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Fabrication of 3-D Curved Microstructures by Constrained Gas Expansion and Photopolymerization Mary B. Chan-Park,*,†,‡ Chun Yang,‡,§ Xun Guo,‡ LQ Chen,† Soon Fatt Yoon,‡,| and Jung-Hoon Chun‡,⊥ School of Chemical and Biomedical Engineering, Nanyang Technological UniVersity, 50 Nanyang AVe, Singapore 639798, Singapore-MIT Alliance, School of Mechanical and Aerospace Engineering, Nanyang Technological UniVersity, 50 Nanyang AVenue, Singapore 639798, Singapore, School of Mechanical and Aerospace Engineering, Nanyang Technological UniVersity, 50 Nanyang AVenue, Singapore 639798, School of Electrical and Electronics Engineering, Nanyang Technological UniVersity, 50 Nanyang AVenue, Singapore 639798, and Laboratory for Manufacturing and ProductiVity, Massachusetts Institute of Technology, 77 Massachusetts AVenue, Cambridge, Massachusetts 02139 ReceiVed NoVember 19, 2007 This paper describes a novel method of fabricating three-dimensional (3-D) curved microstructures with continuous relief through controlled argon gas expansion into a photocurable resin. A microstructured stamp is placed on top of a nonwetting photopolymerizable liquid resin. The setup is heated, and the argon gas in the blind holes of the stamp expands. The expanded gas displaces the resin at the mouth of the microcavities to form 3-D curved indentations in the liquid resin which is subsequently rapidly solidified by photopolymerization. By changing the duration of the preheating, different curvatures can be produced. Arrays of homogeneous 3-D curved microstructures having different cross-sectional geometries and heights were fabricated using various shapes of the blind holes and preheating times, respectively. As a demonstration of applications, high-quality and uniform polydimethylsiloxane microlens arrays were produced. In addition, thorough investigation was carried out to study the factors influencing the fabricated 3-D curved microstructures. Curved microstructures with diameters as small as 2 µm were demonstrated. A simple model was developed, and such a model allows for predicting the curvatures of indentations with different preheating times. It has been found that the predicted curvatures are in good agreement with experimental data.
1. Introduction Convenient and inexpensive fabrication methods for threedimensional (3-D) curved microstructures with variable curvature and cross-sectional geometry are important in many applications including telecommunications,1 miniaturized total analytical systems,2 display and TV projection systems,3 biological devices,4,5 and microelectromechanical systems (MEMS)-based sensors.6 Reported methods for producing continuous relief such as resist reflow of patterns,7 gray mask photolithography,8 ion beam milling or laser writing,8 ink-jet printing,9 self-assembly * To whom correspondence should be addressed. E-mail: mbechan@ ntu.edu.sg. Tel:(65) 6790 6064. Fax: (65) 6794 7553. † School of Chemical and Biomedical Engineering, Nanyang Technological University. ‡ Singapore-MIT Alliance, School of Mechanical and Aerospace Engineering, Nanyang Technological University. § School of Mechanical and Aerospace Engineering, Nanyang Technological University. | School of Electrical and Electronics Engineering, Nanyang Technological University. ⊥ Laboratory for Manufacturing and Productivity, Massachusetts Institute of Technology. (1) Serpe, M. J.; Lyon, J. K, L. A. AdV. Mater. 2004, 16, 184. (2) Meldrum, D. R.; Holl, M. R. Science 2002, 297, 1197. (3) Nussbaum, P. H.; Volkel, R.; Herzig, H. P.; Eisner, M.; Haselbeck, S. Pure Appl. Opt. 1997, 6, 617. (4) Chen, J.; Ko, F.; Chen, H.; Chou, C.; Chen, H.; Chang, F. J. Vac. Sci. Technol. B 2004, 22, 492. (5) He, M.; Yuan, X.; Bu, J.; Cheong, W. C. Opt. Lett. 2004, 29, 1007. (6) Healey, B. G.; Walt, D. R. Anal. Chem. 1997, 69, 2213. (7) Popovic, Z. D.; Sprague, R. A.; Neville Connell, G. A. Appl. Opt. 1998, 27, 1281. (8) Herzig, H. P.; Micro-optics: Elements, Systems and Applications; Taylor and Francis: London, 1997. (9) Biehl, S.; Danzebrink, R.; Oliveira, P.; Aegerter, M. A. J. Sol-Gel Sci. Technol. 1998, 13, 177.
of polymer beads,10 and electrowetting11 suffer from a variety of disadvantages. Some techniques such as ink-jet printing or electrowetting can produce lenslike structures with circular crosssections only. Structures produced by ink-jet printing also tend to have large dimensional deviations. Techniques such as ink-jet printing and resist reflow suffer from limitations of fixed or limited range of curvature. Further, the elevated temperature (normally in the range of 150-200 °C3) for the reflow process limits the range of materials with which this technique can be used. Techniques such as gray mask photolithography and beam milling require the use of expensive facilities. Curved microstructures have also been fabricated by a hot intrusion process:12 plastic materials are intruded into a nickel mold made using LIGA (a German-language acronym for lithography, electroplating, and molding) process and curved surfaces are obtained when the intruded materials do not fully fill the high-aspect-ratio cavities in the nickel mold. Limitations include the high cost of the nickel mold, the high temperature needed for heating the intruding material over its Tg, and the high pressure required. To our knowledge, there are limited reported top-down parallel methods for making arrays of curved microstructures with “small” (i.e., few micrometers) diameter or lateral dimensions.13 This paper presents a novel method for fabricating 3-D curved microstructures with continuous relief (Figure 1). Briefly, a 3-D micropatterned polydimethylsiloxane (PDMS) mold is placed over a liquid ultraviolet (UV) curable prepolymer, trapping argon (10) Lu, Y.; Yin, Y. D.; Xia, Y. N. AdV. Mater. 2001, 13, 34. (11) Yang, S.; Krupenkin, T. N.; Mach, P.; Chandross, E. A. AdV. Mater. 2003, 15, 940. (12) Pan, L. W.; Shen, X. J.; Lin, L. W. J. Microelectromech. Syst. 2004, 13, 1063. (13) Xia, Y. N.; Roger, J. A.; Paul, K. E.; Whitesides, G. M. Chem. ReV. 1999, 99(7), 1823.
10.1021/la703608p CCC: $40.75 2008 American Chemical Society Published on Web 04/29/2008
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cross-sections from 10 to 2 µm were also investigated. The resin degas and preheat times were lengthened to minimize capillarywetting induced flow into the blind holes to successfully make curved microstructures as small as 2 µm in diameter. A model for predicting the curvature from the materials and process parameters was also developed.
2. Experimental Section
Figure 1. Schematic procedure for fabrication of continuous relief. The PDMS stamp is placed on the top of the prepolymer. Heating of the entrapped argon gas expands it into the prepolymer, creating continuousrelief indentations, which, after photopolymerization, can be used as a mold to replicate convex features.
gas in the mold microcavities. The mold is heated and the gas expands to form indentations on the liquid preprepolymer surface which are then “frozen” quickly by UV hardening of the prepolymer. Our novel method, based on constrained gas expansion followed by photopolymerization, is parallel, simple, cost-effective, and can be conveniently used for the fabrication of 3-D curved microstructures with tunable curvature and crosssection geometry. Two arrays of “big” (feature scale of tens of micrometers) 3-D curved microstructures having different cross-sectional geometries (round and square) were investigated. The suitability of a transparent PDMS microlens array produced by this method for projecting high-quality images was also investigated. The effect of argon preheating time on microstructure curvature was investigated. “Small” 3-D curved microstructures having square
2.1. Chemicals. The PDMS silicone rubber used was RTV 651 manufactured by GE Silicones; it has two partssthe base and the curing agent. The curing was performed according to the manufacturer’s recommendation. Two UV prepolymer formulations were investigated. Unless otherwise specified, the chemicals used in the UV-curable prepolymer formulations were purchased from UCB Chemicals. The polyurethane (PU) prepolymer formulation consists of four components: EB 270, an aliphatic urethane acrylate aliphatic; trimethylolpropane triacrylate (TMPTA) from Sartomer Chemicals; EB 350, a silicone diacrylate release additive; and Irgacure 651 (i.e., 2,2-dimethoxy-2-phenylacetophenone) supplied by Ciba Chemicals as photoinitiator. The ratio of these components was 68:30:2:0.2 (w/w). Another Bisphenol A (BA) prepolymer formulation consists of EB600 (a Bisphenol A diacrylate), TMPTA and Irgacure 651 blended with a ratio of 68:30:2:0.2 (w/w). 2.2. Fabrication Process. Figure 1 outlines our new fabrication process in its “large” (tens of micrometers in size) micromolding application; additional steps required for the “small” micromoldings are discussed in the Results and Discussion section. A prepolymer holder was made by cutting a 3 cm × 3 cm through-hole in a piece of Teflon film (about 2 mm thick) and gluing it to the bottom of a Petri dish using double-sided tape. The photopolymerizable prepolymer was filled flush into the holder. The prepolymer was degassed in a vacuum oven (0 atm pressure at 25 °C) for 30 min to remove trapped air. In a glovebox filled with argon of gauge pressure of about 0–1 mbar, a piece of the transparent PDMS stamp with an array of ∼200 µm deep blind-holes was gently brought into contact with the prepolymer, starting at one side of the holder and gradually proceeding across the prepolymer surface in order to minimize gas trapped at the interface between the stamp surface and the prepolymer. (The PDMS stamp was prepared by soft lithography from either a deep reactive ion etched silicon master mold or a SU-8 photolithographically formed master mold.14) This process left argon gas trapped in the blind-hole cavities between the stamp and the prepolymer. Subsequently, the stamp/prepolymer/holder assembly was moved into a small photopolymerization chamber below a UV lamp (SUSS MA6 mercury lamp with 365nm wavelength emission of 13 ( 0.5 mW/cm2 intensity and an infrared filter). The UV irradiation time for complete cure was 35 and 50 s for PU and BA formulations, respectively. A hotplate inside the chamber maintained the temperature of the chamber at 65 °C. A piece of glass plate (5 mm in thickness) heated to 65 °C was placed on top of the PDMS stamp to preheat it before the start of UV irradiation; the preheat duration is an important control parameter for the process. Unless otherwise stated, the preheat duration was 15 s before UV irradiation started. The glass plate also prevented warping of the PDMS stamp during irradiation. After UV polymerization, the PDMS stamp was peeled off and the photopolymerized polymer was ultrasonically cleaned in acetone, ethanol, and DI water each for 3 min. Finally, the photopolymerized polymer with 3-D curved microstructures was used as a mold for soft lithography15,16 in order to replicate complementary convex features.
3. Results and Discussion 3.1. Fabricated 3D Curved Microstructures and their Convex Replica. Figure 2 shows the SEM images of the fabri(14) Chan-Park, M. B.; Yan, Y. H.; Neo, W. K.; Zhou, W.; Zhang, J.; Yue, C. Y. Langmuir 2003, 19, 4371. (15) (a) Xia, Y.; Whitesides, G. M. Annu. ReV. Mater. Sci. 1998, 28, 153. (b) Wilbur, J. L.; Jackman, R. J.; Whitesides, G. M. Chem. Mater. 1996, 8, 1380. (16) Kunnavakkam, M. V.; Houlihan, F. M.; Schlax, M.; Liddle, J. A.; Kolodner, P.; Nalamasu, O.; Rogers, J. A. Appl. Phys. Lett. 2003, 82(8), 1152.
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Figure 2. SEM images of fabricated continuous relief. (a) “Big” circular microstructures made using mold with 80 µm (diameter) × 200 µm (depth) cylindrical blind holes: (I) the fabricated concave PU mold with continuous relief from circular pattern; (II) the replicated PDMS convex profile using soft lithography. (b) “Big” square microstructures using mold with 60 µm × 60 µm by 200 µm deep cuboid blind holes: (I) the fabricated convex continuous profile using BA prepolymer from a square pattern; (II) the replicated convex PDMS profile from the mold.
cated 3-D curved microstructures and their convex replications, using PDMS stamps with cylindrical and cuboid blind-holes. Column (a) in Figure 2 was fabricated by a stamp with cylindrical blind-holes (80 µm in diameter) spaced 80 µm apart using a PU-based prepolymer. Figure 2a (Ι) illustrates the generated concave PU microstructure, the diameter of which is 84 µm, slightly larger than that of the blind-hole. Figure 2a (ΙΙ) demonstrates the lenslike convex profiles of PDMS replicated from the PU microstructures in Figure 2a (I). Figure 2b (I) shows the BA photopolymerized curved microstructure from a PDMS stamp with cuboid blind-holes (60 µm × 60 µm). The generated BA relief cross-section is about 65 µm × 65 µm. Figure 2b (II) illustrates the PDMS replica with convex features from Figure 2b (I). The PU and BA microstructured relieves are found to have high homogeneity across the 3 cm × 3 cm area with less than 10% replication error. Thus, this method can generate arrays of homogeneous 3D curved microstructure with both circle and square cross-sections. One possible application of the transparent convex PDMS microstructures (Figure 2a(II)) is a microlens array. The soft lithographically replicated transparent PDMS concave features in Figure 3a were used as a microlens array to form images of a pattern in a transmission microscope (AXiovert 200, Carl Zeiss) with either “C” or “T” characters (∼3 cm × 3 cm) cut on a piece of black paper.10 Shown in Figure 3b and c are images of “C” and “T” patterns projected through the fabricated microlens array. An interesting observation is the fact that the microlens array is able to project sharp and well-focused images with approximately the same high quality. This implies that the curvatures in the convex features have the identical focal length (i.e., the radius of the curvature) in a single array. Kunnavakkam et al. have shown that backing a PDMS mold by rigid glass or quartz plate and using a liquid photocurable epoxy loaded to a high percent
Figure 3. (a) Microscope image of microlens array made from “big” circular curved microstructures (Figure 2a(II)) without projection. (b and c) Images recorded with a transmission optical microscope with projection of objects “C” and “T” (shown in the insets) through the replicated PDMS convex continuous features. The scale bar is 100 µm.
of functionalized silica can produce replicated microlens array with focal-length distribution and loss comparable to the master mold.16 3.2. Effects of Preheating Time, Gas Environment, and Capillary Wetting. Preheating of the PDMS stamp heats the argon gas trapped within the stamp holes, increasing its pressure, which in turn produces indentations into the liquid prepolymer. A series of 3D microstructures with different curvatures was created with BA prepolymer by varying the preheating time. The heights of the convex features on PDMS replica were measured by a stylus profile meter (Z560, Silicon Instrument). As demonstrated in Figure 4, with increasing the heating time, the height of the curved microstructures is found to increase. Figure 5 shows representative SEM pictures of the lenslike microstructures at two different preheating times (13 and 24 s). Argon gas was used instead of air to avoid the inhibition effect of oxygen on the free-radical photopolymerization.17 We initially tried to carry out the experiments in air but, after UV irradiation, there was always a tacky uncured polymer layer on the surface (17) Kloosterboer, J. G. AdV. Polym. Sci. 1998, 84.
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respectively, corresponding to the stamp holes with a diameter of 10, 6, 4, and 2 µm. For small stamp holes, it is found that, during the course of preheating, the prepolymer can fill into the stamp holes under the capillary wetting effect and thus form micropillars with noticeable height after UV curing. Instead of introducing complex analysis here, it is our intention to explain the trend of our experimental observation using a simplified model. With an assumption of an impermeable stamp wall, the final equilibrium height of the pillars, hL due to the capillary-wetting-induced filling effect can be approximately estimated from eq 1 expressed by
hL ) Figure 4. PDMS replica microstructure relief height versus preheat time (line, prediction; 9, measurements). Each plotted point is an average of eight measurements; the error bars are one standard deviation.
of produced polymeric microstructures. Acetone cleaning of this layer caused nonuniformity of the concave features. The argon gas environment facilitates complete cure to improve the uniformity of the features. One issue associated with the fabrication is the capillary wetting at the mouths of the blind holes when the PDMS stamp mold is in contact with the prepolymer. The capillary wetting effect can induce the filling of liquid prepolymer into the stamp blind holes.18 In Figure 5a, the ringlike edge structures are found to occur in a short preheating time (corresponding to relatively low heated temperature/pressure). In Figure 5b, it is shown that, with a longer preheating time, the relatively higher temperature not only can lower down the surface tension of prepolymer but also can increase the argon gas pressure in the stamp hole, thus hindering the filling of prepolymer into the stamp hole. Nonetheless, our experimental results indicate that the capillary wetting effect causes some prepolymer to fill into the stamp hole, but such effect is insignificant (specifically for long preheating times) when the dimension of blind holes in PDMS stamp is large (for example, more than 50 µm). Such capillary-wetting-induced filling is related to the crosssectional diameter of the capillary hole, the surface tension of the liquid prepolymer, and the contact angle between the prepolymer and the stamp hole surface. For fixed dimensions of holes, possible ways of reducing the capillary-wetting effect are (i) using materials with low surface energy for the stamp and low surface tension for the prepolymer and (ii) increasing the preheating time and the resin degassing time. In the present study, PDMS was chosen in the experiments as the stamp material due to its low surface free energy. Both prepolymers contain EB 350, which is a silicone diacrylate widely used as a wetting agent to decrease the surface tension of prepolymers. However, the capillary-wetting-induced filling of prepolymer into the stamp holes can become significant with decreasing the diameter of the stamp holes. The experiment conditions that can prevent prepolymer from filling into large stamp holes are unable to hinder the prepolymer to fill into small stamp holes. Figure 6a–d shows pillars that are formed in the small stamp holes with diameters of 10, 6, 4, and 2 µm, respectively. It can be seen that these pillars are quite high and the height of the pillars increases with decreasing the diameter of the stamp holes. The results show that, for a fixed depth of the PDMS stamp (e.g., 60 µm), the average height of the pillars is 20, 30, 40, and 50 µm, (18) Schueller, O. J. A.; Zhao, X.; Whitesides, G. M.; Smith, S. P.; Prentiss, M. AdV. Mater. 1999, 11, 37.
(
)
2σLcosθ Pg - Pa 2σLcosθ Pa TgH0 -1 ) FLgR FLg FLgR FLg T0hL (1)
where σL and FL and are the surface tension and density of the prepolymer, respectively, θ is the contact angle, g is the gravitational constant, R and H0 are, respectively, the radius and depth of the PDMS blind hole, Pg and Tg are, respectively, the pressure and temperature of the argon gas in the blind stamp hole, and Pa and T0 are, respectively, the ambient pressure and temperature. Equation 1 was obtained from the force balance among the gravity of the liquid polymer pillar, the increased pressure of the argon gas inside the blind stamp hole due to capillary filling, and the surface tension force of the liquid polymer.19 It is noted that if the PDMS stamp hole is open to ambient air, namely Pg ) Pa, eq 1 can be reduced to the wellknown capillary rise expression.19 Without need of solving this quadratic equation (with hL as the variable), it clearly shows that the height of the polymer filling into the PDMS stamp hole is larger for smaller dimensions of the stamp hole, larger surface tension of the liquid polymer, or shorter preheating time (i.e., lower heating temperature). This conclusion is in accordance with our experimental results. To effectively control the capillary-wetting effect, it was found that the height of the pillars can be remarkably lowered by changing experimental conditions under which we were able to fully prevent prepolymer from filling into the stamp holes even for small-diameter stamp holes (for example, 2 µm). Figure 7 shows PU 3-D curved microstructures made from a PDMS stamp with blind holes of diameter 4 and 2 µm. Their main process parameters are degassing at 65 °C for 3 h, preheating at 80 (for 4 µm hole) or 90 °C (for 2 µm hole) for 60 s, and using UV irradiation at a 365 nm wavelength for 60 s. Although the fabrication steps of making the small-diameter curved microstructures are similar to those of making the large-diameter curved microstructures, specific processing parameters are quite different. Special attention should be paid to the resin degassing process. It is known that gas atoms can be adsorbed on the bottom and sidewall of the resin holder and on the interface between the resin and the hydrophobic PDMS stamp. The gas atoms likely gather together to form nano- or even microbubbles; the presence of nanobubbles at a hydrophobic surface such as PDMS was demonstrated by Ishida et al. 20 and Sakamoto et al. 21 using atomic force microscopy (AFM). These authors have noted that special degassing procedures are needed to reduce the nanobubbles. In fact, it is difficult to completely draw out all nano/ (19) Batchelor, G. K., An Introduction to Fluid Dynamics; Cambridge University Press: New York, 1967. (20) Ishida, N.; Sakamoto, M.; Miyahara, M.; Higashitani, K. Langmuir 2000, 16, 5681. (21) Sakamoto, M.; Kanda, Y.; Miyahara, M.; Higashitani, K. Langmuir 2002, 18, 5713.
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Figure 5. “Big” circular curved microstructures obtained with (a) 13 s preheating time. The ringlike structures surrounding the lenslike structure were due to the capillary effect. (b) 24 s preheating time. The prepolymer used was BA.
Figure 6. SEM images of PU pillars that are formed due to capillary-wetting-induced prepolymer filling into PDMS stamp holes. (a) Pillars formed in stamp holes with 10 µm squares. The average height of the pillars is 20 µm. (b) Pillars formed in stamp holes with 6 µm squares. The average height of the pillars is 30 µm. (c) Pillars formed in stamp holes with 4 µm squares. The average height of the pillars is 40 µm. (d) Pillars formed in stamp holes with feature size 2 µm squares. The average height of the pillars is 50 µm.
microbubbles during the degassing. When the resin is preheated, the bubbles expand, and thus, a pressure (due to nano/ microbubble) is built up. Such pressure has a trend of counteracting the expanding pressure in the stamp holes that is engendered by the argon gas trapped in the blind holes, and thus, the nano/microbubble-induced pressure can facilitate the resin filling into the stamp holes. When the diameter of the stamp holes is big (e.g., larger than 50 µm) where the capillary wetting force in the stamp holes is not so strong, the action of the nano/ microbubble pressure itself is not large enough to push the resin into the stamp holes. When the diameter of stamp holes becomes
smaller, the increasing capillary force together with the nano/ microbubble pressure can push the resin into the stamp holes to form pillars. Therefore, it is important to do a full degassing at higher temperature when a stamp with smaller blind holes is used. Higher degassing temperature can also reduce the surface tension of the resin. In addition, using a high preheating temperature is also important for a small holes stamp because a higher preheating temperature can cause a larger pressure inside the stamp holes that pushes resin down to form an indentation. Next, a model will be presented to interpret the formation of the indentation.
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Figure 7. SEM images of “small” PU 3-D curved microstructures made using PDMS stamps with blind holes of 4 and 2 µm diameter, respectively. (a) Curved microstructures of diameter 4 µm. Degassing done at 65 °C for 3 h. Preheat done at 80 °C for 60 s. (b) Curved microstructures of diameter 2 µm. Degassing done at 65 °C for 3 h. Preheating done at 90 °C for 60 s. Table 1. Parameters Used in the Calculation parameter
value
d0 R
1.7 mm 1.59 × 10-7 m2/s 200 µm 1.014 Pa
H Pa
Figure 8. Illustration of the growth of the spherical cap structure due to gas expansion. The initial radius is r0, and the final equilibrium radius is rc. (The angle of 70° is due to the effect of surface tension of prepolymer, which tends to form a curved surface on the rim of blind-hole before gas expansion starts.).
3.3. Simple Model for the Indentation Depth. We here develop a model describing the dependence of the indentation depth on the stamp preheating process parameters for the circular stamp hole case. We make the following simplifying assumptions: (i) there is no mass transfer between the gas and the stamp or the prepolymer; (ii) the work involved in compression and displacement of the prepolymer is inconsequential to the final equilibrium state; (iii) the indentation is spherical, forming a “cap” at the aperture of the stamp hole; (iv) the prepolymer behaves as a Newtonian liquid; and (v) without consideration of the capillary wetting effect. As shown in Figure 8, the dynamics of the gas/prepolymer spherical interface are described by the modified Rayleigh equation:22–25
[
∆P ) FL r
( ) ]+
dr d2r + 1.5 2 dt dt
2
2σL µL dr +4 r r dt
(2)
where r is the radius of the interface of a spherical cap, t is the time, and ∆P is the pressure drop across the gas/prepolymer interface, approximately given by the difference between the pressure of the argon gas, Pg and the ambient pressure, P0. µL is the dynamic viscosity of the prepolymer. Three terms on the right-hand side of eq 2 represent the effects of inertial force, surface tension, and viscous force, respectively.25 The modified (22) Pinczewski, W. V. Chem. Eng. Sci. 1981, 36, 405. (23) Terasaka, K.; Tsuge, H. Chem. Eng. Sci. 1991, 46, 85. (24) Li, H. Z. Chem. Eng. Sci. 1999, 54, 2247. (25) Albalak, R. J. Fundamental of Bubble Growth, Polymer DeVolatilization; M. Dekker: New York, 1996.
parameter value parameter
value
R T0
40 µm 295 K
µL FL
2200 cp 1500 kg/m3
Tc
338 K
σL
33 mN/m
Rayleigh equation is not strictly applicable to our problem because the speed of motion of the “bubble” center is comparable to the rate of change (rate of decline in our case) of the bubble radius. We are not, however, interested in modeling the precise dynamics of the bubble indentation; rather, we use the Rayleigh equation to (1) define the equilibrium condition of the fully indented bubble and (2) define the indentation time scale for comparison with other timescales of the problem (namely the heat transfer time scale). For these purposes, the modified Rayleigh equation is adequate. It is convenient to re-express eq 2 in a dimensionless form. The equilibrium radius of the spherical cap re, defined by re ) 2σL/∆P,22–25 provides a natural length scale. The quantity t0 ≡ 4µL/∆P provides a natural time scale, t0, for the approach to the equilibrium radius; it is also called the time scale for the viscous relaxation effect. Introducing the dimensionless coordinate jr ≡ r/re and the dimensionless time coordinate jt ≡ t/t0, and dividing by ∆P yields a dimensionless form of the modified Rayleigh equation
[
1 ) ξI r¯
( ) ] + 1r¯ + 1r¯ dd r¯¯t
d r¯ d2r¯ + 1.5 2 d ¯t d ¯t
2
(3)
where ξI ≡ σL2PL/4µ2L∆P parametrizes the relative importance of the inertia and viscosity terms. The parameters used in the calculation are summarized in Table 1. In our problem, the ratio of inertia parameter over viscosity, ξI, is small (about 5.11 × 10-5) so that the dynamics is dominated by viscosity.25 The time scale, t0, for the approach of the radius to equilibrium is short, about 5.3 ms. This time scale is much shorter than the heat conduction time scale (to be discussed below). Thus, for our purposes, we can simply treat the polymer indentations as being instantaneously in equilibrium with the temperature-dependent pressure in the stamp holes. We model the heating of the stamp with a standard semi-infinite plane approximation (thus treating the polymer as an extension of the stamp for this purpose). A simple analysis of heat conduction
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theory (see Appendix) gives the instantaneous temperature penetration at depth x in PDMS under such a situation26
( )
Ts(x, t) ) Tc + (T0 - Tc)erf
x 2√Rt
(4)
where Tc is the temperature of oven for heating the prepolymer, T0 is the ambient temperature, erf(x) is the error function, and R is the thermal diffusivity of PDMS. Then, the temperature of the trapped argon gas in the indentation, Tg can be approximately estimated by setting the depth as d0 (the thickness of the PDMS stamp), which yields
( )
Tg(t) ) Tc + (T0 - Tc)erf
d0
(5)
2√Rt
The heat conduction time scale is of order, d02/2R, which is about 9 s, much longer than the viscous relaxation time of the bubble/polymer interface. Thus, the bubble radius is approximately stationary during photopolymerization. The indentation of the bubble into the polymer also increases the volume of the trapped gas, with a consequent reduction in the ideal gas law pressure. Putting all this together (refer to Appendix) gives a model relation for the bubble radius, r(t), as a function of the heating time: r(t) )
[(
2σL
2
)]
πR HTg(t) - 1 Pa 1 [πR H + π(r - √r2 - R2)2(2r + √r2 - R2)]T0 3 2
and preheating processes. Curved microstructures having diameters as small as 2 µm were demonstrated. In addition, theoretical analysis was conducted to analyze the indentation of gas bubbles into the prepolymer during the preheating process. Through the scaling analysis, it was identified that the contribution to the bubble indentation process due to the inertial force effect is negligible compared to that due to the viscous force effect. Furthermore, the obtained short time scale for the viscous relaxation effect suggests that the polymer indentation can be considered instantaneously in equilibrium with the temperature-dependent pressure in the stamp holes. As a result, a simple model to predict the curvature of indentations with varying preheating times was developed, and the theoretical curvatures were found to be in good agreement with experimental data. Due to simplicity of the fabrication and ability to generate continuous features from a variety of patterns, other than for making transparent PDMS microlens arrays, the present method has possible applications in biological sensors and microphotonic devices.
Appendix On the basis of the derivation in ref 26, the governing equation for 1-D transient semi-infinite heat conduction is
∂2Ts(x, t) 2
∂x
-1
(6)
where R and H are the radius of mouth and the depth of blindholes in PDMS stamp, respectively. Equations 5 and 6 provide a model describing the indentation depth as a function of the duration of the preheating temperature prior to photopolymerization. Figure 4 also shows the model predictions. It can be noted that the model captures the trend of the curvatures produced. This suggests that the developed model, though simple and approximate, does a remarkably good job of predicting the process behavior.
4. Conclusion A new method for the fabrication of 3-D curved microstructures with continuous relief is presented; the method has the advantage of low cost, high throughput, and small (less than 10 µm) microstructure capability. The experimental results show that the continuous profiles generated in the photopolymerized polymer are homogeneous and can have different cross-sectional shapes (depending on the shape of the blind holes) and dimensions. As a demonstration of applications, a transparent PDMS microlens array produced by this method was found to be able to form images of high quality and uniformity across the array. The processing parameters identified to have a remarkable impact on the fabricated 3-D curved microstructures are the preheating time (thus the heating temperature), the duration of degassing process, and the cross-sectional dimension of the stamp hole. By changing the duration of the preheating, different curvatures can be produced. The capillary wetting effect was found to be significant for small diameters of the stamp holes but can be minimized by increasing the duration of the degassing (26) Ozisik, M. N. Heat Conduction, 2nd ed.; Wiley: New York, 1993.
)
1 ∂Ts(x, t) R ∂t
(A)
The boundary and initial conditions are conditions are Ts(0,t) ) Tc, at t > 0 and x ) 0, and Ts(x,0) ) T0, at t ) 0 and x > 0. The solution of eq A is found as
( )
Ts(x, t) ) Tc + (T0 - Tc)erf
x 2√Rt
(A-1)
where erf(x) is the error function. The temperature of the trapped gas in the indentation, Tg, can be approximated as the temperature of the bottom layer of PDMS stamp. As the measured thickness of the PDMS stamp is d0 ) 1.7 mm, one can obtain
( )
Tb(t) ≈ Tg(t) ) Tc + (T0 - Tc)erf
d0
2√Rt
(A-2)
By taking into consideration of the effect of volume expansion on pressure drop and assuming that the entrapped argon follows the ideal gas law, one can show an expression for ∆P as
∆P)Pg-Pa)
TbV0 P - Pa VgT0 a
(A-3)
where Pg is the pressure of gas in the cavity, and Vg and V0 are volume of the cavity after and before gas expansion, respectively. (A simple geometric analysis gives Vg ) πR2H + 1 / 3π(r -
√r2-R2)2(2r + √r2-R2) and V0 ) πR2H.)
Substitution of eqs A-2 and A-3 into the expression for the radius of the bubble, re ) 2σL/∆P, leads to re )
[(
2σL
πR2HTg 1 [πR2H + π(r - √r2 - R2)2(2r + √r2 - R2)]T0 3
(A-4)
Notation d0: H:
)]
-1
- 1 Pa
thickness of the PDMS stamp depth of the blind-holes in PDMS stamp
Fabrication of 3-D CurVed Microstructures
Pa: Pg: ∆P: R: r: re: T0: Tb: Ts: Tc: t: x: V0:
atmospheric pressure pressure of argon gas pressure difference applied upon gas/prepolymer interface. radius of mouth of blind-holes in PDMS stamp. radius of curved structure, i.e. radius of spherical cap. radius of curved structure at equilibrium ambient temperature temperature at the bottom layer of PDMS stamp temperature of PDMS stamp temperature of oven time dimensionless valuable for radius coordinate volume of blind-hole
Langmuir, Vol. 24, No. 10, 2008 5499
Vg:
volume of argon gas including blind-hole and curved microstructure
R: µL: FL: σL:
thermal diffusivity of PDMS viscosity of prepolymer density of prepolymer surface tension of prepolymer
Greek Symbols
Acknowledgment. This research was supported by an A-STAR (Singapore) grant (Project No. 022 107 0004). X. Guo was supported by an SMA Ph.D. scholarship. We also thank S. Conner for help in the discussions. LA703608P
