Optical components with high accuracy and smooth surface finish are increasingly in demand throughout modem consumer products and industry, such as ultraprecise optical lenses for use in cameras and illumination devices, etc. Likewise, metallic components with accurately formed and highly polished surfaces are in demand, for example for use in the production of moulds for producing moulded optical components. There is a need to improve activity in these areas by improving the throughput of manufacture and shortening processing times.

Optical fabrication methods can be divided into two approaches, from the perspective of using rigid or non-rigid tools. Rigid tool based machining methods such as diamond turning and precision grinding are deterministic and repeatable, but highly dependent on structural stiffness, machine positioning accuracy, and machine vibrations. Besides, tool marks are unavoidable and fracture can be easily generated due to the hard contact when processing brittle optical materials such as glass. On the contrary, non-rigid tools can achieve soft contact with the workpiece because of their compliant properties. This technology is applied in the bonnet polishing process, with which a nanometre smooth surface finish can be achieved on brittle optics. In this process, a hemispherical resilient tool is covered with a compliant polishing pad. Under supply of loose abrasive slurry, the tool is rotated to draw abrasives between the pad and the workpiece in order to remove material. The material removal rate (MRR) is subject to the traditional Preston's law, whereby

MRR~k P V where k is the Preston coefficient (Pa ¹ );

P the pressure in the contact region; and

V the velocity of the tool surface in the contact region relative to the workpiece surface.

According to this law, increasing the speed V of the tool surface over the workpiece is an option to improve productivity, as it will increase the rate of material removal and thus the time taken to bring a part of the workpiece surface to the required form will be reduced. The speed V is increased by increasing the rate of rotation of the tool. However, a number of phenomena occur when the tool is rotated at high speed, which cause Preston's law to break down and adversely affect productivity and output quality. Firstly, centrifugal force at high speed will stretch the resilient tool material and cause deformation that changes the shape of the tool, and affects the position and size of the tool/workpiece contact footprint, and the amount and distribution of pressure exerted by the tool on the workpiece.

Secondly, large slurry inflow between the tool and the workpiece caused by the high-speed rotation of the tool induces a hydroplaning effect, where a film of slurry forms between at least a part of the tool and the workpiece, and results in reduced contact pressure P on the workpiece surface and/or a reduction in the contact area between the tool and the workpiece surface. This inherently lowers the material removal rate and degrades polishing performance in terms of accuracy and uniformity. Under extreme conditions, the tool may completely "float" over the workpiece if the film of slurry extends over all of the contact area between the tool and the workpiece, which will considerably reduce the material removal rate since the tool contact pressure is effectively reduced to zero. The fluid nature of the slurry in such a layer between the tool and the workpiece allows the slurry adjacent the workpiece to move much more slowly than the rotation of the tool surface, significantly reducing the material removal rate.

The present invention aims to address these problems and provide a new class of high-speed polishing (HSP) tool and processes, which have improved productivity, deterministic and accurate control, and can produce a highly polished surface finish.

The present invention provides, in a first aspect, a high-speed polishing method in which the tool is formed with a surface which reduces or eliminates the formation of a continuous layer of slurry between the tool and the workpiece. In this aspect, a resilient compliant tool is urged against a workpiece surface so that the tool deforms and a contact footprint between the tool and the workpiece surface is created, the tool is rotated with a precession angle between the tool’s rotation axis and the normal to the workpiece surface, so that the tool surface moves across the tool footprint relatively to the workpiece surface, and abrasive slurry is provided in the nip between the tool surface and the workpiece so as to be drawn between the tool and the workpiece by the rotation of the tool, and wherein the tool surface is formed with grooves such that, within the tool footprint, the grooves form a "V"-shaped point which passes through the tool footprint as the tool is rotated.

Preferably the tool is rotated such that the "V"-shaped point is oriented such that the limbs of the "V"-shape converge in the direction of movement through the tool footprint relative to the workpiece, in other words the apex of the“V” points in the direction of tool surface movement. However, it is foreseen that the tool may be rotated in the opposite direction such that the limbs of the "V"-shape diverge in the direction of movement through the tool footprint. The precession angle is preferably arranged so that the "V"-shaped point passes through or close to the centre of the tool footprint.

The convex tool surface may be formed with two or more concentric annular contact regions corresponding to different precession angles of the tool, such that at a first precession angle a first annular region of the tool contacts the workpiece to form the tool footprint and at a second precession angle of the tool a second annular region of the tool contacts the workpiece to form the tool footprint. The tool may be formed with one or more radially-extending grooves, each groove having a first V-shaped point situated in the first annular region of the tool, and a second V-shaped point situated in the second annular region of the tool. The V- shaped points in different annular regions of the tool may be arranged in different circumferential directions relative to the axis of rotation of the tool.

Preferably, the point of the "V" passes substantially through the centre of the tool footprint as the tool is rotated. However, it is foreseen that the point of the "V" may not pass through the exact centre of the footprint. The point may advantageously pass within about half the radius of the footprint from the centre of the footprint.

Preferably at least one of the limbs of the "V" extends to, or beyond, the perimeter of the tool footprint. Most preferably, both limbs of the "V" extend from the point to or beyond the perimeter of the tool footprint. The limbs of the "V" may be straight, or may be curved. If the limbs are curved, they may be concave or convex in the direction of travel of the tool surface through the workpiece. A plurality of V grooves may be simultaneously present within the tool footprint.

A second aspect of the invention provides a compliant polishing tool having an axisymmetric convex polishing surface, the convex surface being formed in an annular region of the polishing surface with one or more grooves of "V" formation oriented so as to point in a circumferential direction of the polishing surface.

A third aspect of the invention provides a polishing apparatus in which a polishing tool having an axisymmetric convex working surface is held in contact with a workpiece to form a tool footprint between the tool and the workpiece surface, and the apparatus includes means to rotate the tool such that an annular region of the working surface of the tool passes through the tool footprint, the polishing apparatus including means to provide an abrasive slurry into the nip between the tool and the workpiece, and wherein the tool surface is formed with grooves such that, within the tool footprint, the grooves form a "V" which passes through the tool footprint as the tool is rotated. Preferably the "V" is oriented in the direction of movement of the tool surface relative to the workpiece.

In a fourth aspect of the invention, it has been observed that the use of an abrasive slurry with non-Newtonian viscosity properties can mitigate the reduction in material removal rate caused by hydroplaning. The viscosity properties of the slurry may be altered by adding polymers and/or starch, particularly cornstarch, to the slurry mix. In such a slurry, the viscosity of the slurry varies as shear force is applied to the slurry, so that in the region between the workpiece and the footprint of the rotating tool, large shear forces are applied to the slurry and its viscosity markedly increases. The“thickened” slurry in this region effectively acts as an extension of the polishing tool. This increased viscosity causes the slurry to move across the surface of the workpiece more rapidly than would be the case with a conventional Newtonian fluid slurry in similar circumstances, and increases the material removal rate.

A non-Newtonian fluid (NNF) slurry may be used in addition to, or as an alternative to, forming“V” grooves in the regions of the tool working surface which will form the tool footprint. It is further foreseen in this aspect that a non- Newtonian slurry may be used in a shaping method in which the rotating shaping tool is held out of contact with the workpiece so as to define a narrow clearance into which slurry may be drawn by rotation of the tool. The fluid within the clearance is subjected to shear and its viscosity increases, drawing the fluid across the workpiece and removing material from the workpiece.

A fifth aspect of the invention provides a polishing or shaping method in which a resilient tool is rotated at high speed, and the positioning of the tool to control the offset takes account of deformation of the tool caused by the high speed rotation. In a polishing method utilising this technique, the tool path may be calculated on the basis of the shape of the un-deformed (stationary) tool, and then a correction to the tool path may be applied based on the change in shape of the tool which occurs as the tool is accelerated in rotation from rest up to its operating speed. Alternatively, the deformed shape of the tool may be measured or calculated, and the tool path may be then determined on the basis of the deformed shape of the tool. Embodiments of the invention will now be described in detail with reference to the accompanying drawings, in which:

Figure l is a perspective view of a polishing apparatus according to the invention;

Figure 2 is a schematic perspective view of a polishing tool according to the invention;

Figure 3 is a projection of the pattern of grooves formed on the tool surface, seen in the direction of the rotation axis of the tool;

Figure 4 is a schematic illustration of the relative movement of the grooves on the tool surface through a tool footprint;

Figure 5 is a schematic illustration of an alternative pattern of grooves moving through a tool footprint;

Figures 5a to 5f show further examples of polishing tools;

Figure 6 is a schematic side view illustrating a rotary tool forming a tool footprint with a workpiece, and entraining a slurry;.

Figure 7 is a schematic view showing a change in shape between the stationary profile of the tool and the profile of the deformed tool when rotating at high speed.

Figure 8 is a schematic view similar to Figure 6 illustrating the lifting of the tool from the workpiece surface by an entrained layer of slurry.

Figure 9 is a schematic illustration of a tool having three annular contact regions corresponding to three different precession angles.

Figure 10 is a schematic sectional view illustrating the behaviour of a non- Newtonian fluid (NNF) slurry in a gap or clearance between a rotating tool and a workpiece.

Figure 11 A is a schematic view illustrating the action of the abrasive grains in an NNF slurry. Figure 1 IB illustrates the shear stress profiles between the tool and the workpiece.

Figure 1 is a perspective diagram of a polishing machine according to one embodiment of the present invention. The polishing machine comprises a table 1, on which there is mounted an X-slide mechanism 2 for movement in the x- direction as seen in the Figure. On the X-slide mechanism 2 there is mounted a Y-slide mechanism 3 for movement in the y-direction. On the Y-slide mechanism 3 there is mounted a turntable 4 for rotation in the direction indicated by arrow c about a vertical axis (as seen in the Figure) extending in the Z-direction.

The turntable 4 is mounted on the Y-slide mechanism 3 via a Z-movement mechanism (not shown) for movement of the turntable 4 in the z direction (the vertical direction as seen in the Figure). The turntable 4 has a support surface onto which a workpiece 5 is mounted and held. Thus, this arrangement provides for motion of the workpiece 5 in four axes - namely x, y, z and c.

The polishing machine is also provided with a back member 6 on which is mounted a pivot arrangement for pivotally moving a polishing head 7. The polishing head 7 is arranged to contact the workpiece 5 with a tool 7a, for polishing or abrading the workpiece 5.

The pivot mechanism mounted on the back member 6 comprises a first pivot member 700 mounted in an arm for pivoting the tool head 7 about a pivot point in a first plane, so that the tool head 7 may be placed in contact with the workpiece 5 and moved across the surface of the workpiece 5 in a predetermined pattern. The first pivot mechanism 700 is mounted on a second pivot mechanism 800 which provides for the pivoting of the tool head 7 about a pivot point in a plane perpendicular to the plane of pivoting of the first pivot mechanism 700 in the arm. Thus these two orthogonal pivoting mechanisms provide two further axes of control, namely a and b, for controlling the angle at which the tool head 7 contacts the workpiece 5.

The tool head 7 has mounted to it a shaping or polishing tool 7a which has a convex working surface adapted to be placed in contact with the workpiece to define an area of contact known as the tool footprint. The convex working surface may be spherical, part-spherical, or cylindrical or may have portions formed any combination of two or more of these shapes. The working surface may be homogeneous, such as a flexible cloth, or the working surface may be formed from a flexible elastomer material. The tool 7a is rotated about an axis which is inclined at a precession angle relative to the surface of the workpiece, so that the part of the working surface in contact with the workpiece is drawn across the surface of the workpiece within the tool footprint. The tool head 7 is moved relative to the workpiece so that the tool footprint will follow a tool path 7b shown as a dotted line in Figure 1. The tool path shown is an example only, and is in the form of a raster. The tool path may be of any convenient form, such as a raster, a spiral, or a pseudo-random tool-path. The use of a pseudo-random tool path can avoid the formation of polishing artefacts that may be generated using a periodic spiral or raster tool-path.

A nozzle 8, which in the illustrated embodiment is mounted to the back member 6 of the polishing machine, delivers an abrasive slurry into the nip where the rotating tool contacts the workpiece, so that rotation of the tool draws the slurry between the tool and the workpiece. As the slurry is drawn across the surface, abrasive polishing or shaping of the workpiece takes place within the tool footprint.

The back member 6 of the polishing machine also houses a computer control system 9 which includes a display 10 and control inputs 11. This allows a user to input data or commands to control the motion of the workpiece 5 and of the tool head 7 and to view displayed information regarding the abrading, polishing or shaping process being performed.

Each of the axes of motion x, y, z, c, a and b is driven by a respective drive actuator (not shown). Sensors (not shown) are also provided for sensing the positions of the actuators to provide position information for use by the computer control system 9 to control the polishing or abrading process.

Thus, the computer control system 9 operates an algorithm to control these axes (or parameters) in order to clean, abrade or polish the workpiece 5 mounted on the turntable 4, to achieve the desired surface form and/or surface quality (such as smoothness).

As will be apparent to those skilled in the art, the process and apparatus of the invention can be used to treat any desired surface profile including a surface profile containing both concave and convex areas and is not limited to the parallelipipedal form of workpiece illustrated in Figure 1.

The computer control system 9 is arranged to control the relative movement of the tool head 7 and the workpiece 5 so that the tool 7a is arranged to impinge on the workpiece at a treatment region or footprint which is moved relative to the workpiece so as to follow a tool-path 7b over the surface of the workpiece 5, and controls the amount of shaping or polishing over the surface by: (i) varying the time spent at each point along the tool-path (the dwell time), (ii) varying pressure of the tool 7a at points along the tool-path, and/or (iii) varying spin speed of the tool 7a at different points along the tool path. The relative movement may be produced by keeping the workpiece 5 stationary and moving the tool head 7, by keeping the tool head 7 stationary and moving the workpiece 5, or by a combination of movement by the tool head 7 and the workpiece 5. The tool path may be a raster, a spiral, or any other convenient regular or irregular tool path. In some embodiments, the use of a pseudo-random tool-path can avoid the formation of polishing or shaping artefacts that may be generated using a periodic spiral or raster tool-path.

In compliant polishing, a resilient tool comprising for example an inflated membrane or solid elastomeric material such as rubber is utilized to achieve compliance between the tool 10 and workpiece W, while spinning with a tilted precession angle (Yt) as shown in Fig. 6. The tool is positioned with an offset O so that the tool surface is deformed and a contact area or footprint F is formed between the tool and the workpiece. In low speed operation, the slurry with fine grits is entrained in the contact region with the workpiece such that abrasives become embedded into the pad asperities and remove material. A thin and steady layer of slurry forms around the contact region.

When the tool is rotated at very high speeds, the working surface of the tool passes through the tool footprint at high speeds and may entrain so much of the slurry that a layer of slurry is formed between the tool and the workpiece over part or all of the tool footprint, and the tool surface is partially or entirely lifted out of contact with the workpiece. When this occurs, the amount of material removed from the workpiece is reduced, and the form and surface finish of the workpiece becomes unpredictable. The present invention in one aspect overcomes this difficulty by providing a grooved surface to the part of the tool which will pass through the tool footprint, which can control the amount of slurry present between the tool and the working surface. Preferably, the working surface includes a series of grooves on the working surface of the tool, as can be seen in Figure 2.

The tool illustrated in Figure 2 comprises an axi symmetrical convex working surface 10, and is intended to rotate, in use, in a clockwise direction as seen in the Figure, indicated by the arrow A. The tool surface 10 is formed with a pattern of radially-extending grooves 11, each groove 11 having a radially inner part 11a which extends radially outwards and circumferentially in the clockwise direction, and a radially outer part l ib which extends in the radially outwards and counterclockwise directions. The two parts of the radial groove meet at a point 12 which forms a generally "V" shape oriented in the direction of the intended rotation of the tool. The angle a formed between the inner and outer parts 11a and l ib at the point 12 is preferably about 60°, but may range from about 20° to about 150°.

In the tool illustrated in Figure 2, the outer and inner parts 11a and l ib of each groove are inclined at substantially the same angle to a radial line passing through the point 12 of the groove, i.e. the outer and inner parts diverge from the point 12 at the same angle relative to the direction of movement of the tool across the workpiece. It is however foreseen that the outer and inner parts 11a and l ib of each groove 11 may meet at the point 12 forming different angles b and g to the radial line passing through the point 12, illustrated in Figure 3.

As can be seen from the Figure, the tool includes a circumferentially-spaced series of grooves, with the points 12 arranged equidistantly from the tool centre. When the tool is rotated in contact with a workpiece at a particular range of precession angles of the tool, the points 12 will pass through the footprint of the tool. Preferably, the angle of precession is arranged such that the points 12 pass through the centre of the tool footprint, with the radially outer 1 lb and radially inner 11a parts of the groove "trailing" behind the point 12.

Preferably the widths of the grooves are from about 0.1 mm to about 1.5 mm, and the depth of the grooves is about 0.1 mm to about 1 mm. The cross-section of the grooves may be V-shaped or U-shaped, or any other convenient shape for forming. The grooves may be formed by a routing operation, or by laser ablation, or by any other suitable technique.

Figure 3 is a view of the tool of Figure 2 seen in the axial direction of the tool, illustrating the form of the grooves 11. It will be seen that the points 12 of the grooves all lie on a circle concentric with the tool surface. The tool is preferably designed to be operated at a particular precession angle, such that when the tool is in contact with the workpiece at the particular precession angle, the points 12 lie on a circular path which passes through the centre of the tool footprint. In use, the precession angle may be slightly different from the design precession angle, so that the points 12 do not pass through the centre of the tool footprint, but pass close to the centre. The precession angle may be arranged so that the path of the points 12 passes within about one half of the footprint diameter of the centre of the footprint.

Figure 4 and Figure 5 illustrate alternative groove formations, schematically shown in projection in relation to the tool footprint F. Figure 4 illustrates a series of grooves meeting in a "V" shape at a point 12, and passing through the tool footprint F with the points 12 of the grooves passing through the centre of the circular tool footprint F. Figure 5 illustrates a different arrangement of grooves, in which the grooves are interlinked, but still comprise a "V" shaped point 12, from which radially inner and outer parts 11a and 1 lb of each groove 11 extend.

Figures 4 and 5 also illustrate alternative forms of groove 11 in which the outer and inner parts of the groove are curved, either concave in the direction of movement as shown at 11c, or convex in the direction of movement as shown at l id.

In the embodiment shown in Figure 2, the grooves are configured so that a single continuous routing or other forming operation may form all of the grooves, by connecting the radially outer ends of the outer parts l ib of adjacent grooves, and connecting the radially inner ends of the inner parts 11a of adjacent grooves to form a single pathway for the groove-forming tool. This arrangement of the grooves provides a universal path for the groove-forming apparatus to follow over the working surface of the tool while forming all of the radially-extending grooves.

Figures 5a to 5f show alternative forms of polishing tools for use with the method of the present invention. Figure 5a illustrates a tool having a conical polishing surface 10a, mounted to a cylindrical base B with a coaxial spindle S. The tool is mounted to the apparatus by means of the spindle S, and rotated with a part of the conical surface 10a in contact with the workpiece to form a tool footprint. In the illustrated embodiment, grooves extend from the apex of the conical surface along generators of the cone, to meet with obliquely-arranged grooves extending up from the base B of the tool to form a series of points 12. The tool is held in relation to the workpiece so that the part of the conical surface 10a which contacts the workpiece to form the tool footprint is the region where the points 12 are located.

Figure 5b illustrates a tool having a cylindrical working surface 10b, formed with a series of "V"-shaped grooves with their points 12 arranged in a circumferential region centrally positioned along the axis of the cylindrical working surface of the tool this tool shape may be advantageous for polishing an edge of a thin sheet of material such as glass, by placing the tool in contact with the edge of the glass sheet such that the points 12 of the grooves in the tool pass through the contact area between the tool and the sheet edge as the tool is rotated on its spindle S. It will be appreciated that the grooves may have more than one "V"-shaped point spaced axially along the length of the working surface, as is illustrated by the exemplary groove l ie in Figure 5b, to enable different axial areas of the working surface to be in contact with an edge of the workpiece, while a V-shaped point passes through the contact area on rotation of the tool. This will have the advantage of maximising the area of the working surface of the tool which may be used in polishing operations

Figure 5c shows an alternative possibility for the formation of oblique intersecting grooves on a cylindrical working surface of a tool, such that the intersection points form "V"-shaped points at various locations along the axial length of the working surface of the tool. The grooves may be formed by generating one or more pairs of oppositely-handed helical grooves which extend from one axial end of the working surface to the other.

Figures 5d and 5e illustrate, in diametral section and end view respectively, a "cup"-shaped tool which has an annular body AB mounted to a spindle S with an annular and convex working surface WS mounted to an end face of the annular body AB. A series of V-shaped grooves are formed on the working surface WS, so that when the tool is pressed axially against a workpiece, an annular tool footprint is created and rotation of the tool causes the "V"-shaped points to move around the tool footprint.

Figure 5f illustrates an alternative version of the cylindrical tool, of increased diameter and reduced axial length as compared to the cylindrical tools shown in Figures 5b and 5c.

Figure 6 is a schematic view illustrating a polishing operation in which an axisymmetrical resilient tool having a part-spherical working surface 10 is rotated about a tool axis T in the direction of the arrow A. The tool axis T is angled at a precession angle j _t to a direction N perpendicular to the surface of the workpiece W. As the tool moves along its predetermined tool path, at each point along the path the tool is positioned with an offset O, such that the workpiece surface will deform the surface 10 of the tool to create a contact area or tool footprint F where the surface 10 of the tool is in contact with the surface of the workpiece W. The offset distance O is essentially the difference between the distance from the workpiece surface to the centre of the spherical surface of the tool, and the radius of the spherical surface of the tool. An abrasive slurry AS is provided around the tool, and at lower rotation speeds the rotation of the tool causes the abrasive grains in the slurry to be entrained between the tool and the workpiece, as indicated by the arrows E.

Figure 7 illustrates the change in the profile of a resilient tool when rotating at high speed. The view is a diametral section of an axisymmetrical tool having a part-spherical working surface shown by the solid line S. When the tool is rotated at high speed, centrifugal forces deform the resilient material of the tool so that its surface adopts the shape shown by the line D. As illustrated in the Figure, the curved part of the surface is deformed to a flattened curve, as compared to the tool when it is stationary. This change in the outline of the tool will alter the contact area or footprint F formed between the tool and the workpiece, as compared to the tool footprint that would be formed if the tool were rotating at a lower speed and were therefore underformed.

When operating a resilient tool at high rotational speeds, with the consequent changing shape of the tool, the tool path data with which the position and movement of the tool is controlled will require modification to compensate for the changing shape of the tool. This can be done by rotating the tool at the proposed high speed at which it will be used, and measuring the change in the tool profile and thereafter calculating the tool path data on the basis of this new, measured, tool profile. The change in the profile of the tool may be measured by directing laser beams onto the tool working surface as the tool is rotated at high speed, to determine the position of the tool surface in the axial direction, at different radii from the rotation axis. Other measurement techniques may alternatively be used, such as photographic techniques or physical probing of the tool working surface as the tool is rotated.

Figure 8 illustrates the hydroplaning phenomenon which is encountered at high rotation speeds of the tool. As the tool entrains slurry into the nip between the tool and the workpiece at high speed, as indicated by the arrows E, upward hydrodynamic pressure is generated on the tool, causing the compliant tool to deform from its original profile P to the deformed profile DP and the contact area to shrink laterally. This is represented in the Figure by the original contact area F, which is reduced on the slurry inflow side by the amount As to form a new tool footprint DF. This shifts the centre of the contact area to the right (as seen in the Figure), and displaces and alters the material removal profile achieved by the tool.

In order to compensate for this change in the position of the tool footprint and the different removal profile achieved by the tool, the tool path parameters may be altered so that the required material removal is achieved along the length of the tool path. The change in the position and size of the tool footprint may be determined experimentally or by theoretical calculation, and introduced into the tool path calculations so as to calculate a modified tool path for the higher rotation speed of the tool.

Figure 9 illustrates an alternative form of grooved tool, seen in the axial direction along its axis of rotation. The convex curved working surface 10 has three concentric annular zones 13a, 13b and 13c, which each correspond to the area of the tool which is in contact with the workpiece when the tool is at a predetermined first, second or third precession angle.

The tool working surface is formed with, in this example, six radially-extending grooves, each of which has a first V-shaped point 12A situated at the mid-region of the first annular zone 13a and pointing in a first circumferential direction corresponding to a first direction of rotation R1 of the tool. Each of the grooves also has a second V-shaped point 12B situated at the mid-region of the second annular zone 13b and pointing in a second circumferential direction corresponding to a second direction of rotation R2 of the tool. Each of the grooves also has a third V-shaped point 12C situated at the mid-region of the first annular zone 13c and pointing in the first circumferential direction R1 of the tool.

In use, the tool of Figure 9 may be held in a tool fixture and rotated about its axis in the first rotation direction R1. The tool working surface may then be brought into contact with a workpiece, with the tool axis at a precession angle such that the tool contact footprint is within the first annular zone 13a. As the tool rotates in contact with the workpiece, the first V-shaped points 12A will pass substantially through the centre of the tool footprint, controlling the flow of slurry between the tool and the workpiece and avoiding aquaplaning.

In order to increase the usage of the working surface of the tool, the tool may be operated at a second precession angle such that the tool contact footprint is within the second annular zone 13b, and the V-shaped points 12B pass substantially through the centre of the tool footprint as the tool is rotated. When operating in this mode, it is preferred that the tool be rotated in the second rotation direction R2 so that the V-shaped point 12B points in the direction in which it moves through the tool footprint.

The usage of the working surface of the tool may be further increased by operating the tool at a third precession angle such that the tool contact footprint is within the third annular zone 13c, and the V-shaped points 12C pass substantially through the centre of the tool footprint as the tool is rotated. When operating in this mode, it is preferred that the tool be rotated in the first rotation direction R1 so that the V-shaped point 12C points in the direction in which it moves through the tool footprint.

It will be appreciated that there may be more than six radially-extending grooves, and there may be more than three V-shaped points on each groove, in order fully to utilise the working surface of the tool. Preferably there is an even number of radially-extending grooves, so that pairs of adjacent grooves are formed, and all of the grooves may be formed by a routing or laser ablation tool in a single unicursal pass, by connecting together the radially inner ends of the grooves in each pair, and the radially outer ends of adjacent grooves from adjacent pairs.

In the embodiment shown in Figure 9, the outer ends of adjacent grooves from adjacent pairs are joined by respective circumferentially-extending grooves 14, and the inner ends of the grooves in each respective pair are joined by a circumferentially-extending groove 15, so that the entire pattern of grooves may be formed by a single pass of a routing or other shaping tool.

In the foregoing examples, a preference has been expressed for rotating the tool in such a direction that the V-shaped points of the grooves point in the circumferential direction of rotation. Although this is a preferred option, it is foreseen that the tool may be operated in the reverse rotation direction.

Tests have shown that material removal rates (MRR) are significantly higher with V-grooved tools than are obtainable with smooth tools, particularly at speeds of rotation above 15,000 rpm. In comparative tests, the MRR achieved by a conventional tool operated at between 15,000 and 25,000 rpm was significantly lower than the MRR achieved by a conventional tool whose tool path data compensated for the deformation of the tool at these high rotation speeds. However, the MRR achieved by a V-grooved tool at these high rotation speeds was significantly greater than both the normal tool and the normal tool compensated for deformation. Furthermore, in the rotational speed range from 15,000 to 25,000 rpm, the MRR for the V-grooved tool increased with increasing rotation speed, whereas the MRR for the conventional tool and for the conventional tool with compensation both exhibited a marked decrease with increasing rotation speed. This indicates that the grooved tool is effective in reducing or preventing the hydroplaning phenomenon at high rotation speeds of the tool, and may be effective up to 100,000 rpm.

The grooved configuration of the tool, with V-shaped points passing through the tool footprint, thus significantly improves the predictability and controllability of the MRR at high rotation speeds.

In another aspect of the invention, in a rotating-tool shaping or polishing process using a compliant tool, where a layer of slurry is formed between the tool and the workpiece surface, an abrasive slurry which is a non-Newtonian fluid (NNF) is proposed as a means to improve material removal rate.

The layer of slurry between the rotating tool and the workpiece may be a result of the hydroplaning effect in high-speed tools where slurry is drawn into the nip between the workpiece and the tool.

Alternatively, the layer of slurry may be formed when a tool is positioned slightly away from the workpiece so as to form a clearance gap between the workpiece and the tool, and is rotated at low speed in order to draw the slurry into the clearance to form a layer of moving slurry. Such a spacing will correspond to a “negative offset”.

To form an NNF slurry, a polymer and/or starch are added as thickening agents to an aqueous suspension of abrasive particles. Preferably, starch serves as the primary thickening agent and a lesser amount of polymer is included to help stabilise the slurry and improve the process behaviour (without it, the removal behaviour may be sub-optimal). Typically, the abrasive slurry will comprise between 100 and 2000 g/L of thickening agents (preferably around 500 g/L) and between 10 to 40 g/L of abrasive particles. In an advantageous embodiment, the slurry may contain about 500 g/L of starch, about 2.5 g/L of polymer, and 30 g/L of abrasive particles. .

The polymer addition to the slurry may be long chained polymers such as Polyethylene Oxide (PEO), Polyacrylamide (PAM), Anionic Polyacrylamide (HP AM) and Cationic Polyacrylamide (PAMA).

The Zero-Shear (static) Viscosity of the slurry is preferably low (~1.0 mPa.s), while Dynamic-Shear Viscosity should be higher (200 to 500 mPa.s). This depends on the wt% of the thickening agent used in the slurry and can be measured with a Cannon-Fenske capillary tube viscometer. Examples of starches which may be added to the slurry include cornstarch, rice flour, potato flour, wheat flour and synthetic starch.

One example of an NNF slurry mixture may be obtained by mixing between 400 and 500 g starch into between 400 and 500 ml of water in which is suspended between 1.0 and 1.5 g of a polymer (-CH2CHCONH2-, e.g.2.5 g/1) and stirring energetically. The obtained fluid is about 700 to 900 ml in volume, and then between 20 and 30 g alumina powder (type FO #6000, Fujimi Japan) may be mixed into the fluid to get an abrasive concentration of around 30 g/L.

When a low-speed tool is used, the rotation of the tool is insufficient to produce any appreciable change in the shape of the tool, but is effective to draw a layer of the NNF slurry between the tool and the workpiece. The tool is held relative to the workpiece with a negative offset, so that a clearance exists between the tool and the workpiece, the slurry being drawn into this clearance by the rotation of the tool. The clearance between the tool and the workpiece may be from 0.01 mm up to and exceeding 0.15 mm and effectively regulates the thickness of the layer of slurry.

At very low rotation speed and with the tool“floating” above the surface, non- Newtonian fluid (NNF) behaves like regular water based slurry and slowly removes material through erosion of the workpiece surface by the abrasive particles contacting the workpiece as they flow through the clearance between the tool and the workpiece. However, with increasing tool speed the viscosity of the NNF locally increases, i.e. the slurry locally thickens. This is because molecular tumbling/colliding occurs and causes agglomeration of hydro clusters HC, which are formed when the long chained polymers (which are hook shaped) temporarily link together to form a sheath around an abrasive grain, resulting in a remarkable increase in viscosity and static pressure. This greatly enhances the interaction between abrasives and workpiece to quickly remove material without needing direct tool contact. Under these circumstances, the NNF’s viscosity change can be regulated by adjusting the shear force in the narrowest part of the clearance gap, so as to stably form a circular domain of thickened fluid in the tool footprint TF between the tool and workpiece, as illustrated in Figure 10. The hydro clusters reduce or dissolve again once they exit the high shear region in the working gap , so that the slurry forms a "thickened" high- viscosity region at the narrower, high-shear, part of the tool footprint surrounded by a "thinner" lower- viscosity region in the areas where the clearance between the workpiece and the tool surface is greater and thus the shear forces are less.. In the course of polishing with a high-speed tool, the slurry agglomerates into a steady shape in the working gap C between the tool 10 and the workpiece W, as shown in Fig. 10. In the Figure, the tool is rotated about the axis T in the direction shown by the arrow R. The axis T may be tilted so that its upper end (as shown in the Figure) is tilted either towards or away from the viewer. Rotation of the tool 10 in the direction of the arrow R draws the fluid between the tool 10 and the workpiece W in the direction from left to right in the Figure. It has been found that by using an NNF slurry, the intermediate fluid/solid status of the slurry greatly facilitates stable polishing accompanied by considerable material removal rate.

Figures 11A and 11B are schematic illustrations of the behaviour of the NNF slurry in the region between the rotating tool 10 and the workpiece W. In Figure 11 A, there is shown a hydro cluster HC of molecules of the thickening agent which surround an abrasive grain G. The hydrostatic pressure P _s in the slurry generates a downward (as shown in the Figure) force Fn on the hydro cluster HC which urges the abrasive grain G in a direction normal to the surface of the workpiece W, while the shear stresses in the slurry also generate a horizontal (as shown in the Figure) force Ft tangential to the surface of the workpiece W. It is thought that the combination of this force Fn urging the grains onto the workpiece surface, and the force Ft moving the grains across the surface, is what contributes to the increased material removal rates observed.

Figure 1 IB illustrates the variation of the shear stress profile in the gap h between the tool 10 and the workpiece W. The magnitudes of the shear forces r _s at various distances from the surface of the tool 10 are represented by the horizontal lines within the three shear force profiles. At the edges of the tool footprint TF the distance between the tool 10 and the workpiece W is greater than the distance between the tool 10 and the workpiece W at the centre of the tool footprint. The maximum shear forces experienced at these edge regions are smaller than the maximum shear forces generated at the centre of the tool footprint, and thus abrasive grains at the centre of the footprint experience a higher tangential force Ft than those at the edges of the footprint. The same effect is observed as regards the static pressure, which is increased at the centre of the tool footprint TF as compared to the static pressure at the edge, and thus the normal force Fn acting on the abrasive grains at the centre of the footprint is greater than the normal force acting on grains at the edge of the tool footprint to urge the grains against the surface of the workpiece W. This combination of forces leads to a greater material removal at the centre than at the edges of the tool footprint, resulting in a Gaussian-like material removal profile RP. It is presently thought that the shearing action applied to the fluid between the tool 10 and the workpiece W leads to the highly increased viscosity, which in turn leads to large hydrodynamic pressure Ps on the hydro cluster HC, resulting in the increased normal penetration force Fn at the centre of the tool footprint TF, as illustrated in Fig. 11 A. Meanwhile, when the tool 10 spins at high speed, high shear stress rs will be induced on the workpiece W. This phenomenon contributes to the raised tangential force Ft, thereby driving the cluster-enfolded abrasive grain G across the workpiece material W. This results in enhanced material removal at the centre of the circular tool footprint TF as the grains are pushed hard into the workpiece surface and moved with force across the workpiece surface. An enhanced smoothing effect is observed at the edge region SR adjacent the tool footprint TF, as shown in Fig. 11B, from slurry moving across the workpiece surface but at lower viscosities than are generated by the shear forces are applied in the tool footprint region TF. Consequently, a

Gaussian-like removal profile RP can be generated without tool contact occurring in the tool footprint TF, and a smoothing effect can be generated in the region surrounding the tool footprint TF.