Abrasive particle with at most three surfaces and one corner

Abstract

An abrasive particle includes at most three surfaces and at least one edge which has a corner at at least one end. The abrasive particle may contain a ceramic material, particularly polycrystalline -Al.sub.2O.sub.3. Abrasive particles as a whole, methods for producing abrasive particles, moulds, abrasive articles, methods for producing abrasive articles, and methods for abrading a surface are also disclosed.

Claims

1. An abrasive grain comprising: two opposite base faces each defining an outline that has at least one concave section, wherein the abrasive grain is in the shape of a straight prism, wherein a tangent to at least one point on the at least one concave section runs at an angle of between 5 and +5 relative to a perpendicular plane that is orthogonal to a support plane at which the abrasive grain is configured to be supported.

2. The abrasive grain as claimed in claim 1, wherein the outline of each of the two opposite base faces has at least one corner.

3. The abrasive grain as claimed in claim 2, wherein the at least one corner of each of the two opposite base faces is formed at an edge of the at least one concave section and defines an internal angle of between 85 and 95.

4. The abrasive grain as claimed in claim 3, wherein the internal angle is 90.

5. The abrasive grain as claimed in claim 2, wherein the outline of each of the two opposite base faces has at least one linear section.

6. The abrasive grain as claimed in claim 5, wherein the at least one linear section runs at an angle of less than or equal to 20 with respect to the support plane.

7. The abrasive grain as claimed in claim 5, wherein the at least one linear section runs at an angle of less than or equal to 10 with respect to the support plane.

8. The abrasive grain as claimed in claim 5, wherein the at least one linear section runs at an angle of less than or equal to 5 with respect to the support plane.

9. The abrasive grain as claimed in claim 1, wherein the angle is 0.

10. The abrasive grain as claimed in claim 1, wherein: the at least one concave section includes at least three concave sections, each end of each of the at least three concave sections forming a corner of the outline, and the outline has a linear section connecting each respective end of a respective one of the at least three concave sections to an adjacent end of a respective other of the at least three concave sections.

11. The abrasive grain as claimed in claim 1, wherein: the at least one concave section includes exactly three concave sections, each end of the three concave sections forming a corner of the outline; and the outline has exactly three linear sections, each end of the three linear sections forming one of the corners with a respective one of the ends of one of the three concave sections.

12. The abrasive grain as claimed in claim 2, wherein the at least one point that defines the tangent is located at the at least one corner.

13. The abrasive grain as claimed in claim 5, wherein the at least one corner is formed at an intersection of the at least one point that defines the tangent and the at least one linear section.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The invention is illustrated hereinafter with reference to several working examples and drawings. These show:

(2) FIG. 1A: a perspective view of a first embodiment of an inventive abrasive grain;

(3) FIG. 1B: a front view of the abrasive grain from FIG. 1A;

(4) FIG. 1C: a side view of the abrasive grain from FIG. 1A;

(5) FIG. 1D: a top view of the abrasive grain from FIG. 1A;

(6) FIG. 1E: a side view of the abrasive grain from FIG. 1A aligned on an underlayer;

(7) FIG. 2A: a perspective view of a modified abrasive grain from FIG. 1A;

(8) FIG. 2B: a side view of the abrasive grain from FIG. 2A;

(9) FIG. 2C: a section through the abrasive grain from FIG. 2A along a plane through two opposite corners;

(10) FIG. 2D: a side view of the abrasive grain from FIG. 2A aligned on an underlayer;

(11) FIG. 3A: a perspective view of a further embodiment of an inventive abrasive grain;

(12) FIG. 3B: a top view of the abrasive grain from FIG. 3A;

(13) FIG. 3C: a front view of the abrasive grain from FIG. 3A;

(14) FIG. 4: a perspective view of a further embodiment of an inventive abrasive grain;

(15) FIG. 5: a perspective view of a further embodiment of an inventive abrasive grain;

(16) FIG. 6: a further embodiment of an inventive abrasive grain in a perspective view;

(17) FIG. 7: a further embodiment of an inventive abrasive grain in a perspective view;

(18) FIG. 8: a further embodiment of an inventive abrasive grain in a perspective view;

(19) FIG. 9: a further embodiment of an inventive abrasive grain in a perspective view;

(20) FIG. 10: a schematic lateral section view of an extruder for production of abrasive grains;

(21) FIG. 11A: a top view of the extruder according to FIG. 10 having a first nozzle;

(22) FIG. 11B: a top view of an extruder according to FIG. 10 having a second nozzle;

(23) FIG. 11C: a top view of an extruder according to FIG. 10 having a third nozzle;

(24) FIG. 11D: a top view of an extruder according to FIG. 10 having a fourth nozzle;

(25) FIG. 11E: a top view of an extruder according to FIG. 10 having a fifth nozzle;

(26) FIG. 12A: an abrasive grain produced with the first nozzle according to FIG. 11A;

(27) FIG. 12B: an abrasive grain produced with the second nozzle according to FIG. 12B;

(28) FIG. 13: an abrasive grain with the shape of a twisted cuboid;

(29) FIG. 14A: a further embodiment of an inventive abrasive grain in perspective view;

(30) FIG. 14B: a schematic section diagram of the embodiment according to FIG. 14A of the inventive abrasive grain;

(31) FIG. 15A: a further embodiment of an inventive abrasive grain in perspective view;

(32) FIG. 15B: a schematic side view of the embodiment according to FIG. 15A of the inventive abrasive grain;

(33) FIG. 15C: a further schematic side view of the embodiment according to FIG. 15A of the inventive abrasive grain;

(34) FIG. 15D: a schematic top view of the embodiment according to FIG. 15A of the inventive abrasive grain;

(35) FIG. 16A: a further embodiment of an inventive abrasive grain;

(36) FIG. 16B: a schematic section diagram of the embodiment according to FIG. 16A;

(37) FIG. 17: a further embodiment of an inventive abrasive grain;

(38) FIG. 18: a further embodiment of an inventive abrasive grain;

(39) FIG. 19: a further embodiment of an inventive abrasive grain;

(40) FIG. 20A: a further embodiment of an inventive abrasive grain in perspective view;

(41) FIG. 20B: a schematic section diagram of the embodiment according to FIG. 20A of the inventive abrasive grain;

(42) FIG. 21: a further embodiment of an inventive abrasive grain;

(43) FIG. 22: a further embodiment of an inventive abrasive grain;

(44) FIG. 23: a further embodiment of an inventive abrasive grain;

(45) FIG. 24: a side view of a known abrasive grain in the processing of a surface;

(46) FIG. 25A: a first view of a further embodiment of an inventive abrasive grain;

(47) FIG. 25B: a second view of the abrasive grain of FIG. 25A;

(48) FIG. 26: a side view of a further embodiment of an inventive abrasive grain;

(49) FIG. 27: a side view of a further embodiment of an inventive abrasive grain;

(50) FIG. 28: a side view of a further embodiment of an inventive abrasive grain;

(51) FIG. 29A: a first view of a further embodiment of an inventive abrasive grain;

(52) FIG. 29B: a second view of the abrasive grain of FIG. 29A;

(53) FIG. 30A: a first view of a further embodiment of an inventive abrasive grain;

(54) FIG. 30B: a second view of the abrasive grain of FIG. 30A;

(55) FIG. 31: a die for production of an abrasive grain of the inventive embodiment according to FIG. 28.

DETAILED DESCRIPTION

(56) FIGS. 1A to 1E show an inventive abrasive grain 10 in a first embodiment. FIG. 1A shows a perspective view, FIG. 1B shows a front view, FIG. 1C shows a side view, and FIG. 1D shows a top view. FIG. 1E shows the abrasive grain 10 in a side view, the abrasive grain 10 being disposed on an underlayer 20. The abrasive grain 10 has a virtually square outline with four corners 1 to 4 and four edges 4 to 8 that join the corners 1 to 4. The edges 5 to 8 are straight, although curved edges are also conceivable. Proceeding from a flat square, however, two mutually opposite corners 2, 4 are shifted upward out of the plane, and the two other mutually opposite corners 1, 3 are shifted downward out of the plane. The abrasive grain 10 therefore has only two faces 11, 12. These two faces 11, 12 in the working example shown are identical, i.e. form merely the top side and bottom side of a single face. Alternatively, it is also conceivable that both faces have a different curvature profile and therefore enclose a volume between the two faces 11, 12.

(57) As is apparent in FIG. 1E, the abrasive grain 10 lines up on an underlayer 20 such that three corners 1 to 3 lie on the underlayer 20, while the remaining corner 4 protrudes from the underlayer 20. The plane formed from the three corners 1, 3 and 4 is at an angle from the underlayer 20. This angle is simultaneously the angle between the two planes formed by the corners 1, 3 and 4 or 1, 2 and 3. In the embodiment shown, the angle is obtuse.

(58) FIGS. 2A to 2D show the abrasive grain 10 from FIG. 1A in slightly modified form. FIG. 2A shows a perspective view, FIG. 2B a side view, FIG. 2C a section through the abrasive grain 10 from FIG. 2A along a plane through two opposite corners 1, 3, and FIG. 2D another side view, with the abrasive grain 10 aligned on an underlayer 20. The respectively opposite corners 1, 3 and 2, 4 are respectively shifted further upward and downward out of the plane with respect to FIG. 1A. As shown in FIG. 2D, a smaller angle is correspondingly formed between the underlayer 20 and the plane formed from the corners 1, 2 and 3. The angle is acute. According to the configuration of the angle , the abrasion characteristics of the abrasive grain 10 can be significantly influenced. In the section in FIG. 2C, it is clearly apparent that the two faces 11, 12 of the abrasive grain 10 enclose a volume.

(59) FIGS. 3A to 3C show a further embodiment of an inventive abrasive grain 10. FIG. 3A shows a perspective view, FIG. 3B a top view and FIG. 3C a front view of the abrasive grain 10. This abrasive grain 10 has two mutually skewed edges 5, 7, the respective ends of which are bounded by a corner 1, 2 and 3, 4 respectively. The abrasive grain 10 has four corners 1 to 4. The two edges 5, 7 are arranged at an angle of 90 relative to one another. The abrasive grain 10 has a single face 11 which takes the form of a continuous curved face. For easier understanding of the shape, it is possible to imagine a cylindrical piece of tube which is closed off and flattened at either end, the two closures being arranged at right angles to one another and the abrasive grain 10 being formed by the cavity surrounded by the closed-off piece of tube. In FIGS. 3A and 3C, the common perpendicular is drawn in for schematic purposes, which simultaneously defines the distance D between the skewed edges 5, 7. The length of the edges 5, 7 is the same as the distance D. Typically, the ratio of the edge length to the distance D is between 0.5 and 2.0, preferably between 0.7 and 1.4, more preferably between 0.9 and 1.1. In the embodiment shown, the edges 5, 7 are arranged such that the common perpendicular in each case is in the middle of the edges 5, 7. Also conceivable, of course, are abrasive grains in which the common perpendicular has been shifted from the middle. There is also no need for the edges 5, 7, as shown in the working example, to run straight; curved edges are also entirely conceivable.

(60) FIGS. 4 and 5 show, in perspective view, two further embodiments of an inventive abrasive grain 10. These abrasive grains 10 again have four corners 1 to 4. The corners 1 to 4 are each joined to one another via the edges 5 to 6, similarly to FIG. 1A. In addition, however, two mutually opposite corners 1, 3 are joined by a further edge 9. The edges 5 to 9 are all curved lines, but it is also conceivable that the edges are straight. Curved faces 11, 12 are formed between the corners 1, 2 and 3 or 1, 3 and 4, these being concave. Flat faces are also conceivable. A primary face 13 is formed between all four corners 1 to 4, this again being curved. A vertex line 14 of the primary face 13 is concave in the embodiment according to FIG. 4, whereas the vertex line 14 is convex in FIG. 5. However, the vertex line 14 does not itself form an edge of the abrasive grain. The two embodiments according to FIGS. 4 and 5 also differ by an angle which is formed between the planes formed from the corners 1, 2 and 3 or 1, 3 and 4. In FIG. 4 this angle is 90; in FIG. 5 the angle is 135. It will be apparent that these angles may each also have other values. The two abrasive grains 10 according to FIGS. 4 and 5 are also symmetric with respect to a plane at right angles to a straight line connecting the corners 1 and 3 or 2 and 4. Such a symmetry facilitates the production of corresponding dies. In addition, the abrasive grains 10 according to FIGS. 4 and 5 may lie in a first orientation on corners 1, 3 and 4 on an abrasive underlayer, or in a second orientation on corners 1, 2 and 3. In both orientations, the corners 2 or 4 pointing away from the underlayer in each case are similar to one another because of the symmetry; the abrasive action is thus independent of which of the two orientations is present. Moreover, the abrasive grains 10 in FIGS. 4 and 5 are symmetric about an axis of symmetry which runs at the midpoint between the two corners 1 and 3 and intersects the edge 9.

(61) The inventive abrasive grains can be produced, for example, by a process described hereinafter: first of all, a dispersion of 200 g of -Al.sub.2O.sub.3, 0.4 g of MgO, 90 g of water as dispersion medium and 0.5 g of dispersant is produced. The MgO functions here as a nucleating agent. The dispersant used may, for example, be the Dolapix CE64 product obtainable from Zschimmer & Schwarz, 56108 Lahnstein, Germany. The dispersion thus obtained is ground in a planetary ball mill at 200 revolutions per minute for 30 minutes, for example a PM400 planetary ball mill obtainable from Retsch GmbH, 42781 Haan, Germany. Subsequently, the ground dispersion is introduced into a silicone die containing depressions in the shape of the desired abrasive grains. For some embodiments of the abrasive grain, it is possible to use an additional shaping element as described above, for example a further die, with which, in addition to the surface shaped in the die, it is possible to shape at least a portion of the rest of the surface of the abrasive grain. Thereafter, the volatile component, i.e. the water, is removed from the dispersion. This gives rise to an abrasive grain precursor which is removed from the die. In a final step, the precursor is sintered as bulk material at 1550 C. for 5 minutes. The dispersant is burnt out in the course of sintering.

(62) An inventive abrasive article can be produced, for example, as follows: on an underlayer made from vulcanized fiber having a thickness of 0.8 mm, a phenol resin dispersion as make coat precursor is applied in an amount of 120 g/m.sup.2. Subsequently, 600 g/m.sup.2 of the inventive abrasive grains are applied by means of electrostatic scattering. Thereafter, the make coat precursor is cured to give a make coat. On top of the make coat and the abrasive grains, a phenol resin dispersion is applied in an amount of 800 g/m.sup.2 as size coat precursor, which is likewise cured.

(63) FIG. 6 shows, in perspective view, a first embodiment of an inventive abrasive grain 110. The abrasive grain 110 contains a base element 120 and a top element 125. The base element 120 contains a bottom side 121 and a parallel opposite top side 122, on which is disposed a bottom side 126 of the top element 125. Both the base element 120 and the top element 125 have the shape of a twisted cuboid, i.e. of a specific twisted straight prism. The base element 120 and the top element 125 are twisted with respect to the theoretical cuboids about a common twist axis V which runs at right angles to the bottom side 121 and to the top side 122 of the base element 120 and at right angles to the bottom side 126 and to a top side 127 of the top element 125.

(64) The angle by which the cross sections of the base element 120 which run at right angles to the twist axis V have been rotated compared to the theoretical cuboid has a linear dependence on the distance of the sectional plane from the bottom side 121 of the base element 120. Likewise, the angle by which the cross sections of the top element 125 which run at right angles to the twist axis V have been rotated compared to the theoretical cuboid has a linear dependence on the distance of the sectional plane from the bottom side 126 of the top element 125. Both for the base element 120 and for the top element 125, the angle differential between the respective bottom side 121 or 126 and the respective top side 122 or 127 here is 90. In a departure from the working example shown here, this angle for the base element 120 may also be about 45, because corners of the underside 121 of the base element 120 then project beneath the top side 122 of the base element 120 as a result, which can assure particularly good anchoring in a make coat.

(65) The effect of the twisting of the base element 120 is that the base element 120 can be anchored better in a make coat of an abrasive article not shown here. The result of the twisting of the top element 125 is that swarf formed in the course of grinding can be transported away from a surface being processed.

(66) In the second embodiment shown in FIG. 7, the abrasive grain 210 contains an untwisted base element 220 in the form of an untwisted frustopyramid and a top element 225 in the form of a twisted pyramid.

(67) The abrasive grain 310 according to FIG. 8 consists of only a single twisted component 320; it is thus not composed of a base element and a top element like the abrasive grains 110 and 210. The component 320 has the shape of a twisted cylinder with an elliptical base face 321 and an elliptical top face 322.

(68) The abrasive grain 410 shown in FIG. 9 likewise consists only of a single twisted component 420 which has the shape of a twisted polyhedron. The theoretical polyhedron has the shape of a prism with two triangular end faces 421, only one of which can be seen here, and three rectangular lateral faces 422, 422 and 422. The component 420 is twisted with respect to the theoretical polyhedron about a twist axis V which runs at right angles to the center of the lateral face 422.

(69) The inventive abrasive grains can be produced, for example, by a process described hereinafter: first of all, a dispersion of 200 g of -Al.sub.2O.sub.3, 0.4 g of MgO, 90 g of water as dispersion medium and 0.5 g of dispersant is produced. The MgO functions here as a sintering aid, in order to suppress grain growth. The dispersant used may, for example, be the Dolapix CE64 product obtainable from Zschimmer & Schwarz, 56108 Lahnstein, Germany. The dispersion thus obtained is ground in a planetary ball mill at 200 revolutions per minute for 30 minutes, for example a PM400 planetary ball mill obtainable from Retsch GmbH, 42781 Haan, Germany. Subsequently, the ground dispersion is introduced into a silicone die containing depressions in the shape of the desired abrasive grains. Thereafter, the volatile component, i.e. the water, is removed from the dispersion. This gives rise to an abrasive grain precursor which is removed from the die. If the abrasive grain contains a base element as described above with a bottom side, this bottom side can form from the upper free surface of the dispersion which is not in contact with the die. In a final step, the precursor is sintered as bulk material at 1550 C. for 5 minutes. The dispersant is burnt out in the course of sintering.

(70) FIG. 10 shows, in a schematic lateral section view, an extruder 500 for production of inventive abrasive grains. The extruder 500 contains a hollow cylinder 501 with a screw 502 arranged therein. The screw 502 can be driven in a rotating manner with the aid of a motor 503. By means of an introduction funnel 504, a dispersion is introduced into the interior of the hollow cylinder 501. This dispersion contains -alumina particles and/or particles that can be converted to -alumina, and at least one volatile dispersion medium, preferably water, and preferably at least one organic additive, for example the plasticizer Zusoplast C21, available from Zschimmer & Schwarz, 56108 Lahnstein, Germany. With the aid of the screw 502, the dispersion is conveyed to a constricted exit end 505 and forced through an exit orifice of a nozzle 506, 506, so as to form an extrudate which is not shown here. The exit orifice has the shape of a twisted cylinder. With the aid of rotating blades which are likewise not shown here, the extrudate is severed to form individual abrasive grain precursors which can subsequently be calcined and then sintered.

(71) FIGS. 11A to 11E show top views of five nozzles 506, 506, 506, 506, 506. These nozzles 506, 506, 506, 506, 506 each have exit orifices 507, 507, 507, 507 and 507 respectively. The exit orifice 507 has the shape of an equilateral triangle, the exit orifice 507 the shape of a square, the exit orifice 507 the shape of a rectangle with rounded corners, the exit orifice 507 the shape of an ellipse, and the exit orifice 507 the shape of a lens. FIGS. 11C to 11E show the length 1 and the width b of the exit orifice.

(72) The extrudate obtained with the nozzle 506 according to FIG. 11A has the shape of a twisted prism having a base face of an equilateral triangle with side length a. With the aid of the rotating blades, individual abrasive grain precursors are obtained, which, after calcining and sintering, give rise to abrasive grains 510, one of which is reproduced in FIG. 12A. This abrasive grain 510 has a base face 521 and a top face 522 and has a height h measured along the twist axis V, where the ratio of side length a to height h may be in the range from 1:3 to 8:1, preferably from 1:1 to 7:1 and more preferably from 4:1 to 6:1. In the abrasive grain 510 shown in FIG. 12A, the overall twist angle (i.e. the angle differential between the base face 521 and the top face 522) is 60. Even more preferred than the abrasive grain 510 shown in FIG. 12A are abrasive grains in the shape of a twisted prism having base face and top face in the shape of equilateral triangles when the overall twist angle is in the range from 5 to 30 and is most preferably about 10.

(73) An abrasive grain 510 which has been obtained with the nozzle 506 according to FIG. 11B and which has the shape of a twisted cuboid with side length a is shown in FIG. 12B. This abrasive grain 510 has a height h measured along a twist axis V, where the ratio of side length a to height h may be in the range from 1:3 to 8:1, preferably from 1:1 to 7:1 and more preferably from 4:1 to 6:1. In this abrasive grain 510, the overall twist angle between base face 521 and top face 522 is 90, which is particularly preferred for abrasive grains in the shape of twisted cuboids having rectangular, especially square, base and top faces.

(74) An inventive abrasive article can be produced, for example, as follows: on an underlayer made from vulcanized fiber having a thickness of 0.8 mm, a phenol resin dispersion as make coat precursor is applied in an amount of 120 g/m.sup.2. Subsequently, 600 g/m.sup.2 of the inventive abrasive grains are applied by means of electrostatic scattering. Thereafter, the make coat precursor is cured to give a make coat. On top of the make coat and the abrasive grains, a phenol resin dispersion is applied in an amount of 800 g/m.sup.2 as size coat precursor, which is likewise cured.

(75) The abrasive grain 810 shown in FIG. 13 has the shape of a twisted cuboid. The base face 821 and the top face 822 each have a length a and a width b, the ratio of which is preferably in the range from 10:2 to 10:3. In addition, the abrasive grain 810 has a height h measured along the twist axis V, which is preferably in a ratio to the length a of 4:1 to 6:1. The overall twist angle of this abrasive grain 810 is 90.

(76) FIG. 14A shows a first embodiment of an inventive abrasive grain 100 in perspective view. The abrasive grain has a structure having six faces 110 having concave curvature 111.

(77) The concave curvature 111 becomes clear in FIG. 14B, which shows a schematic section diagram of the first embodiment of the inventive abrasive grain 100. The convex corners 112 of the structure correspond to the corners of an imaginary cube.

(78) The faces 110 are curved in two spatial directions 114, 115.

(79) The edges 113 likewise have concave curvature.

(80) The edges 113 which run toward the corners 112 form, by virtue of the face 110 pulled inward, more acute angles at the corners 112 than the edges of a cube.

(81) The base face of the cube has high symmetry, and so the face on which the abrasive grain 100 comes to rest when scattered is immaterial. This is particularly advantageous in the case of mechanical scattering, with which the abrasive grains, in contrast to electrostatic scattering, cannot be aligned with the aid of an electrical field.

(82) FIG. 15A shows a second embodiment of an inventive abrasive grain 200 in perspective view. The abrasive grain 200 has a structure with three reentrant corners 216, which becomes particularly clear in FIG. 15D, which shows a schematic top view of the second embodiment of the inventive abrasive grain 200.

(83) The convex corners 212 of the structure correspond to the corners of an imaginary tetrahedron, i.e. of a Platonic solid.

(84) The reentrant corner 216 is shifted inward compared to the tetrahedral face. There is therefore a tangential plane, i.e. a plane which includes the reentrant corner 216, on which a circle with the corner 216 as the center can be defined, the circumference of which is entirely within the solid.

(85) The effect of the corners 216 pulled inward is that the faces 210 meet at the edges 213 at a more acute angle than in a tetrahedron. The abrasive grain 200 therefore has sharp edges and good cutting power.

(86) FIGS. 15B and 15C show schematic side views of the second embodiment of the inventive abrasive grain 200.

(87) There is no reentrant corner on the base face of the abrasive grain, which is not shown explicitly. The base face therefore contains more material. When scattered, the abrasive grain 200 will come to rest preferentially on the heavier side, i.e. the base face. Ideally, the base face is flat. However, it is possible and covered by the invention that the base face, as a result of production, has slight concave curvature which can arise because of the above-described shrinkage during the removal of the volatile components.

(88) FIG. 16A shows a third embodiment of an inventive abrasive grain 300 in perspective view. The abrasive grain 300 has a structure with reentrant corners 316, which becomes particularly clear in FIG. 16B, which shows a schematic section view of the third embodiment of the inventive abrasive grain 300.

(89) The convex corners 312 of the structure correspond to the corners of an imaginary cube, i.e. of a Platonic solid.

(90) The edges 317 that run toward the convex corners 312 are curved.

(91) The faces 310 of the abrasive grain 300 that meet at the edges 317 form a more acute angle than in the case of a cube.

(92) FIG. 17 shows a fourth embodiment of an inventive abrasive grain 400 in perspective view. The convex corners 412 correspond to the corners of an imaginary frustotetrahedron, i.e. of an Archimedean solid. The faces 410 have concave curvature.

(93) FIG. 18 shows a fifth embodiment of an inventive abrasive grain 500 in perspective view. The convex corners 512 correspond to the corners of an imaginary tetrahedron. Three tetrahedral edges are capped in such a way as to give rise to faces 510 with concave curvature.

(94) FIG. 19 shows a sixth embodiment of an inventive abrasive grain 600 in perspective view. The convex corners 612 correspond to the corners of an imaginary frustotetrahedron, i.e. of an Archimedean solid. The edges are capped in such a way that I give rise to faces 610 with concave curvature.

(95) FIG. 20A shows a seventh embodiment of an inventive abrasive grain 700 in perspective view. The convex corners of the solid correspond to the corners of an imaginary tetrakis hexahedron, i.e. of a Catalan solid.

(96) The concave curvature 711 of the faces 710 becomes clear in FIG. 20B, which shows a schematic section view of the seventh embodiment of the inventive abrasive grain 700.

(97) The edges 713 likewise have concave curvature.

(98) The edges 713 that run toward the corners 712, by virtue of the face 710 pulled inward, form more acute angles at the corners 712 than the edges of a tetrakis hexahedron.

(99) FIG. 21 shows an eighth embodiment of an inventive abrasive grain 800 in perspective view. The convex corners 812 correspond to the corners of an imaginary prism. The faces 810 have concave curvature, and so the faces 810 at the edges 813 form more acute angles than in a prism.

(100) FIG. 22 shows a ninth embodiment of an inventive abrasive grain 900 in perspective view. The convex corners 912 correspond to the corners of an imaginary prism. The faces 910 have concave curvature, and so the faces 910 at the edges 913 form more acute angles than in an antiprism.

(101) FIG. 23 shows a tenth embodiment of an inventive abrasive grain 1000 in perspective view. The abrasive grain 1000 has a structure with faces 1010 having concave curvature. At the edge of the curved faces, sharp edges 1013 are formed.

(102) The inventive abrasive grains can be produced, for example, by a process described hereinafter: first of all, a dispersion of 200 g of -Al.sub.2O.sub.3, 0.4 g of MgO, 90 g of water as dispersion medium and 0.5 g of dispersant is produced. The MgO functions here as a nucleating agent. The dispersant used may, for example, be the Dolapix CE64 product obtainable from Zschimmer & Schwarz, 56108 Lahnstein, Germany. The dispersion thus obtained is ground in a planetary ball mill at 200 revolutions per minute for 30 minutes, for example a PM400 planetary ball mill obtainable from Retsch GmbH, 42781 Haan, Germany. Subsequently, the ground dispersion is introduced into a silicone die containing depressions in the shape of the desired abrasive grains. For some embodiments of the abrasive grain, it is possible to use an additional shaping element as described above, for example a further die, with which, in addition to the surface shaped in the die, it is possible to shape at least a portion of the rest of the surface of the abrasive grain. Thereafter, the volatile component, i.e. the water, is removed from the dispersion. This gives rise to an abrasive grain precursor which is removed from the die. In a final step, the precursor is sintered as bulk material at 1550 C. for 5 minutes. The dispersant is burnt out in the course of sintering.

(103) An inventive abrasive article can be produced, for example, as follows: on an underlayer made from vulcanized fiber having a thickness of 0.8 mm, a phenol resin dispersion as make coat precursor is applied in an amount of 120 g/m.sup.2. Subsequently, 600 g/m.sup.2 of the inventive abrasive grains are applied by means of electrostatic scattering. Thereafter, the make coat precursor is cured to give a make coat. On top of the make coat and the abrasive grains, a phenol resin dispersion is applied in an amount of 800 g/m.sup.2 as size coat precursor, which is likewise cured.

(104) FIG. 24 shows, in schematic form, the processing of a surface 50 with an abrasive article 40 having a known abrasive grain 10. For simplification of the diagram, only a single abrasive grain 10 is shown here, even though the abrasive article 40 of course does in fact contain a multitude of such abrasive grains 10. The abrasive grain 10 has been fixed on a substrate 41 with the aid of a binder 42. It has the shape of a straight cylinder having a base face 11 in the shape of an equilateral triangle. A flat section of the abrasive grain 10 which is in contact with a linear section 14 of the outline 12 lies against the underlayer 41, such that a tip 17 of the abrasive grain 10 is directed toward the surface 50. Between a cutting face 16 and a perpendicular S to the surface 50, an angle of engagement of +30 is formed. This is given a positive sign here, since the perpendicular S runs within the abrasive grain 10. Such an angle of engagement leads only to a comparatively minor cutting effect.

(105) FIGS. 25A and 25B show a first inventive abrasive grain 10. According to the perspective view in FIG. 25A, the abrasive grain 10 has the shape of a cylinder having two opposite base faces 11, only one of which can be seen here. A shell face 18 extends between the two base faces 11. The cylinder has a height h.

(106) FIG. 25B shows a top view of one of the base faces 11. The abrasive grain 10 has a diameter corresponding to the diameter of a circle enveloping the base face 11 which is not shown here. The outline 12 has three concave section 13. The concave sections 13 are each formed by two linear component sections 19 which converge at a reflex corner 20. All component section 19 have the same length, and all reflex corners 20 have the same angle. The concave section 13 end at corners 15. Since the lines connecting two adjacent corners 15 run outside the base face 11, the section 13 that run in between are referred to as concave in the context of the invention. In addition, straight sections 14 extend between adjacent corners 15, which likewise have the same length as one another. The internal angle at each corners 15 may, for example, be 105 (this does not correspond to the angle shown in the merely schematic drawing).

(107) FIG. 25B also shows an imaginary support plane E of the abrasive grain 10. The linear section 14 shown at the top of FIG. 25B runs parallel to this support plane E, i.e. at an angle of 0 to the support plane E. If the abrasive grain 10 lies on an abrasive underlayer 41 such that the support plane E corresponds to the underlayer 41, the abrasive grain 10 is stable to tipping over. In this orientation, the linear section 14 shown at the top runs parallel a surface 50 being processed, and the part-sections 19 of the concave sections 13 form cutting faces.

(108) For the abrasive grain 10 shown in FIGS. 25A and 25B, an angle of engagement of =90=15 is found. In the working example shown here, this angle of engagement corresponds to the angle formed between the part-section 19 and a perpendicular S which runs at right angles to the support plane E. The angle of engagement of this inventive abrasive grain 10 is smaller than the angle of engagement of 30 in the conventional abrasive grain 10 shown in FIG. 24. This results in increased grinding action.

(109) The base face 11 of the abrasive grain 10 shown in FIGS. 25A and 25B has three-fold symmetry. The base face is thus invariable under a rotation by 120 in the plane of the base face 11. The above-described increase in the cutting effect is therefore independent of this orientation.

(110) FIG. 26 shows a top view of a base face 11 of a further inventive abrasive grain 10. The outline 12 thereof likewise contains six corners 15 and three linear sections 14. In contrast to the working example according to FIGS. 25A and 25B, the outline 12 here, however, has three sections 13 having strictly concave curvature. In addition, the internal angle at the corners 15 here is 90. The consequence of this is an angle of engagement of 0, which leads to another improvement in cutting action. In addition, there is only a comparatively slight change in the cross section of the abrasive grain 10 when the abrasive grain 10 is worn away toward the center of the abrasive grain 10 in the direction of a linear section 14.

(111) In the third working example of an inventive abrasive grain 10 shown in FIG. 27, the internal angle is actually less than 90, which leads to a negative angle of engagement , since the above-described perpendicular S here runs outside the abrasive grain.

(112) Finally, FIG. 28 shows a fourth working example of an inventive abrasive grain 10, the outline 12 of which contains only concave sections 13 and corners, but no linear sections.

(113) The inventive abrasive grains can be produced, for example, by a process described hereinafter: first of all, a dispersion of 200 g of -Al.sub.2O.sub.3, 0.4 g of MgO, 90 g of water as dispersion medium and 0.5 g of dispersant is produced. The MgO functions here as a nucleating agent. The dispersant used may, for example, be the Dolapix CE64 product obtainable from Zschimmer & Schwarz, 56108 Lahnstein, Germany. The dispersion thus obtained is ground in a planetary ball mill at 200 revolutions per minute for 30 minutes, for example a PM400 planetary ball mill obtainable from Retsch GmbH, 42781 Haan, Germany. Subsequently, the ground dispersion is introduced into a silicone die as is described in more detail below, containing depressions in the shape of the desired abrasive grains. Thereafter, the volatile component, i.e. the water, is removed from the dispersion. This gives rise to an abrasive grain precursor which is removed from the die. In a final step, the precursor is sintered as bulk material at 1550 C. for 5 minutes. The dispersant is burnt out in the course of sintering.

(114) FIGS. 29A and 29B show two views of a fifth embodiment. This abrasive grain 10 differs from the idealized cylinder shape since the two base faces 11 have slight concave curvature. The abrasive grain 10 shown in FIGS. 30A and 30B contains two base faces 11 having slight convex curvature. Deviations of this kind can arise, for example, by virtue of the production tolerances described in detail above.

(115) FIG. 31 shows, in schematic form, a part of a die 30 with which an abrasive grain 10 according to FIG. 28 can be produced. The die 30 may consist, for example, of silicone. It contains a top side 32 and a multitude of identical depressions 31, only one single example of which is shown here for simplification of the illustration. The depression 31 has an open top face 35 through which a dispersion as described above can be introduced. The surface of the depression 31 contains a base face 34 which is identical to the shape of the base face 11 of the abrasive grain 10 and runs parallel to the top side 32 of the die 30. A lateral wall 36 which consists of a plurality of sections and is complementary to the shell face of the abrasive grain 10 extends from the base face 34.

(116) An inventive abrasive article can be produced, for example, as follows: on an underlayer made from vulcanized fiber having a thickness of 0.8 mm, a phenol resin dispersion as make coat precursor is applied in an amount of 120 g/m.sup.2. Subsequently, 600 g/m.sup.2 of the inventive abrasive grains are applied by means of electrostatic scattering. Thereafter, the make coat precursor is cured to give a make coat. On top of the make coat and the abrasive grains, a phenol resin dispersion is applied in an amount of 800 g/m.sup.2 as size coat precursor, which is likewise cured.

(117) Concept 1. An abrasive grain (10) having not more than three faces (11, 12, 13), characterized in that the abrasive grain (10) contains at least one edge (5, 6, 7, 8, 9) having a corner (1, 2, 3, 4) at at least one end.

(118) Concept 2. The abrasive grain (10) as described in any of the preceding concepts, characterized in that the abrasive grain (10) has at least four corners (1, 2, 3, 4), especially exactly four corners (1, 2, 3, 4).

(119) Concept 3. The abrasive grain (10) as described in concept 2, characterized in that the abrasive grain (10) has at least two edges (5, 7) that are not in contact with one another, the two edges (5, 7) each being bounded by two corners (1, 2, 3, 4) and a first straight connecting line formed between the corners (1, 2) that bound a first edge (5) being arranged at an angle of 9050, preferably 9030, more preferably 9010, with respect to a second straight connecting line formed between the corners (3, 4) that bound a second edge (7).

(120) Concept 4. The abrasive grain (10) as described in concept 2 or 3, characterized in that two planes that are formed between three corners (1, 2, 3, 4) each of the abrasive grain (10) form an angle between 70 and 140, preferably between 80 and 130, more preferably between 90 and 120.

(121) Concept 5. The abrasive grain (10) as described in any of concepts 2 to 4, characterized in that the abrasive grain (10) has at least one feature of symmetry, especially at least one plane of symmetry and/or at least one axis of symmetry.

(122) Concept 6. The abrasive grain (10) as described in any of the preceding concepts, characterized in that the abrasive grain (10) has exactly two faces (11, 12), the two faces (11, 12) of the abrasive grain being curved faces.

(123) Concept 7. The abrasive grain (10) as described in any of concepts 1 to 5, characterized in that the abrasive grain (10) has exactly three faces (11, 12, 13), at least two and especially all three faces (11, 12, 13) of the abrasive grain (10) being curved faces.

(124) Concept 8. The abrasive grain (10) as described in any of the preceding concepts, characterized in that it comprises or consists of a ceramic material, especially a polycrystalline ceramic material, preferably alumina, more preferably -Al.sub.2O.sub.3.

(125) Concept 9. A collective of abrasive grains (10), characterized in that it comprises at least 20% by weight, preferably at least 50% by weight, more preferably at least 90% by weight, of abrasive grains (10) as described in any of the preceding concepts.

(126) Concept 10. A process for producing at least one abrasive grain (10) or a collective of abrasive grains (10) as described in any of the preceding concepts, characterized by the following steps: (a) producing or providing a dispersion comprising -alumina particles and/or particles that can be converted to -alumina, and at least one volatile dispersion medium, preferably water; (b) introducing the dispersion into at least one depression of a die; (c) optionally squeegeeing an upper face of the die in order to remove at least a portion of the dispersion which stands above the upper face of the die; (d) removing a portion of the volatile components of the dispersion, so as to form at least one abrasive grain precursor; (e) removing the abrasive grain precursor from the die; (f) optionally calcining the abrasive grain precursor; and (g) sintering the abrasive grain precursor in order to obtain at least one abrasive grain (10).

(127) Concept 11. A casting mold for producing at least one abrasive grain (10) as described in any of concepts 1 to 8, wherein the casting mold comprises at least one die having at least one depression, preferably a multitude of depressions, having a particular surface, the surface being complementary to the shape of at least part of the surface of the abrasive grain (10).

(128) Concept 12. An abrasive article comprising a collective of abrasive grains (10) as described in concept 9.

(129) Concept 13. A process for producing an abrasive article as described in concept 12, comprising a step in which a collective of abrasive grains (10) as described in concept 9 is fixed on and/or in a substrate (20), especially by means of a binder.

(130) Concept 14. A process for grinding a surface, especially a painted surface, with an abrasive article as described in concept 12.

(131) Concept 15. An abrasive grain (110; 210; 510; 510; 810) comprising or consisting of at least one component (120, 125; 220, 225; 510; 510; 810) having at least essentially the shape of a twisted geometric elementary body.

(132) Concept 16. The abrasive grain (110; 210; 510; 510; 810) as described in concept 15, characterized in that at least one geometric elementary body is a polyhedron, especially a prism, an antiprism, a pyramid or a frustopyramid.

(133) Concept 17. The abrasive grain (210) as described in either of concepts 15 and 16, characterized in that at least one geometric elementary body is a cone, especially a pyramid.

(134) Concept 18. The abrasive grain (210) as described in either of concepts 15 and 16, characterized in that at least one geometric elementary body is a frustocone, especially a frustopyramid.

(135) Concept 19. The abrasive grain (110; 310; 510; 510; 810) as described in either of concepts 15 and 16, characterized in that at least one geometric elementary body is a cylinder, especially a prism.

(136) Concept 20. The abrasive grain (110; 210) as described in any of the preceding concepts, characterized in that the abrasive grain (110; 210) has a base element (120; 220) having a bottom side (121; 221) and an opposite top side (122, 222), and also at least one top element (125; 225) having a bottom side (126; 226) disposed upon the top side (122; 222) of the base element (120; 220), the base element (120; 220) and/or at least one top element (125; 225) constituting one of the components (120, 125; 220, 225).

(137) Concept 21. The abrasive grain (110; 210; 510; 510; 810) as described in any of the preceding concepts, characterized in that it comprises or consists of a ceramic material, especially a polycrystalline ceramic material, preferably alumina, more preferably -Al.sub.2O.sub.3.

(138) Concept 22. A collective of abrasive grains (110; 210; 510; 510; 810), characterized in that it comprises at least 20% by weight, preferably at least 50% by weight, more preferably at least 90% by weight, of abrasive grains (110; 210) as described in any of the preceding concepts.

(139) Concept 23. A process for producing at least one abrasive grain (110; 210; 510; 510; 810) or a collective of abrasive grains (110; 210; 510; 510; 810) as described in any of the preceding concepts, characterized by the following steps: (a) producing or providing a dispersion comprising -alumina particles and/or particles that can be converted to -alumina, and at least one volatile dispersion medium, preferably water; (b) introducing the dispersion into at least one depression (31) of a die; (c) optionally squeegeeing an upper face of the die in order to remove at least a portion of the dispersion which stands above the upper face of the die; (d) removing a portion of the volatile components of the dispersion, so as to form at least one abrasive grain precursor; (e) removing the abrasive grain precursor from the die; (f) optionally calcining the abrasive grain precursor; and (g) sintering the abrasive grain precursor in order to obtain at least one abrasive grain (110; 210; 510; 510; 810).

(140) Concept 24. A process for producing at least one abrasive grain (510; 510; 810) or a collective of abrasive grains (510; 510; 810), especially at least one abrasive grain (510; 510; 810) or a collective of abrasive grains (510; 510; 810) as described in any of concepts 15 to 21, characterized by the following steps: (a) producing or providing a dispersion comprising -alumina particles and/or particles that can be converted to -alumina, and at least one volatile dispersion medium, preferably water, and optionally at least one organic additive; (b) extruding the dispersion through an exit orifice (507; 507; 507; 507; 507) of a nozzle (506; 506; 506; 506; 506), the exit orifice having at least essentially the shape of a twisted cylinder, such that an extrudate is obtained; (c) severing the extrudate to obtain abrasive grain precursors; (d) optionally calcining the abrasive grain precursors; and (e) sintering the abrasive grain precursors in order to obtain at least one abrasive grain (510; 510; 810).

(141) Concept 25. A process for producing at least one abrasive grain or a collective of abrasive grains, especially at least one abrasive grain or a collective of abrasive grains as described in any of concepts 15 to 21, characterized by the following steps: (a) producing or providing a dispersion comprising -alumina particles and/or particles that can be converted to -alumina, and at least one volatile dispersion medium, preferably water, and optionally at least one organic additive; (b) producing a film from the dispersion; (c) severing the film produced in step b) to form film sections; (d) shaping the film sections produced in step c) to obtain abrasive grain precursors; (e) optionally calcining the abrasive grain precursors; and (f) sintering the abrasive grain precursors in order to obtain at least one abrasive grain.

(142) Concept 26. A casting mold for producing at least one abrasive grain (110; 210; 510; 510; 810) as described in any of concepts 15 to 20, wherein the casting mold comprises at least one die having at least one depression, preferably a multitude of depressions, having a particular surface, the surface being complementary to the shape of at least part of the surface of the abrasive grain (110; 210; 510; 510; 810).

(143) Concept 27. An abrasive article comprising a collective of abrasive grains (110; 210; 510; 510; 810) as described in concept 22.

(144) Concept 28. A process for producing an abrasive article as described in concept 12, comprising a step in which a collective of abrasive grains (110; 210; 510; 510; 810) as described in concept 22 is fixed on and/or in a substrate, especially by means of a binder.

(145) Concept 29. A process for grinding a surface, especially a painted surface, with an abrasive article as described in concept 12.

(146) Concept 30. An abrasive grain (100; 400; 500; 600; 700; 800; 900; 1000) having at least six faces (110; 410; 510; 610; 710; 810; 910; 1010), characterized in that at least one of the faces (110; 410; 510; 610; 710; 810; 910; 1010) has concave curvature (111; 711).

(147) Concept 31. The abrasive grain (100; 400; 500; 600; 700; 800; 900) as described in concept 30, characterized in that at least one dished face (110) is curved in at least two directions (114, 115).

(148) Concept 32. An abrasive grain (200, 300), especially as described in concept 30 or 31, characterized in that the abrasive grain (200; 300) has a structure with at least one reentrant corner (216; 316).

(149) Concept 33. The abrasive grain (100; 300; 400; 500; 600; 700; 800; 900) as described in concept 30, 31 or 32, characterized in that the abrasive grain (100; 300; 400; 500; 600; 700; 800; 900) has at least one curved edge (113; 313; 713; 813), especially a concave curved edge (113; 713; 813).

(150) Concept 34. The abrasive grain as described in any of the preceding concepts, characterized in that the abrasive grain (100; 200; 300; 400; 500; 600; 700; 800; 900) has a structure with corners (112; 312; 412; 512; 612; 712; 812; 912) and at least some of the corners, especially all the convex corners (112; 312; 412; 512; 612; 712; 812; 912), correspond to the corners of an imaginary polyhedron, especially a Platonic solid, an Archimedean solid, a Catalan solid, a prism, an antiprism, or a Platonic solid, Archimedean solid, Catalan solid, prism or antiprism with linear distortion.

(151) Concept 35. The abrasive grain (100; 200; 300; 400; 500; 600; 700; 800; 900; 1000) as described in any of the preceding concepts, characterized in that it comprises or consists of a ceramic material, especially a polycrystalline ceramic material, preferably alumina, more preferably -Al.sub.2O.sub.3.

(152) Concept 36. A collective of abrasive grains (100; 200; 300; 400; 500; 600; 700; 800; 900; 1000), characterized in that it comprises at least 20% by weight, preferably at least 50% by weight, more preferably at least 90% by weight, of abrasive grains (100; 200; 300; 400; 500; 600; 700; 800; 900; 1000) as described in any of the preceding concepts.

(153) Concept 37. A process for producing at least one abrasive grain (100; 200; 300; 400; 500; 600; 700; 800; 900; 1000) or a collective of abrasive grains (100; 200; 300; 400; 500; 600; 700; 800; 900; 1000) as described in any of the preceding concepts, characterized by the following steps: (a.) producing or providing a dispersion comprising -alumina particles and/or particles that can be converted to -alumina, and at least one volatile dispersion medium, preferably water; (b.) introducing the dispersion into at least one depression of a die; (c.) optionally squeegeeing an upper face of the die in order to remove at least a portion of the dispersion which stands above the upper face of the die; (d.) removing a portion of the volatile components of the dispersion, so as to form at least one abrasive grain precursor; (e.) removing the abrasive grain precursor from the die; (f.) optionally calcining the abrasive grain precursor; and (g.) sintering the abrasive grain precursor in order to obtain at least one abrasive grain (100; 200; 300; 400; 500; 600; 700; 800; 900; 1000).

(154) Concept 38. A casting mold for producing at least one abrasive grain (100; 200; 300; 400; 500; 600; 700; 800; 900; 1000) as described in any of concepts 30 to 35, wherein the casting mold comprises at least one die having at least one depression, preferably a multitude of depressions, having a particular surface, the surface being complementary to the shape of at least part of the surface of the abrasive grain (100; 200; 300; 400; 500; 600; 700; 800; 900; 1000).

(155) Concept 39. The casting mold as described in concept 38, characterized in that the casting mold has at least one further shaping element, especially a further die or a ram element, with which, in addition to the face shaped in the die, at least some of the rest of the surface of the abrasive grain (100; 200; 300; 400; 500; 600; 700; 800; 900; 1000) can be shaped.

(156) Concept 40. An abrasive article comprising a collective of abrasive grains (100; 200; 300; 400; 500; 600; 700; 800; 900; 1000) as described in concept 36.

(157) Concept 41. A process for producing an abrasive article as described in concept 40, comprising a step in which a collective of abrasive grains (100; 200; 300; 400; 500; 600; 700; 800; 900; 1000) as described in concept 36 is fixed on and/or in a substrate, especially by means of a binder.

(158) Concept 42. A process for grinding a surface, especially a painted surface, with an abrasive article as described in concept 40.

(159) Concept 43. An abrasive grain (10; 10; 10; 10; 10; 10) in the shape especially of a straight cylinder having two opposite base faces (11), characterized in that the base faces (11) have an outline (12) including at least one concave section (13).

(160) Concept 44. The abrasive grain (10; 10; 10; 10; 10; 10) as described in concept 43, characterized in that the outline (12) has at least one corner (15).

(161) Concept 45. The abrasive grain (10; 10; 10; 10; 10; 10) as described in concept 44, characterized in that the outline (12) at the edge (15) defines an inner angle () of the base face (11) within the range from 65 to 120, preferably from 65 to 115, further preferably from 75 to 105, more preferably from 85 to 95 and most preferably 90.

(162) Concept 46. The abrasive grain (10; 10; 10; 10; 10; 10) as described in any of the preceding concepts, characterized in that the outline (12) has at least one linear section (14).

(163) Concept 47. The abrasive grain (10; 10; 10; 10; 10; 10) as described in concept 46, characterized in that at least one linear section (14) runs at an angle with respect to a support plane (E) of the abrasive grain (10; 10; 10; 10; 10; 10) of at most 20, preferably at most 10, more preferably at most 5.

(164) Concept 48. The abrasive grain (10; 10; 10; 10; 10; 10) as described in any of the preceding concepts, characterized in that a tangent (T) to at least one point on the outline (12) runs at an angle to a perpendicular (S) that runs at right angles to the support plane (E) of the abrasive grain (10; 10; 10; 10; 10; 10), this angle being in the range from 30 to +30, preferably from 25 to +25, further preferably from 15 to +15, more preferably from 5 to +5 and most preferably 0.

(165) Concept 49. The abrasive grain (10; 10; 10; 10; 10; 10) as described in any of the preceding concepts, characterized in that the outline (12) has at least three, preferably exactly three, concave sections (13) whose respective ends are corners (15) of the outline (12), and in that the outline (12) has a linear section (14) between any two concave sections (13).

(166) Concept 50. The abrasive grain (10; 10; 10; 10; 10; 10) as described in any of the preceding concepts, characterized in that the ratio of the height (h) of the abrasive grain (10; 10; 10; 10; 10; 10) and the diameter of a circle enveloping the base faces (11) is not more than 1, preferably not more than 0.8, more preferably not more than 0.6.

(167) Concept 51. The abrasive grain (10; 10; 10; 10; 10; 10) as described in any of the preceding concepts, characterized in that it comprises or consists of a ceramic material, especially a polycrystalline ceramic material, preferably alumina, more preferably -Al.sub.2O.sub.3.

(168) Concept 52. A collective of abrasive grains (10; 10; 10; 10; 10; 10), characterized in that it comprises at least 20% by weight, preferably at least 50% by weight, more preferably at least 90% by weight, of abrasive grains (10; 10; 10; 10; 10; 10) as described in any of the preceding concepts.

(169) Concept 53. A process for producing at least one abrasive grain (10; 10; 10; 10; 10; 10) or a collective of abrasive grains (10; 10; 10; 10; 10; 10) as described in any of the preceding concepts, characterized by the following steps: (a.) producing or providing a dispersion comprising -alumina particles and/or particles that can be converted to -alumina, and at least one volatile dispersion medium, preferably water; (b.) introducing the dispersion into at least one depression (31) of a die (30); (c.) optionally squeegeeing an upper face (32) of the die (30) in order to remove at least a portion of the dispersion which stands above the upper face (32) of the die (30); (d.) removing a portion of the volatile components of the dispersion, so as to form at least one abrasive grain precursor; (e.) removing the abrasive grain precursor from the die (30); (f.) optionally calcining the abrasive grain precursor; and (g.) sintering the abrasive grain precursor in order to obtain at least one abrasive grain (10; 10; 10; 10; 10; 10).

(170) Concept 54. A casting mold (30) for producing at least one abrasive grain (10; 10; 10; 10; 10; 10) as described in any of concepts 43 to 51, wherein the casting mold (30) comprises at least one depression (31), preferably a multitude of depressions (31), having a particular surface, the surface being complementary to the shape of at least part of the surface of the abrasive grain (10; 10; 10; 10; 10; 10).

(171) Concept 55. An abrasive article (40) comprising a collective of abrasive grains (10; 10; 10; 10; 10; 10) as described in concept 52.

(172) Concept 56. A process for producing an abrasive article (40) as described in concept 55, comprising a step in which a collective of abrasive grains (10; 10; 10; 10; 10; 10) as described in concept 52 is fixed on and/or in a substrate, especially by means of a binder (42).

(173) Concept 57. A process for grinding a surface, especially a painted surface, with an abrasive article (40) as described in concept 55.