**5. Conclusions**

In this study, we analyzed the non-uniform strain distribution in a web in the roll-to-roll process. In addition, to mitigate the uneven strain distribution, we designed an optimized roller. According to the experimental results, strain uniformity is improved by approximately 48% with the proposed design compared to the conventional roller.

If the strain distribution in the transported web is non-uniform, a register error may occur in the machine direction (MD or transfer direction) of the web, reducing the shape accuracy or position precision of the print pattern. The distribution of the strain generated in a web being transferred under uniform tension is symmetrical along the width direction (CD) of the web with respect to the centerline of the web in the MD. Around this centerline, the strain is high and decreases toward both sides of the web. This deviation decreases as the web moves away from the roller and increases with the increasing web width. Improving this uniformity is important for productivity and precision.

In this study, we designed a novel roller to minimize the strain deviation in the MD. This roller consists of a rigid core and an elastic outer skin. The core is barrel-shaped and covered with the elastic skin. The assembly is a cylinder with a constant diameter. This configuration reduces the strain deviation by changing the contact sti ffness between the web and the roller.

First, we used an FE model composed of a roller and web to simulate the strain due to tension in an actual process; based on this model, we optimized the roller design. For this purpose, we varied the thickness gradient of the skin, the total diameter of the roller, and the physical properties of the skin. Using DOE and ANOVA, we optimized the design to minimize the MD strain distribution at the printing line 50 mm from the roller central axis. The total diameter of the roller interacts with the thickness gradient; in addition, the mechanical properties of the skin interact with the diameter. The barrel roller with a thickness gradient of 8 mm, a roller diameter of 60 mm, and an elastic modulus of 2.85 MPa minimizes the strain distribution.

To compare the strain distributions of the conventional and optimized barrel rollers in the MD, we conducted a verification experiment. The rollers optimized with the FEA and DOE were employed, and the experiment was repeated three times for tensions of 3–5 MPa. The conventional and barrel rollers produce deviations of 8.54 and 4.42 μm, respectively. Thus, the compensation performance is improved by 48% by the barrel roller.

Our experiments had certain limitations; the center area could not be measured due to the mechanical interference between the vision cameras. This should be fixed with a jig in the future.

Despite the experimental limitations, the strain distribution is more uniform with the optimized than with the conventional roller. This is expected to improve the printing precision of fine patterns and to mitigate register errors in the multilayer printing process. The results of this study can promote the introduction of the roll-to-roll process into the industrial fabrication processes of FPCBs and displays.

**Author Contributions:** Conceptualization, funding acquisition, writing—review and editing, M.G.L.; writing—original draft, data curation, formal analysis, methodology, visualization, Y.K.; investigation, Y.K. and G.E.K.; supervision, project administration, M.G.L. and Y.J.; resources, S.K.; validation, M.G.L., Y.J., and H.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study is based on work supported by the Ministry of Trade, Industry, and Energy (MOTIE, Korea) under the Industrial Technology Innovation Program No. 20000665. Development of an ecofriendly and highly durable surface treatment for super-omniphobic substrate on the large area over 4 m2.

**Conflicts of Interest:** The authors declare no conflict of interest.
