Physics-Informed Displacement Control for Variable Pattern Printing with V-shaped PDMS Stamps in Roll-to-Roll Microcontact Printing

Physics-Informed Displacement Control for Variable Pattern Printing with V-shaped PDMS Stamps in Roll-to-Roll Microcontact Printing | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Physics-Informed Displacement Control for Variable Pattern Printing with V-shaped PDMS Stamps in Roll-to-Roll Microcontact Printing Xian Du, Jingyang Yan, Huarui Du This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6780648/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 11 Dec, 2025 Read the published version in Communications Engineering → Version 1 posted You are reading this latest preprint version Abstract Roll-to-roll microcontact printing enables high-throughput production of flexible electronic devices by continuously transferring inks onto substrates via polydimethylsiloxane (PDMS) stamps. Traditional rectangular or cylindrical PDMS stamps yield uniform pattern sizes, limiting manufacturing versatility. This study introduces V-shaped PDMS stamps for variable pattern printing using a single stamp geometry. A physics-based deformation model was developed by combining finite element simulations and experiments to characterize the out-of-plane behavior of V-shaped PDMS under displacement. Leveraging this model, we implemented a neural network-based Model Predictive Control (MPC) system to precisely regulate vertical displacement and achieve desired pattern dimensions. Experimental results demonstrate that a single V-shaped PDMS stamp can reliably produce variable pattern sizes with high repeatability, significantly improving the adaptability and process efficiency of microcontact printing for flexible electronics manufacturing. Physical sciences/Engineering/Mechanical engineering Physical sciences/Engineering/Electrical and electronic engineering Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 I. INTRODUCTION and colleagues in the 1990s, is a soft lithography technique that utilizes elastomeric stamps—typically composed of polydimethylsiloxane (PDMS)—to transfer molecular patterns onto a variety of substrates via conformal contact 2,3 . Its simplicity, low cost, and compatibility with diverse materials have made µCP a widely adopted method in fields such as molecular electronics 4 , surface chemistry 5–7 , and biosensing 8–11 . The initial demonstrations of µCP employed rectangular PDMS stamps to pattern microscale gold features. Since then, most µCP studies and applications have used stamps that are vertically symmetrical, such as rectangular or cylindrical shapes 12,13 , whose deformation behaviors have been extensively characterized. For example, prior work 14 has shown that for rectangular stamps, the size of the printed pattern remains constant regardless of the applied load, meaning that a single stamp can only produce one pattern size. Given that PDMS stamp fabrication is typically time-consuming, this constraint introduces significant complexity and delays in the development cycle for application-specific designs. To address this limitation and enable more flexible printing capabilities, several strategies have been explored to allow a single PDMS stamp to generate variable pattern sizes. One approach involves multi-pass printing on the same substrate 15 ; by rotating the stamp between successive printings, diverse shapes and sizes can be achieved. However, this method suffers from poor alignment accuracy and is generally restricted to simple geometries, such as lines or arrays—thus limiting its utility for fabricating complex layouts in flexible electronics. An alternative strategy involves altering the stamp geometry, with pyramidal PDMS stamps being the most widely studied 16 . These stamps enable variable pattern sizes by exploiting their deformation under different applied loads: increasing the load expands the tip contact area with the substrate, and vice versa. Initially developed to print dot-array patterns for protein patterning, pyramidal stamps are typically fabricated via photolithography and anisotropic etching. Subsequent studies have investigated their deformation behavior both experimentally and analytically. For instance, pyramidal PDMS stamps with a base size of 6 μm and height of 4.24 μm have been fabricated and tested under varying pressures 17 . The experimentally measured contact areas closely matched finite element analysis (FEA) predictions, validating their potential for tunable patterning. More recent work 18 has incorporated dynamic modeling using the Johnson-Kendall-Roberts (JKR) contact mechanics framework to describe the deformation during printing. Despite these advances, pyramidal PDMS stamps are still limited to producing arrays of simple shapes (e.g., rectangular dots), and are therefore unsuitable for applications that require complex or arbitrary patterns. Inspired by the concept of pyramidal PDMS stamps, which enable variable contact areas through controllable deformation, we explore an alternative geometry: the V-shaped PDMS stamp. This structure offers the potential for continuous modulation of contact area through its load-dependent out-of-plane deformation, making it a promising candidate for variable pattern printing in µCP. Unlike pyramidal PDMS stamps—which are typically limited to printing dot arrays or regular geometric patterns—the V-shaped PDMS stamp is capable of printing complex and arbitrarily-shaped circuit layouts, thereby offering greater flexibility for advanced applications in flexible electronics. Notably, V-shaped PDMS stamps have previously been used for µCP 19 , where a two-layer configuration was used: a 2–4 mm thick Sylgard 184 PDMS substrate supported a 30 μm thick hard PDMS (h-PDMS) film with a V-shaped surface profile. This study demonstrated sub-50 nm feature replication in a plate-to-plate µCP setup under a static load of ~20 g. However, the primary focus of that work was on achieving high-resolution static pattern transfer, rather than enabling continuous pattern size variation through controlled deformation. Furthermore, to date, there has been no systematic investigation into the deformation dynamics of V-shaped PDMS stamps, nor their application in roll-to-roll (R2R) µCP. In this work, we investigate the feasibility of using V-shaped PDMS stamps to achieve variable pattern sizes in R2R µCP as shown in Fig. 1. The key to enabling tunable printing lies in precisely controlling the out-of-plane deformation of the stamp, which governs the contact area between the stamp and the substrate. In most existing R2R µCP systems, stable contact between the PDMS stamp and the substrate is typically maintained through force-based control schemes, where actuators such as voice coils or stepper motors regulate the vertical contact force—commonly measured by load cells due to the difficulty of directly sensing interface pressure. For example, a flexure-guided R2R system achieved force variation control within 0.05 N using voice coil actuation 20 . Another system integrated air dampers and step motors in a hybrid configuration, maintaining force stability with a root-mean-square error (RMSE) below 0.25 N through load-cell-based feedback 21 . However, our findings reveal that even under precise force control—achieving a RMSE of 0.05 N—significant variations in printed pattern size still occur when using V-shaped PDMS stamps (Supplemental Fig. S1). A likely explanation for this variability lies in the limitations of load cell measurements: in addition to the actual contact force, load cells also capture vertical forces induced by web tension. Due to assembly imperfections and coupling effects, the controlled web tension can fluctuate within a range of ±1 N, effectively doubling the measured force and introducing unintended deformation in the PDMS stamp. This deformation results in inconsistent contact areas and thus undermines pattern transfer fidelity. To overcome these limitations, we propose a physics-informed displacement control framework that directly targets out-of-plane deformation rather than indirectly inferring it from force. Displacement provides a more robust and physically meaningful control input, as it directly determines the contact geometry between the stamp and the substrate. While some R2R µCP systems incorporate displacement sensing—for purposes such as register alignment or web tracking 22,23 —these capabilities have not yet been applied to actively regulate stamp deformation for variable pattern printing. This represents a critical gap in existing process control strategies. The proposed physics-informed displacement control framework consists of two main steps (Fig. 1b). The first step is to derive a model that captures the relationship between the displacement of the PDMS stamp and the resulting contact area. A straightforward approach to obtaining this model would be to record displacement and contact area data and apply system identification techniques. However, direct observation of the contact region is challenging in typical R2R systems, where the impression and print rollers are metallic and opaque. Previous studies have addressed this by embedding cameras within hollow rollers or using transparent substrates in simplified setups 24 . In this work, we adopt a hybrid method to capture the nonlinear contact mechanics of the stamp under varying displacements, combining a physics-based FEA simulation with a single-roller experimental system. The FEA simulation incorporates the hyperelastic properties of PDMS to accurately represent its mechanical behavior under compression. In the experimental setup, a V-shaped PDMS stamp is wrapped around a rotating roller and brought into contact with a transparent glass or plastic plate mounted on a motorized stage. This configuration enables direct visualization of the contact region and facilitates the acquisition of displacement and contact area data, which are used to validate the FEA simulation results. The second step involves implementing a neural network-based model predictive control (MPC) strategy (Supplementary S1-S2). A neural network is trained to model the nonlinear dynamics of the R2R system, mapping control inputs to the resulting PDMS displacement. This learned model is then integrated into an MPC framework to regulate the input commands and achieve precise displacement control. Experimental results show that this displacement-driven control system significantly reduces pattern size variability and enables tunable printing using a single PDMS stamp geometry. These findings demonstrate the benefits of displacement-based compression using V-shaped PDMS stamps in enhancing process stability, adaptability, and precision—advancing the capabilities of R2R µCP for next-generation flexible electronics manufacturing. II. METHODS A. V-shaped PDMS Stamp Manufacturing The analysis and results in this paper examine a V-shaped stamp with periodic lines on its surface, as shown in the cross-section depicted in Fig. 2a. The stamp features a larger rectangular height 𝑡, integrated with a triangular feature defined by a width 𝑤, and height ℎ. The ratio A represents the relationship between the width and height of the triangular feature. Different ratios result in varying contact areas, corresponding to displacement and force. The design and fabrication process of the V-shaped PDMS is illustrated in Fig. 2. We first use SolidWorks to create the CAD drawing of the master mold for the PDMS, as shown in Fig. 2b. The master mold is then fabricated using a Profluidics P285D 3D printer. The use of Profluidics P285D significantly reduces fabrication time and eliminates the need for a cleanroom. The real mold after 3D printing is shown in Fig. 2e. After printing the master mold, the silicone elastomer base was mixed with the silicone elastomer curing agent in 10:1 weight ratio. The components were thoroughly combined in a clean, dry container for 5 minutes, ensuring a homogeneous mixture by scraping the sides and bottom to incorporate all material. Once mixed, the mixture was degassed to remove trapped air bubbles using a vacuum chamber. Vacuum pressure was applied until bubbles were no longer visible. Following degassing, the mixture was carefully poured into the prepared mold, as shown in Fig. 2c. It was poured slowly and from a low height in a thin stream, starting at one corner, to reduce surface tension and avoid introducing additional bubbles. The PDMS was then cured at 60°C for 4 hours. Once the PDMS was fully cured, the solid V-shaped stamp was gently demolded (Fig. 2d) and inspected for any defects, such as bubbles or incomplete curing, to ensure a high-quality final product. The completed product is shown in Fig. 2f, and Fig. 2g displays microscope images of V-shaped PDMS for accurate measurements. In the current design, the base thickness t of the rectangular section is 1 mm, while the triangular features have a width w of 146 µm and a height h of 73 µm. B. Physics-informed Deformation Model Derivation As the nature limitation of R2R manufacturing system, the contact area between the PDMS and the substrate cannot be directly monitored in-situ. It is necessary to predict the contact area from other indirect methods. Here we use both simulated and experimental data to derive the physical relationship between the displacement applied to the PDMS and the contact area. Thus it is possible to control the contact area through manipulating the applied displacement during R2R µCP. For the FEA simulation, a hyperelastic Neo-Hookean material model was used to represent the PDMS properties 25 . The simulations were conducted using Abaqus, a commercial FEA software widely used for modeling nonlinear materials and complex geometries 26 . The advantage of FEA simulation is that the geometry of the V-shaped PDMS can be changed easily. In this study, we focus on the aspect ratio A of the triangular cross-section, defined as the ratio between the base and the height. For simplicity, the aspect ratio is varied by keeping the base constant while adjusting the height. For the experimental part, we built a single-roller system to mimic the R2R µCP with the ability to monitor the contact area with a camera (Supplementary S4). The primary components of the single-roller system include a force measuring roller, a camera, a motion stage, and a glass plate. The glass plate was positioned beneath the print pattern and connected to a motion stage, emulating the continuous motion of the R2R system. The high-resolution camera is strategically positioned underneath the glass plate to capture detailed images of the contact area. The displacement was controlled by inserting different pieces of tape under the roller. The acquired contact area images are processed using custom MATLAB code to calculate the contact width for each displacement setting. C. R2R µCP with Displacement Sensors and Printing Quality Monitoring System A customized R2R µCP system was developed to evaluate displacement control for printing variable pattern sizes 27 . In contrast to conventional systems, this setup integrates two displacement sensors specifically designed to measure the vertical displacement of the V-shaped PDMS stamp (Supplementary S3). Each sensor comprises a metal plate and a capacitive sensor (CS3, Micro-Epsilon): the metal plates are affixed to the print roller and move vertically with it, while the capacitive sensors are mounted on the frame using an adaptor. Since the capacitance between the metal plate and the sensor varies linearly with their separation, this configuration enables accurate measurement of the PDMS displacement. The measurement resolution is approximately 5 nm. Prior to each experiment, the system is zeroed to ensure that the displacement readings reflect only the deformation of the PDMS stamp. This is achieved by gradually raising the impression roller until a change in the load cell output is detected—indicating contact between the PDMS and the print roller—which is then marked as the zero-displacement position. To monitor the printing results, a condensation figure (CF) monitoring system (Supplementary Fig.S5) was integrated into the R2R platform. During printing, areas covered by ink exhibit higher surface energy than unprinted gold regions, leading to different droplet formation behavior when exposed to water vapor 28 . Specifically, droplets formed on gold surfaces are denser and larger than those on printed areas, enabling in-situ visual differentiation of the printed patterns under CFs prior to the etching step. The captured CF images are subsequently segmented to extract the linewidth information, as illustrated in Supplementary Fig. S6. C. Neural Network-based MPC Control To achieve high-precision control in R2R µCP, our system incorporates a flexure-based mechanism. Unlike traditional mechanical linkages that depend on sliding or rolling contact between components, flexure mechanisms achieve motion through the elastic deformation of compliant elements. This design enables superior positioning accuracy due to the elimination of backlash and friction. Consequently, flexures are widely employed in high-resolution R2R printing systems to enhance motion and force control. However, accurately modeling the dynamics of complex flexure mechanisms poses significant challenges. The physical modeling process is often time-consuming and computationally intensive, requiring the identification of numerous parameters and their nonlinear interdependencies 29 . Moreover, the intrinsic nonlinearities of flexure systems hinder the development of precise analytical models, which in turn limits the effectiveness of traditional controllers. Prior studies have demonstrated that simple PID controllers are inadequate for managing the high dynamics and nonlinear behavior inherent to R2R µCP 27 . To address this issue, we propose a neural network-based MPC strategy for flexure-based position control. Specifically, we employ a feedforward neural network to approximate the nonlinear system dynamics. The network takes as input a sequence of delayed control signals (actuator voltages) and system outputs (measured displacements), and outputs a prediction of the displacement. This data-driven model is then embedded into an MPC framework to optimize control inputs in real-time, ensuring accurate tracking of the desired displacement trajectory (See Supplementary S1 and S2 for detailed description). D. R2R Microcontact Printing To evaluate the performance of the V-shaped PDMS stamps under realistic operating conditions, a standard R2R µCP process was employed. The three essential components involved in the printing process were the ink, the stamp, and the substrate. The ink was prepared by dissolving hexadecanethiol (HDT) in ethanol to a concentration of 15 mM. A custom-designed ink reservoir was positioned beneath the print roller to facilitate consistent ink delivery. The V-shaped PDMS stamps were fabricated following the procedure described earlier. The substrate used for printing was a 0.1 mm thick polyethylene terephthalate (PET) film coated with a 30 nm gold layer, deposited via R2R sputtering process. The printing operation was fully automated. After wrapping the V-shaped PDMS stamp around the print roller, taping the PET substrate to the web, and loading the ink into the reservoir, the system was initialized through software. Web tension, web speed, and stamp displacement were configured in advance. Upon starting the run, the ink reservoir was lifted using a motorized stage to transfer ink to the stamp surface. The voice coil actuators, controlled by the neural network-based MPC, were activated to apply the desired compressive displacement to the stamp. The target displacement was derived from the desired pattern width using the physics-informed displacement model that relates stamp deformation to contact width, followed by the activation of web transport. Throughout the process, the inline monitoring system is used to assess printing quality in real time. After printing, the patterned substrate underwent wet chemical etching to finalize the printed structures. The etching process selectively removed gold from areas not protected by HDT. The etchant solution consisted of 0.1 M thiourea and 0.01 M ferric nitrate in deionized water, with an etching duration of 8 minutes. III. REULSTS AND DISCCUSIONS A. V-shaped PDMS Stamp Deformation Due to the nature of the R2R µCP process, the contact area between the stamp and substrate cannot be directly monitored in real time. To address this challenge, we propose an indirect approach by estimating the contact area through measurements of the stamp's displacement. To establish the relationship between displacement and contact area, we employed a combined numerical and experimental method. In the simulation, we examined PDMS stamps with various aspect ratios, and the results are presented in Fig. 3b. The data reveal a strong correlation between displacement and contact area, which is expected because the primary deformation occurs at the triangular tip of the stamp rather than in the bulk of the PDMS body during compression. To validate the simulation, we fabricated a PDMS stamp with an aspect ratio of 0.5—matching one of the simulated geometries—using our custom-built single-roller system. This system allows precise control and measurement of contact area under varying displacements. The experimental results, overlaid in the same figure, show strong agreement with the simulation across the full displacement range. This consistency confirms the accuracy of the numerical model and demonstrates its practical value in guiding the R2R µCP process. These collected data will be used to derive the dynamics between the displacement and contact area. Specifically, for a target line width required by a given application, the model enables determination of the appropriate stamp displacement, thereby improving process predictability and control. B. Training of the Neural Network Based on the physics-informed model, the required stamp displacement for a given target line width can be determined in advance. However, in the actual printing process, achieving this displacement with high precision depends on the mechanical performance of the R2R printing system. To characterize and capture the dynamic behavior of stamp displacement within our system, we developed a neural network-based model. This data-driven approach enables accurate modeling of the system's displacement response, facilitating closed-loop control for precise pattern transfer during R2R µCP. To generate the training dataset for the neural network model, we employed a pulse-train excitation method. Figures 4a and 4b show the input pulse sequence with randomly varying amplitudes and the corresponding displacement response of the system. Each pulse lasts for one second, allowing the system to reach a steady-state response within each interval. The sampling interval is set to 10 ms, consistent with the control loop rate. The input voltages to the voice coil actuators are randomly selected within the range of [1.5 V, 2.5 V]. These bounds were established by first identifying the minimum voltage at which a measurable force is detected by the load cells and then adding an additional 1 V to ensure that the resulting displacement remains within approximately 150 μm. To sufficiently span the dynamic range of the system, a total of 60,000 samples were collected, with 80% of the data used for training and the remaining 20% reserved for testing. Since the displacement behavior depends on both historical output values and input control signals, the system dynamics are approximated using a Nonlinear AutoRegressive model with eXogenous inputs (NARX), as detailed in the Supplementary S1. The input and output time delays are set to 2. The neural network contains 7 hidden nodes, selected as a trade-off between inference time and prediction accuracy. A tangent-sigmoid activation function is used, as the input data are normalized within the range [−1, 1]. The network is trained using the Levenberg–Marquardt optimization algorithm in MATLAB R2023b. The prediction performance on the testing dataset is illustrated in Fig. 4c and Fig. 4d , achieving an RMSE of 0.88 μm. Additional comparisons with Long Short-Term Memory (LSTM) networks are provided in the Supplementary Fig. S3. C. Displacement Control Performance After developing the neural network model of the displacement behavior of the R2R system, we implemented a neural network-based MPC to control the displacement of the V-shaped PDMS stamp. Initial simulations were conducted to tune key MPC parameters, including the control horizon M, prediction horizon P, and the weighting factor λ, as detailed in the Supplementary S2. To determine suitable values for M and P, both were initially set to one sampling interval and incrementally increased from 1 to 20 intervals. A stable control performance was observed when M≥2. Through iterative tuning, M=4 and P=11 were selected as the final values, yielding desirable real-time force control (Supplementary Table S1). Given that the control loop operates at 10 ms, the total computation time for the MPC must remain below this threshold. Timing tests confirmed that the MPC runtime with M=4 and P=11 is approximately 7 ms, satisfying the real-time constraint (Supplementary Table S2). The optimized neural network-based MPC controller was evaluated using sinusoidal reference signals as displacement setpoints. The control results are presented in Fig. 4e. It is important to note that the length of a single V-shaped PDMS stamp is shorter than the circumference of the print roller, requiring two stamps to be stitched together. Consequently, signal artifacts appear at the junction between the two stamps. Nevertheless, as shown in the enlarged view in Fig. 4e, the actual displacement closely follows the reference trajectory, achieving an RMSE of 5 μm. This demonstrates the high accuracy and responsiveness of the neural network-based MPC controller in displacement regulation for R2R µCP. D. Variable Line Width Printing Results To demonstrate the versatility of the V-shaped PDMS stamp, we printed three distinct line widths by applying different displacement values to a single stamp. The same stamp, with an aspect ratio of 0.5, was used throughout the experiments. To monitor printing quality in a non-destructive and substrate-efficient manner, CFs were captured using our custom-developed imaging system, enabling direct line width measurement without the need for etching or post-processing. The printing was performed using controlled displacements of 10 µm, 50 µm, and 100 µm. The resulting CFs of the corresponding printed patterns are shown in Fig. 5a. The measured line widths are approximately 6 μm, 24 μm, and 40 μm from left to right, respectively, illustrating the stamp’s capability to produce variable features through displacement control. Additionally, we performed continuous printing of variable line widths by dynamically adjusting the displacement setpoint (Supplementary Movie 1). In this experiment, a stepwise varying displacement profile was applied to the controller, and the corresponding line widths were predicted based on the previously established displacement-to-width relationship. The printed results and corresponding CFs are presented in Fig. 5b. The predicted widths were computed from the recorded displacement, while the measured widths were extracted from the CFs. Due to the 2 Hz frame rate of the imaging system, the measured widths exhibit higher fluctuations compared to the predicted values. Nevertheless, the predicted and measured widths show strong agreement, confirming the system's capability for real-time control of variable-width patterning using a single V-shaped PDMS stamp. E. Antenna printing using V-shaped PDMS Finally, to demonstrate the feasibility and versatility of our approach, we used the V-shaped PDMS stamp to print more complex geometries—specifically, two different antenna designs—on flexible substrates. First, we designed a Planar Inverted-F Antenna (PIFA) consisting solely of straight-line segments. Two different line widths were printed using a single stamp with linewidths of 34 µm and 22 µm as shown in Fig. 5c. The results show that the patterns can be proportionally scaled by adjusting the displacement, which is advantageous for parametric studies. To further demonstrate the practical utility of the V-shaped PDMS stamp, we also designed and printed a spiral-shaped antenna (Supplementary Fig. S9). The mold for the spiral pattern was created using the Cut Sweep function in SolidWorks. When different displacements were applied during printing, the linewidth of the spiral pattern changed accordingly, validating the controllable feature size achievable with our displacement-based printing approach. C. Limitations and future work The V-shaped PDMS stamps used in this study were fabricated using a commercial printer with a resolution of 30 µm, which imposes a fundamental limit on the resolution of the resulting printed patterns. In the literature, alternative fabrication techniques have been demonstrated to achieve higher-resolution molds for PDMS casting. For instance, V-shaped molds can be produced via photolithography followed by wet etching of silicon, enabling sub-100 nm apex dimensions at the base of the V-shaped trenches 30 . When such nanoscale features are employed in printing, the displacement control system must also offer commensurate resolution to maintain pattern fidelity. In our current setup, the displacement control resolution is approximately 5 µm, which corresponds to about 2 µm in line width variation for the V-shaped PDMS stamp used in our experiments. This limitation is partly due to the mechanical components of the R2R system being fabricated in an academic machine shop, where precision machining tolerances are difficult to achieve. Future work will focus on enhancing the mechanical precision of the R2R platform, potentially through the integration of high-accuracy, professionally machined components to improve displacement control and overall printing resolution. IV. CONCLUSIONS In this work, we demonstrate that V-shaped PDMS stamps can be used in R2R µCP to enable variable pattern size printing. Since force-based control cannot precisely regulate the deformation of PDMS stamps, we adopt a displacement-based control strategy to achieve stable and tunable line width printing. The physical relationship between stamp displacement and contact area is established through a combination of simulation and experimental data. The simulation employs a Neo-Hookean model to represent the hyperelastic properties of PDMS, and stamps with varying aspect ratios are analyzed. To validate the simulation results, we utilize a single-roller system that mimics the actual R2R printing process. This setup allows for simultaneous measurement of displacement and contact area, enabling direct comparison with simulation outputs. Together, the experimental and simulation data form the physics-informed model of PDMS deformation. Building on this model, we develop a neural network-based MPC framework to regulate the displacement of the PDMS stamp in the R2R system. The displacement can be precisely controlled using the proposed control algorithm, allowing us to print lines with different widths using a single PDMS stamp and demonstrating the effectiveness of the proposed control method. Finally, we print two types of antenna patterns—linear and spiral—further illustrating the flexibility and broad application potential of the system. Declarations Acknowledgements This work was supported by NSF Foundation CMMI-1942185. Author contribution J.Y. conducted all experiments and wrote the initial draft of the manuscript. H.D. performed the finite element analysis and contributed to part of the experiments. X.D. provided project supervision, funding acquisition, and revised the manuscript. Competing interests The authors declare no competing interests. References Kumar, A. & Whitesides, G. M. Features of 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. Applied Physics Letters 63, 2002–2004 (1993). Pallab, N. et al. Polymer brush-assisted microcontact printing: using a tailor-made polydimethylsiloxane (PDMS) stamp for precise patterning of rough surfaces. Polymer Chemistry 15, 853–867 (2024). Mukherjee, A. et al. Synthesis and Microcontact Printing of novel nanoflowers of ZnO_Ag_rGO nanocomposite to check microbial adhesion. Nano-Structures & Nano-Objects 38, 101140 (2024). Torres-Cavanillas, R., Gavara-Edo, M. & Coronado, E. Bistable Spin-Crossover Nanoparticles for Molecular Electronics. Advanced Materials 36, 2307718 (2024). Du, B. et al. Combined laser-induced graphene and microcontact printing for processing scalable and stackable micro-stripe patterns toward multifunctional electronic devices. Carbon 225, 119148 (2024). Gödrich, S., Brodoceanu, D., Kuznetsov, V., Kraus, T. & Papastavrou, G. Defined Transfer of Colloidal Particles by Electrochemical Microcontact Printing. Advanced Materials Interfaces 11, 2400202 (2024). Larrañaga, E. et al. Long-range organization of intestinal 2D-crypts using exogenous Wnt3a micropatterning. Nat Commun 16, 382 (2025). Sun, M., Zhang, J., Xuanyuan, T., Liu, X. & Liu, W. Facile and Rapid Microcontact Printing of Additive-Free Polydimethylsiloxane for Biological Patterning Diversity. ACS Appl. Mater. Interfaces 16, 20132–20142 (2024). Khan, S. et al. Comprehensive fluorescence profiles of contamination-prone foods applied to the design of microcontact-printed in situ functional oligonucleotide sensors. Sci Rep 14, 8277 (2024). Hager, R. et al. Microcontact Printing of Biomolecules on Various Polymeric Substrates: Limitations and Applicability for Fluorescence Microscopy and Subcellular Micropatterning Assays. ACS Appl. Polym. Mater. 4, 6887–6896 (2022). Shiwarski, D. J. et al. Fibronectin-based nanomechanical biosensors to map 3D surface strains in live cells and tissue. Nat Commun 11, 5883 (2020). Garibyan, M. et al. Engineering programmable material-to-cell pathways via synthetic notch receptors to spatially control differentiation in multicellular constructs. Nat Commun 15, 5891 (2024). Xue, X. et al. A patterned human neural tube model using microfluidic gradients. Nature 628, 391–399 (2024). Petrzelka, J. E. & Hardt, D. E. Static load-displacement behavior of PDMS microfeatures for soft lithography. J. Micromech. Microeng. 22, 075015 (2012). Xia, Y. & Whitesides, G. M. Extending Microcontact Printing as a Microlithographic Technique. Langmuir 13, 2059–2067 (1997). Filipponi, L., Livingston, P., Kašpar, O., Tokárová, V. & Nicolau, D. V. Protein patterning by microcontact printing using pyramidal PDMS stamps. Biomed Microdevices 18, 9 (2016). Hong, J. M., Ozkeskin, F. M. & Zou, J. A micromachined elastomeric tip array for contact printing with variable dot size and density. J. Micromech. Microeng. 18, 015003 (2008). Jin, C. & Qiao, Q. Deformation of Pyramidal PDMS Stamps During Microcontact Printing. Journal of Applied Mechanics 83, (2016). Li, H.-W., Muir, B. V. O., Fichet, G. & Huck, W. T. S. Nanocontact Printing: A Route to Sub-50-nm-Scale Chemical and Biological Patterning. Langmuir 19, 1963–1965 (2003). Zhou, X., Xu, H., Cheng, J., Zhao, N. & Chen, S.-C. Flexure-based Roll-to-roll Platform: A Practical Solution for Realizing Large-area Microcontact Printing. Sci Rep 5, 10402 (2015). Choi, Y.-M., Kang, D., Lim, S., Lee, M. G. & Lee, S.-H. High-Precision Printing Force Control System for Roll-to-Roll Manufacturing. IEEE/ASME Transactions on Mechatronics 22, 2351–2358 (2017). Lee, J., Shin, K. & Jung, H. Control Scheme for Rapidly Responding Register Controller Using Response Acceleration Input in Industrial Roll-To-Roll Manufacturing Systems. IEEE Transactions on Industrial Electronics 69, 5215–5224 (2022). Chen, Z., Zheng, Y., Zhang, T., Wong, D. S.-H. & Deng, Z. Modeling and Register Control of the Speed-Up Phase in Roll-to-Roll Printing Systems. IEEE Transactions on Automation Science and Engineering 16, 1438–1449 (2019). Nill, S. T. (Scott T. Integrated hardware, software, and sensor design for control of a scalable, continuous roll-to-roll microcontact printing process. (Massachusetts Institute of Technology, 2014). Petrzelka, J. E. Contact region fidelity, sensitivity, and control in roll-based soft lithography. (Massachusetts Institute of Technology, 2012). Peng, J., Tomsia, A. P., Jiang, L., Tang, B. Z. & Cheng, Q. Stiff and tough PDMS-MMT layered nanocomposites visualized by AIE luminogens. Nat Commun 12, 4539 (2021). Yan, J. & Du, X. Neural-Network-Based Adaptive Model Predictive Control for a Flexure-Based Roll-to-Roll Contact Printing System. IEEE/ASME Transactions on Mechatronics 27, 5084–5094 (2022). Yan, J., Ma, R. & Du, X. Consistent optical surface inspection based on open environment droplet size-controlled condensation figures. Meas. Sci. Technol. 32, 105405 (2021). Zhou, X., Wang, D., Wang, J. & Chen, S.-C. Precision design and control of a flexure-based roll-to-roll printing system. Precision Engineering 45, 332–341 (2016). Wu, J., Tao, K. & Miao, J. Production of centimeter-scale sub-wavelength nanopatterns by controlling the light path of adhesive photomasks. J. Mater. Chem. C 3, 6796–6808 (2015). Additional Declarations There is NO Competing Interest. Supplementary Files video.avi Movie 1 supplementrary.docx Supplementary Information Cite Share Download PDF Status: Published Journal Publication published 11 Dec, 2025 Read the published version in Communications Engineering → Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6780648","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":475161153,"identity":"42b811a5-295d-4067-8d91-25fbd83689d1","order_by":0,"name":"Xian Du","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAu0lEQVRIiWNgGAWjYDACZiDmYbBB5hKnJY0ULQxgLYdJ0GJwnPnZg7dt5xPXth9//IGhwjqxgZAWyWY2c8O5bbcTt53JMZNgOJNOWAs/M4OZNC9Iy4EcNgbGtsOEtbAxs38DajmXuO3888cfGP8RoYWfmQdky4HEbTcSDCQYG4jQItnMUyY551yy8bYbb8wkEo6lGxPUYnD++DaJN2V2stvOpz/+8KHGWpagFlSQQJryUTAKRsEoGAW4AAC7XD2SxSF6TgAAAABJRU5ErkJggg==","orcid":"","institution":"University of Massachusetts","correspondingAuthor":true,"prefix":"","firstName":"Xian","middleName":"","lastName":"Du","suffix":""},{"id":475161154,"identity":"7857067b-84b1-4547-b2bc-44c1e310af41","order_by":1,"name":"Jingyang Yan","email":"","orcid":"https://orcid.org/0000-0001-8330-6272","institution":"UMass Amherst","correspondingAuthor":false,"prefix":"","firstName":"Jingyang","middleName":"","lastName":"Yan","suffix":""},{"id":475161155,"identity":"83f1d5b0-75bc-454c-b3da-d856395029d5","order_by":2,"name":"Huarui Du","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Huarui","middleName":"","lastName":"Du","suffix":""}],"badges":[],"createdAt":"2025-05-30 03:40:10","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6780648/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6780648/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s44172-025-00553-9","type":"published","date":"2025-12-11T05:00:00+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":85377477,"identity":"bcda3878-6a7d-4830-8af2-cb7adac4a762","added_by":"auto","created_at":"2025-06-25 08:43:16","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":1770080,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eOverview of the work. a\u003c/strong\u003e Illustration of the R2R µCP system with V-shaped PDMS stamps. \u003cstrong\u003eb\u003c/strong\u003e Overview of the proposed physics-informed displacement control based on neural network-based MPC.\u003c/p\u003e","description":"","filename":"image1.png","url":"https://assets-eu.researchsquare.com/files/rs-6780648/v1/547ad2a20df849dcf86016c5.png"},{"id":85377479,"identity":"b3333463-8e6b-4d15-86e4-29f49efa2c5b","added_by":"auto","created_at":"2025-06-25 08:43:16","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":3805348,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eDesign and fabrication of the V-shaped PDMS stamp.\u003c/strong\u003e \u003cstrong\u003ea\u003c/strong\u003e Cross section of V-shaped stamp configuration, labeled with characteristic dimensions. \u003cstrong\u003eb\u003c/strong\u003e The CAD model of the mold. \u003cstrong\u003ec\u003c/strong\u003e PDMS casting. \u003cstrong\u003ed\u003c/strong\u003e V-shaped PDMS stamp after curing. \u003cstrong\u003ee\u003c/strong\u003eImage of the 3D printed mold. \u003cstrong\u003ef\u003c/strong\u003e Overview of V-shaped PDMS stamp. \u003cstrong\u003eg\u003c/strong\u003eMicroscope image of fabricated V-shaped PDMS stamp.\u003c/p\u003e","description":"","filename":"image2.png","url":"https://assets-eu.researchsquare.com/files/rs-6780648/v1/e197dbdb34d17201f1759ab6.png"},{"id":85377482,"identity":"5661419e-df23-4b2c-8ece-d1889f9be561","added_by":"auto","created_at":"2025-06-25 08:43:16","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":1770121,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eDerivation of the physics-informed deformation model. a\u003c/strong\u003e Schematic diagram and experimental setup of the single-roller system, along with simulated deformation of the V-shaped PDMS stamp under different applied displacements. \u003cstrong\u003eb\u003c/strong\u003e Simulated contact linewidth as a function of displacement for five different aspect ratios 𝐴 of the V-shaped PDMS stamp. Experimental results of contact linewidth versus displacement are also plotted for comparison.\u003c/p\u003e","description":"","filename":"image3.png","url":"https://assets-eu.researchsquare.com/files/rs-6780648/v1/883dd947c9a49c2ccb161130.png"},{"id":85377483,"identity":"085c8eaa-bef4-4435-affa-23f0f769f5f4","added_by":"auto","created_at":"2025-06-25 08:43:16","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":490827,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eTraining of the neural network model and the performance of the neural network-based MPC controller.\u003c/strong\u003e \u003cstrong\u003ea\u003c/strong\u003ePulse train inputs to the voice coil actuator. \u003cstrong\u003eb\u003c/strong\u003e Responses of the V-shaped PDMS displacement. \u003cstrong\u003ec\u003c/strong\u003e Actual displacement and predicted displacement by the neural network model. \u003cstrong\u003ed\u003c/strong\u003e The corresponding errors of the predicted displacement. \u003cstrong\u003ee\u003c/strong\u003e Experimental results of the neural network-based MPC controller using sinusoidal signals as the setpoint.\u003c/p\u003e","description":"","filename":"image4.png","url":"https://assets-eu.researchsquare.com/files/rs-6780648/v1/5fc44e318b69dce87907e1aa.png"},{"id":85377484,"identity":"2ab2cb03-ab86-486c-93df-e4657bbf9c0f","added_by":"auto","created_at":"2025-06-25 08:43:16","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":6854122,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eResults of the proposed method.\u003c/strong\u003e \u0026nbsp;\u003cstrong\u003ea\u003c/strong\u003e Different pattern sizes were printed using a single V-shaped PDMS stamp, with linewidths of 6 µm, 24 µm, and 40 µm from left to right, respectively. \u003cstrong\u003eb\u003c/strong\u003e The size of the printed patterns can be also derived from the measured displacement according to the physical model. Here we printed a pattern with changing linewidth and the predicted line width from the physical model and the measured line width from the CFs are compared. \u003cstrong\u003ec\u003c/strong\u003e PIFA patterns of varying scales were printed using a single V-shaped PDMS stamp, resulting in linewidths of 34 µm (top) and 22 µm (bottom).\u003c/p\u003e","description":"","filename":"image5.png","url":"https://assets-eu.researchsquare.com/files/rs-6780648/v1/95f799622940cf8eb16811a8.png"},{"id":98927992,"identity":"206fa140-3962-4a0f-857c-429c67211b5e","added_by":"auto","created_at":"2025-12-24 08:10:22","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":17047922,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6780648/v1/bdbdb28b-da3a-4330-8ff5-914b605afbee.pdf"},{"id":85377486,"identity":"7e0c2311-d3e9-47b1-864d-277ff60955fe","added_by":"auto","created_at":"2025-06-25 08:43:17","extension":"avi","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":10176168,"visible":true,"origin":"","legend":"Movie 1","description":"","filename":"video.avi","url":"https://assets-eu.researchsquare.com/files/rs-6780648/v1/debc68cbd3306b372874f1df.avi"},{"id":85377485,"identity":"d16f8ac4-2edc-400d-8799-e062285d1722","added_by":"auto","created_at":"2025-06-25 08:43:16","extension":"docx","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":20028132,"visible":true,"origin":"","legend":"Supplementary Information","description":"","filename":"supplementrary.docx","url":"https://assets-eu.researchsquare.com/files/rs-6780648/v1/c07713b1a17d43182a7d2d72.docx"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Physics-Informed Displacement Control for Variable Pattern Printing with V-shaped PDMS Stamps in Roll-to-Roll Microcontact Printing","fulltext":[{"header":"I. INTRODUCTION","content":"\u003cp\u003eand colleagues in the 1990s, is a soft lithography technique that utilizes elastomeric stamps\u0026mdash;typically composed of polydimethylsiloxane (PDMS)\u0026mdash;to transfer molecular patterns onto a variety of substrates via conformal contact\u003csup\u003e2,3\u003c/sup\u003e. Its simplicity, low cost, and compatibility with diverse materials have made \u0026micro;CP a widely adopted method in fields such as molecular electronics\u003csup\u003e4\u003c/sup\u003e, surface chemistry\u003csup\u003e5\u0026ndash;7\u003c/sup\u003e, and biosensing\u003csup\u003e8\u0026ndash;11\u003c/sup\u003e. The initial demonstrations of \u0026micro;CP employed rectangular PDMS stamps to pattern microscale gold features. Since then, most \u0026micro;CP studies and applications have used stamps that are vertically symmetrical, such as rectangular or cylindrical shapes\u003csup\u003e12,13\u003c/sup\u003e, whose deformation behaviors have been extensively characterized. For example, prior work\u003csup\u003e14\u003c/sup\u003e has shown that for rectangular stamps, the size of the printed pattern remains constant regardless of the applied load, meaning that a single stamp can only produce one pattern size. Given that PDMS stamp fabrication is typically time-consuming, this constraint introduces significant complexity and delays in the development cycle for application-specific designs.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTo address this limitation and enable more flexible printing capabilities, several strategies have been explored to allow a single PDMS stamp to generate variable pattern sizes. One approach involves multi-pass printing on the same substrate\u003csup\u003e15\u003c/sup\u003e; by rotating the stamp between successive printings, diverse shapes and sizes can be achieved. However, this method suffers from poor alignment accuracy and is generally restricted to simple geometries, such as lines or arrays\u0026mdash;thus limiting its utility for fabricating complex layouts in flexible electronics.\u003c/p\u003e\n\u003cp\u003eAn alternative strategy involves altering the stamp geometry, with pyramidal PDMS stamps being the most widely studied\u003csup\u003e16\u003c/sup\u003e. These stamps enable variable pattern sizes by exploiting their deformation under different applied loads: increasing the load expands the tip contact area with the substrate, and vice versa. Initially developed to print dot-array patterns for protein patterning, pyramidal stamps are typically fabricated via photolithography and anisotropic etching. Subsequent studies have investigated their deformation behavior both experimentally and analytically. For instance, pyramidal PDMS stamps with a base size of 6 \u0026mu;m and height of 4.24 \u0026mu;m have been fabricated and tested under varying pressures\u003csup\u003e17\u003c/sup\u003e. The experimentally measured contact areas closely matched finite element analysis (FEA) predictions, validating their potential for tunable patterning. More recent work\u003csup\u003e18\u003c/sup\u003e has incorporated dynamic modeling using the Johnson-Kendall-Roberts (JKR) contact mechanics framework to describe the deformation during printing. Despite these advances, pyramidal PDMS stamps are still limited to producing arrays of simple shapes (e.g., rectangular dots), and are therefore unsuitable for applications that require complex or arbitrary patterns.\u003c/p\u003e\n\u003cp\u003eInspired by the concept of pyramidal PDMS stamps, which enable variable contact areas through controllable deformation, we explore an alternative geometry: the V-shaped PDMS stamp. This structure offers the potential for continuous modulation of contact area through its load-dependent out-of-plane deformation, making it a promising candidate for variable pattern printing in \u0026micro;CP. Unlike pyramidal PDMS stamps\u0026mdash;which are typically limited to printing dot arrays or regular geometric patterns\u0026mdash;the V-shaped PDMS stamp is capable of printing complex and arbitrarily-shaped circuit layouts, thereby offering greater flexibility for advanced applications in flexible electronics.\u003c/p\u003e\n\u003cp\u003eNotably, V-shaped PDMS stamps have previously been used for \u0026micro;CP\u003csup\u003e19\u003c/sup\u003e, where a two-layer configuration was used: a 2\u0026ndash;4 mm thick Sylgard 184 PDMS substrate supported a 30 \u0026mu;m thick hard PDMS (h-PDMS) film with a V-shaped surface profile. This study demonstrated sub-50 nm feature replication in a plate-to-plate \u0026micro;CP setup under a static load of ~20 g. However, the primary focus of that work was on achieving high-resolution static pattern transfer, rather than enabling continuous pattern size variation through controlled deformation. Furthermore, to date, there has been no systematic investigation into the deformation dynamics of V-shaped PDMS stamps, nor their application in roll-to-roll (R2R) \u0026micro;CP.\u003c/p\u003e\n\u003cp\u003eIn this work, we investigate the feasibility of using V-shaped PDMS stamps to achieve variable pattern sizes in R2R \u0026micro;CP as shown in Fig. 1. The key to enabling tunable printing lies in precisely controlling the out-of-plane deformation of the stamp, which governs the contact area between the stamp and the substrate. In most existing R2R \u0026micro;CP systems, stable contact between the PDMS stamp and the substrate is typically maintained through force-based control schemes, where actuators such as voice coils or stepper motors regulate the vertical contact force\u0026mdash;commonly measured by load cells due to the difficulty of directly sensing interface pressure. For example, a flexure-guided R2R system achieved force variation control within 0.05 N using voice coil actuation\u003csup\u003e20\u003c/sup\u003e. Another system integrated air dampers and step motors in a hybrid configuration, maintaining force stability with a root-mean-square error (RMSE) below 0.25 N through load-cell-based feedback\u003csup\u003e21\u003c/sup\u003e.\u003c/p\u003e\n\u003cp\u003eHowever, our findings reveal that even under precise force control\u0026mdash;achieving a RMSE of 0.05 N\u0026mdash;significant variations in printed pattern size still occur when using V-shaped PDMS stamps (Supplemental Fig. S1). A likely explanation for this variability lies in the limitations of load cell measurements: in addition to the actual contact force, load cells also capture vertical forces induced by web tension. Due to assembly imperfections and coupling effects, the controlled web tension can fluctuate within a range of \u0026plusmn;1 N, effectively doubling the measured force and introducing unintended deformation in the PDMS stamp. This deformation results in inconsistent contact areas and thus undermines pattern transfer fidelity.\u003c/p\u003e\n\u003cp\u003eTo overcome these limitations, we propose a physics-informed displacement control framework that directly targets out-of-plane deformation rather than indirectly inferring it from force. Displacement provides a more robust and physically meaningful control input, as it directly determines the contact geometry between the stamp and the substrate. While some R2R \u0026micro;CP systems incorporate displacement sensing\u0026mdash;for purposes such as register alignment or web tracking\u003csup\u003e22,23\u003c/sup\u003e\u0026mdash;these capabilities have not yet been applied to actively regulate stamp deformation for variable pattern printing. This represents a critical gap in existing process control strategies.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe proposed physics-informed displacement control framework consists of two main steps (Fig. 1b). The first step is to derive a model that captures the relationship between the displacement of the PDMS stamp and the resulting contact area. A straightforward approach to obtaining this model would be to record displacement and contact area data and apply system identification techniques. However, direct observation of the contact region is challenging in typical R2R systems, where the impression and print rollers are metallic and opaque. Previous studies have addressed this by embedding cameras within hollow rollers or using transparent substrates in simplified setups\u003csup\u003e24\u003c/sup\u003e. In this work, we adopt a hybrid method to capture the nonlinear contact mechanics of the stamp under varying displacements, combining a physics-based FEA simulation with a single-roller experimental system. The FEA simulation incorporates the hyperelastic properties of PDMS to accurately represent its mechanical behavior under compression. In the experimental setup, a V-shaped PDMS stamp is wrapped around a rotating roller and brought into contact with a transparent glass or plastic plate mounted on a motorized stage. This configuration enables direct visualization of the contact region and facilitates the acquisition of displacement and contact area data, which are used to validate the FEA simulation results.\u003c/p\u003e\n\u003cp\u003eThe second step involves implementing a neural network-based model predictive control (MPC) strategy (Supplementary S1-S2). A neural network is trained to model the nonlinear dynamics of the R2R system, mapping control inputs to the resulting PDMS displacement. This learned model is then integrated into an MPC framework to regulate the input commands and achieve precise displacement control. Experimental results show that this displacement-driven control system significantly reduces pattern size variability and enables tunable printing using a single PDMS stamp geometry. These findings demonstrate the benefits of displacement-based compression using V-shaped PDMS stamps in enhancing process stability, adaptability, and precision\u0026mdash;advancing the capabilities of R2R \u0026micro;CP for next-generation flexible electronics manufacturing.\u003c/p\u003e"},{"header":"II. METHODS","content":"\u003ch2\u003e\u003cbr\u003e\u0026nbsp;A. V-shaped PDMS Stamp Manufacturing\u003c/h2\u003e\n\u003cp\u003eThe analysis and results in this paper examine a V-shaped stamp with periodic lines on its surface, as shown in the cross-section depicted in Fig. 2a. The stamp features a larger rectangular height 𝑡, integrated with a triangular feature defined by a width 𝑤, and height ℎ. The ratio A represents the relationship between the width and height of the triangular feature. Different ratios result in varying contact areas, corresponding to displacement and force. The design and fabrication process of the V-shaped PDMS is illustrated in Fig. 2. We first use SolidWorks to create the CAD drawing of the master mold for the PDMS, as shown in Fig. 2b. The master mold is then fabricated using a Profluidics P285D 3D printer. The use of Profluidics P285D significantly reduces fabrication time and eliminates the need for a cleanroom. The real mold after 3D printing is shown in Fig. 2e. After printing the master mold, the silicone elastomer base was mixed with the silicone elastomer curing agent in 10:1 weight ratio. The components were thoroughly combined in a clean, dry container for 5 minutes, ensuring a homogeneous mixture by scraping the sides and bottom to incorporate all material. Once mixed, the mixture was degassed to remove trapped air bubbles using a vacuum chamber. Vacuum pressure was applied until bubbles were no longer visible. Following degassing, the mixture was carefully poured into the prepared mold, as shown in Fig. 2c. It was poured slowly and from a low height in a thin stream, starting at one corner, to reduce surface tension and avoid introducing additional bubbles. The PDMS was then cured at 60°C for 4 hours. Once the PDMS was fully cured, the solid V-shaped stamp was gently demolded (Fig. 2d) and inspected for any defects, such as bubbles or incomplete curing, to ensure a high-quality final product. The completed product is shown in Fig. 2f, and Fig. 2g displays microscope images of V-shaped PDMS\u003cbr\u003efor accurate measurements. In the current design, the base thickness \u003cem\u003et\u003c/em\u003e of the rectangular section is 1 mm, while the triangular features have a width \u003cem\u003ew\u003c/em\u003e of 146 µm and a height \u003cem\u003eh\u003c/em\u003e of 73 µm.\u003c/p\u003e\n\u003ch2\u003eB. Physics-informed Deformation Model Derivation\u003c/h2\u003e\n\u003cp\u003eAs the nature limitation of R2R manufacturing system, the contact area between the PDMS and the substrate cannot be directly monitored in-situ. It is necessary to predict the contact area from other indirect methods. Here we use both simulated and experimental data to derive the physical relationship between the displacement applied to the PDMS and the contact area. Thus it is possible to control the contact area through manipulating the applied displacement during R2R µCP. For the FEA simulation, a hyperelastic Neo-Hookean material model was used to represent the PDMS properties\u003csup\u003e25\u003c/sup\u003e. The simulations were conducted using Abaqus, a commercial FEA software widely used for modeling nonlinear materials and complex geometries\u003csup\u003e26\u003c/sup\u003e. The advantage of FEA simulation is that the geometry of the V-shaped PDMS can be changed easily. In this study, we focus on the aspect ratio \u003cem\u003eA\u003c/em\u003e of the triangular cross-section, defined as the ratio between the base and the height. For simplicity, the aspect ratio is varied by keeping the base constant while adjusting the height. For the experimental part, we built a single-roller system to mimic the R2R µCP with the ability to monitor the contact area with a camera (Supplementary S4). The primary components of the single-roller system include a force measuring roller, a camera, a motion stage, and a glass plate. The glass plate was positioned beneath the print pattern and connected to a motion stage, emulating the continuous motion of the R2R system. The high-resolution camera is strategically positioned underneath the glass plate to capture detailed images of the contact area. The displacement was controlled by inserting different pieces of tape under the roller. The acquired contact area images are processed using custom MATLAB code to calculate the contact width for each displacement setting.\u003c/p\u003e\n\u003ch2\u003eC. R2R µCP\u0026nbsp;with Displacement Sensors and Printing Quality Monitoring System\u003c/h2\u003e\n\u003cp\u003eA customized\u0026nbsp;R2R µCP\u0026nbsp;system was developed to evaluate displacement control for printing variable pattern sizes\u003csup\u003e27\u003c/sup\u003e. In contrast to conventional systems, this setup integrates two displacement sensors specifically designed to measure the vertical displacement of the V-shaped PDMS stamp (Supplementary S3). Each sensor comprises a metal plate and a capacitive sensor (CS3, Micro-Epsilon): the metal plates are affixed to the print roller and move vertically with it, while the capacitive sensors are mounted on the frame using an adaptor. Since the capacitance between the metal plate and the sensor varies linearly with their separation, this configuration enables accurate measurement of the PDMS displacement. The measurement resolution is approximately 5 nm. Prior to each experiment, the system is zeroed to ensure that the displacement readings reflect only the deformation of the PDMS stamp. This is achieved by gradually raising the impression roller until a change in the load cell output is detected—indicating contact between the PDMS and the print roller—which is then marked as the zero-displacement position.\u003c/p\u003e\n\u003cp\u003eTo monitor the printing results, a condensation figure (CF) monitoring system (Supplementary Fig.S5) was integrated into the R2R platform. During printing, areas covered by ink exhibit higher surface energy than unprinted gold regions, leading to different droplet formation behavior when exposed to water vapor\u003csup\u003e28\u003c/sup\u003e. Specifically, droplets formed on gold surfaces are denser and larger than those on printed areas, enabling in-situ visual differentiation of the printed patterns under CFs prior to the etching step. The captured CF images are subsequently segmented to extract the linewidth information, as illustrated in Supplementary Fig. S6.\u003c/p\u003e\n\u003ch2\u003eC. Neural Network-based MPC Control\u003c/h2\u003e\n\u003cp\u003eTo achieve high-precision control in R2R µCP, our system incorporates a flexure-based mechanism. Unlike traditional mechanical linkages that depend on sliding or rolling contact between components, flexure mechanisms achieve motion through the elastic deformation of compliant elements. This design enables superior positioning accuracy due to the elimination of backlash and friction. Consequently, flexures are widely employed in high-resolution R2R printing systems to enhance motion and force control. However, accurately modeling the dynamics of complex flexure mechanisms poses significant challenges. The physical modeling process is often time-consuming and computationally intensive, requiring the identification of numerous parameters and their nonlinear interdependencies\u003csup\u003e29\u003c/sup\u003e. Moreover, the intrinsic nonlinearities of flexure systems hinder the development of precise analytical models, which in turn limits the effectiveness of traditional controllers. Prior studies have demonstrated that simple PID controllers are inadequate for managing the high dynamics and nonlinear behavior inherent to R2R µCP\u003csup\u003e27\u003c/sup\u003e. To address this issue, we propose a neural network-based MPC strategy for flexure-based position control. Specifically, we employ a feedforward neural network to approximate the nonlinear system dynamics. The network takes as input a sequence of delayed control signals (actuator voltages) and system outputs (measured displacements), and outputs a prediction of the displacement. This data-driven model is then embedded into an MPC framework to optimize control inputs in real-time, ensuring accurate tracking of the desired displacement trajectory (See Supplementary S1 and S2 for detailed description).\u003c/p\u003e\n\u003ch2\u003eD. R2R Microcontact Printing \u0026nbsp;\u003c/h2\u003e\n\u003cp\u003eTo evaluate the performance of the V-shaped PDMS stamps under realistic operating conditions, a standard\u0026nbsp;R2R µCP\u0026nbsp;process was employed. The three essential components\u0026nbsp;\u003cbr\u003einvolved in the printing process were the ink, the stamp, and the substrate. The ink was prepared by dissolving hexadecanethiol (HDT) in ethanol to a concentration of 15 mM. A custom-designed ink reservoir was positioned beneath the print roller to facilitate consistent ink delivery. The V-shaped PDMS stamps were fabricated following the procedure described earlier. The substrate used for printing was a 0.1 mm thick polyethylene terephthalate (PET) film coated with a 30 nm gold layer, deposited via R2R sputtering process.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe printing operation was fully automated. After wrapping the V-shaped PDMS stamp around the print roller, taping the PET substrate to the web, and loading the ink into the reservoir, the system was initialized through software. Web tension, web speed, and stamp displacement were configured in advance. Upon starting the run, the ink reservoir was lifted using a motorized stage to transfer ink to the stamp surface. The voice coil actuators, controlled by the neural network-based MPC, were activated to apply the desired compressive displacement to the stamp. The target displacement was derived from the desired pattern width using the physics-informed displacement model that relates stamp deformation to contact width, followed by the activation of web transport. Throughout the process, the inline monitoring system is used to assess printing quality in real time.\u003c/p\u003e\n\u003cp\u003eAfter printing, the patterned substrate underwent wet chemical etching to finalize the printed structures. The etching\u0026nbsp;\u003cbr\u003eprocess selectively removed gold from areas not protected by HDT. The etchant solution consisted of 0.1 M thiourea and 0.01 M ferric nitrate in deionized water, with an etching duration of 8 minutes.\u003c/p\u003e"},{"header":"III. REULSTS AND DISCCUSIONS ","content":"\u003ch2\u003eA. V-shaped PDMS Stamp Deformation\u003c/h2\u003e\n\u003cp\u003eDue to the nature of the R2R µCP process, the contact area between the stamp and substrate cannot be directly monitored in real time. To address this challenge, we propose an indirect approach by estimating the contact area through measurements of the stamp's displacement. To establish the relationship between displacement and contact area, we employed a combined numerical and experimental method. In the simulation, we examined PDMS stamps with various aspect ratios, and the results are presented in Fig. 3b. The data reveal a strong correlation between displacement and contact area, which is expected because the primary deformation occurs at the triangular tip of the stamp rather than in the bulk of the PDMS body during compression.\u003c/p\u003e\n\u003cp\u003eTo validate the simulation, we fabricated a PDMS stamp with an aspect ratio of 0.5—matching one of the simulated geometries—using our custom-built single-roller system. This system allows precise control and measurement of contact area under varying displacements. The experimental results, overlaid in the same figure, show strong agreement with the simulation across the full displacement range. This consistency confirms the accuracy of the numerical model and demonstrates its practical value in guiding the R2R µCP process. These collected data will be used to derive the dynamics between the displacement and contact area. Specifically, for a target line width required by a given application, the model enables determination of the appropriate stamp displacement, thereby improving process predictability and control.\u003c/p\u003e\n\u003ch2\u003eB. Training of the Neural Network\u003c/h2\u003e\n\u003cp\u003eBased on the physics-informed model, the required stamp displacement for a given target line width can be determined in advance. However, in the actual printing process, achieving this displacement with high precision depends on the mechanical performance of the R2R printing system. To characterize and capture the dynamic behavior of stamp displacement within our system, we developed a neural network-based model. This data-driven approach enables accurate modeling of the system's displacement response, facilitating closed-loop control for precise pattern transfer during R2R µCP. To generate the training dataset for the neural network model, we employed a pulse-train excitation method. Figures 4a and 4b show the input pulse sequence with randomly varying amplitudes and the corresponding displacement response of the system. Each pulse lasts for one second, allowing the system to reach a steady-state response within each interval. The sampling interval is set to 10 ms, consistent with the control loop rate. The input voltages to the voice coil actuators are randomly selected within the range of [1.5 V, 2.5 V]. These bounds were established by first identifying the minimum voltage at which a measurable force is detected by the load cells and then adding an additional 1 V to ensure that the resulting displacement remains within approximately 150 μm. To sufficiently span the dynamic range of the system, a total of 60,000 samples were collected, with 80% of the data used for training and the remaining 20% reserved for testing.\u003c/p\u003e\n\u003cp\u003eSince the displacement behavior depends on both historical output values and input control signals, the system dynamics are approximated using a Nonlinear AutoRegressive model with eXogenous inputs (NARX), as detailed in the Supplementary S1. The input and output time delays are set to 2. The neural network contains 7 hidden nodes, selected as a trade-off between inference time and prediction accuracy. A tangent-sigmoid activation function is used, as the input data are normalized within the range [−1, 1]. The network is trained using the Levenberg–Marquardt optimization algorithm in MATLAB R2023b. The prediction performance on the testing dataset is illustrated in Fig. 4c and Fig. 4d\u0026nbsp;, achieving an RMSE of 0.88 μm. Additional comparisons with Long Short-Term Memory (LSTM) networks are provided in the Supplementary Fig. S3.\u003c/p\u003e\n\u003ch2\u003eC. Displacement Control Performance\u003c/h2\u003e\n\u003cp\u003eAfter developing the neural network model of the displacement behavior of the R2R system, we implemented a neural network-based MPC to control the displacement of the V-shaped PDMS stamp. Initial simulations were conducted to tune key MPC parameters, including the control horizon M, prediction horizon P, and the weighting factor λ, as detailed in the Supplementary S2.\u003c/p\u003e\n\u003cp\u003eTo determine suitable values for M and P, both were initially set to one sampling interval and incrementally increased from 1 to 20 intervals. A stable control performance was observed when M≥2. Through iterative tuning, M=4 and P=11 were selected as the final values, yielding desirable real-time force control (Supplementary Table S1). Given that the control loop operates at 10 ms, the total computation time for the MPC must remain below this threshold. Timing tests confirmed that the MPC runtime with M=4 and P=11 is approximately 7 ms, satisfying the real-time constraint (Supplementary Table S2).\u003c/p\u003e\n\u003cp\u003eThe optimized neural network-based MPC controller was evaluated using sinusoidal reference signals as displacement setpoints. The control results are presented in Fig. 4e. It is important to note that the length of a single V-shaped PDMS stamp is shorter than the circumference of the print roller, requiring two stamps to be stitched together. Consequently, signal artifacts appear at the junction between the two stamps. Nevertheless, as shown in the enlarged view in Fig. 4e, the actual displacement closely follows the reference trajectory, achieving an RMSE of 5 μm. This demonstrates the high accuracy and responsiveness of the neural network-based MPC controller in displacement regulation for R2R µCP.\u003c/p\u003e\n\u003ch2\u003eD. Variable Line Width Printing Results \u0026nbsp;\u003c/h2\u003e\n\u003cp\u003eTo demonstrate the versatility of the V-shaped PDMS stamp, we printed three distinct line widths by applying different displacement values to a single stamp. The same stamp, with an aspect ratio of 0.5, was used throughout the experiments. To monitor printing quality in a non-destructive and substrate-efficient manner, CFs were captured using our custom-developed imaging system, enabling direct line width measurement without the need for etching or post-processing. The printing was performed using controlled displacements of 10 µm, 50 µm, and 100 µm. The resulting CFs of the corresponding printed patterns are shown in Fig. 5a. The measured line widths are approximately 6 μm, 24 μm, and 40 μm from left to right, respectively, illustrating the stamp’s capability to produce variable features through displacement control.\u003c/p\u003e\n\u003cp\u003eAdditionally, we performed continuous printing of variable line widths by dynamically adjusting the displacement setpoint (Supplementary Movie 1). In this experiment, a stepwise varying displacement profile was applied to the controller, and the corresponding line widths were predicted based on the previously established displacement-to-width relationship. The printed results and corresponding CFs are presented in Fig. 5b. The predicted widths were computed from the recorded displacement, while the measured widths were extracted from the CFs. Due to the 2 Hz frame rate of the imaging system, the measured widths exhibit higher fluctuations compared to the predicted values. Nevertheless, the predicted and measured widths show strong agreement, confirming the system's capability for real-time control of variable-width patterning using a single V-shaped PDMS stamp.\u003c/p\u003e\n\u003ch2\u003eE. Antenna printing using V-shaped PDMS\u003c/h2\u003e\n\u003cp\u003eFinally, to demonstrate the feasibility and versatility of our approach, we used the V-shaped PDMS stamp to print more complex geometries—specifically, two different antenna designs—on flexible substrates. First, we designed a Planar Inverted-F Antenna (PIFA) consisting solely of straight-line segments. Two different line widths were printed using a single stamp with linewidths of 34 µm and 22 µm as shown in Fig. 5c. The results show that the patterns can be proportionally scaled by adjusting the displacement, which is advantageous for parametric studies.\u003c/p\u003e\n\u003cp\u003eTo further demonstrate the practical utility of the V-shaped PDMS stamp, we also designed and printed a spiral-shaped antenna (Supplementary Fig. S9). The mold for the spiral pattern was created using the Cut Sweep function in SolidWorks. When different displacements were applied during printing, the linewidth of the spiral pattern changed accordingly, validating the controllable feature size achievable with our displacement-based printing approach.\u003c/p\u003e\n\u003ch2\u003eC. Limitations and future work\u003c/h2\u003e\n\u003cp\u003eThe V-shaped PDMS stamps used in this study were fabricated using a commercial printer with a resolution of 30 µm, which imposes a fundamental limit on the resolution of the resulting printed patterns. In the literature, alternative fabrication techniques have been demonstrated to achieve higher-resolution molds for PDMS casting. For instance, V-shaped molds can be produced via photolithography followed by wet etching of silicon, enabling sub-100 nm apex dimensions at the base of the V-shaped trenches\u003csup\u003e30\u003c/sup\u003e. When such nanoscale features are employed in printing, the displacement control system must also offer commensurate resolution to maintain pattern fidelity. In our current setup, the displacement control resolution is approximately 5 µm, which corresponds to about 2 µm in line width variation for the V-shaped PDMS stamp used in our experiments. This limitation is partly due to the mechanical components of the R2R system being fabricated in an academic machine shop, where precision machining tolerances are difficult to achieve. Future work will focus on enhancing the mechanical precision of the R2R platform, potentially through the integration of high-accuracy, professionally machined components to improve displacement control and overall printing resolution.\u003c/p\u003e"},{"header":"IV. CONCLUSIONS","content":"\u003cp\u003eIn this work, we demonstrate that V-shaped PDMS stamps can be used in R2R µCP to enable variable pattern size printing. Since force-based control cannot precisely regulate the deformation of PDMS stamps, we adopt a displacement-based control strategy to achieve stable and tunable line width printing. The physical relationship between stamp displacement and contact area is established through a combination of simulation and experimental data. The simulation employs a Neo-Hookean model to represent the hyperelastic properties of PDMS, and stamps with varying aspect ratios are analyzed. To validate the simulation results, we utilize a single-roller system that mimics the actual R2R printing process. This setup allows for simultaneous measurement of displacement and contact area, enabling direct comparison with simulation outputs. Together, the experimental and simulation data form the physics-informed model of PDMS deformation. Building on this model, we develop a neural network-based MPC framework to regulate the displacement of the PDMS stamp in the R2R system. The displacement can be precisely controlled using the proposed control algorithm, allowing us to print lines with different widths using a single PDMS stamp and demonstrating the effectiveness of the proposed control method. Finally, we print two types of antenna patterns—linear and spiral—further illustrating the flexibility and broad application potential of the system.\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eAcknowledgements\u003c/span\u003e \u003c/h2\u003e \u003cp\u003eThis work was supported by NSF Foundation CMMI-1942185.\u003c/p\u003e \u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eAuthor contribution\u003c/span\u003e \u003c/p\u003e \u003cp\u003eJ.Y. conducted all experiments and wrote the initial draft of the manuscript. H.D. performed the finite element analysis and contributed to part of the experiments. X.D. provided project supervision, funding acquisition, and revised the manuscript.\u003c/p\u003e \u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eCompeting interests\u003c/span\u003e \u003c/p\u003e \u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eKumar, A. \u0026amp; Whitesides, G. M. Features of gold having micrometer to centimeter dimensions can be formed through a combination of stamping with an elastomeric stamp and an alkanethiol \u0026lsquo;\u0026ldquo;ink\u0026rdquo;\u0026rsquo; followed by chemical etching. \u003cem\u003eApplied Physics Letters\u003c/em\u003e 63, 2002\u0026ndash;2004 (1993).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePallab, N. \u003cem\u003eet al.\u003c/em\u003e Polymer brush-assisted microcontact printing: using a tailor-made polydimethylsiloxane (PDMS) stamp for precise patterning of rough surfaces. \u003cem\u003ePolymer Chemistry\u003c/em\u003e 15, 853\u0026ndash;867 (2024).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMukherjee, A. \u003cem\u003eet al.\u003c/em\u003e Synthesis and Microcontact Printing of novel nanoflowers of ZnO_Ag_rGO nanocomposite to check microbial adhesion. \u003cem\u003eNano-Structures \u0026amp; Nano-Objects\u003c/em\u003e 38, 101140 (2024).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eTorres-Cavanillas, R., Gavara-Edo, M. \u0026amp; Coronado, E. Bistable Spin-Crossover Nanoparticles for Molecular Electronics. \u003cem\u003eAdvanced Materials\u003c/em\u003e 36, 2307718 (2024).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eDu, B. \u003cem\u003eet al.\u003c/em\u003e Combined laser-induced graphene and microcontact printing for processing scalable and stackable micro-stripe patterns toward multifunctional electronic devices. \u003cem\u003eCarbon\u003c/em\u003e 225, 119148 (2024).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eG\u0026ouml;drich, S., Brodoceanu, D., Kuznetsov, V., Kraus, T. \u0026amp; Papastavrou, G. Defined Transfer of Colloidal Particles by Electrochemical Microcontact Printing. \u003cem\u003eAdvanced Materials Interfaces\u003c/em\u003e 11, 2400202 (2024).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLarra\u0026ntilde;aga, E. \u003cem\u003eet al.\u003c/em\u003e Long-range organization of intestinal 2D-crypts using exogenous Wnt3a micropatterning. \u003cem\u003eNat Commun\u003c/em\u003e 16, 382 (2025).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSun, M., Zhang, J., Xuanyuan, T., Liu, X. \u0026amp; Liu, W. Facile and Rapid Microcontact Printing of Additive-Free Polydimethylsiloxane for Biological Patterning Diversity. \u003cem\u003eACS Appl. Mater. Interfaces\u003c/em\u003e 16, 20132\u0026ndash;20142 (2024).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKhan, S. \u003cem\u003eet al.\u003c/em\u003e Comprehensive fluorescence profiles of contamination-prone foods applied to the design of microcontact-printed in situ functional oligonucleotide sensors. \u003cem\u003eSci Rep\u003c/em\u003e 14, 8277 (2024).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eHager, R. \u003cem\u003eet al.\u003c/em\u003e Microcontact Printing of Biomolecules on Various Polymeric Substrates: Limitations and Applicability for Fluorescence Microscopy and Subcellular Micropatterning Assays. \u003cem\u003eACS Appl. Polym. Mater.\u003c/em\u003e 4, 6887\u0026ndash;6896 (2022).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eShiwarski, D. J. \u003cem\u003eet al.\u003c/em\u003e Fibronectin-based nanomechanical biosensors to map 3D surface strains in live cells and tissue. \u003cem\u003eNat Commun\u003c/em\u003e 11, 5883 (2020).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eGaribyan, M. \u003cem\u003eet al.\u003c/em\u003e Engineering programmable material-to-cell pathways via synthetic notch receptors to spatially control differentiation in multicellular constructs. \u003cem\u003eNat Commun\u003c/em\u003e 15, 5891 (2024).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eXue, X. \u003cem\u003eet al.\u003c/em\u003e A patterned human neural tube model using microfluidic gradients. \u003cem\u003eNature\u003c/em\u003e 628, 391\u0026ndash;399 (2024).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePetrzelka, J. E. \u0026amp; Hardt, D. E. Static load-displacement behavior of PDMS microfeatures for soft lithography. \u003cem\u003eJ. Micromech. Microeng.\u003c/em\u003e 22, 075015 (2012).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eXia, Y. \u0026amp; Whitesides, G. M. Extending Microcontact Printing as a Microlithographic Technique. \u003cem\u003eLangmuir\u003c/em\u003e 13, 2059\u0026ndash;2067 (1997).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eFilipponi, L., Livingston, P., Kašpar, O., Tok\u0026aacute;rov\u0026aacute;, V. \u0026amp; Nicolau, D. V. Protein patterning by microcontact printing using pyramidal PDMS stamps. \u003cem\u003eBiomed Microdevices\u003c/em\u003e 18, 9 (2016).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eHong, J. M., Ozkeskin, F. M. \u0026amp; Zou, J. A micromachined elastomeric tip array for contact printing with variable dot size and density. \u003cem\u003eJ. Micromech. Microeng.\u003c/em\u003e 18, 015003 (2008).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJin, C. \u0026amp; Qiao, Q. Deformation of Pyramidal PDMS Stamps During Microcontact Printing. \u003cem\u003eJournal of Applied Mechanics\u003c/em\u003e 83, (2016).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLi, H.-W., Muir, B. V. O., Fichet, G. \u0026amp; Huck, W. T. S. Nanocontact Printing: A Route to Sub-50-nm-Scale Chemical and Biological Patterning. \u003cem\u003eLangmuir\u003c/em\u003e 19, 1963\u0026ndash;1965 (2003).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eZhou, X., Xu, H., Cheng, J., Zhao, N. \u0026amp; Chen, S.-C. Flexure-based Roll-to-roll Platform: A Practical Solution for Realizing Large-area Microcontact Printing. \u003cem\u003eSci Rep\u003c/em\u003e 5, 10402 (2015).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eChoi, Y.-M., Kang, D., Lim, S., Lee, M. G. \u0026amp; Lee, S.-H. High-Precision Printing Force Control System for Roll-to-Roll Manufacturing. \u003cem\u003eIEEE/ASME Transactions on Mechatronics\u003c/em\u003e 22, 2351\u0026ndash;2358 (2017).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLee, J., Shin, K. \u0026amp; Jung, H. Control Scheme for Rapidly Responding Register Controller Using Response Acceleration Input in Industrial Roll-To-Roll Manufacturing Systems. \u003cem\u003eIEEE Transactions on Industrial Electronics\u003c/em\u003e 69, 5215\u0026ndash;5224 (2022).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eChen, Z., Zheng, Y., Zhang, T., Wong, D. S.-H. \u0026amp; Deng, Z. Modeling and Register Control of the Speed-Up Phase in Roll-to-Roll Printing Systems. \u003cem\u003eIEEE Transactions on Automation Science and Engineering\u003c/em\u003e 16, 1438\u0026ndash;1449 (2019).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eNill, S. T. (Scott T. Integrated hardware, software, and sensor design for control of a scalable, continuous roll-to-roll microcontact printing process. (Massachusetts Institute of Technology, 2014).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePetrzelka, J. E. Contact region fidelity, sensitivity, and control in roll-based soft lithography. (Massachusetts Institute of Technology, 2012).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePeng, J., Tomsia, A. P., Jiang, L., Tang, B. Z. \u0026amp; Cheng, Q. Stiff and tough PDMS-MMT layered nanocomposites visualized by AIE luminogens. \u003cem\u003eNat Commun\u003c/em\u003e 12, 4539 (2021).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eYan, J. \u0026amp; Du, X. Neural-Network-Based Adaptive Model Predictive Control for a Flexure-Based Roll-to-Roll Contact Printing System. \u003cem\u003eIEEE/ASME Transactions on Mechatronics\u003c/em\u003e 27, 5084\u0026ndash;5094 (2022).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eYan, J., Ma, R. \u0026amp; Du, X. Consistent optical surface inspection based on open environment droplet size-controlled condensation figures. \u003cem\u003eMeas. Sci. Technol.\u003c/em\u003e 32, 105405 (2021).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eZhou, X., Wang, D., Wang, J. \u0026amp; Chen, S.-C. Precision design and control of a flexure-based roll-to-roll printing system. \u003cem\u003ePrecision Engineering\u003c/em\u003e 45, 332\u0026ndash;341 (2016).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eWu, J., Tao, K. \u0026amp; Miao, J. Production of centimeter-scale sub-wavelength nanopatterns by controlling the light path of adhesive photomasks. \u003cem\u003eJ. Mater. Chem. C\u003c/em\u003e 3, 6796\u0026ndash;6808 (2015).\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nature-portfolio","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"","title":"Nature Portfolio","twitterHandle":"","acdcEnabled":false,"dfaEnabled":false,"editorialSystem":"ejp","reportingPortfolio":"","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-6780648/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6780648/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eRoll-to-roll microcontact printing enables high-throughput production of flexible electronic devices by continuously transferring inks onto substrates via polydimethylsiloxane (PDMS) stamps. Traditional rectangular or cylindrical PDMS stamps yield uniform pattern sizes, limiting manufacturing versatility. This study introduces V-shaped PDMS stamps for variable pattern printing using a single stamp geometry. A physics-based deformation model was developed by combining finite element simulations and experiments to characterize the out-of-plane behavior of V-shaped PDMS under displacement. Leveraging this model, we implemented a neural network-based Model Predictive Control (MPC) system to precisely regulate vertical displacement and achieve desired pattern dimensions. Experimental results demonstrate that a single V-shaped PDMS stamp can reliably produce variable pattern sizes with high repeatability, significantly improving the adaptability and process efficiency of microcontact printing for flexible electronics manufacturing.\u003c/p\u003e","manuscriptTitle":"Physics-Informed Displacement Control for Variable Pattern Printing with V-shaped PDMS Stamps in Roll-to-Roll Microcontact Printing","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-06-25 08:43:12","doi":"10.21203/rs.3.rs-6780648/v1","editorialEvents":[],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"communications-engineering","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"commseng","sideBox":"Learn more about [Communications Engineering](http://link.springer.com/journal/44172)","snPcode":"44172","submissionUrl":"https://mts-commseng.nature.com/cgi-bin/main.plex","title":"Communications Engineering","twitterHandle":"@commseng","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"448a39b4-aab8-447f-be6b-2f2183502f4d","owner":[],"postedDate":"June 25th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":50453666,"name":"Physical sciences/Engineering/Mechanical engineering"},{"id":50453667,"name":"Physical sciences/Engineering/Electrical and electronic engineering"}],"tags":[],"updatedAt":"2025-12-24T08:10:04+00:00","versionOfRecord":{"articleIdentity":"rs-6780648","link":"https://doi.org/10.1038/s44172-025-00553-9","journal":{"identity":"communications-engineering","isVorOnly":false,"title":"Communications Engineering"},"publishedOn":"2025-12-11 05:00:00","publishedOnDateReadable":"December 11th, 2025"},"versionCreatedAt":"2025-06-25 08:43:12","video":"","vorDoi":"10.1038/s44172-025-00553-9","vorDoiUrl":"https://doi.org/10.1038/s44172-025-00553-9","workflowStages":[]},"version":"v1","identity":"rs-6780648","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6780648","identity":"rs-6780648","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}