CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to copending U .S. provisional application entitled "POLISHI NG TECHN IQUE" having serial no. 61 /357,734, filed June 23, 2010, the entirety of which is hereby incorporated by reference.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] This invention was made with government support under agreement CMMI-

1000380 awarded by the National Science Foundation. The Government has certain rights in the invention.

BACKGROUND

[0003] Larger mirrors made for telescopes and the like are used to reflect light to obtain images. I mperfections in such mirrors can degrade the quality of the images obtained.

BRIEF DESCRIPTION OF THE DRAWINGS

[0004] Many aspects of the present disclosure can be better understood with reference to the following drawings. The components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the disclosure. Moreover, in the drawings, like reference numerals designate corresponding parts throughout the several views.

[0005] FIG. 1 is a drawing that illustrates various finishing techniques in which the magnetic flux created by the electromagnets is not parallel to the target surface according to various embodiments of the present disclosure. [0006] FIG. 2 is a drawing that illustrates an approach to finishing large curved and/or planar surfaces according to various embodiments of the present disclosure.

[0007] FIG. 3 is a drawing that illustrates an approach to finishing large curved and/or planar surfaces disposed on thick substrates according to various embodiments of the present disclosure.

[0008] FIG. 4 is a drawing that illustrates one approach for dual-sided finishing according to various embodiments of the present disclosure.

[0009] FIG. 5 is a drawing that illustrates another approach for dual-sided finishing according to various embodiments of the present disclosure.

[0010] FIG. 6 is a drawing that illustrates various examples of electrical waveforms that are supplied to electromagnets according to various embodiments of the present disclosure.

[0011] FIG. 7 is a drawing that illustrates additional examples of electrical waveforms that are applied to electromagnets according to various embodiments of the present disclosure.

[0012] FIG. 8 is a drawing that illustrates additional configurations for dual-sided finishing according to various embodiments of the present disclosure.

[0013] FIG. 9 is a drawing that illustrates other configurations for finishing large curved and/or planar surfaces according to various embodiments of the present disclosure.

[0014] FIG. 10 is a drawing that illustrates another approach for finishing large curved and/or planar surfaces according to various embodiments of the present disclosure.

[0015] FIGS. 11-13 are drawings illustrating other approaches for finishing surfaces according to various embodiments of the present disclosure.

[0016] FIG. 14 is a flow chart illustrating an approach for finishing surfaces according to various embodiments of the present disclosure. DETAILED DESCRIPTION

[0017] According to various embodiments, various approaches are described for finishing (e.g. , polishing) a surface of relatively large mirrors such as those that are employed in telescopes and like devices. Alternatively, the embodiments described here may be applicable to the finishing of surfaces other than mirrors. In one embodiment, the target surfaces to be finished may be very thin and fragile, which makes them difficult to finish and susceptible to breakage or other damage.

[0018] In the various approaches, two or more magnetic elements may be used to finish a target surface. The magnetic elements may include combinations of static magnetic field generator(s), variable magnetic field generator(s), and/or magnetic bulk(s). A static magnetic field generator may be, e.g. , an electromagnet supplied with a constant voltage or a permanent magnet. A variable magnetic field generator may be, e.g. , an electromagnet supplied with a time varying voltage or a mechanical system that varies the position (e.g. , rotates) a permanent magnet. For example, two or more electromagnets may be used to finish a target surface. Each of the electromagnets includes a coil and a pole tip that is oriented in a position relative to the target surface to be finished.

[0019] A magnetic abrasive fluid mixture (MAFM) is disposed on the target surface. According to one embodiment, the target surface is much larger than the area covered by the MAFM . Stated another way, the target surface extends beyond the area covered by the MAFM. Thus, this finishing technique may be employed for relatively large mirrors such as, for example, those employed in telescopes in which the surface area of the target surface is much greater than the surface area covered by the MAFM that is employed to finish the target surface.

[0020] According to one embodiment, electromagnets are used as variable magnetic field generators. A time-varying electrical signal is applied to each of the electromagnets, thereby causing the electromagnets to generate one or more time- varying magnetic fields. Exposure of the MAFM to such time-varying magnetic fields agitates the MAFM. Due to such agitation, abrasive particles within the MAFM effectively finish the target surface by bumping, scraping, indenting, and imparting other physical action against the target surface.

[0021] According to one embodiment, the surface area covered by the MAFM is less than the area of the target surface as shown in the various figures. However, it is understood that the target surfaces described below may be completely covered by the MAFM as well. To this end, the volume of the MAFM can be varied regardless of the size of the target area. To this end, the agitation of the MAFM is localized to the areas where time varying magnetic flux is located. In one embodiment, the effective areas near the target surfaces that experience the time-varying magnetic flux may be smaller than the surface area of the target surfaces themselves. This allows the use of multiple groupings of electromagnets, for example, to effect simultaneous finishing of multiple areas of a given target surface at a time.

[0022] Referring to FIGS. 1 A-1 C, shown are three finishing configurations that illustrate the finishing technique according to various embodiments. As shown in the examples of FIGS. 1A-1 C, the target surface 103 is positioned between two variable magnetic field generators such as, e.g. , electromagnets 106a and 106b. While the examples of FIGS. 1 A-1 C disclose the use of electromagnets 106 as variable magnetic field generators, other types of variable magnetic field generators are equally applicable as can be understood. The pole tips 109 are positioned such that the lines of magnetic flux 1 2 generated by the electromagnets 106 are not parallel to the target surface 103 to be finished. The lines of magnetic flux 1 12 propagate through the substrate or workpiece 1 15 upon which the target surface 103 is disposed. The pole tips 109 of the electromagnets 106 may be positioned such that the lines of magnetic flux 1 12 are perpendicular to the target surface 103 as depicted in FIG. 1A. Alternatively, the pole tips 109 of the electromagnets 106 may be positioned at some other angle β with respect to the target surface as shown in FIG. 1 B.

[0023] The pole tips 109 may be positioned to face each other directly as depicted in FIG. 1 B, or they may be offset with respect to each other as illustrated in FIG. 1 C. In one embodiment, the pole tips 109 are in alignment with a centerline (not shown) that is in alignment with the lines of magnetic flux 1 12. In another embodiment, the pole tips 109 are offset with respect to each other, thereby creating lines of magnetic flux 1 12 that fall at a predefined angle with respect to the target surface 103, where the predefined angle is not perpendicular to the target surface 103.

[0024] The MAFM 1 18 may comprise a fluid mixture that includes magnetic particles, abrasive particles, one or more surfactants, a dispersive media, and potentially other components. In addition , other types of MAFMs may be employed.

[0025] Magnetic particles comprise ferromagnetic particles with a small diameter (e.g. , in the range from tens of nanometers to tens of millimeters). For example, the magnetic particles can include magnetite, which could be made of any ferromagnetic material. The magnetic particles are highly susceptible to magnetic fields and cause the MAFM to respond to an applied field. Abrasive particles comprise particles intended for material removal from the target surface. Abrasive particles may include, but are not limited to, diamond, alumina (Al ₂ 0 ₃ ), silicon carbide (SiC), cubic boron nitride (CBN), or other material that can cause mechanical material removal and/or deformation by cutting, plowing , or indenting. Such materials typically have a very high hardness compared to the material of the target surface and can be characterized by sharp cutting edges. Abrasive particles may also include particles that exhibit chemical effects on the target surface. For example, the particles may not have high hardness compared to target surface material(s), but may bond to surface atoms of the target surface and remove bonded surface atoms by mechanical action . Such particles may be utilized for fine finishing applications (e.g. , final polishing of a surface). [0026] Surfactant(s) prevent agglomeration of the magnetic and/or abrasive particles, allowing for even distribution of the particles within the MAFM, by bonding to the solid particles, effectively coating them. More than one kind of surfactant may be present in the MAFM as the magnetic and abrasive particles may require different surfactants. Dispersive media, which may be called a vehicle, carrier, or base fluid or liquid, is a fluid that suspends the mixed magnetic and abrasive particles. Dispersive media may include, but is not limited to, water, oil (e.g. , hydrocarbon, silicon , or other), glycol, or other appropriate fluid. The type of dispersive media used may affect which surfactants may be utilized. For example, when an oil dispersive media is used , cis- oleic acid may be utilized as the surfactant. Water may use N(CH ₃ ) ₄ OH as the surfactant.

[0027] In some implementations, the MAFM may be prepared by mixing a magnetic fluid (e.g. , ferrofluid) and an abrasive fluid (or slurry), which may be commercially available. For example, the magnetic fluid may be based in either oil or water and may comprise magnetite particles with diameters of about 10 nm, which are made by chemical precipitation. When obtained commercially, the dispersion quality of the magnetic fluid may be selectable while the surfactants are proprietary and not selectable. The abrasive fluid are available with a large variety of abrasive particles (e.g. , with selectable material, size, shape, and grain structure) and a variety of fluids (e.g. , water, oil, or universal that is soluble with water or oil) , which may include selectable pH levels.

[0028] According to various embodiments, a time-varying electrical signal is applied to each of the electromagnets 106. Specifically, a respective time-varying electrical signal is applied to each respective coil 121 associated with the electromagnets 106 that causes the generation of one or more time-varying magnetic fields 1 12 as can be appreciated . [0029] According to one embodiment, separate time-varying electric signals are applied to respective electromagnets 106. Such time-varying electric signals may comprise two or more switched time-varying signals. Switched time-varying signals effectively apply one of multiple time-varying electrical signals to one of the

electromagnets 106a or 106b at a given instant. As an alternative, the time-varying electrical signals may comprise two non-switched time-varying electrical signals that are applied to the electromagnets 106a and 106b simultaneously. The time-varying electric signals may comprise any one of many different kinds of electrical signals. The various types of time-varying electrical signals that may be employed will be discussed in greater detail with reference to later figures.

[0030] As shown with reference to FIGS. 1A-1 C, the target surface 103 is situated on a substrate or workpiece 1 15 that is positioned between the pole tips 109 of the electromagnets 106 in the various orientations shown.

[0031] The various approaches at finishing target surfaces as described herein are effective at finishing surfaces to a very fine smoothness, given the small size of the abrasive particles in the MAFM. As a consequence, the approaches described herein are effective at gently finishing (e.g. , polishing) various fragile surfaces without potential damage. Such fragile surfaces may comprise, for example, thin film-coated surfaces and other types of surfaces.

[0032] With reference to FIG. 2, shown is an example configuration in which variable magnetic field generators are all positioned on one side of the substrate 1 15. The target surface 103 to be finished is disposed on a side of the substrate 115.

According to this embodiment, the electromagnets 106 are positioned on a side of the substrate 115 such as a mirror that is opposite the side upon which the target surface 103 is situated. The pole tips 109 may be positioned relative to the one side of the target surface 103 at an angle Θ relative to a tangent 224 drawn from the target surface 103 at the midpoint between the pole tips 109. Thus, the shape of the mirror or other substrate 1 15 that includes the target surface 103 could be convex, concave, flat, freeformed, or other profile as can be appreciated.

[0033] Ultimately, the lateral dimensions of the substrate 1 15 that includes the target surface 103 can be unlimited, where the substrate 1 15 may be moved relative to the electromagnets 106 to cause the finishing of the entire surface 103 as can be appreciated. The MAFM 1 18 moves with the variable magnetic field generators (e.g. , electromagnets 106) given the magnetic components of the MAFM 1 8 that are attracted to the magnetic fields generated by the variable magnetic field generators as can be appreciated. Further, the MAFM 1 18 finishes the target surface 103 given the agitation of the magnetic components of the MAFM 1 18. The finishing is applicable to metallic and non-metallic materials.

[0034] Also, the pole tips 09 may be placed in a symmetrical or asymmetrical configuration relative to the target surface 103 and the MAFM 1 18. For example, the pole tip angles Θ can vary with respect to each other. As shown, for example, in FIG. 2, the pole tips 109 are co-planar. However, the pole tips 109 can be positioned in different planes. To this end, the position of the pole tips 109 as shown in the various figures herein can be placed in various symmetrical or asymmetrical configurations and in the same or different planes with the aim of causing the location of the magnetic fields they create to be positioned in a desirable location.

[0035] With reference to FIG. 3, shown is a further configuration for finishing a target surface 103. According to one embodiment, the workpiece or substrate 15 upon which the target surface 103 is disposed is very thick and may only be accessed from the side of the substrate or workpiece 1 15 upon which the target surface 103 is disposed. For example, the configuration is applicable for die and mold finishing. Also, variable magnetic field generators such as, e.g. , electromagnets 106 are positioned such that the pole tips 109 are oriented at an angle or with respect to the target surface 103. According to one embodiment, the target surface 103 may be convex, concave, flat, freeform, or some other profile as can be appreciated.

[0036] In addition, a static magnetic field generator 327, e.g. , a permanent magnet or an electromagnet is disposed near the MAFM 118. In one embodiment, the static magnetic field generator 327 is positioned between the pole tips 109 of the

electromagnets 106, although the static magnetic field generator 327 may be located in other positions relative to the pole tips 109. The static magnetic field generator 327 creates a static magnetic field that further enhances the agitation of the MAFM 1 18. The presence of the static magnetic field generator 327 distributes alternating field gradients and may cause the MAFM 118 to exert a normal force on the target surface 103. Specifically, the magnetic particles in the MAFM 118 have the tendency to follow the magnetic flux lines of the static magnetic field when not otherwise pulled by or attracted to a time-varying magnetic field generated by either of the variable magnetic field generators. Thus, the static magnetic field generator 327 provides for an additional point of attraction for the magnetic components of the MAFM 8 that further agitates the MAFM 118 to enhance the finishing of the target surface 103. The finishing is applicable to ferrous and non-ferrous workpieces due to the magnetic field alternation.

[0037] According to one embodiment, a mechanical actuator may be coupled to the static magnetic field generator 327, where the actuator is configured to subject the static magnetic field generator 327 to a predefined motion. Similarly, in all of the various embodiments described herein, actuators may be coupled to each of the electromagnets 106 to cause the electromagnets 106 to move according to a predefined motion. To this end, an actuator may be attached to any variable magnetic field generator or static magnetic field generator described herein to impart a predefined mechanical motion to the attached magnetic field generator. [0038] This mechanical motion causes the magnetic fields generated by the variable magnetic field generator or static magnetic field generator to move in accordance with the movement of the attached magnetic field generators as can be appreciated. Given such mechanical agitation, the MAFM 1 18 is thus subjected to greater agitation given that the magnetic components of the MAFM 1 18 will be affected by the movement of the respective magnetic fields. The predefined motion imparted to the variable magnetic field generator or static magnetic field generator by a respective actuator may comprise a vibration, oscillatory motion, rotation, random motion, or any other type of motion as can be appreciated.

[0039] With reference to FIG. 4, shown is a dual-sided finishing configuration according to the embodiments of the present disclosure. Two target surfaces 103a and 103b are positioned on either side of a substrate or workpiece 1 15. An amount of MAFM 1 18 is placed on each of the target surfaces 103, and a pole tip 109 of a respective electromagnet 106 (or other variable magnetic field generator) is positioned adjacent to the respective amounts of MAFM 1 18 as shown. In one embodiment, the pole tips 109 of the electromagnets 106 are aligned as shown in FIG. 4. Alternatively, the pole tips 109 of the electromagnets 106 may be disposed at an angle β (FIG. 1 ) with respect to the target surfaces 103, or may be offset as described with respect to FIG. 1 .

[0040] Referring next to FIG. 5, shown is an alternative dual-sided finishing approach according to various embodiments. As shown, variable magnetic field generators such as, e.g. , electromagnets 106 are employed, as was described with respect to FIG. 4, relative to two amounts of MAFM 1 18 disposed on the respective target surfaces 103. In addition, static magnetic field generators 327 are situated around the variable magnetic field generators (electromagnets 106) on both sides of the substrate 1 15. The placement of the static magnetic field generators 327 relative to the MAFM 1 18 as such will impart greater amounts of agitation given the time-varying magnetic field generated by the respective electromagnets 106. In addition, the variable magnetic field generators and static magnetic field generators may be coupled to actuators. Such actuators may cause the variable magnetic field generators and/or static magnetic field generators to move in a predefined motion, as described above, thereby further agitating the MAFM 118.

[0041] Referring then to FIG. 6, shown are some examples of switched time-varying electrical signals that are applied to the electromagnets 106 as described above. The examples of Fig. 6 illustrate electrical signals (e.g., voltage) applied to a first coil 121 (curve 603) and a second coil 121 (curve 606). In switched configurations, only one of the electromagnets 106 (FIGS. 1-5) is active at a given time. This may help achieve a large amount of dynamic motion in the MAFM 118. As such, various types of time- varying electrical signals may be employed such as, for example, square waves as in FIG. 6(a), saw tooth waves as in FIG. 6(b), half waves as in FIG. 6(c), or other types of electrical waves as can be appreciated. In addition, pulse width modulation (PWM) may be employed with square waves as in FIG. 6(d) or other appropriate waves as can be appreciated.

[0042] With reference to FIG. 7, shown are non-switching electrical wave forms in which two or more electrical signals are applied to respective electromagnets 106 simultaneously. Non-switching approaches are likely to be employed in configurations that use static magnetic field generators 327 to impart greater agitation against the MAFM 1 18. For the most dynamic fluid motion, according to one embodiment in which two separate time-varying signals such as sine waves are employed, phase shifts of 180°between the respective time-varying signals may be imposed to provide the most dynamic agitation of the MAFM 1 18, although other phase shifts may be employed. This is especially the case where more than two variable magnetic field generators are used to agitate the MAFM 118. To this end, there may be any phase shift angle between the respective time-varying electrical signals. According to one embodiment, the magnetic flux density ranges between 0.01 Tesla to 0.2 Tesla for optimal effect, although it is possible that other magnetic flux densities would be appropriate in various applications. The dimensions of the coils 121 are determined based upon the desired configuration. The various types of time-varying electrical signals (e.g. , voltage) such as that applied to a first coil 121 (curve 703) and a second coil 121 (curve 706) may comprise, for example, square waves as in FIG. 7(a), sinusoids as in FIG. 7(b), saw tooth as in FIG. 7(c), a randomized signal, or any other type of time-varying electrical signal as can be appreciated.

[0043] I n addition, with respect to FIG. 6 and FIG. 7, only two time-varying electrical signals are shown. However, it is understood that more than two electromagnets 106 may be employed , where a corresponding number of electrical signals are applied to respective ones of the electromagnets 106.

[0044] FIG. 8 shows a configuration for dual-sided finishing of two target surfaces 103a and 103b, where two target surfaces 103 are positioned on either side of a substrate or workpiece 1 15. As shown in the example of FIG. 8, the pole tips 109 associated with the electromagnets 106a and 106b can be oriented at any angle (e.g. , γ and φ) relative to the target surfaces 103a and 103b, respectively. In the embodiment of FIG. 8, the variable magnetic field generators (electromagnets 106) are

asymmetrically arranged with different angles of orientation (y≠ φ). Both target surfaces 1 03 are simultaneously finished by agitation of the MAFM 1 18 located on both target surfaces 103.

[0045] FIG. 9 further illustrates an embodiment with both variable magnetic field generators such as, e.g. , electromagnets 106 and pole tips 109 located on the side of the substrate or workpiece 1 15 where the target surface 103 is located. In some embodiments, additional electromagnets 106 may be utilized. Six-axis positioning of the electromagnets 106 (or other variable magnetic field generator) is possible. As shown in the example of FIG. 9, the pole tips 109 may be oriented at any of the angles Θ shown. For example, the angles ^ and Q ₂ of the centerlines 903a and 903b of electromagnets 106 with respect to the target surface 103 (x-y plane) may be the same (e.g. , 45°) or they may be different values in the range from 0° to 180°. In addition, the angle between the centerlines 903 of the electromagnets 106a and 106b ft may be 90° or in a range from -180° to +180°. In the example of FIG. 9, the centerline 903a of electromagnet 106a projects onto line 906 and the centerline 903b of electromagnet 106b projects onto the y-axis, with the angle ft relative to the y-axis less than 90°

[0046] FIG. 10 shows yet another embodiment in which a single variable magnetic field generator (e.g. , an electromagnet 106) is used along with a static magnetic field generator 327 (e.g. , a permanent magnet) or some other magnetic bulk or mass 1030 such as, e.g. , a steel plate or block, ere. As shown, the static magnetic field generator 327 or other magnetic mass 1030 is positioned on a side of the substrate or workpiece 1 15 opposite the target surface 103. Where a static magnetic field generator 327 is employed, the static magnetic field pulls the MAFM 1 18 toward the target surface 03 as shown. The time varying magnetic field generated by the electromagnet 106 causes agitation of the MAFM 1 18. Where a magnetic mass 1030 such as a steel block or plate is employed, the flux of the time-varying magnetic field flows into the magnetic mass 1030 and also agitates the MAFM 1 18.

[0047] FIGS. 1 1 -13 illustrate other implementations that may be used for finishing (e.g. , polishing) a target surface 103. The target surface 103 may be convex, concave, flat, freeform, or some other profile as can be appreciated. The target surface 103 may also be either continuous or discontinuous (e.g., a stepped surface). In the

embodiment of FIG. 1 1 , a first magnetic element 1 33 such as, e.g. , a variable magnetic field generator, a static magnetic field generator 327, or a magnetic bulk or mass 1030 (e.g. , ferromagnetic bulk) is fixed in position over (or adjacent to) the target surface 103. On a side of the substrate or workpiece 1 15 opposite the target surface 103, a second magnetic element 1 136 such as, e.g. , a variable magnetic field generator, static magnetic field generator 327, or magnetic bulk or mass 1030 is allowed to move. The combination of the first magnetic element 1 133 and the second magnetic element 1136 establish lines of magnetic flux 112 that pass through the MAFM 118. Actuators may cause the second magnetic element 1 136 to move in a predefined motion thereby agitating the MAFM (MAFM) 1 18 upon the target surface 103. The movement of the second magnetic element 1 136 may be a linear vibration or translation, a free motion, a revolution, a planetary motion (e.g., rotation about a point that is rotating), or other motion and combinations thereof as can be understood.

Variation in the magnetic flux produced by a variable magnetic field generator such as an electromagnet 106 further agitates the MAFM 118.

[0048] Referring next to the embodiment of FIG. 12, a first magnetic element 1233 such as, e.g., a variable magnetic field generator, a static magnetic field generator 327, or a magnetic bulk 1030 positioned over (or adjacent to) the target surface 103 is allowed to pivot about a fixing point. As in FIG. 11 , a second magnetic element 1136 is allowed to move on a side of the substrate or workpiece 115 opposite the target surface 103. The combination of the first magnetic element 1233 and the second magnetic element 1 136 establish lines of magnetic flux 112 that pass through the MAFM 118. Actuators may cause the second magnetic element 1 36 to move in a predefined motion, as described above, thereby agitating the MAFM 118 upon the target surface 103. In some embodiments, the first magnetic element 1233 pivots in response to changes the magnetic flux lines 112 caused by the movement of the second magnetic element 1136 on the opposite side of the substrate 115. Variation in the magnetic flux produced by a variable magnetic field generator such as an electromagnet 106 further agitates the MAFM 118.

[0049] With respect to the embodiment of FIG. 13, both a first magnetic element 1333 such as, e.g. , a variable magnetic field generator, a static magnetic field generator 327, or magnetic bulk 1030 positioned over (or adjacent to) the target surface 103 and a second magnetic element 1 136 positioned on a side of the substrate or workpiece 1 15 opposite the target surface 103 are allowed to move. An actuator can move the first magnetic element 1 333 in a linear vibration or translation or in a free motion .

Actuators may cause the second magnetic element 1 136 to move in a linear vibration or translation , a free motion, a revolution, a planetary motion (e.g. , rotation about a point that is rotating), or other motion and combinations thereof as can be understood. The combination of the first magnetic element 1 333 and the second magnetic element 1 136 establish lines of magnetic flux 1 12 that pass through the MAFM 1 18. The motion and/or variation in the magnetic flux 1 12 produced by an electromagnet 106 agitate the MAFM 1 18 to finish the target surface 103. Variation in the magnetic flux produced by a variable magnetic field generator such as an electromagnet 06 further agitates the MAFM 1 18.

[0050] Referring now to FIG. 14, shown is a flow chart illustrating an approach for finishing surfaces according to various embodiments of the present disclosure.

Beginning with block 1403, a MAFM 1 18 is placed on a target surface 103. The target surface 103 may extend beyond a surface area covered by the MAFM 1 18. In some implementations, the target surface is many times the surface area covered by the MAFM 1 18. A magnetic element such as a variable magnetic field generator (e.g. , an electromagnet 106), a static magnetic field generator 327 (e.g. , a permanent magnet or an electromagnet), or magnetic bulk or mass 030 is positioned relative to the target surface 103 in block 1406. I n some embodiments, a plurality of magnetic elements may be positioned relative to the target surface 103. For example, combinations of variable magnetic field generator such as electromagnet(s) 106, static magnetic field

generator(s) 327, and/or magnetic bulk(s) 1030 may be positioned over (or adjacent to) the target surface 103. In other implementations, combinations of variable magnetic field generator such as electromagnet(s) 106, static magnetic field generator(s) 327, and/or magnetic bulk(s) 1030 may be positioned on opposite sides of a substrate or workpiece 115 upon which the target surface 103 is disposed. In some embodiments, target surfaces 103 are disposed on opposite sides of the substrate 1 15.

[0051] In block 1409, magnetic flux associated with the magnetic element is varied to agitate the MAFM 118, which results in finishing of the target surface 103. For example, an electromagnet 106 may be driven with a time-varying electrical signal to cause the electromagnet 106 to generate a time-varying magnetic field, thereby agitating the MAFM 118. In some embodiments, multiple electromagnets 106 are driven with the same or different time-varying electrical signals. In other

implementations, the magnetic element is moved (e.g., by an actuator) to vary the magnetic flux. Combinations of movement of one or more magnetic elements and time- varying electrical signals applied to one or more electromagnets may also be utilized.

[0052] It should be noted that ratios, concentrations, amounts, and other numerical data may be expressed herein in a range format. It is to be understood that such a range format is used for convenience and brevity, and thus, should be interpreted in a flexible manner to include not only the numerical values explicitly recited as the limits of the range, but also to include all the individual numerical values or sub-ranges encompassed within that range as if each numerical value and sub-range is explicitly recited. To illustrate, a concentration range of "about 0.1 % to about 5%" should be interpreted to include not only the explicitly recited concentration of about 0.1 wt% to about 5 wt%, but also include individual concentrations (e.g., 1 %, 2%, 3%, and 4%) and the sub-ranges (e.g., 0.5%, 1.1 %, 2.2%, 3.3%, and 4.4%) within the indicated range. The term "about" can include traditional rounding according to significant figures of numerical values. In addition, the phrase "about 'x' to 'y'" includes "about 'x' to about V".

[0053] It should be emphasized that the above-described embodiments of the present disclosure are merely possible examples of implementations set forth for a clear understanding of the principles of the disclosure. Many variations and modifications may be made to the above-described embodiment(s) without departing substantially from the spirit and principles of the disclosure. All such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims.