A thermomechanical approach to mold filling inroll-to-roll nanoimprinting lithography
Juan P. Gomez-Constante, Prabhakar R. Pagilla and Kumbakonam R. Rajagopal{juanpablogomezc, ppagilla, krajagopal}@tamu.edu
Department of Mechanical Engineering, Texas A&M University
College Station, TX USA 77843

In a typical Roll-to-Roll NanoImprinting Lithography (R2RNIL) process, a substrate coated with afluid film is continuously transported through the nip region between the mold roller and the backuproller. The process involves the following sequential steps: filling of the mold pattern with the fluid
film on the substrate (mold filling), the phase change from liquid to solid of the fluid in the moldpattern, and the debonding of the patterned surface from the mold roller; see the illustration
provided in Figure 1. One critical aspect of large area patterning using R2RNIL is to obtain quality
imprinted features at reasonable web transport speeds (> 10 m/min). This has been a challenge dueto the lack of appropriate models of the process which could be employed to evaluate the effect of
key process and transport parameters, such as web speed, pattern geometry, film thickness,viscosity, stress relaxation time, etc. From a mechanics viewpoint, most existing models have one
or more of the following deficiencies: lack of physical compliance with conservation laws andmaterial properties, consideration of the free surface that is formed inside the mold pattern, and theevolutionary nature of the problem.
A preliminary model that addresses the aforementioned deficiencies with regard to moldfilling was described in [1]. In this paper, we will improve this model by applying thethermomechanical framework developed in [2]. Employing this framework and considering
specific forms of the stored energy and rate dissipations functions, we derive the constitutive
equations which along with the conservation laws of continuum mechanics describe the mold filling
process of viscoelastic fluid films at the nanoscale. Kinematic equations associated with thegeometry of the mold pattern, rollers, and the substrate with the fluid film are derived and integrated
into the governing equations; see the illustration provided in Figure 2. The available time for moldfilling is defined based on these kinematic equations. To show the qualitative behavior of theevolution of mold filling within the pattern via numerical simulations, we assume that thedisplacement gradient is small to derive linearized governing equations with dynamic boundary
conditions. A numerical scheme based on the Marker and Cell method first proposed in [3] isderived for the R2RNIL process to solve the governing equations and obtain the evolution of thedynamic free surface of the fluid film in the mold cavity. A number of numerical simulations wereconducted with this numerical scheme, and Figure 3 provides the non-dimensional pressure andstrain for one pattern feature at the end of the mold filling phase. The proposed model is capableof predicting the evolution of the free surface profile within the pattern feature. The model may be
employed to study the effects of various key process, material and transport parameters, which
could be utilized for the process and control design. Although the focus in this paper is on the moldfilling process only, the problem formulation will set the theoretical background and generalassumptions that can be applied to phase change and demolding aspects of the process.
References:[1] Gomez-Constante, J. P., Pagilla, P. R., and Rajagopal, K. R., 2019, “Effect of the Processing

and Transport Parameters on Mold Filling in Roll-to-Roll Nanoimprint Lithography,” Proc. of the15th Intl. Web Handling Conference, Stillwater, OK.
[2] Rajagopal, K.R. and Srinivasa, A.R., 2000. “A thermodynamic frame work for rate type fluidmodels”. Journal of Non-Newtonian Fluid Mechanics, 88(3), pp.207-227.
[3] Harlow, F.H. and Welch, J.E., 1965. Numerical calculation of time‐ dependent viscousincompressible flow of fluid with free surface. The physics of fluids, 8(12), pp.2182-2189.

Figure 1. Mechanical sub-processes of R2RNIL process.

Figure 2. Kinematics and spatial domain for one feature.

Figure 3. Non-dimensional pressure and strain for one feature at the end of mold filling process.