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Circular Halbach array integrated using an abrasive circulating system during the ultra-precision machining of polymethyl methacrylate optical material

2026-08-17
Circular Halbach array integrated using an abrasive circulating system during the ultra-precision machining of polymethyl methacrylate optical material

Circular Halbach array integrated using an abrasive circulating system during the ultra-precision machining of polymethyl methacrylate optical material

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Abstract

A novel approach to enhancing the efficacy and surface quality of magnetic polishing involves the incorporation of a magnetic liquid circulation system for abrasive particle regeneration in conjunction with a circular Halbach array. The continuous renewal of abrasive particles within the polishing zone is realised through a conveyor belt that transports new abrasive particles into the polishing liquid solution. This formation of a continuously circulating polishing system ensures uninterrupted magnetic finishing processes and maintains stability throughout the polishing operation. This study extensively explores polishing force distribution, magnetic field distribution and abrasive grain behaviour in the polishing area facilitated by the magnetic liquid solution. The application of the proposed polishing processes to polymethyl methacrylate, an optical lens material, aims to comprehend the characteristics and validate the feasibility of the polishing method. Key influencing factors in the magnetic polishing process, including abrasive grain size, magnetic particle, polishing distance and conveyor speed to surface quantity, are examined through experimental analysis. Results of the experimental polishing processes demonstrate that the utilisation of circular Halbach arrays with circulating abrasives produces a nanometric surface finish. Even in the polishing of polymethyl methacrylate with an initial rough surface (Ra = 464.895 nm), the process achieves an ultra-fine level with Ra below 9 nm without disruption in the material polishing processes of optical lenses.

Keywords

Ultra-precision
Polymethyl methacrylate
Circular Halbach array
Polishing
Magnetorheological finishing

1. Introduction

In recent years, the pursuit of enhanced surface quality in machined workpieces has led to the emergence of magnetic ribbon-structured cutting tools, which are produced through magnetorheological finishing (MRF) as an advanced and effective means of achieving ultra-smooth surfaces [1,2]. This innovative approach controls the influence of a magnetic field to create a flexible magnetic chain polishing tool by combining magnetic and abrasive particles. This unique tool enhances contact with the workpiece surface, increasing its adaptability to a diverse range of surface shapes, including complex free, flat or grooved and gear surfaces, amongst others [3,4]. The MRF polishing process generates a ribbon that applies minimal pressure to the workpiece surface during polishing, mitigating damage and defects typically associated with machining processes. Consequently, the application of this technology contributes to a substantial improvement in surface quality. Notably, the efficacy of this technology extends to non-magnetic materials such as optical lens materials, titanium and its alloys, stainless steels, ceramics and various other materials [[5], [6], [7]]. The versatility of MRF magnetic polishing technology has led to its increasing adoption in diverse surface finishing applications, particularly in optical equipment, medical devices, and aerospace equipment [[8], [9], [10]].
The explanation of the MRF polishing process characteristics and fundamental principles has been carefully outlined through the research endeavours of [11,12]. Their investigations revealed a direct correlation between material removal and abrasive grain diameter, highlighting that an increase in abrasive grain diameter is accompanied by a proportional decrease in surface quality under established polishing conditions. Furthermore, Shinmuda et al. have demonstrated that increasing magnetic field density, in combination with the formation of high-quality planes, contributes to enhanced polishing efficiency and improved surface quality. The current research endeavours to integrate ultrasound, vibration and chemical elements into the existing MRF polishing process to improve surface quality and polishing efficacy [13,14]. Notably, when subjected to magnetic fields during flat surface polishing, MRF ribbons follow specific orbits that exert a uniform influence on the machined surface. This orbital movement, which is controlled by a meticulously planned trajectory, enhances precision and accuracy in the polishing process [[15], [16], [17]]. Zuo et al. [18] have introduced a model that systematically calculates the polishing trajectory of the MRF ribbon. This model proposes a novel approach by utilising rare earth magnet coils instead of traditional bead chains, particularly beneficial for the application of magnetic forces to the inner lumens of thick tube surfaces comprising non-magnetic materials. The aforementioned studies also investigated the characteristics and principles of MRF polishing under various magnetic strengths, offering valuable insights into the versatility and adaptability of the MRF polishing process across different applications.
The Halbach array comprises magnets organised radially or in parallel according to specific spatial rules [19,20]. Conducting initial investigations on the spatial magnetic field model of a single permanent magnet under static conditions is necessary for the development of a robust magnetic field model for the polishing zone in Halbach array-based magnetic polishing processes involving permanent magnets. Drawing on the modelling approaches described in the literature [21,22], the internal molecular currents counterbalance each other with the uniform magnetisation of a permanent magnet. Consequently, the magnetic field at any point in the space outside a permanent magnet can be excited by a circulating molecular current on its surface. In magnetorheological fluid polishing processes, the Halbach array can notably enhance and expand the magnetic field, thereby improving the polishing performance. Halbach arrays have recently gained considerable interest from researchers in physics and mathematics due to their exceptional capability to enhance magnetic field distribution in electromagnetic systems [22]. The Halbach array, which comprises alternating primary and transition permanent magnets, can be configured either linearly or circularly [23]. The distinctive feature of this array lies in its capability to generate a strong, focused magnetic field on one side (referred to as the active side) whilst reducing it on the opposite side (referred to as the quiet side) [24]. The performance of systems utilising these arrays is substantially improved by this feature. In the context of MRF polishing processes, the Halbach array plays a crucial role by augmenting the magnetic force in the polishing area [25,26]. The Halbach array enhances the polishing performance by intensifying the magnetic force in the active region, thus reducing the required time and resources to achieve the desired outcome [26]. This enhancement can lead to high operational efficiency and reduced machining time in industrial settings [27]. Additionally, the Halbach array contributes to the improvement of the polishing process by minimising magnetic field disturbances on the quiet side [28]. This minimisation is critical for enhancing process accuracy and reliability, particularly in applications demanding high precision, such as MRF polishing of optical and electronic components [27]. The capability to produce a strong, concentrated magnetic field on the active side whilst minimising the magnetic field intensity on the quiet side not only boosts MRF polishing efficiency but also introduces numerous potential applications in various high-tech fields.
Previous research has synthesised key factors that impact the performance of the MRF process [29,30], indicating vital improvements to enhance its effectiveness. This synthesis positively contributed to the advancement of the MRF polishing model. However, the MRF process encounters notable challenges when applied in practical manufacturing settings. A key aspect lies in the polishing performance generated by the MRF ribbon, which is crucial in the surface finishing procedure. The quantity of abrasives within the MRF ribbon engaged in the polishing operation is restricted in the traditional MRF process. Additionally, the positions of these abrasives in the MRF ribbon remain relatively fixed, resulting in the erosion of abrasive grain edges and a subsequent deterioration in MRF polishing effectiveness over time. Furthermore, the initial structure of the MRF ribbon undergoes alterations due to deformations caused by pressure during contact with the workpiece, adversely affecting the precision of the MRF process [31,32]. These issues also contribute to the degradation of the MRF finishing process, necessitating frequent replacement or replenishment of the MRF in the polishing zone and consequent modifications to the polishing tool structure over time. QED Company has successfully developed polishing equipment utilising MRF polishing technology to address these challenges [33,34]. This equipment can adeptly manipulate the shape and angle of optical components with diverse configurations, providing remarkable contributions to the precision machining of optical elements. However, the application of this technology in conventional laboratories encounters obstacles due to the requirements for magnetic fluid measurement and instrumentation.
This study introduces an innovative MRF technique that employs a circular Halbach magnetic field array to address the challenges outlined above and establish a robust magnetic field for influencing magnetic particles in the MRF process. Additionally, the study integrates a conveyor system to regenerate abrasive particles. The main objective is to investigate the unique aspects of the MRF polishing process that utilises circular Halbach arrays in conjunction with regenerated abrasives. The focus is on analysing the magnetic field distribution within the polishing area generated by circular Halbach arrays and understanding critical factors such as polishing force and machining behaviour in the MRF polishing process. The proposed model is validated by constructing a test apparatus and performing a series of experiments. These experiments specifically examine the influence of key parameters in the proposed MRF polishing approach, including abrasive grain size, conveyor speed and working distance. The aim is to evaluate their impact on the uniformity and quality of the workpiece surface. The results are expected to provide insights into the optimisation of the MRF polishing process for enhanced performance and efficiency.

2. Methodology polishing with circular Halbach array and regenerative abrasive particles

The methodology involves the application of the MRF polishing process, incorporating circular Halbach arrays and regenerated abrasives. The execution of this process is demonstrated by a specialised structure and polishing model, as depicted in Fig. 1. The figure offers a visual explanation of the essential components and their spatial arrangement within the MRF polishing setup.
Fig. 1
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Fig. 1. Schematic of the MRF polishing equipment with a circular Halbach array and regenerative abrasives.

Fig. 2 shows the principle of the material removal rate (MRR) of the MRF process with carbonyl iron particles (CIPs) and abrasive particles. The polishing process with regenerative abrasives and Halbach arrays is established through the following sequence: firstly, after receiving a rotation from the motor, the rotation of the active wheel of the conveyor (wheel number 1) causes the passive wheel of the conveyor belt (wheel number 2). This number 2 wheel is a circular Halbach array, providing a magnetic field during the machining process. When wheel number 2 is brought close to the machining area, the polishing liquid is brought from the tank to the surface of the conveyor belt through the peristaltic pump. This polishing liquid mixture comprises CIP, AP and deionized water mixed together. Immediately under the strong impact of the magnetic field, the CIP and AP in the machining area are tightly held on the conveyor surface and arranged into particle clusters along the magnetic field lines. At this time, after the workpiece is provided with rotational motion, CIP and AP clusters are brought to the surface through the support of the conveyor belt to remove excess material. After performing its task, the MRF polishing solution is then taken out of the processing area through a conveyor belt and collected into the storage tank. This process is performed in repeated cycles until the material surface reaches the required quality.
Fig. 2
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Fig. 2. MRF polishing principle with circular Halbach array and regenerative abrasives.

Fig. 2 shows the magnetic induction lines, delineating the trajectories along which magnetic particles are attracted and mobilised. Particles within the machining zone are subjected to forces FX and FY, which are oriented along the magnetic force lines in the x and y directions, respectively. These forces constitute the components of the magnetic force exerted on the magnetic particles located on the conveyor belt. The expression for these forces is presented in the following equation [35]:(1)(2)where V represents the volume of the magnetic particle;  denotes the magnetic susceptibility of the magnetic particle; H and  represent magnetic field density and vacuum magnetic permeability, respectively. The gradient in the x, y direction is determined by , . Fig. 2 shows the magnetic force acting on magnetic particles [36] with the formula as described below:(3)where  denotes the tilt angle of the magnetic induction line. The magnetic force is determined by FM [25] with mathematically expressed as follows:(4)where R represents the radius of the magnetic particle, XR is the specific susceptibility of the magnetic particle and μ0 is the magnetic permeability of the vacuum. The interaction between magnetic particles and the magnetic field force facilitates the tight adherence of the MRF slurry to the belt and its aligned movement with the belt direction, effectively serving as machining tools to complete the process, as depicted in Fig. 2.
This magnetic field is realised through a circular Halbach magnetic field array, comprising Nd–Fe–B permanent magnets embedded in the grinding wheel. The grinding wheel features an outer diameter and thickness of 82 and 20 mm, respectively, and is equipped with 18 magnets, each possessing a 12 mm diameter. The magnetic strength of each magnet is measured at 0.25 T. The grinding disc is constructed from plastic and is mounted with a shaft made of SUS304 to ensure precision and minimise magnetic interference within the polishing region. This design choice effectively reduces unnecessary magnetic influences in the vicinity of the polishing process. The drive wheel, which comprises aluminium, imparts motion to the grinding wheel and the Halbach array. The conveyor belt, which is constructed from polyurethane material with dimensions of 20 mm × 1.5 mm, acts as the polishing pad between the circular Halbach array and the MRF carrier solution, contributing to the formation of ribbons, as illustrated in Fig. 2. These MRF ribbons exhibit flexibility during the polishing process. During the rotation of the grinding wheel, the conveyor belt carrying the MRF ribbon forms a flexible polishing tool due to the magnetic field. In the presence of relative movement to the workpiece surface, the abrasive particles within the semi-solid ribbon layer efficiently remove residues with minimal impact on the machined surface. As shown in Fig. 2, specific arrangements are defined on the grinding wheel to characterise the force components in the x, y and z directions within the Cartesian coordinate system Oxyz.

3. Analysis of magnetic field distribution and polishing force

3.1. Analysis of the magnetic field distribution in the polishing area

An MRF mixture, which comprises magnetic particles, abrasive particles and a polishing solution, is initially used for the surface finishing of polymethyl methacrylate (PMMA) workpieces. This MRF mixture becomes flexible under the influence of a magnetic field, ensuring a uniform impact on the surface of the workpiece. Subsequently, a regeneration mechanism for abrasive particles is implemented via the conveyor system, which serves as a guiding tool in the magnetic fluid mixture. The magnetic finishing fluid facilitates the formation of an MRF ribbon on the conveyor surface when passing through the applied magnetic field. The application of COMSOL Multiphysics software in the current study offered critical insights into the magnetic field dynamics integral to the MRF process. Various configuration options were investigated to evaluate the role and enhancement capabilities of the Halbach array. Fig. 3 illustrates the magnetic field distribution and strength for three different magnet arrangement methods. Specifically, Fig. 3a–c depict the circle ring magnet, the alternating magnet array and the arrangement method of the Halbach array, respectively. The results show that the ring magnet generates a uniform magnetic field over the entire magnet surface; however, the magnetic field strength is lower than that of the alternating magnet array. By contrast, Halbach arrays concentrate on magnetic field clusters at the outer surface, especially in areas with a higher magnetic field intensity than those of alternating magnet arrays and circle ring magnets.
Fig. 3
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Fig. 3. Different magnet arrangement methods applied to circular polishing wheels.

Fig. 4 provides a detailed description of the magnetic field strength in a cross-section of the magnet placement methods in the polishing area. Notably, the magnetic field intensity at the polishing area with the Halbach array is always greater than 0.2 T, ensuring MRF polishing efficiency, as researched by Xie et al. [37]. In addition, Fig. 4 shows that the magnetic field strength at the polishing area is higher with the Halbach array than with the sequential and ring magnet arrays [38]. This finding confirms the outstanding effectiveness of the Halbach array in creating a strong and concentrated magnetic field in the area required for the polishing process. With the effect of the circular Halbach array, the magnetic field capacity is higher than that of conventional circular magnets, thereby creating an MRF ribbon with higher hardness and providing better material removal in the polishing area. Furthermore, with the proposed polishing pattern, the MRF ribbon layer moves synchronously with the conveyor belt, allowing the circulation and reuse of the fluid mixture after each polishing cycle. This continuous circulation ensures the stability of the polishing process, eliminating interruptions and issues associated with abrasive wear and deformation during polishing. Compared to the traditional MRF polishing method, wherein stable abrasive particles are confined within the MRF volume and polishing area, the proposed MRF process with regenerated abrasive particles demonstrates enhanced capabilities in achieving a smooth and uniformly polished surface. This improvement optimises the overall polishing performance and addresses concerns related to abrasive wear and deformation during the polishing process.
Fig. 4
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Fig. 4. Magnetic field measurement and simulation results at the polishing area with different magnet arrangement methods.

The simulation results in Fig. 5 offer a comprehensive analysis of the magnetic field intensity of the circular Halbach array across different perspectives: the coordinate plane perpendicular to the axis (Oxz plane), the axial cross-section and the direction of the magnetic field vector aligned with the Halbach array. The magnetic force is notably concentrated towards the outer region of the Halbach array. This concentration results in a robust magnetic force, enhancing its efficacy in various polishing processes. Notably, Fig. 5 provides valuable insights into the orientation of the magnetic field vector, specifically in the context of the polishing wheel equipped with a circular Halbach array and conveyor belt. A close examination of the left side reveals a distinct direction of the magnetic field vector on the conveyor belt. The simulation illustrates that the magnetic field lines originate from one side of the conveyor belt, run parallel in the middle and then alter their course to move towards the opposite side of the conveyor belt. Specifically, the magnetic particles in the magnetic abrasive fluid organise into a dome structure, creating a magnetic chain region on the surface of the conveyor belt under the influence of the magnetic field lines. The feasibility and effectiveness of the polishing operation are enhanced by this formation.
Fig. 5
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Fig. 5. Simulation results of magnetic field vector direction with circular Halbach array.

Magnetisation stabilises when the external magnetic field increases to a point where magnetic particles align in a straight chain. The intrinsic magnetic properties of the material are demonstrated by this saturated magnetic state. Fig. 6 illustrates the magnetic domain characteristics of CIP under various conditions, revealing the formation of magnetic beads in the presence or absence of an applied magnetic field. Additionally, this figure depicts the arrangement of abrasive and magnetic particles in the MRF process under an external magnetic field.
Fig. 6
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Fig. 6. Formation of a magnetic polishing chain occurs under the influence of an external magnetic field.

The MRF polishing process is distinguished by the strategic distribution of magnetic forces within the polishing area, realised through a methodical arrangement of magnetic particles. This arrangement is crucial for producing a holding force that acts on abrasive particles, facilitating the removal of machining residue in the presence of a relative movement between the abrasive grain and the workpiece surface. Simulation results are leveraged and complemented by validation through the YF-801EXP gauss metre magnetic field density measuring device to determine the force distribution within the machining area. As shown in Fig. 7b, the polishing area corresponds to the region where magnetic field density measurements are conducted. The magnetic field distribution is systematically explored at various positions to analyse the distribution of magnetic field density across the surface of the grinding wheel. This detailed investigation aims to understand the nuanced variations in magnetic forces and their implications for the efficiency and effectiveness of the MRF polishing process.
Fig. 7
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Fig. 7. Method for determining magnetic field density in the polished area with circular Halbach array.

As illustrated in Fig. 7b, at various predefined measuring positions, measurements are conducted from the centre of the grinding wheel on the conveyor surface, specifically at a distance of 1 mm. Compared to the simulated results, the measurement results are used to validate the applied magnetic force on the polishing area. The measurement direction is established perpendicular to the conveyor surface, with the sensor head oriented downward towards the conveyor surface. Fig. 8 shows the measurement and analysis results at various positions. A comparative analysis between the simulation results and the measured values reveals a notable proximity, affirming the reliability of the measurement theory and simulation outcomes. However, the smaller measured magnetic field intensity than the simulation results may be attributed to the imperfect simulation of the manufacturing process of the magnets. The structural configuration induces a high magnetic field flow, creating a series connection to the magnetic field flow. The grinding wheel carrying the magnetic Halbach array facilitates this configuration, which is initiated and finalised by the conveyor positions during the magnetic liquid mixture operation in the magnetic field area. The Halbach array plays a pivotal role in fostering connectivity between magnetic and abrasive particles within the polishing solution, culminating in the formation of a flexible and adaptable polishing tool.
Fig. 8
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Fig. 8. Magnetic field measurement and simulation results at the polishing area with circular Halbach array.

3.2. MRF polishing force with circular Halbach array

The cutting force emerges as a pivotal factor within the framework of MRF processes, which are dedicated to unravelling the material removal mechanism and its remarkable impact on surface quality. This study carefully examines and analyses the influences of cutting force components on surface finish and working distance. The subsequent discussions contribute to a nuanced comprehension of the intricate polishing mechanism facilitated by the circular Halbach array in the MRF polishing process. This investigation aims to enhance our understanding of the multifaceted interactions governing polishing dynamics and their ultimate implications for surface quality.
In the MRF polishing process using circular Halbach arrays, the increase in applied magnetic field intensity to the MRF mixture leads to a stiff MRF ribbon due to the increased magnetic force (FM), as described by Equation (4). A stiff MRF ribbon is directly correlated with increased magnetic field intensity, which is facilitated by the circular Halbach arrays. High normal (FN) and tangential (FT) components of the applied force to the workpiece surface can be attributed to such an enhancement in magnetic force, thereby improving the polishing efficiency. Fig. 9 presents a schematic illustration of the measurement process of the force components (FN) and (FT). As illustrated in Fig. 9, the force measurement system utilises the Kistler Type 9139AA three-component force sensor from the Swiss company Kistler. This sensor is employed to measure the three-component cutting forces with a sensitivity of up to 0.1 N within a measuring range of −3 to 3 kN. The generated signals are processed and collected through a 3160-B-042 signal processor, coupled with a computer. The DynoWare software (Kistler) is employed to streamline the collection and analysis of force data. The polishing distance (K) between the machined surface and the established polishing belt is also depicted. The integration of the CNC programme into the proposed polishing model enables the facile control of various working distances, as detailed in Table 1, which outlines the corresponding measuring conditions. This comprehensive force measurement system, combined with precise control over working distances, increases the accuracy and reliability of the collected data during the MRF process.
Fig. 9
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Fig. 9. Three-component force measurement system applied to MRF polishing process with circular Halbach array.

Table 1. Setting parameters.

Numerical order Characteristic Note
Magnetic iron carbonyl particles Diameter
1 μm		 Diameter
3 μm		 Diameter
12 μm		 200 ml MRF
1 Polishing solution Deionized water
2 Al2O3 abrasive particles Average diameter 3 μm 25 g
3 Polishing distance 0.1 mm; 0.2 mm; 0.3 mm; 0.4 mm; 0.5 mm
4 Conveyor speed 50 rpm
5 MRF solution flow 20 ml/min
Fig. 10 shows distinct polishing forces due to various polishing distances when different circular Halbach arrays are employed. The analysis of force outcomes reveals a notable difference between the tangential and normal force components. Specifically, the tangential force component is substantially smaller compared to the normal force component, and both force values experience a pronounced decline with the increase in the polishing distance. The impact of a magnetic field induces the movement of magnetic particles, dragging abrasive particles in the solution along the distribution of the magnetic field lines. Consequently, the distribution of the magnetic field magnitude plays a pivotal role in influencing the polishing performance in MRF. The pressure exerted by the MRF magnetic ribbon on the workpiece surface decreases as the polishing distance increases, leading to a reduction in normal and tangential forces. In scenarios involving large-sized magnetic particles, the suction force acting on these particles intensifies, resulting in a substantial impact force on the workpiece surface.
Fig. 10
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Fig. 10. Impact of magnetic particle size and polishing distance on polishing force.

4. MRF polishing experiment with circular Halbach arrays and regenerated abrasives

As depicted in Fig. 11, the experimental device, in which the MRF ribbon participates in the polishing process, is located in the centre of the grinding wheel and moves along the conveyor belt away from the polishing area under the conveyor belt effect. Above the MRF ribbon surface, under the effect of a circular Halbach array with abrasive particles held tightly by magnetic particles and pushed to move away from the circular Halbach array surface, the abrasive particles tend to gradually move towards the machined surface. Consequently, the abrasive particles exhibit a tendency to advance towards the machined surface, establishing direct contact and affecting efficient material removal in the polishing processes.
Fig. 11
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Fig. 11. Experimental apparatus and the associated MRF polishing area with a circular Halbach array and regenerative abrasives.

The flow of the MRF fluid under the influence of a magnetic field controls the characteristic polishing behaviour of the MRF method, resulting in the creation of a semi-solid ribbon. Abrasive particles tightly adhere to the magnetic particles through the interaction between the MRF fluid and the workpiece surface, creating a cutting force that introduces scratches on the machined surface in a specific direction. Subsequently, the strip of MRF polishing fluid immediately exits the polishing area with the wearing of the abrasive particles due to prolonged contact with the workpiece surface. A replenishment of fresh magnetic polishing fluid promptly occurs as the abrasive particles undergo wear, ensuring the uninterrupted continuation of the polishing process. Additionally, the MRF polishing fluid, which is responsive to the influence of a magnetic field, exhibits flexibility, enabling continuous and adaptive interaction with the machined surfaces of various shapes. The strategic alteration of the MRF liquid solution throughout the polishing process facilitates the renewal and regeneration of abrasive particles, further enhancing the stability and performance of the MRF finishing process. This dynamic fluidity and adaptability contribute to the uninterrupted nature of the polishing process, refining its stability and overall effectiveness.
Fig. 11 depicts the experimental apparatus set up with a circular Halbach array with regenerative abrasive particles. The PMMA optical material has a diameter of 10 mm and a height of 12 mm. The workpiece, which is mounted on a three-component force-measuring device, can move in the X and Y directions of the machine table. The grinding wheel carries a circular Halbach array that can move up and down in the Z direction, controlled by a stepper motor, thereby easily determining the polishing distance K according to the CNC control programme set up on the computer and the template. The stepper motor transmits torque to the active shaft to facilitate movement for the conveyor belt and easily adjust its speed.
The experimental methodology for MRF polishing, which incorporates circular Halbach arrays and regenerative abrasives, was implemented on PMMA workpieces to perform a systematic examination of the impact of process parameters. Parameters such as abrasive size, conveyor speed and polishing distance to the workpiece surface were investigated to assess their influence on surface quality. A maintenance procedure was also implemented to ensure the stability of the MRF liquid mixture throughout the polishing process. Specifically, 1 ml of MRF solution was added after 30 min of polishing to replenish the liquid lost during the polishing process. Surface quality was quantified through surface roughness measurements performed using a Zygo 7100 profilometer. Surface roughness was determined based on the arithmetic mean roughness (Ra) parameter, in accordance with ISO 4287:1997 and ISO 4288:1996 standards. The equipment was calibrated to these standards to ensure accuracy and consistency in workpiece measurements.

5. Results and discussion of the polishing process with circular Halbach array with regenerative abrasives in polishing PMMA optical lens materials

5.1. Influence of magnetic particle size on surface quality during MRF polishing with circular Halbach arrays and regenerative abrasives

In the MRF polishing process utilising circular Halbach arrays and regenerative abrasives, the role of magnetic particles is crucial in shaping the polishing tool. The arrangement of magnetic particles on the conveyor surface is a critical contributing factor to the formation of a robust MRF ribbon during polishing. Fig. 12 illustrates the impact of different abrasive grain sizes on surface quality, and Table 2 outlines the experimental parameters. A magnetic particle diameter of 12 μm provides high polishing performance, revealing a rapidly decreasing polishing quality from 476.192 nm to 176.096 nm after 15 min of polishing, continuing polishing for up to 30 min of surface quality down to 85.188 nm. However, if polishing continues, then the surface quality does not substantially improve. The critical surface roughness level with 12 μm diameter abrasive particles is obtained at 75 nm when polished for 45 min. A notable improvement with a magnetic particle diameter of 3 μm demonstrates optimal polishing performance. The machining surface roughness of the 3 μm magnetic particle is notably higher than the 12 μm magnetic particle after 45 min of polishing. However, with the continuous increase in the polishing time, the surface quality is substantially improved compared to the 12 μm magnetic particle. The obtained surface corresponds to a surface roughness of 15 nm after 90 min of polishing.
Fig. 12
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Fig. 12. Surface roughness according to different abrasive particle sizes during MRF polishing of PMMA optical lens material with circular Halbach array and regenerative abrasives.

Table 2. Experimental parameters.

Numerical order Characteristic Technology parameters Unit
1 Workpiece speed 60 rpm
2 Flow rate from peristaltic pump 20 ml/min
3 Polishing time 90 min
4 Polishing distance 0.2 mm
5 Conveyor belt speed 30 rpm
6 Al2O3 abrasive particles (weight 25 g) Average diameter 1 μm
7 Carbonyl iron powder weight 200.0 g Average diameter 1 μm
Average diameter 3 μm
Average diameter 12 μm
Fig. 13 shows the surface quality before and after polishing with different-sized magnetic particles. Compared to 12 μm diameter magnetic particles, the surface quality of 1 μm diameter magnetic particles is substantially improved after 90 min of polishing. However, the capability to reduce gloss and surface quality over time is less effective than magnetic particles with a diameter of 3 μm. Equation (1) explains this phenomenon, in which the increase in magnetic force acting on the particles can be attributed to the increasing magnetic particle diameter whilst maintaining other technological parameters constant. The increase in hardness of the MRF tape formed on the conveyor surface is due to the enhanced interaction force between magnetic particles and the impact magnetic field source. Therefore, slippage between particles in the MRF ribbon is reduced, promoting highly efficient movement on the conveyor belt and creating high material removal. In addition, magnetic particles with large diameters contribute to the scratching process of the machined surface, thereby reducing surface quality when the diameter is excessively large. By contrast, a substantially small diameter weakens the magnetic force and reduces the material removal capability, thereby deteriorating polishing performance.
Fig. 13
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Fig. 13. Surface morphology when polished with various magnetic particle sizes.

5.2. Abrasive particle size in the polishing process with circular Halbach array

In the field of MRF processes, the size of abrasive particles assumes a critical role, directly influencing the quality of the final product surface. Based on the polishing conditions outlined in Table 3, Fig. 14 depicts diverse surface qualities corresponding to distinct abrasive grain sizes. Notably, substantial variations in surface quality are observed when altering the abrasive grain size after a 90 min polishing duration. Surface quality metrics, which are measured in terms of surface roughness, reveal values of 28.892, 15.052, 52.371 and 112.729 nm for abrasive particle sizes of 3, 1, 8 and 15 μm, respectively. After 90 min of polishing, the most favourable surface quality is realised with 1 μm abrasive particles. Furthermore, the fastest reduction in surface roughness is associated with 3 μm abrasive particles, whilst the least efficient polishing is observed with 15 μm abrasive particles. These findings highlight the critical influence of abrasive grain size on the overall efficacy and quality outcomes of the MRF polishing process.

Table 3. Experimental parameters.

Numerical order Characteristic Technology parameters Unit
1 Workpiece speed 200 rpm
2 Flow rate from peristaltic pump 20 ml/min
3 Polishing time 90 min
4 Polishing distance 0.1 mm
5 Conveyor belt speed 30 rpm
6 Al2O3 abrasive particles (weight 25 g) Average diameter 1 μm
Average diameter 3 μm
Average diameter 8 μm
Average diameter 15 μm
7 Carbonyl iron powder weight 200.0 g Average diameter 3 μm
Fig. 14
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Fig. 14. Surface quality according to grain size by MRF polishing process with circular Halbach array with regenerated abrasives.

From the perspective elucidated below, the rationale behind the aforementioned phenomenon is expounded as follows: the magnetic particles undergo integration and arrangement with the guidance of the magnetic force flow, moving synchronously in the conveyor belt direction and exerting a close impact on the product surface. Within this context, the critical relationship between magnetic and abrasive particle sizes in MRF polishing processes becomes vital. The supportive capacity of the magnetic field contributes to the movement of abrasive particles along the magnetic force flow. This support diminishes in instances where the abrasive grain size is excessively large, resulting in reduced movement towards the polishing area. Consequently, the number of actively participating abrasive particles in the polishing process decreases, intensified by the weakened holding forces exerted by the magnetic field on the abrasive particles. This phenomenon leads to an ineffective cutting effect. Furthermore, aggregation occurs upon contact with the workpiece surface due to the insufficient force applied by the magnetic particles on the abrasive particle and the large abrasive particle size. This aggregation induces heightened friction between the abrasive grain and the workpiece surface, causing localised and deep scratches. Consequently, the overall polishing efficacy is compromised.
MRF polishing encounters limitations in efficiency when utilising excessively small abrasive particles. This constraint arises from the inherent limitations in cutting capability associated with diminutive abrasive particles. Experimental data indicate that the largest reduction in surface roughness occurs when employing abrasive particles with a size of 15 μm. Conversely, superior surface quality is achieved by utilising abrasive particles smaller than 3 μm. The outcomes of the experiments reveal that abrasive particles measuring 1 and 3 μm exhibit commendable magnetic field support and possess well-defined cutting edges. However, the 3 μm abrasive initially provides a superior surface quality; after 60 min of polishing, the 1 μm abrasive demonstrates superior performance in surface refinement, resulting in a smooth finish and enhanced overall quality. Consequently, abrasive particles with a size of 1 μm are highly suitable for incorporation into the proposed experimental procedures. This selection is based on their optimal balance between effective surface roughening and superior surface quality improvement.
Simultaneously, acknowledging that the utilisation of 15 μm abrasive particles results in a simultaneous decline in polishing efficacy and the ultimate surface quality is crucial. This finding highlights the impracticality of optimising the voluminous 15 μm abrasive particles with magnetic particles, leading to a coiling tendency in localised regions. Consequently, this coiling phenomenon hinders the effective operation of the cutting edge of the abrasive. Conversely, the application of abrasive particles measuring 1 and 3 μm yields a consistently uniform final surface quality. This finding is attributable to the ease with which the 1 and 3 μm abrasive particles can be optimised by magnetic particles, thereby minimising undesirable coiling tendencies. This optimisation facilitates uniform contact of abrasive particles with the product surface, preventing accumulation at specific locations. Therefore, the cutting edges of 1 and 3 μm abrasive particles are prepared to realise a more efficient operation compared to their 8 and 15 μm counterparts, respectively. Thus, the polishing performance of 1 and 3 μm abrasive particles for the surface is almost the same with negligible difference.

5.3. Surface quality according to polishing distance with circular Halbach array and regenerative abrasives

A crucial facet of MRF finishing using circular Halbach arrays and regenerative abrasives lies in conducting experiments to determine the impact of the polishing distance on the quality of the machined surface. In the MRF polishing process employing a circular Halbach array with an extensive polishing distance, under the influence of the magnetic field, the MRF ribbon may not optimally engage with the workpiece surface in the polishing area. Conversely, if the working distance is excessively small, then excessive liquid pressure can be generated on the workpiece surface, leading to undesirable scratches and a lack of improvement in surface quality. Therefore, the establishment of varying polishing distances is necessary to identify the most effective distance. As illustrated in Fig. 15, surface quality exhibits variations corresponding to changes in the polishing distance, with detailed polishing conditions outlined in Table 4. Alterations in the polishing area result in enhanced polishing performance at a polishing time of 90 min, as reflected in reduced surface roughness. The hardness of the MRF ribbon in contact with the workpiece surface experiences variations at different polishing distances due to the distinct effects of polishing force and magnetic field on material removal capacity, as elucidated by the proposed formula derived from Refs. [31,39]. Initiating from an initial surface roughness of 500 nm, the values decrease to 52.751 and 13.657 nm, corresponding to polishing distances of 0.1 and 0.2 mm, respectively. The results unequivocally demonstrate that a polishing distance set at 0.2 mm yields superior surface quality and polishing performance.
Fig. 15
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Fig. 15. Surface quality according to polishing distance by MRF polishing process with circular Halbach array.

Table 4. Experimental parameters.

Numerical order Characteristic Technology parameters Unit
1 Workpiece speed 200 rpm
2 Flow rate from peristaltic pump 20 ml/min
3 Polishing time 90 min
4 Polishing distance 0.1 mm
0.2 mm
0.3 mm
0.4 mm
0.5 mm
5 Conveyor belt speed 30 rpm
6 Al2O3 abrasive particles (weight 25 g) Average diameter 1 μm
7 Carbonyl iron powder weight 200.0 g Average diameter 3 μm

5.4. Conveyor speed to surface quality when polished by circular Halbach array

Fig. 16 reveals alterations in surface quality with varying conveyor speeds, and Table 5 outlines experimental parameters. The findings indicate that changes in the working intervals exert a more substantial influence on surface roughness compared to the impact of conveyor speed. Surface roughness exhibits fluctuations with different polishing durations, ranging from 20.206 nm to 28.571 nm. In the MRF polishing process employing circular Halbach arrays with regenerative abrasives, the increase in conveyor speed amplifies the relative velocity between the workpiece and the MRF ribbon. Simultaneously, the contact distance between the MRF ribbon and the workpiece expands with each unit of time, strengthening the removal capacity of excess machining material and consequently accelerating the reduction in surface roughness. Based on the formula proposed by Preston equation [40], the velocity of the MRF ribbon in motion also increases as the conveyor speed escalates, thereby enhancing the polishing efficiency on the workpiece. However, the observed variance in surface quality with changes in conveyor speed does not precisely align with the model proposed by Preston.
Fig. 16
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Fig. 16. Surface quality according to conveyor speed by MRF polishing process with circular Halbach array and regenerative abrasives.

Table 5. Experimental parameters.

Numerical order Characteristic Technology parameters Unit
1 Workpiece speed 200 rpm
2 Flow rate from peristaltic pump 20 ml/min
3 Polishing time 90 min
4 Polishing distance 0.2 mm
5 The motor speed controls the belt 30 rpm
50 rpm
70 rpm
90 rpm
6 Al2O3 abrasive particles (weight 25 g) Average diameter 1 μm
7 Carbonyl iron powder weight 200.0 g Average diameter 3.5 μm
This phenomenon can be explained by reflecting the generation of the circulating MRF mixture on the conveyor belt by the motor through the nozzle tube and its transmission to the conveyor belt whilst maintaining a constant MRF volume. However, the quantity of the MRF mixture per unit length of the conveyor belt diminishes with an increase in conveyor speed. Notably, the amount of MRF traversing the polishing zone per unit time remains nearly constant when a specific MRF flow rate is established during the polishing processes. Fig. 17 outlines the normal and tangential forces acquired in MRF polishing processes to explain the rationale behind the minimal variation in surface quality, as illustrated in Fig. 16. Under different polishing conditions, the normal and tangential forces in the polishing area decrease with an increase in conveyor belt speed. This reduction can be attributed to the heightened conveyor speed, resulting in a decline in the MRF fluid quantity within the polishing area. Consequently, this reduction diminishes the number of particles engaging in the polishing area when examined at a microscopic level, leading to a decrease in the polishing force components within the polishing area. Subsequently, the diminished polishing force results in a decline in material removal efficiency, thereby reducing the overall polishing effectiveness, aligning with the aforementioned outcomes.
Fig. 17
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Fig. 17. Polishing force variation with conveyor speed by MRF polishing process with circular Halbach array and regenerative abrasives.

The contact pressure between abrasive particles and the workpiece surface is mainly controlled by the normal force. This pressure is crucial because it determines the penetration depth of abrasive particles into the material surface, directly influencing the MRR and the surface quality post-polishing. A decrease in the normal force reduces the contact pressure, leading to low polishing efficiency. By contrast, an increase in the normal force enhances the impact of the abrasive particles on the surface, thereby improving the MRR and surface quality. Meanwhile, the tangential force is essential in generating relative motion between abrasive particles and the workpiece surface. This motion is necessary for the required cutting action to remove material. The quantity of slurry in the processing area decreases as the conveyor speed increases, reducing the number of active abrasives. This phenomenon leads to a reduction in the tangential force, subsequently decreasing cutting efficiency and material removal. Therefore, maintaining a sufficiently strong tangential force is crucial for high polishing efficiency. The overall performance of the polishing process is determined by the interaction between normal and tangential forces. The normal force provides the necessary pressure for effective contact between the abrasive particles and the workpiece surface, whilst the tangential force ensures adequate movement of the abrasive particles for the cutting action. The magnetic field is enhanced in the polishing area during magnetic polishing with circular Halbach arrays, thereby helping maintain the required pressure and improving the uniformity of the polishing process.
The experiments were conducted under the following specific conditions: magnetic particles with a size of 3.5 μm, abrasive particles with a size of 1 μm, a belt speed of 50 rpm and a polishing duration of 120 min. Fig. 18 reveals numerous dense scratches across the entire surface before polishing the PMMA surface. Particularly evident from the 3D image, the surface exhibited rough, scratch-ridden topography with a surface roughness measured at Ra = 464.895 nm. As evidenced by the image shown in Fig. 18, the results indicate that the polished surface is nearly free of scratches. As detailed in Fig. 14, surface roughness (Ra) was measured at four distinct positions on the sample surface to assess the level of surface finish. The measured surface roughness values at these positions are Ra = 7.141, 8.230, 8.426 and 8.838 nm. Compared to recent MRF polishing models, the proposed approach in this study offers considerable improvements. Some recent studies include the study by Ref. [16], wherein the surface roughness values achieved Ra = 34.50 nm. Similarly, Kumar et al. and Zhai et al. [41,42] reported surface roughness values around Ra = 20 nm to Ra = 45 nm, indicating more variability and higher roughness compared to the obtained results.
Fig. 18
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Fig. 18. Workpiece surface roughness before and after polishing with circular Halbach array and regenerative abrasives.

The enhanced performance of the proposed model can be attributed to the innovative use of circular Halbach arrays, which provide a consistent magnetic field distribution, and the employment of regenerative abrasives, which maintain their cutting efficiency over long polishing durations. These advancements lead to a highly controlled and effective polishing process, resulting in superior surface finish and uniformity. These values demonstrate the high uniformity of the surface after polishing. The surface appeared flat, bright and smooth, revealing a considerable reduction of the nearly eradicated rough peaks with the surface roughness. The efficiency of this process can be attributed to several reasons. Firstly, the abrasive replacement system continuously renews the CIPs and abrasive particles within the machining area throughout the process. Secondly, the entire process always has a strong magnetic field due to the use of the Halbach array in polishing, increasing the capability to interact with particles in the solution. Thirdly, CIPs and abrasive particles can be directly controlled by the magnetic field without the need for indirect control through the connection of any other type of particle, eliminating the weakening of the bond between particles whilst polishing realises the process. Herein, the effectiveness of CIPs and APs in the MRF polishing process can be confirmed. Using CIPs and common abrasives (Al2O3) scan facilitate the creation of an extremely bright and smooth nanometre-level material surface without damaging the surface. The consistency in surface roughness measurements across different positions highlights the effectiveness of the MRF polishing process utilising circular Halbach arrays and regenerative abrasives. This method not only enhances surface quality but also ensures high uniformity across the entire polished surface. Consequently, this process holds remarkable potential for wide application in industries that demand high precision and superior surface finishes for materials.

6. Conclusions

This study introduces a magnetic polishing approach employing circular Halbach arrays in conjunction with regenerative abrasives facilitated by a conveyor belt. The findings of this investigation can be outlined as follows:
  • ❖
    The outcomes derived from polishing processes utilising circular Halbach arrays with regenerative abrasives reveal a notable discrepancy between the tangential and normal forces, with the tangential force being considerably smaller than the normal force. A notable observation is the substantial augmentation of normal and tangential forces with a reduction in polishing distance.
  • ❖
    Surface quality in MRF polishing processes employing circular Halbach arrays and regenerative abrasives is enhanced through the considerable effects of polishing distance, abrasive grain size and magnetic particle size.
  • ❖
    The experimental outcomes of the MRF polishing process utilising a circular Halbach array and regenerative abrasives on the optical lens material PMMA reveal a substantial improvement in the initial rough surface, reducing the surface roughness from approximately 500 nm to below 9 nm after 120 min. The surface roughness measurements obtained at different positions on the polished surface indicate a high degree of uniformity, revealing a smooth finish and virtually no observable scratches. These results emphasise the effectiveness of the circular Halbach array with regenerated abrasive grains in achieving a refined surface finish for PMMA optical lens material.
  • ❖
    High-precision sensors and real-time imaging systems will be used in future work to ensure uniform distribution of the MRF solution on the conveyor. Experiments to measure the reduction impact of the cross-sectional area on particle density and distribution, along with the effect of different polishing wheel speeds on the accumulation of abrasive particles and the uniformity of the MRF solution, will also be comprehensively investigated. The integration of these methods in the future aims to establish a clear correlation between operating parameters and surface finish quality, as well as polishing efficiency with circular Halbach arrays and regenerative abrasive.

CRediT authorship contribution statement

Nguyen Duy Trinh: Writing – original draft, Software, Resources, Conceptualization. Dung Hoang Tien: Writing – review & editing, Supervision, Project administration, Methodology, Formal analysis. Pham Thi Thieu Thoa: Writing – original draft, Resources, Formal analysis, Data curation. Nguyen Van Que: Software, Investigation, Data curation, Conceptualization. Kieu Van Quang: Software, Resources, Data curation. Nguyen Trong Mai: Writing – original draft, Formal analysis, Data curation, Conceptualization.

Conflicts of interest

The authors declare that there is no conflicts of interest.

Acknowledgements

The authors would like to thank Hanoi University of Industry, Vietnam, for assistance and support during the data collection in the paper. We also appreciate all the graduate students at Hanoi University of Industry, who have sought our assistance over the years, and in this particular case, those students who have included us among advisors, mentors, family, and friends in their acknowledgements.

References

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