3.1. Fabrication of the Polymeric Stencil with Microdot Apertures with the Variation of Structural Aspect Ratio
Controlling the distance between the PDMS surfaces, which is adjusted by the externally applied pressure from the compressor in the system, is more important in the process of fabricating the polymeric stencil than applying pressure by hand to achieve reproducibility, reliability, uniformity, and accuracy. When the applied pressure is insufficient to secure the proper distance between the PDMS surfaces, the PUA layers, where the apertures should be formed, would be fully cured, resulting in the blocking of the apertures in the PUA stencil. However, if excessive pressure is applied, the pillars in the patterned PDMS in contact with the flat PDMS would be buckled or fractured, which is considered a process failure. The previous studies for fabricating a UV-cured membrane with apertures exhibit the need for investigation of properly applied pressure depending on the structural dimension as summarized in
Table S2. Therefore, it is essential to establish a design rule for the
Pcr depending on the geometrical features of the pillars to prevent buckling-induced process failure. This approach would provide concrete fundamentals for the fabrication criteria progressed from an empirical approach. To investigate the structural stability, six patterned PDMS molds with micropillars having different
D and
H were set. The detailed information on the fabrication process of PDMS molds and patterned silicon masters is provided in the
Section 2.2.
The scanning electron microscopy (SEM) images in
Figure 2 show the geometry of each experimental sample. The samples have geometrical features as listed below: (1)
D = 50 μm,
H = 80 μm, and AR = 1.6; (2)
D = 40 μm,
H = 80 μm, and AR = 2.0; (3)
D = 50 μm,
H = 100 μm, and AR = 2.0; (4)
D = 30 μm,
H = 120 μm, and AR = 4.0; (5)
D = 40 μm,
H = 160 μm, and AR = 4.0; (6)
D = 30 μm,
H = 160 μm, and AR = 5.3. The samples were mainly categorized into four kinds of AR (1.6, 2.0, 4.0, and 5.3). For the case of AR = 2.0 and 4.0, the samples with different
D and
H while fixing the AR were prepared to confirm if AR is the only factor in geometry, which determine the mechanical stability herein. With the prepared patterned PDMS mold, the fabrication of the polymeric stencil with microdot apertures was conducted with the sandwich-like assembly as mentioned above. The applied vertical pressure to the assembly was increased gradually and discretely through the imprint machine. For each case, the applied imprint force was converted into the pressure per single pillar. The pressure that induces the buckling phenomenon is denoted as the experimental buckling pressure.
3.2. Buckling Phenomena of the Microdot Apertures in Stencils Depending on Aspect Ratio
First, to confirm if the other geometrical features, such as
D and
H, affect the buckling phenomena in the same AR condition, the experiment was performed using two pillar structures with same AR = 2.0 (
Figure 3). The results show that the shape of the aperture in the polymeric stencils, depending on the applied pressure, can be categorized into three cases: (1) clogging, (2) stable, and (3) buckled. Without applying external pressure to the assembly, the apertures in the PUA stencil are blocked, which is called clogging. If the appropriate pressure, which is 0.091 MPa to the single pillar in this case, is applied to the assembly, the upstanding apertures without slanting and bending in the PUA stencil are constructed stably. Also, when the pressure over 0.17 MPa is applied, the apertures in the PUA stencil are slanted, which is a buckling phenomenon. This pressure is denoted as
Pcr. From the summarized results in
Table 1, the pressures that transit the state of the apertures from stable to buckled were comparable even with different
D and
H of the pillars, indicating that the critical factor for buckling is the AR, not
D or
H.
The same trend is observed in the other case, as displayed in
Figure 4 and
Table 2. The experiments using two different samples with same AR = 4.0 were conducted, and the structural features depending on the pressure are also classified as clogging, stable, and buckled, which is the same with the previous result. The
Pcr of the two samples was the same, with a value of ~0.229 MPa for both. From the results, it is also confirmed that the only geometric factor affecting the buckling phenomenon is AR. Based on this fact, structures with four ARs (1.6, 2.0, 4.0, and 5.3) were prepared, and the stencils were fabricated with each structure (
Figure 5). The pressures at the transition from the stable to the buckled state, which are
Pcr, were 0.225 MPa (AR = 1.6), 0.171 MPa (AR = 2.0), 0.229 MPa (AR = 4.0), and 0.140 MPa (AR = 5.3) (
Table 3). Note that the shapes in the buckling state appear differently depending on the AR; the structures can be classified according to the buckled shapes into two groups: low (AR = 1.6 and 2.0) and high (AR = 4.0 and 5.3) AR groups. The structures with relatively low AR were slanted without a curved wall and point of inflection, while those with relatively high AR were bent with a curved wall. Obviously,
Pcr decreases in each group as the AR increases since the slender and slim structures would be easily deformed and unstable. However, when comparing the results of the high AR group and low AR group, the critical buckling pressure is expected to be much larger in the case of the low AR group compared to the high AR group, however, experimental results were not. The overall experimental pressures at which the membranes are in stable and buckling states are plotted in
Figure S2. To investigate the previous experimental results theoretically, the comparative analysis of
Pcr based on the commonly known Euler’s buckling theory was conducted. Two main issues were encountered in the analysis—first, a comparison of the theoretical and experimentally obtained values of
Pcr, and second, the criteria for classification into low or high AR.
3.3. Theoretical Approach to Determine the Critical Buckling Pressure
To address both issues, the proposed simple analytical model is illustrated in
Figure 6. To calculate the theoretical value of
Pcr, the end conditions of the pillar in the model should be set reflecting the behavior of the experimentally used PDMS pillar. In this analytical model, the bottom and the vertical line connected as one body represents the patterned PDMS, while the top cover represents the flat PDMS. The bottom-end condition of the pillar is fixed at the base, and the upper-end condition of the pillar is assumed to be rotation-fixed and translation-free.
Figure 6a indicates the state of the pillar with a height of
L without an applied load.
Figure 6b demonstrates the buckling shape of the pillar when the critical buckling load (
Fcr) is applied to the pillar, which corresponds to the shape of the apertures in
Figure 5g,e. In the schematic model, the point of injection (
c′) is defined as the center point of the deflected pillar where the bending moment is zero. The theoretical
Fcr can be obtained by solving the bending-moment equation considering the zero-moment at the point of injection (
c′) in Equation (1), which is called Euler’s buckling equation (see detailed derivation for
Fcr in
Equation S1 from Euler’s Buckling Equation).
Pcr can be calculated by dividing
Fcr by the top surface area of the pillar of the patterned PDMS mold as expressed in Equation (2). The
Pcr in this equation corresponds to the pressure applied to a single pillar among numerous pillars in the patterned PDMS mold with an area of 9 cm
2.
Here,
I is the moment of inertia with respect to the principal axis of buckling, which is
,
A is the top surface area of the pillar, which is
, and the
L is the height of the pillar (
H).
E* =
E/(1 −
v2), where
v is the Poisson’s ratio and
E is the Young’s modulus.
E* in Equations (1) and (2) is the combined elastic modulus, which is commonly used for large deformations considering the material’s Poisson’s ratio [
11,
12,
13]. After substituting the Poisson’s ratio of PDMS as 0.5,
E* is expressed as 4
E/3 [
14,
15].
To elucidate the applicability of this equation to the cases in this study, the criteria for applying Euler’s buckling theory were investigated. Since Euler’s buckling equation is derived assuming that the pillars under the external load are extremely thin like a thread or string, it is not that suitable to assume such for the pillars used herein. In particular, the low-AR structures are likely to be excluded from the applicable range of Euler’s buckling equation. To set the criteria for the applicability of Euler’s buckling equation, the slenderness ratio was introduced and used, which is defined as the ratio of the length of the structure to its least radius of gyration
r. Based on the definition, the slenderness ratio can be expressed as follows:
where the radius of gyration (
r) describes the cross-sectional area distribution in a pillar around its centroidal axis, which is
. This result indicates that the slenderness ratio equals four fold of the AR. The critical slenderness ratio (
L/
r)
c, which determines the applicability of Euler’s buckling equation, is obtained by assuming that the material’s critical stress (
) is equal to its proportional limit (
) [
16]. Euler’s buckling equation is applicable in the range above the critical slenderness ratio; however, it is not applicable in the range below the critical slenderness ratio. Determining the critical slenderness ratio requires calculating
and
. The critical stress (
) can be expressed by dividing the critical load by the top surface area of the single pillar (i.e., equal to the
Pcr), and the relationship between
and (
L/
r)
c can be re-expressed by substituting Equation (2) into Equation (3) as follows:
After rearranging the equation for (
L/
r)
c and assuming that
is the same as
, the equation is expressed as follows:
In Equation (5),
is generally considered half of the material’s yield strength,
[
16,
17,
18,
19]. In a previous study, the yield strength of PDMS was reported as 700 kPa [
20].
E* is calculated as 3.48 MPa using the
E-value of PDMS (2.61 MPa) [
21]. Based on the information, (
L/
r)
c was calculated as 9.906. By considering the relationship between the critical slenderness ratio and AR in Equation (3), the critical AR (
ARc) as an indicator of applicability of Euler’s buckling equation was calculated as 2.4765. Thus, Euler’s buckling equation is only valid for the pillar structure with AR over ~2.5 in the case of using PDMS as molds. From the results, the calculated critical slenderness ratio, the theoretical
Pcr, and the experimentally obtained buckling pressure are plotted in
Figure 7. As shown in the graph, in the applicable range of Euler’s buckling equation, the theoretical (green line) and the experimental (gray bar) values are incompletely matched, indicating that despite applying higher pressure than
Pcr, the polymeric stencil was stably made without the buckling phenomenon. This result can be attributed to the inaccuracy in the following cases, which is not considered while analyzing the buckling phenomenon using Euler’s buckling equation [
22,
23]:
Inhomogeneity due to the inherent non-uniformity of the material and imperfect mixing process of PDMS base and curing agent of PDMS.
The imperfect flatness of the UV-imprinting machine used in the experiment.
Slightly different distances from the substrate to the top surface of each pillar due to the finely inclined PDMS mold.
To correct the inaccuracy, the correction factor was considered, the value of which is 1.67 as derived from the comparison between the experimental and theoretical results. Consequently, in the applicable range of Euler’s buckling equation in
Figure 7, which are the cases of AR = 4.0 and 5.3, the experimental results (Gray bar) and the correction-factor-calibrated theoretical values (blue line) fit well. However, in the non-applicable range of Euler’s buckling equation, which are the cases of AR = 1.6 and 2.0, the experimentally obtained values are more than thrice lower the theoretical values, even when the correction factor is considered. This result indicates that the experiment-based approach should be used to anticipate the structural deformation of the structures with low AR below
ARc due to the mismatch of the experimental result and Euler’s buckling theorem. Apparently, the resultant shapes of the low-AR structures were slanted without the curved wall, whereas the high-AR structures were bent with a curved wall. Also, the lateral shift of the pillar top surface of the low-AR structures was more alleviated than that of the high-AR structures under the comparable applied pressure. This trend implies that although the deformation of the low-AR structures occurs below the anticipated theoretical buckling pressure, the amount of deformation was restrained compared to the high-AR structures. It is well known that the large deformation for a short pillar is restrained, and since the structural stress is accumulated due to less deformation, it would be fractured rather than deformed. In this aspect, ideally, the low-AR structures under compression are likely to be expanded laterally (
Figure 8), which is slightly different from the resultant shape based on observation. This trend is ascribed to the inevitably and unintentionally applied shear stress from the process inaccuracy, including the material’s inhomogeneity and imperfect flatness. Hence, to manufacture completely upstanding apertures without deformation in the polymeric stencil, the design rules should be considered as follows. First, in the case of the pillars with higher AR than AR
c, the pressure on a single pillar should be lower than the predicted
Pcr based on Euler’s buckling theory. Meanwhile, in the case of the pillars with lower AR than AR
c, an experimental approach to obtain
Pcr is required rather than the theoretical approach based on Euler’s buckling equation. Second, in both cases, it is extremely important to minimize the aforementioned inaccuracy, including material inhomogeneity and imperfect flatness in the process.