# Simulating the Residual Layer Thickness in Roll-to-Plate Nanoimprinting with Tensioned Webs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Model Development

#### 2.1.1. Elastic Deformation

#### 2.1.2. Hydrodynamic Lubrication

#### 2.1.3. Web Tension

#### 2.1.4. Load Balance

#### 2.1.5. Numerical Implementation

^{®}[42]. The Reynolds equation in Equation (6) and web tension equation in Equation (13) are discretized using second-order (quadratic) Lagrangian finite elements. The linear elasticity equations in Equation (3) are discretized using third-order (cubic) Lagrangian finite elements, to allow for a smooth second-order derivative of the curvature in Equation (13). The load balance equation is a simple ordinary integral equation, which is associated with the unknown constant film thickness gap ${h}_{0}$. It is directly added to the system of equations as formed by Equations (3), (6) and (13), together with the Fischer–Burmeister constraint function.

#### 2.2. Experimental

## 3. Results

#### 3.1. Model

^{−1}, and an imprint load of 2000 N m

^{−1}. The resulting pressure profiles for the hydrodynamic film pressure and tensioned web contact pressure are shown in Figure 5a. It also shows the Hertz dry contact pressure, for reference. The Hertz pressure and contact half-width are equal to $1.53\times {10}^{5}$ $\mathrm{Pa}$ and $8.3\mathrm{mm}$, respectively. Starting from the inlet of the roller contact, the hydrodynamic film pressure smoothly increases up to the peak pressure in the center $\left(\right)$, after which it decreases to ambient pressure again. The finite thickness of the elastomeric layer results in smaller contact widths and larger peak pressures in the resin film, compared to the Hertz solution. A similar phenomenon can be identified when web tension is included in the model. The tensioned web restricts the elastic deformation of the elastomeric layer material, thereby increasing the effective stiffness of the roller contact. For zero web tension, the tensioned web contact pressure is equal to the hydrodynamic film pressure, as also indicated by Equation (13). Increasing values of the web tension result in smaller contact widths and increased peak pressures in the thin film of resin. Outside the roller contact zone, the tensioned web contact pressure approaches a constant value of $T/R$. This can be explained by an absence of hydrodynamic film pressure, while the second-order derivative of the elastic deformation in Equation (13) approaches zero.

#### 3.2. Experimental Validation

^{−1}is taken into account. This behavior is also shown in Figure 5b. When taking a closer look at the layer height for a 1000 N m

^{−1}imprint load, it can be seen that there is good agreement between the measured layer heights and the minimum layer height from the numerical model with web tension. Contrary to the hypothesis, the minimum layer thickness seems to be the best predictor of the RLT, instead of the central layer thickness. For clarity, the central layer heights for the other imprint loads, which show similar behavior, are not shown. The layer heights for a varying resin viscosity and two different imprint loads are shown in Figure 7. The imprint velocity is kept constant at 6.7 mm s

^{−1}. The results are comparable to the results for a varying imprint velocity in Figure 6. The layer height increases with increasing resin viscosity and decreasing imprint load. Again, good agreement is found between the measured layer heights and the minimum layer height from the numerical model. The layer height for a varying imprint load is shown in Figure 8. The experimental data in this graph are in fact deduced from the measurements in Figure 6 and Figure 7. The resin viscosity and imprint velocity are kept constant at 38 $\mathrm{m}$$\mathrm{Pa}$ $\mathrm{s}$ and 6.7 mm s

^{−1}, respectively. The layer height decreases with increasing imprint load, as expected. Again, good agreement is found between the measured layer heights and the minimum layer height from the numerical model.

## 4. Discussion

^{−1}, the maximum elastic deformation is equal to 0.26 mm. This corresponds to a linear strain of 0.034, which is considered to be small. The resin is the other relevant material in the imprint process. It is assumed to be isoviscous. In practice, the resin viscosity can depend on pressure, shear rate, and temperature. The EHL contact is part of the soft EHL regime, which is characterized by relatively low contact pressures [43,44], as can also be seen in the typical film pressure profiles in Figure 5a. This confirms the assumption that any piezoviscous effects can be neglected. The Newtonian fluid behavior, which assumes a shear-rate-independent viscosity, is confirmed by viscosity measurements for a varying shear rate. Moreover, because the roller and substrate move with a similar velocity, the shear rate in the thin film of resin will be relatively low. Lastly, the process is assumed to be isothermal. It is known that the resin viscosity depends on temperature, similar as with other fluids and lubricants [37]. However, because the location of curing is relatively far way from the imprint roller (see Figure 1), any heating due to the UV source or the exothermal curing process can be neglected. This is confirmed by monitoring the imprint roller temperature during the experiments.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Abbreviations | |

EHL | Elastohydrodynamic Lubrication |

FEM | Finite Element Method |

NIL | Nanoimprint Lithography |

RLT | Residual Layer Thickness |

UV | Ultraviolet |

Symbols | |

a_{H} | Hertz contact half-width (m) |

E | Elastic modulus (Pa) |

E′ | Effective elastic modulus (Pa) |

f | Liquid volume fraction (-) |

h | Film/layer height (m) |

h_{0} | Gap between roller and substrate at x = 0 (m) |

h_{C} | Central layer height (m) |

h_{F} | Final layer height (m) |

h_{M} | Minimum layer height (m) |

p | Hydrodynamic film pressure (Reynolds) (Pa) |

p_{C} | Tensioned web contact pressure (Pa) |

p_{H} | Hertz contact pressure (Pa) |

p_{n} | Normal pressure on the tensioned web (Pa) |

R | Roller radius (m) |

T | Web tension (N m^{−}^{1}) |

t | Elastomeric layer thickness (m) |

u | Elastic deformation in x (m) |

u_{1} | Roller surface imprint velocity (m s^{−}^{1}) |

u_{2} | Substrate surface imprint velocity (m s^{−}^{1}) |

w | Elastic deformation in z (m) |

$\overline{W}$ | Effective imprint load per unit length (N m^{−}^{1}) |

x | Space coordinate in horizontal direction (m) |

z | Space coordinate in vertical direction (m) |

z_{roller} | Roller height profile (m) |

η | Resin dynamic viscosity (Pa s) |

θ | Cavity fraction (1 − f ) (-) |

$\kappa $ | Curvature of the tensioned web (m^{−}^{1}) |

λ | Lamé’s first parameter (Pa) |

μ | Lamé’s second parameter (Pa) |

v | Poisson ratio (-) |

σ_{n} | Normal component of the stress tensor (Pa) |

σ_{t} | Tangential component of the stress tensor (Pa) |

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**Figure 1.**Schematic of a roll-to-plate imprint system. Picture is adapted from [20].

**Figure 2.**(

**a**) Schematic of the imprint roller with tensioned web. The elastic deformation of the elastomeric layer is highly exaggerated for illustrative purposes. (

**b**) Equivalent geometry of the imprint roller with tensioned web.

**Figure 4.**(

**a**) Morphotonics Portis NIL1100 roll-to-plate nanoimprint equipment [20]. (

**b**) Detailed view of the rollers inside the Morphotonics Portis NIL1100 nanoimprint tool.

**Figure 5.**(

**a**) The hydrodynamic film pressure $\left(p\right)$ and tensioned web contact pressure $\left(\right)$ along the x-coordinate for varying web tension values. The Hertz dry contact pressure profile from Equation (15) is shown for reference. (

**b**) The layer height $\left(h\right)$ along the x-coordinate for varying web tension values.

**Figure 6.**Numerical and experimental results for the layer height for varying imprint loads and imprint velocities. The modeled results include a $\pm 10\%$ variation in both elastic modulus of the elastomeric layer and resin viscosity. The simulations and imprints are performed with Resin B from Table 1 (viscosity of 38 $\mathrm{m}$$\mathrm{Pa}$ $\mathrm{s}$).

**Figure 7.**Numerical and experimental results for the layer height for varying imprint loads and resin viscosities. The modeled results include a $\pm 10\%$ variation in both elastic modulus of the elastomeric layer and resin viscosity. The simulations and imprints are performed with a constant imprint velocity of 6.7 mm s

^{−1}. The imprints are performed with the resins as listed in Table 1.

**Figure 8.**Numerical and experimental results for the layer height for a varying imprint load. The modeled results include a $\pm 10\%$ variation in both elastic modulus of the elastomeric layer and resin viscosity. The simulations and imprints are performed with Resin B from Table 1 (viscosity of 38 $\mathrm{m}$$\mathrm{Pa}$ $\mathrm{s}$) and a constant imprint velocity of 6.7 mm s

^{−1}.

Resin | Viscosity (mPa s) | Volumetric Shrinkage (%) |
---|---|---|

A | 6.3 | 12.5 |

B | 38 | 8.1 |

C | 134 | 7.2 |

D | 181 | 8.8 |

E | 349 | 7.8 |

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**MDPI and ACS Style**

Snieder, J.; Dielen, M.; van Ostayen, R.A.J.
Simulating the Residual Layer Thickness in Roll-to-Plate Nanoimprinting with Tensioned Webs. *Micromachines* **2022**, *13*, 461.
https://doi.org/10.3390/mi13030461

**AMA Style**

Snieder J, Dielen M, van Ostayen RAJ.
Simulating the Residual Layer Thickness in Roll-to-Plate Nanoimprinting with Tensioned Webs. *Micromachines*. 2022; 13(3):461.
https://doi.org/10.3390/mi13030461

**Chicago/Turabian Style**

Snieder, Jelle, Marc Dielen, and Ron A. J. van Ostayen.
2022. "Simulating the Residual Layer Thickness in Roll-to-Plate Nanoimprinting with Tensioned Webs" *Micromachines* 13, no. 3: 461.
https://doi.org/10.3390/mi13030461