Multiphysics machine learning framework for on-demand multi-functional nano pattern design by light-controlled capillary force lithography

Predictable UV dose

As stated in the preceding section, lower UV dose values up to the forbidden gap were ignored. The UV dose at the forbidden gap can correspond to a wide array of attained height values, as can be seen in Fig. 2d. Therefore, we opted to use the UV dose at the forbidden gap (from experimental data) as the threshold UV dose, since the attained height values for UV doses beyond this point are more reliable for prediction purposes. Details on the process of selection of the threshold UV dose is given in Supplementary Note 5. From Fig. 2d, it was apparent that the threshold UV dose reduced with increasing grating height. Consequently, we chose to use 1st to 3rd order polynomial fitting to find the best-fitting equation. Although 2nd and 3rd order polynomial fittings showed greater R-squared values (0.999 and 1, respectively), the linear fitting provided a more conservative prediction with an R-squared value of 0.982. Therefore, the equation f(x) = − 1129x + 1.745 × 10⁴ was chosen to represent the relationship between grating height (μm) and UV dose, where x is the grating height (μm), and f(x) is the threshold UV dose (Jm⁻¹). The grating height used in this experiment is 2.5 μm. For more detailed information on why a linear fitting was chosen, refer to Supplementary Fig. 6a, b in Supplementary Note 6.

General steps for surface properties modulation based on height of nano-array

To start, we collected experimental data for nano gratings and nano pillars with varying heights^25,32, which showed the link between the obtained height and UV dosage. We then used artificial and human intelligence combined with a technique known as hybrid intelligence to find a rule explaining this link (details in Methods). We then used this strategy to train a multi-physics surrogate deep learning (DL) model, where the UV dose is the output and the grating (h in Fig. 3d and L in Fig. 3e) and attained heights (x(t) in Fig. 3d, e) are the input parameters (described in Methods). Next, we defined a probability density function for the height, which helps with color reproduction and bactericidal characteristics. The required UV dose is then determined using the learned DL model. By modulating UV dose for each pixel, the desired surface properties can be achieved. The algorithm employed in this paper can be summarized in Fig. 2a−e.

Color reproduction

The intensity of zeroth order diffracted light^32,33 is given by

In this context, W represents the width of the nanostructure (In case of nanoridge, it is equal to d, but in case of nanopillars it is equal to 2r), Λ denotes the pitch (the repetition interval between nanostructures, see Fig. 2d, e), n₁ is the refractive index of NOA 73 (1.56), n₂ is the refractive index of air (1.00), λ is the wavelength of the incident light, and h signifies the final height of the grating. A visual representation of this equation is given in Fig. 4d.

Figure 4d gives the wavelength and corresponding attained height of nanopillars in visible wavelength, i.e., 380 to 750 nm. From Eq. (1), we can formulate the relationship between attained height and wavelength for highest intensity. Using parameters W = 1500 nm, Λ = 3000 nm, n₁ = 1.56, n₂ = 1, and I₀ = 1 in Eq. (1), we obtain two solutions for h: h = 0 and \(h=\frac{\lambda }{0.56}\). The second solution implies that h is a multiple of the wavelength λ, which is consistent with interference patterns in wave phenomena. Using this relationship, we can determine the attained height corresponding to the visible wavelength range. The attained height that yields light in the visible wavelength, with the highest intensity, ranges from 680 nm to 1340 nm. We assume that the heights within this range follow a normal distribution. Therefore, we selected 10,000 random attained heights from a normal distribution (shown in Fig. 4a) to represent a grid of 100 × 100, with each position on the grid corresponding to a different height within a single grating configuration. Since the grating height in our study is 2.5 μm, attained heights above 1.3 μm fall into the forbidden gap (i.e., UV dose is less than the threshold), which we ignore. These newly generated points are illustrated in Fig. 4a.

We now arrange the random heights of nanopillars in a 100 × 100 grid, as depicted in a 3D surface plot in Fig. 4b, c shows the height variation across the surface’s cross-section. Figure 4f shows how this arrangement enables the reproduction of colors based on their corresponding wavelengths. Furthermore, Fig. 4e demonstrates how we can predict the UV dose required to achieve these colors.

Bactericidal properties

Dragonfly wings possess nanopillars—tall pillars (height: 311 ± 62 nm) among short pillars (height: 189 ± 67 nm), which imbue the wing’s surface with bactericidal properties³⁴. It is assumed that maintaining the same difference between the mean heights of the two pillar types, with unchanged standard deviations for each, is necessary to preserve the bactericidal feature. The overall distribution of tall and short nanopillars’ height along with its interaction with bacteria is depicted in Fig. 5b. In our research, we sought to emulate the bactericidal properties of dragonfly wings. To this end, we sought to create a surface with alternate short and tall nanopillars (Fig. 5a).

In Fig. 5a, each grid point represents a nanopillar with varying height: the grids occupied by tall nanopillars are designated in black, whereas shorter ones are in white. The height distribution for tall nanopillars, represented by the yellow curve in Fig. 5b, follows a random distribution given by \({h}_{{{{{{{{\rm{tall}}}}}}}}}={{{{{{{\mathcal{N}}}}}}}}({\mu }_{{{{{{{{\rm{t}}}}}}}}},{\sigma }_{{{{{{{{\rm{t}}}}}}}}}^{2})={{{{{{{\mathcal{N}}}}}}}}(311,6{2}^{2}).\) To determine the heights of shorter nanopillars, we used a 5-point molecule concept, as illustrated in Fig. 5c. The height of short nanopillars is determined by using

where A_s denotes the set of tall nanopillars surrounding the central shorter nanopillar. The first term of Eq. (2) means the average height of tall nanopillars surrounding the smaller one, h_k,l is the height of each tall nanopillars. d is the difference between the mean heights of tall and short nanopillars (122 nm), and u is an uncertainty. This uncertainty, dependent on the height distribution of short nanopillars, is introduced as \(u={{{{{{{\mathcal{N}}}}}}}}({\mu }_{{{{{{{{\rm{s}}}}}}}}},{\sigma }_{{{{{{{{\rm{s}}}}}}}}}^{2})={{{{{{{\mathcal{N}}}}}}}}(0,6{7}^{2}),\) where σ_s is the standard deviation in the height of short nanopillars. Using this two-step process, we generated nanograting heights that mimic the distribution of nanopillars in dragonfly wings. The distribution of short and long pillars, along with randomly generated attained heights, is shown in Fig. 5b.

In this research, we generated 2500 data points, assuming the ability to modulate the UV dose for each attained height. For grating height of 2500 nm, and randomly generated required attained height, we could predict the corresponding UV dose from DL model. For a nanograting array of size 50 × 50, covering 150 μm × 150 μm of surface area, a possible surface configuration achieving bactericidal properties is shown in Fig. 5d. The height variation across the mentioned surface section is depicted in Fig. 5f. To achieve this surface, the UV dose given in Fig. 5e is required.

Color of the bactericidal surface

Using a similar approach to the one described in the previous section, we determined the color that the bactericidal surface would exhibit. However, since the heights of the nanopillars on the bactericidal surface correspond to wavelengths in the UV range, the color reproduction is not straightforward. Due to the lower height of nanopillars on bactericidal surfaces, the surface predominantly falls within the ultraviolet region. Therefore, to obtain a Bluish color for the surface, the obtained height, using methods described in preceding sections, should be increased by a constant of 680 nm. This results in the surface depicted in Fig. 6a. Similarly, to obtain a Reddish surface, the obtained heights should be increased by 840 nm. This results in a surface color shown in Fig. 6b.

Frictional coefficient of bactericidal surface

The surface generated using this methodology possesses certain frictional properties when in contact with another surface, but these need to be verified, which will be addressed in later works.

Nanopillar’s geometry consists of top hemisphere and cylindrical pillars (Fig. 6c). Due to this, a three-staged approach is adopted to compute the relationship between normal force (P) and frictional force (F). Case 1 stands for the initial hemispherical contact when a small deformation takes place. Case 2 corresponds to cylindrical deformation when the nanopillars deform substantially but prior to compressive buckling. Case 3 means post buckling stage (Fig. 6c).

For case 1, the following derivation was performed. For the Young’s modulus E and Poisson’s ratio ν of PDMS, its composite elastic modulus is computed as \({E}^{* }=\frac{E}{1-{\nu }^{2}}\). Under shear, the contact area of top hemisphere changes from a₀ to a_f, where \(B=\frac{{a}_{{{{{{{{\rm{f}}}}}}}}}}{{a}_{{{{{{{{\rm{0}}}}}}}}}}\) is used to describe the change in contact area under shear. On a microscopic scale, for each hemisphere (of each nanopillar), the area of the i-th microcontact is given by a_0,i = π R(h_i − d) if h_i≥ d, and a_0,i = 0 if h_i < d, where d is the separation between the plane of contact and the base of the hemisphere, h_i represents the height, and R is the radius of curvature of each hemisphere. a_f,i denotes the final contact area, while f_i represents the frictional force experienced by each hemispherical top. The value d is computed as the average height between the preceding and succeeding heights of nanopillars, as shown in Fig. 6c. The friction force F is proportional to the contact area A, represented as f = s a_f, where s is the frictional strength of the PDMS-glass interface. On a macroscopic scale, for N asperities, the total initial contact area is \({A}_{0}={\sum }_{i = 1}^{N}{a}_{0,i}\), and the total final contact area is \({A}_{{{{{{{{\rm{f}}}}}}}}}=\mathop{\sum }_{i = 1}^{N}{a}_{{{{{{{{\rm{f}}}}}}}},i}\), with a_f,i = B × a_0,i. The total normal force (P) is calculated using Hertz’s model, which is given by \(P=\mathop{\sum }_{i = 1}^{N}\left(\frac{4{E}^{* }}{3{\pi }^{3/2}R}{a}_{0,i}^{3/2}\right)\), which gives:

and the total friction force (F) is \(F=\mathop{\sum }_{i = 1}^{N}s{a}_{{{{{{{{\rm{f}}}}}}}},i}\), which gives:

This case is valid until the contact surface is in contact with the hemispherical top (i.e., h_i − d < h_asperity), where h_asperity is the height of the top hemisphere. The next stage, case 2, occurs when the contact surface reaches the cylindrical pillars.

In case 2, when h_i − d > h_asperity, axial compression is observed in the cylindrical part of the nanopillars. To calculate the axial force (Normal force) P_i for each cylindrical part, the Hooke’s law is applied

Additionally, the lateral resistance offered by each cylindrical pillar is determined by

In case 3, this normal force (P_i) could cause the nanopillars to buckle, therefore the potential for buckling needs to be assessed. This assessment utilizes Euler’s buckling formula for a circular section, where the boundary condition is fixed support at the bottom of the nanopillars and hinged support at the top is presented as

where I is the second-moment area of circular nanopillars. If any nanopillars buckle, they are assumed to be redundant in friction computation, which is case 3. For case 3, nanopillars buckle and therefore, P_i = 0 and F_i = 0. Finally, the normal force (P) across a surface containing N nanopillars, along with the corresponding frictional force (F), can be calculated as the sum of normal and frictional forces from each individual nanopillars i.e., \(P=\mathop{\sum }_{i = 1}^{N}{P}_{i}\) and \(F=\mathop{\sum }_{i = 1}^{N}{F}_{i}\). The degree of nanopillar buckling with each incremental descent of the contact surface is illustrated in Supplementary Fig. 8, located in Supplementary Note 8.

To understand the impact of buckling on the surface, we computed the friction coefficient (μ), with and without considering buckling in our formulation. As shown in Fig. 6e, the model without buckling stage offers the upper bound of the nano surface’s friction forces. With the buckling effect, the nano surface friction exhibits highly nonlinear weakening. As the contact surface descends on the nanopillar surface, nanopillars buckle given \({P}_{i}\le {P}_{{{{{{{{\rm{cr}}}}}}}}}^{i}\). The detailed explanation on the phenomenon is given in the Supplementary Note 8. The number of buckled nanopillars with each descending step (i) of contact surface is depicted in Supplementary Fig. 8. The effects of different parameters involved in the formulation of P and F are given in Supplementary Note 1. While the effects of each parameter are straightforward, the effect of r is more complex due to its interactions with other parameters, as illustrated in Supplementary Fig. 1. Figure 6d gives the effect of R on μ. Additional fine-tuning to accurately capture the complex behavior of the surface’s non-linear frictional properties with nanopillars will be addressed in future work. The impact of R of asperities is significant and warrants careful consideration. Although modifying the radius necessitates changes to the PDMS mold pattern, which is not feasible through the CFL technique, its importance remains undiminished. Further research is needed to address this challenge, and future iterations will aim to explore viable solutions. Our current algorithm includes an option to alter this radius, and with additional development, it can be effectively adapted to incorporate these modifications.