# Web Unevenness Due to Thermal Deformation in the Roll-to-Roll Manufacturing Process
## 2 Finite-Element Analysis: Thermal-Structural Coupling Analysis
### Simulation Result  
Figure 2 presents the distribution of deformation (unit: mm) in the tensioned web along the y and z directions, expressed as contour plots. To plot Figure 2a-d, web speed and tension of 1 m/min and 1 kgf were used, respectively. Drying-temperature levels were set to 30, 60, 90, and 120 \({}^{\circ}\)C. The length of the web model, that is, the distance between the slot-die coater and the inlet of the dryer, was 0.5 m. Deformations were magnified 20 times to clearly observe their distributions in the tensioned web according to drying temperature. The legend at the right of each contour plot shows the web deformation with respect to the color of the contour plot. The white dotted line in all figures represents the inlet of the dryer. Figure 2a shows that many fewer deformations in the web occurred at the drying temperature of 30 \({}^{\circ}\)C compared to those at 60 \({}^{\circ}\)C. The web was compressed by tensile stress in the transverse direction before the dryer inlet; however, the web expanded due to thermal expansion after passing through the dryer inlet. Thermal deformation in the z direction increased more rapidly than that in the y direction as drying temperature increased. In particular, one can clearly see sine-wave-shaped web unevenness after the inlet of the dryer at the 60 \({}^{\circ}\)C drying temperature, as shown in Figure 2a-d. Table 3 shows the maximal deformation in the y and z directions according to drying temperature. Results show that drying temperature dominantly affected web deformation in the z direction compared to in the y direction. Web unevenness was generated by the thermal deformation caused by conduction and convection heat from the dryer.  
Figure 1: Schematic of flexible substrate in roll-to-roll (R2R) process. (**a**) Dryer; (**b**) detailed view of dryer inlet; (**c**) tension distribution on flexible substrate.  
Figure 3 indicates the deformation of the web in the y and z directions according to drying temperature. The x and y axes of the plots indicate drying temperature (\({}^{\circ}\)C) and web deformation (mm), respectively. Deformation in the y direction increased and remained constant after 90 \({}^{\circ}\)C. However, web deformation in the z direction steadily rose with increasing drying temperature. In particular,  
\begin{table}
\begin{tabular}{c c c} \hline
**Temperature (\({}^{\circ}\)C)** & **Y Deformation (\(\times\)10\({}^{-3}\); mm)** & **Z Deformation (\(\times\)10\({}^{-3}\); mm)** \\ \hline
30 & 5.4697 & 0.008 \\
60 & 38.869 & 77.428 \\
90 & 85.126 & 628.46 \\
120 & 85.299 & 780.39 \\ \hline \end{tabular}
\end{table}
Table 3: Mechanical and thermal deformations of PET flexible substrate.  
Figure 2: Y and Z axis deformation behaviors of flexible substrate at: (**a**) 30 \({}^{\circ}\)C, (**b**) 60 \({}^{\circ}\)C, (**c**) 90 \({}^{\circ}\)C, and (**d**) 120 \({}^{\circ}\)C.  
this deformation rapidly increased in the drying region of 60-90 \({}^{\circ}\)C. Given that thermal deformation generated web unevenness, severe web unevenness could occur above a drying temperature of 60 \({}^{\circ}\)C. Web unevenness spread to the coating section, which generated coating defects due to the unstable coating bead and nonuniform web surface.  
Figure 4a,b show the web moving distance according to web moving time in the dryer, and the distribution of the consequent web unevenness. Web speed, tension, and drying temperature were 1 m/min, 1 kgf, and 120 \({}^{\circ}\)C, respectively. Web unevenness was generated 3.05 s after the web entered the dryer. The amplitudes and positions of the peaks and valleys of web unevenness changed with respect to web moving time. Results suggested that the dried brittle layer on which the coating defect had occurred was prone to cracking by bending due to the change in web unevenness according to web moving time.  
Figure 4: (**a**) Web moving distance over time; (**b**) distribution of consequent web unevenness (cross-sectional view).  
Figure 3: Deformation at temperature domain in (**a**) y and (**b**) z directions.