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Effect of Stamp Deformation on the Quality of Microcontact Printing: Theory and Experiment Kenneth G. Sharp,† Gregory S. Blackman,† Nicholas J. Glassmaker,*,‡ Anand Jagota,†,§ and Chung-Yuen Hui‡ DuPont Central Research and Development, Experimental Station E323/309, Wilmington, Delaware 19880, and Department of Theoretical and Applied Mechanics, 212 Kimball Hall, Cornell University, Ithaca, New York 14850 Received December 10, 2003. In Final Form: April 30, 2004 Microcontact printing (µCP) is an effective way to generate micrometer- or submicrometer-sized patterns on a variety of substrates. However, the fidelity of the final pattern depends critically on the coupled phenomena of stamp deformation, fluid transfer between surfaces, and the ability of the ink to selfassemble on the substrate. In particular, stamp deformation can produce undesirable effects that limit the practice and precision of µCP. Experimental observations and comparison with theoretical predictions are presented here for three of the most undesirable consequences of stamp deformation: (1) roof collapse of low aspect ratio recesses, (2) buckling of high aspect ratio plates, and (3) lateral sticking of high aspect ratio plates. Stamp behavior was observed visually with an inverted optical microscope while loaddisplacement data were collected during compression and retraction of stamps. Additionally, a “robotic stamper” was used to deliver ink patterns in precise locations on substrates. These monomolecular ink patterns were then observed in high contrast using the surface potential scanning mode of an atomic force microscope. Theoretical models based on continuum mechanics were used to accurately predict both physical deformation of the stamp and the resultant inking patterns. The close agreement between these models and the experimental data presented clearly demonstrates the essential considerations one must weigh when designing stamp geometry, material, and loading conditions for optimal pattern fidelity.
1. Introduction In microcontact printing (µCP),1-3 a molecular ink is applied to an elastomeric stamp having a pattern of surface relief. Ink is transferred from the stamp to the substrate by bringing the raised areas into contact with the substrate surface. A schematic drawing of the surface relief pattern used in this work is shown in Figure 1, where the raised regions of the stamp surface are plates with a rectangular cross section ranging in width from 5 to 50 µm. [The word “plates” is used in this paper to refer to the raised features that compose the pattern of the µCP stamps discussed in this paper. (See Figure 1.) A mechanician might refer to these features as flat punches, while a more accurate mathematical term is rectangular prisms.] The separation between the plates, 2w, is equal to the plate width 2a. The stamps are typically fabricated by molding an elastomer, commonly poly(dimethylsiloxane) (PDMS), on a photoresist master fabricated using traditional optical lithography. The low adhesion of the PDMS to the master allows the stamp to be peeled undamaged from the master. Under ideal conditions, the peeled PDMS stamp is an exact negative replica of the master. Several advantages of µCP and related methods over conventional optical lithography have been noted.4 For example, they are not subject to diffraction limitations and are operationally simpler and less expensive. The * Corresponding author. E-mail:
[email protected]. † DuPont Central Research and Development. ‡ Cornell University. § Current address: Department of Chemical Engineering, Iacocca Hall, Lehigh University, Bethlehem, PA 18015. (1) Kumar, A.; Whitesides, G. M. Appl. Phys. Lett. 1993, 63, 2002. (2) Kumar, A.; Biebuyck, H. A.; Whitesides, G. M. Langmuir 1994, 10, 1498. (3) Biebuyck, H. A.; Larsen, N. B.; Delamarche, E.; Michel, B. IBM J. Res. Dev. 1997, 41, 159. (4) Mirkin, C. A.; Rogers, J. A. MRS Bull. 2001, 26, 506.
Figure 1. Stamp with rectangular cross section “roof” recesses.
Figure 2. Idealized depictions of the µCP failure modes considered in this paper.
flexibility of the stamp enables patterning of curved surfaces. Also, the methods are compatible with a wider range of organic and biological materials. There are limitations to µCP, several of which arise as a result of stamp deformation during the ink transfer. As pointed out by Delamarche et al.,5 because of the low modulus of PDMS (shear modulus, G e 1 MPa), not all of the features accessible by microfabrication of patterns in silicon form stable, useful structures on the surface of the stamp. For example, Biebuyck et al.3 have demonstrated experimentally that if the aspect ratio h/2a is too large, the plates can collapse when loaded or even under their own weight (buckling). (See Figure 2a.) Also, lateral collapse of neighboring plates can occur during the inking process, (5) Delamarche, E.; Schmid, H.; Michel, B.; Biebuyck, H. Adv. Mater. 1997, 9, 741.
10.1021/la036332+ CCC: $27.50 © 2004 American Chemical Society Published on Web 06/24/2004
Effect of Stamp Deformation on µCP Quality
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a situation where the capillary forces due to retained liquid on the plates are sufficiently large to cause them to contact. Once contact occurs, plates may adhere to each other as a result of surface adhesive forces.5,6 (See Figure 2b.) On the other hand, when the aspect ratio is too small, recessed surfaces of the stamp, along with the raised plates, can be deformed into contact with the substrate as illustrated in Figure 2c. This mode of failure will be referred to as roof collapse. The mechanics of deformation and contact associated with several failure modes, including those mentioned above, have been analyzed in detail by Hui et al.7 To prevent the three failure modes mentioned above for the geometry drawn in Figure 1, the conditions that must be satisfied are
(
)
[ (
-4σ∞w wπ a 1 + cosh-1 sec πE*h w 2(w + a)
)]
|z| and 1. Because the function
η2n+1xη2 - 1 dη ) 2i η-z
The contour integral
2n+1
∫Γη
φ(η) )
F(z) ) IΓ
η2n+1xη2 - 1 η-z ∞
η2n+1
2 1/2
where
)
η2n+1x1 - η-2
bmη-2m ∑ m)0
) 1 - (z/η) z z2 z3 1 + + + + ... (B.12) η η2 η3
(
)
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Sharp et al.
∞
x1 - η-2 ) ∑ bmη-2m
(B.13)
region, -a e x e a. The solution of eq C.3 is given by Johnson;14 it is
m)0
p(x) )
From eq B.12, the coefficient of the η-1 term of the Laurent series is
bn+1 + bnz2 + bn-1z4 + ... + bn+1-kz2k + ... + b0z2n+2 ) n+1
z2n+2
bkz-2k ∑ k)0
(B.14)
∫-1
η2n+1x1 - η2 dη η-z
xa
P E*B + x ( π 2 ) -x
(C.4)
2
where P is the total load on the punch. This pressure distribution results in a moment on the half space, given by
M)
Combining eqs B.9-11 and B.14 gives 1
1 2
2
B ∫-aaxp(x) dx ) πE*a 4
(C.5)
Now B ) tan θ, where θ is the angle between the punch edge and the half space before deformation. Thus, θ ) arctan B ≈ B, for small B. Thus,
) n+1
-πz2n+2
∑ bkz-2k + πη2n+1xη2 - 1
(B.15)
M≈
k)0
The principal value of eq B.15 can be found using the Plemjl formula, that is,
∫-1 1
η2n+1x1 - η2 dη η-ξ
πE*a2 θ 4
(C.6)
such that, for small angles of deformation, the elastic half space may be replaced with a rigid half space and a torsional spring of stiffness
n+1
) -πξ2n+2
bkξ-2k ∑ k)0
(B.16)
Note that the last term on the right-hand side of eq B.15 does not matter because the sum of the square root function as η approaches from the positive and negative sides of the cut is identically 0. Appendix C Comparison of Torsional Stiffnesses of an Elastic Support and a Plate Loaded in Bending. In this section, the relative influences to plate buckling of elastic end conditions and overall bending flexibility are compared. To make this comparison, we treat an elastic attachment as a torsional spring connecting a rigid rod to a rigid half space. Then, the overall torsional stiffness of a flexible cantilever is found by computing the end rotation as a function of an applied end moment. For the elastic end condition, consider an elastic half space indented by a rigid, frictionless punch with an angled, straight edge. Using the contact mechanics conventions of Johnson,14 the profile of the punch is
z ) Bx
j z(0) - Bx u j z(x) ) u
)
πE*B 2
(C.3)
where p(x) is the pressure distribution in the contact
(C.7)
That is, a rigid rod of width a, with an applied end moment M, will rotate by θ ) M/kend. Next, considering the flexible cantilever plate with an applied end moment M, one can easily obtain the rotation of the end. It is approximately equal to the slope of the plate at the end for small rotations. Thus, for a plate of length h,
θ ≈ u′(h) )
Mh E*I
(C.8)
where I is the moment of inertia, equal to a3/12. Rearranging,
M≈
E*a3 θ 12h
(C.9)
so that the overall torsional stiffness of the flexible plate is
kflex )
E*a3 12h
(C.10)
As a result, one has
(C.2)
This contact condition is converted to an integral equation for the pressure under the punch by integrating the solution for a point load on a half space over the contact region. Details may be found in Johnson’s book.14 Note that because the problem is two-dimensional, the contact condition must be differentiated to avoid the necessity of specifying a datum for the displacements. The resulting equation is
p(s) ds
πE*a2 4
(C.1)
where B is the slope of the punch angle. Such a punch will deform the surface of the half space, with displacements given by
∫-aa x - s
kend )
kflex/kend )
a 3πh
(C.11)
During the buckling event, the end of the plate rotates by a large amount. For slender plates (h/2a . 1), the torsional stiffness due to bending is much smaller than that due to the elastic end, and instability due to bending will be the dominant cause of buckling. In this case, the theoretical Euler buckling load should approximate the experimental buckling load closely. However, for very stout plates (h/2a f 1), the torsional stiffness due to bending approaches that due to the elastic end. In this case, the deformation of the elastic end likely triggers the buckling event. LA036332+
