Information
-
Patent Grant
-
6792856
-
Patent Number
6,792,856
-
Date Filed
Tuesday, July 16, 200222 years ago
-
Date Issued
Tuesday, September 21, 200420 years ago
-
Inventors
-
Original Assignees
-
Examiners
Agents
- F. Chau & Associates, LLC
-
CPC
-
US Classifications
Field of Search
US
- 101 327
- 101 287
- 101 322
- 101 297
- 101 319
- 101 133
- 101 214
- 101 368
-
International Classifications
-
Abstract
Disclosed is a printing apparatus, having a print surface lying in a print plane defined by an imaginary x-axis and y-axis, the print surface having an outward normal pointing in the positive direction along an imaginary z-axis, such that the x-axis, y-axis, and z-axis are substantially orthogonal to one another, a lower stamp clamp disposed adjacent to the negative-x edge of the print surface, an upper stamp clamp, moveable in two dimensions in a trajectory plane defined by the x-axis and z-axis, a stamp comprising a flexible material, the stamp having a first end attached to the lower stamp clamp and a second end attached to the upper stamp clamp, such that a cross section of the stamp parallel to the trajectory plane forms an arc extending from an origin point Q on the lower stamp clamp having (x,z) coordinates (0,0) to point E on the upper stamp clamp, this arc being described by the mathematical function θ(s), where s is the curvilinear distance along the arc measured from point Q, and θ is the angle between the print plane and an imaginary line, the imaginary line being tangent to the cross section of the stamp at s, and wherein, during a print operation, the upper stamp clamp is moved in a trajectory comprising a plurality of xz positions of the upper clamp stamp that blend into a substantially continuous motion, the trajectory being effective in laying the stamp down smoothly and flat upon the print surface in a manner such that a moving contact front between the stamp and the print surface is created, the contact front being disposed substantially along a line characterized by a contact-front coordinate s0 x0 that increases as the trajectory progresses, the trajectory also being effective in causing the curvature ⅆθⅆsof the stamp at or near the contact front to be substantially constant throughout the motion. The preferred embodiment also has either or both of two additional systems that move along the x axis in coordination with the motion of the upper stamp clamp: a print-force-application system effective in pressing the stamp against the print surface, and a stamp control system for helping to control the curvature of the stamp near the contact front.
Description
FIELD OF THE INVENTION
The invention relates to printing, particularly to micro-contact printing, also called “soft lithography”, in which a flexible stamp transfers an “inked” pattern to a receiving surface by mechanical contact, the pattern often having very small features normally associated with optical lithography and other expensive methods. More generally, the invention relates to a precise and controlled way of bringing two surfaces into contact, and subsequently separating them.
BACKGROUND OF THE INVENTION
A number of printing techniques collectively known as “soft lithography” have been recently developed, spurred by the 1993 discovery of micro-contact printing, as described in A. Kumar and G. M. Whitesides, FEATURES ON GOLD HAVING MICROMETER TO CENTIMETER DIMENSIONS CAN BE FORMED THROUGH A COMBINATION OF STAMPING WITH AN ELASTOMERIC STAMP AND AN ALKANETHIOL INK FOLLOWED BY CHEMICAL ETCHING,
Appl. Phys. Lett
., 63, 2002 (1993), the disclosure of which is incorporated by reference herein in their entirety. Typically, in such a printing technique, a flexible, polymeric stamp, embossed with a pattern and coated with a chemical “ink”, is brought into contact with a receiving surface and then separated from it, thereby transferring the image to the receiving surface in the form of a molecular monolayer of the ink. A full review of the techniques of soft lithography has recently been given in Y. Xia and G. M. Whitesides, SOFT LITHOGRAPHY,
Angew. Chem. Int. Ed
., 37, 550 (1998) and in B. Michel, et al., PRINTING MEETS LITHOGRAPHY: SOFT APPROACHES TO HIGH RESOLUTION PATTERNING, to be published in
IBM Journal of Research and Development
(special issue on lithography), the disclosures of both of which are incorporated by reference herein in their entirety.
Soft lithography promises to deliver printing that is less costly than that available with conventional techniques, such as optical lithography, used routinely in semiconductor processing. Soft lithography's lower cost is possible because the per-print process is simpler than conventional techniques—there are fewer steps and fewer costly machines. Moreover, soft lithography can print large areas quickly, whereas traditional, optical techniques can print only small areas at a time, and must build up large areas by “stitching” (step and repeat), a slow process requiring an extremely precise and expensive machine known as a lithographic stepper.
To enable soft lithography, a printing method and apparatus are required to bring the stamp and the receiver into intimate contact, in a controlled and repeatable manner, such that the pattern on the stamp is transferred to the receiver with the greatest possible fidelity (i.e., with minimal distortion). To insure intimate contact everywhere, the printing method must prevent the trapping of gaseous bubbles (e.g., air bubbles) between the opposing surfaces of the stamp and the receiver. To insure repeatability, the printing apparatus must be automated. To achieve high fidelity, two requirements must be met. Firstly, the stamp itself should resist distortions in its own plane; such resistance is provided, for example, by the two-layer “hybrid stamps” described by B. Michel et. al., supra. Secondly, the printing apparatus must provide, when the stamp and the receiver come into contact, uniform contact pressure and uniform geometric conditions over the entire printed area, lest the stamp be non-uniformly strained and therefore the printed pattern distorted.
Several prior-art methods and machines attempt to provide the printing requirements needed for soft lithography. However, these prior-art methods are deficient in several respects. One such method is described in U.S. Pat. No. 5,669,303 entitled APPARATUS AND METHOD FOR STAMPING A SURFACE, issued Sep. 23, 1997. This apparatus brings a circular stamp, held at its edges, into gradual contact with a receiver. The stamp is treated as a membrane under variable pressure: the convex (lower-pressure) side of the curving stamp being gradually flattened against the receiver while the periphery of the stamp is held fixed. Although the gradual contact successfully eliminates the trapping of air bubbles, this method and apparatus clearly produces non-uniform strain in the stamp as the varying pressure stretches the membrane, thereby distorting the pattern. Acknowledging this distortion, various schemes were proposed to compensate it, but the manufacturing practicality of these schemes is doubtful, and it would clearly be preferable if the method did not engender the non-uniform strain in the first place.
Another prior-art apparatus and method are described by B. Michel et al., supra, as the “rocker cylinder printing tool”. In this method, the stamp is wrapped on a partial drum of radius R, and then “rocked” upon the receiving surface in a manner somewhat analogous to the motion of a rocking chair upon a floor. In other words, the method is like a printing press in which the receiver remains stationary while the axis of the rotating drum translates over it. The problems with this method are three-fold. Firstly, the embossed pattern on the stamp is stretched in the print direction due to the drum's curvature, introducing systematic distortion. Secondly, over the print cycle, the peak contact pressure between the stamp and the receiver is spatially non-uniform because it depends critically on the drum-to-receiver gap, which varies as the mechanism moves on account of unavoidable mechanical tolerances such as bearing runout and machining inaccuracies on the drum's surface. Attempting to minimize variations in peak contact pressure by introducing a compliant layer (known as a “soft pad”) behind the stamp simply trades peak-pressure non-uniformity for geometric non-uniformity; that is, as the soft pad compresses to accommodate gap changes, the local curvature of the stamp near the line of contact varies, and thus the tangential strain of the embossed pattern varies—this complex variation being superimposed on the systematic strain due to the drum's curvature. Thirdly, because the drum is both translating and rotating, the accuracy of printing depends critically on precisely matching the drum's translational speed ν with its rotational speed ω; ideally, to roll without slipping and without straining the compliant stamp by frictional forces, the drum's velocity ν should be exactly equal to ωR. However, this ideal matching is nearly impossible to accomplish to the tolerance (˜1 ppm) required for high-accuracy, large-area applications—exactly the applications where soft lithography seeks to replace optical lithogtaphy. Thus the rocker-style printer is ill-suited to the task of soft lithography. In fact, a controlled experiment was performed in which feature-placement errors on two prints from the same stamp were measured—one print made with a well-engineered rocker printer, the other with an alternative scheme (such as the current invention), where the three problems discussed above are absent. The results demonstrate roughly a factor-of-three advantage in feature-placement accuracy for the latter method.
All three shortcomings of the rocker printer, of course, are shared by the “printing press” style of machine. In particular, the printing press shares the third shortcoming mentioned above (print accuracy dependent on precise matching of ν to ωR): although the printing-press's drum rotates without translating, the receiver instead translates beneath it, so speed matching is still an issue. Although the printing press is, of course, suitable for images to be observed by the human eye, where feature-placement accuracy need not be better than about 10 to 20 μm, it appears to be unsuitable for the applications of soft lithography (e.g., printing patterns for electronic circuitry), where feature-placement accuracy on the order of 1 μm or better is required.
Accordingly, there is a need for an improved method and apparatus for transferring patterns from a stamp to a receiver with great fidelity, the method and apparatus being scalable to large-size receivers and amenable of various types of stamps.
SUMMARY OF THE INVENTION
Disclosed is a printing apparatus, comprising a print surface lying in a print plane defined by an imaginary x-axis and y-axis, the print surface having an outward normal pointing in the positive direction along an imaginary z-axis, such that the x-axis, y-axis, and z-axis are substantially orthogonal to one another, a lower stamp clamp disposed adjacent to the negative-x edge of the print surface, an upper stamp clamp, moveable in two dimensions in a trajectory plane defined by the x-axis and z-axis, a stamp comprising a flexible material, the stamp having a first end attached to the lower stamp clamp and a second end attached to the upper stamp clamp, such that a cross section of the stamp parallel to the trajectory plane forms an arc extending from an origin point Q on the lower stamp clamp having (x,z) coordinates (0,0) to point E on the upper stamp clamp, this arc being described by the mathematical function θ(s), where s is the curvilinear distance along the arc measured from point Q, and θ is the angle between the print plane and an imaginary line, the imaginary line being tangent to the cross section of the stamp at s, and wherein, during a print operation, the upper stamp clamp is moved in a trajectory comprising a plurality of xz positions of the upper clamp stamp that blend into a substantially continuous motion, the trajectory being effective in laying the stamp down smoothly and flat upon the print surface in a manner such that a moving contact front between the stamp and the print surface is created, the contact front being disposed substantially along a line characterized by a contact-front coordinate s
0
x
0
that increases as the trajectory progresses, the trajectory also being effective in causing the curvature
of the stamp at or near the contact front to be substantially constant throughout the motion.
Another aspect of the printer comprises a print-force-application system effective in pressing the stamp against the print surface, and defining an approximate contact front disposed substantially along a line l
B
parallel to the y-axis in the xy plane, the line l
B
intersecting the trajectory plane at (x, z)=(x
B
, 0), the approximate-contact-front x-coordinate x
B
increasing as the trajectory progresses and being substantially equal, at any stage of the trajectory, to the arc-length coordinate s
B
of point B, inasmuch as the arc of the stamp is assumed to be substantially flat over the segment from point Q to point B.
Another aspect of the printer further comprises a stamp-control system movable along the x-axis, wherein, throughout the trajectory, each xz position of the upper stamp clamp is a function of the displacement x
C
of the stamp-control system along the x-axis; the trajectory being effective in laying the stamp down upon the print surface such that the stamp is in continuous contact with a contact surface of the stamp-control system throughout the trajectory, the location of the contact surface being characterized by an arc-length coordinate s
C
that increases as the trajectory progresses.
In another aspect of the printer, the stamp-control system is disposed along a line l
C
parallel to the y-axis, line l
C
intersecting the trajectory plane at point C having coordinates x
C
and z
C
, where z
C
is a fixed, positive z-coordinate during any one printing operation, whereas x
C
increases as the trajectory progresses, in coordination with the contact-front coordinate x
0
.
In another aspect of the printer, the contact surface of the stamp-control system is a plane delimited in the x direction by two lines l
C
and l
D
separated by a fixed distance W
CD
, these lines being parallel to the y-axis and intersecting the trajectory plane at points C and D respectively, these points having coordinates (x
C
, z
C
) and (x
D
, z
D
) respectively, such that the contact surface is defined by the three parameters (x
C
, z
C
, θ
CD
), where
is the angle between the contact surface and the print plane, and such that the stamp angle θ(s) between arc-length coordinates s=s
C
and s=s
D
is substantially equal to θ
CD
; that is,
θ(
s
)≈θ
CD
for
s
C
≦s≦s
D
.
In another aspect of the printer, the upper stamp clamp is pivoted about a pivot line l
P
parallel to the y axis and intersecting the xz plane at point P having coordinates x
P
and z
P
; the stamp attaching to the upper stamp clamp along an upper-clamp line l
E
parallel to the y axis and intersecting the xz plane at point E having coordinates x
E
and z
E
; the upper-clamp line l
E
being disposed on the upper stamp clamp at a radius R
S
from the pivot line l
P
, such that the total arc length s
E
from the lower stamp clamp to the line l
E
is s
E
&Quadbond;L, where L is the known, free length of the stamp; and wherein the stamp attaches to the upper-clamp line l
E
at an angle θ
E
≡θ(L).
In another aspect of the printer, the trajectory comprises a plurality of configurations, each configuration described by the coordinate s
0
&Quadbond;x
0
of the contact front and by corresponding coordinates x
P
, z
P
of the pivot line given by the equations
x
P
=x
E
+R
s
cos θ
E
z
P
=z
E
+R
s
sin θ
E
,
where
and where the mathematical function θ(s) describing the shape of the arc for a given configuration is assumed to be
θ(
s
)=0 for 0
≦s≦s
0
,
whereas for s>s
0
, θ(s) is determined by solution of the differential equations
the lower-end boundary conditions
and the upper-end boundary condition
κ
0
is a specified curvature at point O, the parameter β≡F
X0
, unknown a priori, is the internal x-directed force acting on the stamp's cross section at s=s
0
per unit depth of the stamp in the y direction, E is Young's modulus of the stamp, I is the area moment of inertia of the stamp's cross section per unit depth in the y-direction, and w is the weight per unit area of the stamp; and
wherein for each configuration the solution for x
P
and z
P
is derived by means of the “shooting method”, whereby an initial value β
(0)
of β is guessed, the differential equations are solved to yield T(β
(0)
) and
Newton iteration
is applied to obtain a refined value β
(1)
of the unknown parameter β, whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary condition T(β)=0 is achieved to within some tolerance.
In another aspect of the printer the upper stamp clamp is pivoted about a pivot line l
P
parallel to the y axis and intersecting the xz plane at point P having coordinates x
P
and z
P
; the stamp attaching to the upper stamp clamp along an upper-clamp line l
E
parallel to the y axis and intersecting the xz plane at point E having coordinates x
E
and z
E
; the upper-clamp line l
E
being disposed on the upper stamp clamp at a radius R
S
from the pivot line l
P
, such that the total arc length s
E
from the lower stamp clamp to the line l
E
is s
E
&Quadbond;L, where L is the known, free length of the stamp; and wherein the stamp attaches to the upper-clamp line l
E
at an angle θ≡θ(L).
In another aspect of the printer, the trajectory comprises a plurality of configurations, each configuration described by the coordinate s
B
&Quadbond;x
B
of the approximate contact front and by corresponding coordinates x
P
, z
P
of the pivot line given by the equations
x
P
=x
E
+R
s
cos θ
E
z
P
=z
E
+R
s
sin θ
E
,
where
and where the mathematical function θ(s)
describing the shape of the arc for a given configuration is assumed to be
θ(
s
)=0 for 0
≦s≦s
B
,
whereas for s>s
B
, θ(s) is determined by solution of the differential equations
the lower-end boundary conditions
and the upper-end boundary condition
κ
B
is a specified curvature at point B, the parameter β≡F
XB
, unknown a priori, is the internal x-directed force acting on the stamp's cross section at s=s
B
per unit depth of the stamp in the y direction, E is Young's modulus of the stamp, I is the area moment of inertia of the stamp's cross section per unit depth in the y-direction, and w is the weight per unit area of the stamp; and
wherein for each configuration the solution for x
P
and z
P
is derived by means of the “shooting method”, whereby an initial value β
(0)
of β is guessed, the differential equations are solved to yield T (β
(0)
) and
Newton iteration
is applied to obtain a refined value β
(1)
of the unknown parameter β, whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary condition T(β)=0 is achieved to within some tolerance.
In another aspect of the printer, the upper stamp clamp is pivoted about a pivot line l
P
parallel to the y axis and intersecting the xz plane at point P having coordinates x
P
and z
P
; the stamp attaching to the upper stamp clamp along an upper-clamp line l
E
parallel to the y axis and intersecting the xz plane at point E having coordinates x
E
and z
E
, the upper-clamp line l
E
being disposed on the upper stamp clamp at a radius R
S
from the pivot line l
P
, such that the total arc length s
E
from the lower stamp clamp to the line l
E
is s
E
&Quadbond;L, where L is the known, free length of the stamp; and wherein the stamp attaches to the upper-clamp line l
E
at an angle θ
E
≡θ(L).
In another aspect of the printer, the trajectory comprises a plurality of configurations, each configuration described by the coordinate s
0
&Quadbond;x
0
of the contact front and by corresponding coordinates x
P
, z
P
of the pivot line given by the equations:
x
P
=x
E
+R
s
cos θ
E
z
P
=z
E
+R
s
sin θ
E
,
where
and where the mathematical function θ(s) describing the shape of the arc for a given configuration is assumed to be
θ(
s
)=0 for 0
≦s≦s
0
,
whereas for s>s
0
, θ(s) is determined by solution of the differential equations
the lower-end boundary conditions
and the auxiliary boundary conditions
T
(β)=0,
wherein
and wherein F
X
(s) and F
Z
(s) are functions of s describing the internal x-directed and z-directed forces acting on the stamp's cross section at s per unit depth of the stamp in the y direction, F
XE
≡F
X
(s
E
), F
ZE
(s
E
), β is a vector of parameters that are unknown a priori,
s
0
is the aforementioned arc-length coordinate of the contact front, F
X0
≡F
X
(s
0
), F
Z0
≡F
Z
(s
0
), E is Young's modulus of the stamp, I is the area moment of inertia of the stamp's cross section per unit depth in the y-direction, w is the weight per unit area of the stamp, p(s) and f(s) are functions of s describing forces applied normal to the stamp and tangential to the stamp respectively by the print-force-application system, the stamp-control system and the print surface, s
C
is the value of arc-length coordinate a at point C, θ
C
≡θ(s
C
) is the angle of the arc at point C, and θ
CD
is the aforementioned angle of the stamp-control system's contact surface; and
wherein for each configuration the solution for x
P
and z
P
is derived by means of the “shooting method”, whereby an initial value β
(0)
of β is guessed, the differential equations are solved to yield T(β
(0)
) and
Newton-Raphson iteration
is applied to obtain a refined vector β
(1)
whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary conditions T(β)=0 are achieved to within some tolerance.
In another aspect of the printer, the trajectory comprises a plurality of configurations, each configuration described by the coordinate s
0
&Quadbond;x
0
of the contact front and by corresponding coordinates x
P
, z
P
of the pivot line given by the equations:
x
P
=x
E
+R
s
cos θ
E
z
P
=z
E
+R
s
sin θ
E
,
where
and where the mathematical function θ(s) describing the shape of the arc for a given configuration is assumed to be
θ(
s
)=0 for 0
≦s≦s
0
,
whereas for s>s
0
, θ(s) is determined in stamp segments OC and DE by solution of the differential equations
the lower-end boundary conditions
and the upper-end boundary condition
κ
0
is a specified curvature at point O, E is Young's modulus of the stamp, I is the area moment of inertia of the stamp's cross section per unit depth in the y-direction, w is the weight per unit area of the stamp, F
x
(s) and F
z
(s) are the x-directed and z-directed stamp forces per unit length of stamp in the y direction, given by
in which F
z0
≡F
x
(s
0
)≡β is a parameter that is unknown a priori, and the differences ΔF
x
and ΔF
z
are respectively the differences
Δ
F
x
≡F
x
(
s
D
)−
F
x
(
s
C
)
Δ
F
z
≡F
z
(
s
D
)−
F
z
(
s
C
)
that occur across stamp segment CD where the stamp-control system contacts the stamp, the values of which differences, along with the value of the difference
may be calculated from the three equations of static equilibrium for the stamp under the action of forces applied to the stamp by the stamp-control system, these three differences together with θ
D
=θ
C
permitting numerical integration for stamp segment DE to proceed immediately from the numerical-integration result obtained at the final point C in stamp segment OC, and wherein for each configuration the solution for x
P
and z
P
is derived by means of the “shooting method”, whereby an initial value β
(0)
of β is guessed, the differential equations are solved to yield T(β
(0)
) and
Newton iteration
is applied to obtain a refined vector β
(1)
, whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary conditions T(β)=0 are achieved to within some tolerance.
In another aspect of the printer, a stamp-control system movable along the x-axis, wherein, throughout the trajectory, each xz position of the upper stamp clamp is a function of the displacement x
C
of the stamp-control system along the x-axis; the trajectory being effective in laying the stamp down upon the print surface such that the stamp is in continuous contact with a contact surface of the stamp-control system, the location of the contact surface being characterized by an arc-length coordinate s
C
that increases as the trajectory progresses.
In another aspect of the printer, the stamp-control system is disposed along a line l
C
parallel to the y-axis, line l
C
intersecting the trajectory plane at point C having coordinates x
C
and z
C
, where z
C
is a fixed, positive z-coordinate during any one printing operation, whereas x
C
increases as the trajectory progresses, in coordination with the contact-front coordinate x
0
.
In another aspect of the printer the contact surface of the stamp-control system is a plane delimited in the x direction by two lines l
C
and l
D
separated by a fixed distance W
CD
, these lines being parallel to the y-axis and intersecting the trajectory plane at points C and D respectively, these points having coordinates (x
C
, z
C
) and (x
D
, z
D
) respectively, such that the contact surface is defined by the three parameters (x
C
, z
C
, θ
CD
), where
is the angle between the contact surface and the print plane, and such that the stamp angle θ(s) between arc-length coordinates s=s
C
and s=s
D
is substantially equal to θ
CD
; that is,
θ(
s
)≈θ
CD
for
s
C
≦s≦s
D
.
In another aspect of the printer, the upper stamp clamp is pivoted about a pivot line l
P
parallel to the y axis and intersecting the xz plane at point P having coordinates x
P
and z
P
; the stamp attaching to the upper stamp clamp along an upper-clamp line l
E
parallel to the y axis and intersecting the xz plane at point E having coordinates x
E
and z
E
; the upper-clamp line l
E
being disposed on the upper stamp clamp at a radius R
S
from the pivot line l
P
, such that the total arc length s
E
from the lower stamp clamp to the line l
E
is s
E
&Quadbond;L, where L is the known, free length of the stamp; and wherein the stamp attaches to the upper-clamp line l
E
at an angle θ
E
≡θ(L).
In another aspect of the printer, the trajectory comprises a plurality of configurations, each configuration described by the coordinate s
0
&Quadbond;x
0
of the contact front and by corresponding coordinates x
P
, z
P
of the pivot line given by the equations:
x
P
=x
E
+R
s
cos θ
E
z
P
=z
E
+R
s
sin θ
E
,
where
and where the mathematical function θ(s) describing the shape of the arc for a given configuration is assumed to be
θ(
s
)=0 for 0
≦s≦s
0
,
whereas for s>s
0
, θ(s) is determined by solution of the differential equations
the lower-end boundary conditions
and the auxiliary boundary conditions
T
(β)=0,
wherein
and wherein F
X
(s) and F
Z
(s) are functions of s describing the internal x-directed and z-directed forces acting on the stamp's cross section at s per unit depth of the stamp in the y direction, F
XE
≡F
X
(s
E
), F
ZE
≡F
Z
(s
E
), β is a vector of parameters that are unknown a priori,
s
0
is the aforementioned arc-length coordinate of the contact front, F
X0
≡F
X
(0), F
Z0
≡F
Z
(0), E is Young's modulus of the stamp, I is the area moment of inertia of the stamp's cross section per unit depth in the y-direction, w is the weight per unit area of the stamp, p(s) and f(s) are functions of s describing forces applied normal to the stamp and tangential to the stamp respectively by the print-force-application system, the stamp-control system and the print surface, s
C
is the value of arc-length coordinate s at point C, θ
C
≡θ(s
C
) is the angle of the arc at point C, and θ
CD
is the aforementioned angle of the stamp-control system's contact surface; and
wherein for each configuration the solution for x
P
and z
P
is derived by means of the “shooting method”, whereby an initial value β
(0)
of β is guessed, the differential equations are solved to yield T(β
(0)
) and
Newton-Raphson iteration
is applied to obtain a refined vector β
(1)
, whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary conditions T(β)=0 are achieved to within some tolerance.
In another aspect of the printer, the trajectory comprises a plurality of configurations, each configuration described by the coordinate s
B
&Quadbond;x
B
of the approximate contact front and by corresponding coordinates x
P
, z
P
of the pivot line given by the equations
x
P
=x
E
+R
s
cos θ
E
z
P
=z
E
+R
s
sin θ
E
,
where
and where the mathematical function θ(s) describing the shape of the arc for a given configuration is assumed to be
θ(
s
)=0 for 0
≦s≦s
B
,
whereas for s>s
B
, θ(s) is determined in stamp segments OC and DE by solution of the differential equations
the lower-end boundary conditions
and the upper-end boundary condition
κ
B
is a specified curvature at point B, E is Young's modulus of the stamp, I is the area moment of inertia of the stamp's cross section per unit depth in the y-direction, w is the weight per unit area of the stamp, F
x
(s) and F
z
(s) are the x-directed and z-directed stamp forces per unit length of stamp in the y direction, given by
in which F
x0
≡F
x
(s
0
)≡β is a parameter that is unknown a priori, and the differences ΔF
x
and ΔF
z
are respectively the differences
Δ
F
x
≡F
x
(
s
D
)−
F
x
(
s
C
)
Δ
F
z
≡F
z
(
s
D
)−
F
z
(
s
C
)
that occur across stamp segment CD where the stamp-control system contacts the stamp, the values of which differences, along with the value of the difference
may be calculated from the three equations of static equilibrium for the stamp under the action of forces applied to the stamp by the stamp-control system, these three differences together with θ
D
=θ
C
permitting numerical integration for stamp segment DE to proceed immediately from the numerical-integration result obtained at the final point C in stamp segment OC, and wherein for each configuration the solution for x
P
and z
P
is derived by means of the “shooting method”, whereby an initial value β
(0)
of β is guessed, the differential equations are solved to yield T(β
(0)
) and
Newton iteration
is applied to obtain a refined vector β
(1)
whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary conditions T(β)=0 are achieved to within some tolerance.
In another aspect of the printer, the print-force-application system comprises a flat-iron.
In another aspect of the printer, the stamp-control system comprises a vacuum bar.
Disclosed is a printing apparatus, comprising a receiver means whose receiving surface lies in an xy plane, the normal to the surface defining a z-axis direction, a lower stamp clamp means for fixing a first edge of a stamp, an upper stamp clamp means for holding a second edge of a stamp for movement in the xz directions, a flexible stamp means for printing to the receiver, said flexible stamp in substantially the form of a sheet defining edges, the first edge of which is affixed to the lower stamp clamp, and the opposing second edge of which is affixed to the upper stamp clamp, thereby allowing the stamp to hang in a curve under gravity and the sheet's own stiffness, such that every normal to the stamp's curved surface lies substantially parallel to the xz plane, and a trajectory-producing means for moving the upper stamp clamp along a prescribed trajectory in the xz plane, such that the stamp is draped upon the receiving surface in a manner that causes the curvature of the stamp near a contact front at a point B to be constant throughout the trajectory.
Another aspect of the apparatus further comprises print-force application means for applying pressure upon the stamp means against the receiver means and for defining the contact front.
Another aspect of the apparatus further comprises stamp-control means for defining a point C through which the curvature of the sheet will pass throughout the trajectory.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1
is a schematic drawing of a micro-contact printing process.
FIG. 2
is a schematic drawing of a two-layer stamp of the type used by this invention.
FIG. 3
is a block diagram specifying the components of a printing machine.
FIG. 4
is a three-dimensional view of an embodiment of the printing machine of the invention, prior to loading the stamp.
FIG. 5
is a three-dimensional view of an embodiment of the printing machine of the invention, after loading the stamp.
FIGS. 6
a
,
6
b
, and
6
c
are three-dimensional views of the printing machine of the invention at the start, the middle, and the end of the printing process, respectively.
FIG. 7
is a three-dimensional view of a base table.
FIG. 8
a
is a three-dimensional view of a print table before a receiver is loaded.
FIGS. 8
b
and
8
c
are orthographic projections of the print table.
FIG. 9
is a three-dimensional view of the print table after the receiver is loaded.
FIG. 10
is a three-dimensional view of a lower stamp clamp.
FIG. 11
is cross-sectional diagram of the print table, the lower stamp clamp, the stamp and the receiver.
FIG. 12
is a three-dimensional view of the bottom surface of a vacuum plate that is part of the lower stamp clamp.
FIG. 13
is a close-up of an adjustment mechanism for the lower stamp clamp.
FIG. 14
is a close-up of another adjustment mechanism for the lower stamp clamp.
FIG. 15
is a cross-sectional diagram of screws that locate and attach the stamp to the lower stamp clamp.
FIG. 16
is a three-dimensional view of the upper stamp clamp as it appears when mounted in the printing machine.
FIG. 17
is a cross-sectional diagram of the upper stamp clamp.
FIG. 18
is a three-dimensional view of the upper stamp clamp as it appears, remote from the printer, while attaching a stamp.
FIG. 19
is a three-dimensional view of a carriage.
FIG. 20
is another three-dimensional view of the carriage.
FIG. 21
is yet another three-dimensional view of the carriage.
FIG. 22
is a schematic diagram of the system, showing rationale for a flat-iron and a vacuum bar.
FIG. 23
a
and
23
b
are, respectively, a three-dimensional and an orthographic view of the flat-iron.
FIG. 24
a
is a cross-sectional schematic diagram of the flat-iron;
FIG. 24
b
shows the pressure profile in the air bearing.
FIG. 25
is a three-dimensional view of the vacuum-bar assembly.
FIG. 26
is a three-dimensional close-up view of the front end of the vacuum-bar assembly.
FIG. 27
is a three-dimensional close-up view of the rear end of the vacuum-bar assembly.
FIG. 28
shows various orthographic projections of the rear end of the vacuum-bar assembly.
FIG. 29
shows orthographic projections of the vacuum bar.
FIG. 30
is a schematic diagram showing the various stamp segments used in the mathematical analysis of the invention.
FIG. 31
is a free-body diagram of a differential element of the stamp.
FIG. 32
is a diagram showing forces and dimensions on the lower surface of the vacuum bar.
FIG. 33
is a close-up diagram of stamp segment AB.
FIG. 34
is a close-up diagram of stamp segment CD.
FIG. 35
is a close-up diagram illustrating the stamp's top-end boundary condition.
FIG. 36
shows various mathematically generated solutions for the shape of a stamp.
FIG. 37
is a schematic diagram explaining the graphical methodology used in
FIGS. 38 through 42
.
FIGS. 38 through 45
are various measurements of feature placement errors obtained with a prototype of the invention.
FIG. 46
shows the meaning of forward and reverse printing directions.
DETAILED DESCRIPTION OF THE INVENTION
1. Overall Description
FIG. 1
shows a schematic of the printing process accomplished by the invention. Apart from the printing machine itself, the printing process requires three elements, as shown in
FIG. 1
a
: a stamp
5
having a printing surface
10
embossed with a raised pattern
15
; ink
20
that coats the raised pattern
15
; and a receiver
25
to whose receiver surface
30
the ink
20
will be transferred during the printing process by mechanical contact of raised pattern
15
and receiver surface
30
, as shown in
FIG. 1
b
. The printing machine (not shown in
FIG. 1
) must accomplish this transfer in such a way that after separation of the stamp
5
from the receiver
25
(
FIG. 1
c
), the placement of transferred ink on the receiver surface
30
faithfully replicates the raised pattern
15
. The invention provides such a printing machine having a number of advantages over prior-art machines.
Although the printing machine may be used with any type of stamp consistent with the following detailed description, the preferred embodiment of the machine uses the type of stamp described in commonly assigned Biebuyck et al., U.S. Pat. No. 5,817,242, entitled STAMP FOR LITHOGRAPHIC PROCESS, issued Oct. 6, 1998, the disclosures of which are incorporated by reference herein in their entirety. As shown in
FIG. 2
, this type of stamp comprises two bonded layers, a polymer layer
35
, upon whose printing surface
10
the raised pattern
15
is embossed, and a back-plane layer
40
, typically composed of metal or other material having a high modulus of elasticity. The back-plane layer
40
provides high lateral stiffness, and therefore high feature-placement accuracy of the printed pattern on receiver surface
30
. Such accuracy is difficult to achieve—particularly for large-area stamps—using the low-modulus polymer layer
35
alone. As drawn in
FIG. 2
, the area of the back-plane layer
40
is typically somewhat larger than that of the polymer layer
35
, for reasons explained subsequently.
Referring to
FIG. 3
, the printing machine
100
described in this invention will preferably comprise six subsystems, including a base-table assembly
105
, a print-table assembly
110
, a linear-motion system
115
, a lower stamp clamp
120
, a carriage
125
, and an upper stamp clamp
130
.
As also shown in
FIG. 3
, four of these six subsystems may be further divided into more detailed subsystems. The base-table assembly
105
comprises a horizontal base table
135
, a vertical base table
140
, and a vibration-control system
145
. The print-table assembly
110
comprises the print-table itself,
150
, as well as pneumatics
155
. The linear-motion system
115
comprises three axes of motion (
160
,
180
, and
200
), requiring three sets of linear-motion stages (
165
,
185
,
205
); motors (
170
,
190
, and
210
) and motor drivers (
175
,
195
,
215
); a multi-axis motor controller
220
; a computer
225
utilizing standard hardware
230
but specialized software
235
; and a joystick
240
or similar input device. The lower stamp clamp
120
comprises a vacuum chuck assembly
245
that may be micro-positioned with respect to the print table
150
, as well as pneumatics
250
to operate the vacuum chuck. The carriage
125
comprises a mechanical frame
255
; a print-force-application system
260
for applying the force of contact between the stamp
5
and the receiver
25
during printing; and a stamp-control system
265
to insure that the critical geometry of the stamp is maintained throughout the printing process, thereby to achieve the best possible printed replica of the embossed pattern
15
on the receiver
25
.
The latter two systems may be further subdivided into components. The print-force-application system
260
comprises an air-bearing-supported flat-iron
270
to apply the force of contact between stamp
5
and receiver
25
during printing, and pneumatics
275
to control the flow of air to the flat-iron. The stamp-control system comprises a vacuum-bar assembly
280
to control the geometry of the stamp during printing, and pneumatics
285
to operate the vacuum bar.
FIG. 4
depicts the entire printing machine
100
without the stamp installed. For reference, an xyz coordinate system
290
is shown; on subsequent figures, the xyz orientation of this coordinate system (but not the origin) remains consistent. The base-table assembly
105
(from
FIG. 3
) comprises a horizontal base table
135
and a vertical base table
140
, affixed to each other at right angles, forming an “L” shape. The print table
150
rests on the horizontal base table
135
, and the lower stamp clamp
120
rests on the print table's stepped-down surface
295
, which is preferably machined lower than its elevated receiver surface
300
, for reasons clarified later. Pneumatic controls represented by
155
,
250
,
275
, and
285
are shown at the right end of the horizontal base table
135
.
The system of three computer-driven stages (
165
,
185
,
205
) are affixed to the vertical base table
140
. Two of these stages,
165
and
185
, which correspond to axes (x
1
,z), form a two-axis set: horizontal stage
165
is affixed to the vertical base table
140
; stage
185
is affixed to faceplate
305
of stage
165
. Thus faceplate
310
of z stage
185
is capable of executing two-dimensional motion parallel to the xz plane. The third computer-driven stage,
205
, corresponding to axis x
2
, is affixed to the vertical base table, parallel to stage
165
and below it, in such a way that the z stage
185
can pass over the body of the x
2
stage
205
without interference.
The upper stamp clamp
130
is affixed to faceplate
310
of z-stage
185
; thus, the upper stamp clamp
130
can move in two dimensions parallel to the xz plane. Likewise, the carriage
125
is affixed to faceplate
315
of x
2
-stage
205
; thus, the carriage can move in one dimension parallel to the x-axis. Because the motors (
170
,
190
,
210
) driving all three stages are computer controlled via the same controller
220
(from FIG.
3
), it is possible to provide coordinated triple-axis movement of the upper stamp clamp and the carriage simultaneously.
Referring to
FIGS. 5 and 6
, the receiver
25
is held fixed, throughout the printing process, to the print table's elevated receiver surface
300
. As shown in
FIG. 6
b
, one end of the stamp's back-plane layer
40
, hereafter called the stationary end
320
, is attached to the lower stamp clamp
120
, such that the stamp's polymer layer
35
faces downward. This stationary end
320
remains fixed during printing. The opposite end of the stamp's back-plane layer
40
, hereafter called the movable end
325
, is attached to the upper stamp clamp
130
. Because the stamp is flexible, when it is mounted in the printing machine, suspended between the upper stamp clamp
130
and the lower stamp clamp
120
, it hangs in a natural curve, as shown.
FIG. 6
shows how the printing process proceeds. Specifically,
FIGS. 6
a
,
6
b
, and
6
c
show the beginning, middle, and end of the printing process respectively. As shown by comparison of the three figures, the upper stamp clamp
130
moves in two dimensions (xz) during printing—downward and to the right, thereby laying the stamp's polymer layer
35
gradually upon the receiver
25
. Simultaneously, in a motion coordinated with that of the upper stamp clamp
130
, the carriage
125
moves to the right, so that the systems it carries—the print-force-application system
255
and the stamp-control system
260
—can apply the contact force and geometrical control necessary to achieve uniform, accurate printing. When the end of the printing process is reached (i.e., when the stamp is fully in contact with the receiver), or perhaps after an intervening delay, the stamp is peeled from the receiver by reversing the three-axis motions used for printing. When peeling is done—when the stamp is completely separated from the receiver
25
—the receiver, now patterned with the ink
20
, may be removed from the printing machine.
2. Detailed Description of Sub-Systems
2.1 Base-Table Assembly
FIG. 7
shows the base-table assembly
105
(from
FIG. 3
) in more detail. In addition to the horizontal base table
135
and the vertical base table
140
, it preferably comprises two diagonal struts
330
or other suitable reinforcing means, to hold the vertical base table perpendicular to the horizontal base table. These may be attached by six strut blocks
335
for attachment of the diagonal struts
330
to the base tables
135
and
140
as well as for attachment of the tables to each other. Also provided is a vibration-control system
145
. In a prototype of the invention constructed to generate the print images of
FIGS. 38 through 45
, the horizontal base table
135
, vertical base table
140
, and vibration-isolation system
145
were purchased from Newport Corporation (Fountain Valley, Calif.). The thickness of the tables (
135
,
140
) and load-bearing capacity of the vibration-control system
145
depend on the x and y dimensions of the receiver
25
(from FIG.
3
). In the prototype of the invention, the receiver dimensions are 381 mm×381 mm (15″×15″). For this case the horizontal base table
135
is 914 mm×1,524 mm×102 mm thick (36″×60″×4″ thick), the vertical base table
140
is 914 mm×1,524 mm×51 mm thick (36″×60″×2″ thick), and the vibration-isolation system
145
has a load capacity of 5,780 N (1300 lbs).
2.2 Print-Table Assembly
FIGS. 8
a
,
8
b
, and
8
c
show an embodiment of the print table
150
in more detail.
FIG. 8
a
is a three-dimensional view of the print table
150
with a lower stamp clamp
120
mounted on it.
FIGS. 8
b
and
8
c
are respectively a top view and a rear view of the print table
150
alone. As shown in
FIG. 8
a
, a preferred top surface of the print table
150
comprises a step
340
, which divides the surface into an elevated receiver surface
300
and, parallel to it, the stepped-down surface
295
. Preferably, both of these surfaces will be flat to a high degree of precision. Stepped-down surface
295
may have threaded holes
345
(
FIG. 8
b
) or other mounting means as will be described below in conjunction with the lower stamp clamp
120
. Elevated receiver surface
300
will preferably have a connected series of narrow, shallow grooves
350
or other suitable vacuum carrying means, which provide a vacuum chuck to hold receiver
25
fixed to elevated receiver surface
300
during the printing process, and also provide an air-pressure chuck to break the vacuum seal when the printing process is finished. The grooves
350
in elevated receiver surface
300
are alternately exposed to vacuum and air pressure, during and after the printing process respectively, by means of one or more vacuum openings
355
, drilled perpendicular to receiver surface
300
, and which communicate with grooves
350
. The vacuum openings
355
may intersect a pressure-supply hole
360
(
FIG. 8
c
) drilled at right angles to it, parallel to receiver surface
300
, from the rear surface
365
of print table
150
. Where pressure-supply hole
360
exits print table
150
at rear surface
365
, it is terminated with a common pneumatic fitting
370
, which is connected to pneumatic controls
155
, of a type well known in the art, that can alternately supply vacuum or air pressure, as described above, to grooves
350
.
FIG. 9
shows that, before receiver
25
is affixed to elevated receiver surface
300
using vacuum pressure applied to grooves
350
, the receiver will preferably be located precisely on the surface by alignment means
375
such as, for example, a plurality of banking pins threaded into the three threaded holes
380
shown on
FIG. 8
b
. The banking pins
375
protrude above receiver surface
300
by an amount just slightly less than the thickness of the receiver
25
. Receiver
25
is banked to the cylindrical surfaces of banking pins
375
by application of lateral forces
385
applied in the -x and -y directions. The banking pins
375
are located such that, when the receiver
25
is registered against the pins' side surfaces, the receiver's leading edge
390
is substantially perpendicular to the line formed by the print table's step
340
.
Preferably, the print table
150
is made of a material, such as granite, whose surfaces may be made very flat. Custom block of granite—comprising step
340
, threaded holes
345
, grooves
350
, vacuum hole
355
, pressure-supply hole
360
, pneumatic fitting
370
, and threaded holes
380
—may be ordered from various commercial sources, such as L. S. Starrett Co., Granite Surface Plate Division, Mt. Airy, N.C. Solid granite is a preferred material for this application because the flatness of the receiver surface
300
, upon which receiver
25
rests during printing, significantly influences the accuracy of the transfer of the pattern
15
from the stamp
5
to the receiver
25
. As is well-known in the art of tool-making, a lapped granite surface provides a durable, high-precision flat surface.
2.3 Lower Stamp Clamp
FIG. 10
shows the vacuum-chuck assembly
245
(from
FIG. 3
) of lower stamp clamp
120
(from FIG.
3
). This assembly, which rests upon stepped-down surface
295
of print table
150
, comprises a vacuum plate
395
; a stamp attachment means
400
such as, for example, one or more locating screws that mate with threaded locating holes
405
(shown in
FIG. 9
) for attachment of the stamp to the vacuum plate
395
; and three micrometer assemblies
425
,
430
, and
435
. In the embodiment shown in the drawings, a first micrometer assembly
425
is affixed to front surface
440
of vacuum plate
395
. Likewise, a second micrometer assembly
430
—a mirror image of
425
—is affixed to the rear surface of vacuum plate
395
. Yet a third micrometer assembly
435
is affixed to the stepped-down surface
295
of print table
150
using two threaded holes
345
. The three micrometer assemblies
425
,
430
, and
435
are used to adjust the position and orientation of the vacuum plate
395
with respect to the print table
150
.
As shown in
FIG. 11
, the vacuum plate
395
is machined to thickness h
395
satisfying the following equation:
h
395
=h
340
+h
25
+h
35
,
where h
340
is the height of step
340
, h
25
is the thickness of receiver
25
, and h
35
is the thickness of the stamp's polymer layer
35
. In this way, when the polymer layer
35
is brought into contact with the receiver during the printing process, the stamp's back-plane layer
40
lies flat, as shown in FIG.
11
.
As further shown in
FIG. 11
, and also in
FIG. 12
(which is a bottom view of vacuum plate
395
), the bottom surface
445
of vacuum plate
395
is machined with a number of shallow grooves
450
or other suitable vacuum carrying means. These grooves provide a vacuum chuck to hold vacuum plate
395
fixed to surface
295
during the printing process, but also provide, as needed during machine setup, an air-pressure chuck by means of which the vacuum plate
395
is floated on an air cushion to facilitate translational and rotational adjustment of vacuum plate
395
with respect to the print table
150
, as discussed further below in connection with the micrometer assemblies
425
,
430
, and
435
. The grooves
455
in surface
450
may be alternately exposed to vacuum and air pressure, as needed, by means of a bleed hole
455
drilled perpendicular to surface
445
and communicates with grooves
450
. The bleed hole
455
intersects a pressure-supply hole
460
drilled at right angles to it, parallel to surface
445
, as shown in FIG.
11
. The pressure-supply hole
460
is terminated with a thread suitable for a pneumatic fitting
465
. Pneumatic controls
250
, of a type well know in the art, are connected to the fitting
465
to alternately supply the vacuum and air pressure as described above.
FIG. 13
is a close-up of first micrometer assembly
425
, which comprises a micrometer head
470
, and a mounting means, such as an L-bracket
475
, and screws
480
that affix L-bracket
475
to front surface
440
of vacuum plate
395
. The body of the micrometer head
470
is affixed to the L-bracket
475
using, for example, a slotted-hole clamp arrangement, wherein a clamping screw
485
forces together the legs
490
on either side of the slot
495
, thereby gripping the micrometer head
475
, such that the axis of the micrometer head is parallel to front surface
440
of vacuum plate
395
, and rigidly connected to it. Non-rotating spindle
500
of micrometer head
475
, outfitted with ball-bearing tip
505
, is extended so that the ball-bearing tip
505
bears against step
340
of print table
150
, such that, by further extension of the spindle
500
, the distance
510
(between step
340
and vacuum plate
395
near the front of the print table
150
) is increased. Conversely, distance
510
may be decreased by retraction of the spindle, provided only that a force (manual, spring-loaded, etc.) is applied to vacuum plate
395
, in the positive x direction, to keep the ball-bearing tip
505
and the step
340
in intimate contact. The ball-bearing tip assures a rolling, single-point contact between the micrometer and the step
340
.
As shown in
FIG. 14
, there is preferably provided a second micrometer assembly
430
, affixed to the rear surface of vacuum plate
395
, a mirror image of first micrometer assembly
425
. Thus, by analogy to the previous paragraph, second micrometer assembly
430
is used to modulate the distance
515
between step
340
and vacuum plate
395
near the rear of the print table
150
. Thus, the two micrometer assemblies
425
and
430
, by modulating distances
510
and
511
respectively, move vacuum plate
395
in the two degrees of freedom x and θ.
As further shown in
FIG. 14
, there is preferably provided a third micrometer assembly
435
comprising a micrometer head
520
, a straight bracket
525
, and screws
530
or similar means to affix straight bracket
525
or other mount means to the stepped-down surface
295
of the print table
150
. The body of the micrometer head
520
may be affixed to the straight bracket
525
, using the slotted-hole clamp arrangement described in connection with
FIG. 13
, such that the axis of the micrometer head is perpendicular to the front surface
440
of vacuum plate
395
, and rigidly connected to surface
295
of print table
150
. Non-rotating spindle
535
of micrometer head
520
, outfitted with ball-bearing tip
536
(analogous to tip
505
of micrometer head
470
) is extended to bear against the front surface
440
of vacuum plate
395
, such that, by further extension of the spindle
535
, the distance
537
(between the front surface
440
of vacuum plate
395
and the straight bracket
525
) is increased. Conversely, distance
537
may be decreased by retraction of spindle
535
, provided only that a manual force is applied to vacuum plate
395
, in the negative y direction, to keep the spindle
535
and surface
440
in intimate contact.
In the prototype of the invention, the micrometer heads used in micrometer assemblies
425
,
430
, and
435
are sold under the tradename “Model 262ML Micrometer Head”, and are commercially available from The L. S. Starrett Company, Athol, Mass. The ball-bearing tips
505
and
536
are sold under the tradename “Model 247MA ball attachment” and are likewise available from L. S. Starrett.
In summary, referring to
FIGS. 13 and 14
, the three micrometer assemblies
425
,
430
, and
435
, because they allow adjustment of the three distances
510
,
515
, and
537
, provide complete, three-degree-of-freedom (xyθ) adjustment of the location and orientation of vacuum plate
395
with respect to the print table
150
. Thus, the xy location and θ orientation of the stamp
5
can be adjusted with respect to the receiver
25
, because, as previously shown in
FIG. 11
, the stamp's back-plane
40
is affixed to the vacuum plate
395
, whereas, as shown in
FIG. 9
, the receiver
25
is affixed to the print table
150
.
This xyθ adjustability of the stamp
5
with respect to the receiver
25
is achieved, however, without sacrificing stiffness of the stamp-to-receiver connection during the printing process. During printing, the vacuum chuck
455
locks the vacuum plate
395
(and hence the stationary end
320
of the stamp's back-plane
40
, as described in the next paragraph) to the print-table's stepped-down surface
295
, while the vacuum chuck
350
locks the receiver
25
to the print-table's elevated receiver surface
300
. Thus during printing, the connection between the stamp
5
and the receiver
25
is extremely stiff—much stiffer than could be attained by alternative, xyθ-adjustable arrangements such as conventional stages. Such high stiffness contributes substantially to faithful transfer of the stamp's raised pattern
15
to the receiver
25
.
FIG. 15
shows a cross-sectional, exploded detail of the locating screws
400
(introduced in FIG.
10
and repeated in
FIG. 14
) and the mating locating holes
405
(shown in FIG.
9
), that are used in the prototype of this invention. Of course, there are many means by which a back-plane layer
40
may be simultaneously located and attached to a vacuum plate
395
, but the means shown are simple yet effective. Near its stationary end
320
, the back-plane
40
is provided with two or more holes
540
, separated by a precise center-to-center distance
545
(shown on FIG.
14
). Locating screws
400
are inserted through the stamp holes
540
and screwed into the threaded portion
550
of locating holes
405
. The unthreaded portion
555
of locating holes
405
are precisely machined to match the distances between and diameters of the holes
540
in the back-plane
40
. By these means, the two screws
400
serve not only to locate the stamp's back-plane
40
with respect to the vacuum plate
395
, but also to rigidly attach the one to the other.
As further shown in
FIG. 15
, the locating screw
400
has a threaded portion
560
that mates with the threaded portion
550
of locating hole
405
; a chamfered portion
565
that facilitates the entry of screw
400
into locating hole
405
; a locating portion
570
that mates with the unthreaded portion
555
of locating hole
405
; and a head
575
whose under-surface comprises a relieved area
580
and a bearing area
585
, and whose top surface comprises a screw-driver slot
590
or other screw-torquing means. The locating portion
570
of locating screw
400
has an axial length shorter than that of the hole's unthreaded portion
555
, and a diameter that is larger than the clearance diameter of the locating screw's threaded portion
560
but slightly less than the diameter of the hole's unthreaded portion
555
, such that a slip fit of the screw's locating portion
570
into the hole's locating portion
555
is achieved. The diameter of screw's locating portion
570
is also slightly less than the diameter of the stamp hole
540
in the back-plane
40
, such that a slip fit of the screw's locating portion
570
through stamp hole
540
is achieved. Regarding the head
575
of screw
400
, the relieved area
580
insures that the critical diameter of the screw's locating portion
570
continues toward the top of the screw far enough to avoid any unwanted interference with the aforementioned slip-fit clearances, and the bearing area
585
affixes back-plane
40
to vacuum plate
395
when locating screw
400
is tightened.
2.4 Upper Stamp Clamp
FIG. 16
shows an embodiment of an upper stamp clamp
130
(from
FIG. 3
) looking in the positive x direction (see
FIG. 4
for definition of the x axis). The upper stamp clamp shown comprises a cantilevered rod
600
, two rod clamps
605
and
610
, a base plate
615
, two side plates
620
and
625
, a precision shaft
630
, two ball-bushings assemblies
635
and
640
, two T-bars
645
and
650
, and two clamp strips
655
and
660
.
Cantilevered rod
600
, such as Model 45 available from Newport Corp. as used in the prototype, is rigidly affixed at its root end to faceplate
310
of z-stage
185
. Rod clamp
605
, such as Newport's Model 340C also used in the prototype, with its faceplate facing down, is clamped to rod
600
, but is not attached to base plate
615
. Rod clamp
610
(identical to
605
), its faceplate also facing down, is rigidly affixed to base plate
615
. Side plates
620
and
625
, which comprise coaxial holes
670
and
675
respectively, are also rigidly affixed to base plate
615
, such that precision shaft
630
(such as Thompson Model LRS-8-SS as used in the prototype) may be suspended, parallel to the y direction, between the coaxial holes
670
and
675
, and affixed there by set screws. Ball-bushing assemblies
635
and
640
ride on shaft
630
, such that each assembly is free to rotate about the axis of shaft
630
, and also to slide along the axis.
With the exception of the cantilevered rod
600
and the rod clamp
605
, the remaining parts of the upper stamp clamp (
610
,
615
,
620
,
625
,
630
,
635
,
640
,
645
,
650
,
655
,
660
), hereafter referred to as the stamp carrier
680
, may be removed from the printing machine
100
as a single unit, simply by releasing the clamp
610
and sliding it off the cantilevered rod
600
.
Referring to
FIG. 17
, there is shown a schematic, cross-sectional view of the upper-stamp-clamp assembly, perpendicular to the axis of shaft
630
. (The rod
600
and rod clamps
605
and
610
are not shown in this Figure). Ball-bushing assembly
635
referred to in
FIG. 16
, actually comprises ball bushing
685
, such as a Thompson Model A-81420-SS as used in the prototype, and housing
690
, into which ball bushing
685
is affixed using C rings, press fit, or any other suitable manner. Housing
690
is provided with attachment surfaces
695
on opposite sides of its cylindrical surface. One end of the T-bars
645
and
650
is rigidly attached to attachment surfaces
695
by means of small bolts. The other ends of the two T-bars
645
and
650
are likewise attached to similar flats in the ball-bushing housing belonging to the other ball-bushing assembly
640
. Thus T-bars
645
and
650
are affixed parallel to shaft
630
, on opposite sides of it, and the whole assembly (ball-bushing assemblies
635
and
640
plus T-bars
645
and
650
) can freely rotate about the axis of shaft
630
, like a paddle wheel with two paddles, and can also freely translate along the axis of shaft
630
. The latter degree of freedom prevents shear in the stamp
5
, in the y direction, as it is laid down upon receiver
25
during the printing process.
Referring to
FIG. 18
, the back-plane layer
40
of the stamp
5
is attached to the stamp carrier
680
. This attachment is most conveniently done prior to the attachment of the back-plane layer
40
to lower stamp clamp
120
, on some flat surface
705
. For attaching the stamp to the stamp carrier, only one of the T-bars (either
645
or
650
) and its associated clamp strips (
655
or
660
) need to be used. The unused T-bar and clamp strip exist merely for rotational balance. As shown in
FIG. 18
, suppose that T-bar
645
(and associated clamp strip
655
) are selected for attaching the stamp. The stamp's back-plane
40
has been pre-punched, near its right end
320
, with a plurality of holes
710
that align with threaded holes
715
in T-bar
645
, and also with clearance holes
720
in clamp strip
655
. To begin the attachment, the back-plane's holes
710
are manually aligned with the T-bar's threaded holes
715
. Then the clamp strip
655
is laid on top of back-plane
40
, thereby sandwiching the back-plane between T-bar
645
and clamp strip
655
(as also shown in FIG.
17
), and the clamp-strip's holes
720
are also aligned. Bolts
725
(shown in
FIGS. 16 and 18
) are then inserted manually through holes
720
and
710
and tightened into threaded holes
715
.
Referring back to
FIG. 16
, the assembled stamp and the stamp carrier
680
may now be mounted on the printing machine by sliding rod clamp
610
onto cantilevered rod
600
. To successfully complete the latter operation, surface
730
of base plate
615
must slide underneath faceplate
665
of rod clamp
605
, which can only occur when the angular orientation of rod clamp
610
matches that of clamp
605
. This insures that the stamp carrier is always returned to the same angular orientation about the axis of the cantilevered rod
600
, because the orientation of clamp
605
is set once and never changed.
2.5 Carriage
As disclosed above with respect to
FIG. 3
, the carriage
125
comprises three components: a mechanical frame
255
, a print-force-application system
260
, and a stamp-control system
265
. These three components are now described in detail.
2.5.1 Carriage: Mechanical Frame
Referring to
FIGS. 19 through 21
, the mechanical frame
255
(from
FIG. 3
) is shown in detail from various viewpoints. In
FIG. 19
, the entire assembly is cantilevered from faceplate
315
of x
2
-stage
205
by means of two stiff rods (such as Newport Corp. Model 45 rods as used in the prototype), including the upper cantilevered rod
735
and the lower cantilevered rod
740
. Two rods rather than one are used for improved stiffness, and also to insure that the assembly is aligned parallel to the print table
150
, as explained presently. Rod clamps
745
,
750
, and
755
(such as Newport Corp. Model 340C as used in the prototype) are used to grip the rods and allow the position of the entire assembly to be adjusted in the y direction.
Referring to
FIG. 20
, there is provided a T-shaped plate
760
, shown to better advantage in
FIG. 20
, which binds the two cantilevered rods (
735
and
740
) together into one, stiff unit by virtue of attaching to all the rod clamps. In the embodiment shown, to allow for the staggered arrangement of the rods, a spacer plate
770
is interposed between the descending tongue
765
of T-shaped plate
760
and rod clamp
755
. If the bolt that ties the descending tongue
765
to spacer plate
770
is removed and the rod clamps
745
and
750
are loosened, the T-shaped plate
760
may be freely rotated about the axis of the upper cantilevered rod
735
. This rotation is useful for servicing, because the carriage's entire front-end assembly
775
, described in more detail below, is suspended from front surface
780
of T-shaped plate
760
, such that the rotation provides access to the underside of assembly
775
. After servicing, the T-shaped plate
760
and the assembly
775
may be easily returned to the original position, because the spacer plate
770
acts as a stop that limits rotation in the clockwise direction: the thickness of spacer plate
770
has been machined such that, when the tongue
765
of T-plate
760
rests against the spacer plate
770
, the assembly
775
is accurately aligned parallel to the print table
150
.
Affixed to the front of the T-shaped plate
760
is a micrometer-driven stage
785
(such as that sold under the tradename Melles Griot Model 07 XSC 007 as used in the prototype), the faceplate of which,
790
, may be adjusted in the z direction by means of micrometer
795
. Affixed to the faceplate
790
of the stage
785
(and thereby adjustable by its micrometer
795
) is a three-sided frame
800
, shown to better advantage in
FIG. 21
, that resembles the frame of a sofa. The frame
800
comprises a back plate
805
and two J-slotted side plates
810
and
815
that are rigidly affixed to back plate
805
at right angles.
2.5.2 Carriage: Need for Print-Force-Application System and Stamp-Control System
The side plates
810
and
815
carry two systems important for printing; namely, the print-force-application system
260
and the stamp-control system
265
.
Referring to
FIG. 22
, there is shown an embodiment wherein the print-force-application system
260
is embodied in a flat-iron
270
, and the stamp-control system
265
is embodied in a vacuum-bar
820
, which is part of the vacuum-bar assembly
280
.
The simplest embodiment of the invention comprises neither the print-force-application system
260
nor the stamp-control system
265
. In such a simplified system, the upper stamp clamp
130
shown in
FIG. 22
would simply move in a trajectory
821
downward and to the right during the printing process, causing the inked stamp
5
to be draped upon the receiver
25
in a manner such that the curvature of the stamp at or near the contact front is substantially constant throughout the trajectory. However, it has been found experimentally that, when the stamp's back-plane layer
40
is 150-μm thick metal (e.g., Invar 36) and the lateral stamp dimensions are 381×381 mm, such a simplified system does not reliably produce uniform, intimate contact between the stamp
5
and the receiver
25
, and thus does not reliably provide uniform transfer of the ink from stamp to receiver. Apparently, although the stiffness of the stamp's back-plane layer
40
does induce some vertical reaction force between stamp
5
and receiver
25
as they come into contact, for the dimensions stated, this reaction force is insufficient for reliable printing.
Thus, to obtain intimate contact reliably between stamp
5
and receiver
25
, it is necessary to introduce the print-force application system
260
comprising the flat-iron
270
depicted in FIG.
22
. Riding atop the stamp during the printing process, the flat-iron
270
is essentially a bar that moves horizontally in coordination with the motion of the upper stamp clamp
130
, as described below in great detail. By virtue of its weight, the flat-iron
270
“irons” the stamp
5
onto the receiver
25
. The y dimension of the flat-iron is substantially equal to that of the polymer layer
35
of stamp
5
, such that the stamp is “ironed” by a uniform weight across it's entire width. The preferred embodiment of the print-force application system
260
, of which the flat-iron
270
is the main constituent, is described in detail below in connection with
FIGS. 23 through 25
.
Regarding the vacuum bar
820
in
FIG. 22
a
, the invention will function without it, but printing accuracy is improved by its addition. The vacuum bar
820
, having a vacuum chuck in its lower surface
825
, attracts the backplane layer
40
of stamp
5
upward against this vacuum surface
825
, thereby creating, in the critical vicinity where the stamp
5
and the receiver
25
meet, a segment of the stamp BC, shown in
FIG. 22
b
, having a well determined geometry that is constant both in time (throughout the printing process) and in space (across the width of the stamp). As long as the vacuum bar
820
retains its vacuum hold on the back-plane
40
, the geometry of segment BC is guaranteed to remain constant throughout the printing process because it is determined by the fixed geometrical relationship between the flat-iron
270
and the vacuum bar
820
. The geometry of segment BC is also guaranteed to be uniform across the width of the stamp (in the y direction) because the rigidity of the short segment BC prevents stamp sag across the width. By thus controlling stamp geometry, the stamp-control system
265
, whose payload is the vacuum bar
820
, has been found to improve markedly the accurate (undistorted) placement of the stamp's raised pattern
15
onto receiver surface
30
of receiver
25
. Details of the stamp-control system are described below in connection with
FIGS. 26-30
.
2.5.3 Carriage: Print-Force Application System
Although contact force between stamp
40
and receiver
25
may be applied in a variety of ways (such as by a roller), in the preferred embodiment force is applied using the flat-iron
270
, which is constructed as shown in
FIGS. 23
a
and
23
b.
Referring to
FIG. 23
a
, the disembodied flat-iron is shown three-dimensionally and inverted; in
FIG. 23
b
, it is shown as an orthographically projected side view (also inverted). During printing and peeling, the bottom surface
830
of the flat-iron
270
floats over the surface of the stamp's back-plane
40
on a cushion of compressed gas (e.g., air or clean nitrogen) emanating from one or more double-blind grooves
832
cut along the centerline of surface
830
. Thus, the weight of the flat-iron is supported by gaseous pressure. The compressed gas may be supplied to the groove
832
via a series of small, equally spaced bleed holes
835
that connect the groove
832
to a plenum chamber
840
, a hole drilled parallel to the long axis of the flat-iron
270
. Compressed gas may be supplied to the plenum chamber
840
via an annular pin
845
whose proximal end is pressed into a precisely bored hole
850
that terminates one end of the plenum chamber
840
. The distal end of the annular pin
845
is fitted with a pneumatic fitting
855
suitable for attachment of a flexible hose
860
. The opposite end of the plenum is also terminated with a precisely bored hole
865
, which is precisely co-axial with hole
850
. Pressed into hole
865
is a solid pin
870
that seals the plenum chamber
840
.
Referring back to
FIG. 21
, the flat-iron's pins
845
and
870
, described above, will preferably ride in J-shaped slots
875
cut into side plates
810
and
815
. Thus, during printing the flat-iron's motion is not constrained in the vertical direction; rather, its lower surface
830
is free to float over the surface of the stamp's back-plane
40
as the stamp is laid upon the receiver
25
. The flat-iron floats on the cushion of compressed gas as described above. At the same time, by virtue of being captured in the front and rear J-shaped slots
875
, the flat-iron
270
is forced, during printing, to move with the carriage in the +x direction at the print speed ν. During times other than printing, the short vertical segments
876
of J-shaped slots
875
provide a resting place for the flat-iron's pins
845
and
870
, so that the flat-iron's lapped surface
830
is protected from inadvertent contact with other surfaces. In an alternative, more sophisticated, embodiment, the flat-iron would be moved automatically between the active (down) position and inactive (up) position by means of a computer-controller cam or other effective mechanism.
Referring to
FIG. 24
, the print-force application system
260
insures that the peak contact pressure between the stamp
5
and the receiver
25
is uniform.
FIG. 24
a
shows a typical cross section of the flat-iron
270
as it moves over the stamp's back-plane
40
at velocity ν (the print velocity). A cushion of compressed gas
880
emanates from double-blind groove
832
and flows in both the positive and negative x directions as shown. We assume simple Couette flow (and therefore, uniform pressure gradient) in each direction, as described, for example, in
Boundary Layer Theory
, by H. Schlichting, McGraw Hill (1968), the disclosures of which are incorporated by reference herein in their entirety. Thus, the pressure distribution
885
under the flat-iron is substantially triangular in shape as shown in
FIG. 24
b
: the gage pressure is p
0
at the slit (x=0) and decreases to zero (atmospheric pressure) at the edges of the flat-iron (i.e., at x=±W
AB
/2). The pressure causes the flat-iron to float a distance b above the surface of the stamp's back-plane
40
. The integral of the gage pressure over x (i.e., the area of the triangle in
FIG. 24
b
) equals the weight-per-unit-length of the flat-iron in the y direction, denoted σ, which is uniform because the flat-iron's height H and width W
AB
are uniform. Thus,
From Couette theory, it may easily be shown that {dot over (Q)}, the volumetric flow rate of gas per unit length in the y direction, is related to p
0
and to the gap size b by
where μ
g
is the viscosity of the gas. Comparing Equations (1) and (2) yields
which implies that the gap size b may be controlled by the volumetric flow rate{dot over (Q)}. In the preferred embodiment, {dot over (Q)} is controlled directly by a flow-control device, such as a rotameter (e.g., Omega Engineering Inc., Model FL-102, as used in the prototype). In the prototype of the invention, the flat-iron was made of hardenable stainless steel such as 17-4 PH, the air-bearing surface
830
was surface ground, the width of groove
832
was 1.5 mm, the flow rate per unit depth was {dot over (Q)}=0.93 cm
2
/sec, and the cross-sectional dimensions of the flat-iron
270
were H=19 mm and W
AB
=31 mm, such that the weight per unit length was σ=4.5×10
4
dyne/cm. If the gas is air, then μ
g
=2×10
−4
dyne-s/cm
2
, so the air-bearing gap for this set of parameters, according to Equation (3), is b=49.2 μm.
2.5.4 Carriage: Stamp-Control System
The rationale for the stamp-control system
265
has been stated above in connection with FIG.
22
. The stamp may be urged upward against the surface
825
shown in
FIG. 22
b
by a variety of means, such as magnetically if the stamp's back-plane
40
is ferromagnetic. In the preferred embodiment it is held by vacuum, and thus the bar
820
, which spans the width of the stamp in the y direction, we refer to herein as the “vacuum bar”.
Referring to
FIGS. 25 through 29
, an aluminum L-beam
890
or equivalent spans the distance between side plates
810
and
815
of the three-sided frame
800
. The L-beam
890
may be, attached to these side plates by means of bolts
895
and
900
, which pass through slots
905
and
910
(visible in
FIGS. 27 and 28
respectively) into threaded holes in side plates
810
and
815
, thereby allowing coarse height and parallelism adjustment of vacuum bar
820
with respect to receiver surface
300
of print table
150
. Finer (and more convenient) adjustment of height and parallelism is preferably provided by micrometer heads
915
and
920
(such as Model 262 from The L. S. Starrett Co. as used in the prototype) that are mounted in the horizontal flange of L-beam
890
, near each end of the L-beam and fixed by setscrews
925
and
930
or equivalent means. As shown in
FIG. 26
, the non-rotating spindle
935
of the first micrometer head
915
may be affixed to a first axle clamp
940
by means of clamp screw
945
. That is, after spindle
935
is inserted into hole
950
, clamp screw
945
is tightened, causing the width of slot
955
—and hence the diameter of hole
950
—to be reduced, thereby clamping spindle
935
firmly to the first axle clamp
940
. Likewise, as shown in
FIG. 27
(and also in
FIGS. 29
a
and
29
d
), the non-rotating spindle
960
of the second micrometer head
920
is affixed to the second axle clamp
965
by means of clamp screw
970
, which reduces the width of slot
975
—and hence the diameter of hole
980
, thereby clamping spindle
960
firmly to the second axle clamp
965
.
Referring to
FIG. 28
, axle clamps
940
and
965
, in addition to grasping the micrometer spindles
935
and
960
also serve to hold the vacuum bar
820
in such a way as to allow rotational adjustment of the vacuum bar about an axis parallel to the y direction. As shown in
FIG. 28
a
, a flanged ball-bearing
985
is pressed into the rear axle clamp
940
. This ball bearing
985
faces the vacuum bar
820
, and its inner race receives an annular pin
990
that forms the rear end of the vacuum-bar's axis. Likewise, a similar ball bearing (not shown) is pressed into the front axle clamp, and receives a similar (although solid rather than annular) pin
995
(not shown) that forms the front end of the vacuum-bar's axis. Thus the vacuum bar
820
, suspended between the two axle clamps
940
and
965
, rotates freely about the axis formed by pins
990
and
995
, thereby allowing angular adjustment of the vacuum bar
820
about an axis parallel to the y direction. After such an adjustment, the vacuum bar's angular position θ
CD
may be locked, such as by tightening nylon-tipped set screws
1000
and
1005
that are threaded into the front and rear axle clamps, respectively. The setscrews' nylon tips insure that pins
990
and
995
are not scored when the setscrews are tightened. Slots
1010
(
FIG. 26
) and
1015
(
FIGS. 28 and 29
) provide access to the setscrews
1000
and
1005
, so that they may be tightened or loosened.
Referring to
FIGS. 29
a
,
29
c
, and
30
, the vacuum bar
820
comprises two pieces: the main body
1020
and the wear layer
1025
. As described above in connection with
FIG. 22
, the bottom surface
825
of the vacuum bar
820
(i.e., the bottom surface of the wear layer
1025
) is in sliding contact with the stamp's back-plane
40
during printing, thereby holding the stamp up, against the force of gravity, by means of vacuum suction. The wear layer
1025
will preferably be made of a low-friction material (such as polyethylene terephthalate, fluoropolymer, or the like) in order to minimize friction during the aforementioned sliding contact, thus eliminating “stick-slip” vibration that would compromise the accuracy of the printing process. Moreover, from a manufacturing viewpoint, the wear layer
1025
is a sacrificial medium that protects the main body
1020
from sliding-contact wear; after many printing cycles, the easy-to-fabricate wear layer
1025
, preferably attached to the main body
1020
with double-stick tape, may be simply and economically replaced.
Referring to
FIGS. 28 and 29
a
, a vacuum hose
1030
is threaded through slot
1015
and is attached to fitting
1035
, thereby providing vacuum to cavity
1040
of rear-axle clamp
965
, thence to the axial hole of annular pin
990
, and thence to plenum
1045
of the vacuum-bar's main body
1020
. The plenum
1045
—an axial hole that runs the length of the vacuum-bar's main body
1020
—is bored precisely coaxial near the two ends, such that, when the plenum is sealed by pins
990
and
995
pressed into the precisely bored ends, these pins form an axis that is precisely straight. As shown in
FIG. 29
, the vacuum-bar's main body
1020
has a flat surface
1050
along whose centerline is drilled a plurality of bleed holes
1055
, each of which connects the plenum
1045
to the flat surface
1050
. Wear layer
1025
, which has the same rectangular dimensions as flat surface
1050
, also comprises a plurality of bleed holes
1060
disposed in the same pattern as the holes
1055
in the main body
1020
, such that when surface
1065
of the wear layer
1025
is aligned and affixed to flat surface
1050
of the main body
1020
using double-stick tape (the tape having been perforated at the location of each of the bleed holes
1060
) or other suitable attachment means, bleed holes
1055
align with bleed holes
1060
, and the vacuum pressure is thus communicated from the main-body's plenum
1045
to the wear layer's bleed holes
1060
. A shallow, double-blind vacuum groove
1070
is cut along the centerline of wear-layer
1025
's bottom surface
825
. The vacuum groove
1070
is said to be “double-blind” because it does not extend all the way to the ends of surface
825
. Because the vacuum groove
1070
intersects the bleed holes
1060
, vacuum is communicated to this groove, and thus vacuum pressure acts on the whole area of the groove to lift the stamp's backplane
40
against the vacuum surface
825
.
In the preferred embodiment, the vacuum pressure thus supplied to groove
1070
in surface
825
is variable, so that the force that lifts the stamp's backplane
40
against surface
825
may be adjusted high enough to hold the backplane reliably, yet low enough to avoid excessive drag as the vacuum bar's wear layer slides across the backplane
40
. Toward this end, the vacuum-bar pneumatics
285
, mentioned above in connection with
FIG. 3
, will preferably comprise a venturi-based vacuum pump (such as that used in the prototype, an Edco C4M10N-A), whose vacuum pressure output can be controlled by the venturi's supply pressure, which in turn is controlled by a conventional pressure regulator.
2.6 Linear-Motion System: Hardware
Referring again to
FIGS. 3 and 4
, stages
165
,
185
, and
205
, which convert the rotary motion of motors
170
,
190
, and
210
into linear motion along axes x
1
, z, and x
2
respectively, are high quality, medium-precision stages. Suitable stages for use with the invention, such as used in the prototype, are manufactured by Parker/Daedal, with the following model numbers:
Stage
165
(x
1
axis): 406600XR-MS-D3H3L2C4M4E1B1R1P2
Stage
185
(z axis): 406600XR-MS-D3H3L2C4M4E1B2R1P2
Stage
205
(x
2
axis): 406600XR-MS-D3H3L2C4M4E1B1R1P1.
Details about these stages are given in Parker/Daedal Manual No. 100-9313-01, the disclosures of which are incorporated by reference herein in their entirety.
Absolute accuracy of the stages is not critical in the quest for printed accuracy, because the placement accuracy of printed features of the stamp's raised pattern
15
, although critically dependent on the integrity of the stamp's back-plane layer
40
as it is laid upon the receiver
25
, does not depend on the precise motion of the stages. For example, the Parker stages cited above have an absolute accuracy of only 50 microns, yet the embodiment shown in
FIG. 4
can faithfully replicate printed features over an area of 271 by 203 mm with an absolute, three-sigma accuracy of less than 3 microns.
It is desirable to avoid vibration in the stages. For this reason, in the preferred embodiment, motors
170
,
190
and
210
are high-quality DC-servo motors rather than stepper motors. DC-servo motors have been found to produce much smoother motion of the stamp and the carriage as they execute the three-axis coordinated motion required to lay the stamp on the receiver. Motors suitable for use with the invention are Parker Gemini Series motors, model SM233AE-NGSN, driven by Parker Gemini drivers, model GV-L3E, such as are used in the prototype. Detailed instructions for operating the motors are in Parker manual 88-017790-10C; instructions for operating the drivers are in Parker manuals 88-017778-10B and 88-017791-10C, the disclosures of which are incorporated by reference herein in their entirety.
The motor controller
220
directs the motor drivers to turn the motors as required to create the coordinated, three-axis motions of the upper end of the stamp (axes x
1
and z) and the carriage (axis x
2
), as required for printing and peeling. Such motor controllers are well known in the art. A suitable motor controller for use with the invention is that used in the prototype, a Parker model 6K8, which is described in Parker manual 88-017547-10A, the disclosures of which are incorporated by reference herein in their entirety. The Parker 6K8 controller may be programmed using the 6K Programming Language, as documented in two Parker publications “6K Programmer's Guide” (88-017137-10A) and “6000 Series Software Reference” (88-012966-01), the disclosures of which are incorporated by reference herein in their entirety. For manual control of the stacked pair of stages
165
and
185
, it is convenient to use a hardware joystick
240
, such as that used in the prototype, a Parker model JS-6000.
The information required to create the three-axis coordinated motions of axes x
1
, z, and x
2
(i.e.,
160
,
180
,
200
) originates in the computer
225
, which for example may be an IBM-compatible Personal Computer, or any other suitable computer system. The required information is generated by software
235
, which will be described in detail presently. Typically, the entire series of coordinates required to specify the x
1
, z, and x
2
motions are calculated once by the software
235
, and then downloaded to the motor controller
220
. Subsequently, the motors may be repeatedly driven directly from the controller
220
without further computation in the computer
225
.
The computer hardware
230
comprises, in addition to the usual components (microprocessor, memory, storage, I/O), a communications port such as an RS-232 port or other I/O means by which the computer
225
communicates with the motor controller
220
.
2.7 Linear-Motion-System Software
The software
235
, provides mathematical means to specify—for virtually any type and size of stamp
5
—the proper, three-axis coordinated motions that must be executed along axes x
1
, z, and x
2
during the printing process to lay the stamp
5
progressively upon the receiver
25
and subsequently to peel the stamp from the receiver. These three-axis coordinated motions will hereinafter be called “the printing trajectory” and “the peeling trajectory” respectively.
2.7.1 Software: Motivation for a Mathematical Solution
Referring to
FIG. 30
, the printing trajectory must be arranged such that the curved shapes that the stamp assumes throughout the trajectory, such as those shown in
FIGS. 6
a
,
6
b
, and
6
c
, allow the lower surface
825
of the vacuum bar
820
(from
FIG. 22
) to retain its vacuum hold upon the stamp
5
. That is, as the carriage
125
moves the flat-iron
270
and the vacuum bar
820
left-to-right along axis x
2
at a uniform rate, the upper stamp clamp
130
must move the upper end of the stamp along axes x
1
and z smoothly downward and to the right so that, at every instant, the curve of the stamp “kisses” the vacuum bar.
It may be appreciated intuitively that if the upper end of the stamp is moved downward too slowly, the vacuum bar, moving rightward, will eventually stretch and kink the stamp. Conversely, if the upper end of the stamp is moved-downward too quickly, it will eventually, by virtue of the stamp's stiffness, pull way from the vacuum bar's hold. Preventing the latter is typically the more difficult requirement, because the vacuum-bar's vacuum chuck (i.e., the double-blind groove
1070
in wear layer
1065
from
FIG. 29
) is deliberately weak so as to minimize sliding friction between the stamp
5
and the vacuum-bar's wear surface
1025
. Therefore, the printing trajectory will be specified rather carefully to insure that the stamp's natural shape remains tangent to the vacuum bar throughout the motion.
Retaining this vacuum attachment enhances printed accuracy because holding the stamp's shape constant in the segment BC ensures constant curvature near the contact line (where the stamp and receiver meet, near or at point B in FIG.
22
), and thereby provide uniform tensile strain in the stamp's raised pattern
15
. If the printing trajectory is incorrectly specified, the stamp will assume incorrect shapes during the printing process, eventually causing the stamp to break away suddenly from the vacuum bar's vacuum hold. If this happens, the accuracy of printing is ruined for two reasons: first, the printing process is violently disturbed when the vacuum suddenly breaks and the stamp suddenly changes its shape and, second, the stamp segment BC will thereafter assume unknown and variable geometry throughout the remainder of the printing process such that the curvature near the contact line, and therefore the tensile strain induced there in the stamp's raised pattern
15
, will not be constant.
Correct specification of the printing trajectory requires a mathematical understanding of the bending mechanics of the stamp. The analysis given below is based on Euler's famous work on the problem of “the elastica”, as described in articles 262 and 263 of
A Treatise on the Mathematical Theory of Elasticity
, by A. E. H. Love, ISBN: 0486601749, the disclosures of which are incorporated by reference herein in their entirety. Euler's analytical solution predicts the static shape assumed by a uniform thin elastic object, such as the stamp
5
, when forces are applied to its ends only. The problem considered here is more general, because two additional elements are included: (1) the force of gravity on the stamp, and (2) the mid-span forces applied to the stamp by the flat-iron and vacuum bar. Consequently, the solutions obtained here must be numerical rather than analytical, but such numerical solutions are rapidly generated by the computer
225
.
2.7.2 Software: Formulation of the Mathematical Solution
The stamp
5
is considered, like Euler's elastica, to be a thin, elastic object of uniform cross section, as suggested by FIG.
30
. Thus, each labeled point on
FIG. 30
actually represents a line in the y direction. Point Q, which is fixed in space, is the lower edge of the stamp that is attached to the vacuum plate
395
. Point E is the upper edge of the stamp that is attached to the upper stamp clamp
130
along an upper-clamp line
E
parallel to the y axis, which translates via crossed stages
165
and
185
along axes x
1
and z, as previously described. Point P is the axis of rotation of the upper stamp clamp's shaft
630
. Points A and B are the points on the stamp that at time t are nearest, respectively, to the edges of the air-bearing-supported flat-iron
270
; while at the same time points C and D touch the edges of the vacuum bar's wear layer
1025
. As explained above, the flat-iron
270
and the vacuum-bar assembly
280
move in unison along axis x
2
. The vacuum-bar angle θ
CD
and the height of the vacuum bar above z=0, although adjustable, are fixed throughout a trajectory. Hereinafter, we refer to the various segments of the stamp as follows: QA is the “flat segment”, AB is the “air-bearing segment”, BC is the “lower segment”, CD is the “vacuum-bar segment”, and DE is the “upper segment.” By referring to QA as the “flat segment”, we implicitly assume that the flat-iron
270
is heavy enough to force all of segment QA flat onto the receiver
25
. In practice, some or all of segment AB may also be flattened onto the stamp, as discussed presently.
During the printing process, point A (together with points B, C, and D, whose geometry is fixed relative to A) moves along the x
2
axis at a constant rate by virtue of the motor-driven x
2
stage
205
. Each position x
A
of point A, together with coordinates (x
P
,z
P
) of point P (the upper pivot), will be referred to as a “configuration”. That is, a configuration is specified by the three numbers (x
A
,x
P
,z
P
). If a configuration satisfies all the constraints of the problem—the differential equation of the stamp and boundary conditions, then it is considered a “viable configuration”. A printing trajectory is simply an ensemble of viable configurations having monotonically increasing values of x
A
. That is, the printing trajectory is formed by “connecting the dots” between viable configurations—by moving stages (x
1
,z,x
2
) (i.e.,
165
,
185
,
205
) in coordination with each other, with point A moving continually rightward and point P moving continually rightward and downward, so as to perform piecewise linear interpolation between a set of viable configurations. Thus, the objective of the following mathematical analysis is to specify, for each given value of x
A
, the required location (x
P
,z
P
) of pivot P to produce a viable configuration.
The differential equation of the stamp, as well as the various boundary conditions, are specified in the ensuing analysis. Following Euler, we treat the stamp as a bending beam that transmits an internal bending moment, a shear force, and an axial (tangential) force. Referring to
FIG. 31
, we use curvilinear coordinates s and θ, where s is arc length along the stamp, measured rightward from point Q. There is shown in the drawing a differential element of the stamp, of infinitesimal length ds and unit depth normal to the plane of the drawing. The angle θ, measured counterclockwise from horizontal, is the angle of this differential element ds. Thus, θ is a function of s; that is, the stamp hangs in a curve. The weight per unit area of the stamp is denoted w, and so the force w ds acts downward on the differential element.
Also, acting internally on the stamp's cross-sectional face at each end of the differential element, are the forces-per-unit-depth F
x
and F
z
, as well as the internal bending-moment-per-unit-depth M. These internal loads are the resultants of internal, tensile/compressive stresses tangential to the stamp and shear stresses normal to it. To simplify the mathematics, however, the loads F
x
and F
z
are resolved along the x and z axes rather than tangential and normal to the stamp.
Surface stresses acting transverse to the surface of the stamp are also included in this analysis. The stress normal to the face, positive downward, is denoted p (i.e., the normal force per unit depth is p ds); the stress tangential to the surface, positive toward positive s, is denoted f (i.e., the tangential force per unit depth is f ds). For segments BC and DE, p=f=0, because no surface stresses act in these regions.
For the air-bearing segment AB, the tangential stress f is negligible because the air-bearing's coefficient of friction is nearly zero. The normal pressure distribution p
AB
(s) acting in the air-bearing segment AB comprises both a positive (i.e., downward) pressure p
AIR
(s) from the air bearing and a negative (i.e., upward) reaction pressure p
R
(s) from the receiver
25
. The air-pressure profile, shown in
FIG. 24
b
, may assumed to be
where the relationship between p
0
and the weight per unit length σ the flat-iron
270
is given by Equation (1) above. The reaction pressure p
R
(s) may be assumed to be equal and opposite to p
AIR
(s) over the segment AO where, as shown in
FIG. 33
, the stamp
5
is assumed to be in contact with the receiver
25
. Thus
The net pressure in the air-bearing segment AB is therefore
p
AB
(
s
)=
p
AIR
(
s
)−
p
R
(
s
). (4e)
Referring to
FIG. 32
, the normal-pressure distribution p
CD
(s) acting in the vacuum-bar segment CD comprises both the negative (i.e., upward) vacuum pressure, −p
vac
, acting over the area C
1
D
1
of the double-blind groove
1070
in wear layer
1025
, as well as the positive (i.e., downward) reaction pressure p
react
acting over the lower surface
825
(CC
1
and D
1
D) of wear layer
1025
. Thus, assuming that p
react
is a uniform load,
Assuming further that the positive and negative normal forces due to these pressures balance each other,
where, as shown in
FIG. 32
, W
CD
is the width of segment CD, and W
C
1
D
1
is the width of the double-blind groove
1070
. The tangential stress distribution f
CD
(s) acting in the vacuum-bar segment CD is the frictional stress associated with the reaction forces p
react
. Thus
where μ is the coefficient of friction between the stamp
5
and the wear layer
1025
.
Although the stamp is in motion during the printing process, mathematically we assume that the process is quasi-static; that is, that inertia forces are negligible compared to other forces. Therefore, the equations that describe the differential stamp element in
FIG. 31
at any time t during the printing process (i.e., for any configuration) are the equations of static equilibrium:
ΣHorizontal Forces=0 (6a)
ΣVertical Forces=0 (6b)
ΣCounterclockwise moments (about left end of element ds)=0 (6c)
By inspection of
FIG. 31
, ignoring terms of order ds
2
, these equations are:
The relation between moment M and curvature
in a beam is
as derived in many standard engineering texts; for example, in Section 3-5 of
The Analysis of Stress and Deformation
, by George W. Housner and Thad Vreeland, Jr., Library of Congress Catalog Card Number 65-22615 (1975), the disclosures of which are incorporated by reference herein in their entirety. In Equation (8), E is the Young's modulus of the stamp, and I is the area moment of inertia of the stamp's cross section per unit depth in the y direction (see FIG.
30
). Although the stamp
5
may comprise two layers, the back-plane layer
40
is typically so much stiffer than the polymer layer
35
that, to an excellent approximation, E and I may be taken to be the modulus and the moment of inertia of the back-plane layer
40
only.
Introducing Equation (8) into Equation (7c) yields:
The solution for the shape of the stamp, θ(s), is obtained by piecewise (i.e., segment-wise) numerical integration of Equations (7a), (7b) and (9). The numerical solution is obtained by defining the vector u of unknowns as
Then Equations (7a), (7b) and (9) may be written as
Given the stamp's physical properties E, I, and w, and the surface-loading conditions p and f for each segment, Equations (11) describe how θ (as well as F
x
and F
z
) vary as functions of arc-length s throughout the various segments. The surface-loading conditions p and f in the various segments were discussed previously in connection with Equations (4) and (5). To solve for the shape of the stamp in a given configuration, therefore, it remains only to specify the initial values of the variables u
1
at some starting point s
0
, and then to integrate numerically the differential equations (11a) starting at s=s
0
. Numerical integration of these nonlinear differential equations may be carried out using various methods, such as fourth-order Runge-Kutta integration. Such methods are described, for example, in
Numerical Recipes in C
, by William H. Press et al., the disclosures of which are incorporated by reference herein in their entirety. When the right end of a segment is reached, the values of the u
1
obtained at the final point (e.g., point B in segment AB) are used as the initial conditions for integration of the next segment (e.g., segment BC).
The arc-length coordinates of points A, B, C, D, and E, denoted s
A
, s
B
, s
C
, s
D
, and s
E
respectively, must be known a priori for straightforward integration of the differential equations (11a). Of course, because segment QA is flat,
s
A
=x
A
. (12a)
Although the exact arc-lengths of segments AB, BC, and CD are not precisely known a priori, these segments are typically short and the stamp's curvature therein is low. Therefore, to an excellent approximation,
s
B
≈x
A
+W
AB
; (12b)
that is, the arc length s
B
−s
A
(i.e. s
B
−x
A
) of segment AB is assumed to be equal to the width W
AB
of the flat-iron itself (see FIG.
30
). The arc length of segment BC, s
C
−s
B
, is approximately that of the straight line connecting points B and C, thus
s
C
≈s
B
+{square root over ((
x
C
−x
B
)
2
+(
z
C
−z
B
)
2
)}. (12c)
where the differences under the radical in Equation (12) are known by direct measurement of the apparatus. The arc length of segment CD is assumed to be equal to the width W
CD
of the vacuum bar, thus
s
D
≈s
C
+W
CD
. (12d)
Finally,
s
E
≈L, (12e)
where L is the total length L of the stamp, from point Q to point E which is known by direct measurement of the stamp.
As shown in
FIG. 32
, point O, with arc-length coordinate s
0
, is the point where the stamp first departs from the flat condition θ=0, and thus is the starting point for integration of the differential equations (11a). Unfortunately, the location of point O is unknown a priori. However, as suggested by
FIG. 32
, it is typically in the segment AB. For integration of the differential equations, the initial conditions at point O (i.e., the values of the u
1
at s=s
0
) are:
On the right side of Equation (13), the first two components are zero because, by hypothesis, segment AO is completely flat.
Thus there are three unknown parameters, s
0
, F
x0
, and F
z0
, that must be determined for each integration of the differential equations (11a) (i.e., for each configuration). To operate the printing machine
100
, it is necessary, for each configuration, to determine values of these three parameters that satisfy the following three conditions:
1. Stamp passes through point C. To guarantee that the stamp remains attached to the vacuum bar throughout the trajectory, each stamp configuration should pass through the point C (i.e., the trailing edge of the vacuum bar's wear layer
1025
). As shown in
FIG. 34
, the coordinates (x
C
,z
C
) of point C are given in terms of the coordinates (x
V
,z
V
) of the vacuum-bar's center of rotation and the hardware-adjustable rotation angle θ
CD
by
where, in Equation (14a), x
V
has been replaced by its equivalent x
A
+x
VA
, where x
VA
is the constant distance shown in FIG.
30
.
To determine whether a candidate stamp shape θ(s) passes through point C, the stamp's curvilinear coordinates (θ,s) must be converted to Cartesian coordinates (x,z) using the relations dx=cos θds and dz=sin θds. First, s
C
is determined either approximately using Equation (12c) or more accurately as the value for which
Once s
C
is determined, the curve θ(s) is deemed to pass through point C if, to some tolerance,
The integrations in Equations (15a) and (15b) must be done numerically using, for example, the well-known trapezoidal rule, which is described in many standard texts such as
Numerical Recipes in C
, previously cited.
2. Stamp angle at point C equals θ
CD
. To guarantee that the stamp remains attached to the vacuum bar, the stamp's angle at point C, θ
C
, should equal the preset value θ
CD
discussed above in connection with Equations (14). That is,
θ(
s
C
)≡θ
C
=θ
CD
(16)
where s
C
is determined either by Equation (14c) or more accurately by Equation (15a).
3. Stamp rotates freely at point E. The upper stamp clamp
130
, discussed previously in connection with
FIG. 17
, imposes a free-rotation condition at point E. The appropriate mathematical condition is derived from FIG.
35
. As illustrated in
FIG. 35
a
, the ball-bushing assemblies
635
and
640
(e.g., ball bushing
685
and ball-bushing housing
690
), as well as T-bars
645
and
650
, rotate freely about axis P of shaft
630
, like a paddle wheel with two paddles. Because the stamp
5
exerts forces-per-unit-y-length F
xE
, F
zE
, and bending-moment-per-unit-y-length M
E
upon this assembly, the condition for free rotation demands that the sum of the moments exerted by these loads about axis P must be zero. That is, by inspection of
FIG. 35
b
, which is a mathematical representation of the situation:
M
E
+F
xE
R
s
sin θ
E
−F
zE
R
s
cos θ
E
=0. (17a)
Substituting Equation (8) for bending-moment-per-unit-length M
E
yields
which is a relationship between the stamp's angle and its curvature at point E.
To summarize, the three conditions (15b), (16), and (17b) are used to determine the three unknown parameters s
0
, F
x0
, and F
z0
. This determination can only be made by numerical iteration. To describe this process, let β be the vector of the three unknown parameters,
and let the three conditions (15b), (16), and (17) be expressed as the vector condition
T=0, (19)
where T is defined as
As indicated in Equation (20), T is a function of β because the solution θ(s) to the differential Equations (11a), and hence θ
C
, θ
E
, and
depend on the components of β.
To determine the correct value of β—that is, the values of s
0
, F
x0
, and F
z0
that will yield a viable configuration—an initial guess, β
(0)
, is made, the differential Equations (11a) are integrated, and the resulting value of T in Equation (20) is evaluated. In general, the initial guess β
(0)
will not produce the desired result T=0, but the correct value of β may be found from the initial guess (provided the initial guess is close enough to the correct value) by means of the well-known root-finding method known as Newton-Raphson. This entire scenario—seeking a solution satisfying given conditions (that can not be evaluated a priori) by guessing initial parameters, integrating, evaluating the conditions, and then iterating using the Newton-Raphson method—is a technique called the “shooting method”, which is well known in the art of numerical solutions for differential equations. The shooting method, as well as the Newton-Raphson method which it employs, are described more fully, for example, in
Numerical Recipes in C
, previously cited.
In terms of the notation introduced above, the Newton-Raphson iteration step that produces the new approximation for β, β
(n+1)
, from the existing one β
(n)
, is
and the iteration is assumed to have converged when two consecutive iterates are sufficiently close to each other; that is, when
for some small number ε
C
. In Equation (21a),
is the inverse of the Jacobian matrix of partial derivatives
evaluated at β=β
(n)
. These partial derivatives must be approximated numerically by finite differences:
where ε
D
is some small number. The above procedure—applying the shooting method to find β that satisfies the conditions T=0—specifies a “viable stamp configuration” in the sense described previously in connection with FIG.
30
. Thus, the correct shape of the stamp, θ(s), is determined for one particular position of the carriage
125
; that is, for one particular value of s
A
=x
A
. As stated above, the objective of the mathematical analysis above is to specify, for each given value of x
A
, the required location (x
P
, z
P
) of pivot P to produce a viable configuration. The sought-after Cartesian coordinates (x
P
, z
P
) may be determined from the stamp shape θ(s), using dx=cos θds and dz=sin θds, as follows:
x
P
=x
E
+R
s
cos θ
E
(23c)
z
P
=z
E
+R
s
sin θ
E
, (23d)
where L is the total length of the stamp (from point Q to point E), and where Equations (23c) and (23d) make use of the geometry drawn in
FIG. 35
b
. The integrations in Equations (23a) and (23b) are performed using the well-known trapezoidal rule, previously cited in connection with Equations (15).
To generate a printing trajectory, the above mathematical solution must be developed numerous times, once for each configuration, where a configuration is characterized by its value of x
A
(the position of the flat-iron
270
). The array of configurations comprising a printing trajectory has the property that x
A
increases monotonically throughout the array. Let this array of x
A
values be denoted {x
A1
, x
A2
, x
A3
, . . . , x
AN
}. Once the first viable configuration has been found, for the first value x
A1
, the associated value of β—upon which the Newton-Raphson method converged—is used as the initial guess for the next configuration x
A2
. As long as x
A2
−x
A1
is not too large, this strategy guarantees that the Newton-Raphson method will easily converge for the x
A2
configuration. The same strategy is continued throughout the array: the value of β found for configuration x
Ai
is used as the initial guess for configuration x
A,i+1
, where i=1, . . . N−1.
2.7.3 Software: Simplification of the Mathematical Solution
Experience with the prototype of the invention has shown that the goal of the mathematical analysis above—providing a printing trajectory in which the stamp
5
remains attached to the vacuum-bar's wear layer
1025
throughout the trajectory—can be achieved by a simpler embodiment of the mathematical solution above. Three simplifying assumptions are made:
1. Known location of point O. First, assume that the contact point O is coincident with B (i.e., all of the air-bearing segment AB is flat). Then
s
0
=s
B
. (24)
Integration of the differential equations can thus proceed from a known starting point, point B.
2. Known conditions at point B. Further assume that the vertical component of stamp force at B is zero, and that the curvature of the stamp at B may be specified a priori to be a known value κ
B
. That is,
F
zB
=0 (25)
and
3. Stamp segment CD lies flat on vacuum bar. Finally, assume that over the vacuum-bar segment CD, the stamp is not curved, but rather lies flat on the surface of the vacuum-bar's wear layer
1025
; that is,
θ(
s
)=θ
CD
=constant,
s
C
≦s≦s
D
. (27)
The first two simplifying assumptions above, embodied in Equations (24) through (26), are the most essential ones. They imply that the initial conditions for integrating the differential equations (11b) are, in place of Equation (13),
That is, because κ
B
is known, there is now only one unknown initial parameter, F
xB
, instead of three as there were in Equation (13). Thus we may impose only one of the three conditions (15b), (16), and (17). We choose to impose (17b), a natural boundary condition that is essential. The rest of the analysis proceeds as described previously, excepting that the three-component vectors β and T, previously defined by Equations (18) and (20), here reduce to scalars β and T:
β≡F
xB
(29a)
and the Newton-Raphson iteration (21) reduces to a simple, scalar Newton iteration:
Thus the advantage of the first two simplifying assumptions above, embodied in Equations (24) through (26), is that the guesswork involved in initiating the Newton iteration, as well as the subsequent calculation, is minimized. Nevertheless, as is typical with finding the roots of nonlinear equations, if the initial guess for F
xB
is too far away from the solution, the Newton iteration will not converge, or worse, will converge to a bizarre stamp shape θ(s) that is physically impossible. Examples of such bizarre shapes are shown in FIG.
36
. Blindly commanding the servomotors
170
,
190
, and
210
to execute a trajectory composed of such bizarre stamp shapes is extremely unwise. Therefore, in the preferred embodiment, the various stamp shapes θ(s) in a trajectory are first plotted graphically, thereby simulating the trajectory visually before executing it in hardware.
The third simplifying assumption above, embodied in Equation (27), allows the differential equations (11a) to be integrated analytically in the vacuum bar segment CD. The scalar form of these equations is Equations (7). Integrating the latter:
By virtue of assumption (27), the stamp angle θ between points C and D is a constant, θ
CD
, and s
D
−s
C
=W
CD
. Thus Equations (31) become
In connection with Equations (5), the positive and negative normal forces due to pressure p
CD
(s) were assumed to balance each other, implying
Further, with the help of Equations (5b) and (5c),
Thus, Equations (32a) and (32b) reduce to
F
xD
−F
xC
=−μW
C
1
D
1
p
vac
cos θ
CD
(34a)
F
zD
−F
zC
=−μW
C
1
D
1
p
vac
sin θ
CD
+wW
CD
(34b)
Moreover, because segment CD is fairly short, the integrals in (32c) may be approximated by the trapezoidal rule, using only the values of F
x
(s) and F
z
(s) at the endpoints C and D. Using the moment-curvature relation (8), Equation (32c) reduces to
which is a statement of how the stamp's curvature at D relates to that at C. We also recall Equation (27), which implies
θ
D
=θ
C
≡θ
CD
. (34d)
By virtue of Equations (34), all four components of u, defined in Equation (10) as θ,
F
x
, and F
z
, are known at point D as soon as they are known at C. Therefore, no integration of the differential equations (11a) is required in segment CD; rather, the difference conditions (34) are applied.
An additional advantage of the simplifications above is that only two stamp segments, BC and DE, remain over which the differential equations (11a) must be numerically integrated. Because p=f=0 for both these segments, the third and fourth components of Equations (11a) may be written
whence
u
3
=F
x
=constant=
F
xB
(36a)
u
4
=F
z
=w
(
s−s
B
) (36b)
where Equation (36b) uses the simplifying assumption (25). Thus Equations (10) and (11b) may be replaced by the simpler equations
That is, the dimensionality of the system of differential equations (11a) has been reduced from four to two, thereby eliminating unnecessary calculations.
Of course, the simplifications (24) through (27) above are approximations, and reality differs from the assumptions. First, for a prototype of the invention, the stamp under segment AB is seen to violate slightly the simplifying assumption, stated in Equation (24), that segment AB is entirely flat. That is, in reality the stamp's stiffness often raises point B slightly off the surface of the receiver
25
, and the contact front is actually somewhere in the segment AB, as assumed by the unsimplified, full theory (see FIG.
33
). Moreover, because the simplified theory does not impose the conditions (15b) and (16), there is no guarantee that the various stamp configurations throughout the trajectory will accommodate the fixed geometrical relationship between points B, C, and D. In other words, there is no guarantee that the stamp will remain attached to the vacuum bar, because the stamp's stiffness may break the vacuum seal if the stamp's shape θ(s) is too far wrong. However, because the simplified solution does insist, via Equation (26), that the theoretical curvature of the stamp at B, κ
B
, remains constant throughout the trajectory, the shape of the stamp near B tends to remain constant from one configuration to another. Experience has shown that, because C is relatively near B for the prototype system, the simplified solution works—the stamp does remain attached to the vacuum bar throughout the printing trajectory, even though only a partial vacuum (e.g., p
vac
=0.2 to 0.4 atmospheres) is used, and even though the vacuum bar is explicitly adjusted to “kiss” the stamp's shape only at the beginning of the trajectory.
The “kissing” adjustment proceeds as follows. After a printing trajectory is calculated for the first time using the simplified solution given in Equations (24)-(26), the stamp
5
and the carriage
125
are moved to their calculated starting positions with the vacuum-bar's micrometer heads
915
and
920
initially adjusted all the way up, to insure that the vacuum bar does not initially hit the stamp. Then the height z
C
and the angle θ
CD
of the vacuum bar are adjusted so that, at the starting position, the vacuum bar is tangent to the natural curve of the stamp. The height z
C
is adjusted via the micrometer heads
915
and
920
; in the prototype, θ
CD
is adjusted by loosening the nylon-tipped set screws
1000
and
1005
, rotating the vacuum bar manually until its lower surface is tangent to the stamp, and then re-tightening the set screws. Once this is done, these adjustments never have to be altered unless a new trajectory is computed—whenever the vacuum-bar's vacuum suction is turned on at the beginning of the print cycle, the suction naturally “grabs” the stamp as desired. Thus printing and peeling can be done time after time, as they would be in a manufacturing environment.
A typical set of parameters used in the prototype was as follows:
E
=147×10
9
N/m
2
I
=2.8125×10
−13
m
3
w
=19.55
N/m
2
κ
B
=8.333
m
−1
s
C
−s
B
=0.048
m
s
E
−s
B
=0.605
m
(at start of print process)
W
CD
=0.016 m (38a)
W
C
1
D
1
=0.00159 m
μ=0.10
p
vac
=20,000
N/m
2
R
s
=0.037 m
ε
C
=0.001
ε
D
=0.0001 (38a)
For this set of parameters, the associated solution F
xB
upon which the Newton iteration (30) converges is
F
xB
=1.67
N/m.
(38b)
Thus at the beginning of the trajectory, the stamp near the contact line B is under slight tension (i.e., F
xB
is positive). Further computation shows that this tension at B gradually reduces as the printing process proceeds (i.e., as B moves rightward), becoming, for example, F
xB
=1.27 N/m when 57 percent the stamp is flattened. These levels of tension have negligible effect on the raised pattern
15
of the stamp's polymer layer
35
. For example, the thickness h of the stamp implied by the conditions given in Equation (38a) is h={square root over (12I)}=150×10
−6
m, so the associated tensile strain ε for the level of tension in Equation (38b) is ε=F
xB
/hE=0.076×10
−6
(i.e., 76 parts per billion), which is truly negligible.
The computed stamp shape associated with the solution associated with Equations (38) is plotted in
FIG. 36
a
. For Newton's method to converge on this desired (i.e., physically realizable) answer, the initial guess for F
xB
, denoted (F
xB
)
guess
must be in the range
0.0162≦(
F
xB
)
guess
≦0.0171 (39)
Otherwise, undesired solutions will be converged upon instead. For example,
FIGS. 37
b
and
37
c
show the undesired solutions converged upon when (F
xB
)
guess
=0.0161 and (F
xB
)
guess
=0.0172 respectively, these values of (F
xB
)
guess
being just outside the narrow range given in Equation (39b). Thus, finding a desirable solution for F
xB
the first time can be tedious—which is why Equations (38) and (39) are given herein. However, once a desirable solution is found for some set of parameters, other solutions (for other sets of parameters) may be found by perturbation, wherein one of the parameters is perturbed, and (F
xB
)
guess
is taken to be the known solution F
xB
for the unperturbed set of parameters. This very process is used in developing a print trajectory, wherein the only parameter that changes from configuration to configuration is the stamp length s
E−s
B
ahead of the flat-iron
270
.
3. Results: Printed Accuracy
One of the objectives of the printing machine
100
is to obtain the greatest possible printing accuracy. Results shown below demonstrate the level of accuracy that has been achieved with a prototype of this invention. For reasons discussed below, it is anticipated that even greater accuracy is possible by using superior raw materials for the stamp's back-plane layer
40
.
Perfect accuracy would be attained if the location of each printed feature on the receiver
25
were identical to that of the corresponding feature on a reference substrate. This reference substrate may be the stamp
5
; alternatively, it may be the “master” from which the stamp is made. In reality, feature locations on the receiver
25
differ slightly, in both the x and y directions, from those on the reference substrate. Experimentally, these feature-placement errors may be measured using a high-precision, coordinate measuring machine such as a Nikon-3i, which is well known in the art of semiconductor lithography. Results of such measurements may be presented as an array of arrows, such as arrows
1100
,
1105
, and
1110
on
FIG. 37
a
, that compare various feature locations on a printed receiver to their counterparts on the reference substrate. Such plots, well known in the art of semiconductor lithography, are described, for example, in J. D. Armitage Jr. and J. P. Kirk,
Proc. SPIE
, 921 (1988). The arrows' tails show the locations of features (such as
1115
,
1120
, and
1125
) on the reference substrate. If drawn to scale as in
FIG. 37
a
, the arrows' heads show the location of the corresponding features (such as
1130
,
1135
, and
1140
) on the printed receiver. Thus the magnitude and direction of the arrows represent feature-placement error. In practice, however, the errors are so small compared to the size of the array that, in order to see the arrows graphically, it is necessary to exaggerate their size greatly, as suggested in
FIG. 37
b
. In such a plot, the arrows only give the direction of the feature-placement error; the magnitude is given by comparison to a reference arrow
1145
, which is annotated by the length L
1
that the reference arrow represents. In contrast, the actual scale of the substrate is given by the reference scale
1150
, which is delimited by two tick marks and annotated with the dimension L
2
between the ticks.
If the stamp itself is used as the reference substrate, then measured errors are attributable solely to the printing process. However, if the reference substrate is the “master” from which the stamp is made, then measured errors are attributable to both the stamp-making process and the printing process together. In all of the results below, the latter technique was used, because the stamp was too large to measure in the Nikon-3i machine. Thus, stamp-making and printing errors are inextricably combined in the results below.
The prototype of the invention has been used to make a number of prototype prints of size 381×381 mm. In these prototypical prints, the receiver
25
is a piece of glass sputter-coated with a thin film of titanium/gold, the ink
20
is hexadecanethiol, and the stamp's polymer layer
35
is polydimethylsiloxane (PDMS). This combination of materials for receiver, ink and polymer layer is well-known in the art, of the type having been pioneered by Kumar and Whitesides, previously cited. The prototype stamps' polymer layer
35
is 750 μm thick; the back-plane layer
40
, made of Invar 36, is 150 μm thick.
For each prototypical print, error measurements such as those described above have been made over an array of approximately 2200 features regularly distributed over a 271-by-203-mm rectangle that is centered on the print. The results of such measurements on eight such prints, made from two different stamps designated A and B, are shown as
FIGS. 39 through 46
. Occasionally there are missing data points (i.e., missing arrows) on these plots. This missing data has no significance except that the Nikon-3i could not optically interpret the features there. The data is summarized by statistics given in Table 1. For each of the prototypical prints listed, the print velocityν, defined above in connection with
FIG. 24
a
, was 10 mm/s. However, other prints having ν=40 mm/s were made and measured. At this higher print speed, the placement accuracy for a given stamp and print direction is nearly identical to that shown in Table 1.
TABLE 1
|
|
Measured Accuracy of Printing with a Prototype of the Invention
|
Std.
|
Deviations
|
of Placement
|
Print
Vacuum
FIG.
Receiver
Print
Error (μm)
|
Stamp
Table
Bar?
No.
No.
Direction
σ
x
σy
|
|
A
Al
No
38
1
Forward
2.58
2.07
|
Ground
39
2
Forward
2.46
1.36
|
Al
40
3
Forward
1.61
0.65
|
Granite
Yes
41
4
Forward
1.70
0.69
|
42
5
Reverse
1.42
0.80
|
B
Granite
Yes
43
6
Reverse
0.99
0.93
|
44
7
Reverse
1.08
0.97
|
45
8
Forward
1.78
2.08
|
|
Column 2 of Table 1 specifies the type of print table used, thereby giving a qualitative assessment of its flatness, the importance of which has previously been discussed in connection with FIG.
9
. The print table for receiver No. 1 was a piece of cast aluminum, commonly known as “jig plate”. The improved, flatter print table used for receivers No. 2 and No. 3 was the same piece of aluminum after surface grinding. For receivers No. 4 through No. 8, a lapped granite print table, as described above for the preferred embodiment, was used to obtain the ultimate in flatness. Column 3 of Table 1 specifies whether or not the vacuum bar, described above in connection with
FIGS. 26 through 30
, was used during printing.
Column 6 of Table 1 specifies the “Print Direction”, the meaning of which is explained by FIG.
43
. As shown, let the two x-facing ends of the stamp
5
be denoted S and T respectively. Receivers 1, 2, 3, 4, and 8 in Table 1 were printed with their respective stamps mounted in the printing machine as shown in
FIG. 43
a
—with the S end (at left in the drawing) mounted on the lower stamp clamp's vacuum plate
395
, and the T end (at right in the drawing) mounted in the stamp carrier
680
. This configuration is designated as the “forward” print direction. Receiver
5
,
6
, and
7
in Table 1 were printed with the stamps mounted in the reverse direction, as shown in
FIG. 43
b
—with the T and S ends reversed. This reversal was possible because the stamps were made symmetrically left to right in the prototype, with appropriate mounting holes in both ends, and the appropriate extra length of back-plane (only required on the top end) on both ends. On
FIGS. 39-46
, the orientation of the printed pattern is presented as invariant from drawing to drawing, regardless of print direction. However, as shown on the figures, the direction of printing (motion of flat-iron
270
and vacuum bar
820
) is left to right (forward) on
FIGS. 39
,
40
,
41
,
42
and
46
, whereas it is right to left (reverse) on
FIGS. 43
,
44
, and
45
. Of course, in reality the motion and operation of the printing machine is always the same. It is only the stamp that is turned around.
Columns 7 and 8 of Table 1 give standard deviations of the placement-error measurements over the entire array of features. As is common in the art of such error measurements (see J. Kirk, previously cited), the rigid-body (x,y) displacements of the array as a whole, the rigid-body rotation of the array, and a uniform “magnification” error have all been statistically removed from the plots and from the standard deviations reported in Table 1, because these components of error, being easily compensated, are of little interest in practice. Inasmuch as the rigid-body components of error have been removed, the random variables whose standard deviations are reported in Table 1 have zero mean, such that “three sigma error bars” are simply ±3σ.
For the best print, receiver No. 6 (FIG.
43
), these ±3σ error bars are slightly less than ±3 μm. To our knowledge, this is the most accurate, large-scale mechanical printing known in the art. Moreover, even greater accuracy should be attainable with improvements in stamp-making, as described below.
Furthermore, the error statistics quoted in Table 1 include a significant component of error called differential magnification (see J. Kirk, previously cited). If this component is removed for receiver No. 6, for example, the statistics are reduced to σ
x
=0.83 μm and σ
y
=0.61 μm. For some applications, the latter statistics are more relevant than those in Table 1. For example, when a pattern printed with the printing machine
100
is subsequently overlaid with another layer by means of standard step-and-repeat optical lithography, it has been found that the ±3σ overlay errors closely track the smaller numbers, such as those above, where the differential magnification error has been removed. The reason for this is that the optical stepper measures magnification anisotropy via three or more fiducials, and corrects for the differentiation magnification error by stepping anisotropically in the two directions. This correction works best if the stepper's shot size is small; worse if it is large. For the prototype prints (271×203 mm), the differential magnification error was found to be virtually eliminated from the overlay error when the optical stepper has a shot size of 32×32 mm.
Based on experience with several hundred prints, four conclusions, illustrated by
FIGS. 39-46
, may be drawn:
1. The flatness of the print table is significant. This conclusion is illustrated by comparing
FIGS. 39 and 40
, which show errors for receiver
1
and
2
respectively. The only difference between these two cases is the flatness of the print table. As shown by the associated statistics in Table 1, the flatter, surface-ground aluminum print table, although it produced only a slight improvement in σ
x
, produced a significant, 34-percent improvement in σ
y
(2.07 μm reduced to 1.36 μm). The granite table appears to have produced no further improvement vis-à-vis the ground aluminum, as shown by a comparison of
FIGS. 41 and 42
(as well as the associated statistics), which are quite similar.
2. The stamp-control system, embodied in the vacuum bar, is important. This conclusion is illustrated by comparing
FIGS. 40 and 41
, where the only difference is the absence or presence of the vacuum bar. As shown by the associated statistics in Table 1, the vacuum bar produced a 35-percent improvement in σ
x
(2.46 μm reduced to 1.61 μm), and a 52-percent improvement in σ
y
(1.36 μm reduced to 0.65 μm).
3. With fixed conditions, printing repeatedly with the same stamp in the same direction always yields nearly identical results. This conclusion is illustrated by comparing
FIGS. 44 and 45
, which were both obtained by printing with stamp B in the “Reverse” direction. Notice that the “fingerprint” of errors—the shape of the vector map—is very similar for these two plots, and the statistics differ by only a few percent. Such repeatability (and often better) was observed consistently over dozens of prints, even for prints made months apart with the same stamp.
4. With fixed conditions, printing with different stamps, or even with the same stamp in opposite directions, yields significantly different results. This conclusion is illustrated by the following paired comparisons:
FIG. 41
vs.
42
(same stamp A, opposite directions);
FIG. 44
vs.
45
(same stamp B, opposite directions);
FIG. 41
vs.
45
(different stamps, both printed in the forward direction); and
FIG. 42
vs.
43
(different stamps, both printed in the reverse direction). In none of these cases do the error maps compare in any simple way. For example, when the print direction is reversed, the fingerprint is not simply rotated 180 degrees, as it would be if the errors were attributable to the printing process alone.
The most likely explanation for observations Nos. 3 and 4 above is that, for the prototype system, the sheet-metal material (Invar 36, 150-μm thick) from which the stamp's back-plane layer
40
is made is not perfectly planar. Rather, the metal has slight bumps caused by anomalies in its manufacture. With prototypical Invar sheets resting on a granite surface plate, bump amplitudes up to 2.8 mm and wavelengths from 90 to 200 mm have been measured. During printing, these bumps are “ironed out” by the flat-iron
270
, thereby causing lateral displacement of material points on the surface of the polymer layer, since the polymer follows the much-stiffer metal.
This mechanism easily explains the observation that different stamps have different error fingerprints, inasmuch as each stamp's back-plane is a unique piece of sheet metal having a unique distribution of bumps that are “ironed out” in unique ways. This mechanism also explains the excellent print-to-print repeatability when a given stamp is repeatedly printed in the same direction—apparently, the bumps “iron out” the same way every time. Less obviously, this mechanism also explains the observation that, with the same stamp, the error fingerprint depends on print direction. The reason for this is shown in FIG.
46
. Suppose there is a single bump
1155
located near the T end of the stamp. When printing is performed from S to T, as in
FIG. 46
a
, the bump
1155
will not affect printed errors except near the right end of the receiver
25
, because the bump is not encountered by the flat-iron
270
until late in the printing process. However, if the same stamp is printed in the reverse direction, from T to S, as shown in
FIG. 46
b
, bump
1155
is flattened by flat-iron
270
nearly at the outset of the printing process, and thus its effect can spread throughout the printed pattern on the receiver. With many bumps participating and interacting on a real stamp, it can be imagined that complex patterns of errors, such as those observed in
FIGS. 42-46
, may result, and may differ markedly for the two print directions.
For the above reasons, the accuracy of printing with the current invention may be improved, beyond that given for receiver No. 6 in Table 1, by improving the quality of the back-plane material
40
. The prototype stamps were made with “off-the-shelf” material. However, as is well known in the art of sheet-metal manufacture, post-processing called “leveling” may be applied to reduce the bumpiness of metal sheets. The basic idea is to pull the material beyond its yield point in order to stretch out the extra arc length of material in the bumps. Various types of this procedure are described, for example, in
The Metals Handbook, Ninth Edition, Volume
14
: Forming and Forging
, ASM International, S. L. Semiatin, Joseph R. Davis, et al., editors (1988), the disclosures of which are incorporated by reference herein in their entirety.
Although not mentioned in
The Metals Handbook
, a non-dimensional measure of bumpiness used by the sheet-metal industry is
where Δs is the extra arc length in a sinusoidal bump of amplitude A and wavelength λ. This extra arc length may easily be shown mathematically to be related to the other parameters by
By this measure, the off-the-shelf material referred to above—typical of that from which prototype stamps A and B were made—has worst-case bumps having I=10 to I=50. In contrast, industrial leveling of sheet-metal material is available to reduce such worst-case bumps by an order of magnitude, to I=1. Thus, although the current invention already has been demonstrated to yield the most accurate large-scale mechanical printing known in the art, further improvement in printed accuracy is available via a well-known industrial process.
Alternatively, the invention may be used with other back-plane materials, such as thin sheets of glass or ceramic, whose inherent flatness may be superior to sheet metal. Although the use of such alternative materials may require slight changes in the above detailed description, these changes would not depart from the spirit and scope of the invention.
It is to be understood that all physical quantities disclosed herein, unless explicitly indicated otherwise, are not to be construed as exactly equal to the quantity disclosed, but rather about equal to the quantity disclosed. Further, the mere absence of a qualifier such as “about” or the like, is not to be construed as an explicit indication that any such disclosed physical quantity is an exact quantity, irrespective of whether such qualifiers are used with respect to any other physical quantities disclosed herein.
While preferred embodiments have been shown and described, various modifications and substitutions may be made thereto without departing from the spirit and scope of the invention. Accordingly, it is to be understood that the present invention has been described by way of illustration only, and such illustrations and embodiments as have been disclosed herein are not to be construed as limiting to the claims.
Claims
- 1. A printing apparatus, comprising:a print surface lying in a print plane defined by an imaginary x-axis and y-axis, the print surface having an outward normal pointing in the positive direction along an imaginary z-axis, such that the x-axis, y-axis, and z-axis are substantially orthogonal to one another; a lower stamp clamp disposed adjacent to the negative-x edge of the print surface; an upper stamp clamp, moveable in two dimensions in a trajectory plane defined by the x-axis and z-axis; a stamp comprising a flexible material, the stamp having a first end attached to the lower stamp clamp and a second end attached to the upper stamp clamp, such that a cross section of the stamp parallel to the trajectory plane forms an arc extending from an origin point Q on the lower stamp clamp having (x, z) coordinates (0, 0) to point E on the upper stamp clamp, this arc being described by the mathematical function θ(s), where s is the curvilinear distance along the arc measured from point Q, and θ is the angle between the print plane and an imaginary line, the imaginary line being tangent to the cross section of the stamp at s; and a stamp-control system movable along the x-axis, wherein, during a print operation, the upper stamp clamp is moved in a trajectory comprising a plurality of xz positions of the upper stamp clamp that blend into a substantially continuous motion, the trajectory being effective in laying the stamp down smoothly and flat upon the print surface in a manner such that a moving contact front between the stamp and the print surface is created, the contact front being disposed substantially along a line characterized by a contact-front coordinate s0≡x0 that increases as the trajectory progresses, the trajectory also being effective in causing the curvature ⅆθⅆsof the stamp at or near the contact front to be substantially constant throughout the motion; andwherein, throughout the trajectory, each xz position of the upper stamp clamp is a function of the displacement xC of the stamp-control system along the x-axis; the trajectory being effective in laying the stamp down upon the print surface such that the stamp is in continuous contact with a contact surface of the stamp-control system throughout the trajectory, the location of the contact surface being characterized by an arc-length coordinate sC that increases as the trajectory progresses.
- 2. The apparatus of claim 1 further comprising a print-force-application system effective in pressing the stamp against the print surface, and defining an approximate contact front disposed substantially along a line B parallel to the y-axis in the xy plane, the line lB intersecting the trajectory plane at (x, z)=(xB, 0), the approximate-contact-front x-coordinate xB increasing as the trajectory progresses and being substantially equal, at any stage of the trajectory, to the arc-length coordinate sB of point B, inasmuch as the arc of the stamp is assumed to be substantially flat over the segment from point Q to point B.
- 3. The apparatus of claim 2 wherein the upper stamp clamp is pivoted about a pivot line lP parallel to the y axis and intersecting the xz plane at point P having coordinates xP and zP; the stamp attaching to the upper stamp clamp along an upper-clamp line lP parallel to the y axis and intersecting the xz plane at point E having coordinates xE and zE; the upper-clamp line lE being disposed on the upper stamp clamp at a radius RS from the pivot line lP, such that the total arc length sE from the lower stamp clamp to the line lP is sE&Quadbond;L, where L is the known, free length of the stamp; and wherein the stamp attaches to the upper-clamp line lE at an angle θEθ(L).
- 4. The apparatus of claim 3 wherein the trajectory comprises a plurality of configurations, each configuration described by the coordinate sB&Quadbond;xB of the approximate contact front and by corresponding coordinates xP, zP of the pivot line given by the equationsxP=xE+Rs cos θE zP=zE+Rs sin θE where xE=∫0Lcos θ(s)ⅆs and zE=∫0Lsin θ(s)ⅆs,and where the mathematical function θ(s)describing the shape of the arc for a given configuration is assumed to beθ(s)=0 for 0≦s≦sB, whereas for s>sB, θ(s) is determined by solution of the differential equations ⅆuⅆs=F(u),the lower-end boundary conditions uB≡{u1Bu2B}≡{θBⅆθⅆs|B}={0κB},and the upper-end boundary condition T(β)≡EIⅆθⅆs|E+FXBRSsin θE-w(s-sB)RScos θE=0,whereinu≡{u1u2}≡{θⅆθⅆs},F(u)≡{u2FXBEIsin u1-w(s-sB)EIcos u1},κB is a specified curvature at point B, the parameter β≡FXB, unknown a priori, is the internal x-directed force acting on the stamp's cross section at s=sB per unit depth of the stamp in the y direction, E is Young's modulus of the stamp, I is the area moment of inertia of the stamp's cross section per unit depth in the y-direction, and w is the weight per unit area of the stamp; andwherein for each configuration the solution for xP and zP is derived by means of the “shooting method”, whereby an initial value β(0) of β is guessed, the differential equations are solved to yield T(β(0)) and [∂T∂β]β=β(0)-1,Newton iteration β(n+1)=β(n)-[∂T∂β]β=β(n)-1T(β(n))is applied to obtain a refined value β(1) of the unknown parameter β, whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary condition T(β)=0 is achieved to within some tolerance.
- 5. The apparatus of claim 2 wherein the print-force-application system comprises a flat-iron.
- 6. The apparatus of claim 2 wherein the stamp-control system is disposed along a line lC parallel to the y-axis, line lC intersecting the trajectory plane at point C having coordinates xC and zC, where zC is a fixed, positive z-coordinate during any one printing operation, whereas xC increases as the trajectory progresses, in coordination with the contact-front coordinate x0.
- 7. The apparatus of claim 6 wherein the contact surface of the stamp-control system is a plane delimited in the x direction by two lines lC and lD separated by a fixed distance WCD, these lines being parallel to the y-axis and intersecting the trajectory plane at points C and D respectively, these points having coordinates (xC, zC) and (xD, zD) respectively, such that the contact surface is defined by the three parameters (xC, zC, θCD), where θCD≡tan-1(zD-zCxD-xC)is the angle between the contact surface and the print plane, and such that the stamp angle β(s) between arc-length coordinates s=sC and s=sD is substantially equal to θCD; that is,θ(s)≈θCD for sC≦s≦sD.
- 8. The apparatus of claim 7 wherein the upper stamp clamp is pivoted about a pivot line lP parallel to the y axis and intersecting the xz plane at point P having coordinates xP and zP; the stamp attaching to the upper stamp clamp along an upper-clamp line lE parallel to the y axis and intersecting the xz plane at point E having coordinates xE and zE; the upper-clamp line lE being disposed on the upper stamp clamp at a radius RS from the pivot line lP, such that the total arc length sE from the lower stamp clamp to the line lE is sE&Quadbond;L, where L is the known, free length of the stamp; and wherein the stamp attaches to the upper-clamp line lE at an angle θE≡θ(L).
- 9. The apparatus of claim 8 wherein the trajectory comprises a plurality of configurations, each configuration described by the coordinate s0&Quadbond;x0 of the contact front and by corresponding coordinates xP, zP of the pivot line given by the equations:xP=xE+Rs cos θE zP=zE+Rs sin θE where xE=∫0Lcos θ(s)ⅆs and zE=∫0Lsin θ(s)ⅆs,and where the mathematical function θ(s) describing the shape of the arc for a given configuration is assumed to beθ(s)=0 for 0≦s≦s0, whereas for s>s0, θ(s) is determined by solution of the differential equations ⅆuⅆs=F(u),the lower-end boundary conditions u0≡{u10u20u30u40}≡{θ0ⅆθⅆs&LeftBracketingBar;0FX0FZ0}={00FX0FZ0},and the auxiliary boundary conditionsT(β)=0, wherein u0≡{u1u2u3u4}≡{θ0ⅆθⅆsFx(s)Fz(s)},F(u)≡{u2u3EIsin u1-u4EIcos u1-p(s)sin u1-f(s)cos u1w+p(s)cos u1-f(s)sin u1},T(β)≡{zC-∫0sCsin θ(s) ⅆsθC-θCDEIⅆθⅆs&LeftBracketingBar;E+FXERSsin θE-FZERScos θE},and wherein FX(s) and FZ(s) are functions of a describing the internal x-directed and z-directed forces acting on the stamp's cross section at a per unit depth of the stamp in they direction, FXE≡FX(sE), FZE≡FZ(sE), β is a vector of parameters that are unknown a priori, β={s0FX0FZ0},s0 is the aforementioned arc-length coordinate of the contact front, FX0≡FX(0), FZ0≡FZ(0), E is Young's modulus of the stamp, I is the area moment of inertia of the stamp's cross section per unit depth in the y-direction, w is the weight per unit area of the stamp, p(s) and f(s) are functions of s describing forces applied normal to the stamp and tangential to the stamp respectively by the print-force-application system, the stamp-control system and the print surface, sC is the value of arc-length coordinate sat point C, θC≡θ(sC) is the angle of the arc at point C, and θCD is the aforementioned angle of the stamp-control system's contact surface; andwherein for each configuration the solution for xP and zP is derived by means of the “shooting method”, whereby an initial value β(0) of β is guessed, the differential equations are solved to yield T(β(0)) and [∂T∂β]β=β(0)-1,Newton-Raphson iteration β(n+1)=β(n)-[∂T∂β]β=β(n)-1T(β(n))is applied to obtain a refined vector β(1), whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary conditions T(β)=0 are achieved to within some tolerance.
- 10. The apparatus of claim 8 wherein the trajectory comprises a plurality of configurations, each configuration described by the coordinate sB&Quadbond;xB of the approximate contact front and by corresponding coordinates xP, zP of the pivot line given by the equationsxP=xE+Rs cos θE zP=zE+Rs sin θE, where xE=∫0Lcos θ(s)ⅆs and zE=∫0Lsin θ(s)ⅆs,and where the mathematical function θ(s) describing the shape of the arc for a given configuration is assumed to beθ(s)=0 for 0≦s≦sB, whereas for s>sB, θ(s) is determined in stamp segments OC and DE by solution of the differential equations ⅆuⅆs=F(u),the lower-end boundary conditions uB≡{u1Bu2B}≡{θBⅆθⅆs&LeftBracketingBar;B}={0κB},and the upper-end boundary condition T(β)≡EIⅆθⅆs&LeftBracketingBar;E+FXERSsin θE-FZERScos θE=0,whereinu≡{u1u2}≡{θⅆθⅆs},F(u)≡{u2Fx(s)EIsin u1-Fz(s)EIcos u1},κB is a specified curvature at point B, E is Young's modulus of the stamp, I is the area moment of inertia of the stamp's cross section per unit depth in the y-direction, w is the weight per unit area of the stamp, Fx(s) and Fz(s) are the x-directed and z-directed stamp forces per unit length of stamp in the y direction, given by Fx(s)={Fx0, 0≤s≤sCFx0+Δ Fx,sD≤s≤sE,andFz(s)={0, 0≤s≤s0w(s-s0), s0≤s≤sCw(s-s0)+Δ Fz,sD≤s≤sE,in which Fx0≡Fx(s0)≡β is a parameter that is unknown a priori, and the differences ΔFx and ΔFz are respectively the differencesΔFx≡Fx(sD)−Fx(sC) ΔFz≡Fz(sD)−Fz(sC) that occur across stamp segment CD where the stamp-control system contacts the stamp, the values of which differences, along with the value of the difference Δ κ≡ⅆθⅆs&LeftBracketingBar;D-ⅆθⅆs&RightBracketingBar;C,may be calculated from the three equations of static equilibrium for the stamp under the action of forces applied to the stamp by the stamp-control system, these three differences together with θD=θC permitting numerical integration for stamp segment DE to proceed immediately from the numerical-integration result obtained at the final point C in stamp segment OC; andwherein for each configuration the solution for xP and zP is derived by means of the “shooting method”, whereby an initial value β(0) of β is guessed, the differential equations are solved to yield T(β(0)) and [∂T∂β]β=β(0)-1,Newton iteration β(n+1)=β(n)-[∂T∂β]β=β(n)-1T(β(n))is applied to obtain a refined vector β(1), whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary conditions T(β)=0 are achieved to within some tolerance.
- 11. The apparatus of claim 1 wherein the stamp-control system is disposed along a line lC parallel to the y-axis, line lC intersecting the trajectory plane at point C having coordinates xC and zC, where zC is a fixed, positive z-coordinate during any one printing operation, whereas xC increases as the trajectory progresses, in coordination with the contact-front coordinate x0.
- 12. The apparatus of claim 11 wherein the contact surface of the stamp-control system is a plane delimited in the x direction by two lines 1C and lD separated by a fixed distance WCD, these lines being parallel to the y-axis and intersecting the trajectory plane at points C and D respectively, these points having coordinates (xC, zC) and (xD, zD) respectively, such that the contact surface is defined by the three parameters (xC, zC, θCD), where θCD≡tan-1(zD-zCxD-xC)is the angle between the contact surface and the print plane, and such that the stamp angle θ(s) between arc-length coordinates s=sC and s=sD is substantially equal to θCD; that is,θ(s)≈θCD for sC≦s≦sD.
- 13. The apparatus of claim 12 wherein the upper stamp clamp is pivoted about a pivot line lP parallel to the y axis and intersecting the xz plane at point P having coordinates xP and zP; the stamp attaching to the upper stamp clamp along an upper-clamp line lE parallel to the y axis and intersecting the xz plane at point E having coordinates xE and zE; the upper-clamp line lE being disposed on the upper stamp clamp at a radius RS from the pivot line lP, such that the total arc length sE from the lower stamp clamp to the line lE is sE&Quadbond;L, where L is the known, free length of the stamp; and wherein the stamp attaches to the upper-clamp line lE at an angle θEθ(L).
- 14. The apparatus of claim 13 wherein the trajectory comprises a plurality of configurations, each configuration described by the coordinate s0&Quadbond;x0 of the contact front and by corresponding coordinates xP, zP of the pivot line given by the equations:xP=xE+Rs cos θE zP=zE+Rs sin θE, where xE=∫0Lcos θ(s)ⅆs and zE=∫0Lsin θ(s)ⅆs,and where the mathematical function θ(s) describing the shape of the arc for a given configuration is assumed to beθ(s)=0 for 0≦s≦s0, whereas for s>s0, θ(s) is determined by solution of the differential equations ⅆuⅆs=F(u),the lower-end boundary conditions u0≡{u10u20u30u40}≡{θ0ⅆθⅆs|0FX0FZ0}={00FX0FZ0},and the auxiliary boundary conditionsT(β)=0, wherein u≡{u1u2u3u4}≡{θⅆθⅆsFx(s)Fz(s)},F(u)≡{u2u3EIsin u1-u4EIcos u1-p(s)sin u1-f(s)cos u1w+p(s)cos u1-f(s)sin u1},T(β)≡{zC-∫0sCsin θ(s)ⅆsθC-θCDEIⅆθⅆs|E+FXERSsin θE-FZERScos θE},and wherein FX(s) and FZ(s) are functions of s describing the internal x-directed and z-directed forces acting on the stamp's cross section at s per unit depth of the stamp in they direction, FXE≡FX(sE), FZE≡FZ(sE), β is a vector of parameters that are unknown a priori, β={s0FX0FZ0},s0 is the aforementioned arc-length coordinate of the contact front, FX0≡FX(s0), FZ0≡FZ(s0), E is Young's modulus of the stamp, I is the area moment of inertia of the stamp's cross section per unit depth in the y-direction, w is the weight per unit area of the stamp, p(s) and f(s) are functions of s describing forces applied normal to the stamp and tangential to the stamp respectively by the print-force-application system, the stamp-control system and the print surface, sC is the value of arc-length coordinate s at point C, θC≡θ(sC) is the angle of the arc at point C, and θCD is the aforementioned angle of the stamp-control system's contact surface; andwherein for each configuration the solution for xP and zP is derived by means of the “shooting method”, whereby an initial value ⊕(0) of β is guessed, the differential equations are solved to yield T(β(0)) and [∂T∂β]β=β(0)-1,Newton-Raphson iteration β(n+1)=β(n)-[∂T∂β]β=β(n)-1T(β(n))is applied to obtain a refined vector β(1), whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary conditions T(β)=0 are achieved to within some tolerance.
- 15. The apparatus of claim 12 wherein the stamp-control system comprises a vacuum bar.
- 16. The apparatus of claim 13 wherein the trajectory comprises a plurality of configurations, each configuration described by the coordinate s0&Quadbond;x0 of the contact front and by corresponding coordinates xP, zP of the pivot line given by the equations:xP=xE+Rs cos θE zP=zE+Rs sin θE, where xE=∫0Lcos θ(s)ⅆs and zE=∫0Lsin θ(s)ⅆs,and where the mathematical function θ(s) describing the shape of the arc for a given configuration is assumed to beθ(s)=0 for 0≦s≦s0, whereas for s>s0, θ(s) is determined in stamp segments OC and DE by solution of the differential equations ⅆuⅆs=F(u),the lower-end boundary conditions u0≡{u10u20}≡{θ0ⅆθⅆs&LeftBracketingBar;0}={0κ0},and the upper-end boundary condition T(β)≡EIⅆθⅆs&LeftBracketingBar;E+FXERSsin θE-FZERScos θE=0,whereinu≡{u1u2}≡{θⅆθⅆs},F(u)≡{u2Fx(s)EIsin u1-Fz(s)EIcos u1},κ0 is a specified curvature at point O, E is Young's modulus of the stamp, I is the area moment of inertia of the stamp's cross section per unit depth in the y-direction, w is the weight per unit area of the stamp, Fx(s) and Fz(s) are the x-directed and z-directed stamp forces per unit length of stamp in the y direction, given by Fx(s)={Fx0,0≤s≤sCFx0+Δ Fx,sD≤s≤sE,andFz(s)={0,0≤s≤s0w(s-s0),s0≤s≤sCw(s-s0)+Δ Fz,sD≤s≤sE,in which Fx0≡Fx(s0)≡β is a parameter that is unknown a priori, and the differences ΔFx and ΔFz are respectively the differencesΔFx≡Fx(sD)−Fx(sC) ΔFz≡Fz(sD)−Fz(sC) that occur across stamp segment CD where the stamp-control system contacts the stamp, the values of which differences, along with the value of the difference Δ κ≡ⅆθⅆs&LeftBracketingBar;D-ⅆθⅆs&LeftBracketingBar;C,may be calculated from the three equations of static equilibrium for the stamp under the action of forces applied to the stamp by the stamp-control system, these three differences together with θD=θC permitting numerical integration for stamp segment DE to proceed immediately from the numerical-integration result obtained at the final point C in stamp segment OC; andwherein for each configuration the solution for xP and zP is derived by means of the “shooting method”, whereby an initial value β(0) of β is guessed, the differential equations are solved to yield T(β(0)) and [∂T∂β]β=β(0)-1,Newton iteration β(n+1)=β(n)-[∂T∂β]β=β(n)-1T(β(n))is applied to obtain a refined vector β(1), whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary conditions T(β)=0 are achieved to within some tolerance.
- 17. The apparatus of claim 1 wherein the upper stamp clamp is pivoted about a pivot line lP parallel to the y axis and intersecting the xz plane at point P having coordinates xP and zP; the stamp attaching to the upper stamp clamp along an upper-clamp line lE parallel to the y axis and intersecting the xz plane at point E having coordinates xE and zE; the upper-clamp line lE being disposed on the upper stamp clamp at a radius RS from the pivot line lP, such that the total arc length sE from the lower stamp clamp to the line lE is sE&Quadbond;L, where L is the known, free length of the stamp; and wherein the stamp attaches to the upper-clamp line lE at an angle θE≡θ(L).
- 18. The apparatus of claim 17 wherein the trajectory comprises a plurality of configurations, each configuration described by the coordinate s0&Quadbond;x0 of the contact front and by corresponding coordinates xP, zP of the pivot line given by the equationsxP=xE+Rs cos θE zP=zE+Rs sin θE, where xE=∫0Lcos θ(s)ⅆs and zE=∫0Lsin θ(s)ⅆs,and where the mathematical function θ(s) describing the shape of the arc for a given configuration is assumed to beθ(s)=0 for 0≦s≦s0, whereas for s>s0, θ(s) is determined by solution of the differential equations ⅆuⅆs=F(u),the lower-end boundary conditions u0≡{u10u20}≡{θ0ⅆθⅆs|0}={0κ0},and the upper-end boundary condition T(β)≡EIⅆθⅆs|E+FX0RSsin θE-w(s-s0)Rscos θE=0,whereinu≡{u1u2}≡{θⅆθⅆs},F(u)≡{u2FX0EIsin u1-w(s-s0)EIcos u1},κ0 is a specified curvature at point O, the parameter β=FX0, unknown a priori, is the internal x-directed force acting on the stamp's cross section at s=s0 per unit depth of the stamp in the y direction, E is Young's modulus of the stamp, us the area moment of inertia of the stamp's cross section per unit depth in the y-direction, and w is the weight per unit area of the stamp; andwherein for each configuration the solution for xP and zP is derived by means of the “shooting method”, whereby an initial value β(0) of β is guessed, the differential equations are solved to yield T(β(0)) and [∂T∂β]β=β(0)-1,Newton iteration β(n+1)=β(n)-[∂T∂β]β=β(n)-1T(β(n))is applied to obtain a refined value β(1) of the unknown parameter β, whereupon the differential equations are solved again; this iteration procedure being applied repeatedly until the correct auxiliary boundary condition T(β)=0 is achieved to within some tolerance.
- 19. A printing apparatus, comprising:a receiver means whose receiving surface lies in an xy plane, the normal to the surface defining a z-axis direction; a lower stamp clamp means for fixing a first edge of a stamp; an upper stamp clamp means for holding a second edge of a stamp for movement in the xz directions; a flexible stamp means for printing to the receiver, said flexible stamp in substantially the form of a sheet defining edges, the first edge of which is affixed to the lower stamp clamp, and the opposing second edge of which is affixed to the upper stamp clamp, thereby allowing the stamp to hang in a curve under gravity and the sheet's own stiffness, such that every normal to the stamp's curved surface lies substantially parallel to the xz plane; and a trajectory-producing means for moving the upper stamp clamp along a prescribed trajectory in the xz plane, such that the stamp is draped upon the receiving surface in a manner that causes the curvature of the stamp near a contact front at a point B to be constant throughout the trajectory.
- 20. The apparatus of claim 19 further comprising print-force application means for applying pressure upon the stamp means against the receiver means and for defining the contact front.
- 21. The apparatus of claim 19 further comprising stamp-control means for defining a point C through which the curvature of the sheet will pass throughout the trajectory.
- 22. A printing apparatus, comprising:a print surface lying in a print plane defined by an imaginary x-axis and y-axis, the print surface having an outward normal pointing in the positive direction along an imaginary z-axis, such that the x-axis, y-axis, and z-axis are substantially orthogonal to one another; a lower stamp clamp disposed adjacent to the negative-x edge of the print surface; an upper stamp clamp, moveable in two dimensions in a trajectory plane defined by the x-axis and z-axis; and a stamp comprising a flexible material, the stamp having a first end attached to the lower stamp clamp and a second end attached to the upper stamp clamp, such that a cross section of the stamp parallel to the trajectory plane forms an arc extending from an origin point Q on the lower stamp clamp having (x, z) coordinates (0, 0) to point E on the upper stamp clamp, this arc being described by the mathematical function θ(s), where s is the curvilinear distance along the arc measured from point Q, and θ is the angle between the print plane and an imaginary line, the imaginary line being tangent to the cross section of the stamp at s, wherein the upper stamp clamp is pivoted about a pivot line lP parallel to the y axis and intersecting the xz plane at point P having coordinates xP and zP.
- 23. A printing apparatus, comprising:a print surface lying in a print plane defined by an imaginary x-axis and y-axis, the print surface having an outward normal pointing in the positive direction along an imaginary z-axis, such that the x-axis, y-axis, and z-axis are substantially orthogonal to one another; a lower stamp clamp disposed adjacent to the negative-x edge of the print surface; an upper stamp clamp, moveable in a trajectory comprising a plurality of xz positions; a stamp comprising a flexible material, the stamp having a first end attached to the lower stamp clamp and a second end attached to the upper stamp clamp, such that a cross section of the stamp parallel to a trajectory plane defined by the x-axis and z-axis forms an arc extending from an origin point Q on the lower stamp clamp having (x, z) coordinates (0, 0) to point E on the upper stamp clamp, this arc being described by the mathematical function θ(s), where s is the curvilinear distance along the arc measured from point Q, and θ is the angle between the print plane and an imaginary line, the imaginary line being tangent to the cross section of the stamp at s; and a stamp-control system movable along the x-axis, wherein each xz position of the upper stamp clamp is a function of the displacement xC of the stamp-control system along the x-axis.
- 24. A method for printing, comprising:lying a print surface in a print plane defined by an imaginary x-axis and y-axis, the print surface having an outward normal pointing in the positive direction along an imaginary z-axis, such that the x-axis, y-axis, and z-axis are substantially orthogonal to one another; disposing a lower stamp clamp adjacent to the negative-x edge of the print surface; attaching a first end of a stamp comprising a flexible material to the lower stamp clamp and a second end of the stamp to an upper stamp clamp, such that a cross section of the stamp parallel to a trajectory plane defined by the x-axis and z-axis forms an arc extending from an origin point Q on the lower stamp clamp having (x, z) coordinates (0, 0) to point E on the upper stamp clamp, this arc being described by the mathematical function θ(s), where s is the curvilinear distance along the arc measured from point Q, and θ is the angle between the print plane and an imaginary line, the imaginary line being tangent to the cross section of the stamp at s; and moving the upper stamp clamp in a trajectory comprising a plurality of xz positions of the upper stamp clamp that blend into a substantially continuous motion, the trajectory being effective in laying the stamp down smoothly and fiat upon the print surface in a manner such that a moving contact front between the stamp and the print surface is created, the contact front being disposed substantially along a line characterized by a contact-front coordinate s0≡x0 that increases as the trajectory progresses, the trajectory also being effective in causing the curvature ⅆθⅆsof the stamp at or near the contact front to be substantially constant throughout the motion.
US Referenced Citations (4)