U.S. patent application number 10/668799 was filed with the patent office on 2005-03-24 for structured abrasive article.
This patent application is currently assigned to 3M Innovative Properties Company. Invention is credited to Culler, Scott R., Haas, John D., Simons, Jeffrey R..
Application Number | 20050064805 10/668799 |
Document ID | / |
Family ID | 34313576 |
Filed Date | 2005-03-24 |
United States Patent
Application |
20050064805 |
Kind Code |
A1 |
Culler, Scott R. ; et
al. |
March 24, 2005 |
Structured abrasive article
Abstract
An abrasive article includes an array of protruding abrasive
units. Each unit has a base and a distal apex that is off-center
from the base when projected on to a plane that is coplanar with
the base. The abrasive article includes a backing bonded to an
abrasive coating formed to include the aforementioned abrasive
units. Methods for making the abrasive article include nipping a
production tool, an abrasive slurry, and the backing. A binder
within the slurry is cured during fabrication. The abrasive article
may be used to abrade a workpiece.
Inventors: |
Culler, Scott R.;
(Burnsville, MN) ; Haas, John D.; (Roseville,
MN) ; Simons, Jeffrey R.; (Hudson, WI) |
Correspondence
Address: |
3M INNOVATIVE PROPERTIES COMPANY
PO BOX 33427
ST. PAUL
MN
55133-3427
US
|
Assignee: |
3M Innovative Properties
Company
|
Family ID: |
34313576 |
Appl. No.: |
10/668799 |
Filed: |
September 23, 2003 |
Current U.S.
Class: |
451/527 |
Current CPC
Class: |
B24D 11/00 20130101;
B24D 2203/00 20130101 |
Class at
Publication: |
451/527 |
International
Class: |
B24D 011/00 |
Claims
What is claimed is:
1. An abrasive array of a plurality of protruding units, each unit
having a body composed of at least abrasive grains and a binder,
each body having a base and a substantially linear region most
distal from the base, the abrasive array comprising: an at least
two-by-two array of protruding units, wherein each protruding unit
has a base that has a first side and an oppositely disposed second
side, wherein, for each unit, its respective distal linear region,
when projected on to a plane that is coplanar with its respective
base, extends between a non-central point on the first side of the
base and a non-central point on the second side of the base.
2. The abrasive array of claim 1, wherein each base is a rectangle
having length and a width.
3. The abrasive array of claim 2, wherein the length of the
rectangle is between 1 and 150 mils and the width of the rectangle
is between 1 and 150 mils.
4. The abrasive array of claim 2, wherein the distance between each
base and its respective distal linear region is at most 60
mils.
5. The abrasive array of claim 2, wherein the distal linear regions
are substantially parallel with at least one side of their
corresponding rectangular bases.
6. The abrasive array of claim 2, wherein the distal linear regions
are not parallel with at least one side of their corresponding
rectangular bases.
7. The abrasive array of claim 1, wherein each base has
substantially the same geometry.
8. The abrasive array of claim 1, wherein each base has a different
geometry.
9. The abrasive array of claim 1, wherein each base is the same
size.
10. The abrasive array of claim 1, wherein at least one base is a
different size than another base.
11. The abrasive array of claim 1, wherein the distal linear
regions are substantially parallel with one another.
12. The abrasive array of claim 1, wherein at least one distal
linear region is not parallel with another distal linear
region.
13. The abrasive array of claim 1, wherein each the distance
between each base and its respective distal linear region is
substantially constant.
14. The abrasive array of claim 1, wherein the distance between
each base and its respective distal linear region varies.
15. The abrasive array of claim 1, wherein each of the bases are
substantially coplanar.
16. The abrasive array of claim 1, wherein the body of each
protruding unit is formed at least in part from the group
consisting of aluminum oxide, silicon carbide, alumina zirconia,
garnet, diamond, cubic boron nitride, oxide cerium, and mixtures
thereof.
17. An abrasive article comprising: a backing having a front and
back surface; and an abrasive coating bonded to the front surface
of the backing, wherein the abrasive coating includes an at least
two-by-two array of protruding units, wherein each protruding unit
has a base that has a first side and an oppositely disposed second
side, wherein, for each unit, its respective distal linear region,
when projected on to a plane that is coplanar with its respective
base, extends between a non-central point on the first side of the
base and a non-central point on the second side of the base.
18. The abrasive article of claim 17, wherein each base is a
rectangle having length and a width.
19. The abrasive article of claim 18, wherein the length of the
rectangle is between 1 and 150 mils and the width of the rectangle
is between 1 and 150 mils.
20. The abrasive article of claim 18, wherein the distance between
each base and its respective distal linear region is at most 60
mils.
21. The abrasive article of claim 18, wherein the distal linear
regions are substantially parallel with at least one side of their
corresponding rectangular bases.
22. The abrasive article of claim 18, wherein the distal linear
regions are not parallel with at least one side of their
corresponding rectangular bases.
23. The abrasive article of claim 17, wherein each base has
substantially the same geometry.
24. The abrasive article of claim 17, wherein each base has a
different geometry.
25. The abrasive article of claim 17, wherein each base is the same
size.
26. The abrasive article of claim 17, wherein at least one base is
a different size than another base.
27. The abrasive article of claim 17, wherein the distal linear
regions are substantially parallel with one another.
28. The abrasive article of claim 17, wherein at least one distal
linear region is not parallel with another distal linear
region.
29. The abrasive article of claim 17, wherein each the distance
between each base and its respective distal linear region is
substantially constant.
30. The abrasive article of claim 17, wherein the distance between
each base and its respective distal linear region varies.
31. The abrasive article of claim 17, wherein the body of each
protruding unit is formed at least in part from the group
consisting of aluminum oxide, silicon carbide, alumina zirconia,
garnet, diamond, cubic boron nitride, oxide cerium, and mixtures
thereof.
Description
FIELD OF THE INVENTION
[0001] This invention pertains to an abrasive array, an abrasive
article, a method of making such an abrasive article and a method
of using such an abrasive article. The abrasive article comprises a
backing having an abrasive coating bonded to at least one surface
of the backing. The abrasive coating is shaped to include
protruding units exhibiting useful geometries.
BACKGROUND
[0002] Abrasive articles have been utilized to abrade and finish
workpieces surfaces for well over a hundred years. These
applications have ranged from high stock removal, high pressure
metal grinding processes to fine polishing of ophthalmic lenses. In
general abrasive articles comprise a plurality of abrasive
particles bonded either together (e.g., a bonded abrasive or
grinding wheel) or to a backing (e.g., a coated abrasive). For a
coated abrasive there is typically a single, or sometimes two
layers of abrasive particles. Once these abrasive particles are
worn, the coated abrasive is essentially worn out and is typically
discarded.
[0003] A structured abrasive is taught by U.S. Pat. No. 5,152,917
(Pieper et al.). Importantly, the structured abrasive taught by
Peiper results in a relatively high rate of cut and a relatively
fine surface finish on the workpiece surface. The structured
abrasive comprises non-random, precisely shaped abrasive composites
that are bonded to a backing.
[0004] Although structured abrasives, such as the one taught by
Pieper exhibit desirable characteristics, such as a high cut rate,
structured abrasives still tend to lose their effectiveness over
time. Thus, a structured abrasive may yield a particular cut rate
(expressed, for example, in grams per cycle) in its initial three
or four cycles of abrasion, but may yield a cut rate of only a
fraction of its initial value after 5 or 10 cycles. Such
deterioration in cut rate is inimical to the goal of providing
efficient abrasion technology.
[0005] As is evident from the foregoing, there exists a need for a
scheme by which a structured abrasive may be made to prolong its
life span and minimize its cut-rate deterioration.
SUMMARY
[0006] This invention pertains to an abrasive array, an abrasive
article, a method of making an abrasive article and a method of
using an abrasive article. According to one embodiment of the
present invention, an abrasive array of a plurality of protruding
units may be structure such that each unit has a body composed of
at least abrasive grains and a binder. Each body may have a base
and a substantially linear region most distal from the base. The
abrasive array may include an at least two-by-two array of
protruding units. Each protruding unit may have a base that has a
first side and an oppositely disposed second side. For each unit,
its respective distal linear region, when projected on to a plane
that is coplanar with its respective base, may extend between a
non-central point on the first side of the base and a non-central
point on the second side of the base.
[0007] According to another embodiment of the present invention, an
abrasive article may include a backing having a front and back
surface. An abrasive coating may be bonded to the front surface of
the backing. The abrasive array may include an at least two-by-two
array of protruding units. Each protruding unit may have a base
that has a first side and an oppositely disposed second side. For
each unit, its respective distal linear region, when projected on
to a plane that is coplanar with its respective base, may extend
between a non-central point on the first side of the base and a
non-central point on the second side of the base.
BRIEF DESCRIPTION OF THE DRAWINGS
[0008] FIG. 1 is a section view, enlarged, representing another
abrasive article embodiment of this invention;
[0009] FIG. 2 is a schematic of a process of making the abrasive
article of FIG. 1; and
[0010] FIG. 3 is a schematic of another process of making the
abrasive article of FIG. 1.
[0011] FIG. 4A depicts a top view of a protruding unit in
accordance with an embodiment of the present invention.
[0012] FIG. 4B depicts a top view of a protruding unit in
accordance with an embodiment of the present invention.
[0013] FIG. 4C depicts a top view of an abrasive article in
accordance with an embodiment of the present invention.
[0014] FIG. 4D depicts another top view of an abrasive article in
accordance with an embodiment of the present invention.
[0015] FIG. 4E depicts another top view of a protruding unit in
accordance with an embodiment of the present invention.
[0016] FIG. 4F depicts another top view of a protruding unit in
accordance with an embodiment of the present invention.
[0017] FIG. 4G depicts another top view of a protruding unit in
accordance with an embodiment of the present invention.
[0018] FIG. 4H depicts another top view of a protruding unit in
accordance with an embodiment of the present invention.
[0019] FIG. 5 depicts another abrasive article in accordance with
an embodiment of the present invention.
[0020] FIG. 6A depicts an array of protruding units in accordance
with an embodiment of the present invention.
[0021] FIG. 6B depicts another array of protruding units in
accordance with an embodiment of the present invention.
DETAILED DESCRIPTION
[0022] This invention pertains to an abrasive array, an abrasive
article, a method of making an abrasive article and a method of
using an abrasive article.
[0023] Referring to FIG. 1, the abrasive article 20 comprises
abrasive composites 22 separated by boundary 25. The abrasive
composites are bonded to a surface of a backing 21. The boundary or
boundaries associated with the composite shape result in one
abrasive composite being separated to some degree from another
adjacent abrasive composite. To form an individual abrasive
composite, a portion of the boundaries forming the shape of the
abrasive composite must be separated from one another. Note that in
FIG. 2, the base or a portion of the abrasive composite closest to
the backing can abut with its neighboring abrasive composite. (Note
that "neighboring" does not necessarily mean "adjacent".) Abrasive
composites 22 comprise a plurality of abrasive particles 24 that
are dispersed in a binder 23 and a grinding aid 26. It is also
within the scope of this invention to have a combination of
abrasive composites bonded to a backing in which some of the
abrasive composites abut, while other abrasive composites have open
spaces between them.
[0024] Backing
[0025] The backing of this invention has a front and back surface
and can be any conventional abrasive backing. Examples of useful
backings include polymeric film, primed polymeric film, cloth,
paper, vulcanized fiber, nonwovens, and combinations thereof. Other
useful backings include a fibrous reinforced thermoplastic backing
as disclosed in U.S. Pat. No. 5,316,812 and an endless seamless
backing as disclosed in World Patent Application No. WO 93/12911
published. The backing may also contain a treatment or treatments
to seal the backing and/or modify some physical properties of the
backing. These treatments are well known in the art.
[0026] The backing may also have an attachment means on its back
surface to enable securing the resulting coated abrasive to a
support pad or back-up pad. This attachment means can be a pressure
sensitive adhesive, one surface of a hook and loop attachment
system, or a threaded projection as disclosed in the
above-mentioned U.S. Pat. No. 5,316,812. Alternatively, there may
be an intermeshing attachment system as described in the assignee's
U.S. Pat. No. 5,201,101 incorporated herein after by reference.
[0027] The backside of the abrasive article may also contain a slip
resistant or frictional coating. Examples of such coatings include
an inorganic particulate (e.g., calcium carbonate or quartz)
dispersed in an adhesive.
[0028] Abrasive Coating
[0029] Abrasive Particles
[0030] The abrasive particles typically have a particle size
ranging from about 0.1 to 1500 micrometers, usually between about
0.1 to 400 micrometers, preferably between 0.1 to 100 micrometers
and most preferably between 0.1 to 50 micrometers. It is preferred
that the abrasive particles have a Mohs' hardness of at least about
8, more preferably above 9. Examples of such abrasive particles
include fused aluminum oxide (which includes brown aluminum oxide,
heat treated aluminum oxide and white aluminum oxide), ceramic
aluminum oxide, green silicon carbide, silicon carbide, chromia,
alumina zirconia, diamond, iron oxide, ceria, cubic boron nitride,
boron carbide, garnet and combinations thereof.
[0031] The term "abrasive particle" also encompasses when single
abrasive particles are bonded together to form an abrasive
agglomerate. Abrasive agglomerates are further described in U.S.
Pat. Nos. 4,311,489; 4,652,275 and 4,799,939 incorporated herein by
reference.
[0032] It is also within the scope of this invention to have a
surface coating on the abrasive particles. The surface coating may
have many different functions. In some instances the surface
coatings increase adhesion of abrasive particles to the binder,
alter the abrading characteristics of the abrasive particle, and
the like. Examples of surface coatings include coupling agents,
halide salts, metal oxides including silica, refractory metal
nitrides, refractory metal carbides and the like.
[0033] In the abrasive composite there may also be diluent
particles. The particle size of these diluent particles may be on
the same order of magnitude as the abrasive particles. Examples of
such diluent particles include gypsum, marble, limestone, flint,
silica, glass bubbles, glass beads, aluminum silicate, and the
like.
[0034] Binder
[0035] The abrasive particles are dispersed in an organic binder to
form the abrasive composite. The binder is derived from a binder
precursor which comprises an organic polymerizable resin. During
the manufacture of the inventive abrasive articles, the binder
precursor is exposed to an energy source which aids in the
initiation of the polymerization or curing process. Examples of
energy sources include thermal energy and radiation energy, the
latter including electron beam, ultraviolet light, and visible
light. During this polymerization process, the resin is polymerized
and the binder precursor is converted into a solidified binder.
Upon solidification of the binder precursor, the abrasive coating
is formed. The binder in the abrasive coating is also generally
responsible for adhering the abrasive coating to the backing.
[0036] There are two preferred classes of resins for use in the
present invention, condensation curable and addition polymerizable
resins. The preferred binder precursors comprise additional
polymerizable resins because these resins are readily cured by
exposure to radiation energy. Addition polymerizable resins can
polymerize through a cationic mechanism or a free radical
mechanism. Depending upon the energy source that is utilized and
the binder precursor chemistry, a curing agent, initiator, or
catalyst is sometimes preferred to help initiate the
polymerization.
[0037] Examples of typical and preferred organic resins include
phenolic resins, ureaformaldehyde resins, melamine formaldehyde
resins, acrylated urethanes, acrylated epoxies, ethylenically
unsaturated compounds, aminoplast derivatives having pendant
unsaturated carbonyl groups, isocyanurate derivatives having at
least one pendant acrylate group, isocyanate derivatives having at
least one pendant acrylate group, vinyl ethers, epoxy resins, and
mixtures and combinations thereof. The term "acrylate" encompasses
acrylates and methacrylates.
[0038] Phenolic resins are widely used in abrasive article binders
because of their thermal properties, availability, and cost. There
are two types of phenolic resins, resole and novolac. Resole
phenolic resins have a molar ratio of formaldehyde to phenol of
greater than or equal to one to one, typically between 1.5:1.0 to
3.0:1.0. Novolac resins have a molar ratio of formaldehyde to
phenol of less than one to one. Examples of commercially available
phenolic resins include those known by the trade names "Durez" and
"Varcum" from Occidental Chemicals Corp.; "Resinox" from Monsanto;
"Aerofene" from Ashland Chemical Co. and "Aerotap" from Ashland
Chemical Co.
[0039] Acrylated urethanes are diacrylate esters of
hydroxy-terminated, isocyanate NCO extended polyesters or
polyethers. Examples of commercially available acrylated urethanes
include those known under the trade designations "UVITHANE 782",
available from Morton Thiokol Chemical, and "CMD 6600", "CMD 8400",
and "CMD 8805", available from Radcure Specialties.
[0040] Acrylated epoxies are diacrylate esters of epoxy resins,
such as the diacrylate esters of bisphenol A epoxy resin. Examples
of commercially available acrylated epoxies include those known
under the trade designations "CMD 3500", "CMD 3600", and "CMD
3700", available from Radcure Specialities.
[0041] Ethylenically unsaturated resins include both monomeric and
polymeric compounds that contain atoms of carbon, hydrogen, and
oxygen, and optionally, nitrogen and the halogens. Oxygen or
nitrogen atoms or both are generally present in ether, ester,
urethane, amide, and urea groups.
[0042] Ethylenically unsaturated compounds preferably have a
molecular weight of less than about 4,000 and are preferably esters
made from the reaction of compounds containing aliphatic
monohydroxy groups or aliphatic polyhydroxy groups and unsaturated
carboxylic acids, such as acrylic acid, methacrylic acid, itaconic
acid, crotonic acid, isocrotonic acid, maleic acid, and the like.
Representative examples of acrylate resins include methyl
methacrylate, ethyl methacrylate styrene, divinylbenzene, vinyl
toluene, ethylene glycol diacrylate, ethylene glycol methacrylate,
hexanediol diacrylate, triethylene glycol diacrylate,
trimethylolpropane triacrylate, glycerol triacrylate,
pentaerythritol triacrylate, pentaerythritol methacrylate,
pentaerythritol tetraacrylate and pentaerythritol tetraacrylate.
Other ethylenically unsaturated resins include monoallyl,
polyallyl, and polymethallyl esters and amides of carboxylic acids,
such as diallyl phthalate, diallyl adipate, and
N,N-diallyladkipamide. Still other nitrogen containing compounds
include tris(2-acryloyloxyethyl)isocyanurat- e,
1,3,5-tri(2-methyacryloxyethyl)-triazine, acrylamide,
methylacrylamide, N-methylacrylamide, N,N-dimethylacrylamide,
N-vinylpyrrolidone, and N-vinylpiperidone.
[0043] The aminoplast resins have at least one pendant alpha, beta
unsaturated carbonyl group per molecule or oligomer. These
unsaturated carbonyl groups can be acrylate, methacrylate, or
acrylamide type groups. Examples of such materials include
N-(hydroxymethyl)acrylamide, N,N'-oxydimethylenebisacrylamide,
ortho and para acrylamidomethylated phenol, acrylamidomethylated
phenolic novolac, and combinations thereof. These materials are
further described in U.S. Pat. Nos. 4,903,440 and 5,236,472 both
incorporated herein by reference.
[0044] Isocyanurate derivatives having at least one pendant
acrylate group and isocyanate derivatives having at least one
pendant acrylate group are further described in U.S. Pat. No.
4,652,274 incorporated herein after by reference. The preferred
isocyanurate material is a triacrylate of tris(hydroxy ethyl)
isocyanurate.
[0045] Epoxy resins have an oxirane and are polymerized by the ring
opening. Such epoxide resins include monomeric epoxy resins and
oligomeric epoxy resins. Examples of some preferred epoxy resins
include 2,2-bis[4-(2,3-epoxypropoxy)-phenyl propane] (diglycidyl
ether of bisphenol) and commercially available materials under the
trade designations "Epon 828", "Epon 1004", and "Epon 1001F"
available from Shell Chemical Co., "DER-331", "DER-332", and
"DER-334" available from Dow Chemical Co. Other suitable epoxy
resins include glycidyl ethers of phenol formaldehyde novolac
(e.g., "DEN-431" and "DEN-428" available from Dow chemical
Co.).
[0046] The epoxy resins of the invention can polymerize via a
cationic mechanism with the addition of an appropriate cationic
curing agent. Cationic curing agents generate an acid source to
initiate the polymerization of an epoxy resin. These cationic
curing agents can include a salt having an onium cation and a
halogen containing a complex anion of a metal or metalloid. Other
cationic curing agents include a salt having an organometallic
complex cation and a halogen containing complex anion of a metal or
metalloid which are further described in U.S. Pat. No. 4,751,138
incorporated here in after by reference (column 6, line 65 to
column 9, line 45). Another example is an organometallic salt and
an onium salt is described in U.S. Pat. No. 4,985,340 (column 4,
line 65 to column 14, line 50); and European Patent Application
Nos. 306,161 and 306,162, both published Mar. 8, 1989, all
incorporated by reference. Still other cationic curing agents
include an ionic salt of an organometallic complex in which the
metal is selected from the elements of Periodic Group IVB, VB,
VIIB, VIIB and VMB which is described in European Patent
Application No. 109,581, published Nov. 21, 1983, incorporated by
reference.
[0047] Regarding free radical curable resins, in some instances it
is preferred that the abrasive slurry further comprises a free
radical curing agent. However in the case of an electron beam
energy source, the curing agent is not always required because the
electron beam itself generates free radicals.
[0048] Examples of free radical thermal initiators include
peroxides, e.g., benzoyl peroxide, azo compounds, benzophenones,
and quinones. For either ultraviolet or visible light energy
source, this curing agent is sometimes referred to as a
photoinitiator. Examples of initiators, that when exposed to
ultraviolet light generate a free radical source, include but are
not limited to those selected from the group consisting of organic
peroxides, azo compounds, quinones, benzophenones, nitroso
compounds, acryl halides, hydrozones, mercapto compounds, pyrylium
compounds, triacrylimdazoles, bisimidazoles, chloroalkytriazines,
benzoin ethers, benzil ketals, thioxanthones, and acetophenone
derivatives, and mixtures thereof. Examples of initiators that when
exposed to visible radiation generate a free radical source, can be
found in U.S. Pat. No. 4,735,632, entitled Coated Abrasive Binder
Containing Ternary Photoinitiator System, incorporated herein by
reference. The preferred initiator for use with visible light is
"Irgacure 369" commercially available from Ciba Geigy
Corporation.
[0049] Grinding Aid
[0050] A grinding aid is defined as a material, preferably a
particulate material, the addition of which to an abrasive article
has a significant effect on the chemical and physical processes of
abrading which results in improved performance. Typically and
preferably the grinding aid is added to the slurry as a
particulate, however it may be added to the slurry as a liquid. The
presence of the grinding aid will increase the grinding efficiency
or cut rate (defined as weight of work piece removed per weight of
abrasive article lost) of the corresponding abrasive article in
comparison to an abrasive article that does not contain a grinding
aid. In particular, it is believed in the art that the grinding aid
will either 1) decrease the friction between the abrasive grains
and the workpiece being abraded, 2) prevent the abrasive grain from
"capping", i.e., prevent metal particles (in the case of a metal
workpiece) from becoming welded to the tops of the abrasive grains,
3) decrease the interface temperature between the abrasive grains
the workpiece, 4) decreases the grinding force required, or 5)
prevents oxidation of the metal workpiece. In general, the addition
of a grinding aid increases the useful life of the abrasive
article.
[0051] Grinding aids useful in the invention encompass a wide
variety of different materials and can be inorganic or organic
based. Examples of chemical groups of grinding aids include waxes,
organic halide compounds, halide salts and metals and their alloys.
The organic halide compounds will typically break down during
abrading and release a halogen acid or a gaseous halide compound.
Examples of such materials include chlorinated waxes like
tetrachloronaphtalene, pentachloronaphthalene; and polyvinyl
chloride. Examples of halide salts include sodium chloride,
potassium cryolite, sodium cryolite, ammonium cryolite, potassium
tetrafluoroborate, sodium tetrafluoroborate, silicon fluorides,
potassium chloride, magnesium chloride. Examples of metals include,
tin, lead, bismuth, cobalt, antimony, cadmium, iron titanium, other
miscellaneous grinding aids include sulfur, organic sulfur
compounds, graphite and metallic sulfides. It is also within the
scope of this invention to use a combination of different grinding
aids and in some instances this may produce a synergistic
effect.
[0052] The above-mentioned examples of grinding aids are meant to
be representative only. A preferred grinding aid for use in the
invention is cryolite, and the most preferred is potassium
tetrafluoroborate (KBF.sub.4).
[0053] The grinding aid is considered to be non-abrasive, that is,
the Moh hardness of the grinding aid is less than 8. The grinding
aid may also contain impurities; these impurities should not
significantly adversely affect performance of the abrasive
article.
[0054] The grinding aid particle size preferably ranges from about
0.1 to 100 micrometers, more preferably between 10 to 70
micrometers. In general the particle size of the grinding aid is
preferably equal to or less than the size of the abrasive
particles.
[0055] The abrasive coating comprises generally at least about 1%
by weight, typically at least about 2.5% by weight, preferably at
least about 5% by weight, more preferably at least about 10% by
weight grinding aid and most preferably at least about 20% by
weight grinding aid. More than about 50 weight % grinding aid may
be detrimental since it is theorized that grinding performance
would decrease (since there are less abrasive particles present).
It was surprising that as the amount of grinding aid was increased,
the relative grinding performance as measured by cut rate is also
increased. This was unexpected since as the amount of grinding aid
in the abrasive coating is increased, the relative amount of
abrasive particles is decreased. The abrasive particles are
responsible for cutting the workpiece surface, not the grinding
aid. In general, the abrasive coating comprises from 5 to 90% by
weight, preferably from 20 to 80% by weight abrasive particles,
from 5 to 80% by weight, preferably from 5 to 40% by weight binder,
and from 5 to 60% by weight, preferably from 10 to 40% by weight
grinding aid.
[0056] Optional Additives
[0057] Slurries useful in the invention may further comprise
optional additives, such as, for example, fillers, fibers,
lubricants, wetting agents, thixotropic materials, surfactants,
pigments, dyes, antistatic agents, coupling agents, plasticizers,
and suspending agents. The amounts of these materials are selected
to provide the properties desired. The use of these can affect the
erodability of the abrasive composite. In some instances an
additive is purposely added to make the abrasive composite more
erodable, thereby expelling dulled abrasive particles and exposing
new abrasive particles.
[0058] Examples of antistatic agents useful in the invention
include graphite, carbon black, vanadium oxide, humectants, and the
like. These antistatic agents are disclosed in U.S. Pat. Nos.
5,061,294; 5,137,542, and 5,203,884 incorporated herein after by
reference.
[0059] A coupling agent can provide an association bridge between
the binder precursor and the filler particles or abrasive
particles. Examples of useful coupling agents include silanes,
titanates, and zircoaluminates. Useful slurries preferably contain
from about 0.01 to 3% by weight coupling agent.
[0060] An example of a suspending agent useful in the invention is
an amorphous silica particle having a surface area less than 150
meters square/gram that is commercially available from DeGussa
Corp., under the trade name "OX-50".
[0061] Abrasive Coating Comprising Abrasive Composites
[0062] In one preferred aspect of the invention, the abrasive
coating is in the form of a plurality of abrasive composites bonded
to the backing. It is generally preferred that each abrasive
composites have a precise shape. The precise shape of each
composite is determined by distinct and discernible boundaries.
These distinct and discernible boundaries are readily visible and
clear when a cross section of the abrasive article is examined
under a microscope such as a scanning electron microscope. In
comparison, in an abrasive coating comprising composites that do
not have precise shapes, the boundaries are not definitive and may
be illegible. These distinct and discernible boundaries form the
outline or contour of the precise shape. These boundaries separate
to some degree one abrasive composite from another and also
distinguish one abrasive composite from another.
[0063] Referring to FIG. 1, the abrasive article 10 comprises
abrasive composites 22 separated by boundary 25. The boundary or
boundaries associated with the composite shape result in one
abrasive composite being separated to some degree from another
adjacent abrasive composite. To form an individual abrasive
composite, a portion of the boundaries forming the shape of the
abrasive composite must be separated from one another. Note that in
FIG. 1, the base or a portion of the abrasive composite closest to
the backing can abut with its neighboring abrasive composite. (Note
that "neighboring" does not necessarily mean "adjacent".) Abrasive
composites 22 comprise a plurality of abrasive particles 24 that
are dispersed in a binder 23 and a grinding aid 26. It is also
within the scope of this invention to have a combination of
abrasive composites bonded to a backing in which some of the
abrasive composites abut, while other abrasive composites have open
spaces between them.
[0064] In some instances the boundaries forming the shape are
planar. For shapes that have planes, there are at least three
planes. The number of planes for a given shape can vary depending
upon the desired geometry, for instance the number of planes can
range from three to over 20. Generally, there are between three to
ten planes, preferably between three to six planes. These planes
intersect to form the desired shape and the angles at which these
planes intersect will determine the shape dimensions.
[0065] In another aspect of this invention, a portion of the
abrasive composites have a neighboring abrasive composite of a
different dimension. In this aspect of the invention, at least 10%,
preferably at least 30%, more preferably at least 50% and most
preferably at least 60% of the abrasive composites have an adjacent
abrasive composite that has a different dimension. These different
dimensions can pertain to the abrasive composite shape, angle
between planar boundaries or dimensions of the abrasive composite.
The result of these different dimensions for neighboring abrasive
composites results in an abrasive article that produces a
relatively finer surface finish on the workpiece being abraded or
refined. This aspect of the invention is further described in the
assignee's co-pending patent application U.S. Ser. No. 08/120,300
filed Sep. 13, 1993.
[0066] The abrasive composite shape can be any shape, but it is
preferably a geometric shape such as a rectangle, cone, semicircle,
circle, triangle, square, hexagon, pyramid, octagon and the like.
Embodiments of preferred shapes are presented below in a section
entitled "GEOMETRIES." An individual abrasive composite shape may
be referred to herein as a "protruding unit." The preferred shape
is a pyramid and the base of this pyramid can be a three or four
sided. It is also preferred that the abrasive composite cross
sectional surface area decreases away from the backing or decreases
along its height. This variable surface area results in a
non-uniform pressure as the abrasive composite wears during use.
Additionally, during manufacture of the abrasive article, this
variable surface area results in easier release of the abrasive
composite from the production tool. In general there are at least 5
individual abrasive composites per square cm. In some instances,
there may be at least 500 individual abrasive composites/square
cm.
[0067] Method of Making the Abrasive Article
[0068] An essential step to make any of the inventive abrasive
articles is to prepare the slurry. The slurry is made by combining
together by any suitable mixing technique the binder precursor, the
grinding aid, the abrasive particles and the optional additives.
Examples of mixing techniques include low shear and high shear
mixing, with high shear mixing being preferred. Ultrasonic energy
may also be utilized in combination with the mixing step to lower
the abrasive slurry viscosity. Typically, the abrasive particles
and grinding aid are gradually added into the binder precursor. The
amount of air bubbles in the slurry can be minimized by pulling a
vacuum during the mixing step. In some instances it is preferred to
heat, generally in the range of 30.degree. to 70.degree. C., the
slurry to lower the viscosity. It is important the slurry have
theological properties that allow the slurry to coat well and in
which the abrasive particles and grinding aid do not settle out of
the slurry.
[0069] Energy Source
[0070] After the slurry is coated onto the backing, such as via
transfer from a production tool (discussed below), the slurry may
be exposed to an energy source to initiate the polymerization of
the resin in the binder precursor. Examples of energy sources
include thermal energy and radiation energy. The amount of energy
depends upon several factors such as the binder precursor
chemistry, the dimensions of the abrasive slurry, the amount and
type of abrasive particles and the amount and type of the optional
additives. For thermal energy, the temperature can range from about
30.degree. to 150.degree. C., generally from 40.degree. to
120.degree. C. The exposure time can range from about 5 minutes to
over 24 hours.
[0071] Suitable radiation energy sources include electron beam,
ultraviolet light, or visible light. Electron beam radiation, which
is also known as ionizing radiation, can be used at an energy level
of about 0.1 to about 10 Mrad, preferably at an energy level of
about 1 to about 10 Mrad. Ultraviolet radiation refers to
non-particulate radiation having a wavelength within the range of
about 200 to about 400 nanometers, preferably within the range of
about 250 to 400 nanometers. Visible radiation refers to
non-particulate radiation having a wavelength within the range of
about 400 to about 800 nanometers, preferably in the range of about
400 to about 550 nanometers. It is preferred that 300 to 600
Watt/inch visible lights are used.
[0072] After this polymerization process is complete, the binder
precursor is converted into a binder and the slurry is converted
into an abrasive coating. The resulting abrasive article is
generally ready for use. However, in some instances other processes
may still be necessary such as humidification or flexing. The
abrasive article can be converted into any desired form such as a
cone, endless belt, sheet, disc, and the like, before the abrasive
article is used.
[0073] Production Tool
[0074] Regarding the third and fourth aspects of the invention, in
some instances it is preferred that the abrasive coating be present
as precisely shaped abrasive composites. In order to make this type
of abrasive article, a production tool is generally required.
[0075] The production tool contains a plurality of cavities. These
cavities are essentially the inverse shape of the abrasive
composite and are responsible for generating the shape of the
abrasive composites. The dimensions of the cavities are selected to
provide the desired shape and dimensions of the abrasive
composites. If the shape or dimensions of the cavities are not
properly fabricated, the resulting production tool will not provide
the desired dimensions for the abrasive composites.
[0076] The cavities can be present in a dot like pattern with
spaces between adjacent cavities or the cavities can butt up
against one another. It is preferred that the cavities butt up
against one another. Additionally, the shape of the cavities is
selected such that the cross-sectional area of the abrasive
composite decreases away from the backing.
[0077] The production tool can be a belt, a sheet, a continuous
sheet or web, a coating roll such as a rotogravure roll, a sleeve
mounted on a coating roll, or die. The production tool can be
composed of metal, (e.g., nickel), metal alloys, or plastic. The
metal production tool can be fabricated by any conventional
technique such as engraving, bobbing, electroforming, diamond
turning, and the like. One preferred technique for a metal
production tool is diamond turning.
[0078] A thermoplastic tool can be replicated off a metal master
tool. The master tool will have the inverse pattern desired for the
production tool. The master tool can be made in the same manner as
the production tool. The master tool is preferably made out of
metal, e.g., nickel and is diamond turned. The thermoplastic sheet
material can be heated and optionally along with the master tool
such that the thermoplastic material is embossed with the master
tool pattern by pressing the two together. The thermoplastic can
also be extruded or cast onto the master tool and then pressed. The
thermoplastic material is cooled to solidify and produce the
production tool. Examples of preferred thermoplastic production
tool materials include polyester, polycarbonates, polyvinyl
chloride, polypropylene, polyethylene and combinations thereof. If
a thermoplastic production tool is utilized, then care must be
taken not to generate excessive heat that may distort the
thermoplastic production tool.
[0079] The production tool may also contain a release coating to
permit easier release of the abrasive article from the production
tool. Examples of such release coatings for metals include hard
carbide, nitrides or borides coatings. Examples of release coatings
for thermoplastics include silicones and fluorochemicals.
[0080] One method to make the abrasive article of the invention
illustrated in FIG. 2 is illustrated in FIG. 2. Backing 41 leaves
an unwind station 42 and at the same time the production tool 46
leaves an unwind station 45. Production tool 46 is coated with
slurry by means of coating station 44. It is possible to heat the
slurry and/or subject the slurry to ultrasonics prior to coating to
lower the viscosity. The coating station can be any conventional
coating means such as drop die coater, knife coater, curtain
coater, vacuum die coater or a die coater. During coating the
formation of air bubbles should be minimized. The preferred coating
technique is a vacuum fluid-bearing die, such as disclosed in U.S.
Pat. Nos. 3,594,865, 4,959,265, and 5,077,870, all incorporated
herein by reference. After the production tool is coated, the
backing and the slurry are brought into contact by any means such
that the slurry wets the front surface of the backing. In FIG. 2,
the slurry is brought into contact with the backing by means of
contact nip roll 47. Next, contact nip roll 47 also forces the
resulting construction against support drum 43. A source of energy
48 (preferably a source of visible light) transmits a sufficient
amount of energy into the slurry to at least partially cure the
binder precursor. The term partial cure is meant that the binder
precursor is polymerized to such a state that the slurry does not
flow from an inverted test tube. The binder precursor can be fully
cured once it is removed from the production tool by any energy
source. Following this, the production tool is rewound on mandrel
49 so that the production tool can be reused again. Optionally, the
production tool may be removed from the binder precursor prior to
any curing of the precursor at all. After removal, the precursor
may be cured, and the production tool may be rewound on mandrel 49
for reuse. Additionally, abrasive article 120 is wound on mandrel
121. If the binder precursor is not fully cured, the binder
precursor can then be fully cured by either time and/or exposure to
an energy source. Additional steps to make abrasive articles
according to this first method are further described in U.S. Pat.
No. 5,152,917 and U.S. Ser. No. 08/004,929, filed Jan. 14, 1993,
both incorporated herein by reference. Randomly shaped abrasives
composites may be made by the tooling and procedures described in
copending Ser. No. 08/120,300, filed Sep. 13, 1993, incorporated
herein by reference.
[0081] It is preferred that the binder precursor is cured by
radiation energy. The radiation energy can be transmitted through
the production tool so long as the production tool does not
appreciably absorb the radiation energy. Additionally, the
radiation energy source should not appreciably degrade the
production tool. It is preferred to use a thermoplastic production
tool and ultraviolet or visible light.
[0082] The slurry can be coated onto the backing and not into the
cavities of the production tool. The slurry coated backing is then
brought into contact with the production tool such that the slurry
flows into the cavities of the production tool. The remaining steps
to make the abrasive article are the same as detailed above.
[0083] Another method is illustrated in FIG. 3. Backing 51 leaves
an unwind station 52 and the slurry 54 is coated into the cavities
of the production tool 55 by means of the coating station 53. The
slurry can be coated onto the tool by any one of many techniques
such as drop die coating, roll coating, knife coating, curtain
coating, vacuum die coating, or die coating. Again, it is possible
to heat the slurry and/or subject the slurry to ultrasonics prior
to coating to lower the viscosity. During coating the formation of
air bubbles should be minimized. Then, the backing and the
production tool containing the abrasive slurry are brought into
contact by a nip roll 56 such that the slurry wets the front
surface of the backing. Next, the binder precursor in the slurry is
at least partially cured by exposure to an energy source 57. After
this at least partial cure, the slurry is converted to an abrasive
composite 59 that is bonded or adhered to the backing. The
resulting abrasive article is removed from the production tool by
means of nip rolls 58 and wound onto a rewind station 60.
Optionally, the production tool may be removed from the binder
precursor prior to any curing of the precursor at all. After
removal of the production tool, the precursor may be cured. In
either event, the energy source can be thermal energy or radiation
energy. If the energy source is either ultraviolet light or visible
light, it is preferred that the backing be transparent to
ultraviolet or visible light. An example of such a backing is
polyester backing.
[0084] The slurry can be coated directly onto the front surface of
the backing. The slurry coated backing is then brought into contact
with the production tool such that the slurry wets into the
cavities of the production tool. The remaining steps to make the
abrasive article are the same as detailed above.
[0085] Method of Refining a Workpiece Surface
[0086] Another aspect of this invention pertains to a method of
abrading a metal surface. This method involves bringing into
frictional contact the abrasive article of this invention with a
workpiece having a metal surface. The term "abrading" means that a
portion of the metal workpiece is cut or removed by the abrasive
article. Additionally, the surface finish associated with the
workpiece surface is typically reduced after this refining process.
One typical surface finish measurement is Ra; Ra is the arithmetic
surface finish generally measured in microinches or micrometers.
The surface finish can be measured by a profilometer, such as a
Perthometer or Surtronic.
[0087] Workpiece
[0088] The metal workpiece can be any type of metal such as mild
steel, stainless steel, titanium, metal alloys, exotic metal alloys
and the like. The workpiece may be flat or may have a shape or
contour associated with it.
[0089] Depending upon the application, the force at the abrading
interface can range from about 0.1 kg to over 1000 kg. Generally
this range is from 1 kg to 500 kg of force at the abrading
interface. Also depending upon the application, there may be a
liquid present during abrading. This liquid can be water and/or an
organic compound. Examples of typical organic compounds include
lubricants, oils, emulsified organic compounds, cutting fluids,
soaps, or the like. These liquids may also contain other additives
such as defoamers, degreasers, corrosion inhibitors, or the like.
The abrasive article may oscillate at the abrading interface during
use. In some instances, this oscillation may result in a finer
surface on the workpiece being abraded.
[0090] The abrasive articles of the invention can be used by hand
or used in combination with a machine. At least one or both of the
abrasive article and the workpiece is moved relative to the other
during grinding. The abrasive article can be converted into a belt,
tape roll, disc, sheet, and the like. For belt applications, the
two free ends of an abrasive sheet are joined together and a splice
is formed. It is also within the scope of this invention to use a
spliceless belt like that described in the assignee's co-pending
patent application U.S. Ser. No. 07/919,541, filed Jul. 24, 1992,
incorporated herein after by reference. Generally the endless
abrasive belt traverses over at least one idler roll and a platen
or contact wheel. The hardness of the platen or contact wheel is
adjusted to obtain the desired rate of cut and workpiece surface
finish. The abrasive belt speed depends upon the desired cut rate
and surface finish. The belt dimensions can range from about 5 mm
to 1,000 mm wide and from about 5 mm to 10,000 mm long. Abrasive
tapes are continuous lengths of the abrasive article. They can
range in width from about 1 mm to 1,000 mm, generally between 5 mm
to 250 mm. The abrasive tapes are usually unwound, traverse over a
support pad that forces the tape against the workpiece and then
rewound. The abrasive tapes can be continuously feed through the
abrading interface and can be indexed. The abrasive disc can range
from about 50 mm to 1,000 mm in diameter. Typically abrasive discs
are secured to a back-up pad by an attachment means. These abrasive
discs can rotate between 100 to 20,000 revolutions per minute,
typically between 1,000 to 15,000 revolutions per minute.
[0091] Geometries
[0092] As alluded to in the section of this disclosure entitled
"ABRASIVE COATING COMPRISING ABRASIVE COMPOSITES," the abrasive
composites may be shaped into units that protrude from the backing
to which they are bonded. The individual shaped composite abrasives
are referred to herein as "protruding units." The particular
geometry chosen for the protruding units may impact the performance
of the structured abrasive article in which they are situated. The
geometry schemes presented below are chosen to provide elevated
initial cut rates (as measured in mass per cycle) and to exhibit
minimal deterioration in cut rates with each successive abrasion
cycle.
[0093] The protruding units shown in FIGS. 4A-H, 5, and 6A and 6B,
and the other protruding units discussed herein may be structured
from materials described above, making use of fabrication methods
described above. Although FIGS. 4A-H, 5, and 6A and 6B do not
depict abrasive grains and binder within the protruding units, it
is understood that such grains and binder exist, as the protruding
units have abrasive grains and binder as a constituent
material.
[0094] FIG. 4A depicts a top view of a protruding unit 400. The
protruding unit has a base 401, which is in the shape of a square.
Other than the base 401, the protruding unit 400 has four sides,
which extend from each of the various sides of the base 401 to a
linear apex 406. Due to the perspective of FIG. 4A, only sides 403
and 405 are visible.
[0095] As can be seen from FIG. 4A, the linear apex 406, when
projected on to a plain that is coplanar with the base 401, extends
between oppositely disposed sides of the base 401. When referring
to the projection of an apex, such as apex 406, on to a plain that
is coplanar with a base of a protruding unit, the terms "projection
of the apex" or "projection of the linear apex" may be used herein.
The center points of the oppositely disposed sides between which
the projection of the linear apex 406 extends are identified with
small hashings. The projection of the linear apex 406 does not
extend between the center points of the oppositely disposed
sides.
[0096] The protruding unit of FIG. 4A may be arranged into a
two-dimensional array, as shown in FIG. 4B. FIG. 4B depicts an
array of substantially identical protruding units 400 disposed such
that the base of each protruding unit 400 abuts the base of an
adjacent protruding unit 400. The protruding units 400 are shown as
being bonded to a backing 408, creating an abrasive article.
Although the array depicted in FIG. 4B is shown as being
two-by-two, the array may be of any size in principle. Furthermore,
as shown in FIG. 4C, the array may be constructed so that the bases
of adjacent protruding units 400 do not abut one another.
[0097] FIG. 4D depicts a protruding unit 410. As can be seen
therein, the protruding unit has a linear apex 412 that has a
length that is insufficient for its projection to extend from one
side of the base 414 to the other. Thus, each of the sides tapers
inwardly from the base 414 toward the distal linear apex. Notably,
if the projection of the linear apex 412 is extrapolated, its
extrapolation does not meet with a center point of either
oppositely disposed side of the base 414. In this way, it can be
said that the projection of the linear apex 412 does not extend
"between" center points of oppositely disposed sides of the base
414.
[0098] FIG. 4E depicts yet another protruding unit 416. The
protruding units depicted in FIGS. 4A, 4B, 4C, and 4D exhibit the
characteristic that their respective linear apexes 412 do not
extend between center points of oppositely disposed sides of their
respective bases via employment of a similar scheme: the linear
apexes have been askew to all of the sides of their respective
bases. As shown in FIG. 4E, the projection of the linear apex 418
may be parallel to some of the sides of the base 420, and yet not
extend between center points of oppositely disposed sides of the
base 420.
[0099] FIG. 4F depicts yet another protruding unit 422. FIG. 4F
demonstrates that while the apex of a protruding unit may be
linear, it is not essential that it be rectilinear. The protruding
unit 422 has a curvilinear (as opposed to rectilinear) apex 424.
The projection of the curvilinear apex 424 does not extend between
center points of oppositely disposed sides of the base 426.
[0100] The bases that have been presented in FIGS. 4A, 4B, 4C, 4D,
4E, and 4F have all been in the shape of a square. Such a
restriction is not essential. In principle, the base may be any
closed shape. For example, the base may be any regular or irregular
polygon, may be a parallelogram, rectangle, or any for of
quadrilateral. The base may be circular or elliptical. The sides of
the base may be rectilinear or curvilinear. For example, the
protruding unit 428 depicted in FIG. 4G has four sides, two of
which are curvilinear 430 and 432. The center point of the
oppositely disposed curvilinear sides of the base may be found by
dividing the curvilinear sides into two segments, wherein the
length of the first segment is equal to the length of the second
segment. For example, side 430 has been divided into two segments:
segments AB and BC. Point B, the center point, is positioned so
that the length of segment AB is equal to the length of segment BC.
One skilled in the art understands that other measures of
centrality may be used to identify the center point of a line that
is not rectilinear. Again, the projection of the linear apex 434
extends between oppositely disposed sides 430 and 432, but not at
their respective center points.
[0101] FIG. 4H depicts a protruding unit 436 that has a base 438
that is in the shape of a pentagon. The center point of side AB is
identified with a small hash mark. Notably, protruding unit 436
does not, at first glance, appear to have a side disposed opposite
of side AB. For the sake of orienting the linear apex 440 so as to
extend between non-central points on oppositely disposed sides of a
base, one may consider the side opposite AB to be the compound
segment ACDEB. The center point of side ACDEB is point D, because
the length of segment ACD equals the length of segment DEB. Thus,
it is plain to see that linear apex 440 does not extend between
center point of oppositely disposed sides of the base 438.
[0102] The general principle to be extracted from the discussion
associated with FIG. 4H is that a particular scheme can be used to
find a side that is disposed opposite a given side of a base. In
short, a set of sides that subtends a given side of a base may
collectively be considered a single side that is disposed opposite
the given side (e.g., side ACDEB subtends side AB, and may be
considered to be disposed opposite side AB).
[0103] FIG. 5 depicts a perspective view of an abrasive article 500
including a two dimensional array of protruding units, some of
which have been identified with reference numeral 502. Each
protruding unit 502 has a base that is rectangular. According to
one embodiment of the present invention, the length and width of
the base may be between 1 and 150 mils. Each base has a linear apex
504. According to one embodiment of the present invention, the
linear apex may be oriented up to 60 mils above the base. Although
each of the bases in FIG. 5 is depicted as having the same size and
geometry, neither condition is essential. The bases may be of
varying size and/or geometry. Also, although each of the linear
apexes 504 is depicted as being parallel with one another, this
condition is not essential. The linear apexes 504 may be
non-parallel to one another. Finally, although each of the linear
apexes is depicted as being a constant distance from their
respective bases, this condition is also not essential. The
distance between the bases and their respective linear apexes 504
may vary from protruding unit 502 to protruding unit 502.
[0104] FIG. 6A depicts an array of protruding units 600-606. Each
of the protruding units 600-606 has an apex 608-614 that is
substantially in the shape of a point. Any of the linear apexes in
any of the preceding examples may be embodied as a point, as
opposed to being embodied as a linear segment. Returning the
discussion to FIG. 6A, in each of the protruding units 600-606, the
apex 608-614 is located remote from the center. The projection of
each apex 608-614 defines an offset vector v.sub.1, v.sub.2,
v.sub.3, and v.sub.4 extending from the center and/or center of
mass of the respective base to the projection of the apex 608-614.
Notably, the sum of the offset vectors v.sub.1, v.sub.2, v.sub.3,
and v.sub.4 does not equal zero. For example, assuming that each of
the offset vectors v.sub.1, v.sub.2, v.sub.3, and v.sub.4 is a unit
vector, the sum of the vectors is 2y. For a large array of
protruding units, the sum of the offset vectors should not approach
a limit of zero as the number of vectors summed-together approaches
infinity:
lim.sub.n.fwdarw..infin.(.SIGMA..nu..sub.n).noteq.0
[0105] Stated another way, when viewed in totality, the array
should exhibit net directionality with respect to the positioning
of the apexes 608-614.
[0106] FIG. 6B shows the idea of net proportionality as it applies
to protruding units having linear apexes 616-622. As can be seen
from FIG. 6B, the projection of the linear apexes 616-622 define an
offset vector v.sub.1, v.sub.2, v.sub.3, and v.sub.4 extending from
the centers and/or center of mass of the respective base to the
center of the projection of the apexes 616-622. Once again, for a
large array of protruding units, the sum of the offset vectors
should not approach a limit of zero as the number of vectors summed
together approaches infinity:
lim .sub.n.fwdarw..infin.(.SIGMA..nu..sub.n).noteq.0
[0107] Stated another way, when viewed in totality, the array
should exhibit net directionality with respect to the positioning
of the apexes 608-614.
[0108] Various modifications and alterations of this invention will
become apparent to those skilled in the art without departing from
the scope and spirit of this invention, and it should be understood
that this invention is not to be unduly limited to the illustrative
embodiments set forth herein.
* * * * *