> has  NHSE, its energy spectrum is no longer that of the periodic boundary conditions. At this point all the energy spectrums have a non-zero energy gap (red part of [[Zhong Li 2024a#img-1|Figure 1]]), therefore, we expect an exponential long-time damping of <math>\tilde{\Delta }(t)</math>. The numerical simulation in [[Zhong Li 2024a#img-3|Figure 3]] confirms this exponential behaviour with <math display="inline"> R(t)</math> having a period of algebraic damping before entering into the exponential damping. The time of the algebraic damping increases with the size <math display="inline"> L </math> ([[Zhong Li 2024a#img-3|Figure 3]](a)). To better understand this feature, we plot the damping in several unit cells in the same x dimension (<math display="inline">ix = 1</math>), as shown in [[Zhong Li 2024a#img-3|Figure 3]](b). It can be seen that the left end () enters the exponential damping immediately, and the other sites enter the exponential damping in turn according to their different distances to the left end.Due to a process of algebraic damping that occurs before entering the exponential stage,there is a "damping wavefront" from left (<math>ix=1,iy=1</math>) to right (<math>ix=1,iy=N_y</math>). This phenomenon is known as "chiral damping".<div id="img-3"></div>	+	|}
−	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:80%;"	+	====2.3.2 Determination of electrochemical processing method====
		+	In the electrochemical machining process, generally keep the processing current or processing voltage constant, with the increase of workpiece aperture, the electrolyte resistance increases gradually. When the current is constant, the voltage increases with the increase of the electrolyte resistance, and the current density decreases with the increase of the processed surface area when; when the voltage is constant, the current decreases with the increase of the electrolyte resistance, and the current density decreases rapidly with the decrease of the current and the increase of the processed surface area. In summary, it can be found that, when the voltage is constant, the decreasing trend of current density is greater than that of constant current, which is conducive to ensuring the processing efficiency in the early stage while ensuring the surface quality in the later stage of processing, so this paper adopts constant voltage for electrolytic processing.
		+	====2.3.3 Selection of electrolyte in electrochemical processing====
		+	Electrolytic processing, the electrode potential applied at the interface between the workpiece and the solution is different, the workpiece processing surface is in a different state, NaNO<sub>3</sub> solution is a passive electrolyte, the polarization curve of the anode of the workpiece into a non-linear distribution, passivation area of the metal surface to form an adsorbed oxygen layer or an oxide layer, i.e., passivation film; and precipitation of the oxygen area of the workpiece and the interface of the solution at the OH- ions easy to oxidize, precipitation of oxygen, precipitation of O<sub>2</sub> easy to oxidation Metal surface, so that the workpiece processing surface at the passivation film becomes thicker. The passivation film is less conductive, which in turn reduces the processing current density and lowers the dissolution rate of the metal. The NaCl solution is an active electrolyte, and the polarization curve of the anode of the workpiece is linearly distributed. The activation capacity of Cl- in the solution is very strong, the anode surface in a variety of conditions are in the active state of low-value dissolution, does not produce passivation and its side reactions, so the metal dissolution rate is higher, but at the same time there is a serious stray corrosion, which seriously affects the machining accuracy. In order to improve the nonlinear dissolution characteristics of the electrolyte, improve the current efficiency, soften the passivation salt film, this paper adopts a certain concentration of NaNO<sub>3</sub> solution mixed with a certain proportion of NaCl solution, in order to improve the machining efficiency at the same time easier to achieve the uneven dissolution or uneven super-passivation of the dissolution state, so as to improve the machining quality of the workpiece.
		+	==3. Results and analysis of removal of recast layer and orifice reshaping based on electrochemical light leveling==
		+	===3.1 Effect of electrolysis parameters on the removal of recast layers===
		+	When constant voltage electrolysis is used, the appropriate voltage should be selected. Theoretically, when the electrolysis voltage increases, the amount of electricity passing through the surface increases and the electrolysis rate increases. The electrolysis products and heat in the processing gap also increase, increasing the flow resistance in the gap, resulting in a decrease in the flow of electrolyte, so that the ability to exclude electrolysis products and heat is reduced. When there is a serious imbalance between the two, abnormal phenomena such as evaporation, boiling, and cavitation will occur in the processing gap, leading to serious failures such as short circuits and scarring, resulting in the interruption of processing. In addition, with the increase in current density, the electric field strength around the electrode wire increases, the stray corrosion around the orifice is enhanced, and the orifice is not rounded after electrolysis. Therefore, the voltage increase should not destroy the above balance as a prerequisite for the given electrolytic processing conditions, need to choose a suitable voltage value. Take the electrolysis time of 30s, the electrolyte is the conductivity of 100.4ms/cm NaNO<sub>3</sub> and NaCl mixed solution solution, respectively, with a voltage of 3-18V (interval 3V) for the recast layer removal experiment.
		+
		+	After the completion of the experiment, the experimental results are processed: take the amount of hole expansion after processing and surface roughness change value as the reference amount; different voltage electrolysis, the expansion of the hole diameter is different, take the median value of the hole diameter in the process of electrolysis (with the minimum voltage electrolysis hole median value) as the benchmark, and take the change in the quality of the processing before and after the processing as an auxiliary reference for processing rate.
		+
		+	From the [[Draft Seryazdan 514767809#img-6|Figure 6]], it can be seen that after the same time of electrolysis, the pore radius and electrolysis rate increased with the increase of electrolysis voltage. When the voltage is 15V, the recast layer is completely removed. At this time, the electrolysis current is 0.5 A. Calculation gives a minimum current density of 10.09 A/cm<sup>2</sup>. The mass change and pore radius change shown in [[Draft Seryazdan 514767809#img-6|Figure 6]] reflect that the electrolysis rate is approximately proportional to the voltage. When electrolysis removes the recast layer, the volume electrochemical equivalent <math>\omega</math> of the material is a constant value, and the oxidation reaction of Ni, Fe, Cr, and other elements in the material occurs, in which Ni, Fe, and Cr will be oxidized with multiple valences according to the current density, and when the current density is greater than, all three elements are oxidized to the cation with the highest valence, i.e., the current efficiency <math>\eta</math> is constant. When the electrolysis voltage is 6 V, the extension of the fitted primary curve can be found that the electrolysis rate is larger at this time. This is because when the current density is small, Ni, Fe, Cr elements are partially oxidized to low valence cations, at this time the current efficiency <math>\eta</math> is larger, and the proportion of high valence oxidation of these three elements increases with the increase of current density, that is, the current efficiency decreases with the increase of current density.<div id="img-6"></div>
		+	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
	|- style="background:white;"		|- style="background:white;"
−	| style="text-align: center;padding:10px;" | [[Image:Draft_Zhong_847600978-image98.png|700px|link=https://www.scipedia.com/public/File:Draft_Zhong_847600978-image98.png]]	+	| style="text-align: center;padding:10px;" | [[Image:Draft_Seryazdan_636830651-image9.png|400px|link=https://www.scipedia.com/public/File:Draft_Seryazdan_636830651-image9.png]]
	|-		|-
−	| style="background:#efefef;text-align:justify;padding:10px;font-size: 85%;" | '''Figure 3'''. (a) Site-averaged particle number damping under periodic boundary conditions (solid line) and open boundary conditions (dashed line) for several sizes <math display="inline"> L </math>. The long-time damping of <math display="inline">  R(t)</math> follows a power law under periodic boundary condition, while the damping follows an exponential law after an initial power law stage under open boundary condition. (b) Particle number damping on several sites. The system size is <math display="inline">  30\times 30</math>, and the left end (<math>ix=1,iy=1</math>) enters the exponential phase from the beginning, followed by the other sites in turn. For (a) and (b), the initial state is completely filled state, <math>m=1.5,\lambda =0.1</math>	+	| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;" | '''Figure 6'''. Plot of the effect of voltage parameters on the removal of recast layer processing
−	|}The phenomenon of chiral damping can be observed more intuitively as shown in Fig. 4(a) where the colour shades indicate the value of <math display="inline">  R(t)</math>. Under the periodic boundary condition, the time evolution follows a slow power law while under the open boundary condition, a wavefront moving to the upper right is observed. This can be intuitively linked to the phenomenon that all eigenstates of are localized in the upper right corner, which arises from the non-Hermitian skin effect of the damping matrix . If the matrix  does not have NHSE under the open boundary condition, the fermion number of the system should have a similar behaviour of damping under different boundary conditions. Therefore, the non-Hermitian skin effect plays an important role in open quantum systems and significantly affects the dynamical behaviour of open quantum systems.<div id="img-4"></div>	+	|}[[Draft Seryazdan 514767809#img-7|Figure 7]] shows the SEM and metallographic images of the control holes and after removal of the recast layer by light-needle processing, from which it can be seen that the holes are rounded after light-finishing processing, and the recast layer is completely removed and the hole wall is smooth. When the constant pressure finishing process, the higher the voltage, the higher the current density, the higher the electrolysis rate. But at the same time as the voltage increases due to the hole entrance surface near the tool cathode, the electric field effect than other parts of the General Assembly serious stray corrosion, and because the entrance surface to increase the auxiliary anode, and by providing a potential difference to change the direction of the electric field of the anode entrance surface to reduce the entrance of the stray corrosion, effective control of electrochemical reaction of the hole entrance near the upper surface of the hole inlet, practical electrolytic machining, generally according to the cut-off In actual electrolytic processing, the voltage is generally determined according to the cut-off voltage and the electrolytic processing area. Under the condition of ensuring the processing surface quality, the electrolytic voltage can be increased in order to obtain a higher electrolytic rate, and the electrolyte flow rate can be increased at the same time in order to meet the exclusion of electrolytic processing products and heat.<div id="img-7"></div>
−	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:60%;"	+	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
	|- style="background:white;"		|- style="background:white;"
−	| style="text-align: center;padding:10px;" | [[Image:Draft_Zhong_847600978-image101-c.png|456px|link=https://www.scipedia.com/public/File:Draft_Zhong_847600978-image101-c.png]]	+	| style="text-align: center;padding:10px;" | [[Image:Draft_Seryazdan_636830651-image10.png|600px|link=https://www.scipedia.com/public/File:Draft_Seryazdan_636830651-image10.png]]
	|-		|-
−	| style="background:#efefef;text-align:justify;padding:10px;font-size: 85%;" | '''Figure 4'''. Evolution of <math display="inline">  R(t)</math> at each lattice site under open boundary conditions (a) and periodic boundary conditions (b)	+	| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;" | '''Figure 7'''. SEM and metallographic images after removal of the recast layer
	|}		|}
−	==5. Experiment realized==	+	===3.2 Effect of pulsed power supply on the removal of recast layers===
−	Next we give the scheme of topolectrical circuits to simulate the damping matrix. Based on the similarity between the Kirchhoff equation and the Schrödinger equation, it is possible to simulate the Hamiltonian of the system using different circuit components, and the different parameters in the Hamiltonian can be adjusted independently by various components. The circuit Laplacian corresponding to the Hamiltonian can be written as	+	The traditional DC electrolytic machining has a strong ability to scatter etching and a weak ability to centralized etching, which affects the accuracy of electrolytic machining, the general three-dimensional surface molding accuracy of 0.2 to 0.5 mm, and the hole machining molding accuracy of 0.025 to 0.05 mm. High-frequency, narrow pulse electrolytic machining can achieve high precision (dimensional accuracy less than 5<math>\mu</math>m, surface roughness less than 0.03mm), small gap (10 to 50<math>\mu</math>m) machining, machining quality is greatly improved. The principle of pulse electrolytic processing is to replace the continuous DC power supply with periodic intermittent power supply, and the anode undergoes periodic intermittent dissolution in the electrolyte. It uses the intermittent depolarization of the pulse gap with power failure to dissipate heat from the workpiece, and the electrochemical properties, flow field, and electric field of the machining gap are restored to the starting state. Pulse current electrolysis, the gap produces hydrogen pressure wave synchronized with the pulse current, the frequency and intensity with the increase of pulse frequency and enhancement, which strengthens the stirring effect on the electrolyte, so that the flow field distribution in the gap tends to be uniform, and to improve the heat dissipation conditions in the gap, so as to make the minimum stable machining gap is greatly reduced, and thus to enhance the fixed-domain nature of electrolytic machining. Therefore, this paper compares the difference between the processing quality of pulse power supply and DC power supply, and finally chooses the processing method of pulse current electrolysis to get a higher electrolysis rate under the premise that the processing quality can be guaranteed.
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>J=D-C+W,</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (15)	+
−	|}where  and <math display="inline"> D </math> are diagonal matrices containing the total conductance from each node to the ground and to the rest of the circuit, respectively. <math display="inline"> C </math> is the adjacency matrix of conductances [44].<div id="img-5"></div>	+
−	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:70%;"	+
−	|- style="background:white;"	+
−	| style="text-align: center;padding:10px;" | [[Image:Draft_Zhong_847600978-image103.png|600px|link=https://www.scipedia.com/public/File:Draft_Zhong_847600978-image103.png]]	+
−	|-	+
−	| style="background:#efefef;text-align:justify;padding:10px;font-size: 85%;" | '''Figure 5'''. Structure of topolectrical circuit to realize damping matrix under periodic boundary conditions. (a) Connection relations between the nodes. The blue solid line box containing two “sublattice” nodes A (red) and B (blue) simulates a unit cell of <math display="inline"> X</math>. The black (grey) solid line indicates the coupling between nodes in the <math display="inline"> x(y) </math>-direction. (b) Circuit element structure is detailed for the green dashed framed rectangle in (a).   (c) Internal circuit diagram of the INIC element, consisting of an operational amplifier and impedances  . The impedance  <math>Z</math> is the target element, and different conductance in different directions of  <math>V_{i,r}</math> can be achieved by connecting the INIC in series.   satisfies  <math>Z_+=Z_{-}</math>. (d) Grounding module of the nodes. The resistances <math>R_{A,}R_B</math> and capacitance <math display="inline"> C </math> are used to simulate the onsite potential, and inductance L allows the Laplacian eigenvalue spectrum to be shifted uniformly as desired	+
−	|}Considering the periodic boundary conditions first, the topolectrical circuit for realizing the damping matrix X is illustrated in [[Zhong Li 2024a#img-5|Figure 5]].  [[Zhong Li 2024a#img-5|Figure 5]] depicts the schematic diagram of the overall circuit structure, which gives the connection relationship between the nodes. [[Zhong Li 2024a#img-5|Figure 5]] shows the detailed circuit component of the unit which is the green dashed box in [[Zhong Li 2024a#img-5|Figure 5]](a). The blue box in [[Zhong Li 2024a#img-5|Figure 5]](a) represents a unit cell in the system, and the two nodes inside it correspond to sublattices A (red) and B (blue). The circuit connections in the x and y directions are distinguished by black and gray. From [[Zhong Li 2024a#img-5|Figure 5]](b) we can obtain the matrices C and D in Eq. (9), so that	+
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>\begin{array}{c}	+
−	D-C=-i\omega \left(\begin{array}{cc}	+
−	-i\displaystyle\frac{2}{\omega R_2}\cos k_x-i\displaystyle\frac{2}{\omega R_4}\cos k_y-C_1-i\displaystyle\frac{1}{\omega R_0} & i2C_2\sin k_x+\displaystyle\frac{2}{\omega R_6}\sin k_y-i\displaystyle\frac{1}{\omega R_1}+C_1\\	+
−	i2C_2\sin k_x+\displaystyle\frac{2}{\omega R_5}\sin k_y+i\displaystyle\frac{1}{\omega R_1}+C_1 & -i\displaystyle\frac{2}{\omega R_3}\cos k_x-i\displaystyle\frac{2}{\omega R_7}\cos k_y-C_1+i\frac{1}{\omega R_0}	+
−	\end{array}\right),\\	+
−	\mbox{    }	+
−	\end{array}</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (16)	+
−	|}with <math>C_1=-\lambda</math>, <math>C_2=l_x/2</math>, , <math>R_1=-1/\left(\omega \lambda \right)</math>, <math>R_2=-R_3=-2/t_x</math>,  ,	+
−	Comparing it with the damping matrix, we need to add grounding elements to match the onsite potential. The grounding elements of nodes A and B are shown in  [[Zhong Li 2024a#img-5|Figure 5]](d), where the resistors  <math>R_{A,}R_B</math> and capacitors  <math>C</math> simulate the lattice potential, and <math>R_{A,}R_B</math> satisfies  <math>R_A=-R_B</math>. So the diagonal matrix <math display="inline"> W </math> is	+	As can be seen from [[Draft Seryazdan 514767809#img-8|Figure 8]](a), the surface quality obtained after processing with DC as the processing power source is relatively rough, the recast layer is not completely removed, and the electrochemical dissolution effect is small, and because of the DC electrolytic processing scattering corrosion ability is strong, so that the stray corrosion area is larger and the corrosion effect is more obvious, and at the same time, due to the fact that by-products can't be eliminated in time during the DC processing and electrolytic solution can't be renewed in time, which makes the local resistance increase, resulting in the inner wall of the small holes in part of the recast layer was not removed. As can be seen from [[Draft Seryazdan 514767809#img-8|Figure 8]](b), the use of pulsed current as the power supply processing after the surface is more smooth, the recast layer is completely removed and the processing quality is relatively stable, effectively solving the problems arising from the DC power supply, and further improving the quality and stability of the processing of electrolytic removal of the recast layer.<div id="img-8"></div>
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>W=\left(\begin{array}{cc}	+
−	i\omega C+\displaystyle\frac{1}{R_A}+\displaystyle\frac{1}{i\omega L} & \\	+
−	& i\omega C+\displaystyle\frac{1}{R_B}+\displaystyle\frac{1}{i\omega L}	+
−	\end{array}\right).</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (17)	+
−	|}From Eq. (10) we get the conductance matrix of the circuit of [[Zhong Li 2024a#img-5|Figure 5]](a) at  <math>\omega </math> frequency	+
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>\begin{array}{c}	+
−	J(\omega )=-i\omega \left(\begin{array}{cc}	+
−	-i\displaystyle\frac{2}{\omega R_2}\cos k_x-i\displaystyle\frac{2}{\omega R_4}\cos k_y-(C+C_1)-i\left(\displaystyle\frac{1}{\omega R_0}+\displaystyle\frac{1}{\omega R_A}\right) & i2C_2\sin k_x+\displaystyle\frac{2}{\omega R_6}\sin k_y-i\displaystyle\frac{1}{\omega R_1}+C_1\\	+
−	i2C_2\sin k_x+\displaystyle\frac{2}{\omega R_5}\sin k_y+i\displaystyle\frac{1}{\omega R_1}+C_1 & -i\displaystyle\frac{2}{\omega R_3}\cos k_x-i\displaystyle\frac{2}{\omega R_7}\cos k_y-(C+C_1)+i\left(\displaystyle\frac{1}{\omega R_0}+\displaystyle\frac{1}{\omega R_A}\right)	+
−	\end{array}\right)+\displaystyle\frac{1}{i\omega L}\epsilon \\	+
−	=-i\omega J_P+\displaystyle\frac{1}{i\omega L}\epsilon ,	+
−	\end{array}</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (18)	+
−	|}where  <math>C=(1-\sqrt{2})\lambda </math>, <math>\frac{1}{R_A}=-\frac{1}{R_B}=-\omega m-\frac{1}{R_0}</math>. Comparing this Laplacian matrix with the damping matrix, the mapping relationship can be established by  <math>J_P\Leftrightarrow X</math>.<div id="img-6"></div>	+
−	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:70%;"	+
	|- style="background:white;"		|- style="background:white;"
−	| style="text-align: center;padding:10px;" | [[Image:Draft_Zhong_847600978-image123-c.png|600px|link=https://www.scipedia.com/public/File:Draft_Zhong_847600978-image123-c.png]]	+	| style="text-align: center;padding:10px;" | [[Image:Draft_Seryazdan_636830651-image11.png|600px|link=https://www.scipedia.com/public/File:Draft_Seryazdan_636830651-image11.png]]
−	|-	+
−	| style="background:#efefef;text-align:justify;padding:10px;font-size: 85%;" | '''Figure 6'''. Negative impedance module [22]. (a) A single-port circuit to ground. The input impedance is  <math>Z_g=-Z</math>. (b) Free-port circuit. Its input impedance at both ends is  <math>Z_{ij}=Z_{ji}=-Z</math>.  The markings on the ideal amplifier indicate the output voltage versus the input voltage	+
−	|}Notice that the circuit requires a negative component,which is implemented as shown in [[Zhong Li 2024a#img-6|Figure 6]]. [[Zhong Li 2024a#img-6|Figures 6]](a) and (b) show the equivalent negative impedance modules for a single port to ground and a free two-terminal port, respectively. They achieve the equivalent negative impedance through an amplifier. According to Kirchhoff's law, the input impedance of the single-port circuit to ground ([[Zhong Li 2024a#img-6|Figure 6]](a)) can be obtained as	+
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
	|-		|-
−	| <math>Z_g=\frac{V_g}{I_g}=\frac{V_g}{(V_g-2V_g)/Z}=-Z.</math>	+	| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;" | '''Figure 8'''. Comparison chart of AC power supply and DC power supply
	|}		|}
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (19)	+	===3.3 Influence of cathodic lifting method on orifice reshaping===
−	|}The input impedance at both ends of the free port circuit ([[Zhong Li 2024a#img-6|Figure 6]](b)) are	+	In order to further improve the machining effect of micro small hole workpieces, a staged combination of processing methods is proposed, based on the electrolytic removal of the recast layer, in the same station to further complete the small hole electrolytic repair of the machining method, i.e., cathode lifting method. The method is as follows: in the first stage, the surface quality of the workpiece is rapidly improved and the surface roughness is reduced by relatively large voltage processing to ensure the rapid removal of the recast layer; in the second stage, after the completion of the removal of the recast layer, the electric field distribution is influenced by changing the electrode position and the relatively small voltage is used for the trimming process to obtain a better surface quality and to realize the goal of the trimming of the micro-hole aperture. Due to the use of hybrid electrolyte, as mentioned above, the electrolyte has the characteristic of cutting the gap, when using small voltage processing, this time the inner wall of the hole although the current through, but the anode does not dissolve, the current efficiency is 0, which has been processed to protect the wall of the hole at the same time as the hole inlet trimming the shape of the hole to make the micro-hole has a better to meet the demand for the performance of the use of the micro-hole. Based on the above research, the experimental process parameters are as follows: the electrolyte used is a mixed solution of NaNO<sub>3</sub> and NaCl with a conductivity of 100.4ms/cm, the electrolysis voltage is 4v, and the electrolysis time is 20s.
−	{| class="formulaSCP" style="width: 100%; text-align: center;"	+
−	|-	+
−	|	+
−	{| style="text-align: center; margin:auto;"	+
−	|-	+
−	| <math>\begin{array}{c}	+
−	Z_{ij}=\displaystyle\frac{V_i-V_j}{I_i}=\displaystyle\frac{V_i-V_j}{[V_i-2(V_i-V_j)]/Z}=-Z,\\	+
−	Z_{ji}=\displaystyle\frac{V_j-V_i}{I_j}=\displaystyle\frac{V_j-V_i}{[V_j-2(V_j-V_i)]/Z}=-Z.	+
−	\end{array}</math>	+
−	|}	+
−	| style="width: 5px;text-align: right;white-space: nowrap;" | (20)	+
−	|}That is <math>Z_{ij}=Z_{ji}=-Z</math>.	+
−	Under the open boundary condition, the hopping amplitude of the cells located at the boundary weakens, leading to fewer branches connected to the boundary nodes in the circuit model, as shown in [[Zhong Li 2024a#img-7|Figure 7]](a).  [[Zhong Li 2024a#img-7|Figure 7]](a) gives the connection relationship between the nodes of the circuit under the open boundary condition, and the circuit nodes can be classified into body nodes (in the black dashed box), edge nodes (in yellow) and corner nodes (in green). Changes in the branch circuit of the nodes at the boundary will cause variations of the matrices <math display="inline"> C </math> and <math display="inline"> D </math>. The matrix <math display="inline"> C </math> corresponds to the hopping amplitude between the lattice points, which is allowed to change. Whereas the change of D is not desired due to the same onsite potential under different boundary condition.	+	From the [[Draft Seryazdan 514767809#img-9|Figure 9]](a) can be seen, without improving the processing method workpiece due to the over-processing phase of the surface obtained on the more rough and micro-hole expansion obvious, poor machining locality, taper relative to the [[Draft Seryazdan 514767809#img-9|Figure 9]](b) is greater, at the same time, the effect of repairing the shape of the micro-hole can not meet the demand for the use of the performance of the micro-hole is not obvious; from the [[Draft Seryazdan 514767809#img-9|Figure 9]](b) can be seen, after the elevation of the method of low-pressure combination of machining, the inner wall of the hole in the cut off gap characteristics of the end of the end of the protection of the dissolution occurred, the machining of locality Higher, the height of the surface bulge is reduced, the surface of the workpiece becomes relatively flat, the taper relative to the [[Draft Seryazdan 514767809#img-9|Figure 9]](a) has obvious improvement, while the effect of the repair effect is good, can meet the functional requirements of the micro-hole.<div id="img-9"></div>
−		+	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
−	Therefore, we need to design specific grounding elements to eliminate the effects of variations in <math display="inline"> D </math>. Owing to the asymmetry of the coupling strengths under periodic boundary condition, the types of the edge and corner nodes are different for each of the four orientations, so there are a total of 16 different grounding modules, as shown in [[Zhong Li 2024a#img-7|Figure 7]](b). The additional grounding elements keep the diagonal matrix D+W unchanged, i.e., the onsite potential is unchanged, which achieves the mapping of the circuit Laplacian in [[Zhong Li 2024a#img-7|Figure 7]] to the damping matrix <math display="inline"> X </math> under the open boundary condition.<div id="img-7"></div>	+
−	{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:75%;"	+
	|- style="background:white;"		|- style="background:white;"
−	| style="text-align: center;padding:10px;" | [[Image:Draft_Zhong_847600978-image129-c.png|750px|link=https://www.scipedia.com/public/File:Draft_Zhong_847600978-image129-c.png]]	+	| style="text-align: center;padding:10px;" | [[Image:Draft_Seryazdan_636830651-image12.png|600px|link=https://www.scipedia.com/public/File:Draft_Seryazdan_636830651-image12.png]]
	|-		|-
−	| style="background:#efefef;text-align:justify;padding:10px;font-size: 85%;" | '''Figure 7'''. Schematic diagram of the circuit of the damping matrix <math display="inline"> X </math> under open boundary conditions. (a) Schematic diagram of the connection relations among the nodes. The black, yellow and green dashed boxes correspond to the body, edge and corner nodes, respectively. The circuit connections of the body node are the same as those of the periodic boundary, while the edge and corner nodes require additional grounding elements to regulate the onsite potential. (b) Grounding modules for edge and corner nodes. The grounding elements for the edge and corner nodes are different for each of the four orientations, where the negative impedance elements can be realized by [[Zhong Li 2024a#img-6|Figure 6]](a). Note that in addition to these grounding elements, all nodes need to be connected to the elements in [[Zhong Li 2024a#img-5|Figure 5]](d)	+	| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;" | '''Figure 9'''. Comparison of the results of the cathodic lift method
	|}		|}
−	==6. Conclusion==	+	==4. Conclusions==
−	In summary, we study the dynamical properties of a two-dimensional open system. The open quantum system is constructed by introducing appropriate gain and loss to the Chern insulator, and then using the damping matrix derived from the Liouvillian superoperator explore its long-time evolution. It is found that  the site-averaged fermion number deviation from the steady state under periodic boundary conditions shows a slow algebraic damping when the energy gap closes and an exponential damping when the energy gap opens. Under open boundary conditions, due to the non-Hermitian skin effect of the damping matrix, the system exhibits the phenomenon of chiral damping that the fermion number at each site undergoes a period of algebraic damping before entering an exponential damping, and the transition time that is proportional to the distance from that site to the boundary. Finally, we map the damping matrix in terms of the circuit Laplacian to give a model diagram of the topolectrical circuit implementation of the system.	+	In this paper, the use of electrochemical finishing processing method to remove the heavy casting layer left on the hole wall by laser processing and at the same time complete the orifice trimming processing can better meet the demand of micro-hole performance. Theoretical analysis and simulation of the formation mechanism of the recast layer and the surface geometrical features during laser processing were carried out, and experiments were carried out for the removal of the recast layer and orifice shaping, and small holes with the recast layer completely removed and the orifice rounded were obtained. Based on the experimental results, the following conclusions were obtained:
−	==References==	+
−	<div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">	+
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−		+	==Acknowledgements==
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Latest revision as of 04:39, 10 November 2025

Abstract

This research is aimed at Laser processing of micro holes there are recast layer and microcracks, analyzed the laser processing of recast layer formation mechanism and surface geometry features not, and for the laser processing of the formation of the recast layer in the process of analog simulation, for electrochemical finishing processing to remove the recast layer to provide a theoretical basis. The laser-electrolysis asynchronous composite processing experimental system was developed, and the electrolytic removal of recast layer on the wall of laser perforated holes and the process research of aperture trimming were carried out, which mainly analyzed the effects of electrolysis voltage, electrolyte and pulse electrolysis parameters on the rate of electrolysis and the effect of the removal of the recast layer. The cathode lifting method is also proposed to lift the tool cathode for orifice shaping after effective removal of the recast layer. The experimental results show that: when the current density is greater than 10.09 A/cm², the electrolysis rate is proportional to the voltage; when the constant voltage electrolysis, the mixed solution of NaNO₃ and NaCl enhances the effect of the recast layer removal significantly, and relative to the single solution with the same electrical conductivity, the value of the change in the surface roughness of the workpiece before and after the finishing process is enhanced more significantly, and there is no significant difference in the processing efficiency; the low-frequency wide pulse and high-frequency narrow pulse in the pulse electrolysis processing is favorable for the removal of recast layer. Pulse, high-frequency narrow pulse is conducive to the improvement of electrolysis rate, and the frequency of 1 kHz, the duty cycle of 40% to 60% of the electrolysis rate is the largest; at the same time, the use of cathodic enhancement method of aperture trimming processing, the use of the truncation effect, through the appropriate electrolytic processing parameters, to achieve the complete removal of the laser hole hole wall recast layer at the same time to complete the aperture trimming processing, to meet the demand for the use of performance of the small holes to further enhance the processing efficiency and small holes, and to improve the processing efficiency of the laser hole. Enhance the processing efficiency and functionality of small holes.

Keywords: Laser processing, recast layer simulation, electrochemical finishing, cathodic lift

1. Introduction

In the field of modern machinery and equipment, the application of microporous in enhancing the use of equipment performance and reliability and other aspects of the value is increasingly apparent. For example, in the key high-temperature components of aircraft engines, the application of microporous promotes the development of air film cooling technology, and effectively make up for the limitations of high-temperature-resistant materials to enhance the reliability and durability of the engine; in the field of internal combustion engines, the application of microporous design of injector nozzles not only significantly improves the effect of fuel atomization, but also plays an important role in improving combustion efficiency and effectively reducing harmful emissions. Therefore, the application of microporous not only promotes the development of greening technology, but also promotes the innovation of microporous machining technology and process, which makes the special machining technology such as electric discharge technology (EDM) [1], electrochemical machining (ECM) [2], and laser machining (LM) [3] become the mainstream microporous machining methods.

However, micro-hole machining technology based on special machining methods such as EDM, laser machining and electron beam is also facing some technical challenges. Due to the existence of the recast layer, the changes in its composition, structure, hardness and strength affect the performance of micro-holes in use, so scholars at home and abroad have carried out a wide range of white research in the removal of the recast layer and its effects. On the one hand, some scholars have deeply studied the formation mechanism of the recast layer, and while revealing the physical and chemical processes of the formation of the recast layer, they have also proposed the introduction of new processing media (e.g., nano-fluid, carnauba oil, etc.) [4,5] to adjust the processing environment [6,7] (e.g., temperature, pressure, etc.) and the development of new machining tools [8,9] and other means, which have effectively reduced the thickness of the recast layer; on the other hand, some scholars have reduced the formation and thickness of the recast layer by optimizing the process parameters [10,11] (e.g., current, voltage, and processing speed, etc.); in addition, some scholars have utilized post-processing techniques (e.g., chemical etching [12], ion-beam polishing [13], and electron-beam evaporation [14], as well as femtosecond laser technology, etc.) to achieve efficient removal of the recast layer and improve the microporosity. efficient removal and improved processing quality of micro-holes.

Despite this, some major problems remain. The formation of recast layers involves complex physical and chemical processes, including melting, solidification, and phase transformation of the material. These processes are influenced by a variety of factors, such as processing temperature, pressure, material composition etc. Their complexity makes it difficult in controlling the formation of recast layers. During micro-hole machining, the optimization of machining parameters is crucial to reduce the recast layer. However, finding the best combination of parameters is a challenging task due to the diversity of machining parameters and the coupling of interactions. In addition, the optimal parameters may be different for different materials and machining conditions, which further increases the difficulty of parameter optimization. Limitations of post-processing technologies: chemical etching may introduce new surface defects or contaminants, while ion and electron beams may be limited by equipment cost and operability, resulting in restricted form geometry of the processed object. The problem of the balance between processing efficiency and quality. In the process of micro-hole machining, there is a certain contradiction between improving the processing efficiency and guaranteeing the processing quality, in order to reduce the formation of recast layer, it may be necessary to reduce the processing speed or processing steps, but this will sacrifice the processing efficiency. Therefore, based on the characteristics of electrochemical light finishing processing, this paper proposes the concept of using electrochemical anodic dissolution to remove the recast layer, and the process method of electrochemical removal of recast layer is studied. Finally, a two-step lifting process method is proposed to complete the micro-hole orifice trimming processing on the basis of satisfying the high quality removal of the recast layer, so that it can meet the specific use performance requirements.

2. Experimental programming

2.1 Formation mechanism and simulation of recast layer and its surface geometric characteristics

2.1.1 Formation mechanism and simulation of recast layer

Laser processing is the result of the superposition of multiple single molten pools, so the recast layer formed by laser processing is inextricably linked to the formation of the single molten pool recast layer, the basic process of which is shown in Figure 1(a). The material under a single pulsed laser irradiation is rapidly heated to the melting and boiling points, and melting and vaporization occur, forming a solid, liquid, and vapor three-phase coexistence. The melt flows along the walls of the processed area under a combination of vaporizing pressure and thermal influences, resulting in the formation of a concave contour of the melt pool that resembles a replica of the laser energy distribution curve. At the end of a single laser pulse, the vaporized metal in the center of the melt pool releases a large amount of thermal energy, which causes the melt attached to the walls of the melt pool and the curved melt at the bottom to be subjected to the recoil pressure of the metal vapor jet directly, which leads to the splashing of melt in the melt pool. Among them, part of the melt splashes out of the melt pool to form melt spatter, and accumulates at the edge of the melt pool to form a flying edge or burr, the simulation results are shown in Figure 1(b); the other part of the melt will be deposited and attached to the surface of the inner wall of the melt pool,, and under the action of the combined coupling force, the melt will flow downward along the wall surface and reach a static equilibrium to form a smooth and excessive linear melt. After rapid cooling, the surface of the single molten pool forms a recast layer covered by a fused cladding after being subjected to a pressure gradient and a temperature gradient. The final laser process consists of the superposition of individual melt pools in the axial direction of the processing area and the formation of a recast layer with a periodic stream-like distribution on the surface of the hole wall.

Figure 1. Schematic diagram of the simulation of recast layer formation

2.1.2 Surface geometry of the recast layer

As shown in Figure 2(a), at the surface of the initial melt pool, due to the large temperature gradient, the overflowed melt is mainly subject to Marangoni force, surface tension and gravity. The Marangoni force affects the flow of the melt, thus forming the shape of a domed dome; the surface tension and gravity in turn cause the overflowed melt to shrink outside the hole to form a ring-shaped accumulation.

As can be seen in Figure 2(b) below, in the upper part of the inner wall of the micro-hole, the recast layer has a smooth surface and relatively low thickness but is unevenly distributed along the periphery of the hole. Because the steam action is the strongest here, the melt pool is shallow, the metal vapor expansion rate is unevenly distributed, and a large amount of molten metal is randomly discharged, but under the action of the Marangoni force, it shows a smooth and excessive gully shape. In the 1/3 position from the top of the orifice, the recast layer surface formed a turbulent ripple-like undulation, randomly distributed in the circumferential direction, the thickness of a relative increase in the molten metal was a localized layer accumulation. Because with the non-stop stacking of the molten pool, the pressure gradient and the molten liquid velocity increase, so that the molten liquid layer of the explosion, the small size of the fast particles and the large size of the liquid particles with small velocity successively excluded by spattering, the vapor flow still has a high recoil force but weakened in relation to the orifice. In the middle and lower part of the micro-hole, the flow on the surface of the recast layer tends to stabilize, and the flow direction is more consistent, but the thickness reaches the maximum; at the same time, the layer spacing decreases, and a large number of overlapping melts blown down by the gas stream appear in the cross-section. Because here the melt pool flow rate has been lower, the melt by gravity and surface tension to reach equilibrium, the formation of a large number of droplets of solute accumulation, and at the same time because of the convection of the melt pool so that the melt can not be quickly discharged and thus re-melting and condensation to form a thicker recast layer. At the bottom of the micro-hole, the surface of the recast layer shows an obvious insufficient flow, because the steam effect here is weakened, and the melt sputtering becomes the main mode of discharge to increase the viscosity of the melt, which leads to the formation of localized agglomerates. Overall, the surface geometry of the laser-processed recast layer shows a stable and continuous periodic stripe distribution, which is characterized by an orderly wave-like microstructure on the macroscopic level and a spherical dome distribution of "peaks and valleys" on the microscopic level.

Figure 2. Simulation of surface geometric features of recast layer with SEM

2.2 Design of experimental platforms

Experimentally used electrolytic combination processing device shown in Figure 3, the device has electrochemical processing module and three-coordinate displacement platform, the platform is equipped with ${\textstyle X,Y,Z}$ three axes, three-axis linkage can be realized. ${\textstyle z}$-axis is equipped with electrochemical machining device, ${\textstyle XY}$ motion platform is equipped with a work box, electrochemical removal processing in the work box. The working box is made of plexiglass and resin materials, which not only realizes the electrical insulation from the machine tool, but also facilitates the observation of the experimental process.

Figure 3. Electrochemical light finishing processing experimental platform

2.3 Design of experimental methodology

2.3.1 Experimental flow design and preparation of samples

The basic idea of the experiment, the laser was first used to process the prefabricated holes in the 2 mm thick metal plate, and then electrochemical processing was used to remove the recast layer on the hole wall, the specific process is shown in Figure 4. Laser parameters for the optimized parameters: pulse width of 0.2ms, peak power of 16kW, repetition frequency of 70Hz, drilling process along the laser head sprayed 0.6MPa argon to accelerate the exclusion of molten material, to reduce the burr, melt spattering and residual in the hole wall of the recast layer. Electrochemical processing, the use of motion control systems to insert the electrode wire centered in the pre-fabricated holes, detect whether the short circuit and set the pulse power parameters. First turn on the electrolyte pump, wait for the electrolyte to flow evenly through the hole and see a clear and stable liquid column at the exit, turn on the pulse power supply to start electrolysis, observe the experimental phenomena and power supply parameter changes, and turn off the power supply according to the set electrolysis time. The electrolyte is a neutral solution configured with pure water, and the dynamic liquid flushing method is used. In the experiment, considering the experimental error, 3 holes were processed under each parameter, and after the completion of the processing, the samples were cleaned, polished and metallurgically corroded, and the micro-hole morphology was observed with a scanning electron microscope, and the pore diameter was observed and measured with an optical microscope, and the average value of the 3 holes' diameters was finally taken under the same parameter. The micro-hole morphology of the sample is shown in Figure 5.

Figure 4. Micro-hole machining process flowchart

Figure 5. Laser processing sample drawing

2.3.2 Determination of electrochemical processing method

In the electrochemical machining process, generally keep the processing current or processing voltage constant, with the increase of workpiece aperture, the electrolyte resistance increases gradually. When the current is constant, the voltage increases with the increase of the electrolyte resistance, and the current density decreases with the increase of the processed surface area when; when the voltage is constant, the current decreases with the increase of the electrolyte resistance, and the current density decreases rapidly with the decrease of the current and the increase of the processed surface area. In summary, it can be found that, when the voltage is constant, the decreasing trend of current density is greater than that of constant current, which is conducive to ensuring the processing efficiency in the early stage while ensuring the surface quality in the later stage of processing, so this paper adopts constant voltage for electrolytic processing.

2.3.3 Selection of electrolyte in electrochemical processing

Electrolytic processing, the electrode potential applied at the interface between the workpiece and the solution is different, the workpiece processing surface is in a different state, NaNO₃ solution is a passive electrolyte, the polarization curve of the anode of the workpiece into a non-linear distribution, passivation area of the metal surface to form an adsorbed oxygen layer or an oxide layer, i.e., passivation film; and precipitation of the oxygen area of the workpiece and the interface of the solution at the OH- ions easy to oxidize, precipitation of oxygen, precipitation of O₂ easy to oxidation Metal surface, so that the workpiece processing surface at the passivation film becomes thicker. The passivation film is less conductive, which in turn reduces the processing current density and lowers the dissolution rate of the metal. The NaCl solution is an active electrolyte, and the polarization curve of the anode of the workpiece is linearly distributed. The activation capacity of Cl- in the solution is very strong, the anode surface in a variety of conditions are in the active state of low-value dissolution, does not produce passivation and its side reactions, so the metal dissolution rate is higher, but at the same time there is a serious stray corrosion, which seriously affects the machining accuracy. In order to improve the nonlinear dissolution characteristics of the electrolyte, improve the current efficiency, soften the passivation salt film, this paper adopts a certain concentration of NaNO₃ solution mixed with a certain proportion of NaCl solution, in order to improve the machining efficiency at the same time easier to achieve the uneven dissolution or uneven super-passivation of the dissolution state, so as to improve the machining quality of the workpiece.

3. Results and analysis of removal of recast layer and orifice reshaping based on electrochemical light leveling

3.1 Effect of electrolysis parameters on the removal of recast layers

When constant voltage electrolysis is used, the appropriate voltage should be selected. Theoretically, when the electrolysis voltage increases, the amount of electricity passing through the surface increases and the electrolysis rate increases. The electrolysis products and heat in the processing gap also increase, increasing the flow resistance in the gap, resulting in a decrease in the flow of electrolyte, so that the ability to exclude electrolysis products and heat is reduced. When there is a serious imbalance between the two, abnormal phenomena such as evaporation, boiling, and cavitation will occur in the processing gap, leading to serious failures such as short circuits and scarring, resulting in the interruption of processing. In addition, with the increase in current density, the electric field strength around the electrode wire increases, the stray corrosion around the orifice is enhanced, and the orifice is not rounded after electrolysis. Therefore, the voltage increase should not destroy the above balance as a prerequisite for the given electrolytic processing conditions, need to choose a suitable voltage value. Take the electrolysis time of 30s, the electrolyte is the conductivity of 100.4ms/cm NaNO₃ and NaCl mixed solution solution, respectively, with a voltage of 3-18V (interval 3V) for the recast layer removal experiment.

After the completion of the experiment, the experimental results are processed: take the amount of hole expansion after processing and surface roughness change value as the reference amount; different voltage electrolysis, the expansion of the hole diameter is different, take the median value of the hole diameter in the process of electrolysis (with the minimum voltage electrolysis hole median value) as the benchmark, and take the change in the quality of the processing before and after the processing as an auxiliary reference for processing rate.

From the Figure 6, it can be seen that after the same time of electrolysis, the pore radius and electrolysis rate increased with the increase of electrolysis voltage. When the voltage is 15V, the recast layer is completely removed. At this time, the electrolysis current is 0.5 A. Calculation gives a minimum current density of 10.09 A/cm². The mass change and pore radius change shown in Figure 6 reflect that the electrolysis rate is approximately proportional to the voltage. When electrolysis removes the recast layer, the volume electrochemical equivalent ${\displaystyle \omega }$ of the material is a constant value, and the oxidation reaction of Ni, Fe, Cr, and other elements in the material occurs, in which Ni, Fe, and Cr will be oxidized with multiple valences according to the current density, and when the current density is greater than, all three elements are oxidized to the cation with the highest valence, i.e., the current efficiency ${\displaystyle \eta }$ is constant. When the electrolysis voltage is 6 V, the extension of the fitted primary curve can be found that the electrolysis rate is larger at this time. This is because when the current density is small, Ni, Fe, Cr elements are partially oxidized to low valence cations, at this time the current efficiency ${\displaystyle \eta }$ is larger, and the proportion of high valence oxidation of these three elements increases with the increase of current density, that is, the current efficiency decreases with the increase of current density.

Figure 6. Plot of the effect of voltage parameters on the removal of recast layer processing

Figure 7 shows the SEM and metallographic images of the control holes and after removal of the recast layer by light-needle processing, from which it can be seen that the holes are rounded after light-finishing processing, and the recast layer is completely removed and the hole wall is smooth. When the constant pressure finishing process, the higher the voltage, the higher the current density, the higher the electrolysis rate. But at the same time as the voltage increases due to the hole entrance surface near the tool cathode, the electric field effect than other parts of the General Assembly serious stray corrosion, and because the entrance surface to increase the auxiliary anode, and by providing a potential difference to change the direction of the electric field of the anode entrance surface to reduce the entrance of the stray corrosion, effective control of electrochemical reaction of the hole entrance near the upper surface of the hole inlet, practical electrolytic machining, generally according to the cut-off In actual electrolytic processing, the voltage is generally determined according to the cut-off voltage and the electrolytic processing area. Under the condition of ensuring the processing surface quality, the electrolytic voltage can be increased in order to obtain a higher electrolytic rate, and the electrolyte flow rate can be increased at the same time in order to meet the exclusion of electrolytic processing products and heat.

Figure 7. SEM and metallographic images after removal of the recast layer

3.2 Effect of pulsed power supply on the removal of recast layers

The traditional DC electrolytic machining has a strong ability to scatter etching and a weak ability to centralized etching, which affects the accuracy of electrolytic machining, the general three-dimensional surface molding accuracy of 0.2 to 0.5 mm, and the hole machining molding accuracy of 0.025 to 0.05 mm. High-frequency, narrow pulse electrolytic machining can achieve high precision (dimensional accuracy less than 5${\displaystyle \mu }$m, surface roughness less than 0.03mm), small gap (10 to 50${\displaystyle \mu }$m) machining, machining quality is greatly improved. The principle of pulse electrolytic processing is to replace the continuous DC power supply with periodic intermittent power supply, and the anode undergoes periodic intermittent dissolution in the electrolyte. It uses the intermittent depolarization of the pulse gap with power failure to dissipate heat from the workpiece, and the electrochemical properties, flow field, and electric field of the machining gap are restored to the starting state. Pulse current electrolysis, the gap produces hydrogen pressure wave synchronized with the pulse current, the frequency and intensity with the increase of pulse frequency and enhancement, which strengthens the stirring effect on the electrolyte, so that the flow field distribution in the gap tends to be uniform, and to improve the heat dissipation conditions in the gap, so as to make the minimum stable machining gap is greatly reduced, and thus to enhance the fixed-domain nature of electrolytic machining. Therefore, this paper compares the difference between the processing quality of pulse power supply and DC power supply, and finally chooses the processing method of pulse current electrolysis to get a higher electrolysis rate under the premise that the processing quality can be guaranteed.

As can be seen from Figure 8(a), the surface quality obtained after processing with DC as the processing power source is relatively rough, the recast layer is not completely removed, and the electrochemical dissolution effect is small, and because of the DC electrolytic processing scattering corrosion ability is strong, so that the stray corrosion area is larger and the corrosion effect is more obvious, and at the same time, due to the fact that by-products can't be eliminated in time during the DC processing and electrolytic solution can't be renewed in time, which makes the local resistance increase, resulting in the inner wall of the small holes in part of the recast layer was not removed. As can be seen from Figure 8(b), the use of pulsed current as the power supply processing after the surface is more smooth, the recast layer is completely removed and the processing quality is relatively stable, effectively solving the problems arising from the DC power supply, and further improving the quality and stability of the processing of electrolytic removal of the recast layer.

Figure 8. Comparison chart of AC power supply and DC power supply

3.3 Influence of cathodic lifting method on orifice reshaping

In order to further improve the machining effect of micro small hole workpieces, a staged combination of processing methods is proposed, based on the electrolytic removal of the recast layer, in the same station to further complete the small hole electrolytic repair of the machining method, i.e., cathode lifting method. The method is as follows: in the first stage, the surface quality of the workpiece is rapidly improved and the surface roughness is reduced by relatively large voltage processing to ensure the rapid removal of the recast layer; in the second stage, after the completion of the removal of the recast layer, the electric field distribution is influenced by changing the electrode position and the relatively small voltage is used for the trimming process to obtain a better surface quality and to realize the goal of the trimming of the micro-hole aperture. Due to the use of hybrid electrolyte, as mentioned above, the electrolyte has the characteristic of cutting the gap, when using small voltage processing, this time the inner wall of the hole although the current through, but the anode does not dissolve, the current efficiency is 0, which has been processed to protect the wall of the hole at the same time as the hole inlet trimming the shape of the hole to make the micro-hole has a better to meet the demand for the performance of the use of the micro-hole. Based on the above research, the experimental process parameters are as follows: the electrolyte used is a mixed solution of NaNO₃ and NaCl with a conductivity of 100.4ms/cm, the electrolysis voltage is 4v, and the electrolysis time is 20s.

From the Figure 9(a) can be seen, without improving the processing method workpiece due to the over-processing phase of the surface obtained on the more rough and micro-hole expansion obvious, poor machining locality, taper relative to the Figure 9(b) is greater, at the same time, the effect of repairing the shape of the micro-hole can not meet the demand for the use of the performance of the micro-hole is not obvious; from the Figure 9(b) can be seen, after the elevation of the method of low-pressure combination of machining, the inner wall of the hole in the cut off gap characteristics of the end of the end of the protection of the dissolution occurred, the machining of locality Higher, the height of the surface bulge is reduced, the surface of the workpiece becomes relatively flat, the taper relative to the Figure 9(a) has obvious improvement, while the effect of the repair effect is good, can meet the functional requirements of the micro-hole.

Figure 9. Comparison of the results of the cathodic lift method

4. Conclusions

In this paper, the use of electrochemical finishing processing method to remove the heavy casting layer left on the hole wall by laser processing and at the same time complete the orifice trimming processing can better meet the demand of micro-hole performance. Theoretical analysis and simulation of the formation mechanism of the recast layer and the surface geometrical features during laser processing were carried out, and experiments were carried out for the removal of the recast layer and orifice shaping, and small holes with the recast layer completely removed and the orifice rounded were obtained. Based on the experimental results, the following conclusions were obtained:

1. Laser processing is made by a single molten pool superimposed in the axial direction of the processing area, the molten material remaining in the molten pool in the combined force coupling role in the formation of a smooth excess of linear melt, and in the rapid cooling of the single molten pool surface in the pressure gradient and the temperature gradient, the formation of a recast layer covered by the fusion layer. The surface geometry of the laser-processed recast layer shows a stable and continuous periodic stripe distribution, which is characterized by an ordered wave-like microstructure on the macro level and a spherical dome distribution of "peaks and valleys" on the micro level.

2. Current density is an important parameter affecting the electrolysis rate. When using mixed solution finishing processing recast layer, the current density is greater than 10.09A/cm², the current efficiency remains unchanged, and the electrolysis rate is proportional to the electrolysis voltage. When the actual processing, the current density is too large easily lead to processing area scarring, short circuit and other quality defects. Therefore, according to the current density greater than 10.09A/cm² and electrolytic processing area to determine the current voltage, according to the electrolyte flow rate to increase the voltage to obtain a higher efficiency of finishing processing.

3. When the electrolyte with the same conductivity is used under constant pressure processing conditions, the surface quality of the surface processed with the mixed electrolyte of NaNO₃ and NaCl is good, and the recast layer is completely removed; while the surface processed with the electrolyte of NaCl is seriously corroded by stray corrosion, and the recast layer is not completely removed.

4. The pulse effect in pulse finishing, the passivation of the anode of the workpiece, the stirring effect of the hydrogen pressure wave and the maintenance of the stability of the flow and electric fields during the power-off gap play a key role in enhancing the surface quality of the finish after the removal of the recast layer by the finishing process.

5. When the cathodic lifting method is used to process the micro-hole, the truncation effect is utilized so that the inner wall surface of the micro-hole will not be processed twice while the orifice is trimmed to achieve the shape of the process, and ultimately the inner wall is smooth and the orifice is rounded to obtain a micro-hole with better usability.

Acknowledgements

This work was supported by Application of corrosion protection and online monitoring technology based on chlor-alkali industry (ZYYD2023B03).

References