Simulation of multidrop spreading in ultraviolet nanoimprint lithography is performed to study the effects of droplet size, droplet arrangement, droplet placement error, and gas diffusion on filling-time and defects. Simulations are carried out for square, hexagonal, and modified hexagonal arrangements of up to 1024 droplets ink-jetted on a substrate to determine the optimum arrangement for minimum imprint time. The effect of error in droplet placement by the inkjet dispenser on the imprint time for different droplet sizes is also investigated. The square droplet arrangement leads to the shortest fluid filling time for a flat template. The filling time increases significantly for droplet placement errors of more than 0.01% relative to the width of the substrate. A model is presented to study the diffusion of gas encapsulated between droplets into the resist. A dimensionless parameter *α* $\u223c\mu D/kH\gamma Ho$ measures the relative importance gas diffusion to hydrodynamics, where *D* is the gas diffusion constant, *k _{H}* is the Henry's law constant,

*μ*is the resist viscosity, and

*γ*is the surface tension of the imprint resist. For small values of

*α*, gas diffusion is slower than resist spreading and defect size is diffusion-controlled while for larger values, gas diffusion is faster than resist spreading and the defect size is hydrodynamically controlled. Scaling laws are developed to extrapolate predictions on filling time, residual layer thickness, and defects here for hundreds to a thousand droplets to tens and hundreds of thousands of droplets.

## I. INTRODUCTION

UV nanoimprint lithography (UVNIL) is a low cost, high throughput process to replicate high resolution nano- and micropatterns from a template onto a rigid substrate.^{1} UVNIL has been used for fabrication of many nanoenabled devices such as high density bit pattern media,^{2,3} photonics,^{4,5} and displays.^{6} Lithography techniques like immersion and extreme ultraviolet (EUV) lithography also provide high resolution patterning; however, these techniques are more expensive with lower throughput.^{7} Pattern resolution in these techniques is limited by the wavelength of light and the numerical aperture of the optical system. Patterning sub-100 nm structures require complicated optical elements which increases the cost of exposure tools. UVNIL does not have these limitations and has the following competitive advantages over other lithography techniques such as immersion and EUV lithography: (1) pattern resolution in UVNIL is essentially only limited by the resolution of the template pattern;^{8,9} (2) UVNIL has demonstrated sub-100 nm pattern replication for features with high aspect ratios;^{10} and (3) the cost of ownership per wafer is significantly lower for UVNIL compared to photolithography and EUV.^{11,12}

The process for UVNIL includes a transparent template (also called a mask or mold), a rigid (glass or silicon) substrate, an inkjet dispenser, and a UV light source as shown in Fig. 1. The stepper template is patterned with micro- or nanofeatures of the same size as the final pattern.^{13} The nanoimprint process begins with the inkjet dispensing low viscosity photo-curable resist droplets on the substrate. A typical inkjet dispenser is capable of dispensing droplets of volume as low as one picoliter (10^{−15} m^{3}) at a variable pitch ranging from 85 to 150 *μ*m.^{14,15} The droplet density and arrangement depend on the volume of resist required to fill the local cavities on the template and achieve a desired residual layer thickness (RLT). The stepper template is then lowered on the droplets until they merge together to form a uniform resist film. Once the fluid spreading is complete and the entire template is filled, UV light is irradiated on the resist through the transparent template. The UV light causes photo-polymerization of the monomers to create a solid resist film. The exposure dose and time depend on the resist material and RLT.^{16,17} Once the resist is completely cured, the template is peeled off from the resist through controlled movement of the template. The surface energy of the template is low compared to the substrate, which allows the resist to preferentially adhere to the substrate. This process is repeated multiple times on different portions of the substrate to imprint a large wafer.

High throughput and low defectivity are critical for high volume manufacturing of devices such as flash memory using UVNIL.^{18,19} A recent cost of ownership calculation shows that a four-imprint tool system needs to produce 15 wafers per hour per imprint station at a defect density of 0.1 defects/cm^{2} to compete with a single ArF self-aligned quadruple patterning tool.^{18} This cost advantage was calculated for 15 nm half pitch grating. With current UVNIL technology, five wafers of size 300 mm can be imprinted per hour per imprint station.^{19,20} This implies that a threefold increase in process throughput is required. In UVNIL, UV exposure and template/resist separation typically require 0.1–0.2 s for a template of size 6 × 6 in.^{19} UV exposure and template/resist separation are well understood and are not the throughput limiting step. The throughput is limited by fluid filling, which takes about 1–5 s depending on the droplet size and surface energy of the substrate and template.^{20}

Fluid filling involves resist spreading over the entire wafer and the surrounding gas escaping from the features on the template as the resist fills these features. High throughput may lead to defects, namely, unfilled regions, in the final pattern since the resist may not have sufficient time to fill the features or the gas may not escape the feature completely. By optimizing droplet size and allowed spreading time, defectivity of the imprint process can be improved. Khusnatdinov *et al.* conducted nanoimprinting with varying drop volume and spread times to demonstrate their effect on defect density.^{20} They demonstrated a tenfold improvement in defect rate by reducing the drop volume from 1.5 to 0.9 pl when nanoimprinting with a 28 nm half pitch imprint mask. They also demonstrated that the defect density can be reduced from 30 to 0.1 defects/cm^{2} by increasing the filling time from 0.7 to 1 s. Thus, defect density needs to be considered in the context of throughput.^{19}

Conducting experiments to identify optimum droplet size and arrangement for different template patterns can be expensive and laborious. Previously, modeling and simulation has been effective in studying fluid behavior and gas trapping during droplet spreading in UVNIL. Colburn *et al.* studied the spreading of droplets between a template and a substrate for UV imprint lithography under several template control schemes.^{21} They proposed models for pressure distribution and filling time for the droplets as the template is moved with a fixed velocity, fixed pressure or constant applied force. Reddy and Bonnecaze presented a dynamic, multidrop simulation based on lubrication theory and simulated spreading of up to 49 drops with a flat template.^{22} Various mechanisms for interface advancement have been described to predict incomplete feature filling leading to air trapping. Local effect of sharp features on interface advancement with patterned template has been explored.^{23–25} Chauhan *et al*. analyzed the diffusion of gas entrapped in the features through liquid imprint resist and found that the absolute time for diffusion of a gas, entrapped in features is very short and gas diffusion is not a cause for nonfilling of features during the UVNIL.^{26}

We simulate multidrop spreading in UVNIL with a flat template to study the effect of various factors such as droplet size, droplet arrangement, droplet placement error, and gas diffusion that influence filling time and defectivity in the UVNIL process. We explore three droplet arrangements: (a) square, (b) hexagonal, and (c) modified hexagonal (as shown in Fig. 3). We also study the effect of error in droplet placement by the inkjet dispenser on the filling time for different droplet sizes. For the same final RLT, if the size of the droplets is reduced, more droplets need to be dispensed in order to fill the template completely. We perform simulations with up to 1024 droplets. The objective is to understand the effect of droplet arrangement, droplet placement accuracy, and drop size on throughput and identify the optimum droplet arrangement for minimum throughput. We also present a model to study the diffusion of gas encapsulated between droplets into the resist and identify parameters that determine whether the defect size and density are controlled by gas diffusion or hydrodynamic spreading.

## II. SIMULATION METHOD

A schematic of droplets with radius *R* in a gap of height *H* between a template and a substrate is shown in Fig. 2(a). The template can be controlled to move with a constant velocity, constant pressure, or applied force.^{21} As the template approaches the substrate, the liquid–air interface of the droplet exerts capillary force on the template. The fluid flow in the gap results in viscous force. This viscous force in the fluid balances the capillary force due to the liquid–air interface and any externally applied force.

The typical width of the template *L* used in UVNIL is 2.5–15 cm.^{20,27} During UVNIL, the gap *H* between the template and the substrate reduces from a few microns to finally a few nanometers. Thus, the size of the gap for fluid flow is much smaller than the transverse length scale. Reynolds' lubrication theory is a simplified form of Navier–Stokes equations that can be used to describe this flow of thin fluid films.^{28,29} The governing equations describing the pressure and velocity fields in the drop using Reynolds' lubrication theory are

where *H* is the gap between substrate and template, *P* is the pressure in the droplet, **U** is the vertically averaged fluid velocity, and *μ* is the viscosity of the imprint resist. The pressure distribution in the fluid is calculated using Eq. (1). The capillary pressure at the droplet interface is used as boundary condition to solve the pressure field. The imprint material used in UVNIL is highly wetting and it makes small contact angles *θ*_{1} and *θ*_{2} with the template and the substrate as shown in Fig. 2(a). The pressure at the liquid–air interface is given by

where *R _{m}* is the radius of curvature of the liquid meniscus,

*R*is the radius of curvature of the droplet, and

*γ*is the surface tension of the imprint.

*R*can be written in terms of gap

_{m}*H*as $Rm=H/(cos\u2009\theta 1+cos\u2009\theta 2).$ In the imprinting process, the gap

*H*is much smaller than the radius

*R*; therefore, $Rm\u22121\u226bR\u22121$. Therefore, the capillary pressure at the interface depends only on the radius of curvature

*R*and can be written as

where *P*_{atm} is the atmospheric pressure, $\gamma \u0302$ = $\gamma (cos\u2009\theta 1+\u2009cos\u2009\theta 2)/2$, and *γ* is the surface tension of the imprint resist. Initially, we assume that UVNIL is being carried out in vacuum or in a gas which is highly soluble in imprint material, and so, the gas release and gas trapping can be neglected. The rate of gas dissolution will be considered later in this paper. The imprint process begins with the droplets making contact with both the substrate and the template. We assume that the feature filling is practically instantaneous as the feature fill time is negligible compared to the time required for the fluid to spread across the gap.^{26} The typical values of parameters in the UVNIL process are listed in Table I.

Parameter . | Symbol . | Typical value . |
---|---|---|

Surface tension | γ | 30 dyn/cm |

Viscosity | μ | 0.003–0.005 Pa s |

Template width | L | 10 cm |

Substrate width | L | 10 cm |

Feature height | ΔH | 10–100 nm |

Initial gap | H _{o} | 1 μm |

Final gap | H _{f} | 5 nm |

Resist contact angle | θ_{1,2} | 5°–10° |

Initial drop height | H_{drop} | 1 μm |

Initial drop radius | R _{o} | 5 mm (1 droplet, H_{drop} = 1 μm) |

500 μm (100 droplets, H_{drop} = 1 μm) | ||

200 μm (1024 droplets, H_{drop} = 1 μm) |

Parameter . | Symbol . | Typical value . |
---|---|---|

Surface tension | γ | 30 dyn/cm |

Viscosity | μ | 0.003–0.005 Pa s |

Template width | L | 10 cm |

Substrate width | L | 10 cm |

Feature height | ΔH | 10–100 nm |

Initial gap | H _{o} | 1 μm |

Final gap | H _{f} | 5 nm |

Resist contact angle | θ_{1,2} | 5°–10° |

Initial drop height | H_{drop} | 1 μm |

Initial drop radius | R _{o} | 5 mm (1 droplet, H_{drop} = 1 μm) |

500 μm (100 droplets, H_{drop} = 1 μm) | ||

200 μm (1024 droplets, H_{drop} = 1 μm) |

The governing equations and boundary condition [Eqs. (1)–(3)] are nondimensionalized using following characteristic values of the variables: $Pc=2\gamma \u0302/Ho$, $Hc=Ho$, $Tc=6\mu L2/\gamma \u0302Ho$, $Vc=\gamma \u0302Ho2/6\mu L2$, and $Uc=\gamma \u0302Ho/6\mu L$, where *H _{o}* is the initial gap. The

*x-*and

*y-*coordinates are nondimensionalized using the width of the template

*L*. The dimensionless governing equations are then given by

where, *p* = *P*/*P _{c}*,

*h*=

*H*/

*H*,

_{c}*t*=

*T*/

*T*=

_{c}, v*V*/

*V*$\u2202h/\u2202t$, and

_{c}=**u**=

**U**/

*U*.

_{c}*v*is the dimensionless template velocity. The dimensionless pressure boundary condition is

The equations are solved using the volume of fluid (VOF) method.^{30} The domain is discretized into cells, and the fluid content in each cell is tracked based on a characteristic function *f*. This function is defined as 1 for liquid and 0 for gas [as shown in Fig. 2(b)]. A value of *f* between 0 and 1 implies that the cell is partially filled and contains the droplet interface. Several interface reconstruction algorithms were explored to define the droplet interface efficiently and accurately.^{31–34} Efficient least squares VOF interface reconstruction algorithm described by Pilliod and Puckett^{31} is used here because it is second order accurate and fast. In order to solve for the pressure field, Eq. (5) is discretized using a second order finite difference scheme. The typical grid size for one droplet in the simulation is 64 × 64 cells to ensure volume loss is less than 1%. The simulation is stable if the time step is chosen using the Courant–Friedrichs–Lewy condition which states that a fluid particle may not travel further than one cell during one time step, i.e., |*uΔt*| < Δ*x*.

A guess for template velocity *v* is used to initiate the simulation. Generalized minimal residual method (GMRES) method,^{35} an iterative method to solve nonsymmetric linear systems, is used to numerically calculate the pressure field in Eq. (5) at every time step. The total force on the template is calculated by integrating the pressure over the entire substrate. Then, the template velocity is adjusted such that the viscous force balances the capillary force and any externally applied force. The pressure field is then recalculated. Once the pressure field is known, the total fluid fluxes in the *x*- and *y*-directions are determined using Eq. (6). Immersed boundary methods are used to extrapolate the pressure field and accurately determine the fluid flux at the interfacial cells.^{36} Once the velocity field is known, the fluid is advanced using the VOF method and the template is lowered. At every time step, the liquid–air interface is reconstructed based on *f*. The simulation is stopped once the desired gap is reached and the fluid fills the entire domain.

Simulation of large number of droplets in UVNIL is computationally expensive and time consuming since the size of the domain scales as the number of droplets in the process. To accelerate the computations, a parallel implementation of GMRES method is used for fast computation of the pressure field. The simulation also becomes computationally more expensive as the droplet size increases since droplet interface, velocity flux, and pressure need to be calculated on a larger domain. For faster simulations without loss of accuracy, the domain is remeshed with one-fourth of the initial number of cells every time the radius of the droplet doubles. Mesh sizes and times steps were chosen to ensure no more than one-percent variation of volume. The simulations were performed using the high performance computing resources at Texas Advanced Computing Center at The University of Texas at Austin. The computational time varied from 6 h (for 100 drops) to a day (for 1024 drops) on 12 core Intel Xeon processor with 2.6 GHz clock speed and 64 GB RAM. Simulations were validated against the results by Reddy and Bonnecaze.^{22}

Simulations are carried out with droplets dispensed in square, hexagonal, and modified hexagonal arrangements. The droplet placement in a square arrangement is shown in Fig. 3(a). The shaded region in Fig. 3(a) shows the droplet-free region between the droplets and the edge of the substrate. This region is 16% of the total imprint area for 100 drops in square arrangement. Figure 3(b) shows droplets dispensed in a hexagonal arrangement where $lh/lb\u22483/2$ (= 0.866). The figure illustrates that the droplet-free region (shaded) is 26% of the total imprint area. Figure 3(c) shows droplets dispensed in a modified hexagonal arrangement where $lh*/lb*=0.893$. In this arrangement, the droplet-free region at the edge is reduced by increasing the distance between the droplets. This droplet arrangement generally results in unfilled region at the corners and edges. Placing additional droplets at the edge region and corners is necessary to cover the unfilled region and fill the mold edge region completely. Figure 3(c) shows 105 droplets placed in a modified hexagonal arrangement. The droplet-free region (shaded) at the edge reduces from 26% of the substrate in hexagonal arrangement to 16% in the modified hexagonal arrangement. In all the simulations, the gap between the substrate and the template closes from a gap of *h* = 1 to a final gap of *h* = 0.01. The total volume of droplets in all arrangements is equal to the volume required to fill the gap between the template and the substrate to a desired final gap of *h* = *h _{f}* = 0.01.

## III. RESULTS

Figure 4 shows the spreading of nine droplets, including the location of the interface and the pressure field within the droplet. The droplets spread on the substrate as a flat template approaches the substrate. The template is driven by the capillary forces in the droplet and has no external force acting on it. The pressure is negative at the liquid–air interface because of the capillary pressure and positive at the center because of the viscous component of the pressure. The capillary pressure is very low toward the end when the gap is very small. At all times, the viscous force balances the capillary force creating a net zero force on the template.

The simulation is repeated for multidrop UVNIL with flat template with zero force acting on it. Figure 5 shows spreading of about 100 droplets that are dispensed in (a) square, (b) hexagonal, and (c) modified hexagonal arrangement. The desired final gap is *h* = 0.01. The droplets spread as the gap between the template and the substrate closes. At very small gaps, the neighboring droplets merge together and spread to the edge of the substrate forming unfilled edge regions. Figures 5(a)–5(c) show these unfilled edges for the three arrangements. The unfilled edge in the hexagonal arrangement is significantly larger than the unfilled edge in the square or modified hexagonal arrangement. At *h* = 0.01 the gap is completely filled with the resist.

### A. Throughput for square, hexagonal, and modified hexagonal droplet arrangement

Figure 6(a) shows the gap height *h* as a function of time *t* for UVNIL with multiple droplets in a square arrangement. For larger number of droplets, the overall filling time is shorter. Figure 6(b) shows that plots for different number of droplets collapse into a single curve when the *x*-axis is scaled with number of droplets *N*. An analytical expression for the gap height *h* as a function of time *t* was previously derived for multiple droplets spreading on a substrate with net zero force on a flat template,^{21,22} and is given by

The gap *h* from the analytical solution is shown as dashed lines in Fig. 6(b). The simulation results (solid line) match the analytical solution until the droplets come in contact. The analytical expression is only valid for droplets spreading before merging. As the droplets merge, they form a single large droplet which spreads slower than multiple droplets.^{22} Thus, after merging, the simulated filling time starts to deviate from the analytical solution.

Figure 7 shows the gap height *h* as a function of filling time *t* for hexagonal droplet arrangement. The filling time for about the same number of droplets in square arrangement is shown as dashed lines. Similar to square arrangement, the time scales as 1/*Nh*^{2} except at the end of the process when the gap is very small [as shown in Fig. 7(b)]. For UVNIL with substrate length *L* = 10 cm, initial gap height of *H _{o}* = 1

*μ*m, resist viscosity

*μ*= 0.001 Pa s, surface tension $\gamma $ = 70 dyn/cm, and characteristic time

*T*= 857 s. The simulated filling time for 100 droplets in square arrangement is 52 s while the filling time for 101 droplets in hexagonal arrangement is 420 s. Therefore, for about the same number of droplets, the overall filling time for droplets in hexagonal arrangement is longer than droplets in square arrangement. This is attributed to the different size of unfilled edge regions in square arrangement and hexagonal arrangement. Before droplet spreading begins, the area of droplet-free region between the droplets and the edge of the substrate in square arrangement is 16% of the total imprint area as compared to 26% in hexagonal arrangement (Fig. 3). A plot of the unfilled edge area as function of

_{c}*h*for 100 droplets in square and hexagonal droplet arrangement is shown in Fig. 8(a). The unfilled area is larger in hexagonal arrangement compared to square arrangement. Since the resist spreads slowly after droplet merging, larger unfilled edge region takes longer to fill, resulting in longer overall filling time for hexagonal droplet arrangement.

A modified hexagonal arrangement is proposed in which the size of the unfilled region is reduced by spreading out the droplets in hexagonal arrangement toward the edges and adding more droplets at the corner as shown in Fig. 3(c). Before droplet spreading begins, the area of droplet-free region in modified hexagonal arrangement is same as square arrangement (as shown in Fig. 3). Figure 8(a) shows that after droplet merging, the unfilled region in modified hexagonal arrangement is larger than that in square arrangement. Figure 8(b) shows the filling time for UVNIL with about 100 droplets dispensed in square, hexagonal, and modified hexagonal arrangement. The filling time required to close from a gap height of *h* = 1 to *h* = 0.013 is almost the same for all droplet arrangements as shown in Fig. 8(b). However, the filling time required to close from *h* = 0.013 to *h* = 0.01 is considerably longer in hexagonal and modified arrangement compared to square arrangement as shown in Fig. 8(b) inset. The filling time for 105 droplets in modified hexagonal arrangement is 143 s. This is a significant improvement in filling time compared to 420 s required for 101 droplets in hexagonal arrangement. However, imprinting with droplets in square arrangement is still the fastest which requires 52 s. Thus, the unfilled edge region and droplet arrangement has a significant impact on the overall filling time. For flat templates, the square droplet arrangement is found to be the optimum droplet dispensing pattern to achieve fast droplet spreading.

### B. Effect of droplet placement error

There is error in droplet placement during dispensing based on the size of the droplet and distance between the imprint head from the substrate. This error can manifest itself in droplet spreading and overall filling time. The normalized placement error *ε* is defined as

where *L* is the substrate width and *E* is the distance between the target droplet position and the actual droplet position. For a substrate width of 10 cm and a dispensing error of 10 *μ*m, *ε* is 10^{−4}.

The filling time required for the gap height to close from *h* = 1 to *h* = 0.01 is simulated for droplets dispensed in a square arrangement with different placement error. Figure 9 shows spreading of 81 droplets dispensed in a square arrangement with *ε* = 0.02. Unlike square or hexagonal arrangements, the unfilled regions are randomly spread over the substrate due to the droplet placement error. Droplets merge to create larger droplets of different sizes. The resist completely fills the gap between the template and the substrate at *h* = 0.01.

Figure 10 shows gap height as a function of time for different values of droplet placement error. Figure 11(a) shows the overall filling time (*t _{i}*) required for the gap to close from

*h*= 1 to

*h*= 0.01 for different values of

*ε*and number of droplets

*N*. For

*ε*< 10

^{−4}

_{,}increase in

*ε*has negligible effect on filling time. However for

*ε*> 10

^{−4}, the filling time increases significantly as

*ε*increases. As the placement error increases, the droplets start merging sooner creating larger droplets of different sizes. Since larger droplets lead to longer filling time, early droplet merging leads to longer filling time. Figure 11(b) shows the ratio

*t*/

_{i,}_{ε}*t*

_{i,}_{0}as function of

*ε*where

*t*is the filling time with droplet error placement

_{i,}_{ε}*ε. t*

_{i,}_{0}is the filling time with no placement error. This ratio is a measure of sensitivity of filling time to placement error. We find that for a fixed

*ε*,

*t*/

_{i,}_{ε}*t*

_{i,}_{0}is larger for more number of droplets. For example, for

*ε*= 0.01,

*t*/

_{i,}_{ε}*t*

_{i,}_{0}is 1.5 for 9 droplets and 4 for 100 droplets. This indicated that the filling time is more sensitive to

*ε*when droplet size is smaller.

### C. Gas dissolution

When droplets make contact, unfilled regions or gas-filled pockets are formed, as seen, for example, in Fig. 7. Large gas-pockets will remain after UV curing and result in defects in the imprint process. However, small gas-pockets will dissolve away into the photocurable monomer.^{26,37} If the gas dissolution is fast relative to hydrodynamic motion of the interfaces, then one may ignore the gas in modeling. If the gas dissolution rate is relatively slower, then waiting times are necessary to remove the voids.^{20} The simulations presented so far assume that gas dissolution is fast enough that gas dissolution can be ignored. Here, we develop a model to predict the rate of dissolution of gas-pockets, determine the waiting times and the conditions under which it is fast enough to ignore.

Figure 12 shows a schematic for a gas-pocket formed during the UVNIL process. The pocket can be modeled as a cylinder of radius *R _{d}* and height

*H*.

*C*is the gas concentration inside the gas-pocket,

_{d}*m*is the number of moles, and

*C*is the gas concentration at the gas–liquid interface. The gas–liquid interfacial area

_{i}*A*and pocket volume

_{d}*S*are given by

_{d}and

For a final gap height *H _{f}* and substrate width

*L*,

*S*can be written as

_{d}where *n _{d}* is the number of gas-pockets and

*L*

^{2}

*H*is the total liquid volume in the gap at any time. Rearranging, we get

_{f}The number of pockets, their initial pocket volume and time of their formation can be derived from the simulations. The rate at which moles of gas deplete from the pocket can be written as

where *D* is the diffusion constant. The concentration gradient profile in the resist can be written as^{38}

where *C*_{bulk} is the gas concentration in the bulk of the liquid. *C*_{bulk} can be assumed to be negligible. *C _{i}* can be determined using Henry's law

^{26}

where *R* is the gas constant, *T _{s}* is the surrounding temperature and

*k*is the Henry's law constant. Applying Eqs. (16) and (15) into (14), we find

_{H}Using the ideal gas law, the gas pressure *P*_{gas} is given by

We can nondimensionalize Eqs. (11), (13), (18), and (19) using *r _{d}* =

*R*/

_{d}*L, h*=

*H*/

*H*,

_{c}*p*

_{gas}=

*P*

_{gas}/

*P*,

_{c}*t*=

*T*/

*T*, and

_{c}*s*=

_{d}*S*/

_{d}*L*

^{2}

*H*The characteristic values of the variables

_{o}.*H*,

_{c}*P*, and

_{c}*T*are given by $Hc=Ho$, $Pc=2\gamma \u0302/Ho$, and $Tc=6\mu L2/\gamma \u0302Ho$. The dimensionless equations are given by

_{c}where

The dimensionless force *f _{app}* applied on the template taking into account gas pressure is given by

^{22}

*f _{app}* = 0 when there is no force acting on the template. Since the total volume of the droplets

*q*is equal to the volume of resist required to fill the gap at

*h*=

*h*

_{f}Assuming net zero force on the template and using Eq. (26), we find

The unknowns *r _{d}*,

*h*,

*s*,

_{d}*m*, and

*p*

_{gas}are solved numerically using Eqs. (20)–(24) and (27). For UVNIL, the typical value of

*R*= 8.314 Pa m

^{3 }K

^{−1 }mol

^{−1}

*; T*300 K

_{s}=*; D =*10

^{−10}–10

^{−9}m

^{2 }s

^{−1}; and

*k*= 10

_{H}^{4}–10

^{5 }Pa m

^{3 }mol

^{−1}.

^{26,39}For these values,

*α*ranges from 10

^{−6}to 10

^{−4}. If the substrate width is 2.5 cm and desired final RLT is 10 nm, the initial pocket size is about 120

*μ*m when 10 000 droplets are dispensed. The diameter of the pockets at different times as it shrinks is shown in Fig. 13. The initial pocket filling is orders of magnitude faster compared to the remainder of the pocket filling process. Initially, the gas pressure inside the pocket is about the same as the atmospheric pressure and gas diffusion is slow. As the template lowers, the pocket size reduces and the gas pressure and concentration increase. The increase in gas pressure slows down the template and the gap height becomes close very slowly as shown in Fig. 13(b). The high gas concentration leads to a higher gas diffusion rate. As the gas diffuses into the resist, the pocket slowly fills with the resist. We find that, as

*α*increases, the pockets fill faster due to faster gas diffusion. The black line shows simulation results in which gas diffusion is neglected and the pockets fill only by hydrodynamic droplet spreading. For values of

*α*$\u2265$ 10

^{−4}, gas diffuses faster than the time required to fill the pocket by hydrodynamic motion of the interface. This implies that for

*α*$\u2265$ 10

^{−4}, the nonfilling defects in UVNIL are only a result of hydrodynamic nonfilling and not gas diffusion. However for

*α*< 10

^{−4}, the gas diffusion is the rate limiting process for the smallest defects. Thus, for low values of

*α,*defect size is diffusion-controlled while for high values, it is hydrodynamically controlled. Figure 14 shows the time at which the droplets come in contact

*T*

_{contact}, time required for gas to diffuse

*T*

_{diff}, and the total filling time

*T*

_{fill}for different number of droplets and values of

*α*.

*T*

_{fill}is a sum of

*T*

_{contact}and

*T*

_{diff}.

*T*

_{contact}depends only on the number of droplets and is independent of

*α*.

*T*

_{diff}reduces as

*α*increases. The figure suggests that for achieving filling time of less than a second on 2.5 × 2.5 cm substrate,

*α*needs to be greater than 10

^{−5}. For

*α*= 10

^{−4}

_{,}

*T*

_{contact}and

*T*

_{diff}become comparable suggesting that the time required for the droplets to spread and come in contact is about the same as the time required for the entrapped gas to diffuse.

## IV. CONCLUSIONS

Multidrop spreading is simulated for UVNIL with flat template and droplets ink-jetted in square, hexagonal and modified hexagonal arrangements. Lubrication theory is used to describe the pressure and velocity field for the fluid flow. VOF method is used to track fluid interface. No external force is applied on the template during the imprint process. We find that droplet size, droplet arrangement, and droplet placement accuracy are critical to achieving high throughput in UVNIL. Large unfilled edge regions increase the filling time for hexagonal and modified hexagonal droplet arrangement. For the same size of droplets, the square arrangement is found to be the optimum arrangement as it provides lower filling time than both hexagonal and modified hexagonal droplet arrangement. The effect of droplet placement error by inkjet system on the filling time is observed. For *ε* > 10^{−4}, the filling time increases significantly as the droplet placement error increases. Filling time for UVNIL with smaller droplets is more sensitive to the droplet placement error. We also propose a model to study the diffusion of gas encapsulated between droplets into the resist. We find that the defect size can be diffusion-controlled or hydrodynamics-controlled based on a parameter *α* which scales as $\u223c\mu D/kH\gamma Ho$. For values of *α* < 10^{−4}, gas diffusion is slow and defect size is diffusion-controlled, while for higher values, it is hydrodynamically controlled. For achieving filling time of less than a second on a 2.5 × 2.5 cm substrate, *α* needs to be greater than 10^{−5}. The droplet size, droplet arrangement, droplet placement accuracy, and nonfilling are some of the most critical factors affecting throughput and defect rate. We have successfully presented the effect of these factors on UVNIL process. Designing the UVNIL process based on the proposed models can significantly improve the throughput and the defect rate for the process.

## ACKNOWLEDGMENTS

The authors thank S. V. Sreenivasan, Randall Schunk, and Shrawan Singhal for their helpful comments on this work. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing High Performance Computing resources that have contributed to the research results reported within this paper. Funding for this project was provided by NASCENT Engineering Research Center supported by the National Science Foundation under Cooperative Agreement No. EEC-1160494.